jgraph_6.miz

begin



 Lm1: for a,b be Real st b <= 0 & a <= b holds (a * ( sqrt (1 + (b ^2 )))) <= (b * ( sqrt (1 + (a ^2 ))))

 proof

 let a,b be Real;

 assume that

 A1: b <= 0 and

 A2: a <= b;



 A3: (a ^2 ) >= 0 by XREAL_1: 63;

 then

 A4: (( - b) * ( sqrt (1 + (a ^2 )))) = ( sqrt ((( - b) ^2 ) * (1 + (a ^2 )))) by A1, SQUARE_1: 54;



 A5: (( - a) * ( sqrt (1 + (b ^2 )))) = ( sqrt ((( - a) ^2 ) * (1 + (b ^2 )))) by A1, A2, SQUARE_1: 54;

 a < b or a = b by A2, XXREAL_0: 1;

 then (b ^2 ) < (a ^2 ) or a = b by A1, SQUARE_1: 44;

 then

 A6: (((b ^2 ) * 1) + ((b ^2 ) * (a ^2 ))) <= (((a ^2 ) * 1) + ((a ^2 ) * (b ^2 ))) by XREAL_1: 7;

 (b ^2 ) >= 0 by XREAL_1: 63;

 then ( - (a * ( sqrt (1 + (b ^2 ))))) >= ( - (b * ( sqrt (1 + (a ^2 ))))) by A3, A4, A5, A6, SQUARE_1: 26;

 hence thesis by XREAL_1: 24;

 end;



 Lm2: for a,b be Real st a <= 0 & a <= b holds (a * ( sqrt (1 + (b ^2 )))) <= (b * ( sqrt (1 + (a ^2 ))))

 proof

 let a,b be Real;

 assume that

 A1: a <= 0 and

 A2: a <= b;

 now

 per cases ;

 case b <= 0 ;

 hence thesis by A2, Lm1;

 end;

 case

 A3: b > 0 ;

 (b ^2 ) >= 0 by XREAL_1: 63;

 then ( sqrt (1 + (b ^2 ))) > 0 by SQUARE_1: 25;

 then

 A4: (a * ( sqrt (1 + (b ^2 )))) <= 0 by A1;

 (a ^2 ) >= 0 by XREAL_1: 63;

 then ( sqrt (1 + (a ^2 ))) > 0 by SQUARE_1: 25;

 hence thesis by A3, A4;

 end;

 end;

 hence thesis;

 end;



 Lm3: for a,b be Real st a >= 0 & a >= b holds (a * ( sqrt (1 + (b ^2 )))) >= (b * ( sqrt (1 + (a ^2 ))))

 proof

 let a,b be Real;

 assume that

 A1: a >= 0 and

 A2: a >= b;

 ( - a) <= ( - b) by A2, XREAL_1: 24;

 then (( - a) * ( sqrt (1 + (( - b) ^2 )))) <= (( - b) * ( sqrt (1 + (( - a) ^2 )))) by A1, Lm2;

 then ( - (a * ( sqrt (1 + (b ^2 ))))) <= ( - (b * ( sqrt (1 + (a ^2 )))));

 hence thesis by XREAL_1: 24;

 end;

 theorem :: JGRAPH_6:1



 Th1: for a,c,d be Real, p be Point of ( TOP-REAL 2) st c <= d & p in ( LSeg ( |[a, c]|, |[a, d]|)) holds (p `1 ) = a & c <= (p `2 ) & (p `2 ) <= d

 proof

 let a,c,d be Real, p be Point of ( TOP-REAL 2);

 assume that

 A1: c <= d and

 A2: p in ( LSeg ( |[a, c]|, |[a, d]|));

 thus (p `1 ) = a by A2, TOPREAL3: 11;



 A3: ( |[a, c]| `2 ) = c by EUCLID: 52;

 ( |[a, d]| `2 ) = d by EUCLID: 52;

 hence thesis by A1, A2, A3, TOPREAL1: 4;

 end;

 theorem :: JGRAPH_6:2



 Th2: for a,c,d be Real, p be Point of ( TOP-REAL 2) st c < d & (p `1 ) = a & c <= (p `2 ) & (p `2 ) <= d holds p in ( LSeg ( |[a, c]|, |[a, d]|))

 proof

 let a,c,d be Real, p be Point of ( TOP-REAL 2);

 assume that

 A1: c < d and

 A2: (p `1 ) = a and

 A3: c <= (p `2 ) and

 A4: (p `2 ) <= d;



 A5: (d - c) > 0 by A1, XREAL_1: 50;

 reconsider w = (((p `2 ) - c) / (d - c)) as Real;



 A6: (((1 - w) * |[a, c]|) + (w * |[a, d]|)) = ( |[((1 - w) * a), ((1 - w) * c)]| + (w * |[a, d]|)) by EUCLID: 58

 .= ( |[((1 - w) * a), ((1 - w) * c)]| + |[(w * a), (w * d)]|) by EUCLID: 58

 .= |[(((1 - w) * a) + (w * a)), (((1 - w) * c) + (w * d))]| by EUCLID: 56

 .= |[a, (c + (w * (d - c)))]|

 .= |[a, (c + ((p `2 ) - c))]| by A5, XCMPLX_1: 87

 .= p by A2, EUCLID: 53;



 A7: ((p `2 ) - c) >= 0 by A3, XREAL_1: 48;

 ((p `2 ) - c) <= (d - c) by A4, XREAL_1: 9;

 then w <= ((d - c) / (d - c)) by A5, XREAL_1: 72;

 then w <= 1 by A5, XCMPLX_1: 60;

 hence thesis by A5, A6, A7;

 end;

 theorem :: JGRAPH_6:3



 Th3: for a,b,d be Real, p be Point of ( TOP-REAL 2) st a <= b & p in ( LSeg ( |[a, d]|, |[b, d]|)) holds (p `2 ) = d & a <= (p `1 ) & (p `1 ) <= b

 proof

 let a,b,d be Real, p be Point of ( TOP-REAL 2);

 assume that

 A1: a <= b and

 A2: p in ( LSeg ( |[a, d]|, |[b, d]|));

 thus (p `2 ) = d by A2, TOPREAL3: 12;



 A3: ( |[a, d]| `1 ) = a by EUCLID: 52;

 ( |[b, d]| `1 ) = b by EUCLID: 52;

 hence thesis by A1, A2, A3, TOPREAL1: 3;

 end;

 theorem :: JGRAPH_6:4



 Th4: for a,b be Real, B be Subset of I[01] st B = [.a, b.] holds B is closed

 proof

 let a,b be Real, B be Subset of I[01] ;

 assume

 A1: B = [.a, b.];

 reconsider B2 = B as Subset of R^1 by BORSUK_1: 40, TOPMETR: 17, XBOOLE_1: 1;



 A2: B2 is closed by A1, TREAL_1: 1;

 reconsider I1 = [. 0 , 1.] as Subset of R^1 by TOPMETR: 17;



 A3: ( [#] ( R^1 | I1)) = the carrier of I[01] by BORSUK_1: 40, PRE_TOPC:def 5;



 A4: I[01] = ( R^1 | I1) by TOPMETR: 19, TOPMETR: 20;

 B = (B2 /\ ( [#] ( R^1 | I1))) by A3, XBOOLE_1: 28;

 hence thesis by A2, A4, PRE_TOPC: 13;

 end;

 theorem :: JGRAPH_6:5



 Th5: for X be TopStruct, Y,Z be non empty TopStruct, f be Function of X, Y, g be Function of X, Z holds ( dom f) = ( dom g) & ( dom f) = the carrier of X & ( dom f) = ( [#] X)

 proof

 let X be TopStruct, Y,Z be non empty TopStruct, f be Function of X, Y, g be Function of X, Z;

 ( dom f) = the carrier of X by FUNCT_2:def 1;

 hence thesis by FUNCT_2:def 1;

 end;

 theorem :: JGRAPH_6:6



 Th6: for X be non empty TopSpace, B be non empty Subset of X holds ex f be Function of (X | B), X st (for p be Point of (X | B) holds (f . p) = p) & f is continuous

 proof

 let X be non empty TopSpace, B be non empty Subset of X;

 defpred P[ set, set] means (for p be Point of (X | B) holds $2 = $1);



 A1: ( [#] (X | B)) = B by PRE_TOPC:def 5;



 A2: for x be Element of (X | B) holds ex y be Element of X st P[x, y]

 proof

 let x be Element of (X | B);

 x in B by A1;

 then

 reconsider px = x as Point of X;

 set y0 = px;

 P[x, y0];

 hence thesis;

 end;

 ex g be Function of the carrier of (X | B), the carrier of X st for x be Element of (X | B) holds P[x, (g . x)] from FUNCT_2:sch 3( A2);

 then

 consider g be Function of the carrier of (X | B), the carrier of X such that

 A3: for x be Element of (X | B) holds P[x, (g . x)];



 A4: for p be Point of (X | B) holds (g . p) = p by A3;



 A5: for r0 be Point of (X | B), V be Subset of X st (g . r0) in V & V is open holds ex W be Subset of (X | B) st r0 in W & W is open & (g .: W) c= V

 proof

 let r0 be Point of (X | B), V be Subset of X;

 assume that

 A6: (g . r0) in V and

 A7: V is open;

 reconsider W2 = (V /\ ( [#] (X | B))) as Subset of (X | B);

 (g . r0) = r0 by A3;

 then

 A8: r0 in W2 by A6, XBOOLE_0:def 4;



 A9: W2 is open by A7, TOPS_2: 24;

 (g .: W2) c= V

 proof

 let y be object;

 assume y in (g .: W2);

 then

 consider x be object such that

 A10: x in ( dom g) and

 A11: x in W2 and

 A12: y = (g . x) by FUNCT_1:def 6;

 reconsider px = x as Point of (X | B) by A10;

 (g . px) = px by A3;

 hence thesis by A11, A12, XBOOLE_0:def 4;

 end;

 hence thesis by A8, A9;

 end;

 reconsider g1 = g as Function of (X | B), X;

 g1 is continuous by A5, JGRAPH_2: 10;

 hence thesis by A4;

 end;

 theorem :: JGRAPH_6:7



 Th7: for X be non empty TopSpace, f1 be Function of X, R^1 , a be Real st f1 is continuous holds ex g be Function of X, R^1 st (for p be Point of X, r1 be Real st (f1 . p) = r1 holds (g . p) = (r1 - a)) & g is continuous

 proof

 let X be non empty TopSpace, f1 be Function of X, R^1 , a be Real;

 assume f1 is continuous;

 then

 consider g1 be Function of X, R^1 such that

 A1: for p be Point of X, r1 be Real st (f1 . p) = r1 holds (g1 . p) = (r1 + ( - a)) and

 A2: g1 is continuous by JGRAPH_2: 24;

 for p be Point of X, r1 be Real st (f1 . p) = r1 holds (g1 . p) = (r1 - a)

 proof

 let p be Point of X, r1 be Real;

 assume (f1 . p) = r1;

 then (g1 . p) = (r1 + ( - a)) by A1;

 hence thesis;

 end;

 hence thesis by A2;

 end;

 theorem :: JGRAPH_6:8



 Th8: for X be non empty TopSpace, f1 be Function of X, R^1 , a be Real st f1 is continuous holds ex g be Function of X, R^1 st (for p be Point of X, r1 be Real st (f1 . p) = r1 holds (g . p) = (a - r1)) & g is continuous

 proof

 let X be non empty TopSpace, f1 be Function of X, R^1 , a be Real;

 assume f1 is continuous;

 then

 consider g1 be Function of X, R^1 such that

 A1: for p be Point of X, r1 be Real st (f1 . p) = r1 holds (g1 . p) = (r1 - a) and

 A2: g1 is continuous by Th7;

 consider g2 be Function of X, R^1 such that

 A3: for p be Point of X, r1 be Real st (g1 . p) = r1 holds (g2 . p) = ( - r1) and

 A4: g2 is continuous by A2, JGRAPH_4: 8;

 for p be Point of X, r1 be Real st (f1 . p) = r1 holds (g2 . p) = (a - r1)

 proof

 let p be Point of X, r1 be Real;

 assume (f1 . p) = r1;

 then (g1 . p) = (r1 - a) by A1;



 then (g2 . p) = ( - (r1 - a)) by A3

 .= (a - r1);

 hence thesis;

 end;

 hence thesis by A4;

 end;

 theorem :: JGRAPH_6:9



 Th9: for X be non empty TopSpace, n be Nat, p be Point of ( TOP-REAL n), f be Function of X, R^1 st f is continuous holds ex g be Function of X, ( TOP-REAL n) st (for r be Point of X holds (g . r) = ((f . r) * p)) & g is continuous

 proof

 let X be non empty TopSpace, n be Nat, p be Point of ( TOP-REAL n), f be Function of X, R^1 ;

 assume

 A1: f is continuous;

 defpred P[ set, set] means $2 = ((f . $1) * p);



 A2: for x be Element of X holds ex y be Element of ( TOP-REAL n) st P[x, y];

 ex g be Function of the carrier of X, the carrier of ( TOP-REAL n) st for x be Element of X holds P[x, (g . x)] from FUNCT_2:sch 3( A2);

 then

 consider g be Function of the carrier of X, the carrier of ( TOP-REAL n) such that

 A3: for x be Element of X holds P[x, (g . x)];

 reconsider g as Function of X, ( TOP-REAL n);

 for r0 be Point of X, V be Subset of ( TOP-REAL n) st (g . r0) in V & V is open holds ex W be Subset of X st r0 in W & W is open & (g .: W) c= V

 proof

 let r0 be Point of X, V be Subset of ( TOP-REAL n);

 assume that

 A4: (g . r0) in V and

 A5: V is open;



 A6: (g . r0) in ( Int V) by A4, A5, TOPS_1: 23;

 reconsider u = (g . r0) as Point of ( Euclid n) by TOPREAL3: 8;

 consider s be Real such that

 A7: s > 0 and

 A8: ( Ball (u,s)) c= V by A6, GOBOARD6: 5;

 now

 per cases ;

 case

 A9: p <> ( 0. ( TOP-REAL n));

 then

 A10: |.p.| <> 0 by TOPRNS_1: 24;

 set r2 = (s / |.p.|);

 reconsider G1 = ].((f . r0) - r2), ((f . r0) + r2).[ as Subset of R^1 by TOPMETR: 17;



 A11: (f . r0) < ((f . r0) + r2) by A7, A10, XREAL_1: 29, XREAL_1: 139;

 then ((f . r0) - r2) < (f . r0) by XREAL_1: 19;

 then

 A12: (f . r0) in G1 by A11, XXREAL_1: 4;

 G1 is open by JORDAN6: 35;

 then

 consider W2 be Subset of X such that

 A13: r0 in W2 and

 A14: W2 is open and

 A15: (f .: W2) c= G1 by A1, A12, JGRAPH_2: 10;

 (g .: W2) c= V

 proof

 let y be object;

 assume y in (g .: W2);

 then

 consider r be object such that

 A16: r in ( dom g) and

 A17: r in W2 and

 A18: y = (g . r) by FUNCT_1:def 6;

 reconsider r as Point of X by A16;

 ( dom f) = the carrier of X by FUNCT_2:def 1;

 then (f . r) in (f .: W2) by A17, FUNCT_1:def 6;

 then

 A19: |.((f . r) - (f . r0)).| < r2 by A15, RCOMP_1: 1;

 reconsider t = (f . r), t0 = (f . r0) as Real;



 A20: |.(t0 - t).| = |.(t - t0).| by UNIFORM1: 11;

 reconsider v = (g . r) as Point of ( Euclid n) by TOPREAL3: 8;

 (g . r0) = ((f . r0) * p) by A3;



 then

 A21: |.((g . r0) - (g . r)).| = |.(((f . r0) * p) - ((f . r) * p)).| by A3

 .= |.(((f . r0) - (f . r)) * p).| by RLVECT_1: 35

 .= ( |.(t0 - t).| * |.p.|) by TOPRNS_1: 7;

 ( |.((f . r) - (f . r0)).| * |.p.|) < (r2 * |.p.|) by A10, A19, XREAL_1: 68;

 then |.((g . r0) - (g . r)).| < s by A9, A20, A21, TOPRNS_1: 24, XCMPLX_1: 87;

 then ( dist (u,v)) < s by JGRAPH_1: 28;

 then (g . r) in ( Ball (u,s)) by METRIC_1: 11;

 hence thesis by A8, A18;

 end;

 hence thesis by A13, A14;

 end;

 case

 A22: p = ( 0. ( TOP-REAL n));



 A23: for r be Point of X holds (g . r) = ( 0. ( TOP-REAL n))

 proof

 let r be Point of X;



 thus (g . r) = ((f . r) * p) by A3

 .= ( 0. ( TOP-REAL n)) by A22, RLVECT_1: 10;

 end;

 then

 A24: ( 0. ( TOP-REAL n)) in V by A4;

 set W2 = ( [#] X);

 (g .: W2) c= V

 proof

 let y be object;

 assume y in (g .: W2);

 then ex x be object st (x in ( dom g)) & (x in W2) & (y = (g . x)) by FUNCT_1:def 6;

 hence thesis by A23, A24;

 end;

 hence thesis;

 end;

 end;

 hence thesis;

 end;

 then g is continuous by JGRAPH_2: 10;

 hence thesis by A3;

 end;

 theorem :: JGRAPH_6:10



 Th10: ( Sq_Circ . |[( - 1), 0 ]|) = |[( - 1), 0 ]|

 proof

 set p = |[( - 1), 0 ]|;



 A1: (p `1 ) = ( - 1) by EUCLID: 52;



 A2: (p `2 ) = 0 by EUCLID: 52;



 A3: p <> ( 0. ( TOP-REAL 2)) by A1, EUCLID: 52, EUCLID: 54;

 (p `2 ) <= ( - (p `1 )) by A1, EUCLID: 52;

 then ( Sq_Circ . p) = |[((p `1 ) / ( sqrt (1 + (((p `2 ) / (p `1 )) ^2 )))), ((p `2 ) / ( sqrt (1 + (((p `2 ) / (p `1 )) ^2 ))))]| by A1, A2, A3, JGRAPH_3:def 1;

 hence thesis by A2, EUCLID: 52, SQUARE_1: 18;

 end;

 theorem :: JGRAPH_6:11

 for P be compact non empty Subset of ( TOP-REAL 2) st P = { p where p be Point of ( TOP-REAL 2) : |.p.| = 1 } holds ( Sq_Circ . |[( - 1), 0 ]|) = ( W-min P) by Th10, JGRAPH_5: 29;

 theorem :: JGRAPH_6:12



 Th12: for X be non empty TopSpace, n be Nat, g1,g2 be Function of X, ( TOP-REAL n) st g1 is continuous & g2 is continuous holds ex g be Function of X, ( TOP-REAL n) st (for r be Point of X holds (g . r) = ((g1 . r) + (g2 . r))) & g is continuous

 proof

 let X be non empty TopSpace, n be Nat, g1,g2 be Function of X, ( TOP-REAL n);

 assume that

 A1: g1 is continuous and

 A2: g2 is continuous;

 defpred P[ set, set] means (for r1,r2 be Element of ( TOP-REAL n) st (g1 . $1) = r1 & (g2 . $1) = r2 holds $2 = (r1 + r2));



 A3: for x be Element of X holds ex y be Element of ( TOP-REAL n) st P[x, y]

 proof

 let x be Element of X;

 set rr1 = (g1 . x);

 set rr2 = (g2 . x);

 set r3 = (rr1 + rr2);

 for s1,s2 be Point of ( TOP-REAL n) st (g1 . x) = s1 & (g2 . x) = s2 holds r3 = (s1 + s2);

 hence thesis;

 end;

 ex f be Function of the carrier of X, the carrier of ( TOP-REAL n) st for x be Element of X holds P[x, (f . x)] from FUNCT_2:sch 3( A3);

 then

 consider f be Function of the carrier of X, the carrier of ( TOP-REAL n) such that

 A4: for x be Element of X holds for r1,r2 be Element of ( TOP-REAL n) st (g1 . x) = r1 & (g2 . x) = r2 holds (f . x) = (r1 + r2);

 reconsider g0 = f as Function of X, ( TOP-REAL n);



 A5: for r be Point of X holds (g0 . r) = ((g1 . r) + (g2 . r)) by A4;

 for p be Point of X, V be Subset of ( TOP-REAL n) st (g0 . p) in V & V is open holds ex W be Subset of X st p in W & W is open & (g0 .: W) c= V

 proof

 let p be Point of X, V be Subset of ( TOP-REAL n);

 assume that

 A6: (g0 . p) in V and

 A7: V is open;



 A8: (g0 . p) in ( Int V) by A6, A7, TOPS_1: 23;

 reconsider r = (g0 . p) as Point of ( Euclid n) by TOPREAL3: 8;

 consider r0 be Real such that

 A9: r0 > 0 and

 A10: ( Ball (r,r0)) c= V by A8, GOBOARD6: 5;

 reconsider r01 = (g1 . p) as Point of ( Euclid n) by TOPREAL3: 8;

 reconsider G1 = ( Ball (r01,(r0 / 2))) as Subset of ( TOP-REAL n) by TOPREAL3: 8;

 reconsider r02 = (g2 . p) as Point of ( Euclid n) by TOPREAL3: 8;

 reconsider G2 = ( Ball (r02,(r0 / 2))) as Subset of ( TOP-REAL n) by TOPREAL3: 8;



 A11: (g1 . p) in G1 by A9, GOBOARD6: 1, XREAL_1: 215;



 A12: the TopStruct of ( TOP-REAL n) = ( TopSpaceMetr ( Euclid n)) by EUCLID:def 8;

 then

 reconsider GG1 = G1, GG2 = G2 as Subset of ( TopSpaceMetr ( Euclid n));

 GG1 is open by TOPMETR: 14;

 then G1 is open by A12, PRE_TOPC: 30;

 then

 consider W1 be Subset of X such that

 A13: p in W1 and

 A14: W1 is open and

 A15: (g1 .: W1) c= G1 by A1, A11, JGRAPH_2: 10;



 A16: (g2 . p) in G2 by A9, GOBOARD6: 1, XREAL_1: 215;

 GG2 is open by TOPMETR: 14;

 then G2 is open by A12, PRE_TOPC: 30;

 then

 consider W2 be Subset of X such that

 A17: p in W2 and

 A18: W2 is open and

 A19: (g2 .: W2) c= G2 by A2, A16, JGRAPH_2: 10;

 set W = (W1 /\ W2);



 A20: p in W by A13, A17, XBOOLE_0:def 4;

 (g0 .: W) c= ( Ball (r,r0))

 proof

 let x be object;

 assume x in (g0 .: W);

 then

 consider z be object such that

 A21: z in ( dom g0) and

 A22: z in W and

 A23: (g0 . z) = x by FUNCT_1:def 6;



 A24: z in W1 by A22, XBOOLE_0:def 4;

 reconsider pz = z as Point of X by A21;

 ( dom g1) = the carrier of X by FUNCT_2:def 1;

 then

 A25: (g1 . pz) in (g1 .: W1) by A24, FUNCT_1:def 6;

 reconsider aa1 = (g1 . pz) as Point of ( TOP-REAL n);

 reconsider bb1 = aa1 as Point of ( Euclid n) by TOPREAL3: 8;

 ( dist (r01,bb1)) < (r0 / 2) by A15, A25, METRIC_1: 11;

 then

 A26: |.((g1 . p) - (g1 . pz)).| < (r0 / 2) by JGRAPH_1: 28;



 A27: z in W2 by A22, XBOOLE_0:def 4;

 ( dom g2) = the carrier of X by FUNCT_2:def 1;

 then

 A28: (g2 . pz) in (g2 .: W2) by A27, FUNCT_1:def 6;

 reconsider aa2 = (g2 . pz) as Point of ( TOP-REAL n);

 reconsider bb2 = aa2 as Point of ( Euclid n) by TOPREAL3: 8;

 ( dist (r02,bb2)) < (r0 / 2) by A19, A28, METRIC_1: 11;

 then

 A29: |.((g2 . p) - (g2 . pz)).| < (r0 / 2) by JGRAPH_1: 28;



 A30: (aa1 + aa2) = x by A4, A23;

 reconsider bb0 = (aa1 + aa2) as Point of ( Euclid n) by TOPREAL3: 8;



 A31: (g0 . pz) = ((g1 . pz) + (g2 . pz)) by A4;

 (((g1 . p) + (g2 . p)) - ((g1 . pz) + (g2 . pz))) = ((((g1 . p) + (g2 . p)) - (g1 . pz)) - (g2 . pz)) by RLVECT_1: 27

 .= ((((g1 . p) + (g2 . p)) + ( - (g1 . pz))) - (g2 . pz))

 .= ((((g1 . p) + (g2 . p)) + ( - (g1 . pz))) + ( - (g2 . pz)))

 .= ((((g1 . p) + ( - (g1 . pz))) + (g2 . p)) + ( - (g2 . pz))) by RLVECT_1:def 3

 .= (((g1 . p) + ( - (g1 . pz))) + ((g2 . p) + ( - (g2 . pz)))) by RLVECT_1:def 3

 .= (((g1 . p) - (g1 . pz)) + ((g2 . p) + ( - (g2 . pz))))

 .= (((g1 . p) - (g1 . pz)) + ((g2 . p) - (g2 . pz)));

 then

 A32: |.(((g1 . p) + (g2 . p)) - ((g1 . pz) + (g2 . pz))).| <= ( |.((g1 . p) - (g1 . pz)).| + |.((g2 . p) - (g2 . pz)).|) by TOPRNS_1: 29;

 ( |.((g1 . p) - (g1 . pz)).| + |.((g2 . p) - (g2 . pz)).|) < ((r0 / 2) + (r0 / 2)) by A26, A29, XREAL_1: 8;

 then |.(((g1 . p) + (g2 . p)) - ((g1 . pz) + (g2 . pz))).| < r0 by A32, XXREAL_0: 2;

 then |.((g0 . p) - (g0 . pz)).| < r0 by A4, A31;

 then ( dist (r,bb0)) < r0 by A23, A30, JGRAPH_1: 28;

 hence thesis by A30, METRIC_1: 11;

 end;

 hence thesis by A10, A14, A18, A20, XBOOLE_1: 1;

 end;

 then g0 is continuous by JGRAPH_2: 10;

 hence thesis by A5;

 end;

 theorem :: JGRAPH_6:13



 Th13: for X be non empty TopSpace, n be Nat, p1,p2 be Point of ( TOP-REAL n), f1,f2 be Function of X, R^1 st f1 is continuous & f2 is continuous holds ex g be Function of X, ( TOP-REAL n) st (for r be Point of X holds (g . r) = (((f1 . r) * p1) + ((f2 . r) * p2))) & g is continuous

 proof

 let X be non empty TopSpace, n be Nat, p1,p2 be Point of ( TOP-REAL n), f1,f2 be Function of X, R^1 ;

 assume that

 A1: f1 is continuous and

 A2: f2 is continuous;

 consider g1 be Function of X, ( TOP-REAL n) such that

 A3: for r be Point of X holds (g1 . r) = ((f1 . r) * p1) and

 A4: g1 is continuous by A1, Th9;

 consider g2 be Function of X, ( TOP-REAL n) such that

 A5: for r be Point of X holds (g2 . r) = ((f2 . r) * p2) and

 A6: g2 is continuous by A2, Th9;

 consider g be Function of X, ( TOP-REAL n) such that

 A7: for r be Point of X holds (g . r) = ((g1 . r) + (g2 . r)) and

 A8: g is continuous by A4, A6, Th12;

 for r be Point of X holds (g . r) = (((f1 . r) * p1) + ((f2 . r) * p2))

 proof

 let r be Point of X;

 (g . r) = ((g1 . r) + (g2 . r)) by A7;

 then (g . r) = ((g1 . r) + ((f2 . r) * p2)) by A5;

 hence thesis by A3;

 end;

 hence thesis by A8;

 end;

 begin



 Lm4: ( |[( - 1), 0 ]| `1 ) = ( - 1) by EUCLID: 52;



 Lm5: ( |[( - 1), 0 ]| `2 ) = 0 by EUCLID: 52;



 Lm6: ( |[1, 0 ]| `1 ) = 1 by EUCLID: 52;



 Lm7: ( |[1, 0 ]| `2 ) = 0 by EUCLID: 52;



 Lm8: ( |[ 0 , ( - 1)]| `1 ) = 0 by EUCLID: 52;



 Lm9: ( |[ 0 , ( - 1)]| `2 ) = ( - 1) by EUCLID: 52;



 Lm10: ( |[ 0 , 1]| `1 ) = 0 by EUCLID: 52;



 Lm11: ( |[ 0 , 1]| `2 ) = 1 by EUCLID: 52;

 Lm12:

 now



 thus |. |[( - 1), 0 ]|.| = ( sqrt ((( - 1) ^2 ) + ( 0 ^2 ))) by Lm4, Lm5, JGRAPH_3: 1

 .= 1 by SQUARE_1: 18;



 thus |. |[1, 0 ]|.| = ( sqrt ((1 ^2 ) + ( 0 ^2 ))) by Lm6, Lm7, JGRAPH_3: 1

 .= 1 by SQUARE_1: 18;



 thus |. |[ 0 , ( - 1)]|.| = ( sqrt (( 0 ^2 ) + (( - 1) ^2 ))) by Lm8, Lm9, JGRAPH_3: 1

 .= 1 by SQUARE_1: 18;



 thus |. |[ 0 , 1]|.| = ( sqrt (( 0 ^2 ) + (1 ^2 ))) by Lm10, Lm11, JGRAPH_3: 1

 .= 1 by SQUARE_1: 18;

 end;



 Lm13: 0 in [. 0 , 1.] by XXREAL_1: 1;



 Lm14: 1 in [. 0 , 1.] by XXREAL_1: 1;

 reserve p,p1,p2,p3,q,q1,q2 for Point of ( TOP-REAL 2),

i for Nat,

lambda for Real;

 theorem :: JGRAPH_6:14



 Th14: for f,g be Function of I[01] , ( TOP-REAL 2), C0,KXP,KXN,KYP,KYN be Subset of ( TOP-REAL 2), O,I be Point of I[01] st O = 0 & I = 1 & f is continuous one-to-one & g is continuous one-to-one & C0 = { p : |.p.| <= 1 } & KXP = { q1 where q1 be Point of ( TOP-REAL 2) : |.q1.| = 1 & (q1 `2 ) <= (q1 `1 ) & (q1 `2 ) >= ( - (q1 `1 )) } & KXN = { q2 where q2 be Point of ( TOP-REAL 2) : |.q2.| = 1 & (q2 `2 ) >= (q2 `1 ) & (q2 `2 ) <= ( - (q2 `1 )) } & KYP = { q3 where q3 be Point of ( TOP-REAL 2) : |.q3.| = 1 & (q3 `2 ) >= (q3 `1 ) & (q3 `2 ) >= ( - (q3 `1 )) } & KYN = { q4 where q4 be Point of ( TOP-REAL 2) : |.q4.| = 1 & (q4 `2 ) <= (q4 `1 ) & (q4 `2 ) <= ( - (q4 `1 )) } & (f . O) in KXP & (f . I) in KXN & (g . O) in KYP & (g . I) in KYN & ( rng f) c= C0 & ( rng g) c= C0 holds ( rng f) meets ( rng g)

 proof

 let f,g be Function of I[01] , ( TOP-REAL 2), C0,KXP,KXN,KYP,KYN be Subset of ( TOP-REAL 2), O,I be Point of I[01] ;

 assume

 A1: O = 0 & I = 1 & f is continuous & f is one-to-one & g is continuous & g is one-to-one & C0 = { p : |.p.| <= 1 } & KXP = { q1 where q1 be Point of ( TOP-REAL 2) : |.q1.| = 1 & (q1 `2 ) <= (q1 `1 ) & (q1 `2 ) >= ( - (q1 `1 )) } & KXN = { q2 where q2 be Point of ( TOP-REAL 2) : |.q2.| = 1 & (q2 `2 ) >= (q2 `1 ) & (q2 `2 ) <= ( - (q2 `1 )) } & KYP = { q3 where q3 be Point of ( TOP-REAL 2) : |.q3.| = 1 & (q3 `2 ) >= (q3 `1 ) & (q3 `2 ) >= ( - (q3 `1 )) } & KYN = { q4 where q4 be Point of ( TOP-REAL 2) : |.q4.| = 1 & (q4 `2 ) <= (q4 `1 ) & (q4 `2 ) <= ( - (q4 `1 )) } & (f . O) in KXP & (f . I) in KXN & (g . O) in KYP & (g . I) in KYN & ( rng f) c= C0 & ( rng g) c= C0;

 then ex f2 be Function of I[01] , ( TOP-REAL 2) st ((f2 . 0 ) = (f . 1)) & ((f2 . 1) = (f . 0 )) & (( rng f2) = ( rng f)) & (f2 is continuous) & (f2 is one-to-one) by JGRAPH_5: 12;

 hence thesis by A1, JGRAPH_5: 13;

 end;

 theorem :: JGRAPH_6:15



 Th15: for f,g be Function of I[01] , ( TOP-REAL 2), C0,KXP,KXN,KYP,KYN be Subset of ( TOP-REAL 2), O,I be Point of I[01] st O = 0 & I = 1 & f is continuous & f is one-to-one & g is continuous & g is one-to-one & C0 = { p : |.p.| <= 1 } & KXP = { q1 where q1 be Point of ( TOP-REAL 2) : |.q1.| = 1 & (q1 `2 ) <= (q1 `1 ) & (q1 `2 ) >= ( - (q1 `1 )) } & KXN = { q2 where q2 be Point of ( TOP-REAL 2) : |.q2.| = 1 & (q2 `2 ) >= (q2 `1 ) & (q2 `2 ) <= ( - (q2 `1 )) } & KYP = { q3 where q3 be Point of ( TOP-REAL 2) : |.q3.| = 1 & (q3 `2 ) >= (q3 `1 ) & (q3 `2 ) >= ( - (q3 `1 )) } & KYN = { q4 where q4 be Point of ( TOP-REAL 2) : |.q4.| = 1 & (q4 `2 ) <= (q4 `1 ) & (q4 `2 ) <= ( - (q4 `1 )) } & (f . O) in KXP & (f . I) in KXN & (g . O) in KYN & (g . I) in KYP & ( rng f) c= C0 & ( rng g) c= C0 holds ( rng f) meets ( rng g)

 proof

 let f,g be Function of I[01] , ( TOP-REAL 2), C0,KXP,KXN,KYP,KYN be Subset of ( TOP-REAL 2), O,I be Point of I[01] ;

 assume

 A1: O = 0 & I = 1 & f is continuous & f is one-to-one & g is continuous & g is one-to-one & C0 = { p : |.p.| <= 1 } & KXP = { q1 where q1 be Point of ( TOP-REAL 2) : |.q1.| = 1 & (q1 `2 ) <= (q1 `1 ) & (q1 `2 ) >= ( - (q1 `1 )) } & KXN = { q2 where q2 be Point of ( TOP-REAL 2) : |.q2.| = 1 & (q2 `2 ) >= (q2 `1 ) & (q2 `2 ) <= ( - (q2 `1 )) } & KYP = { q3 where q3 be Point of ( TOP-REAL 2) : |.q3.| = 1 & (q3 `2 ) >= (q3 `1 ) & (q3 `2 ) >= ( - (q3 `1 )) } & KYN = { q4 where q4 be Point of ( TOP-REAL 2) : |.q4.| = 1 & (q4 `2 ) <= (q4 `1 ) & (q4 `2 ) <= ( - (q4 `1 )) } & (f . O) in KXP & (f . I) in KXN & (g . O) in KYN & (g . I) in KYP & ( rng f) c= C0 & ( rng g) c= C0;

 then ex g2 be Function of I[01] , ( TOP-REAL 2) st ((g2 . 0 ) = (g . 1)) & ((g2 . 1) = (g . 0 )) & (( rng g2) = ( rng g)) & (g2 is continuous) & (g2 is one-to-one) by JGRAPH_5: 12;

 hence thesis by A1, Th14;

 end;

 theorem :: JGRAPH_6:16



 Th16: for p1,p2,p3,p4 be Point of ( TOP-REAL 2), P be compact non empty Subset of ( TOP-REAL 2), C0 be Subset of ( TOP-REAL 2) st P = { p where p be Point of ( TOP-REAL 2) : |.p.| = 1 } & LE (p1,p2,P) & LE (p2,p3,P) & LE (p3,p4,P) holds for f,g be Function of I[01] , ( TOP-REAL 2) st f is continuous one-to-one & g is continuous one-to-one & C0 = { p8 where p8 be Point of ( TOP-REAL 2) : |.p8.| <= 1 } & (f . 0 ) = p3 & (f . 1) = p1 & (g . 0 ) = p2 & (g . 1) = p4 & ( rng f) c= C0 & ( rng g) c= C0 holds ( rng f) meets ( rng g)

 proof

 let p1,p2,p3,p4 be Point of ( TOP-REAL 2), P be compact non empty Subset of ( TOP-REAL 2), C0 be Subset of ( TOP-REAL 2);

 assume that

 A1: P = { p where p be Point of ( TOP-REAL 2) : |.p.| = 1 } and

 A2: LE (p1,p2,P) and

 A3: LE (p2,p3,P) and

 A4: LE (p3,p4,P);

 let f,g be Function of I[01] , ( TOP-REAL 2);

 assume that

 A5: f is continuous one-to-one and

 A6: g is continuous one-to-one and

 A7: C0 = { p8 where p8 be Point of ( TOP-REAL 2) : |.p8.| <= 1 } and

 A8: (f . 0 ) = p3 and

 A9: (f . 1) = p1 and

 A10: (g . 0 ) = p2 and

 A11: (g . 1) = p4 and

 A12: ( rng f) c= C0 and

 A13: ( rng g) c= C0;



 A14: ( dom f) = the carrier of I[01] by FUNCT_2:def 1;



 A15: ( dom g) = the carrier of I[01] by FUNCT_2:def 1;

 per cases ;

 suppose

 A16: not (p1 <> p2 & p2 <> p3 & p3 <> p4);

 now

 per cases by A16;

 case

 A17: p1 = p2;



 A18: p1 in ( rng f) by A9, A14, Lm14, BORSUK_1: 40, FUNCT_1:def 3;

 p2 in ( rng g) by A10, A15, Lm13, BORSUK_1: 40, FUNCT_1:def 3;

 hence ( rng f) meets ( rng g) by A17, A18, XBOOLE_0: 3;

 end;

 case

 A19: p2 = p3;



 A20: p3 in ( rng f) by A8, A14, Lm13, BORSUK_1: 40, FUNCT_1:def 3;

 p2 in ( rng g) by A10, A15, Lm13, BORSUK_1: 40, FUNCT_1:def 3;

 hence ( rng f) meets ( rng g) by A19, A20, XBOOLE_0: 3;

 end;

 case

 A21: p3 = p4;



 A22: p3 in ( rng f) by A8, A14, Lm13, BORSUK_1: 40, FUNCT_1:def 3;

 p4 in ( rng g) by A11, A15, Lm14, BORSUK_1: 40, FUNCT_1:def 3;

 hence ( rng f) meets ( rng g) by A21, A22, XBOOLE_0: 3;

 end;

 end;

 hence thesis;

 end;

 suppose p1 <> p2 & p2 <> p3 & p3 <> p4;

 then

 consider h be Function of ( TOP-REAL 2), ( TOP-REAL 2) such that

 A23: h is being_homeomorphism and

 A24: for q be Point of ( TOP-REAL 2) holds |.(h . q).| = |.q.| and

 A25: |[( - 1), 0 ]| = (h . p1) and

 A26: |[ 0 , 1]| = (h . p2) and

 A27: |[1, 0 ]| = (h . p3) and

 A28: |[ 0 , ( - 1)]| = (h . p4) by A1, A2, A3, A4, JGRAPH_5: 67;



 A29: h is one-to-one by A23, TOPS_2:def 5;

 reconsider f2 = (h * f) as Function of I[01] , ( TOP-REAL 2);

 reconsider g2 = (h * g) as Function of I[01] , ( TOP-REAL 2);



 A30: ( dom f2) = the carrier of I[01] by FUNCT_2:def 1;



 A31: ( dom g2) = the carrier of I[01] by FUNCT_2:def 1;



 A32: (f2 . 1) = |[( - 1), 0 ]| by A9, A25, A30, Lm14, BORSUK_1: 40, FUNCT_1: 12;



 A33: (g2 . 1) = |[ 0 , ( - 1)]| by A11, A28, A31, Lm14, BORSUK_1: 40, FUNCT_1: 12;



 A34: (f2 . 0 ) = |[1, 0 ]| by A8, A27, A30, Lm13, BORSUK_1: 40, FUNCT_1: 12;



 A35: (g2 . 0 ) = |[ 0 , 1]| by A10, A26, A31, Lm13, BORSUK_1: 40, FUNCT_1: 12;



 A36: f2 is continuous one-to-one by A5, A23, JGRAPH_5: 5, JGRAPH_5: 6;



 A37: g2 is continuous one-to-one by A6, A23, JGRAPH_5: 5, JGRAPH_5: 6;



 A38: ( rng f2) c= C0

 proof

 let y be object;

 assume y in ( rng f2);

 then

 consider x be object such that

 A39: x in ( dom f2) and

 A40: y = (f2 . x) by FUNCT_1:def 3;



 A41: (f2 . x) = (h . (f . x)) by A39, FUNCT_1: 12;



 A42: (f . x) in ( rng f) by A14, A39, FUNCT_1:def 3;

 then

 A43: (f . x) in C0 by A12;

 reconsider qf = (f . x) as Point of ( TOP-REAL 2) by A42;



 A44: ex q5 be Point of ( TOP-REAL 2) st (q5 = (f . x)) & ( |.q5.| <= 1) by A7, A43;

 |.(h . qf).| = |.qf.| by A24;

 hence thesis by A7, A40, A41, A44;

 end;



 A45: ( rng g2) c= C0

 proof

 let y be object;

 assume y in ( rng g2);

 then

 consider x be object such that

 A46: x in ( dom g2) and

 A47: y = (g2 . x) by FUNCT_1:def 3;



 A48: (g2 . x) = (h . (g . x)) by A46, FUNCT_1: 12;



 A49: (g . x) in ( rng g) by A15, A46, FUNCT_1:def 3;

 then

 A50: (g . x) in C0 by A13;

 reconsider qg = (g . x) as Point of ( TOP-REAL 2) by A49;



 A51: ex q5 be Point of ( TOP-REAL 2) st (q5 = (g . x)) & ( |.q5.| <= 1) by A7, A50;

 |.(h . qg).| = |.qg.| by A24;

 hence thesis by A7, A47, A48, A51;

 end;

 defpred Q[ Point of ( TOP-REAL 2)] means |.$1.| = 1 & ($1 `2 ) <= ($1 `1 ) & ($1 `2 ) >= ( - ($1 `1 ));

 { q1 where q1 be Point of ( TOP-REAL 2) : Q[q1] } is Subset of ( TOP-REAL 2) from JGRAPH_2:sch 1;

 then

 reconsider KXP = { q1 where q1 be Point of ( TOP-REAL 2) : |.q1.| = 1 & (q1 `2 ) <= (q1 `1 ) & (q1 `2 ) >= ( - (q1 `1 )) } as Subset of ( TOP-REAL 2);

 defpred Q[ Point of ( TOP-REAL 2)] means |.$1.| = 1 & ($1 `2 ) >= ($1 `1 ) & ($1 `2 ) <= ( - ($1 `1 ));

 { q2 where q2 be Point of ( TOP-REAL 2) : Q[q2] } is Subset of ( TOP-REAL 2) from JGRAPH_2:sch 1;

 then

 reconsider KXN = { q2 where q2 be Point of ( TOP-REAL 2) : |.q2.| = 1 & (q2 `2 ) >= (q2 `1 ) & (q2 `2 ) <= ( - (q2 `1 )) } as Subset of ( TOP-REAL 2);

 defpred Q[ Point of ( TOP-REAL 2)] means |.$1.| = 1 & ($1 `2 ) >= ($1 `1 ) & ($1 `2 ) >= ( - ($1 `1 ));

 { q3 where q3 be Point of ( TOP-REAL 2) : Q[q3] } is Subset of ( TOP-REAL 2) from JGRAPH_2:sch 1;

 then

 reconsider KYP = { q3 where q3 be Point of ( TOP-REAL 2) : |.q3.| = 1 & (q3 `2 ) >= (q3 `1 ) & (q3 `2 ) >= ( - (q3 `1 )) } as Subset of ( TOP-REAL 2);

 defpred Q[ Point of ( TOP-REAL 2)] means |.$1.| = 1 & ($1 `2 ) <= ($1 `1 ) & ($1 `2 ) <= ( - ($1 `1 ));

 { q4 where q4 be Point of ( TOP-REAL 2) : Q[q4] } is Subset of ( TOP-REAL 2) from JGRAPH_2:sch 1;

 then

 reconsider KYN = { q4 where q4 be Point of ( TOP-REAL 2) : |.q4.| = 1 & (q4 `2 ) <= (q4 `1 ) & (q4 `2 ) <= ( - (q4 `1 )) } as Subset of ( TOP-REAL 2);

 reconsider O = 0 , I = 1 as Point of I[01] by BORSUK_1: 40, XXREAL_1: 1;

 ( - ( |[( - 1), 0 ]| `1 )) = 1 by Lm4;

 then

 A52: (f2 . I) in KXN by A32, Lm5, Lm12;



 A53: (f2 . O) in KXP by A34, Lm6, Lm7, Lm12;

 ( - ( |[ 0 , ( - 1)]| `1 )) = 0 by Lm8;

 then

 A54: (g2 . I) in KYN by A33, Lm9, Lm12;

 ( - ( |[ 0 , 1]| `1 )) = 0 by Lm10;

 then (g2 . O) in KYP by A35, Lm11, Lm12;

 then ( rng f2) meets ( rng g2) by A7, A36, A37, A38, A45, A52, A53, A54, Th14;

 then

 consider x2 be object such that

 A55: x2 in ( rng f2) and

 A56: x2 in ( rng g2) by XBOOLE_0: 3;

 consider z2 be object such that

 A57: z2 in ( dom f2) and

 A58: x2 = (f2 . z2) by A55, FUNCT_1:def 3;

 consider z3 be object such that

 A59: z3 in ( dom g2) and

 A60: x2 = (g2 . z3) by A56, FUNCT_1:def 3;



 A61: ( dom h) = the carrier of ( TOP-REAL 2) by FUNCT_2:def 1;



 A62: (g . z3) in ( rng g) by A15, A59, FUNCT_1:def 3;



 A63: (f . z2) in ( rng f) by A14, A57, FUNCT_1:def 3;

 reconsider h1 = h as Function;



 A64: ((h1 " ) . x2) = ((h1 " ) . (h . (f . z2))) by A57, A58, FUNCT_1: 12

 .= (f . z2) by A29, A61, A63, FUNCT_1: 34;



 A65: ((h1 " ) . x2) = ((h1 " ) . (h . (g . z3))) by A59, A60, FUNCT_1: 12

 .= (g . z3) by A29, A61, A62, FUNCT_1: 34;



 A66: ((h1 " ) . x2) in ( rng f) by A14, A57, A64, FUNCT_1:def 3;

 ((h1 " ) . x2) in ( rng g) by A15, A59, A65, FUNCT_1:def 3;

 hence thesis by A66, XBOOLE_0: 3;

 end;

 end;

 theorem :: JGRAPH_6:17



 Th17: for p1,p2,p3,p4 be Point of ( TOP-REAL 2), P be compact non empty Subset of ( TOP-REAL 2), C0 be Subset of ( TOP-REAL 2) st P = { p where p be Point of ( TOP-REAL 2) : |.p.| = 1 } & LE (p1,p2,P) & LE (p2,p3,P) & LE (p3,p4,P) holds for f,g be Function of I[01] , ( TOP-REAL 2) st f is continuous one-to-one & g is continuous one-to-one & C0 = { p8 where p8 be Point of ( TOP-REAL 2) : |.p8.| <= 1 } & (f . 0 ) = p3 & (f . 1) = p1 & (g . 0 ) = p4 & (g . 1) = p2 & ( rng f) c= C0 & ( rng g) c= C0 holds ( rng f) meets ( rng g)

 proof

 let p1,p2,p3,p4 be Point of ( TOP-REAL 2), P be compact non empty Subset of ( TOP-REAL 2), C0 be Subset of ( TOP-REAL 2);

 assume that

 A1: P = { p where p be Point of ( TOP-REAL 2) : |.p.| = 1 } and

 A2: LE (p1,p2,P) and

 A3: LE (p2,p3,P) and

 A4: LE (p3,p4,P);

 let f,g be Function of I[01] , ( TOP-REAL 2);

 assume that

 A5: f is continuous one-to-one and

 A6: g is continuous one-to-one and

 A7: C0 = { p8 where p8 be Point of ( TOP-REAL 2) : |.p8.| <= 1 } and

 A8: (f . 0 ) = p3 and

 A9: (f . 1) = p1 and

 A10: (g . 0 ) = p4 and

 A11: (g . 1) = p2 and

 A12: ( rng f) c= C0 and

 A13: ( rng g) c= C0;



 A14: ( dom f) = the carrier of I[01] by FUNCT_2:def 1;



 A15: ( dom g) = the carrier of I[01] by FUNCT_2:def 1;

 per cases ;

 suppose

 A16: not (p1 <> p2 & p2 <> p3 & p3 <> p4);

 now

 per cases by A16;

 case

 A17: p1 = p2;



 A18: p1 in ( rng f) by A9, A14, Lm14, BORSUK_1: 40, FUNCT_1:def 3;

 p2 in ( rng g) by A11, A15, Lm14, BORSUK_1: 40, FUNCT_1:def 3;

 hence ( rng f) meets ( rng g) by A17, A18, XBOOLE_0: 3;

 end;

 case

 A19: p2 = p3;



 A20: p3 in ( rng f) by A8, A14, Lm13, BORSUK_1: 40, FUNCT_1:def 3;

 p2 in ( rng g) by A11, A15, Lm14, BORSUK_1: 40, FUNCT_1:def 3;

 hence ( rng f) meets ( rng g) by A19, A20, XBOOLE_0: 3;

 end;

 case

 A21: p3 = p4;



 A22: p3 in ( rng f) by A8, A14, Lm13, BORSUK_1: 40, FUNCT_1:def 3;

 p4 in ( rng g) by A10, A15, Lm13, BORSUK_1: 40, FUNCT_1:def 3;

 hence ( rng f) meets ( rng g) by A21, A22, XBOOLE_0: 3;

 end;

 end;

 hence thesis;

 end;

 suppose p1 <> p2 & p2 <> p3 & p3 <> p4;

 then

 consider h be Function of ( TOP-REAL 2), ( TOP-REAL 2) such that

 A23: h is being_homeomorphism and

 A24: for q be Point of ( TOP-REAL 2) holds |.(h . q).| = |.q.| and

 A25: |[( - 1), 0 ]| = (h . p1) and

 A26: |[ 0 , 1]| = (h . p2) and

 A27: |[1, 0 ]| = (h . p3) and

 A28: |[ 0 , ( - 1)]| = (h . p4) by A1, A2, A3, A4, JGRAPH_5: 67;



 A29: h is one-to-one by A23, TOPS_2:def 5;

 reconsider f2 = (h * f) as Function of I[01] , ( TOP-REAL 2);

 reconsider g2 = (h * g) as Function of I[01] , ( TOP-REAL 2);



 A30: ( dom f2) = the carrier of I[01] by FUNCT_2:def 1;



 A31: ( dom g2) = the carrier of I[01] by FUNCT_2:def 1;



 A32: (f2 . 1) = |[( - 1), 0 ]| by A9, A25, A30, Lm14, BORSUK_1: 40, FUNCT_1: 12;



 A33: (g2 . 1) = |[ 0 , 1]| by A11, A26, A31, Lm14, BORSUK_1: 40, FUNCT_1: 12;



 A34: (f2 . 0 ) = |[1, 0 ]| by A8, A27, A30, Lm13, BORSUK_1: 40, FUNCT_1: 12;



 A35: (g2 . 0 ) = |[ 0 , ( - 1)]| by A10, A28, A31, Lm13, BORSUK_1: 40, FUNCT_1: 12;



 A36: f2 is continuous one-to-one by A5, A23, JGRAPH_5: 5, JGRAPH_5: 6;



 A37: g2 is continuous one-to-one by A6, A23, JGRAPH_5: 5, JGRAPH_5: 6;



 A38: ( rng f2) c= C0

 proof

 let y be object;

 assume y in ( rng f2);

 then

 consider x be object such that

 A39: x in ( dom f2) and

 A40: y = (f2 . x) by FUNCT_1:def 3;



 A41: (f2 . x) = (h . (f . x)) by A39, FUNCT_1: 12;



 A42: (f . x) in ( rng f) by A14, A39, FUNCT_1:def 3;

 then

 A43: (f . x) in C0 by A12;

 reconsider qf = (f . x) as Point of ( TOP-REAL 2) by A42;



 A44: ex q5 be Point of ( TOP-REAL 2) st (q5 = (f . x)) & ( |.q5.| <= 1) by A7, A43;

 |.(h . qf).| = |.qf.| by A24;

 hence thesis by A7, A40, A41, A44;

 end;



 A45: ( rng g2) c= C0

 proof

 let y be object;

 assume y in ( rng g2);

 then

 consider x be object such that

 A46: x in ( dom g2) and

 A47: y = (g2 . x) by FUNCT_1:def 3;



 A48: (g2 . x) = (h . (g . x)) by A46, FUNCT_1: 12;



 A49: (g . x) in ( rng g) by A15, A46, FUNCT_1:def 3;

 then

 A50: (g . x) in C0 by A13;

 reconsider qg = (g . x) as Point of ( TOP-REAL 2) by A49;



 A51: ex q5 be Point of ( TOP-REAL 2) st (q5 = (g . x)) & ( |.q5.| <= 1) by A7, A50;

 |.(h . qg).| = |.qg.| by A24;

 hence thesis by A7, A47, A48, A51;

 end;

 defpred P[ Point of ( TOP-REAL 2)] means |.$1.| = 1 & ($1 `2 ) <= ($1 `1 ) & ($1 `2 ) >= ( - ($1 `1 ));

 { q1 where q1 be Point of ( TOP-REAL 2) : P[q1] } is Subset of ( TOP-REAL 2) from JGRAPH_2:sch 1;

 then

 reconsider KXP = { q1 where q1 be Point of ( TOP-REAL 2) : |.q1.| = 1 & (q1 `2 ) <= (q1 `1 ) & (q1 `2 ) >= ( - (q1 `1 )) } as Subset of ( TOP-REAL 2);

 defpred P[ Point of ( TOP-REAL 2)] means |.$1.| = 1 & ($1 `2 ) >= ($1 `1 ) & ($1 `2 ) <= ( - ($1 `1 ));

 { q2 where q2 be Point of ( TOP-REAL 2) : P[q2] } is Subset of ( TOP-REAL 2) from JGRAPH_2:sch 1;

 then

 reconsider KXN = { q2 where q2 be Point of ( TOP-REAL 2) : |.q2.| = 1 & (q2 `2 ) >= (q2 `1 ) & (q2 `2 ) <= ( - (q2 `1 )) } as Subset of ( TOP-REAL 2);

 defpred P[ Point of ( TOP-REAL 2)] means |.$1.| = 1 & ($1 `2 ) >= ($1 `1 ) & ($1 `2 ) >= ( - ($1 `1 ));

 { q3 where q3 be Point of ( TOP-REAL 2) : P[q3] } is Subset of ( TOP-REAL 2) from JGRAPH_2:sch 1;

 then

 reconsider KYP = { q3 where q3 be Point of ( TOP-REAL 2) : |.q3.| = 1 & (q3 `2 ) >= (q3 `1 ) & (q3 `2 ) >= ( - (q3 `1 )) } as Subset of ( TOP-REAL 2);

 defpred P[ Point of ( TOP-REAL 2)] means |.$1.| = 1 & ($1 `2 ) <= ($1 `1 ) & ($1 `2 ) <= ( - ($1 `1 ));

 { q4 where q4 be Point of ( TOP-REAL 2) : P[q4] } is Subset of ( TOP-REAL 2) from JGRAPH_2:sch 1;

 then

 reconsider KYN = { q4 where q4 be Point of ( TOP-REAL 2) : |.q4.| = 1 & (q4 `2 ) <= (q4 `1 ) & (q4 `2 ) <= ( - (q4 `1 )) } as Subset of ( TOP-REAL 2);

 reconsider O = 0 , I = 1 as Point of I[01] by BORSUK_1: 40, XXREAL_1: 1;

 ( - ( |[( - 1), 0 ]| `1 )) = 1 by Lm4;

 then

 A52: (f2 . I) in KXN by A32, Lm5, Lm12;



 A53: (f2 . O) in KXP by A34, Lm6, Lm7, Lm12;

 ( - ( |[ 0 , ( - 1)]| `1 )) = 0 by Lm8;

 then

 A54: (g2 . I) in KYP by A33, Lm10, Lm11, Lm12;

 ( - ( |[ 0 , 1]| `1 )) = 0 by Lm10;

 then (g2 . O) in KYN by A35, Lm8, Lm9, Lm12;

 then ( rng f2) meets ( rng g2) by A7, A36, A37, A38, A45, A52, A53, A54, Th15;

 then

 consider x2 be object such that

 A55: x2 in ( rng f2) and

 A56: x2 in ( rng g2) by XBOOLE_0: 3;

 consider z2 be object such that

 A57: z2 in ( dom f2) and

 A58: x2 = (f2 . z2) by A55, FUNCT_1:def 3;

 consider z3 be object such that

 A59: z3 in ( dom g2) and

 A60: x2 = (g2 . z3) by A56, FUNCT_1:def 3;



 A61: ( dom h) = the carrier of ( TOP-REAL 2) by FUNCT_2:def 1;



 A62: (g . z3) in ( rng g) by A15, A59, FUNCT_1:def 3;



 A63: (f . z2) in ( rng f) by A14, A57, FUNCT_1:def 3;

 reconsider h1 = h as Function;



 A64: ((h1 " ) . x2) = ((h1 " ) . (h . (f . z2))) by A57, A58, FUNCT_1: 12

 .= (f . z2) by A29, A61, A63, FUNCT_1: 34;



 A65: ((h1 " ) . x2) = ((h1 " ) . (h . (g . z3))) by A59, A60, FUNCT_1: 12

 .= (g . z3) by A29, A61, A62, FUNCT_1: 34;



 A66: ((h1 " ) . x2) in ( rng f) by A14, A57, A64, FUNCT_1:def 3;

 ((h1 " ) . x2) in ( rng g) by A15, A59, A65, FUNCT_1:def 3;

 hence thesis by A66, XBOOLE_0: 3;

 end;

 end;

 theorem :: JGRAPH_6:18



 Th18: for p1,p2,p3,p4 be Point of ( TOP-REAL 2), P be compact non empty Subset of ( TOP-REAL 2), C0 be Subset of ( TOP-REAL 2) st P = { p where p be Point of ( TOP-REAL 2) : |.p.| = 1 } & (p1,p2,p3,p4) are_in_this_order_on P holds for f,g be Function of I[01] , ( TOP-REAL 2) st f is continuous one-to-one & g is continuous one-to-one & C0 = { p8 where p8 be Point of ( TOP-REAL 2) : |.p8.| <= 1 } & (f . 0 ) = p1 & (f . 1) = p3 & (g . 0 ) = p2 & (g . 1) = p4 & ( rng f) c= C0 & ( rng g) c= C0 holds ( rng f) meets ( rng g)

 proof

 let p1,p2,p3,p4 be Point of ( TOP-REAL 2), P be compact non empty Subset of ( TOP-REAL 2), C0 be Subset of ( TOP-REAL 2);

 assume that

 A1: P = { p where p be Point of ( TOP-REAL 2) : |.p.| = 1 } and

 A2: (p1,p2,p3,p4) are_in_this_order_on P;

 per cases by A2, JORDAN17:def 1;

 suppose LE (p1,p2,P) & LE (p2,p3,P) & LE (p3,p4,P);

 hence thesis by A1, JGRAPH_5: 68;

 end;

 suppose LE (p2,p3,P) & LE (p3,p4,P) & LE (p4,p1,P);

 hence thesis by A1, JGRAPH_5: 69;

 end;

 suppose LE (p3,p4,P) & LE (p4,p1,P) & LE (p1,p2,P);

 hence thesis by A1, Th17;

 end;

 suppose LE (p4,p1,P) & LE (p1,p2,P) & LE (p2,p3,P);

 hence thesis by A1, Th16;

 end;

 end;

 begin

 notation

 let a,b,c,d be Real;

 synonym rectangle (a,b,c,d) for [.a,b,c,d.];

 end



 Lm15: for a,b,c,d be Real st a <= b & c <= d holds ( rectangle (a,b,c,d)) = { p : (p `1 ) = a & c <= (p `2 ) & (p `2 ) <= d or (p `2 ) = d & a <= (p `1 ) & (p `1 ) <= b or (p `1 ) = b & c <= (p `2 ) & (p `2 ) <= d or (p `2 ) = c & a <= (p `1 ) & (p `1 ) <= b }

 proof

 let a,b,c,d be Real;

 set X = { p : (p `1 ) = a & c <= (p `2 ) & (p `2 ) <= d or (p `2 ) = d & a <= (p `1 ) & (p `1 ) <= b or (p `1 ) = b & c <= (p `2 ) & (p `2 ) <= d or (p `2 ) = c & a <= (p `1 ) & (p `1 ) <= b };

 assume that

 A1: a <= b and

 A2: c <= d;



 A3: ( rectangle (a,b,c,d)) = { p2 : (p2 `1 ) = a & (p2 `2 ) <= d & (p2 `2 ) >= c or (p2 `1 ) <= b & (p2 `1 ) >= a & (p2 `2 ) = d or (p2 `1 ) <= b & (p2 `1 ) >= a & (p2 `2 ) = c or (p2 `1 ) = b & (p2 `2 ) <= d & (p2 `2 ) >= c } by A1, A2, SPPOL_2: 54;

 hereby

 let x be object;

 assume x in ( rectangle (a,b,c,d));

 then ex p2 st x = p2 & ((p2 `1 ) = a & (p2 `2 ) <= d & (p2 `2 ) >= c or (p2 `1 ) <= b & (p2 `1 ) >= a & (p2 `2 ) = d or (p2 `1 ) <= b & (p2 `1 ) >= a & (p2 `2 ) = c or (p2 `1 ) = b & (p2 `2 ) <= d & (p2 `2 ) >= c) by A3;

 hence x in X;

 end;

 let x be object;

 assume x in X;

 then ex p st x = p & ((p `1 ) = a & c <= (p `2 ) & (p `2 ) <= d or (p `2 ) = d & a <= (p `1 ) & (p `1 ) <= b or (p `1 ) = b & c <= (p `2 ) & (p `2 ) <= d or (p `2 ) = c & a <= (p `1 ) & (p `1 ) <= b);

 hence thesis by A3;

 end;

 theorem :: JGRAPH_6:19



 Th19: for a,b,c,d be Real, p be Point of ( TOP-REAL 2) st a <= b & c <= d & p in ( rectangle (a,b,c,d)) holds a <= (p `1 ) & (p `1 ) <= b & c <= (p `2 ) & (p `2 ) <= d

 proof

 let a,b,c,d be Real, p be Point of ( TOP-REAL 2);

 assume that

 A1: a <= b and

 A2: c <= d and

 A3: p in ( rectangle (a,b,c,d));

 p in { p2 : (p2 `1 ) = a & (p2 `2 ) <= d & (p2 `2 ) >= c or (p2 `1 ) <= b & (p2 `1 ) >= a & (p2 `2 ) = d or (p2 `1 ) <= b & (p2 `1 ) >= a & (p2 `2 ) = c or (p2 `1 ) = b & (p2 `2 ) <= d & (p2 `2 ) >= c } by A1, A2, A3, SPPOL_2: 54;

 then

 A4: ex p2 st (p2 = p) & ((p2 `1 ) = a & (p2 `2 ) <= d & (p2 `2 ) >= c or (p2 `1 ) <= b & (p2 `1 ) >= a & (p2 `2 ) = d or (p2 `1 ) <= b & (p2 `1 ) >= a & (p2 `2 ) = c or (p2 `1 ) = b & (p2 `2 ) <= d & (p2 `2 ) >= c);

 per cases by A4;

 suppose (p `1 ) = a & c <= (p `2 ) & (p `2 ) <= d;

 hence thesis by A1;

 end;

 suppose (p `2 ) = d & a <= (p `1 ) & (p `1 ) <= b;

 hence thesis by A2;

 end;

 suppose (p `1 ) = b & c <= (p `2 ) & (p `2 ) <= d;

 hence thesis by A1;

 end;

 suppose (p `2 ) = c & a <= (p `1 ) & (p `1 ) <= b;

 hence thesis by A2;

 end;

 end;

 definition

 let a,b,c,d be Real;

 :: JGRAPH_6:def1

 func inside_of_rectangle (a,b,c,d) -> Subset of ( TOP-REAL 2) equals { p : a < (p `1 ) & (p `1 ) < b & c < (p `2 ) & (p `2 ) < d };

 coherence

 proof

 defpred P[ Point of ( TOP-REAL 2)] means a < ($1 `1 ) & ($1 `1 ) < b & c < ($1 `2 ) & ($1 `2 ) < d;

 { p : P[p] } c= the carrier of ( TOP-REAL 2) from FRAENKEL:sch 10;

 hence thesis;

 end;

 end

 definition

 let a,b,c,d be Real;

 :: JGRAPH_6:def2

 func closed_inside_of_rectangle (a,b,c,d) -> Subset of ( TOP-REAL 2) equals { p : a <= (p `1 ) & (p `1 ) <= b & c <= (p `2 ) & (p `2 ) <= d };

 coherence

 proof

 defpred P[ Point of ( TOP-REAL 2)] means a <= ($1 `1 ) & ($1 `1 ) <= b & c <= ($1 `2 ) & ($1 `2 ) <= d;

 { p : P[p] } c= the carrier of ( TOP-REAL 2) from FRAENKEL:sch 10;

 hence thesis;

 end;

 end

 definition

 let a,b,c,d be Real;

 :: JGRAPH_6:def3

 func outside_of_rectangle (a,b,c,d) -> Subset of ( TOP-REAL 2) equals { p : not (a <= (p `1 ) & (p `1 ) <= b & c <= (p `2 ) & (p `2 ) <= d) };

 coherence

 proof

 defpred P[ Point of ( TOP-REAL 2)] means not (a <= ($1 `1 ) & ($1 `1 ) <= b & c <= ($1 `2 ) & ($1 `2 ) <= d);

 { p : P[p] } c= the carrier of ( TOP-REAL 2) from FRAENKEL:sch 10;

 hence thesis;

 end;

 end

 definition

 let a,b,c,d be Real;

 :: JGRAPH_6:def4

 func closed_outside_of_rectangle (a,b,c,d) -> Subset of ( TOP-REAL 2) equals { p : not (a < (p `1 ) & (p `1 ) < b & c < (p `2 ) & (p `2 ) < d) };

 coherence

 proof

 defpred P[ Point of ( TOP-REAL 2)] means not (a < ($1 `1 ) & ($1 `1 ) < b & c < ($1 `2 ) & ($1 `2 ) < d);

 { p : P[p] } c= the carrier of ( TOP-REAL 2) from FRAENKEL:sch 10;

 hence thesis;

 end;

 end

 theorem :: JGRAPH_6:20



 Th20: for a,b,r be Real, Kb,Cb be Subset of ( TOP-REAL 2) st r >= 0 & Kb = { q : |.q.| = 1 } & Cb = { p2 where p2 be Point of ( TOP-REAL 2) : |.(p2 - |[a, b]|).| = r } holds (( AffineMap (r,a,r,b)) .: Kb) = Cb

 proof

 let a,b,r be Real, Kb,Cb be Subset of ( TOP-REAL 2);

 assume

 A1: r >= 0 & Kb = { q : |.q.| = 1 } & Cb = { p2 where p2 be Point of ( TOP-REAL 2) : |.(p2 - |[a, b]|).| = r };

 reconsider rr = r as Real;



 A2: (( AffineMap (r,a,r,b)) .: Kb) c= Cb

 proof

 let y be object;

 assume y in (( AffineMap (r,a,r,b)) .: Kb);

 then

 consider x be object such that x in ( dom ( AffineMap (r,a,r,b))) and

 A3: x in Kb and

 A4: y = (( AffineMap (r,a,r,b)) . x) by FUNCT_1:def 6;

 consider p be Point of ( TOP-REAL 2) such that

 A5: x = p and

 A6: |.p.| = 1 by A1, A3;



 A7: (( AffineMap (r,a,r,b)) . p) = |[((r * (p `1 )) + a), ((r * (p `2 )) + b)]| by JGRAPH_2:def 2;

 then

 reconsider q = y as Point of ( TOP-REAL 2) by A4, A5;



 A8: (q - |[a, b]|) = |[(((r * (p `1 )) + a) - a), (((r * (p `2 )) + b) - b)]| by A4, A5, A7, EUCLID: 62

 .= (r * |[(p `1 ), (p `2 )]|) by EUCLID: 58

 .= (r * p) by EUCLID: 53;

 |.(r * p).| = ( |.rr.| * |.p.|) by TOPRNS_1: 7

 .= r by A1, A6, ABSVALUE:def 1;

 hence thesis by A1, A8;

 end;

 Cb c= (( AffineMap (r,a,r,b)) .: Kb)

 proof

 let y be object;

 assume y in Cb;

 then

 consider p2 be Point of ( TOP-REAL 2) such that

 A9: y = p2 and

 A10: |.(p2 - |[a, b]|).| = r by A1;

 now

 per cases by A1;

 case

 A11: r > 0 ;

 set p1 = ((1 / r) * (p2 - |[a, b]|));

 |.p1.| = ( |.(1 / rr).| * |.(p2 - |[a, b]|).|) by TOPRNS_1: 7

 .= ((1 / r) * r) by A10, ABSVALUE:def 1

 .= 1 by A11, XCMPLX_1: 87;

 then

 A12: p1 in Kb by A1;



 A13: p1 = |[((1 / r) * ((p2 - |[a, b]|) `1 )), ((1 / r) * ((p2 - |[a, b]|) `2 ))]| by EUCLID: 57;

 then

 A14: (p1 `1 ) = ((1 / r) * ((p2 - |[a, b]|) `1 )) by EUCLID: 52;



 A15: (p1 `2 ) = ((1 / r) * ((p2 - |[a, b]|) `2 )) by A13, EUCLID: 52;



 A16: (r * (p1 `1 )) = ((r * (1 / r)) * ((p2 - |[a, b]|) `1 )) by A14

 .= (1 * ((p2 - |[a, b]|) `1 )) by A11, XCMPLX_1: 87

 .= ((p2 `1 ) - ( |[a, b]| `1 )) by TOPREAL3: 3

 .= ((p2 `1 ) - a) by EUCLID: 52;



 A17: (r * (p1 `2 )) = ((r * (1 / r)) * ((p2 - |[a, b]|) `2 )) by A15

 .= (1 * ((p2 - |[a, b]|) `2 )) by A11, XCMPLX_1: 87

 .= ((p2 `2 ) - ( |[a, b]| `2 )) by TOPREAL3: 3

 .= ((p2 `2 ) - b) by EUCLID: 52;



 A18: (( AffineMap (r,a,r,b)) . p1) = |[((r * (p1 `1 )) + a), ((r * (p1 `2 )) + b)]| by JGRAPH_2:def 2

 .= p2 by A16, A17, EUCLID: 53;

 ( dom ( AffineMap (r,a,r,b))) = the carrier of ( TOP-REAL 2) by FUNCT_2:def 1;

 hence thesis by A9, A12, A18, FUNCT_1:def 6;

 end;

 case

 A19: r = 0 ;

 set p1 = |[1, 0 ]|;



 A20: (p1 `1 ) = 1 by EUCLID: 52;

 (p1 `2 ) = 0 by EUCLID: 52;



 then |.p1.| = ( sqrt ((1 ^2 ) + ( 0 ^2 ))) by A20, JGRAPH_3: 1

 .= 1 by SQUARE_1: 22;

 then

 A21: p1 in Kb by A1;



 A22: (( AffineMap (r,a,r,b)) . p1) = |[(( 0 * (p1 `1 )) + a), (( 0 * (p1 `2 )) + b)]| by A19, JGRAPH_2:def 2

 .= p2 by A10, A19, TOPRNS_1: 28;

 ( dom ( AffineMap (r,a,r,b))) = the carrier of ( TOP-REAL 2) by FUNCT_2:def 1;

 hence thesis by A9, A21, A22, FUNCT_1:def 6;

 end;

 end;

 hence thesis;

 end;

 hence thesis by A2;

 end;

 theorem :: JGRAPH_6:21



 Th21: for P,Q be Subset of ( TOP-REAL 2) st (ex f be Function of (( TOP-REAL 2) | P), (( TOP-REAL 2) | Q) st f is being_homeomorphism) & P is being_simple_closed_curve holds Q is being_simple_closed_curve

 proof

 let P,Q be Subset of ( TOP-REAL 2);

 assume that

 A1: ex f be Function of (( TOP-REAL 2) | P), (( TOP-REAL 2) | Q) st f is being_homeomorphism and

 A2: P is being_simple_closed_curve;

 consider f be Function of (( TOP-REAL 2) | P), (( TOP-REAL 2) | Q) such that

 A3: f is being_homeomorphism by A1;

 consider g be Function of (( TOP-REAL 2) | R^2-unit_square ), (( TOP-REAL 2) | P) such that

 A4: g is being_homeomorphism by A2, TOPREAL2:def 1;



 A5: ( |[1, 0 ]| `1 ) = 1 by EUCLID: 52;

 ( |[1, 0 ]| `2 ) = 0 by EUCLID: 52;

 then

 A6: |[1, 0 ]| in R^2-unit_square by A5, TOPREAL1: 14;



 A7: ( dom g) = ( [#] (( TOP-REAL 2) | R^2-unit_square )) by A4, TOPS_2:def 5;



 A8: ( rng g) = ( [#] (( TOP-REAL 2) | P)) by A4, TOPS_2:def 5;

 ( dom g) = R^2-unit_square by A7, PRE_TOPC:def 5;

 then

 A9: (g . |[1, 0 ]|) in ( rng g) by A6, FUNCT_1: 3;

 then

 A10: (g . |[1, 0 ]|) in P by A8, PRE_TOPC:def 5;

 reconsider P1 = P as non empty Subset of ( TOP-REAL 2) by A9;

 ( dom f) = ( [#] (( TOP-REAL 2) | P)) by A3, TOPS_2:def 5;

 then ( dom f) = P by PRE_TOPC:def 5;

 then (f . (g . |[1, 0 ]|)) in ( rng f) by A10, FUNCT_1: 3;

 then

 reconsider Q1 = Q as non empty Subset of ( TOP-REAL 2);

 reconsider f1 = f as Function of (( TOP-REAL 2) | P1), (( TOP-REAL 2) | Q1);

 reconsider g1 = g as Function of (( TOP-REAL 2) | R^2-unit_square ), (( TOP-REAL 2) | P1);

 reconsider h = (f1 * g1) as Function of (( TOP-REAL 2) | R^2-unit_square ), (( TOP-REAL 2) | Q1);

 h is being_homeomorphism by A3, A4, TOPS_2: 57;

 hence thesis by TOPREAL2:def 1;

 end;

 theorem :: JGRAPH_6:22



 Th22: for P be Subset of ( TOP-REAL 2) st P is being_simple_closed_curve holds P is compact;

 theorem :: JGRAPH_6:23



 Th23: for a,b,r be Real, Cb be Subset of ( TOP-REAL 2) st r > 0 & Cb = { p where p be Point of ( TOP-REAL 2) : |.(p - |[a, b]|).| = r } holds Cb is being_simple_closed_curve

 proof

 let a,b,r be Real, Cb be Subset of ( TOP-REAL 2);

 assume that

 A1: r > 0 and

 A2: Cb = { p where p be Point of ( TOP-REAL 2) : |.(p - |[a, b]|).| = r };



 A3: ( |[r, 0 ]| `1 ) = r by EUCLID: 52;



 A4: ( |[r, 0 ]| `2 ) = 0 by EUCLID: 52;

 |.( |[(r + a), b]| - |[a, b]|).| = |.( |[(r + a), ( 0 + b)]| - |[a, b]|).|

 .= |.(( |[r, 0 ]| + |[a, b]|) - |[a, b]|).| by EUCLID: 56

 .= |.( |[r, 0 ]| + ( |[a, b]| - |[a, b]|)).| by RLVECT_1:def 3

 .= |.( |[r, 0 ]| + ( 0. ( TOP-REAL 2))).| by RLVECT_1: 5

 .= |. |[r, 0 ]|.| by RLVECT_1: 4

 .= ( sqrt ((r ^2 ) + ( 0 ^2 ))) by A3, A4, JGRAPH_3: 1

 .= r by A1, SQUARE_1: 22;

 then |[(r + a), b]| in Cb by A2;

 then

 reconsider Cbb = Cb as non empty Subset of ( TOP-REAL 2);

 set v = |[1, 0 ]|;



 A5: (v `1 ) = 1 by EUCLID: 52;

 (v `2 ) = 0 by EUCLID: 52;



 then |.v.| = ( sqrt ((1 ^2 ) + ( 0 ^2 ))) by A5, JGRAPH_3: 1

 .= 1 by SQUARE_1: 22;

 then

 A6: |[1, 0 ]| in { q : |.q.| = 1 };

 defpred P[ Point of ( TOP-REAL 2)] means |.$1.| = 1;

 { q where q be Element of ( TOP-REAL 2) : P[q] } is Subset of ( TOP-REAL 2) from DOMAIN_1:sch 7;

 then

 reconsider Kb = { q : |.q.| = 1 } as non empty Subset of ( TOP-REAL 2) by A6;



 A7: the carrier of (( TOP-REAL 2) | Kb) = Kb by PRE_TOPC: 8;

 set SC = ( AffineMap (r,a,r,b));



 A8: SC is one-to-one by A1, JGRAPH_2: 44;



 A9: ( dom SC) = the carrier of ( TOP-REAL 2) by FUNCT_2:def 1;



 A10: ( dom (SC | Kb)) = (( dom SC) /\ Kb) by RELAT_1: 61

 .= the carrier of (( TOP-REAL 2) | Kb) by A7, A9, XBOOLE_1: 28;



 A11: ( rng (SC | Kb)) c= ((SC | Kb) .: the carrier of (( TOP-REAL 2) | Kb))

 proof

 let u be object;

 assume u in ( rng (SC | Kb));

 then ex z be object st (z in ( dom (SC | Kb))) & (u = ((SC | Kb) . z)) by FUNCT_1:def 3;

 hence thesis by A10, FUNCT_1:def 6;

 end;

 ((SC | Kb) .: the carrier of (( TOP-REAL 2) | Kb)) = (SC .: Kb) by A7, RELAT_1: 129

 .= Cb by A1, A2, Th20

 .= the carrier of (( TOP-REAL 2) | Cbb) by PRE_TOPC: 8;

 then

 reconsider f0 = (SC | Kb) as Function of (( TOP-REAL 2) | Kb), (( TOP-REAL 2) | Cbb) by A10, A11, FUNCT_2: 2;

 ( rng (SC | Kb)) c= the carrier of ( TOP-REAL 2);

 then

 reconsider f00 = f0 as Function of (( TOP-REAL 2) | Kb), ( TOP-REAL 2) by A10, FUNCT_2: 2;



 A12: ( rng f0) = ((SC | Kb) .: the carrier of (( TOP-REAL 2) | Kb)) by RELSET_1: 22

 .= (SC .: Kb) by A7, RELAT_1: 129

 .= Cb by A1, A2, Th20;



 A13: f0 is one-to-one by A8, FUNCT_1: 52;

 Kb is compact by Th22, JGRAPH_3: 26;

 then ex f1 be Function of (( TOP-REAL 2) | Kb), (( TOP-REAL 2) | Cbb) st f00 = f1 & f1 is being_homeomorphism by A12, A13, JGRAPH_1: 46, TOPMETR: 7;

 hence thesis by Th21, JGRAPH_3: 26;

 end;

 definition

 let a,b,r be Real;

 :: JGRAPH_6:def5

 func circle (a,b,r) -> Subset of ( TOP-REAL 2) equals { p where p be Point of ( TOP-REAL 2) : |.(p - |[a, b]|).| = r };

 coherence

 proof

 defpred P[ Point of ( TOP-REAL 2)] means |.($1 - |[a, b]|).| = r;

 { p where p be Point of ( TOP-REAL 2) : P[p] } c= the carrier of ( TOP-REAL 2) from FRAENKEL:sch 10;

 hence thesis;

 end;

 end

 registration

 let a,b,r be Real;

 cluster ( circle (a,b,r)) -> compact;

 coherence

 proof

 set Cb = ( circle (a,b,r));

 per cases ;

 suppose

 A1: r < 0 ;

 Cb = {}

 proof

 hereby

 let x be object;

 assume x in Cb;

 then ex p be Point of ( TOP-REAL 2) st x = p & |.(p - |[a, b]|).| = r;

 hence x in {} by A1;

 end;

 thus thesis;

 end;

 hence thesis;

 end;

 suppose r > 0 ;

 hence thesis by Th22, Th23;

 end;

 suppose

 A2: r = 0 ;

 Cb = { |[a, b]|}

 proof

 hereby

 let x be object;

 assume x in Cb;

 then

 consider p be Point of ( TOP-REAL 2) such that

 A3: x = p and

 A4: |.(p - |[a, b]|).| = r;

 p = |[a, b]| by A2, A4, TOPRNS_1: 28;

 hence x in { |[a, b]|} by A3, TARSKI:def 1;

 end;

 let x be object;

 assume x in { |[a, b]|};

 then

 A5: x = |[a, b]| by TARSKI:def 1;

 |.( |[a, b]| - |[a, b]|).| = 0 by TOPRNS_1: 28;

 hence thesis by A2, A5;

 end;

 hence thesis;

 end;

 end;

 end

 registration

 let a,b be Real;

 let r be non negative Real;

 cluster ( circle (a,b,r)) -> non empty;

 coherence

 proof

 set Cb = ( circle (a,b,r));



 A1: ( |[r, 0 ]| `1 ) = r by EUCLID: 52;



 A2: ( |[r, 0 ]| `2 ) = 0 by EUCLID: 52;

 |.( |[(r + a), b]| - |[a, b]|).| = |.( |[(r + a), ( 0 + b)]| - |[a, b]|).|

 .= |.(( |[r, 0 ]| + |[a, b]|) - |[a, b]|).| by EUCLID: 56

 .= |.( |[r, 0 ]| + ( |[a, b]| - |[a, b]|)).| by RLVECT_1:def 3

 .= |.( |[r, 0 ]| + ( 0. ( TOP-REAL 2))).| by RLVECT_1: 5

 .= |. |[r, 0 ]|.| by RLVECT_1: 4

 .= ( sqrt ((r ^2 ) + ( 0 ^2 ))) by A1, A2, JGRAPH_3: 1

 .= r by SQUARE_1: 22;

 then |[(r + a), b]| in Cb;

 hence thesis;

 end;

 end

 definition

 let a,b,r be Real;

 :: JGRAPH_6:def6

 func inside_of_circle (a,b,r) -> Subset of ( TOP-REAL 2) equals { p where p be Point of ( TOP-REAL 2) : |.(p - |[a, b]|).| < r };

 coherence

 proof

 defpred P[ Point of ( TOP-REAL 2)] means |.($1 - |[a, b]|).| < r;

 { p where p be Point of ( TOP-REAL 2) : P[p] } c= the carrier of ( TOP-REAL 2) from FRAENKEL:sch 10;

 hence thesis;

 end;

 end

 definition

 let a,b,r be Real;

 :: JGRAPH_6:def7

 func closed_inside_of_circle (a,b,r) -> Subset of ( TOP-REAL 2) equals { p where p be Point of ( TOP-REAL 2) : |.(p - |[a, b]|).| <= r };

 coherence

 proof

 defpred P[ Point of ( TOP-REAL 2)] means |.($1 - |[a, b]|).| <= r;

 { p where p be Point of ( TOP-REAL 2) : P[p] } c= the carrier of ( TOP-REAL 2) from FRAENKEL:sch 10;

 hence thesis;

 end;

 end

 definition

 let a,b,r be Real;

 :: JGRAPH_6:def8

 func outside_of_circle (a,b,r) -> Subset of ( TOP-REAL 2) equals { p where p be Point of ( TOP-REAL 2) : |.(p - |[a, b]|).| > r };

 coherence

 proof

 defpred P[ Point of ( TOP-REAL 2)] means |.($1 - |[a, b]|).| > r;

 { p where p be Point of ( TOP-REAL 2) : P[p] } c= the carrier of ( TOP-REAL 2) from FRAENKEL:sch 10;

 hence thesis;

 end;

 end

 definition

 let a,b,r be Real;

 :: JGRAPH_6:def9

 func closed_outside_of_circle (a,b,r) -> Subset of ( TOP-REAL 2) equals { p where p be Point of ( TOP-REAL 2) : |.(p - |[a, b]|).| >= r };

 coherence

 proof

 defpred P[ Point of ( TOP-REAL 2)] means |.($1 - |[a, b]|).| >= r;

 { p where p be Point of ( TOP-REAL 2) : P[p] } c= the carrier of ( TOP-REAL 2) from FRAENKEL:sch 10;

 hence thesis;

 end;

 end

 theorem :: JGRAPH_6:24



 Th24: for r be Real holds ( inside_of_circle ( 0 , 0 ,r)) = { p : |.p.| < r } & (r > 0 implies ( circle ( 0 , 0 ,r)) = { p2 : |.p2.| = r }) & ( outside_of_circle ( 0 , 0 ,r)) = { p3 : |.p3.| > r } & ( closed_inside_of_circle ( 0 , 0 ,r)) = { q : |.q.| <= r } & ( closed_outside_of_circle ( 0 , 0 ,r)) = { q2 : |.q2.| >= r }

 proof

 let r be Real;

 defpred P[ Point of ( TOP-REAL 2)] means |.($1 - |[ 0 , 0 ]|).| < r;

 defpred Q[ Point of ( TOP-REAL 2)] means |.$1.| < r;

 deffunc F( set) = $1;



 A1: for p holds P[p] iff Q[p] by EUCLID: 54, RLVECT_1: 13;

 ( inside_of_circle ( 0 , 0 ,r)) = { F(p) where p be Point of ( TOP-REAL 2) : P[p] }

 .= { F(p2) where p2 be Point of ( TOP-REAL 2) : Q[p2] } from FRAENKEL:sch 3( A1);

 hence ( inside_of_circle ( 0 , 0 ,r)) = { p : |.p.| < r };

 defpred P[ Point of ( TOP-REAL 2)] means |.($1 - |[ 0 , 0 ]|).| = r;

 defpred Q[ Point of ( TOP-REAL 2)] means |.$1.| = r;



 A2: for p holds P[p] iff Q[p] by EUCLID: 54, RLVECT_1: 13;

 hereby

 assume r > 0 ;

 ( circle ( 0 , 0 ,r)) = { F(p) : P[p] }

 .= { F(p2) where p2 be Point of ( TOP-REAL 2) : Q[p2] } from FRAENKEL:sch 3( A2);

 hence ( circle ( 0 , 0 ,r)) = { p2 : |.p2.| = r };

 end;

 defpred P[ Point of ( TOP-REAL 2)] means |.($1 - |[ 0 , 0 ]|).| > r;

 defpred Q[ Point of ( TOP-REAL 2)] means |.$1.| > r;



 A3: for p holds P[p] iff Q[p] by EUCLID: 54, RLVECT_1: 13;

 ( outside_of_circle ( 0 , 0 ,r)) = { F(p) : P[p] }

 .= { F(p2) where p2 be Point of ( TOP-REAL 2) : Q[p2] } from FRAENKEL:sch 3( A3);

 hence ( outside_of_circle ( 0 , 0 ,r)) = { p3 : |.p3.| > r };

 defpred P[ Point of ( TOP-REAL 2)] means |.($1 - |[ 0 , 0 ]|).| <= r;

 defpred Q[ Point of ( TOP-REAL 2)] means |.$1.| <= r;



 A4: for p holds P[p] iff Q[p] by EUCLID: 54, RLVECT_1: 13;

 ( closed_inside_of_circle ( 0 , 0 ,r)) = { F(p) : P[p] }

 .= { F(p2) where p2 be Point of ( TOP-REAL 2) : Q[p2] } from FRAENKEL:sch 3( A4);

 hence ( closed_inside_of_circle ( 0 , 0 ,r)) = { p : |.p.| <= r };

 defpred P[ Point of ( TOP-REAL 2)] means |.($1 - |[ 0 , 0 ]|).| >= r;

 defpred Q[ Point of ( TOP-REAL 2)] means |.$1.| >= r;



 A5: for p holds P[p] iff Q[p] by EUCLID: 54, RLVECT_1: 13;

 ( closed_outside_of_circle ( 0 , 0 ,r)) = { F(p) : P[p] }

 .= { F(p2) where p2 be Point of ( TOP-REAL 2) : Q[p2] } from FRAENKEL:sch 3( A5);

 hence thesis;

 end;

 theorem :: JGRAPH_6:25



 Th25: for Kb,Cb be Subset of ( TOP-REAL 2) st Kb = { p : ( - 1) < (p `1 ) & (p `1 ) < 1 & ( - 1) < (p `2 ) & (p `2 ) < 1 } & Cb = { p2 where p2 be Point of ( TOP-REAL 2) : |.p2.| < 1 } holds ( Sq_Circ .: Kb) = Cb

 proof

 let Kb,Cb be Subset of ( TOP-REAL 2);

 assume

 A1: Kb = { p : ( - 1) < (p `1 ) & (p `1 ) < 1 & ( - 1) < (p `2 ) & (p `2 ) < 1 } & Cb = { p2 where p2 be Point of ( TOP-REAL 2) : |.p2.| < 1 };

 thus ( Sq_Circ .: Kb) c= Cb

 proof

 let y be object;

 assume y in ( Sq_Circ .: Kb);

 then

 consider x be object such that x in ( dom Sq_Circ ) and

 A2: x in Kb and

 A3: y = ( Sq_Circ . x) by FUNCT_1:def 6;

 consider q be Point of ( TOP-REAL 2) such that

 A4: q = x and

 A5: ( - 1) < (q `1 ) and

 A6: (q `1 ) < 1 and

 A7: ( - 1) < (q `2 ) and

 A8: (q `2 ) < 1 by A1, A2;

 now

 per cases ;

 case

 A9: q = ( 0. ( TOP-REAL 2));

 then

 A10: ( Sq_Circ . q) = q by JGRAPH_3:def 1;

 |.q.| = 0 by A9, TOPRNS_1: 23;

 hence ex p2 be Point of ( TOP-REAL 2) st p2 = y & |.p2.| < 1 by A3, A4, A10;

 end;

 case

 A11: q <> ( 0. ( TOP-REAL 2)) & ((q `2 ) <= (q `1 ) & ( - (q `1 )) <= (q `2 ) or (q `2 ) >= (q `1 ) & (q `2 ) <= ( - (q `1 )));

 then

 A12: ( Sq_Circ . q) = |[((q `1 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))), ((q `2 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 ))))]| by JGRAPH_3:def 1;



 A13: ( |[((q `1 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))), ((q `2 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 ))))]| `1 ) = ((q `1 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) by EUCLID: 52;



 A14: ( |[((q `1 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))), ((q `2 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 ))))]| `2 ) = ((q `2 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) by EUCLID: 52;



 A15: (1 + (((q `2 ) / (q `1 )) ^2 )) > 0 by XREAL_1: 34, XREAL_1: 63;

 A16:

 now

 assume

 A17: (q `1 ) = 0 ;

 then (q `2 ) = 0 by A11;

 hence contradiction by A11, A17, EUCLID: 53, EUCLID: 54;

 end;

 then

 A18: ((q `1 ) ^2 ) > 0 by SQUARE_1: 12;

 ((q `1 ) ^2 ) < (1 ^2 ) by A5, A6, SQUARE_1: 50;

 then

 A19: ( sqrt ((q `1 ) ^2 )) < 1 by A18, SQUARE_1: 18, SQUARE_1: 27;

 ( |. |[((q `1 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))), ((q `2 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 ))))]|.| ^2 ) = ((((q `1 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) ^2 ) + (((q `2 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) ^2 )) by A13, A14, JGRAPH_3: 1

 .= ((((q `1 ) ^2 ) / (( sqrt (1 + (((q `2 ) / (q `1 )) ^2 ))) ^2 )) + (((q `2 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) ^2 )) by XCMPLX_1: 76

 .= ((((q `1 ) ^2 ) / (( sqrt (1 + (((q `2 ) / (q `1 )) ^2 ))) ^2 )) + (((q `2 ) ^2 ) / (( sqrt (1 + (((q `2 ) / (q `1 )) ^2 ))) ^2 ))) by XCMPLX_1: 76

 .= ((((q `1 ) ^2 ) / (1 + (((q `2 ) / (q `1 )) ^2 ))) + (((q `2 ) ^2 ) / (( sqrt (1 + (((q `2 ) / (q `1 )) ^2 ))) ^2 ))) by A15, SQUARE_1:def 2

 .= ((((q `1 ) ^2 ) / (1 + (((q `2 ) / (q `1 )) ^2 ))) + (((q `2 ) ^2 ) / (1 + (((q `2 ) / (q `1 )) ^2 )))) by A15, SQUARE_1:def 2

 .= ((((q `1 ) ^2 ) + ((q `2 ) ^2 )) / (1 + (((q `2 ) / (q `1 )) ^2 ))) by XCMPLX_1: 62

 .= ((((q `1 ) ^2 ) + ((q `2 ) ^2 )) / (1 + (((q `2 ) ^2 ) / ((q `1 ) ^2 )))) by XCMPLX_1: 76

 .= ((((q `1 ) ^2 ) + ((q `2 ) ^2 )) / ((((q `1 ) ^2 ) / ((q `1 ) ^2 )) + (((q `2 ) ^2 ) / ((q `1 ) ^2 )))) by A18, XCMPLX_1: 60

 .= ((((q `1 ) ^2 ) + ((q `2 ) ^2 )) / ((((q `1 ) ^2 ) + ((q `2 ) ^2 )) / ((q `1 ) ^2 ))) by XCMPLX_1: 62

 .= (((q `1 ) ^2 ) * ((((q `1 ) ^2 ) + ((q `2 ) ^2 )) / (((q `1 ) ^2 ) + ((q `2 ) ^2 )))) by XCMPLX_1: 81

 .= (((q `1 ) ^2 ) * 1) by A16, COMPLEX1: 1, XCMPLX_1: 60

 .= ((q `1 ) ^2 );

 then |. |[((q `1 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))), ((q `2 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 ))))]|.| < 1 by A19, SQUARE_1: 22;

 hence ex p2 be Point of ( TOP-REAL 2) st p2 = y & |.p2.| < 1 by A3, A4, A12;

 end;

 case

 A20: q <> ( 0. ( TOP-REAL 2)) & not ((q `2 ) <= (q `1 ) & ( - (q `1 )) <= (q `2 ) or (q `2 ) >= (q `1 ) & (q `2 ) <= ( - (q `1 )));

 then

 A21: ( Sq_Circ . q) = |[((q `1 ) / ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))), ((q `2 ) / ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 ))))]| by JGRAPH_3:def 1;



 A22: ( |[((q `1 ) / ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))), ((q `2 ) / ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 ))))]| `1 ) = ((q `1 ) / ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))) by EUCLID: 52;



 A23: ( |[((q `1 ) / ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))), ((q `2 ) / ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 ))))]| `2 ) = ((q `2 ) / ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))) by EUCLID: 52;



 A24: (1 + (((q `1 ) / (q `2 )) ^2 )) > 0 by XREAL_1: 34, XREAL_1: 63;



 A25: (q `2 ) <> 0 by A20;

 then

 A26: ((q `2 ) ^2 ) > 0 by SQUARE_1: 12;

 ((q `2 ) ^2 ) < (1 ^2 ) by A7, A8, SQUARE_1: 50;

 then

 A27: ( sqrt ((q `2 ) ^2 )) < 1 by A26, SQUARE_1: 18, SQUARE_1: 27;

 ( |. |[((q `1 ) / ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))), ((q `2 ) / ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 ))))]|.| ^2 ) = ((((q `1 ) / ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))) ^2 ) + (((q `2 ) / ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))) ^2 )) by A22, A23, JGRAPH_3: 1

 .= ((((q `1 ) ^2 ) / (( sqrt (1 + (((q `1 ) / (q `2 )) ^2 ))) ^2 )) + (((q `2 ) / ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))) ^2 )) by XCMPLX_1: 76

 .= ((((q `1 ) ^2 ) / (( sqrt (1 + (((q `1 ) / (q `2 )) ^2 ))) ^2 )) + (((q `2 ) ^2 ) / (( sqrt (1 + (((q `1 ) / (q `2 )) ^2 ))) ^2 ))) by XCMPLX_1: 76

 .= ((((q `1 ) ^2 ) / (1 + (((q `1 ) / (q `2 )) ^2 ))) + (((q `2 ) ^2 ) / (( sqrt (1 + (((q `1 ) / (q `2 )) ^2 ))) ^2 ))) by A24, SQUARE_1:def 2

 .= ((((q `1 ) ^2 ) / (1 + (((q `1 ) / (q `2 )) ^2 ))) + (((q `2 ) ^2 ) / (1 + (((q `1 ) / (q `2 )) ^2 )))) by A24, SQUARE_1:def 2

 .= ((((q `1 ) ^2 ) + ((q `2 ) ^2 )) / (1 + (((q `1 ) / (q `2 )) ^2 ))) by XCMPLX_1: 62

 .= ((((q `1 ) ^2 ) + ((q `2 ) ^2 )) / (1 + (((q `1 ) ^2 ) / ((q `2 ) ^2 )))) by XCMPLX_1: 76

 .= ((((q `1 ) ^2 ) + ((q `2 ) ^2 )) / ((((q `1 ) ^2 ) / ((q `2 ) ^2 )) + (((q `2 ) ^2 ) / ((q `2 ) ^2 )))) by A26, XCMPLX_1: 60

 .= ((((q `1 ) ^2 ) + ((q `2 ) ^2 )) / ((((q `1 ) ^2 ) + ((q `2 ) ^2 )) / ((q `2 ) ^2 ))) by XCMPLX_1: 62

 .= (((q `2 ) ^2 ) * ((((q `1 ) ^2 ) + ((q `2 ) ^2 )) / (((q `1 ) ^2 ) + ((q `2 ) ^2 )))) by XCMPLX_1: 81

 .= (((q `2 ) ^2 ) * 1) by A25, COMPLEX1: 1, XCMPLX_1: 60

 .= ((q `2 ) ^2 );

 then |. |[((q `1 ) / ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))), ((q `2 ) / ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 ))))]|.| < 1 by A27, SQUARE_1: 22;

 hence ex p2 be Point of ( TOP-REAL 2) st p2 = y & |.p2.| < 1 by A3, A4, A21;

 end;

 end;

 hence thesis by A1;

 end;

 let y be object;

 assume y in Cb;

 then

 consider p2 be Point of ( TOP-REAL 2) such that

 A28: p2 = y and

 A29: |.p2.| < 1 by A1;

 set q = p2;

 now

 per cases ;

 case

 A30: q = ( 0. ( TOP-REAL 2));

 then

 A31: (q `1 ) = 0 by EUCLID: 52, EUCLID: 54;

 (q `2 ) = 0 by A30, EUCLID: 52, EUCLID: 54;

 then

 A32: y in Kb by A1, A28, A31;



 A33: (( Sq_Circ " ) . y) = y by A28, A30, JGRAPH_3: 28;



 A34: ( dom Sq_Circ ) = the carrier of ( TOP-REAL 2) by FUNCT_2:def 1;

 y = ( Sq_Circ . y) by A28, A33, FUNCT_1: 35, JGRAPH_3: 43;

 hence ex x be set st x in ( dom Sq_Circ ) & x in Kb & y = ( Sq_Circ . x) by A32, A34;

 end;

 case

 A35: q <> ( 0. ( TOP-REAL 2)) & ((q `2 ) <= (q `1 ) & ( - (q `1 )) <= (q `2 ) or (q `2 ) >= (q `1 ) & (q `2 ) <= ( - (q `1 )));

 set px = |[((q `1 ) * ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))), ((q `2 ) * ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 ))))]|;



 A36: (px `1 ) = ((q `1 ) * ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) by EUCLID: 52;



 A37: (px `2 ) = ((q `2 ) * ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) by EUCLID: 52;

 (1 + (((q `2 ) / (q `1 )) ^2 )) > 0 by XREAL_1: 34, XREAL_1: 63;

 then

 A38: ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 ))) > 0 by SQUARE_1: 25;



 A39: (1 + (((px `2 ) / (px `1 )) ^2 )) > 0 by XREAL_1: 34, XREAL_1: 63;



 A40: ((px `2 ) / (px `1 )) = ((q `2 ) / (q `1 )) by A36, A37, A38, XCMPLX_1: 91;



 A41: (q `1 ) = (((q `1 ) * ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) by A38, XCMPLX_1: 89

 .= ((px `1 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) by EUCLID: 52;



 A42: (q `2 ) = (((q `2 ) * ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) by A38, XCMPLX_1: 89

 .= ((px `2 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) by EUCLID: 52;



 A43: ( |.q.| ^2 ) = (((q `1 ) ^2 ) + ((q `2 ) ^2 )) by JGRAPH_3: 1;



 A44: ( |.q.| ^2 ) < (1 ^2 ) by A29, SQUARE_1: 16;

 A45:

 now

 assume that

 A46: (px `1 ) = 0 and

 A47: (px `2 ) = 0 ;



 A48: ((q `1 ) * ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) = 0 by A46, EUCLID: 52;



 A49: ((q `2 ) * ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) = 0 by A47, EUCLID: 52;



 A50: (q `1 ) = 0 by A38, A48, XCMPLX_1: 6;

 (q `2 ) = 0 by A38, A49, XCMPLX_1: 6;

 hence contradiction by A35, A50, EUCLID: 53, EUCLID: 54;

 end;

 (q `2 ) <= (q `1 ) & ( - (q `1 )) <= (q `2 ) or (q `2 ) >= (q `1 ) & ((q `2 ) * ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) <= (( - (q `1 )) * ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) by A35, A38, XREAL_1: 64;

 then

 A51: (q `2 ) <= (q `1 ) & (( - (q `1 )) * ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) <= ((q `2 ) * ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) or (px `2 ) >= (px `1 ) & (px `2 ) <= ( - (px `1 )) by A36, A37, A38, XREAL_1: 64;

 then (px `2 ) <= (px `1 ) & ( - (px `1 )) <= (px `2 ) or (px `2 ) >= (px `1 ) & (px `2 ) <= ( - (px `1 )) by A36, A37, A38, XREAL_1: 64;

 then

 A52: ( Sq_Circ . px) = |[((px `1 ) / ( sqrt (1 + (((px `2 ) / (px `1 )) ^2 )))), ((px `2 ) / ( sqrt (1 + (((px `2 ) / (px `1 )) ^2 ))))]| by A45, JGRAPH_2: 3, JGRAPH_3:def 1;

 (px `2 ) <= (px `1 ) & ( - ( - (px `1 ))) >= ( - (px `2 )) or (px `2 ) >= (px `1 ) & (px `2 ) <= ( - (px `1 )) by A36, A37, A38, A51, XREAL_1: 24, XREAL_1: 64;

 then

 A53: (px `2 ) <= (px `1 ) & (px `1 ) >= ( - (px `2 )) or (px `2 ) >= (px `1 ) & ( - (px `2 )) >= ( - ( - (px `1 ))) by XREAL_1: 24;



 A54: ((px `1 ) / ( sqrt (1 + (((px `2 ) / (px `1 )) ^2 )))) = (q `1 ) by A36, A38, A40, XCMPLX_1: 89;



 A55: ((px `2 ) / ( sqrt (1 + (((px `2 ) / (px `1 )) ^2 )))) = (q `2 ) by A37, A38, A40, XCMPLX_1: 89;



 A56: ( dom Sq_Circ ) = the carrier of ( TOP-REAL 2) by FUNCT_2:def 1;

 not (px `1 ) = 0 by A36, A37, A38, A45, A51, XREAL_1: 64;

 then

 A57: ((px `1 ) ^2 ) > 0 by SQUARE_1: 12;



 A58: ((px `2 ) ^2 ) >= 0 by XREAL_1: 63;

 ((((px `1 ) ^2 ) / (( sqrt (1 + (((px `2 ) / (px `1 )) ^2 ))) ^2 )) + (((px `2 ) / ( sqrt (1 + (((px `2 ) / (px `1 )) ^2 )))) ^2 )) < 1 by A40, A41, A42, A43, A44, XCMPLX_1: 76;

 then ((((px `1 ) ^2 ) / (( sqrt (1 + (((px `2 ) / (px `1 )) ^2 ))) ^2 )) + (((px `2 ) ^2 ) / (( sqrt (1 + (((px `2 ) / (px `1 )) ^2 ))) ^2 ))) < 1 by XCMPLX_1: 76;

 then ((((px `1 ) ^2 ) / (1 + (((px `2 ) / (px `1 )) ^2 ))) + (((px `2 ) ^2 ) / (( sqrt (1 + (((px `2 ) / (px `1 )) ^2 ))) ^2 ))) < 1 by A39, SQUARE_1:def 2;

 then ((((px `1 ) ^2 ) / (1 + (((px `2 ) / (px `1 )) ^2 ))) + (((px `2 ) ^2 ) / (1 + (((px `2 ) / (px `1 )) ^2 )))) < 1 by A39, SQUARE_1:def 2;

 then (((((px `1 ) ^2 ) / (1 + (((px `2 ) / (px `1 )) ^2 ))) + (((px `2 ) ^2 ) / (1 + (((px `2 ) / (px `1 )) ^2 )))) * (1 + (((px `2 ) / (px `1 )) ^2 ))) < (1 * (1 + (((px `2 ) / (px `1 )) ^2 ))) by A39, XREAL_1: 68;

 then (((((px `1 ) ^2 ) / (1 + (((px `2 ) / (px `1 )) ^2 ))) * (1 + (((px `2 ) / (px `1 )) ^2 ))) + ((((px `2 ) ^2 ) / (1 + (((px `2 ) / (px `1 )) ^2 ))) * (1 + (((px `2 ) / (px `1 )) ^2 )))) < (1 * (1 + (((px `2 ) / (px `1 )) ^2 )));

 then (((px `1 ) ^2 ) + ((((px `2 ) ^2 ) / (1 + (((px `2 ) / (px `1 )) ^2 ))) * (1 + (((px `2 ) / (px `1 )) ^2 )))) < (1 * (1 + (((px `2 ) / (px `1 )) ^2 ))) by A39, XCMPLX_1: 87;

 then

 A59: (((px `1 ) ^2 ) + ((px `2 ) ^2 )) < (1 * (1 + (((px `2 ) / (px `1 )) ^2 ))) by A39, XCMPLX_1: 87;

 (1 * (1 + (((px `2 ) / (px `1 )) ^2 ))) = (1 + (((px `2 ) ^2 ) / ((px `1 ) ^2 ))) by XCMPLX_1: 76;

 then ((((px `1 ) ^2 ) + ((px `2 ) ^2 )) - 1) < ((1 + (((px `2 ) ^2 ) / ((px `1 ) ^2 ))) - 1) by A59, XREAL_1: 9;

 then (((((px `1 ) ^2 ) + ((px `2 ) ^2 )) - 1) * ((px `1 ) ^2 )) < ((((px `2 ) ^2 ) / ((px `1 ) ^2 )) * ((px `1 ) ^2 )) by A57, XREAL_1: 68;

 then

 A60: ((((px `1 ) ^2 ) + (((px `2 ) ^2 ) - 1)) * ((px `1 ) ^2 )) < ((px `2 ) ^2 ) by A57, XCMPLX_1: 87;

 (((((px `1 ) ^2 ) * ((px `1 ) ^2 )) + ((((px `1 ) ^2 ) * ((px `2 ) ^2 )) - (((px `1 ) ^2 ) * 1))) - ((px `2 ) ^2 )) = ((((px `1 ) ^2 ) - 1) * (((px `1 ) ^2 ) + ((px `2 ) ^2 )));

 then (((px `1 ) ^2 ) - 1) < 0 or (((px `1 ) ^2 ) + ((px `2 ) ^2 )) < 0 by A60, XREAL_1: 49;

 then

 A61: ((((px `1 ) ^2 ) - 1) + 1) < ( 0 + 1) by A58, XREAL_1: 6;

 then

 A62: (px `1 ) < (1 ^2 ) by SQUARE_1: 48;



 A63: (px `1 ) > ( - (1 ^2 )) by A61, SQUARE_1: 48;

 (px `2 ) < 1 & 1 > ( - (px `2 )) or (px `2 ) >= (px `1 ) & ( - (px `2 )) >= (px `1 ) by A53, A62, XXREAL_0: 2;

 then (px `2 ) < 1 & ( - 1) < ( - ( - (px `2 ))) or (px `2 ) > ( - 1) & ( - (px `2 )) > ( - 1) by A63, XREAL_1: 24, XXREAL_0: 2;

 then (px `2 ) < 1 & ( - 1) < (px `2 ) or (px `2 ) > ( - 1) & ( - ( - (px `2 ))) < ( - ( - 1)) by XREAL_1: 24;

 then px in Kb by A1, A62, A63;

 hence ex x be set st x in ( dom Sq_Circ ) & x in Kb & y = ( Sq_Circ . x) by A28, A52, A54, A55, A56, EUCLID: 53;

 end;

 case

 A64: q <> ( 0. ( TOP-REAL 2)) & not ((q `2 ) <= (q `1 ) & ( - (q `1 )) <= (q `2 ) or (q `2 ) >= (q `1 ) & (q `2 ) <= ( - (q `1 )));

 set px = |[((q `1 ) * ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))), ((q `2 ) * ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 ))))]|;



 A65: (q `1 ) <= (q `2 ) & ( - (q `2 )) <= (q `1 ) or (q `1 ) >= (q `2 ) & (q `1 ) <= ( - (q `2 )) by A64, JGRAPH_2: 13;



 A66: (px `2 ) = ((q `2 ) * ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))) by EUCLID: 52;



 A67: (px `1 ) = ((q `1 ) * ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))) by EUCLID: 52;

 (1 + (((q `1 ) / (q `2 )) ^2 )) > 0 by XREAL_1: 34, XREAL_1: 63;

 then

 A68: ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 ))) > 0 by SQUARE_1: 25;



 A69: (1 + (((px `1 ) / (px `2 )) ^2 )) > 0 by XREAL_1: 34, XREAL_1: 63;



 A70: ((px `1 ) / (px `2 )) = ((q `1 ) / (q `2 )) by A66, A67, A68, XCMPLX_1: 91;



 A71: (q `2 ) = (((q `2 ) * ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))) / ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))) by A68, XCMPLX_1: 89

 .= ((px `2 ) / ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))) by EUCLID: 52;



 A72: (q `1 ) = (((q `1 ) * ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))) / ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))) by A68, XCMPLX_1: 89

 .= ((px `1 ) / ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))) by EUCLID: 52;



 A73: ( |.q.| ^2 ) = (((q `2 ) ^2 ) + ((q `1 ) ^2 )) by JGRAPH_3: 1;



 A74: ( |.q.| ^2 ) < (1 ^2 ) by A29, SQUARE_1: 16;

 A75:

 now

 assume that

 A76: (px `2 ) = 0 and

 A77: (px `1 ) = 0 ;



 A78: ((q `2 ) * ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))) = 0 by A76, EUCLID: 52;

 ((q `1 ) * ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))) = 0 by A77, EUCLID: 52;

 then (q `1 ) = 0 by A68, XCMPLX_1: 6;

 hence contradiction by A64, A68, A78, XCMPLX_1: 6;

 end;

 (q `1 ) <= (q `2 ) & ( - (q `2 )) <= (q `1 ) or (q `1 ) >= (q `2 ) & ((q `1 ) * ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))) <= (( - (q `2 )) * ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))) by A65, A68, XREAL_1: 64;

 then

 A79: (q `1 ) <= (q `2 ) & (( - (q `2 )) * ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))) <= ((q `1 ) * ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))) or (px `1 ) >= (px `2 ) & (px `1 ) <= ( - (px `2 )) by A66, A67, A68, XREAL_1: 64;

 then (px `1 ) <= (px `2 ) & ( - (px `2 )) <= (px `1 ) or (px `1 ) >= (px `2 ) & (px `1 ) <= ( - (px `2 )) by A66, A67, A68, XREAL_1: 64;

 then

 A80: ( Sq_Circ . px) = |[((px `1 ) / ( sqrt (1 + (((px `1 ) / (px `2 )) ^2 )))), ((px `2 ) / ( sqrt (1 + (((px `1 ) / (px `2 )) ^2 ))))]| by A75, JGRAPH_2: 3, JGRAPH_3: 4;

 (px `1 ) <= (px `2 ) & ( - ( - (px `2 ))) >= ( - (px `1 )) or (px `1 ) >= (px `2 ) & (px `1 ) <= ( - (px `2 )) by A66, A67, A68, A79, XREAL_1: 24, XREAL_1: 64;

 then

 A81: (px `1 ) <= (px `2 ) & (px `2 ) >= ( - (px `1 )) or (px `1 ) >= (px `2 ) & ( - (px `1 )) >= ( - ( - (px `2 ))) by XREAL_1: 24;



 A82: ((px `2 ) / ( sqrt (1 + (((px `1 ) / (px `2 )) ^2 )))) = (q `2 ) by A66, A68, A70, XCMPLX_1: 89;



 A83: ((px `1 ) / ( sqrt (1 + (((px `1 ) / (px `2 )) ^2 )))) = (q `1 ) by A67, A68, A70, XCMPLX_1: 89;



 A84: ( dom Sq_Circ ) = the carrier of ( TOP-REAL 2) by FUNCT_2:def 1;

 not (px `2 ) = 0 by A66, A67, A68, A75, A79, XREAL_1: 64;

 then

 A85: ((px `2 ) ^2 ) > 0 by SQUARE_1: 12;



 A86: ((px `1 ) ^2 ) >= 0 by XREAL_1: 63;

 ((((px `2 ) ^2 ) / (( sqrt (1 + (((px `1 ) / (px `2 )) ^2 ))) ^2 )) + (((px `1 ) / ( sqrt (1 + (((px `1 ) / (px `2 )) ^2 )))) ^2 )) < 1 by A70, A71, A72, A73, A74, XCMPLX_1: 76;

 then ((((px `2 ) ^2 ) / (( sqrt (1 + (((px `1 ) / (px `2 )) ^2 ))) ^2 )) + (((px `1 ) ^2 ) / (( sqrt (1 + (((px `1 ) / (px `2 )) ^2 ))) ^2 ))) < 1 by XCMPLX_1: 76;

 then ((((px `2 ) ^2 ) / (1 + (((px `1 ) / (px `2 )) ^2 ))) + (((px `1 ) ^2 ) / (( sqrt (1 + (((px `1 ) / (px `2 )) ^2 ))) ^2 ))) < 1 by A69, SQUARE_1:def 2;

 then ((((px `2 ) ^2 ) / (1 + (((px `1 ) / (px `2 )) ^2 ))) + (((px `1 ) ^2 ) / (1 + (((px `1 ) / (px `2 )) ^2 )))) < 1 by A69, SQUARE_1:def 2;

 then (((((px `2 ) ^2 ) / (1 + (((px `1 ) / (px `2 )) ^2 ))) + (((px `1 ) ^2 ) / (1 + (((px `1 ) / (px `2 )) ^2 )))) * (1 + (((px `1 ) / (px `2 )) ^2 ))) < (1 * (1 + (((px `1 ) / (px `2 )) ^2 ))) by A69, XREAL_1: 68;

 then (((((px `2 ) ^2 ) / (1 + (((px `1 ) / (px `2 )) ^2 ))) * (1 + (((px `1 ) / (px `2 )) ^2 ))) + ((((px `1 ) ^2 ) / (1 + (((px `1 ) / (px `2 )) ^2 ))) * (1 + (((px `1 ) / (px `2 )) ^2 )))) < (1 * (1 + (((px `1 ) / (px `2 )) ^2 )));

 then (((px `2 ) ^2 ) + ((((px `1 ) ^2 ) / (1 + (((px `1 ) / (px `2 )) ^2 ))) * (1 + (((px `1 ) / (px `2 )) ^2 )))) < (1 * (1 + (((px `1 ) / (px `2 )) ^2 ))) by A69, XCMPLX_1: 87;

 then

 A87: (((px `2 ) ^2 ) + ((px `1 ) ^2 )) < (1 * (1 + (((px `1 ) / (px `2 )) ^2 ))) by A69, XCMPLX_1: 87;

 (1 * (1 + (((px `1 ) / (px `2 )) ^2 ))) = (1 + (((px `1 ) ^2 ) / ((px `2 ) ^2 ))) by XCMPLX_1: 76;

 then ((((px `2 ) ^2 ) + ((px `1 ) ^2 )) - 1) < ((1 + (((px `1 ) ^2 ) / ((px `2 ) ^2 ))) - 1) by A87, XREAL_1: 9;

 then (((((px `2 ) ^2 ) + ((px `1 ) ^2 )) - 1) * ((px `2 ) ^2 )) < ((((px `1 ) ^2 ) / ((px `2 ) ^2 )) * ((px `2 ) ^2 )) by A85, XREAL_1: 68;

 then

 A88: ((((px `2 ) ^2 ) + (((px `1 ) ^2 ) - 1)) * ((px `2 ) ^2 )) < ((px `1 ) ^2 ) by A85, XCMPLX_1: 87;

 (((((px `2 ) ^2 ) * ((px `2 ) ^2 )) + ((((px `2 ) ^2 ) * ((px `1 ) ^2 )) - (((px `2 ) ^2 ) * 1))) - ((px `1 ) ^2 )) = ((((px `2 ) ^2 ) - 1) * (((px `2 ) ^2 ) + ((px `1 ) ^2 )));

 then (((px `2 ) ^2 ) - 1) < 0 or (((px `2 ) ^2 ) + ((px `1 ) ^2 )) < 0 by A88, XREAL_1: 49;

 then

 A89: ((((px `2 ) ^2 ) - 1) + 1) < ( 0 + 1) by A86, XREAL_1: 6;

 then

 A90: (px `2 ) < (1 ^2 ) by SQUARE_1: 48;



 A91: (px `2 ) > ( - (1 ^2 )) by A89, SQUARE_1: 48;

 (px `1 ) < 1 & 1 > ( - (px `1 )) or (px `1 ) >= (px `2 ) & ( - (px `1 )) >= (px `2 ) by A81, A90, XXREAL_0: 2;

 then (px `1 ) < 1 & ( - 1) < ( - ( - (px `1 ))) or (px `1 ) > ( - 1) & ( - (px `1 )) > ( - 1) by A91, XREAL_1: 24, XXREAL_0: 2;

 then (px `1 ) < 1 & ( - 1) < (px `1 ) or (px `1 ) > ( - 1) & ( - ( - (px `1 ))) < ( - ( - 1)) by XREAL_1: 24;

 then px in Kb by A1, A90, A91;

 hence ex x be set st x in ( dom Sq_Circ ) & x in Kb & y = ( Sq_Circ . x) by A28, A80, A82, A83, A84, EUCLID: 53;

 end;

 end;

 hence thesis by FUNCT_1:def 6;

 end;

 theorem :: JGRAPH_6:26



 Th26: for Kb,Cb be Subset of ( TOP-REAL 2) st Kb = { p : not (( - 1) <= (p `1 ) & (p `1 ) <= 1 & ( - 1) <= (p `2 ) & (p `2 ) <= 1) } & Cb = { p2 where p2 be Point of ( TOP-REAL 2) : |.p2.| > 1 } holds ( Sq_Circ .: Kb) = Cb

 proof

 let Kb,Cb be Subset of ( TOP-REAL 2);

 assume

 A1: Kb = { p : not (( - 1) <= (p `1 ) & (p `1 ) <= 1 & ( - 1) <= (p `2 ) & (p `2 ) <= 1) } & Cb = { p2 where p2 be Point of ( TOP-REAL 2) : |.p2.| > 1 };

 thus ( Sq_Circ .: Kb) c= Cb

 proof

 let y be object;

 assume y in ( Sq_Circ .: Kb);

 then

 consider x be object such that x in ( dom Sq_Circ ) and

 A2: x in Kb and

 A3: y = ( Sq_Circ . x) by FUNCT_1:def 6;

 consider q be Point of ( TOP-REAL 2) such that

 A4: q = x and

 A5: not (( - 1) <= (q `1 ) & (q `1 ) <= 1 & ( - 1) <= (q `2 ) & (q `2 ) <= 1) by A1, A2;

 now

 per cases ;

 case q = ( 0. ( TOP-REAL 2));

 hence contradiction by A5, EUCLID: 52, EUCLID: 54;

 end;

 case

 A6: q <> ( 0. ( TOP-REAL 2)) & ((q `2 ) <= (q `1 ) & ( - (q `1 )) <= (q `2 ) or (q `2 ) >= (q `1 ) & (q `2 ) <= ( - (q `1 )));

 then

 A7: ( Sq_Circ . q) = |[((q `1 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))), ((q `2 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 ))))]| by JGRAPH_3:def 1;



 A8: not (( - 1) <= (q `2 ) & (q `2 ) <= 1) implies ( - 1) > (q `1 ) or (q `1 ) > 1

 proof

 assume

 A9: not (( - 1) <= (q `2 ) & (q `2 ) <= 1);

 now

 per cases by A9;

 case

 A10: ( - 1) > (q `2 );

 then ( - (q `1 )) < ( - 1) or (q `2 ) >= (q `1 ) & (q `2 ) <= ( - (q `1 )) by A6, XXREAL_0: 2;

 hence thesis by A10, XREAL_1: 24, XXREAL_0: 2;

 end;

 case (q `2 ) > 1;

 then 1 < (q `1 ) or 1 < ( - (q `1 )) by A6, XXREAL_0: 2;

 then 1 < (q `1 ) or ( - ( - (q `1 ))) < ( - 1) by XREAL_1: 24;

 hence thesis;

 end;

 end;

 hence thesis;

 end;



 A11: ( |[((q `1 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))), ((q `2 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 ))))]| `1 ) = ((q `1 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) by EUCLID: 52;



 A12: ( |[((q `1 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))), ((q `2 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 ))))]| `2 ) = ((q `2 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) by EUCLID: 52;



 A13: (1 + (((q `2 ) / (q `1 )) ^2 )) > 0 by XREAL_1: 34, XREAL_1: 63;

 A14:

 now

 assume

 A15: (q `1 ) = 0 ;

 then (q `2 ) = 0 by A6;

 hence contradiction by A6, A15, EUCLID: 53, EUCLID: 54;

 end;

 then

 A16: ((q `1 ) ^2 ) > 0 by SQUARE_1: 12;

 ((q `1 ) ^2 ) > (1 ^2 ) by A5, A8, SQUARE_1: 47;

 then

 A17: ( sqrt ((q `1 ) ^2 )) > 1 by SQUARE_1: 18, SQUARE_1: 27;

 ( |. |[((q `1 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))), ((q `2 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 ))))]|.| ^2 ) = ((((q `1 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) ^2 ) + (((q `2 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) ^2 )) by A11, A12, JGRAPH_3: 1

 .= ((((q `1 ) ^2 ) / (( sqrt (1 + (((q `2 ) / (q `1 )) ^2 ))) ^2 )) + (((q `2 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) ^2 )) by XCMPLX_1: 76

 .= ((((q `1 ) ^2 ) / (( sqrt (1 + (((q `2 ) / (q `1 )) ^2 ))) ^2 )) + (((q `2 ) ^2 ) / (( sqrt (1 + (((q `2 ) / (q `1 )) ^2 ))) ^2 ))) by XCMPLX_1: 76

 .= ((((q `1 ) ^2 ) / (1 + (((q `2 ) / (q `1 )) ^2 ))) + (((q `2 ) ^2 ) / (( sqrt (1 + (((q `2 ) / (q `1 )) ^2 ))) ^2 ))) by A13, SQUARE_1:def 2

 .= ((((q `1 ) ^2 ) / (1 + (((q `2 ) / (q `1 )) ^2 ))) + (((q `2 ) ^2 ) / (1 + (((q `2 ) / (q `1 )) ^2 )))) by A13, SQUARE_1:def 2

 .= ((((q `1 ) ^2 ) + ((q `2 ) ^2 )) / (1 + (((q `2 ) / (q `1 )) ^2 ))) by XCMPLX_1: 62

 .= ((((q `1 ) ^2 ) + ((q `2 ) ^2 )) / (1 + (((q `2 ) ^2 ) / ((q `1 ) ^2 )))) by XCMPLX_1: 76

 .= ((((q `1 ) ^2 ) + ((q `2 ) ^2 )) / ((((q `1 ) ^2 ) / ((q `1 ) ^2 )) + (((q `2 ) ^2 ) / ((q `1 ) ^2 )))) by A16, XCMPLX_1: 60

 .= ((((q `1 ) ^2 ) + ((q `2 ) ^2 )) / ((((q `1 ) ^2 ) + ((q `2 ) ^2 )) / ((q `1 ) ^2 ))) by XCMPLX_1: 62

 .= (((q `1 ) ^2 ) * ((((q `1 ) ^2 ) + ((q `2 ) ^2 )) / (((q `1 ) ^2 ) + ((q `2 ) ^2 )))) by XCMPLX_1: 81

 .= (((q `1 ) ^2 ) * 1) by A14, COMPLEX1: 1, XCMPLX_1: 60

 .= ((q `1 ) ^2 );

 then |. |[((q `1 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))), ((q `2 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 ))))]|.| > 1 by A17, SQUARE_1: 22;

 hence ex p2 be Point of ( TOP-REAL 2) st p2 = y & |.p2.| > 1 by A3, A4, A7;

 end;

 case

 A18: q <> ( 0. ( TOP-REAL 2)) & not ((q `2 ) <= (q `1 ) & ( - (q `1 )) <= (q `2 ) or (q `2 ) >= (q `1 ) & (q `2 ) <= ( - (q `1 )));

 then

 A19: ( Sq_Circ . q) = |[((q `1 ) / ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))), ((q `2 ) / ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 ))))]| by JGRAPH_3:def 1;



 A20: ( |[((q `1 ) / ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))), ((q `2 ) / ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 ))))]| `1 ) = ((q `1 ) / ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))) by EUCLID: 52;



 A21: ( |[((q `1 ) / ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))), ((q `2 ) / ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 ))))]| `2 ) = ((q `2 ) / ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))) by EUCLID: 52;



 A22: (1 + (((q `1 ) / (q `2 )) ^2 )) > 0 by XREAL_1: 34, XREAL_1: 63;



 A23: (q `2 ) <> 0 by A18;

 then

 A24: ((q `2 ) ^2 ) > 0 by SQUARE_1: 12;

 not (( - 1) <= (q `1 ) & (q `1 ) <= 1) implies ( - 1) > (q `2 ) or (q `2 ) > 1

 proof

 assume

 A25: not (( - 1) <= (q `1 ) & (q `1 ) <= 1);

 now

 per cases by A25;

 case

 A26: ( - 1) > (q `1 );

 then (q `2 ) < ( - 1) or (q `1 ) < (q `2 ) & ( - (q `2 )) < ( - ( - (q `1 ))) by A18, XREAL_1: 24, XXREAL_0: 2;

 then ( - (q `2 )) < ( - 1) or ( - 1) > (q `2 ) by A26, XXREAL_0: 2;

 hence thesis by XREAL_1: 24;

 end;

 case

 A27: (q `1 ) > 1;

 ( - ( - (q `1 ))) < ( - (q `2 )) & (q `2 ) < (q `1 ) or (q `2 ) > (q `1 ) & (q `2 ) > ( - (q `1 )) by A18, XREAL_1: 24;

 then 1 < ( - (q `2 )) or (q `2 ) > (q `1 ) & (q `2 ) > ( - (q `1 )) by A27, XXREAL_0: 2;

 then ( - 1) > ( - ( - (q `2 ))) or 1 < (q `2 ) by A27, XREAL_1: 24, XXREAL_0: 2;

 hence thesis;

 end;

 end;

 hence thesis;

 end;

 then ((q `2 ) ^2 ) > (1 ^2 ) by A5, SQUARE_1: 47;

 then

 A28: ( sqrt ((q `2 ) ^2 )) > 1 by SQUARE_1: 18, SQUARE_1: 27;

 ( |. |[((q `1 ) / ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))), ((q `2 ) / ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 ))))]|.| ^2 ) = ((((q `1 ) / ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))) ^2 ) + (((q `2 ) / ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))) ^2 )) by A20, A21, JGRAPH_3: 1

 .= ((((q `1 ) ^2 ) / (( sqrt (1 + (((q `1 ) / (q `2 )) ^2 ))) ^2 )) + (((q `2 ) / ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))) ^2 )) by XCMPLX_1: 76

 .= ((((q `1 ) ^2 ) / (( sqrt (1 + (((q `1 ) / (q `2 )) ^2 ))) ^2 )) + (((q `2 ) ^2 ) / (( sqrt (1 + (((q `1 ) / (q `2 )) ^2 ))) ^2 ))) by XCMPLX_1: 76

 .= ((((q `1 ) ^2 ) / (1 + (((q `1 ) / (q `2 )) ^2 ))) + (((q `2 ) ^2 ) / (( sqrt (1 + (((q `1 ) / (q `2 )) ^2 ))) ^2 ))) by A22, SQUARE_1:def 2

 .= ((((q `1 ) ^2 ) / (1 + (((q `1 ) / (q `2 )) ^2 ))) + (((q `2 ) ^2 ) / (1 + (((q `1 ) / (q `2 )) ^2 )))) by A22, SQUARE_1:def 2

 .= ((((q `1 ) ^2 ) + ((q `2 ) ^2 )) / (1 + (((q `1 ) / (q `2 )) ^2 ))) by XCMPLX_1: 62

 .= ((((q `1 ) ^2 ) + ((q `2 ) ^2 )) / (1 + (((q `1 ) ^2 ) / ((q `2 ) ^2 )))) by XCMPLX_1: 76

 .= ((((q `1 ) ^2 ) + ((q `2 ) ^2 )) / ((((q `1 ) ^2 ) / ((q `2 ) ^2 )) + (((q `2 ) ^2 ) / ((q `2 ) ^2 )))) by A24, XCMPLX_1: 60

 .= ((((q `1 ) ^2 ) + ((q `2 ) ^2 )) / ((((q `1 ) ^2 ) + ((q `2 ) ^2 )) / ((q `2 ) ^2 ))) by XCMPLX_1: 62

 .= (((q `2 ) ^2 ) * ((((q `1 ) ^2 ) + ((q `2 ) ^2 )) / (((q `1 ) ^2 ) + ((q `2 ) ^2 )))) by XCMPLX_1: 81

 .= (((q `2 ) ^2 ) * 1) by A23, COMPLEX1: 1, XCMPLX_1: 60

 .= ((q `2 ) ^2 );

 then |. |[((q `1 ) / ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))), ((q `2 ) / ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 ))))]|.| > 1 by A28, SQUARE_1: 22;

 hence ex p2 be Point of ( TOP-REAL 2) st p2 = y & |.p2.| > 1 by A3, A4, A19;

 end;

 end;

 hence thesis by A1;

 end;

 let y be object;

 assume y in Cb;

 then

 consider p2 be Point of ( TOP-REAL 2) such that

 A29: p2 = y and

 A30: |.p2.| > 1 by A1;

 set q = p2;

 now

 per cases ;

 case q = ( 0. ( TOP-REAL 2));

 hence contradiction by A30, TOPRNS_1: 23;

 end;

 case

 A31: q <> ( 0. ( TOP-REAL 2)) & ((q `2 ) <= (q `1 ) & ( - (q `1 )) <= (q `2 ) or (q `2 ) >= (q `1 ) & (q `2 ) <= ( - (q `1 )));

 set px = |[((q `1 ) * ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))), ((q `2 ) * ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 ))))]|;



 A32: (px `1 ) = ((q `1 ) * ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) by EUCLID: 52;



 A33: (px `2 ) = ((q `2 ) * ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) by EUCLID: 52;

 (1 + (((q `2 ) / (q `1 )) ^2 )) > 0 by XREAL_1: 34, XREAL_1: 63;

 then

 A34: ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 ))) > 0 by SQUARE_1: 25;



 A35: (1 + (((px `2 ) / (px `1 )) ^2 )) > 0 by XREAL_1: 34, XREAL_1: 63;



 A36: ((px `2 ) / (px `1 )) = ((q `2 ) / (q `1 )) by A32, A33, A34, XCMPLX_1: 91;



 A37: (q `1 ) = (((q `1 ) * ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) by A34, XCMPLX_1: 89

 .= ((px `1 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) by EUCLID: 52;



 A38: (q `2 ) = (((q `2 ) * ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) by A34, XCMPLX_1: 89

 .= ((px `2 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) by EUCLID: 52;



 A39: ( |.q.| ^2 ) = (((q `1 ) ^2 ) + ((q `2 ) ^2 )) by JGRAPH_3: 1;



 A40: ( |.q.| ^2 ) > (1 ^2 ) by A30, SQUARE_1: 16;

 A41:

 now

 assume that

 A42: (px `1 ) = 0 and

 A43: (px `2 ) = 0 ;



 A44: ((q `1 ) * ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) = 0 by A42, EUCLID: 52;



 A45: ((q `2 ) * ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) = 0 by A43, EUCLID: 52;



 A46: (q `1 ) = 0 by A34, A44, XCMPLX_1: 6;

 (q `2 ) = 0 by A34, A45, XCMPLX_1: 6;

 hence contradiction by A31, A46, EUCLID: 53, EUCLID: 54;

 end;

 (q `2 ) <= (q `1 ) & ( - (q `1 )) <= (q `2 ) or (q `2 ) >= (q `1 ) & ((q `2 ) * ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) <= (( - (q `1 )) * ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) by A31, A34, XREAL_1: 64;

 then

 A47: (q `2 ) <= (q `1 ) & (( - (q `1 )) * ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) <= ((q `2 ) * ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) or (px `2 ) >= (px `1 ) & (px `2 ) <= ( - (px `1 )) by A32, A33, A34, XREAL_1: 64;

 then (px `2 ) <= (px `1 ) & ( - (px `1 )) <= (px `2 ) or (px `2 ) >= (px `1 ) & (px `2 ) <= ( - (px `1 )) by A32, A33, A34, XREAL_1: 64;

 then

 A48: ( Sq_Circ . px) = |[((px `1 ) / ( sqrt (1 + (((px `2 ) / (px `1 )) ^2 )))), ((px `2 ) / ( sqrt (1 + (((px `2 ) / (px `1 )) ^2 ))))]| by A41, JGRAPH_2: 3, JGRAPH_3:def 1;



 A49: ((px `1 ) / ( sqrt (1 + (((px `2 ) / (px `1 )) ^2 )))) = (q `1 ) by A32, A34, A36, XCMPLX_1: 89;



 A50: ((px `2 ) / ( sqrt (1 + (((px `2 ) / (px `1 )) ^2 )))) = (q `2 ) by A33, A34, A36, XCMPLX_1: 89;



 A51: ( dom Sq_Circ ) = the carrier of ( TOP-REAL 2) by FUNCT_2:def 1;

 not (px `1 ) = 0 by A32, A33, A34, A41, A47, XREAL_1: 64;

 then

 A52: ((px `1 ) ^2 ) > 0 by SQUARE_1: 12;



 A53: ((px `2 ) ^2 ) >= 0 by XREAL_1: 63;

 ((((px `1 ) ^2 ) / (( sqrt (1 + (((px `2 ) / (px `1 )) ^2 ))) ^2 )) + (((px `2 ) / ( sqrt (1 + (((px `2 ) / (px `1 )) ^2 )))) ^2 )) > 1 by A36, A37, A38, A39, A40, XCMPLX_1: 76;

 then ((((px `1 ) ^2 ) / (( sqrt (1 + (((px `2 ) / (px `1 )) ^2 ))) ^2 )) + (((px `2 ) ^2 ) / (( sqrt (1 + (((px `2 ) / (px `1 )) ^2 ))) ^2 ))) > 1 by XCMPLX_1: 76;

 then ((((px `1 ) ^2 ) / (1 + (((px `2 ) / (px `1 )) ^2 ))) + (((px `2 ) ^2 ) / (( sqrt (1 + (((px `2 ) / (px `1 )) ^2 ))) ^2 ))) > 1 by A35, SQUARE_1:def 2;

 then ((((px `1 ) ^2 ) / (1 + (((px `2 ) / (px `1 )) ^2 ))) + (((px `2 ) ^2 ) / (1 + (((px `2 ) / (px `1 )) ^2 )))) > 1 by A35, SQUARE_1:def 2;

 then (((((px `1 ) ^2 ) / (1 + (((px `2 ) / (px `1 )) ^2 ))) + (((px `2 ) ^2 ) / (1 + (((px `2 ) / (px `1 )) ^2 )))) * (1 + (((px `2 ) / (px `1 )) ^2 ))) > (1 * (1 + (((px `2 ) / (px `1 )) ^2 ))) by A35, XREAL_1: 68;

 then (((((px `1 ) ^2 ) / (1 + (((px `2 ) / (px `1 )) ^2 ))) * (1 + (((px `2 ) / (px `1 )) ^2 ))) + ((((px `2 ) ^2 ) / (1 + (((px `2 ) / (px `1 )) ^2 ))) * (1 + (((px `2 ) / (px `1 )) ^2 )))) > (1 * (1 + (((px `2 ) / (px `1 )) ^2 )));

 then (((px `1 ) ^2 ) + ((((px `2 ) ^2 ) / (1 + (((px `2 ) / (px `1 )) ^2 ))) * (1 + (((px `2 ) / (px `1 )) ^2 )))) > (1 * (1 + (((px `2 ) / (px `1 )) ^2 ))) by A35, XCMPLX_1: 87;

 then

 A54: (((px `1 ) ^2 ) + ((px `2 ) ^2 )) > (1 * (1 + (((px `2 ) / (px `1 )) ^2 ))) by A35, XCMPLX_1: 87;

 (1 * (1 + (((px `2 ) / (px `1 )) ^2 ))) = (1 + (((px `2 ) ^2 ) / ((px `1 ) ^2 ))) by XCMPLX_1: 76;

 then ((((px `1 ) ^2 ) + ((px `2 ) ^2 )) - 1) > ((1 + (((px `2 ) ^2 ) / ((px `1 ) ^2 ))) - 1) by A54, XREAL_1: 9;

 then (((((px `1 ) ^2 ) + ((px `2 ) ^2 )) - 1) * ((px `1 ) ^2 )) > ((((px `2 ) ^2 ) / ((px `1 ) ^2 )) * ((px `1 ) ^2 )) by A52, XREAL_1: 68;

 then

 A55: ((((px `1 ) ^2 ) + (((px `2 ) ^2 ) - 1)) * ((px `1 ) ^2 )) > ((px `2 ) ^2 ) by A52, XCMPLX_1: 87;

 (((((px `1 ) ^2 ) * ((px `1 ) ^2 )) + ((((px `1 ) ^2 ) * ((px `2 ) ^2 )) - (((px `1 ) ^2 ) * 1))) - ((px `2 ) ^2 )) = ((((px `1 ) ^2 ) - 1) * (((px `1 ) ^2 ) + ((px `2 ) ^2 )));

 then (((px `1 ) ^2 ) - 1) > 0 or (((px `1 ) ^2 ) + ((px `2 ) ^2 )) < 0 by A55, XREAL_1: 50;

 then ((((px `1 ) ^2 ) - 1) + 1) > ( 0 + 1) by A52, A53, XREAL_1: 6;

 then (px `1 ) > (1 ^2 ) or (px `1 ) < ( - 1) by SQUARE_1: 49;

 then px in Kb by A1;

 hence ex x be set st x in ( dom Sq_Circ ) & x in Kb & y = ( Sq_Circ . x) by A29, A48, A49, A50, A51, EUCLID: 53;

 end;

 case

 A56: q <> ( 0. ( TOP-REAL 2)) & not ((q `2 ) <= (q `1 ) & ( - (q `1 )) <= (q `2 ) or (q `2 ) >= (q `1 ) & (q `2 ) <= ( - (q `1 )));

 set px = |[((q `1 ) * ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))), ((q `2 ) * ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 ))))]|;



 A57: (q `1 ) <= (q `2 ) & ( - (q `2 )) <= (q `1 ) or (q `1 ) >= (q `2 ) & (q `1 ) <= ( - (q `2 )) by A56, JGRAPH_2: 13;



 A58: (px `2 ) = ((q `2 ) * ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))) by EUCLID: 52;



 A59: (px `1 ) = ((q `1 ) * ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))) by EUCLID: 52;

 (1 + (((q `1 ) / (q `2 )) ^2 )) > 0 by XREAL_1: 34, XREAL_1: 63;

 then

 A60: ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 ))) > 0 by SQUARE_1: 25;



 A61: (1 + (((px `1 ) / (px `2 )) ^2 )) > 0 by XREAL_1: 34, XREAL_1: 63;



 A62: ((px `1 ) / (px `2 )) = ((q `1 ) / (q `2 )) by A58, A59, A60, XCMPLX_1: 91;



 A63: (q `2 ) = (((q `2 ) * ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))) / ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))) by A60, XCMPLX_1: 89

 .= ((px `2 ) / ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))) by EUCLID: 52;



 A64: (q `1 ) = (((q `1 ) * ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))) / ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))) by A60, XCMPLX_1: 89

 .= ((px `1 ) / ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))) by EUCLID: 52;



 A65: ( |.q.| ^2 ) = (((q `2 ) ^2 ) + ((q `1 ) ^2 )) by JGRAPH_3: 1;



 A66: ( |.q.| ^2 ) > (1 ^2 ) by A30, SQUARE_1: 16;

 A67:

 now

 assume that

 A68: (px `2 ) = 0 and

 A69: (px `1 ) = 0 ;



 A70: ((q `2 ) * ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))) = 0 by A68, EUCLID: 52;

 ((q `1 ) * ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))) = 0 by A69, EUCLID: 52;

 then (q `1 ) = 0 by A60, XCMPLX_1: 6;

 hence contradiction by A56, A60, A70, XCMPLX_1: 6;

 end;

 (q `1 ) <= (q `2 ) & ( - (q `2 )) <= (q `1 ) or (q `1 ) >= (q `2 ) & ((q `1 ) * ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))) <= (( - (q `2 )) * ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))) by A57, A60, XREAL_1: 64;

 then

 A71: (q `1 ) <= (q `2 ) & (( - (q `2 )) * ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))) <= ((q `1 ) * ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))) or (px `1 ) >= (px `2 ) & (px `1 ) <= ( - (px `2 )) by A58, A59, A60, XREAL_1: 64;

 then (px `1 ) <= (px `2 ) & ( - (px `2 )) <= (px `1 ) or (px `1 ) >= (px `2 ) & (px `1 ) <= ( - (px `2 )) by A58, A59, A60, XREAL_1: 64;

 then

 A72: ( Sq_Circ . px) = |[((px `1 ) / ( sqrt (1 + (((px `1 ) / (px `2 )) ^2 )))), ((px `2 ) / ( sqrt (1 + (((px `1 ) / (px `2 )) ^2 ))))]| by A67, JGRAPH_2: 3, JGRAPH_3: 4;



 A73: ((px `2 ) / ( sqrt (1 + (((px `1 ) / (px `2 )) ^2 )))) = (q `2 ) by A58, A60, A62, XCMPLX_1: 89;



 A74: ((px `1 ) / ( sqrt (1 + (((px `1 ) / (px `2 )) ^2 )))) = (q `1 ) by A59, A60, A62, XCMPLX_1: 89;



 A75: ( dom Sq_Circ ) = the carrier of ( TOP-REAL 2) by FUNCT_2:def 1;

 not (px `2 ) = 0 by A58, A59, A60, A67, A71, XREAL_1: 64;

 then

 A76: ((px `2 ) ^2 ) > 0 by SQUARE_1: 12;



 A77: ((px `1 ) ^2 ) >= 0 by XREAL_1: 63;

 ((((px `2 ) ^2 ) / (( sqrt (1 + (((px `1 ) / (px `2 )) ^2 ))) ^2 )) + (((px `1 ) / ( sqrt (1 + (((px `1 ) / (px `2 )) ^2 )))) ^2 )) > 1 by A62, A63, A64, A65, A66, XCMPLX_1: 76;

 then ((((px `2 ) ^2 ) / (( sqrt (1 + (((px `1 ) / (px `2 )) ^2 ))) ^2 )) + (((px `1 ) ^2 ) / (( sqrt (1 + (((px `1 ) / (px `2 )) ^2 ))) ^2 ))) > 1 by XCMPLX_1: 76;

 then ((((px `2 ) ^2 ) / (1 + (((px `1 ) / (px `2 )) ^2 ))) + (((px `1 ) ^2 ) / (( sqrt (1 + (((px `1 ) / (px `2 )) ^2 ))) ^2 ))) > 1 by A61, SQUARE_1:def 2;

 then ((((px `2 ) ^2 ) / (1 + (((px `1 ) / (px `2 )) ^2 ))) + (((px `1 ) ^2 ) / (1 + (((px `1 ) / (px `2 )) ^2 )))) > 1 by A61, SQUARE_1:def 2;

 then (((((px `2 ) ^2 ) / (1 + (((px `1 ) / (px `2 )) ^2 ))) + (((px `1 ) ^2 ) / (1 + (((px `1 ) / (px `2 )) ^2 )))) * (1 + (((px `1 ) / (px `2 )) ^2 ))) > (1 * (1 + (((px `1 ) / (px `2 )) ^2 ))) by A61, XREAL_1: 68;

 then (((((px `2 ) ^2 ) / (1 + (((px `1 ) / (px `2 )) ^2 ))) * (1 + (((px `1 ) / (px `2 )) ^2 ))) + ((((px `1 ) ^2 ) / (1 + (((px `1 ) / (px `2 )) ^2 ))) * (1 + (((px `1 ) / (px `2 )) ^2 )))) > (1 * (1 + (((px `1 ) / (px `2 )) ^2 )));

 then (((px `2 ) ^2 ) + ((((px `1 ) ^2 ) / (1 + (((px `1 ) / (px `2 )) ^2 ))) * (1 + (((px `1 ) / (px `2 )) ^2 )))) > (1 * (1 + (((px `1 ) / (px `2 )) ^2 ))) by A61, XCMPLX_1: 87;

 then

 A78: (((px `2 ) ^2 ) + ((px `1 ) ^2 )) > (1 * (1 + (((px `1 ) / (px `2 )) ^2 ))) by A61, XCMPLX_1: 87;

 (1 * (1 + (((px `1 ) / (px `2 )) ^2 ))) = (1 + (((px `1 ) ^2 ) / ((px `2 ) ^2 ))) by XCMPLX_1: 76;

 then ((((px `2 ) ^2 ) + ((px `1 ) ^2 )) - 1) > ((1 + (((px `1 ) ^2 ) / ((px `2 ) ^2 ))) - 1) by A78, XREAL_1: 9;

 then (((((px `2 ) ^2 ) + ((px `1 ) ^2 )) - 1) * ((px `2 ) ^2 )) > ((((px `1 ) ^2 ) / ((px `2 ) ^2 )) * ((px `2 ) ^2 )) by A76, XREAL_1: 68;

 then

 A79: ((((px `2 ) ^2 ) + (((px `1 ) ^2 ) - 1)) * ((px `2 ) ^2 )) > ((px `1 ) ^2 ) by A76, XCMPLX_1: 87;

 (((((px `2 ) ^2 ) * ((px `2 ) ^2 )) + ((((px `2 ) ^2 ) * ((px `1 ) ^2 )) - (((px `2 ) ^2 ) * 1))) - ((px `1 ) ^2 )) = ((((px `2 ) ^2 ) - 1) * (((px `2 ) ^2 ) + ((px `1 ) ^2 )));

 then (((px `2 ) ^2 ) - 1) > 0 or (((px `1 ) ^2 ) + ((px `2 ) ^2 )) < 0 by A79, XREAL_1: 50;

 then ((((px `2 ) ^2 ) - 1) + 1) > ( 0 + 1) by A76, A77, XREAL_1: 6;

 then (px `2 ) > (1 ^2 ) or (px `2 ) < ( - 1) by SQUARE_1: 49;

 then px in Kb by A1;

 hence ex x be set st x in ( dom Sq_Circ ) & x in Kb & y = ( Sq_Circ . x) by A29, A72, A73, A74, A75, EUCLID: 53;

 end;

 end;

 hence thesis by FUNCT_1:def 6;

 end;

 theorem :: JGRAPH_6:27



 Th27: for Kb,Cb be Subset of ( TOP-REAL 2) st Kb = { p : ( - 1) <= (p `1 ) & (p `1 ) <= 1 & ( - 1) <= (p `2 ) & (p `2 ) <= 1 } & Cb = { p2 where p2 be Point of ( TOP-REAL 2) : |.p2.| <= 1 } holds ( Sq_Circ .: Kb) = Cb

 proof

 let Kb,Cb be Subset of ( TOP-REAL 2);

 assume

 A1: Kb = { p : ( - 1) <= (p `1 ) & (p `1 ) <= 1 & ( - 1) <= (p `2 ) & (p `2 ) <= 1 } & Cb = { p2 where p2 be Point of ( TOP-REAL 2) : |.p2.| <= 1 };

 thus ( Sq_Circ .: Kb) c= Cb

 proof

 let y be object;

 assume y in ( Sq_Circ .: Kb);

 then

 consider x be object such that x in ( dom Sq_Circ ) and

 A2: x in Kb and

 A3: y = ( Sq_Circ . x) by FUNCT_1:def 6;

 consider q be Point of ( TOP-REAL 2) such that

 A4: q = x and

 A5: ( - 1) <= (q `1 ) and

 A6: (q `1 ) <= 1 and

 A7: ( - 1) <= (q `2 ) and

 A8: (q `2 ) <= 1 by A1, A2;

 now

 per cases ;

 case

 A9: q = ( 0. ( TOP-REAL 2));

 then

 A10: ( Sq_Circ . q) = q by JGRAPH_3:def 1;

 |.q.| = 0 by A9, TOPRNS_1: 23;

 hence ex p2 be Point of ( TOP-REAL 2) st p2 = y & |.p2.| <= 1 by A3, A4, A10;

 end;

 case

 A11: q <> ( 0. ( TOP-REAL 2)) & ((q `2 ) <= (q `1 ) & ( - (q `1 )) <= (q `2 ) or (q `2 ) >= (q `1 ) & (q `2 ) <= ( - (q `1 )));

 then

 A12: ( Sq_Circ . q) = |[((q `1 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))), ((q `2 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 ))))]| by JGRAPH_3:def 1;



 A13: ( |[((q `1 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))), ((q `2 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 ))))]| `1 ) = ((q `1 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) by EUCLID: 52;



 A14: ( |[((q `1 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))), ((q `2 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 ))))]| `2 ) = ((q `2 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) by EUCLID: 52;



 A15: (1 + (((q `2 ) / (q `1 )) ^2 )) > 0 by XREAL_1: 34, XREAL_1: 63;

 A16:

 now

 assume

 A17: (q `1 ) = 0 ;

 then (q `2 ) = 0 by A11;

 hence contradiction by A11, A17, EUCLID: 53, EUCLID: 54;

 end;

 then

 A18: ((q `1 ) ^2 ) > 0 by SQUARE_1: 12;

 ((q `1 ) ^2 ) <= (1 ^2 ) by A5, A6, SQUARE_1: 49;

 then

 A19: ( sqrt ((q `1 ) ^2 )) <= 1 by A18, SQUARE_1: 18, SQUARE_1: 26;

 ( |. |[((q `1 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))), ((q `2 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 ))))]|.| ^2 ) = ((((q `1 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) ^2 ) + (((q `2 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) ^2 )) by A13, A14, JGRAPH_3: 1

 .= ((((q `1 ) ^2 ) / (( sqrt (1 + (((q `2 ) / (q `1 )) ^2 ))) ^2 )) + (((q `2 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) ^2 )) by XCMPLX_1: 76

 .= ((((q `1 ) ^2 ) / (( sqrt (1 + (((q `2 ) / (q `1 )) ^2 ))) ^2 )) + (((q `2 ) ^2 ) / (( sqrt (1 + (((q `2 ) / (q `1 )) ^2 ))) ^2 ))) by XCMPLX_1: 76

 .= ((((q `1 ) ^2 ) / (1 + (((q `2 ) / (q `1 )) ^2 ))) + (((q `2 ) ^2 ) / (( sqrt (1 + (((q `2 ) / (q `1 )) ^2 ))) ^2 ))) by A15, SQUARE_1:def 2

 .= ((((q `1 ) ^2 ) / (1 + (((q `2 ) / (q `1 )) ^2 ))) + (((q `2 ) ^2 ) / (1 + (((q `2 ) / (q `1 )) ^2 )))) by A15, SQUARE_1:def 2

 .= ((((q `1 ) ^2 ) + ((q `2 ) ^2 )) / (1 + (((q `2 ) / (q `1 )) ^2 ))) by XCMPLX_1: 62

 .= ((((q `1 ) ^2 ) + ((q `2 ) ^2 )) / (1 + (((q `2 ) ^2 ) / ((q `1 ) ^2 )))) by XCMPLX_1: 76

 .= ((((q `1 ) ^2 ) + ((q `2 ) ^2 )) / ((((q `1 ) ^2 ) / ((q `1 ) ^2 )) + (((q `2 ) ^2 ) / ((q `1 ) ^2 )))) by A18, XCMPLX_1: 60

 .= ((((q `1 ) ^2 ) + ((q `2 ) ^2 )) / ((((q `1 ) ^2 ) + ((q `2 ) ^2 )) / ((q `1 ) ^2 ))) by XCMPLX_1: 62

 .= (((q `1 ) ^2 ) * ((((q `1 ) ^2 ) + ((q `2 ) ^2 )) / (((q `1 ) ^2 ) + ((q `2 ) ^2 )))) by XCMPLX_1: 81

 .= (((q `1 ) ^2 ) * 1) by A16, COMPLEX1: 1, XCMPLX_1: 60

 .= ((q `1 ) ^2 );

 then |. |[((q `1 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))), ((q `2 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 ))))]|.| <= 1 by A19, SQUARE_1: 22;

 hence ex p2 be Point of ( TOP-REAL 2) st p2 = y & |.p2.| <= 1 by A3, A4, A12;

 end;

 case

 A20: q <> ( 0. ( TOP-REAL 2)) & not ((q `2 ) <= (q `1 ) & ( - (q `1 )) <= (q `2 ) or (q `2 ) >= (q `1 ) & (q `2 ) <= ( - (q `1 )));

 then

 A21: ( Sq_Circ . q) = |[((q `1 ) / ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))), ((q `2 ) / ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 ))))]| by JGRAPH_3:def 1;



 A22: ( |[((q `1 ) / ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))), ((q `2 ) / ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 ))))]| `1 ) = ((q `1 ) / ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))) by EUCLID: 52;



 A23: ( |[((q `1 ) / ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))), ((q `2 ) / ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 ))))]| `2 ) = ((q `2 ) / ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))) by EUCLID: 52;



 A24: (1 + (((q `1 ) / (q `2 )) ^2 )) > 0 by XREAL_1: 34, XREAL_1: 63;



 A25: (q `2 ) <> 0 by A20;

 then

 A26: ((q `2 ) ^2 ) > 0 by SQUARE_1: 12;

 ((q `2 ) ^2 ) <= (1 ^2 ) by A7, A8, SQUARE_1: 49;

 then

 A27: ( sqrt ((q `2 ) ^2 )) <= 1 by A26, SQUARE_1: 18, SQUARE_1: 26;

 ( |. |[((q `1 ) / ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))), ((q `2 ) / ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 ))))]|.| ^2 ) = ((((q `1 ) / ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))) ^2 ) + (((q `2 ) / ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))) ^2 )) by A22, A23, JGRAPH_3: 1

 .= ((((q `1 ) ^2 ) / (( sqrt (1 + (((q `1 ) / (q `2 )) ^2 ))) ^2 )) + (((q `2 ) / ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))) ^2 )) by XCMPLX_1: 76

 .= ((((q `1 ) ^2 ) / (( sqrt (1 + (((q `1 ) / (q `2 )) ^2 ))) ^2 )) + (((q `2 ) ^2 ) / (( sqrt (1 + (((q `1 ) / (q `2 )) ^2 ))) ^2 ))) by XCMPLX_1: 76

 .= ((((q `1 ) ^2 ) / (1 + (((q `1 ) / (q `2 )) ^2 ))) + (((q `2 ) ^2 ) / (( sqrt (1 + (((q `1 ) / (q `2 )) ^2 ))) ^2 ))) by A24, SQUARE_1:def 2

 .= ((((q `1 ) ^2 ) / (1 + (((q `1 ) / (q `2 )) ^2 ))) + (((q `2 ) ^2 ) / (1 + (((q `1 ) / (q `2 )) ^2 )))) by A24, SQUARE_1:def 2

 .= ((((q `1 ) ^2 ) + ((q `2 ) ^2 )) / (1 + (((q `1 ) / (q `2 )) ^2 ))) by XCMPLX_1: 62

 .= ((((q `1 ) ^2 ) + ((q `2 ) ^2 )) / (1 + (((q `1 ) ^2 ) / ((q `2 ) ^2 )))) by XCMPLX_1: 76

 .= ((((q `1 ) ^2 ) + ((q `2 ) ^2 )) / ((((q `1 ) ^2 ) / ((q `2 ) ^2 )) + (((q `2 ) ^2 ) / ((q `2 ) ^2 )))) by A26, XCMPLX_1: 60

 .= ((((q `1 ) ^2 ) + ((q `2 ) ^2 )) / ((((q `1 ) ^2 ) + ((q `2 ) ^2 )) / ((q `2 ) ^2 ))) by XCMPLX_1: 62

 .= (((q `2 ) ^2 ) * ((((q `1 ) ^2 ) + ((q `2 ) ^2 )) / (((q `1 ) ^2 ) + ((q `2 ) ^2 )))) by XCMPLX_1: 81

 .= (((q `2 ) ^2 ) * 1) by A25, COMPLEX1: 1, XCMPLX_1: 60

 .= ((q `2 ) ^2 );

 then |. |[((q `1 ) / ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))), ((q `2 ) / ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 ))))]|.| <= 1 by A27, SQUARE_1: 22;

 hence ex p2 be Point of ( TOP-REAL 2) st p2 = y & |.p2.| <= 1 by A3, A4, A21;

 end;

 end;

 hence thesis by A1;

 end;

 let y be object;

 assume y in Cb;

 then

 consider p2 be Point of ( TOP-REAL 2) such that

 A28: p2 = y and

 A29: |.p2.| <= 1 by A1;

 set q = p2;

 now

 per cases ;

 case

 A30: q = ( 0. ( TOP-REAL 2));

 then

 A31: (q `1 ) = 0 by EUCLID: 52, EUCLID: 54;

 (q `2 ) = 0 by A30, EUCLID: 52, EUCLID: 54;

 then

 A32: y in Kb by A1, A28, A31;



 A33: (( Sq_Circ " ) . y) = y by A28, A30, JGRAPH_3: 28;



 A34: ( dom Sq_Circ ) = the carrier of ( TOP-REAL 2) by FUNCT_2:def 1;

 y = ( Sq_Circ . y) by A28, A33, FUNCT_1: 35, JGRAPH_3: 43;

 hence ex x be set st x in ( dom Sq_Circ ) & x in Kb & y = ( Sq_Circ . x) by A32, A34;

 end;

 case

 A35: q <> ( 0. ( TOP-REAL 2)) & ((q `2 ) <= (q `1 ) & ( - (q `1 )) <= (q `2 ) or (q `2 ) >= (q `1 ) & (q `2 ) <= ( - (q `1 )));

 set px = |[((q `1 ) * ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))), ((q `2 ) * ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 ))))]|;



 A36: (px `1 ) = ((q `1 ) * ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) by EUCLID: 52;



 A37: (px `2 ) = ((q `2 ) * ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) by EUCLID: 52;

 (1 + (((q `2 ) / (q `1 )) ^2 )) > 0 by XREAL_1: 34, XREAL_1: 63;

 then

 A38: ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 ))) > 0 by SQUARE_1: 25;



 A39: (1 + (((px `2 ) / (px `1 )) ^2 )) > 0 by XREAL_1: 34, XREAL_1: 63;



 A40: ((px `2 ) / (px `1 )) = ((q `2 ) / (q `1 )) by A36, A37, A38, XCMPLX_1: 91;



 A41: (q `1 ) = (((q `1 ) * ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) by A38, XCMPLX_1: 89

 .= ((px `1 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) by EUCLID: 52;



 A42: (q `2 ) = (((q `2 ) * ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) by A38, XCMPLX_1: 89

 .= ((px `2 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) by EUCLID: 52;



 A43: ( |.q.| ^2 ) = (((q `1 ) ^2 ) + ((q `2 ) ^2 )) by JGRAPH_3: 1;



 A44: ( |.q.| ^2 ) <= (1 ^2 ) by A29, SQUARE_1: 15;

 A45:

 now

 assume that

 A46: (px `1 ) = 0 and

 A47: (px `2 ) = 0 ;



 A48: ((q `1 ) * ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) = 0 by A46, EUCLID: 52;



 A49: ((q `2 ) * ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) = 0 by A47, EUCLID: 52;



 A50: (q `1 ) = 0 by A38, A48, XCMPLX_1: 6;

 (q `2 ) = 0 by A38, A49, XCMPLX_1: 6;

 hence contradiction by A35, A50, EUCLID: 53, EUCLID: 54;

 end;

 (q `2 ) <= (q `1 ) & ( - (q `1 )) <= (q `2 ) or (q `2 ) >= (q `1 ) & ((q `2 ) * ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) <= (( - (q `1 )) * ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) by A35, A38, XREAL_1: 64;

 then

 A51: (q `2 ) <= (q `1 ) & (( - (q `1 )) * ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) <= ((q `2 ) * ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) or (px `2 ) >= (px `1 ) & (px `2 ) <= ( - (px `1 )) by A36, A37, A38, XREAL_1: 64;

 then (px `2 ) <= (px `1 ) & ( - (px `1 )) <= (px `2 ) or (px `2 ) >= (px `1 ) & (px `2 ) <= ( - (px `1 )) by A36, A37, A38, XREAL_1: 64;

 then

 A52: ( Sq_Circ . px) = |[((px `1 ) / ( sqrt (1 + (((px `2 ) / (px `1 )) ^2 )))), ((px `2 ) / ( sqrt (1 + (((px `2 ) / (px `1 )) ^2 ))))]| by A45, JGRAPH_2: 3, JGRAPH_3:def 1;

 (px `2 ) <= (px `1 ) & ( - ( - (px `1 ))) >= ( - (px `2 )) or (px `2 ) >= (px `1 ) & (px `2 ) <= ( - (px `1 )) by A36, A37, A38, A51, XREAL_1: 24, XREAL_1: 64;

 then

 A53: (px `2 ) <= (px `1 ) & (px `1 ) >= ( - (px `2 )) or (px `2 ) >= (px `1 ) & ( - (px `2 )) >= ( - ( - (px `1 ))) by XREAL_1: 24;



 A54: ((px `1 ) / ( sqrt (1 + (((px `2 ) / (px `1 )) ^2 )))) = (q `1 ) by A36, A38, A40, XCMPLX_1: 89;



 A55: ((px `2 ) / ( sqrt (1 + (((px `2 ) / (px `1 )) ^2 )))) = (q `2 ) by A37, A38, A40, XCMPLX_1: 89;



 A56: ( dom Sq_Circ ) = the carrier of ( TOP-REAL 2) by FUNCT_2:def 1;

 not (px `1 ) = 0 by A36, A37, A38, A45, A51, XREAL_1: 64;

 then

 A57: ((px `1 ) ^2 ) > 0 by SQUARE_1: 12;



 A58: ((px `2 ) ^2 ) >= 0 by XREAL_1: 63;

 ((((px `1 ) ^2 ) / (( sqrt (1 + (((px `2 ) / (px `1 )) ^2 ))) ^2 )) + (((px `2 ) / ( sqrt (1 + (((px `2 ) / (px `1 )) ^2 )))) ^2 )) <= 1 by A40, A41, A42, A43, A44, XCMPLX_1: 76;

 then ((((px `1 ) ^2 ) / (( sqrt (1 + (((px `2 ) / (px `1 )) ^2 ))) ^2 )) + (((px `2 ) ^2 ) / (( sqrt (1 + (((px `2 ) / (px `1 )) ^2 ))) ^2 ))) <= 1 by XCMPLX_1: 76;

 then ((((px `1 ) ^2 ) / (1 + (((px `2 ) / (px `1 )) ^2 ))) + (((px `2 ) ^2 ) / (( sqrt (1 + (((px `2 ) / (px `1 )) ^2 ))) ^2 ))) <= 1 by A39, SQUARE_1:def 2;

 then ((((px `1 ) ^2 ) / (1 + (((px `2 ) / (px `1 )) ^2 ))) + (((px `2 ) ^2 ) / (1 + (((px `2 ) / (px `1 )) ^2 )))) <= 1 by A39, SQUARE_1:def 2;

 then (((((px `1 ) ^2 ) / (1 + (((px `2 ) / (px `1 )) ^2 ))) + (((px `2 ) ^2 ) / (1 + (((px `2 ) / (px `1 )) ^2 )))) * (1 + (((px `2 ) / (px `1 )) ^2 ))) <= (1 * (1 + (((px `2 ) / (px `1 )) ^2 ))) by A39, XREAL_1: 64;

 then (((((px `1 ) ^2 ) / (1 + (((px `2 ) / (px `1 )) ^2 ))) * (1 + (((px `2 ) / (px `1 )) ^2 ))) + ((((px `2 ) ^2 ) / (1 + (((px `2 ) / (px `1 )) ^2 ))) * (1 + (((px `2 ) / (px `1 )) ^2 )))) <= (1 * (1 + (((px `2 ) / (px `1 )) ^2 )));

 then (((px `1 ) ^2 ) + ((((px `2 ) ^2 ) / (1 + (((px `2 ) / (px `1 )) ^2 ))) * (1 + (((px `2 ) / (px `1 )) ^2 )))) <= (1 * (1 + (((px `2 ) / (px `1 )) ^2 ))) by A39, XCMPLX_1: 87;

 then

 A59: (((px `1 ) ^2 ) + ((px `2 ) ^2 )) <= (1 * (1 + (((px `2 ) / (px `1 )) ^2 ))) by A39, XCMPLX_1: 87;

 (1 * (1 + (((px `2 ) / (px `1 )) ^2 ))) = (1 + (((px `2 ) ^2 ) / ((px `1 ) ^2 ))) by XCMPLX_1: 76;

 then ((((px `1 ) ^2 ) + ((px `2 ) ^2 )) - 1) <= ((1 + (((px `2 ) ^2 ) / ((px `1 ) ^2 ))) - 1) by A59, XREAL_1: 9;

 then (((((px `1 ) ^2 ) + ((px `2 ) ^2 )) - 1) * ((px `1 ) ^2 )) <= ((((px `2 ) ^2 ) / ((px `1 ) ^2 )) * ((px `1 ) ^2 )) by A57, XREAL_1: 64;

 then ((((px `1 ) ^2 ) + (((px `2 ) ^2 ) - 1)) * ((px `1 ) ^2 )) <= ((px `2 ) ^2 ) by A57, XCMPLX_1: 87;

 then

 A60: (((((px `1 ) ^2 ) * ((px `1 ) ^2 )) + (((px `1 ) ^2 ) * (((px `2 ) ^2 ) - 1))) - ((px `2 ) ^2 )) <= 0 by XREAL_1: 47;

 (((((px `1 ) ^2 ) * ((px `1 ) ^2 )) + ((((px `1 ) ^2 ) * ((px `2 ) ^2 )) - (((px `1 ) ^2 ) * 1))) - ((px `2 ) ^2 )) = ((((px `1 ) ^2 ) - 1) * (((px `1 ) ^2 ) + ((px `2 ) ^2 )));

 then (((px `1 ) ^2 ) - 1) <= 0 & (((px `1 ) ^2 ) - 1) >= 0 or (((px `1 ) ^2 ) - 1) <= 0 & (((px `1 ) ^2 ) + ((px `2 ) ^2 )) >= 0 or (((px `1 ) ^2 ) + ((px `2 ) ^2 )) <= 0 & (((px `1 ) ^2 ) - 1) >= 0 or (((px `1 ) ^2 ) + ((px `2 ) ^2 )) <= 0 & (((px `1 ) ^2 ) + ((px `2 ) ^2 )) >= 0 by A60, XREAL_1: 129, XREAL_1: 130;

 then

 A61: ((((px `1 ) ^2 ) - 1) + 1) <= ( 0 + 1) by A58, XREAL_1: 7;

 then

 A62: (px `1 ) <= (1 ^2 ) by SQUARE_1: 47;



 A63: (px `1 ) >= ( - (1 ^2 )) by A61, SQUARE_1: 47;

 then (px `2 ) <= 1 & 1 >= ( - (px `2 )) or (px `2 ) >= ( - 1) & ( - (px `2 )) >= (px `1 ) by A53, A62, XXREAL_0: 2;

 then (px `2 ) <= 1 & ( - 1) <= ( - ( - (px `2 ))) or (px `2 ) >= ( - 1) & ( - (px `2 )) >= ( - 1) by A63, XREAL_1: 24, XXREAL_0: 2;

 then (px `2 ) <= 1 & ( - 1) <= (px `2 ) or (px `2 ) >= ( - 1) & ( - ( - (px `2 ))) <= ( - ( - 1)) by XREAL_1: 24;

 then px in Kb by A1, A62, A63;

 hence ex x be set st x in ( dom Sq_Circ ) & x in Kb & y = ( Sq_Circ . x) by A28, A52, A54, A55, A56, EUCLID: 53;

 end;

 case

 A64: q <> ( 0. ( TOP-REAL 2)) & not ((q `2 ) <= (q `1 ) & ( - (q `1 )) <= (q `2 ) or (q `2 ) >= (q `1 ) & (q `2 ) <= ( - (q `1 )));

 set px = |[((q `1 ) * ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))), ((q `2 ) * ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 ))))]|;



 A65: (q `1 ) <= (q `2 ) & ( - (q `2 )) <= (q `1 ) or (q `1 ) >= (q `2 ) & (q `1 ) <= ( - (q `2 )) by A64, JGRAPH_2: 13;



 A66: (px `2 ) = ((q `2 ) * ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))) by EUCLID: 52;



 A67: (px `1 ) = ((q `1 ) * ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))) by EUCLID: 52;

 (1 + (((q `1 ) / (q `2 )) ^2 )) > 0 by XREAL_1: 34, XREAL_1: 63;

 then

 A68: ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 ))) > 0 by SQUARE_1: 25;



 A69: (1 + (((px `1 ) / (px `2 )) ^2 )) > 0 by XREAL_1: 34, XREAL_1: 63;



 A70: ((px `1 ) / (px `2 )) = ((q `1 ) / (q `2 )) by A66, A67, A68, XCMPLX_1: 91;



 A71: (q `2 ) = (((q `2 ) * ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))) / ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))) by A68, XCMPLX_1: 89

 .= ((px `2 ) / ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))) by EUCLID: 52;



 A72: (q `1 ) = (((q `1 ) * ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))) / ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))) by A68, XCMPLX_1: 89

 .= ((px `1 ) / ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))) by EUCLID: 52;



 A73: ( |.q.| ^2 ) = (((q `2 ) ^2 ) + ((q `1 ) ^2 )) by JGRAPH_3: 1;



 A74: ( |.q.| ^2 ) <= (1 ^2 ) by A29, SQUARE_1: 15;

 A75:

 now

 assume that

 A76: (px `2 ) = 0 and

 A77: (px `1 ) = 0 ;



 A78: ((q `2 ) * ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))) = 0 by A76, EUCLID: 52;

 ((q `1 ) * ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))) = 0 by A77, EUCLID: 52;

 then (q `1 ) = 0 by A68, XCMPLX_1: 6;

 hence contradiction by A64, A68, A78, XCMPLX_1: 6;

 end;

 (q `1 ) <= (q `2 ) & ( - (q `2 )) <= (q `1 ) or (q `1 ) >= (q `2 ) & ((q `1 ) * ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))) <= (( - (q `2 )) * ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))) by A65, A68, XREAL_1: 64;

 then

 A79: (q `1 ) <= (q `2 ) & (( - (q `2 )) * ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))) <= ((q `1 ) * ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))) or (px `1 ) >= (px `2 ) & (px `1 ) <= ( - (px `2 )) by A66, A67, A68, XREAL_1: 64;

 then (px `1 ) <= (px `2 ) & ( - (px `2 )) <= (px `1 ) or (px `1 ) >= (px `2 ) & (px `1 ) <= ( - (px `2 )) by A66, A67, A68, XREAL_1: 64;

 then

 A80: ( Sq_Circ . px) = |[((px `1 ) / ( sqrt (1 + (((px `1 ) / (px `2 )) ^2 )))), ((px `2 ) / ( sqrt (1 + (((px `1 ) / (px `2 )) ^2 ))))]| by A75, JGRAPH_2: 3, JGRAPH_3: 4;

 (px `1 ) <= (px `2 ) & ( - ( - (px `2 ))) >= ( - (px `1 )) or (px `1 ) >= (px `2 ) & (px `1 ) <= ( - (px `2 )) by A66, A67, A68, A79, XREAL_1: 24, XREAL_1: 64;

 then

 A81: (px `1 ) <= (px `2 ) & (px `2 ) >= ( - (px `1 )) or (px `1 ) >= (px `2 ) & ( - (px `1 )) >= ( - ( - (px `2 ))) by XREAL_1: 24;



 A82: ((px `2 ) / ( sqrt (1 + (((px `1 ) / (px `2 )) ^2 )))) = (q `2 ) by A66, A68, A70, XCMPLX_1: 89;



 A83: ((px `1 ) / ( sqrt (1 + (((px `1 ) / (px `2 )) ^2 )))) = (q `1 ) by A67, A68, A70, XCMPLX_1: 89;



 A84: ( dom Sq_Circ ) = the carrier of ( TOP-REAL 2) by FUNCT_2:def 1;

 not (px `2 ) = 0 by A66, A67, A68, A75, A79, XREAL_1: 64;

 then

 A85: ((px `2 ) ^2 ) > 0 by SQUARE_1: 12;



 A86: ((px `1 ) ^2 ) >= 0 by XREAL_1: 63;

 ((((px `2 ) ^2 ) / (( sqrt (1 + (((px `1 ) / (px `2 )) ^2 ))) ^2 )) + (((px `1 ) / ( sqrt (1 + (((px `1 ) / (px `2 )) ^2 )))) ^2 )) <= 1 by A70, A71, A72, A73, A74, XCMPLX_1: 76;

 then ((((px `2 ) ^2 ) / (( sqrt (1 + (((px `1 ) / (px `2 )) ^2 ))) ^2 )) + (((px `1 ) ^2 ) / (( sqrt (1 + (((px `1 ) / (px `2 )) ^2 ))) ^2 ))) <= 1 by XCMPLX_1: 76;

 then ((((px `2 ) ^2 ) / (1 + (((px `1 ) / (px `2 )) ^2 ))) + (((px `1 ) ^2 ) / (( sqrt (1 + (((px `1 ) / (px `2 )) ^2 ))) ^2 ))) <= 1 by A69, SQUARE_1:def 2;

 then ((((px `2 ) ^2 ) / (1 + (((px `1 ) / (px `2 )) ^2 ))) + (((px `1 ) ^2 ) / (1 + (((px `1 ) / (px `2 )) ^2 )))) <= 1 by A69, SQUARE_1:def 2;

 then (((((px `2 ) ^2 ) / (1 + (((px `1 ) / (px `2 )) ^2 ))) + (((px `1 ) ^2 ) / (1 + (((px `1 ) / (px `2 )) ^2 )))) * (1 + (((px `1 ) / (px `2 )) ^2 ))) <= (1 * (1 + (((px `1 ) / (px `2 )) ^2 ))) by A69, XREAL_1: 64;

 then (((((px `2 ) ^2 ) / (1 + (((px `1 ) / (px `2 )) ^2 ))) * (1 + (((px `1 ) / (px `2 )) ^2 ))) + ((((px `1 ) ^2 ) / (1 + (((px `1 ) / (px `2 )) ^2 ))) * (1 + (((px `1 ) / (px `2 )) ^2 )))) <= (1 * (1 + (((px `1 ) / (px `2 )) ^2 )));

 then (((px `2 ) ^2 ) + ((((px `1 ) ^2 ) / (1 + (((px `1 ) / (px `2 )) ^2 ))) * (1 + (((px `1 ) / (px `2 )) ^2 )))) <= (1 * (1 + (((px `1 ) / (px `2 )) ^2 ))) by A69, XCMPLX_1: 87;

 then

 A87: (((px `2 ) ^2 ) + ((px `1 ) ^2 )) <= (1 * (1 + (((px `1 ) / (px `2 )) ^2 ))) by A69, XCMPLX_1: 87;

 (1 * (1 + (((px `1 ) / (px `2 )) ^2 ))) = (1 + (((px `1 ) ^2 ) / ((px `2 ) ^2 ))) by XCMPLX_1: 76;

 then ((((px `2 ) ^2 ) + ((px `1 ) ^2 )) - 1) <= ((1 + (((px `1 ) ^2 ) / ((px `2 ) ^2 ))) - 1) by A87, XREAL_1: 9;

 then (((((px `2 ) ^2 ) + ((px `1 ) ^2 )) - 1) * ((px `2 ) ^2 )) <= ((((px `1 ) ^2 ) / ((px `2 ) ^2 )) * ((px `2 ) ^2 )) by A85, XREAL_1: 64;

 then ((((px `2 ) ^2 ) + (((px `1 ) ^2 ) - 1)) * ((px `2 ) ^2 )) <= ((px `1 ) ^2 ) by A85, XCMPLX_1: 87;

 then

 A88: (((((px `2 ) ^2 ) * ((px `2 ) ^2 )) + (((px `2 ) ^2 ) * (((px `1 ) ^2 ) - 1))) - ((px `1 ) ^2 )) <= 0 by XREAL_1: 47;

 (((((px `2 ) ^2 ) * ((px `2 ) ^2 )) + ((((px `2 ) ^2 ) * ((px `1 ) ^2 )) - (((px `2 ) ^2 ) * 1))) - ((px `1 ) ^2 )) = ((((px `2 ) ^2 ) - 1) * (((px `2 ) ^2 ) + ((px `1 ) ^2 )));

 then (((px `2 ) ^2 ) - 1) <= 0 or (((px `2 ) ^2 ) + ((px `1 ) ^2 )) <= 0 by A88, XREAL_1: 129;

 then

 A89: ((((px `2 ) ^2 ) - 1) + 1) <= ( 0 + 1) by A86, XREAL_1: 7;

 then

 A90: (px `2 ) <= (1 ^2 ) by SQUARE_1: 47;



 A91: (px `2 ) >= ( - (1 ^2 )) by A89, SQUARE_1: 47;

 then (px `1 ) <= 1 & 1 >= ( - (px `1 )) or (px `1 ) >= ( - 1) & ( - (px `1 )) >= (px `2 ) by A81, A90, XXREAL_0: 2;

 then (px `1 ) <= 1 & ( - 1) <= ( - ( - (px `1 ))) or (px `1 ) >= ( - 1) & ( - (px `1 )) >= ( - 1) by A91, XREAL_1: 24, XXREAL_0: 2;

 then (px `1 ) <= 1 & ( - 1) <= (px `1 ) or (px `1 ) >= ( - 1) & ( - ( - (px `1 ))) <= ( - ( - 1)) by XREAL_1: 24;

 then px in Kb by A1, A90, A91;

 hence ex x be set st x in ( dom Sq_Circ ) & x in Kb & y = ( Sq_Circ . x) by A28, A80, A82, A83, A84, EUCLID: 53;

 end;

 end;

 hence thesis by FUNCT_1:def 6;

 end;

 theorem :: JGRAPH_6:28



 Th28: for Kb,Cb be Subset of ( TOP-REAL 2) st Kb = { p : not (( - 1) < (p `1 ) & (p `1 ) < 1 & ( - 1) < (p `2 ) & (p `2 ) < 1) } & Cb = { p2 where p2 be Point of ( TOP-REAL 2) : |.p2.| >= 1 } holds ( Sq_Circ .: Kb) = Cb

 proof

 let Kb,Cb be Subset of ( TOP-REAL 2);

 assume

 A1: Kb = { p : not (( - 1) < (p `1 ) & (p `1 ) < 1 & ( - 1) < (p `2 ) & (p `2 ) < 1) } & Cb = { p2 where p2 be Point of ( TOP-REAL 2) : |.p2.| >= 1 };

 thus ( Sq_Circ .: Kb) c= Cb

 proof

 let y be object;

 assume y in ( Sq_Circ .: Kb);

 then

 consider x be object such that x in ( dom Sq_Circ ) and

 A2: x in Kb and

 A3: y = ( Sq_Circ . x) by FUNCT_1:def 6;

 consider q be Point of ( TOP-REAL 2) such that

 A4: q = x and

 A5: not (( - 1) < (q `1 ) & (q `1 ) < 1 & ( - 1) < (q `2 ) & (q `2 ) < 1) by A1, A2;

 now

 per cases ;

 case

 A6: q = ( 0. ( TOP-REAL 2));

 then

 A7: (q `1 ) = 0 by EUCLID: 52, EUCLID: 54;

 (q `2 ) = 0 by A6, EUCLID: 52, EUCLID: 54;

 hence contradiction by A5, A7;

 end;

 case

 A8: q <> ( 0. ( TOP-REAL 2)) & ((q `2 ) <= (q `1 ) & ( - (q `1 )) <= (q `2 ) or (q `2 ) >= (q `1 ) & (q `2 ) <= ( - (q `1 )));

 then

 A9: ( Sq_Circ . q) = |[((q `1 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))), ((q `2 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 ))))]| by JGRAPH_3:def 1;



 A10: not (( - 1) < (q `2 ) & (q `2 ) < 1) implies ( - 1) >= (q `1 ) or (q `1 ) >= 1

 proof

 assume

 A11: not (( - 1) < (q `2 ) & (q `2 ) < 1);

 now

 per cases by A11;

 case

 A12: ( - 1) >= (q `2 );

 then ( - (q `1 )) <= ( - 1) or (q `2 ) >= (q `1 ) & (q `2 ) <= ( - (q `1 )) by A8, XXREAL_0: 2;

 hence thesis by A12, XREAL_1: 24, XXREAL_0: 2;

 end;

 case (q `2 ) >= 1;

 then 1 <= (q `1 ) or 1 <= ( - (q `1 )) by A8, XXREAL_0: 2;

 then 1 <= (q `1 ) or ( - ( - (q `1 ))) <= ( - 1) by XREAL_1: 24;

 hence thesis;

 end;

 end;

 hence thesis;

 end;



 A13: ( |[((q `1 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))), ((q `2 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 ))))]| `1 ) = ((q `1 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) by EUCLID: 52;



 A14: ( |[((q `1 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))), ((q `2 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 ))))]| `2 ) = ((q `2 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) by EUCLID: 52;



 A15: (1 + (((q `2 ) / (q `1 )) ^2 )) > 0 by XREAL_1: 34, XREAL_1: 63;

 A16:

 now

 assume

 A17: (q `1 ) = 0 ;

 then (q `2 ) = 0 by A8;

 hence contradiction by A8, A17, EUCLID: 53, EUCLID: 54;

 end;

 then

 A18: ((q `1 ) ^2 ) > 0 by SQUARE_1: 12;

 ((q `1 ) ^2 ) >= (1 ^2 ) by A5, A10, SQUARE_1: 48;

 then

 A19: ( sqrt ((q `1 ) ^2 )) >= 1 by SQUARE_1: 18, SQUARE_1: 26;

 ( |. |[((q `1 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))), ((q `2 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 ))))]|.| ^2 ) = ((((q `1 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) ^2 ) + (((q `2 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) ^2 )) by A13, A14, JGRAPH_3: 1

 .= ((((q `1 ) ^2 ) / (( sqrt (1 + (((q `2 ) / (q `1 )) ^2 ))) ^2 )) + (((q `2 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) ^2 )) by XCMPLX_1: 76

 .= ((((q `1 ) ^2 ) / (( sqrt (1 + (((q `2 ) / (q `1 )) ^2 ))) ^2 )) + (((q `2 ) ^2 ) / (( sqrt (1 + (((q `2 ) / (q `1 )) ^2 ))) ^2 ))) by XCMPLX_1: 76

 .= ((((q `1 ) ^2 ) / (1 + (((q `2 ) / (q `1 )) ^2 ))) + (((q `2 ) ^2 ) / (( sqrt (1 + (((q `2 ) / (q `1 )) ^2 ))) ^2 ))) by A15, SQUARE_1:def 2

 .= ((((q `1 ) ^2 ) / (1 + (((q `2 ) / (q `1 )) ^2 ))) + (((q `2 ) ^2 ) / (1 + (((q `2 ) / (q `1 )) ^2 )))) by A15, SQUARE_1:def 2

 .= ((((q `1 ) ^2 ) + ((q `2 ) ^2 )) / (1 + (((q `2 ) / (q `1 )) ^2 ))) by XCMPLX_1: 62

 .= ((((q `1 ) ^2 ) + ((q `2 ) ^2 )) / (1 + (((q `2 ) ^2 ) / ((q `1 ) ^2 )))) by XCMPLX_1: 76

 .= ((((q `1 ) ^2 ) + ((q `2 ) ^2 )) / ((((q `1 ) ^2 ) / ((q `1 ) ^2 )) + (((q `2 ) ^2 ) / ((q `1 ) ^2 )))) by A18, XCMPLX_1: 60

 .= ((((q `1 ) ^2 ) + ((q `2 ) ^2 )) / ((((q `1 ) ^2 ) + ((q `2 ) ^2 )) / ((q `1 ) ^2 ))) by XCMPLX_1: 62

 .= (((q `1 ) ^2 ) * ((((q `1 ) ^2 ) + ((q `2 ) ^2 )) / (((q `1 ) ^2 ) + ((q `2 ) ^2 )))) by XCMPLX_1: 81

 .= (((q `1 ) ^2 ) * 1) by A16, COMPLEX1: 1, XCMPLX_1: 60

 .= ((q `1 ) ^2 );

 then |. |[((q `1 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))), ((q `2 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 ))))]|.| >= 1 by A19, SQUARE_1: 22;

 hence ex p2 be Point of ( TOP-REAL 2) st p2 = y & |.p2.| >= 1 by A3, A4, A9;

 end;

 case

 A20: q <> ( 0. ( TOP-REAL 2)) & not ((q `2 ) <= (q `1 ) & ( - (q `1 )) <= (q `2 ) or (q `2 ) >= (q `1 ) & (q `2 ) <= ( - (q `1 )));

 then

 A21: ( Sq_Circ . q) = |[((q `1 ) / ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))), ((q `2 ) / ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 ))))]| by JGRAPH_3:def 1;



 A22: ( |[((q `1 ) / ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))), ((q `2 ) / ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 ))))]| `1 ) = ((q `1 ) / ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))) by EUCLID: 52;



 A23: ( |[((q `1 ) / ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))), ((q `2 ) / ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 ))))]| `2 ) = ((q `2 ) / ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))) by EUCLID: 52;



 A24: (1 + (((q `1 ) / (q `2 )) ^2 )) > 0 by XREAL_1: 34, XREAL_1: 63;



 A25: (q `2 ) <> 0 by A20;

 then

 A26: ((q `2 ) ^2 ) > 0 by SQUARE_1: 12;

 not (( - 1) < (q `1 ) & (q `1 ) < 1) implies ( - 1) >= (q `2 ) or (q `2 ) >= 1

 proof

 assume

 A27: not (( - 1) < (q `1 ) & (q `1 ) < 1);

 now

 per cases by A27;

 case

 A28: ( - 1) >= (q `1 );

 then (q `2 ) <= ( - 1) or (q `1 ) < (q `2 ) & ( - (q `2 )) <= ( - ( - (q `1 ))) by A20, XREAL_1: 24, XXREAL_0: 2;

 then ( - (q `2 )) <= ( - 1) or ( - 1) >= (q `2 ) by A28, XXREAL_0: 2;

 hence thesis by XREAL_1: 24;

 end;

 case

 A29: (q `1 ) >= 1;

 ( - ( - (q `1 ))) <= ( - (q `2 )) & (q `2 ) <= (q `1 ) or (q `2 ) >= (q `1 ) & (q `2 ) >= ( - (q `1 )) by A20, XREAL_1: 24;

 then 1 <= ( - (q `2 )) or (q `2 ) >= (q `1 ) & (q `2 ) >= ( - (q `1 )) by A29, XXREAL_0: 2;

 then ( - 1) >= ( - ( - (q `2 ))) or 1 <= (q `2 ) by A29, XREAL_1: 24, XXREAL_0: 2;

 hence thesis;

 end;

 end;

 hence thesis;

 end;

 then ((q `2 ) ^2 ) >= (1 ^2 ) by A5, SQUARE_1: 48;

 then

 A30: ( sqrt ((q `2 ) ^2 )) >= 1 by SQUARE_1: 18, SQUARE_1: 26;

 ( |. |[((q `1 ) / ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))), ((q `2 ) / ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 ))))]|.| ^2 ) = ((((q `1 ) / ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))) ^2 ) + (((q `2 ) / ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))) ^2 )) by A22, A23, JGRAPH_3: 1

 .= ((((q `1 ) ^2 ) / (( sqrt (1 + (((q `1 ) / (q `2 )) ^2 ))) ^2 )) + (((q `2 ) / ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))) ^2 )) by XCMPLX_1: 76

 .= ((((q `1 ) ^2 ) / (( sqrt (1 + (((q `1 ) / (q `2 )) ^2 ))) ^2 )) + (((q `2 ) ^2 ) / (( sqrt (1 + (((q `1 ) / (q `2 )) ^2 ))) ^2 ))) by XCMPLX_1: 76

 .= ((((q `1 ) ^2 ) / (1 + (((q `1 ) / (q `2 )) ^2 ))) + (((q `2 ) ^2 ) / (( sqrt (1 + (((q `1 ) / (q `2 )) ^2 ))) ^2 ))) by A24, SQUARE_1:def 2

 .= ((((q `1 ) ^2 ) / (1 + (((q `1 ) / (q `2 )) ^2 ))) + (((q `2 ) ^2 ) / (1 + (((q `1 ) / (q `2 )) ^2 )))) by A24, SQUARE_1:def 2

 .= ((((q `1 ) ^2 ) + ((q `2 ) ^2 )) / (1 + (((q `1 ) / (q `2 )) ^2 ))) by XCMPLX_1: 62

 .= ((((q `1 ) ^2 ) + ((q `2 ) ^2 )) / (1 + (((q `1 ) ^2 ) / ((q `2 ) ^2 )))) by XCMPLX_1: 76

 .= ((((q `1 ) ^2 ) + ((q `2 ) ^2 )) / ((((q `1 ) ^2 ) / ((q `2 ) ^2 )) + (((q `2 ) ^2 ) / ((q `2 ) ^2 )))) by A26, XCMPLX_1: 60

 .= ((((q `1 ) ^2 ) + ((q `2 ) ^2 )) / ((((q `1 ) ^2 ) + ((q `2 ) ^2 )) / ((q `2 ) ^2 ))) by XCMPLX_1: 62

 .= (((q `2 ) ^2 ) * ((((q `1 ) ^2 ) + ((q `2 ) ^2 )) / (((q `1 ) ^2 ) + ((q `2 ) ^2 )))) by XCMPLX_1: 81

 .= (((q `2 ) ^2 ) * 1) by A25, COMPLEX1: 1, XCMPLX_1: 60

 .= ((q `2 ) ^2 );

 then |. |[((q `1 ) / ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))), ((q `2 ) / ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 ))))]|.| >= 1 by A30, SQUARE_1: 22;

 hence ex p2 be Point of ( TOP-REAL 2) st p2 = y & |.p2.| >= 1 by A3, A4, A21;

 end;

 end;

 hence thesis by A1;

 end;

 let y be object;

 assume y in Cb;

 then

 consider p2 be Point of ( TOP-REAL 2) such that

 A31: p2 = y and

 A32: |.p2.| >= 1 by A1;

 set q = p2;

 now

 per cases ;

 case q = ( 0. ( TOP-REAL 2));

 hence contradiction by A32, TOPRNS_1: 23;

 end;

 case

 A33: q <> ( 0. ( TOP-REAL 2)) & ((q `2 ) <= (q `1 ) & ( - (q `1 )) <= (q `2 ) or (q `2 ) >= (q `1 ) & (q `2 ) <= ( - (q `1 )));

 set px = |[((q `1 ) * ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))), ((q `2 ) * ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 ))))]|;



 A34: (px `1 ) = ((q `1 ) * ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) by EUCLID: 52;



 A35: (px `2 ) = ((q `2 ) * ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) by EUCLID: 52;

 (1 + (((q `2 ) / (q `1 )) ^2 )) > 0 by XREAL_1: 34, XREAL_1: 63;

 then

 A36: ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 ))) > 0 by SQUARE_1: 25;



 A37: (1 + (((px `2 ) / (px `1 )) ^2 )) > 0 by XREAL_1: 34, XREAL_1: 63;



 A38: ((px `2 ) / (px `1 )) = ((q `2 ) / (q `1 )) by A34, A35, A36, XCMPLX_1: 91;



 A39: (q `1 ) = (((q `1 ) * ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) by A36, XCMPLX_1: 89

 .= ((px `1 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) by EUCLID: 52;



 A40: (q `2 ) = (((q `2 ) * ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) by A36, XCMPLX_1: 89

 .= ((px `2 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) by EUCLID: 52;



 A41: ( |.q.| ^2 ) = (((q `1 ) ^2 ) + ((q `2 ) ^2 )) by JGRAPH_3: 1;



 A42: ( |.q.| ^2 ) >= (1 ^2 ) by A32, SQUARE_1: 15;

 A43:

 now

 assume that

 A44: (px `1 ) = 0 and

 A45: (px `2 ) = 0 ;



 A46: ((q `1 ) * ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) = 0 by A44, EUCLID: 52;



 A47: ((q `2 ) * ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) = 0 by A45, EUCLID: 52;



 A48: (q `1 ) = 0 by A36, A46, XCMPLX_1: 6;

 (q `2 ) = 0 by A36, A47, XCMPLX_1: 6;

 hence contradiction by A33, A48, EUCLID: 53, EUCLID: 54;

 end;

 (q `2 ) <= (q `1 ) & ( - (q `1 )) <= (q `2 ) or (q `2 ) >= (q `1 ) & ((q `2 ) * ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) <= (( - (q `1 )) * ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) by A33, A36, XREAL_1: 64;

 then

 A49: (q `2 ) <= (q `1 ) & (( - (q `1 )) * ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) <= ((q `2 ) * ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) or (px `2 ) >= (px `1 ) & (px `2 ) <= ( - (px `1 )) by A34, A35, A36, XREAL_1: 64;

 then (px `2 ) <= (px `1 ) & ( - (px `1 )) <= (px `2 ) or (px `2 ) >= (px `1 ) & (px `2 ) <= ( - (px `1 )) by A34, A35, A36, XREAL_1: 64;

 then

 A50: ( Sq_Circ . px) = |[((px `1 ) / ( sqrt (1 + (((px `2 ) / (px `1 )) ^2 )))), ((px `2 ) / ( sqrt (1 + (((px `2 ) / (px `1 )) ^2 ))))]| by A43, JGRAPH_2: 3, JGRAPH_3:def 1;



 A51: ((px `1 ) / ( sqrt (1 + (((px `2 ) / (px `1 )) ^2 )))) = (q `1 ) by A34, A36, A38, XCMPLX_1: 89;



 A52: ((px `2 ) / ( sqrt (1 + (((px `2 ) / (px `1 )) ^2 )))) = (q `2 ) by A35, A36, A38, XCMPLX_1: 89;



 A53: ( dom Sq_Circ ) = the carrier of ( TOP-REAL 2) by FUNCT_2:def 1;

 not (px `1 ) = 0 by A34, A35, A36, A43, A49, XREAL_1: 64;

 then

 A54: ((px `1 ) ^2 ) > 0 by SQUARE_1: 12;

 then

 A55: (((px `1 ) ^2 ) + ((px `2 ) ^2 )) > 0 by XREAL_1: 34, XREAL_1: 63;

 ((((px `1 ) ^2 ) / (( sqrt (1 + (((px `2 ) / (px `1 )) ^2 ))) ^2 )) + (((px `2 ) / ( sqrt (1 + (((px `2 ) / (px `1 )) ^2 )))) ^2 )) >= 1 by A38, A39, A40, A41, A42, XCMPLX_1: 76;

 then ((((px `1 ) ^2 ) / (( sqrt (1 + (((px `2 ) / (px `1 )) ^2 ))) ^2 )) + (((px `2 ) ^2 ) / (( sqrt (1 + (((px `2 ) / (px `1 )) ^2 ))) ^2 ))) >= 1 by XCMPLX_1: 76;

 then ((((px `1 ) ^2 ) / (1 + (((px `2 ) / (px `1 )) ^2 ))) + (((px `2 ) ^2 ) / (( sqrt (1 + (((px `2 ) / (px `1 )) ^2 ))) ^2 ))) >= 1 by A37, SQUARE_1:def 2;

 then ((((px `1 ) ^2 ) / (1 + (((px `2 ) / (px `1 )) ^2 ))) + (((px `2 ) ^2 ) / (1 + (((px `2 ) / (px `1 )) ^2 )))) >= 1 by A37, SQUARE_1:def 2;

 then (((((px `1 ) ^2 ) / (1 + (((px `2 ) / (px `1 )) ^2 ))) + (((px `2 ) ^2 ) / (1 + (((px `2 ) / (px `1 )) ^2 )))) * (1 + (((px `2 ) / (px `1 )) ^2 ))) >= (1 * (1 + (((px `2 ) / (px `1 )) ^2 ))) by A37, XREAL_1: 64;

 then (((((px `1 ) ^2 ) / (1 + (((px `2 ) / (px `1 )) ^2 ))) * (1 + (((px `2 ) / (px `1 )) ^2 ))) + ((((px `2 ) ^2 ) / (1 + (((px `2 ) / (px `1 )) ^2 ))) * (1 + (((px `2 ) / (px `1 )) ^2 )))) >= (1 * (1 + (((px `2 ) / (px `1 )) ^2 )));

 then (((px `1 ) ^2 ) + ((((px `2 ) ^2 ) / (1 + (((px `2 ) / (px `1 )) ^2 ))) * (1 + (((px `2 ) / (px `1 )) ^2 )))) >= (1 * (1 + (((px `2 ) / (px `1 )) ^2 ))) by A37, XCMPLX_1: 87;

 then

 A56: (((px `1 ) ^2 ) + ((px `2 ) ^2 )) >= (1 * (1 + (((px `2 ) / (px `1 )) ^2 ))) by A37, XCMPLX_1: 87;

 (1 * (1 + (((px `2 ) / (px `1 )) ^2 ))) = (1 + (((px `2 ) ^2 ) / ((px `1 ) ^2 ))) by XCMPLX_1: 76;

 then ((((px `1 ) ^2 ) + ((px `2 ) ^2 )) - 1) >= ((1 + (((px `2 ) ^2 ) / ((px `1 ) ^2 ))) - 1) by A56, XREAL_1: 9;

 then (((((px `1 ) ^2 ) + ((px `2 ) ^2 )) - 1) * ((px `1 ) ^2 )) >= ((((px `2 ) ^2 ) / ((px `1 ) ^2 )) * ((px `1 ) ^2 )) by A54, XREAL_1: 64;

 then ((((px `1 ) ^2 ) + (((px `2 ) ^2 ) - 1)) * ((px `1 ) ^2 )) >= ((px `2 ) ^2 ) by A54, XCMPLX_1: 87;

 then

 A57: (((((px `1 ) ^2 ) * ((px `1 ) ^2 )) + (((px `1 ) ^2 ) * (((px `2 ) ^2 ) - 1))) - ((px `2 ) ^2 )) >= 0 by XREAL_1: 48;

 (((((px `1 ) ^2 ) * ((px `1 ) ^2 )) + ((((px `1 ) ^2 ) * ((px `2 ) ^2 )) - (((px `1 ) ^2 ) * 1))) - ((px `2 ) ^2 )) = ((((px `1 ) ^2 ) - 1) * (((px `1 ) ^2 ) + ((px `2 ) ^2 )));

 then (((px `1 ) ^2 ) - 1) >= 0 by A55, A57, XREAL_1: 132;

 then ((((px `1 ) ^2 ) - 1) + 1) >= ( 0 + 1) by XREAL_1: 7;

 then (px `1 ) >= (1 ^2 ) or (px `1 ) <= ( - 1) by SQUARE_1: 50;

 then px in Kb by A1;

 hence ex x be set st x in ( dom Sq_Circ ) & x in Kb & y = ( Sq_Circ . x) by A31, A50, A51, A52, A53, EUCLID: 53;

 end;

 case

 A58: q <> ( 0. ( TOP-REAL 2)) & not ((q `2 ) <= (q `1 ) & ( - (q `1 )) <= (q `2 ) or (q `2 ) >= (q `1 ) & (q `2 ) <= ( - (q `1 )));

 set px = |[((q `1 ) * ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))), ((q `2 ) * ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 ))))]|;



 A59: (q `1 ) <= (q `2 ) & ( - (q `2 )) <= (q `1 ) or (q `1 ) >= (q `2 ) & (q `1 ) <= ( - (q `2 )) by A58, JGRAPH_2: 13;



 A60: (px `2 ) = ((q `2 ) * ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))) by EUCLID: 52;



 A61: (px `1 ) = ((q `1 ) * ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))) by EUCLID: 52;

 (1 + (((q `1 ) / (q `2 )) ^2 )) > 0 by XREAL_1: 34, XREAL_1: 63;

 then

 A62: ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 ))) > 0 by SQUARE_1: 25;



 A63: (1 + (((px `1 ) / (px `2 )) ^2 )) > 0 by XREAL_1: 34, XREAL_1: 63;



 A64: ((px `1 ) / (px `2 )) = ((q `1 ) / (q `2 )) by A60, A61, A62, XCMPLX_1: 91;



 A65: (q `2 ) = (((q `2 ) * ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))) / ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))) by A62, XCMPLX_1: 89

 .= ((px `2 ) / ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))) by EUCLID: 52;



 A66: (q `1 ) = (((q `1 ) * ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))) / ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))) by A62, XCMPLX_1: 89

 .= ((px `1 ) / ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))) by EUCLID: 52;



 A67: ( |.q.| ^2 ) = (((q `2 ) ^2 ) + ((q `1 ) ^2 )) by JGRAPH_3: 1;



 A68: ( |.q.| ^2 ) >= (1 ^2 ) by A32, SQUARE_1: 15;

 A69:

 now

 assume that

 A70: (px `2 ) = 0 and

 A71: (px `1 ) = 0 ;



 A72: ((q `2 ) * ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))) = 0 by A70, EUCLID: 52;

 ((q `1 ) * ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))) = 0 by A71, EUCLID: 52;

 then (q `1 ) = 0 by A62, XCMPLX_1: 6;

 hence contradiction by A58, A62, A72, XCMPLX_1: 6;

 end;

 (q `1 ) <= (q `2 ) & ( - (q `2 )) <= (q `1 ) or (q `1 ) >= (q `2 ) & ((q `1 ) * ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))) <= (( - (q `2 )) * ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))) by A59, A62, XREAL_1: 64;

 then

 A73: (q `1 ) <= (q `2 ) & (( - (q `2 )) * ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))) <= ((q `1 ) * ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))) or (px `1 ) >= (px `2 ) & (px `1 ) <= ( - (px `2 )) by A60, A61, A62, XREAL_1: 64;

 then (px `1 ) <= (px `2 ) & ( - (px `2 )) <= (px `1 ) or (px `1 ) >= (px `2 ) & (px `1 ) <= ( - (px `2 )) by A60, A61, A62, XREAL_1: 64;

 then

 A74: ( Sq_Circ . px) = |[((px `1 ) / ( sqrt (1 + (((px `1 ) / (px `2 )) ^2 )))), ((px `2 ) / ( sqrt (1 + (((px `1 ) / (px `2 )) ^2 ))))]| by A69, JGRAPH_2: 3, JGRAPH_3: 4;



 A75: ((px `2 ) / ( sqrt (1 + (((px `1 ) / (px `2 )) ^2 )))) = (q `2 ) by A60, A62, A64, XCMPLX_1: 89;



 A76: ((px `1 ) / ( sqrt (1 + (((px `1 ) / (px `2 )) ^2 )))) = (q `1 ) by A61, A62, A64, XCMPLX_1: 89;



 A77: ( dom Sq_Circ ) = the carrier of ( TOP-REAL 2) by FUNCT_2:def 1;

 not (px `2 ) = 0 by A60, A61, A62, A69, A73, XREAL_1: 64;

 then

 A78: ((px `2 ) ^2 ) > 0 by SQUARE_1: 12;



 A79: ((px `1 ) ^2 ) >= 0 by XREAL_1: 63;

 ((((px `2 ) ^2 ) / (( sqrt (1 + (((px `1 ) / (px `2 )) ^2 ))) ^2 )) + (((px `1 ) / ( sqrt (1 + (((px `1 ) / (px `2 )) ^2 )))) ^2 )) >= 1 by A64, A65, A66, A67, A68, XCMPLX_1: 76;

 then ((((px `2 ) ^2 ) / (( sqrt (1 + (((px `1 ) / (px `2 )) ^2 ))) ^2 )) + (((px `1 ) ^2 ) / (( sqrt (1 + (((px `1 ) / (px `2 )) ^2 ))) ^2 ))) >= 1 by XCMPLX_1: 76;

 then ((((px `2 ) ^2 ) / (1 + (((px `1 ) / (px `2 )) ^2 ))) + (((px `1 ) ^2 ) / (( sqrt (1 + (((px `1 ) / (px `2 )) ^2 ))) ^2 ))) >= 1 by A63, SQUARE_1:def 2;

 then ((((px `2 ) ^2 ) / (1 + (((px `1 ) / (px `2 )) ^2 ))) + (((px `1 ) ^2 ) / (1 + (((px `1 ) / (px `2 )) ^2 )))) >= 1 by A63, SQUARE_1:def 2;

 then (((((px `2 ) ^2 ) / (1 + (((px `1 ) / (px `2 )) ^2 ))) + (((px `1 ) ^2 ) / (1 + (((px `1 ) / (px `2 )) ^2 )))) * (1 + (((px `1 ) / (px `2 )) ^2 ))) >= (1 * (1 + (((px `1 ) / (px `2 )) ^2 ))) by A63, XREAL_1: 64;

 then (((((px `2 ) ^2 ) / (1 + (((px `1 ) / (px `2 )) ^2 ))) * (1 + (((px `1 ) / (px `2 )) ^2 ))) + ((((px `1 ) ^2 ) / (1 + (((px `1 ) / (px `2 )) ^2 ))) * (1 + (((px `1 ) / (px `2 )) ^2 )))) >= (1 * (1 + (((px `1 ) / (px `2 )) ^2 )));

 then (((px `2 ) ^2 ) + ((((px `1 ) ^2 ) / (1 + (((px `1 ) / (px `2 )) ^2 ))) * (1 + (((px `1 ) / (px `2 )) ^2 )))) >= (1 * (1 + (((px `1 ) / (px `2 )) ^2 ))) by A63, XCMPLX_1: 87;

 then

 A80: (((px `2 ) ^2 ) + ((px `1 ) ^2 )) >= (1 * (1 + (((px `1 ) / (px `2 )) ^2 ))) by A63, XCMPLX_1: 87;

 (1 * (1 + (((px `1 ) / (px `2 )) ^2 ))) = (1 + (((px `1 ) ^2 ) / ((px `2 ) ^2 ))) by XCMPLX_1: 76;

 then ((((px `2 ) ^2 ) + ((px `1 ) ^2 )) - 1) >= ((1 + (((px `1 ) ^2 ) / ((px `2 ) ^2 ))) - 1) by A80, XREAL_1: 9;

 then (((((px `2 ) ^2 ) + ((px `1 ) ^2 )) - 1) * ((px `2 ) ^2 )) >= ((((px `1 ) ^2 ) / ((px `2 ) ^2 )) * ((px `2 ) ^2 )) by A78, XREAL_1: 64;

 then ((((px `2 ) ^2 ) + (((px `1 ) ^2 ) - 1)) * ((px `2 ) ^2 )) >= ((px `1 ) ^2 ) by A78, XCMPLX_1: 87;

 then

 A81: (((((px `2 ) ^2 ) * ((px `2 ) ^2 )) + (((px `2 ) ^2 ) * (((px `1 ) ^2 ) - 1))) - ((px `1 ) ^2 )) >= 0 by XREAL_1: 48;

 (((((px `2 ) ^2 ) * ((px `2 ) ^2 )) + ((((px `2 ) ^2 ) * ((px `1 ) ^2 )) - (((px `2 ) ^2 ) * 1))) - ((px `1 ) ^2 )) = ((((px `2 ) ^2 ) - 1) * (((px `2 ) ^2 ) + ((px `1 ) ^2 )));

 then (((px `2 ) ^2 ) - 1) >= 0 & (((px `1 ) ^2 ) + ((px `2 ) ^2 )) >= 0 or (((px `2 ) ^2 ) - 1) <= 0 & (((px `1 ) ^2 ) + ((px `2 ) ^2 )) <= 0 by A81, XREAL_1: 132;

 then ((((px `2 ) ^2 ) - 1) + 1) >= ( 0 + 1) by A78, A79, XREAL_1: 7;

 then (px `2 ) >= (1 ^2 ) or (px `2 ) <= ( - 1) by SQUARE_1: 50;

 then px in Kb by A1;

 hence ex x be set st x in ( dom Sq_Circ ) & x in Kb & y = ( Sq_Circ . x) by A31, A74, A75, A76, A77, EUCLID: 53;

 end;

 end;

 hence thesis by FUNCT_1:def 6;

 end;

 theorem :: JGRAPH_6:29

 for P0,P1,P01,P11,K0,K1,K01,K11 be Subset of ( TOP-REAL 2), P be non empty compact Subset of ( TOP-REAL 2), f be Function of ( TOP-REAL 2), ( TOP-REAL 2) st P = ( circle ( 0 , 0 ,1)) & P0 = ( inside_of_circle ( 0 , 0 ,1)) & P1 = ( outside_of_circle ( 0 , 0 ,1)) & P01 = ( closed_inside_of_circle ( 0 , 0 ,1)) & P11 = ( closed_outside_of_circle ( 0 , 0 ,1)) & K0 = ( inside_of_rectangle (( - 1),1,( - 1),1)) & K1 = ( outside_of_rectangle (( - 1),1,( - 1),1)) & K01 = ( closed_inside_of_rectangle (( - 1),1,( - 1),1)) & K11 = ( closed_outside_of_rectangle (( - 1),1,( - 1),1)) & f = Sq_Circ holds (f .: ( rectangle (( - 1),1,( - 1),1))) = P & ((f " ) .: P) = ( rectangle (( - 1),1,( - 1),1)) & (f .: K0) = P0 & ((f " ) .: P0) = K0 & (f .: K1) = P1 & ((f " ) .: P1) = K1 & (f .: K01) = P01 & (f .: K11) = P11 & ((f " ) .: P01) = K01 & ((f " ) .: P11) = K11

 proof

 let P0,P1,P01,P11,K0,K1,K01,K11 be Subset of ( TOP-REAL 2), P be non empty compact Subset of ( TOP-REAL 2), f be Function of ( TOP-REAL 2), ( TOP-REAL 2);

 assume that

 A1: P = ( circle ( 0 , 0 ,1)) and

 A2: P0 = ( inside_of_circle ( 0 , 0 ,1)) and

 A3: P1 = ( outside_of_circle ( 0 , 0 ,1)) and

 A4: P01 = ( closed_inside_of_circle ( 0 , 0 ,1)) and

 A5: P11 = ( closed_outside_of_circle ( 0 , 0 ,1)) and

 A6: K0 = ( inside_of_rectangle (( - 1),1,( - 1),1)) and

 A7: K1 = ( outside_of_rectangle (( - 1),1,( - 1),1)) and

 A8: K01 = ( closed_inside_of_rectangle (( - 1),1,( - 1),1)) and

 A9: K11 = ( closed_outside_of_rectangle (( - 1),1,( - 1),1)) and

 A10: f = Sq_Circ ;

 set K = ( rectangle (( - 1),1,( - 1),1));



 A11: P0 = { p : |.p.| < 1 } by A2, Th24;



 A12: P01 = { p : |.p.| <= 1 } by A4, Th24;



 A13: P1 = { p : |.p.| > 1 } by A3, Th24;



 A14: P11 = { p : |.p.| >= 1 } by A5, Th24;

 defpred P[ Point of ( TOP-REAL 2)] means ($1 `1 ) = ( - 1) & ($1 `2 ) <= 1 & ($1 `2 ) >= ( - 1) or ($1 `1 ) <= 1 & ($1 `1 ) >= ( - 1) & ($1 `2 ) = 1 or ($1 `1 ) <= 1 & ($1 `1 ) >= ( - 1) & ($1 `2 ) = ( - 1) or ($1 `1 ) = 1 & ($1 `2 ) <= 1 & ($1 `2 ) >= ( - 1);

 defpred Q[ Point of ( TOP-REAL 2)] means ( - 1) = ($1 `1 ) & ( - 1) <= ($1 `2 ) & ($1 `2 ) <= 1 or ($1 `1 ) = 1 & ( - 1) <= ($1 `2 ) & ($1 `2 ) <= 1 or ( - 1) = ($1 `2 ) & ( - 1) <= ($1 `1 ) & ($1 `1 ) <= 1 or 1 = ($1 `2 ) & ( - 1) <= ($1 `1 ) & ($1 `1 ) <= 1;

 deffunc F( set) = $1;



 A15: for p be Element of ( TOP-REAL 2) holds P[p] iff Q[p];



 A16: K = { F(p) : P[p] } by SPPOL_2: 54

 .= { F(q) : Q[q] } from FRAENKEL:sch 3( A15);

 defpred Q[ Point of ( TOP-REAL 2)] means |.$1.| = 1;

 defpred P[ Point of ( TOP-REAL 2)] means |.($1 - |[ 0 , 0 ]|).| = 1;



 A17: for p holds P[p] iff Q[p] by EUCLID: 54, RLVECT_1: 13;

 P = { F(p) : P[p] } by A1

 .= { F(p2) where p2 be Point of ( TOP-REAL 2) : Q[p2] } from FRAENKEL:sch 3( A17);

 then

 A18: (f .: K) = P by A10, A16, JGRAPH_3: 23;



 A19: ( dom f) = the carrier of ( TOP-REAL 2) by FUNCT_2:def 1;



 A20: (f .: K0) = P0 by A6, A10, A11, Th25;

 (f .: K1) = P1 by A7, A10, A13, Th26;

 hence (f .: K) = P & ((f " ) .: P) = K & (f .: K0) = P0 & ((f " ) .: P0) = K0 & (f .: K1) = P1 & ((f " ) .: P1) = K1 by A10, A18, A19, A20, FUNCT_1: 107;



 A21: (f .: K01) = P01 by A8, A10, A12, Th27;

 (f .: K11) = P11 by A9, A10, A14, Th28;

 hence thesis by A10, A19, A21, FUNCT_1: 107;

 end;

 begin

 theorem :: JGRAPH_6:30



 Th30: for a,b,c,d be Real st a <= b & c <= d holds ( LSeg ( |[a, c]|, |[a, d]|)) = { p1 : (p1 `1 ) = a & (p1 `2 ) <= d & (p1 `2 ) >= c } & ( LSeg ( |[a, d]|, |[b, d]|)) = { p2 : (p2 `1 ) <= b & (p2 `1 ) >= a & (p2 `2 ) = d } & ( LSeg ( |[a, c]|, |[b, c]|)) = { q1 : (q1 `1 ) <= b & (q1 `1 ) >= a & (q1 `2 ) = c } & ( LSeg ( |[b, c]|, |[b, d]|)) = { q2 : (q2 `1 ) = b & (q2 `2 ) <= d & (q2 `2 ) >= c }

 proof

 let a,b,c,d be Real;

 assume that

 A1: a <= b and

 A2: c <= d;

 set L1 = { p : (p `1 ) = a & (p `2 ) <= d & (p `2 ) >= c }, L2 = { p : (p `1 ) <= b & (p `1 ) >= a & (p `2 ) = d }, L3 = { p : (p `1 ) <= b & (p `1 ) >= a & (p `2 ) = c }, L4 = { p : (p `1 ) = b & (p `2 ) <= d & (p `2 ) >= c };

 set p0 = |[a, c]|, p01 = |[a, d]|, p10 = |[b, c]|, p1 = |[b, d]|;



 A3: (p01 `1 ) = a by EUCLID: 52;



 A4: (p01 `2 ) = d by EUCLID: 52;



 A5: (p10 `1 ) = b by EUCLID: 52;



 A6: (p10 `2 ) = c by EUCLID: 52;



 A7: L1 c= ( LSeg (p0,p01))

 proof

 let a2 be object;

 assume a2 in L1;

 then

 consider p such that

 A8: a2 = p and

 A9: (p `1 ) = a and

 A10: (p `2 ) <= d and

 A11: (p `2 ) >= c;

 now

 per cases ;

 case

 A12: d <> c;

 reconsider lambda = (((p `2 ) - c) / (d - c)) as Real;

 d >= c by A10, A11, XXREAL_0: 2;

 then d > c by A12, XXREAL_0: 1;

 then

 A13: (d - c) > 0 by XREAL_1: 50;



 A14: ((p `2 ) - c) >= 0 by A11, XREAL_1: 48;

 (d - c) >= ((p `2 ) - c) by A10, XREAL_1: 9;

 then ((d - c) / (d - c)) >= (((p `2 ) - c) / (d - c)) by A13, XREAL_1: 72;

 then

 A15: 1 >= lambda by A13, XCMPLX_1: 60;



 A16: (((1 - lambda) * c) + (lambda * d)) = (((((d - c) / (d - c)) - (((p `2 ) - c) / (d - c))) * c) + ((((p `2 ) - c) / (d - c)) * d)) by A13, XCMPLX_1: 60

 .= (((((d - c) - ((p `2 ) - c)) / (d - c)) * c) + ((((p `2 ) - c) / (d - c)) * d)) by XCMPLX_1: 120

 .= ((c * ((d - (p `2 )) / (d - c))) + ((d * ((p `2 ) - c)) / (d - c))) by XCMPLX_1: 74

 .= (((c * (d - (p `2 ))) / (d - c)) + ((d * ((p `2 ) - c)) / (d - c))) by XCMPLX_1: 74

 .= ((((c * d) - (c * (p `2 ))) + ((d * (p `2 )) - (d * c))) / (d - c)) by XCMPLX_1: 62

 .= (((d - c) * (p `2 )) / (d - c))

 .= ((p `2 ) * ((d - c) / (d - c))) by XCMPLX_1: 74

 .= ((p `2 ) * 1) by A13, XCMPLX_1: 60

 .= (p `2 );

 (((1 - lambda) * p0) + (lambda * p01)) = ( |[((1 - lambda) * a), ((1 - lambda) * c)]| + (lambda * |[a, d]|)) by EUCLID: 58

 .= ( |[((1 - lambda) * a), ((1 - lambda) * c)]| + |[(lambda * a), (lambda * d)]|) by EUCLID: 58

 .= |[(((1 - lambda) * a) + (lambda * a)), (((1 - lambda) * c) + (lambda * d))]| by EUCLID: 56

 .= p by A9, A16, EUCLID: 53;

 hence thesis by A8, A13, A14, A15;

 end;

 case d = c;

 then

 A17: (p `2 ) = c by A10, A11, XXREAL_0: 1;

 reconsider lambda = 0 as Real;

 (((1 - lambda) * p0) + (lambda * p01)) = ( |[((1 - lambda) * a), ((1 - lambda) * c)]| + (lambda * |[a, d]|)) by EUCLID: 58

 .= ( |[((1 - lambda) * a), ((1 - lambda) * c)]| + |[(lambda * a), (lambda * d)]|) by EUCLID: 58

 .= |[(((1 - lambda) * a) + (lambda * a)), (((1 - lambda) * c) + (lambda * d))]| by EUCLID: 56

 .= p by A9, A17, EUCLID: 53;

 hence thesis by A8;

 end;

 end;

 hence thesis;

 end;

 ( LSeg (p0,p01)) c= L1

 proof

 let a2 be object;

 assume a2 in ( LSeg (p0,p01));

 then

 consider lambda such that

 A18: a2 = (((1 - lambda) * p0) + (lambda * p01)) and

 A19: 0 <= lambda and

 A20: lambda <= 1;

 set q = (((1 - lambda) * p0) + (lambda * p01));



 A21: (q `1 ) = ((((1 - lambda) * p0) `1 ) + ((lambda * p01) `1 )) by TOPREAL3: 2

 .= (((1 - lambda) * (p0 `1 )) + ((lambda * p01) `1 )) by TOPREAL3: 4

 .= (((1 - lambda) * (p0 `1 )) + (lambda * (p01 `1 ))) by TOPREAL3: 4

 .= (((1 - lambda) * a) + (lambda * a)) by A3, EUCLID: 52

 .= a;



 A22: (q `2 ) = ((((1 - lambda) * p0) `2 ) + ((lambda * p01) `2 )) by TOPREAL3: 2

 .= (((1 - lambda) * (p0 `2 )) + ((lambda * p01) `2 )) by TOPREAL3: 4

 .= (((1 - lambda) * (p0 `2 )) + (lambda * (p01 `2 ))) by TOPREAL3: 4

 .= (((1 - lambda) * c) + (lambda * d)) by A4, EUCLID: 52;

 then

 A23: (q `2 ) <= d by A2, A20, XREAL_1: 172;

 (q `2 ) >= c by A2, A19, A20, A22, XREAL_1: 173;

 hence thesis by A18, A21, A23;

 end;

 hence L1 = ( LSeg (p0,p01)) by A7;



 A24: L2 c= ( LSeg (p01,p1))

 proof

 let a2 be object;

 assume a2 in L2;

 then

 consider p such that

 A25: a2 = p and

 A26: (p `1 ) <= b and

 A27: (p `1 ) >= a and

 A28: (p `2 ) = d;

 now

 per cases ;

 case

 A29: b <> a;

 reconsider lambda = (((p `1 ) - a) / (b - a)) as Real;

 b >= a by A26, A27, XXREAL_0: 2;

 then b > a by A29, XXREAL_0: 1;

 then

 A30: (b - a) > 0 by XREAL_1: 50;



 A31: ((p `1 ) - a) >= 0 by A27, XREAL_1: 48;

 (b - a) >= ((p `1 ) - a) by A26, XREAL_1: 9;

 then ((b - a) / (b - a)) >= (((p `1 ) - a) / (b - a)) by A30, XREAL_1: 72;

 then

 A32: 1 >= lambda by A30, XCMPLX_1: 60;



 A33: (((1 - lambda) * a) + (lambda * b)) = (((((b - a) / (b - a)) - (((p `1 ) - a) / (b - a))) * a) + ((((p `1 ) - a) / (b - a)) * b)) by A30, XCMPLX_1: 60

 .= (((((b - a) - ((p `1 ) - a)) / (b - a)) * a) + ((((p `1 ) - a) / (b - a)) * b)) by XCMPLX_1: 120

 .= ((a * ((b - (p `1 )) / (b - a))) + ((b * ((p `1 ) - a)) / (b - a))) by XCMPLX_1: 74

 .= (((a * (b - (p `1 ))) / (b - a)) + ((b * ((p `1 ) - a)) / (b - a))) by XCMPLX_1: 74

 .= ((((a * b) - (a * (p `1 ))) + ((b * (p `1 )) - (b * a))) / (b - a)) by XCMPLX_1: 62

 .= (((b - a) * (p `1 )) / (b - a))

 .= ((p `1 ) * ((b - a) / (b - a))) by XCMPLX_1: 74

 .= ((p `1 ) * 1) by A30, XCMPLX_1: 60

 .= (p `1 );

 (((1 - lambda) * p01) + (lambda * p1)) = ( |[((1 - lambda) * a), ((1 - lambda) * d)]| + (lambda * |[b, d]|)) by EUCLID: 58

 .= ( |[((1 - lambda) * a), ((1 - lambda) * d)]| + |[(lambda * b), (lambda * d)]|) by EUCLID: 58

 .= |[(((1 - lambda) * a) + (lambda * b)), (((1 - lambda) * d) + (lambda * d))]| by EUCLID: 56

 .= p by A28, A33, EUCLID: 53;

 hence thesis by A25, A30, A31, A32;

 end;

 case b = a;

 then

 A34: (p `1 ) = a by A26, A27, XXREAL_0: 1;

 reconsider lambda = 0 as Real;

 (((1 - lambda) * p01) + (lambda * p1)) = ( |[((1 - lambda) * a), ((1 - lambda) * d)]| + (lambda * |[b, d]|)) by EUCLID: 58

 .= ( |[((1 - lambda) * a), ((1 - lambda) * d)]| + |[(lambda * b), (lambda * d)]|) by EUCLID: 58

 .= |[(((1 - lambda) * a) + (lambda * b)), (((1 - lambda) * d) + (lambda * d))]| by EUCLID: 56

 .= p by A28, A34, EUCLID: 53;

 hence thesis by A25;

 end;

 end;

 hence thesis;

 end;

 ( LSeg (p01,p1)) c= L2

 proof

 let a2 be object;

 assume a2 in ( LSeg (p01,p1));

 then

 consider lambda such that

 A35: a2 = (((1 - lambda) * p01) + (lambda * p1)) and

 A36: 0 <= lambda and

 A37: lambda <= 1;

 set q = (((1 - lambda) * p01) + (lambda * p1));



 A38: (q `2 ) = ((((1 - lambda) * p01) `2 ) + ((lambda * p1) `2 )) by TOPREAL3: 2

 .= (((1 - lambda) * (p01 `2 )) + ((lambda * p1) `2 )) by TOPREAL3: 4

 .= (((1 - lambda) * (p01 `2 )) + (lambda * (p1 `2 ))) by TOPREAL3: 4

 .= (((1 - lambda) * d) + (lambda * d)) by A4, EUCLID: 52

 .= d;



 A39: (q `1 ) = ((((1 - lambda) * p01) `1 ) + ((lambda * p1) `1 )) by TOPREAL3: 2

 .= (((1 - lambda) * (p01 `1 )) + ((lambda * p1) `1 )) by TOPREAL3: 4

 .= (((1 - lambda) * (p01 `1 )) + (lambda * (p1 `1 ))) by TOPREAL3: 4

 .= (((1 - lambda) * a) + (lambda * b)) by A3, EUCLID: 52;

 then

 A40: (q `1 ) <= b by A1, A37, XREAL_1: 172;

 (q `1 ) >= a by A1, A36, A37, A39, XREAL_1: 173;

 hence thesis by A35, A38, A40;

 end;

 hence L2 = ( LSeg (p01,p1)) by A24;



 A41: L3 c= ( LSeg (p0,p10))

 proof

 let a2 be object;

 assume a2 in L3;

 then

 consider p such that

 A42: a2 = p and

 A43: (p `1 ) <= b and

 A44: (p `1 ) >= a and

 A45: (p `2 ) = c;

 now

 per cases ;

 case

 A46: b <> a;

 reconsider lambda = (((p `1 ) - a) / (b - a)) as Real;

 b >= a by A43, A44, XXREAL_0: 2;

 then b > a by A46, XXREAL_0: 1;

 then

 A47: (b - a) > 0 by XREAL_1: 50;



 A48: ((p `1 ) - a) >= 0 by A44, XREAL_1: 48;

 (b - a) >= ((p `1 ) - a) by A43, XREAL_1: 9;

 then ((b - a) / (b - a)) >= (((p `1 ) - a) / (b - a)) by A47, XREAL_1: 72;

 then

 A49: 1 >= lambda by A47, XCMPLX_1: 60;



 A50: (((1 - lambda) * a) + (lambda * b)) = (((((b - a) / (b - a)) - (((p `1 ) - a) / (b - a))) * a) + ((((p `1 ) - a) / (b - a)) * b)) by A47, XCMPLX_1: 60

 .= (((((b - a) - ((p `1 ) - a)) / (b - a)) * a) + ((((p `1 ) - a) / (b - a)) * b)) by XCMPLX_1: 120

 .= ((a * ((b - (p `1 )) / (b - a))) + ((b * ((p `1 ) - a)) / (b - a))) by XCMPLX_1: 74

 .= (((a * (b - (p `1 ))) / (b - a)) + ((b * ((p `1 ) - a)) / (b - a))) by XCMPLX_1: 74

 .= ((((a * b) - (a * (p `1 ))) + ((b * (p `1 )) - (b * a))) / (b - a)) by XCMPLX_1: 62

 .= (((b - a) * (p `1 )) / (b - a))

 .= ((p `1 ) * ((b - a) / (b - a))) by XCMPLX_1: 74

 .= ((p `1 ) * 1) by A47, XCMPLX_1: 60

 .= (p `1 );

 (((1 - lambda) * p0) + (lambda * p10)) = ( |[((1 - lambda) * a), ((1 - lambda) * c)]| + (lambda * |[b, c]|)) by EUCLID: 58

 .= ( |[((1 - lambda) * a), ((1 - lambda) * c)]| + |[(lambda * b), (lambda * c)]|) by EUCLID: 58

 .= |[(((1 - lambda) * a) + (lambda * b)), (((1 - lambda) * c) + (lambda * c))]| by EUCLID: 56

 .= p by A45, A50, EUCLID: 53;

 hence thesis by A42, A47, A48, A49;

 end;

 case b = a;

 then

 A51: (p `1 ) = a by A43, A44, XXREAL_0: 1;

 reconsider lambda = 0 as Real;

 (((1 - lambda) * p0) + (lambda * p10)) = ( |[((1 - lambda) * a), ((1 - lambda) * c)]| + (lambda * |[b, c]|)) by EUCLID: 58

 .= ( |[((1 - lambda) * a), ((1 - lambda) * c)]| + |[(lambda * b), (lambda * c)]|) by EUCLID: 58

 .= |[(((1 - lambda) * a) + (lambda * b)), (((1 - lambda) * c) + (lambda * c))]| by EUCLID: 56

 .= p by A45, A51, EUCLID: 53;

 hence thesis by A42;

 end;

 end;

 hence thesis;

 end;

 ( LSeg (p0,p10)) c= L3

 proof

 let a2 be object;

 assume a2 in ( LSeg (p0,p10));

 then

 consider lambda such that

 A52: a2 = (((1 - lambda) * p0) + (lambda * p10)) and

 A53: 0 <= lambda and

 A54: lambda <= 1;

 set q = (((1 - lambda) * p0) + (lambda * p10));



 A55: (q `2 ) = ((((1 - lambda) * p0) `2 ) + ((lambda * p10) `2 )) by TOPREAL3: 2

 .= (((1 - lambda) * (p0 `2 )) + ((lambda * p10) `2 )) by TOPREAL3: 4

 .= (((1 - lambda) * (p0 `2 )) + (lambda * (p10 `2 ))) by TOPREAL3: 4

 .= (((1 - lambda) * c) + (lambda * c)) by A6, EUCLID: 52

 .= c;



 A56: (q `1 ) = ((((1 - lambda) * p0) `1 ) + ((lambda * p10) `1 )) by TOPREAL3: 2

 .= (((1 - lambda) * (p0 `1 )) + ((lambda * p10) `1 )) by TOPREAL3: 4

 .= (((1 - lambda) * (p0 `1 )) + (lambda * (p10 `1 ))) by TOPREAL3: 4

 .= (((1 - lambda) * a) + (lambda * b)) by A5, EUCLID: 52;

 then

 A57: (q `1 ) <= b by A1, A54, XREAL_1: 172;

 (q `1 ) >= a by A1, A53, A54, A56, XREAL_1: 173;

 hence thesis by A52, A55, A57;

 end;

 hence L3 = ( LSeg (p0,p10)) by A41;



 A58: L4 c= ( LSeg (p10,p1))

 proof

 let a2 be object;

 assume a2 in L4;

 then

 consider p such that

 A59: a2 = p and

 A60: (p `1 ) = b and

 A61: (p `2 ) <= d and

 A62: (p `2 ) >= c;

 now

 per cases ;

 case

 A63: d <> c;

 reconsider lambda = (((p `2 ) - c) / (d - c)) as Real;

 d >= c by A61, A62, XXREAL_0: 2;

 then d > c by A63, XXREAL_0: 1;

 then

 A64: (d - c) > 0 by XREAL_1: 50;



 A65: ((p `2 ) - c) >= 0 by A62, XREAL_1: 48;

 (d - c) >= ((p `2 ) - c) by A61, XREAL_1: 9;

 then ((d - c) / (d - c)) >= (((p `2 ) - c) / (d - c)) by A64, XREAL_1: 72;

 then

 A66: 1 >= lambda by A64, XCMPLX_1: 60;



 A67: (((1 - lambda) * c) + (lambda * d)) = (((((d - c) / (d - c)) - (((p `2 ) - c) / (d - c))) * c) + ((((p `2 ) - c) / (d - c)) * d)) by A64, XCMPLX_1: 60

 .= (((((d - c) - ((p `2 ) - c)) / (d - c)) * c) + ((((p `2 ) - c) / (d - c)) * d)) by XCMPLX_1: 120

 .= ((c * ((d - (p `2 )) / (d - c))) + ((d * ((p `2 ) - c)) / (d - c))) by XCMPLX_1: 74

 .= (((c * (d - (p `2 ))) / (d - c)) + ((d * ((p `2 ) - c)) / (d - c))) by XCMPLX_1: 74

 .= ((((c * d) - (c * (p `2 ))) + ((d * (p `2 )) - (d * c))) / (d - c)) by XCMPLX_1: 62

 .= (((d - c) * (p `2 )) / (d - c))

 .= ((p `2 ) * ((d - c) / (d - c))) by XCMPLX_1: 74

 .= ((p `2 ) * 1) by A64, XCMPLX_1: 60

 .= (p `2 );

 (((1 - lambda) * p10) + (lambda * p1)) = ( |[((1 - lambda) * b), ((1 - lambda) * c)]| + (lambda * |[b, d]|)) by EUCLID: 58

 .= ( |[((1 - lambda) * b), ((1 - lambda) * c)]| + |[(lambda * b), (lambda * d)]|) by EUCLID: 58

 .= |[(((1 - lambda) * b) + (lambda * b)), (((1 - lambda) * c) + (lambda * d))]| by EUCLID: 56

 .= p by A60, A67, EUCLID: 53;

 hence thesis by A59, A64, A65, A66;

 end;

 case d = c;

 then

 A68: (p `2 ) = c by A61, A62, XXREAL_0: 1;

 reconsider lambda = 0 as Real;

 (((1 - lambda) * p10) + (lambda * p1)) = ( |[((1 - lambda) * b), ((1 - lambda) * c)]| + (lambda * |[b, d]|)) by EUCLID: 58

 .= ( |[((1 - lambda) * b), ((1 - lambda) * c)]| + |[(lambda * b), (lambda * d)]|) by EUCLID: 58

 .= |[(((1 - lambda) * b) + (lambda * b)), (((1 - lambda) * c) + (lambda * d))]| by EUCLID: 56

 .= p by A60, A68, EUCLID: 53;

 hence thesis by A59;

 end;

 end;

 hence thesis;

 end;

 ( LSeg (p10,p1)) c= L4

 proof

 let a2 be object;

 assume a2 in ( LSeg (p10,p1));

 then

 consider lambda such that

 A69: a2 = (((1 - lambda) * p10) + (lambda * p1)) and

 A70: 0 <= lambda and

 A71: lambda <= 1;

 set q = (((1 - lambda) * p10) + (lambda * p1));



 A72: (q `1 ) = ((((1 - lambda) * p10) `1 ) + ((lambda * p1) `1 )) by TOPREAL3: 2

 .= (((1 - lambda) * (p10 `1 )) + ((lambda * p1) `1 )) by TOPREAL3: 4

 .= (((1 - lambda) * (p10 `1 )) + (lambda * (p1 `1 ))) by TOPREAL3: 4

 .= (((1 - lambda) * b) + (lambda * b)) by A5, EUCLID: 52

 .= b;



 A73: (q `2 ) = ((((1 - lambda) * p10) `2 ) + ((lambda * p1) `2 )) by TOPREAL3: 2

 .= (((1 - lambda) * (p10 `2 )) + ((lambda * p1) `2 )) by TOPREAL3: 4

 .= (((1 - lambda) * (p10 `2 )) + (lambda * (p1 `2 ))) by TOPREAL3: 4

 .= (((1 - lambda) * c) + (lambda * d)) by A6, EUCLID: 52;

 then

 A74: (q `2 ) <= d by A2, A71, XREAL_1: 172;

 (q `2 ) >= c by A2, A70, A71, A73, XREAL_1: 173;

 hence thesis by A69, A72, A74;

 end;

 hence L4 = ( LSeg (p10,p1)) by A58;

 end;

 theorem :: JGRAPH_6:31



 Th31: for a,b,c,d be Real st a <= b & c <= d holds (( LSeg ( |[a, c]|, |[a, d]|)) /\ ( LSeg ( |[a, c]|, |[b, c]|))) = { |[a, c]|}

 proof

 let a,b,c,d be Real;

 assume that

 A1: a <= b and

 A2: c <= d;

 for ax be object holds ax in (( LSeg ( |[a, c]|, |[a, d]|)) /\ ( LSeg ( |[a, c]|, |[b, c]|))) iff ax = |[a, c]|

 proof

 let ax be object;

 thus ax in (( LSeg ( |[a, c]|, |[a, d]|)) /\ ( LSeg ( |[a, c]|, |[b, c]|))) implies ax = |[a, c]|

 proof

 assume

 A3: ax in (( LSeg ( |[a, c]|, |[a, d]|)) /\ ( LSeg ( |[a, c]|, |[b, c]|)));

 then

 A4: ax in ( LSeg ( |[a, c]|, |[a, d]|)) by XBOOLE_0:def 4;

 ax in ( LSeg ( |[a, c]|, |[b, c]|)) by A3, XBOOLE_0:def 4;

 then ax in { p2 : (p2 `1 ) <= b & (p2 `1 ) >= a & (p2 `2 ) = c } by A1, Th30;

 then

 A5: ex p2 st p2 = ax & (p2 `1 ) <= b & (p2 `1 ) >= a & (p2 `2 ) = c;

 ax in { p2 : (p2 `1 ) = a & (p2 `2 ) <= d & (p2 `2 ) >= c } by A2, A4, Th30;

 then ex p st p = ax & (p `1 ) = a & (p `2 ) <= d & (p `2 ) >= c;

 hence thesis by A5, EUCLID: 53;

 end;

 assume

 A6: ax = |[a, c]|;

 then

 A7: ax in ( LSeg ( |[a, c]|, |[a, d]|)) by RLTOPSP1: 68;

 ax in ( LSeg ( |[a, c]|, |[b, c]|)) by A6, RLTOPSP1: 68;

 hence thesis by A7, XBOOLE_0:def 4;

 end;

 hence thesis by TARSKI:def 1;

 end;

 theorem :: JGRAPH_6:32



 Th32: for a,b,c,d be Real st a <= b & c <= d holds (( LSeg ( |[a, c]|, |[b, c]|)) /\ ( LSeg ( |[b, c]|, |[b, d]|))) = { |[b, c]|}

 proof

 let a,b,c,d be Real;

 assume that

 A1: a <= b and

 A2: c <= d;

 for ax be object holds ax in (( LSeg ( |[a, c]|, |[b, c]|)) /\ ( LSeg ( |[b, c]|, |[b, d]|))) iff ax = |[b, c]|

 proof

 let ax be object;

 thus ax in (( LSeg ( |[a, c]|, |[b, c]|)) /\ ( LSeg ( |[b, c]|, |[b, d]|))) implies ax = |[b, c]|

 proof

 assume

 A3: ax in (( LSeg ( |[a, c]|, |[b, c]|)) /\ ( LSeg ( |[b, c]|, |[b, d]|)));

 then

 A4: ax in ( LSeg ( |[a, c]|, |[b, c]|)) by XBOOLE_0:def 4;



 A5: ax in ( LSeg ( |[b, c]|, |[b, d]|)) by A3, XBOOLE_0:def 4;

 ax in { q1 : (q1 `1 ) <= b & (q1 `1 ) >= a & (q1 `2 ) = c } by A1, A4, Th30;

 then

 A6: ex p2 st p2 = ax & (p2 `1 ) <= b & (p2 `1 ) >= a & (p2 `2 ) = c;

 ax in { q2 : (q2 `1 ) = b & (q2 `2 ) <= d & (q2 `2 ) >= c } by A2, A5, Th30;

 then ex p st p = ax & (p `1 ) = b & (p `2 ) <= d & (p `2 ) >= c;

 hence thesis by A6, EUCLID: 53;

 end;

 assume

 A7: ax = |[b, c]|;

 then

 A8: ax in ( LSeg ( |[a, c]|, |[b, c]|)) by RLTOPSP1: 68;

 ax in ( LSeg ( |[b, c]|, |[b, d]|)) by A7, RLTOPSP1: 68;

 hence thesis by A8, XBOOLE_0:def 4;

 end;

 hence thesis by TARSKI:def 1;

 end;

 theorem :: JGRAPH_6:33



 Th33: for a,b,c,d be Real st a <= b & c <= d holds (( LSeg ( |[a, d]|, |[b, d]|)) /\ ( LSeg ( |[b, c]|, |[b, d]|))) = { |[b, d]|}

 proof

 let a,b,c,d be Real;

 assume that

 A1: a <= b and

 A2: c <= d;

 for ax be object holds ax in (( LSeg ( |[a, d]|, |[b, d]|)) /\ ( LSeg ( |[b, c]|, |[b, d]|))) iff ax = |[b, d]|

 proof

 let ax be object;

 thus ax in (( LSeg ( |[a, d]|, |[b, d]|)) /\ ( LSeg ( |[b, c]|, |[b, d]|))) implies ax = |[b, d]|

 proof

 assume

 A3: ax in (( LSeg ( |[a, d]|, |[b, d]|)) /\ ( LSeg ( |[b, c]|, |[b, d]|)));

 then

 A4: ax in ( LSeg ( |[b, c]|, |[b, d]|)) by XBOOLE_0:def 4;

 ax in ( LSeg ( |[a, d]|, |[b, d]|)) by A3, XBOOLE_0:def 4;

 then ax in { p : (p `1 ) <= b & (p `1 ) >= a & (p `2 ) = d } by A1, Th30;

 then

 A5: ex p st p = ax & (p `1 ) <= b & (p `1 ) >= a & (p `2 ) = d;

 ax in { p : (p `1 ) = b & (p `2 ) <= d & (p `2 ) >= c } by A2, A4, Th30;

 then ex p2 st p2 = ax & (p2 `1 ) = b & (p2 `2 ) <= d & (p2 `2 ) >= c;

 hence thesis by A5, EUCLID: 53;

 end;

 assume

 A6: ax = |[b, d]|;

 then

 A7: ax in ( LSeg ( |[a, d]|, |[b, d]|)) by RLTOPSP1: 68;

 ax in ( LSeg ( |[b, c]|, |[b, d]|)) by A6, RLTOPSP1: 68;

 hence thesis by A7, XBOOLE_0:def 4;

 end;

 hence thesis by TARSKI:def 1;

 end;

 theorem :: JGRAPH_6:34



 Th34: for a,b,c,d be Real st a <= b & c <= d holds (( LSeg ( |[a, c]|, |[a, d]|)) /\ ( LSeg ( |[a, d]|, |[b, d]|))) = { |[a, d]|}

 proof

 let a,b,c,d be Real;

 assume that

 A1: a <= b and

 A2: c <= d;

 for ax be object holds ax in (( LSeg ( |[a, c]|, |[a, d]|)) /\ ( LSeg ( |[a, d]|, |[b, d]|))) iff ax = |[a, d]|

 proof

 let ax be object;

 thus ax in (( LSeg ( |[a, c]|, |[a, d]|)) /\ ( LSeg ( |[a, d]|, |[b, d]|))) implies ax = |[a, d]|

 proof

 assume

 A3: ax in (( LSeg ( |[a, c]|, |[a, d]|)) /\ ( LSeg ( |[a, d]|, |[b, d]|)));

 then

 A4: ax in ( LSeg ( |[a, c]|, |[a, d]|)) by XBOOLE_0:def 4;

 ax in ( LSeg ( |[a, d]|, |[b, d]|)) by A3, XBOOLE_0:def 4;

 then ax in { p2 : (p2 `1 ) <= b & (p2 `1 ) >= a & (p2 `2 ) = d } by A1, Th30;

 then

 A5: ex p2 st p2 = ax & (p2 `1 ) <= b & (p2 `1 ) >= a & (p2 `2 ) = d;

 ax in { p2 : (p2 `1 ) = a & (p2 `2 ) <= d & (p2 `2 ) >= c } by A2, A4, Th30;

 then ex p st p = ax & (p `1 ) = a & (p `2 ) <= d & (p `2 ) >= c;

 hence thesis by A5, EUCLID: 53;

 end;

 assume

 A6: ax = |[a, d]|;

 then

 A7: ax in ( LSeg ( |[a, c]|, |[a, d]|)) by RLTOPSP1: 68;

 ax in ( LSeg ( |[a, d]|, |[b, d]|)) by A6, RLTOPSP1: 68;

 hence thesis by A7, XBOOLE_0:def 4;

 end;

 hence thesis by TARSKI:def 1;

 end;

 theorem :: JGRAPH_6:35



 Th35: { q : ( - 1) = (q `1 ) & ( - 1) <= (q `2 ) & (q `2 ) <= 1 or (q `1 ) = 1 & ( - 1) <= (q `2 ) & (q `2 ) <= 1 or ( - 1) = (q `2 ) & ( - 1) <= (q `1 ) & (q `1 ) <= 1 or 1 = (q `2 ) & ( - 1) <= (q `1 ) & (q `1 ) <= 1 } = { p : (p `1 ) = ( - 1) & ( - 1) <= (p `2 ) & (p `2 ) <= 1 or (p `2 ) = 1 & ( - 1) <= (p `1 ) & (p `1 ) <= 1 or (p `1 ) = 1 & ( - 1) <= (p `2 ) & (p `2 ) <= 1 or (p `2 ) = ( - 1) & ( - 1) <= (p `1 ) & (p `1 ) <= 1 }

 proof

 thus { q : ( - 1) = (q `1 ) & ( - 1) <= (q `2 ) & (q `2 ) <= 1 or (q `1 ) = 1 & ( - 1) <= (q `2 ) & (q `2 ) <= 1 or ( - 1) = (q `2 ) & ( - 1) <= (q `1 ) & (q `1 ) <= 1 or 1 = (q `2 ) & ( - 1) <= (q `1 ) & (q `1 ) <= 1 } c= { p : (p `1 ) = ( - 1) & ( - 1) <= (p `2 ) & (p `2 ) <= 1 or (p `2 ) = 1 & ( - 1) <= (p `1 ) & (p `1 ) <= 1 or (p `1 ) = 1 & ( - 1) <= (p `2 ) & (p `2 ) <= 1 or (p `2 ) = ( - 1) & ( - 1) <= (p `1 ) & (p `1 ) <= 1 }

 proof

 let x be object;

 assume x in { q : ( - 1) = (q `1 ) & ( - 1) <= (q `2 ) & (q `2 ) <= 1 or (q `1 ) = 1 & ( - 1) <= (q `2 ) & (q `2 ) <= 1 or ( - 1) = (q `2 ) & ( - 1) <= (q `1 ) & (q `1 ) <= 1 or 1 = (q `2 ) & ( - 1) <= (q `1 ) & (q `1 ) <= 1 };

 then ex q st (x = q) & (( - 1) = (q `1 ) & ( - 1) <= (q `2 ) & (q `2 ) <= 1 or (q `1 ) = 1 & ( - 1) <= (q `2 ) & (q `2 ) <= 1 or ( - 1) = (q `2 ) & ( - 1) <= (q `1 ) & (q `1 ) <= 1 or 1 = (q `2 ) & ( - 1) <= (q `1 ) & (q `1 ) <= 1);

 hence thesis;

 end;

 thus { p : (p `1 ) = ( - 1) & ( - 1) <= (p `2 ) & (p `2 ) <= 1 or (p `2 ) = 1 & ( - 1) <= (p `1 ) & (p `1 ) <= 1 or (p `1 ) = 1 & ( - 1) <= (p `2 ) & (p `2 ) <= 1 or (p `2 ) = ( - 1) & ( - 1) <= (p `1 ) & (p `1 ) <= 1 } c= { q : ( - 1) = (q `1 ) & ( - 1) <= (q `2 ) & (q `2 ) <= 1 or (q `1 ) = 1 & ( - 1) <= (q `2 ) & (q `2 ) <= 1 or ( - 1) = (q `2 ) & ( - 1) <= (q `1 ) & (q `1 ) <= 1 or 1 = (q `2 ) & ( - 1) <= (q `1 ) & (q `1 ) <= 1 }

 proof

 let x be object;

 assume x in { p : (p `1 ) = ( - 1) & ( - 1) <= (p `2 ) & (p `2 ) <= 1 or (p `2 ) = 1 & ( - 1) <= (p `1 ) & (p `1 ) <= 1 or (p `1 ) = 1 & ( - 1) <= (p `2 ) & (p `2 ) <= 1 or (p `2 ) = ( - 1) & ( - 1) <= (p `1 ) & (p `1 ) <= 1 };

 then ex p st (p = x) & ((p `1 ) = ( - 1) & ( - 1) <= (p `2 ) & (p `2 ) <= 1 or (p `2 ) = 1 & ( - 1) <= (p `1 ) & (p `1 ) <= 1 or (p `1 ) = 1 & ( - 1) <= (p `2 ) & (p `2 ) <= 1 or (p `2 ) = ( - 1) & ( - 1) <= (p `1 ) & (p `1 ) <= 1);

 hence thesis;

 end;

 end;

 theorem :: JGRAPH_6:36



 Th36: for a,b,c,d be Real st a <= b & c <= d holds ( W-bound ( rectangle (a,b,c,d))) = a

 proof

 let a,b,c,d be Real;

 assume that

 A1: a <= b and

 A2: c <= d;

 set X = ( rectangle (a,b,c,d));

 reconsider Z = (( proj1 | X) .: the carrier of (( TOP-REAL 2) | X)) as Subset of REAL ;



 A3: X = the carrier of (( TOP-REAL 2) | X) by PRE_TOPC: 8;



 A4: for p be Real st p in Z holds p >= a

 proof

 let p be Real;

 assume p in Z;

 then

 consider p0 be object such that

 A5: p0 in the carrier of (( TOP-REAL 2) | X) and p0 in the carrier of (( TOP-REAL 2) | X) and

 A6: p = (( proj1 | X) . p0) by FUNCT_2: 64;

 reconsider p0 as Point of ( TOP-REAL 2) by A3, A5;

 X = { q : (q `1 ) = a & (q `2 ) <= d & (q `2 ) >= c or (q `1 ) <= b & (q `1 ) >= a & (q `2 ) = d or (q `1 ) <= b & (q `1 ) >= a & (q `2 ) = c or (q `1 ) = b & (q `2 ) <= d & (q `2 ) >= c } by A1, A2, SPPOL_2: 54;

 then ex q be Point of ( TOP-REAL 2) st p0 = q & ((q `1 ) = a & (q `2 ) <= d & (q `2 ) >= c or (q `1 ) <= b & (q `1 ) >= a & (q `2 ) = d or (q `1 ) <= b & (q `1 ) >= a & (q `2 ) = c or (q `1 ) = b & (q `2 ) <= d & (q `2 ) >= c) by A3, A5;

 hence thesis by A1, A3, A5, A6, PSCOMP_1: 22;

 end;



 A7: for q be Real st for p be Real st p in Z holds p >= q holds a >= q

 proof

 let q be Real such that

 A8: for p be Real st p in Z holds p >= q;

 |[a, c]| in ( LSeg ( |[a, c]|, |[b, c]|)) by RLTOPSP1: 68;

 then

 A9: |[a, c]| in (( LSeg ( |[a, c]|, |[b, c]|)) \/ ( LSeg ( |[b, c]|, |[b, d]|))) by XBOOLE_0:def 3;

 X = ((( LSeg ( |[a, c]|, |[a, d]|)) \/ ( LSeg ( |[a, d]|, |[b, d]|))) \/ (( LSeg ( |[a, c]|, |[b, c]|)) \/ ( LSeg ( |[b, c]|, |[b, d]|)))) by SPPOL_2:def 3;

 then

 A10: |[a, c]| in X by A9, XBOOLE_0:def 3;



 then (( proj1 | X) . |[a, c]|) = ( |[a, c]| `1 ) by PSCOMP_1: 22

 .= a by EUCLID: 52;

 hence thesis by A3, A8, A10, FUNCT_2: 35;

 end;



 thus ( W-bound X) = ( lower_bound ( proj1 | X)) by PSCOMP_1:def 7

 .= ( lower_bound Z) by PSCOMP_1:def 1

 .= a by A4, A7, SEQ_4: 44;

 end;

 theorem :: JGRAPH_6:37



 Th37: for a,b,c,d be Real st a <= b & c <= d holds ( N-bound ( rectangle (a,b,c,d))) = d

 proof

 let a,b,c,d be Real;

 assume that

 A1: a <= b and

 A2: c <= d;

 set X = ( rectangle (a,b,c,d));

 reconsider Z = (( proj2 | X) .: the carrier of (( TOP-REAL 2) | X)) as Subset of REAL ;



 A3: X = the carrier of (( TOP-REAL 2) | X) by PRE_TOPC: 8;



 A4: for p be Real st p in Z holds p <= d

 proof

 let p be Real;

 assume p in Z;

 then

 consider p0 be object such that

 A5: p0 in the carrier of (( TOP-REAL 2) | X) and p0 in the carrier of (( TOP-REAL 2) | X) and

 A6: p = (( proj2 | X) . p0) by FUNCT_2: 64;

 reconsider p0 as Point of ( TOP-REAL 2) by A3, A5;

 X = { q : (q `1 ) = a & (q `2 ) <= d & (q `2 ) >= c or (q `1 ) <= b & (q `1 ) >= a & (q `2 ) = d or (q `1 ) <= b & (q `1 ) >= a & (q `2 ) = c or (q `1 ) = b & (q `2 ) <= d & (q `2 ) >= c } by A1, A2, SPPOL_2: 54;

 then ex q be Point of ( TOP-REAL 2) st p0 = q & ((q `1 ) = a & (q `2 ) <= d & (q `2 ) >= c or (q `1 ) <= b & (q `1 ) >= a & (q `2 ) = d or (q `1 ) <= b & (q `1 ) >= a & (q `2 ) = c or (q `1 ) = b & (q `2 ) <= d & (q `2 ) >= c) by A3, A5;

 hence thesis by A2, A3, A5, A6, PSCOMP_1: 23;

 end;



 A7: for q be Real st for p be Real st p in Z holds p <= q holds d <= q

 proof

 let q be Real such that

 A8: for p be Real st p in Z holds p <= q;

 |[b, d]| in ( LSeg ( |[b, c]|, |[b, d]|)) by RLTOPSP1: 68;

 then

 A9: |[b, d]| in (( LSeg ( |[a, c]|, |[b, c]|)) \/ ( LSeg ( |[b, c]|, |[b, d]|))) by XBOOLE_0:def 3;

 X = ((( LSeg ( |[a, c]|, |[a, d]|)) \/ ( LSeg ( |[a, d]|, |[b, d]|))) \/ (( LSeg ( |[a, c]|, |[b, c]|)) \/ ( LSeg ( |[b, c]|, |[b, d]|)))) by SPPOL_2:def 3;

 then

 A10: |[b, d]| in X by A9, XBOOLE_0:def 3;



 then (( proj2 | X) . |[b, d]|) = ( |[b, d]| `2 ) by PSCOMP_1: 23

 .= d by EUCLID: 52;

 hence thesis by A3, A8, A10, FUNCT_2: 35;

 end;



 thus ( N-bound X) = ( upper_bound ( proj2 | X)) by PSCOMP_1:def 8

 .= ( upper_bound Z) by PSCOMP_1:def 2

 .= d by A4, A7, SEQ_4: 46;

 end;

 theorem :: JGRAPH_6:38



 Th38: for a,b,c,d be Real st a <= b & c <= d holds ( E-bound ( rectangle (a,b,c,d))) = b

 proof

 let a,b,c,d be Real;

 assume that

 A1: a <= b and

 A2: c <= d;

 set X = ( rectangle (a,b,c,d));

 reconsider Z = (( proj1 | X) .: the carrier of (( TOP-REAL 2) | X)) as Subset of REAL ;



 A3: X = the carrier of (( TOP-REAL 2) | X) by PRE_TOPC: 8;



 A4: for p be Real st p in Z holds p <= b

 proof

 let p be Real;

 assume p in Z;

 then

 consider p0 be object such that

 A5: p0 in the carrier of (( TOP-REAL 2) | X) and p0 in the carrier of (( TOP-REAL 2) | X) and

 A6: p = (( proj1 | X) . p0) by FUNCT_2: 64;

 reconsider p0 as Point of ( TOP-REAL 2) by A3, A5;

 X = { q : (q `1 ) = a & (q `2 ) <= d & (q `2 ) >= c or (q `1 ) <= b & (q `1 ) >= a & (q `2 ) = d or (q `1 ) <= b & (q `1 ) >= a & (q `2 ) = c or (q `1 ) = b & (q `2 ) <= d & (q `2 ) >= c } by A1, A2, SPPOL_2: 54;

 then ex q be Point of ( TOP-REAL 2) st p0 = q & ((q `1 ) = a & (q `2 ) <= d & (q `2 ) >= c or (q `1 ) <= b & (q `1 ) >= a & (q `2 ) = d or (q `1 ) <= b & (q `1 ) >= a & (q `2 ) = c or (q `1 ) = b & (q `2 ) <= d & (q `2 ) >= c) by A3, A5;

 hence thesis by A1, A3, A5, A6, PSCOMP_1: 22;

 end;



 A7: for q be Real st for p be Real st p in Z holds p <= q holds b <= q

 proof

 let q be Real such that

 A8: for p be Real st p in Z holds p <= q;

 |[b, d]| in ( LSeg ( |[b, c]|, |[b, d]|)) by RLTOPSP1: 68;

 then

 A9: |[b, d]| in (( LSeg ( |[a, c]|, |[b, c]|)) \/ ( LSeg ( |[b, c]|, |[b, d]|))) by XBOOLE_0:def 3;

 X = ((( LSeg ( |[a, c]|, |[a, d]|)) \/ ( LSeg ( |[a, d]|, |[b, d]|))) \/ (( LSeg ( |[a, c]|, |[b, c]|)) \/ ( LSeg ( |[b, c]|, |[b, d]|)))) by SPPOL_2:def 3;

 then

 A10: |[b, d]| in X by A9, XBOOLE_0:def 3;



 then (( proj1 | X) . |[b, d]|) = ( |[b, d]| `1 ) by PSCOMP_1: 22

 .= b by EUCLID: 52;

 hence thesis by A3, A8, A10, FUNCT_2: 35;

 end;



 thus ( E-bound X) = ( upper_bound ( proj1 | X)) by PSCOMP_1:def 9

 .= ( upper_bound Z) by PSCOMP_1:def 2

 .= b by A4, A7, SEQ_4: 46;

 end;

 theorem :: JGRAPH_6:39



 Th39: for a,b,c,d be Real st a <= b & c <= d holds ( S-bound ( rectangle (a,b,c,d))) = c

 proof

 let a,b,c,d be Real;

 assume that

 A1: a <= b and

 A2: c <= d;

 set X = ( rectangle (a,b,c,d));

 reconsider Z = (( proj2 | X) .: the carrier of (( TOP-REAL 2) | X)) as Subset of REAL ;



 A3: X = the carrier of (( TOP-REAL 2) | X) by PRE_TOPC: 8;



 A4: for p be Real st p in Z holds p >= c

 proof

 let p be Real;

 assume p in Z;

 then

 consider p0 be object such that

 A5: p0 in the carrier of (( TOP-REAL 2) | X) and p0 in the carrier of (( TOP-REAL 2) | X) and

 A6: p = (( proj2 | X) . p0) by FUNCT_2: 64;

 reconsider p0 as Point of ( TOP-REAL 2) by A3, A5;

 X = { q : (q `1 ) = a & (q `2 ) <= d & (q `2 ) >= c or (q `1 ) <= b & (q `1 ) >= a & (q `2 ) = d or (q `1 ) <= b & (q `1 ) >= a & (q `2 ) = c or (q `1 ) = b & (q `2 ) <= d & (q `2 ) >= c } by A1, A2, SPPOL_2: 54;

 then ex q be Point of ( TOP-REAL 2) st p0 = q & ((q `1 ) = a & (q `2 ) <= d & (q `2 ) >= c or (q `1 ) <= b & (q `1 ) >= a & (q `2 ) = d or (q `1 ) <= b & (q `1 ) >= a & (q `2 ) = c or (q `1 ) = b & (q `2 ) <= d & (q `2 ) >= c) by A3, A5;

 hence thesis by A2, A3, A5, A6, PSCOMP_1: 23;

 end;



 A7: for q be Real st for p be Real st p in Z holds p >= q holds c >= q

 proof

 let q be Real such that

 A8: for p be Real st p in Z holds p >= q;

 |[b, c]| in ( LSeg ( |[b, c]|, |[b, d]|)) by RLTOPSP1: 68;

 then

 A9: |[b, c]| in (( LSeg ( |[a, c]|, |[b, c]|)) \/ ( LSeg ( |[b, c]|, |[b, d]|))) by XBOOLE_0:def 3;

 X = ((( LSeg ( |[a, c]|, |[a, d]|)) \/ ( LSeg ( |[a, d]|, |[b, d]|))) \/ (( LSeg ( |[a, c]|, |[b, c]|)) \/ ( LSeg ( |[b, c]|, |[b, d]|)))) by SPPOL_2:def 3;

 then

 A10: |[b, c]| in X by A9, XBOOLE_0:def 3;



 then (( proj2 | X) . |[b, c]|) = ( |[b, c]| `2 ) by PSCOMP_1: 23

 .= c by EUCLID: 52;

 hence thesis by A3, A8, A10, FUNCT_2: 35;

 end;



 thus ( S-bound X) = ( lower_bound ( proj2 | X)) by PSCOMP_1:def 10

 .= ( lower_bound Z) by PSCOMP_1:def 1

 .= c by A4, A7, SEQ_4: 44;

 end;

 theorem :: JGRAPH_6:40



 Th40: for a,b,c,d be Real st a <= b & c <= d holds ( NW-corner ( rectangle (a,b,c,d))) = |[a, d]|

 proof

 let a,b,c,d be Real;

 assume that

 A1: a <= b and

 A2: c <= d;

 set K = ( rectangle (a,b,c,d));



 A3: ( NW-corner K) = |[( W-bound K), ( N-bound K)]| by PSCOMP_1:def 12;

 ( W-bound K) = a by A1, A2, Th36;

 hence thesis by A1, A2, A3, Th37;

 end;

 theorem :: JGRAPH_6:41



 Th41: for a,b,c,d be Real st a <= b & c <= d holds ( NE-corner ( rectangle (a,b,c,d))) = |[b, d]|

 proof

 let a,b,c,d be Real;

 set K = ( rectangle (a,b,c,d));

 assume that

 A1: a <= b and

 A2: c <= d;



 A3: ( NE-corner K) = |[( E-bound K), ( N-bound K)]| by PSCOMP_1:def 13;

 ( E-bound K) = b by A1, A2, Th38;

 hence thesis by A1, A2, A3, Th37;

 end;

 theorem :: JGRAPH_6:42

 for a,b,c,d be Real st a <= b & c <= d holds ( SW-corner ( rectangle (a,b,c,d))) = |[a, c]|

 proof

 let a,b,c,d be Real;

 set K = ( rectangle (a,b,c,d));

 assume that

 A1: a <= b and

 A2: c <= d;



 A3: ( SW-corner K) = |[( W-bound K), ( S-bound K)]| by PSCOMP_1:def 11;

 ( W-bound K) = a by A1, A2, Th36;

 hence thesis by A1, A2, A3, Th39;

 end;

 theorem :: JGRAPH_6:43

 for a,b,c,d be Real st a <= b & c <= d holds ( SE-corner ( rectangle (a,b,c,d))) = |[b, c]|

 proof

 let a,b,c,d be Real;

 set K = ( rectangle (a,b,c,d));

 assume that

 A1: a <= b and

 A2: c <= d;



 A3: ( SE-corner K) = |[( E-bound K), ( S-bound K)]| by PSCOMP_1:def 14;

 ( E-bound K) = b by A1, A2, Th38;

 hence thesis by A1, A2, A3, Th39;

 end;

 theorem :: JGRAPH_6:44



 Th44: for a,b,c,d be Real st a <= b & c <= d holds ( W-most ( rectangle (a,b,c,d))) = ( LSeg ( |[a, c]|, |[a, d]|))

 proof

 let a,b,c,d be Real;

 set K = ( rectangle (a,b,c,d));

 assume that

 A1: a <= b and

 A2: c <= d;

 K = ((( LSeg ( |[a, c]|, |[a, d]|)) \/ ( LSeg ( |[a, d]|, |[b, d]|))) \/ (( LSeg ( |[a, c]|, |[b, c]|)) \/ ( LSeg ( |[b, c]|, |[b, d]|)))) by SPPOL_2:def 3;

 then

 A3: (( LSeg ( |[a, c]|, |[a, d]|)) \/ ( LSeg ( |[a, d]|, |[b, d]|))) c= K by XBOOLE_1: 7;



 A4: ( LSeg ( |[a, c]|, |[a, d]|)) c= (( LSeg ( |[a, c]|, |[a, d]|)) \/ ( LSeg ( |[a, d]|, |[b, d]|))) by XBOOLE_1: 7;



 A5: ( SW-corner K) = |[( W-bound K), ( S-bound K)]| by PSCOMP_1:def 11;



 A6: ( NW-corner K) = |[a, d]| by A1, A2, Th40;



 A7: ( W-bound K) = a by A1, A2, Th36;



 A8: ( S-bound K) = c by A1, A2, Th39;



 thus ( W-most K) = (( LSeg (( SW-corner K),( NW-corner K))) /\ K) by PSCOMP_1:def 15

 .= ( LSeg ( |[a, c]|, |[a, d]|)) by A3, A4, A5, A6, A7, A8, XBOOLE_1: 1, XBOOLE_1: 28;

 end;

 theorem :: JGRAPH_6:45



 Th45: for a,b,c,d be Real st a <= b & c <= d holds ( E-most ( rectangle (a,b,c,d))) = ( LSeg ( |[b, c]|, |[b, d]|))

 proof

 let a,b,c,d be Real;

 set K = ( rectangle (a,b,c,d));

 assume that

 A1: a <= b and

 A2: c <= d;

 K = ((( LSeg ( |[a, c]|, |[a, d]|)) \/ ( LSeg ( |[a, d]|, |[b, d]|))) \/ (( LSeg ( |[a, c]|, |[b, c]|)) \/ ( LSeg ( |[b, c]|, |[b, d]|)))) by SPPOL_2:def 3;

 then

 A3: (( LSeg ( |[a, c]|, |[b, c]|)) \/ ( LSeg ( |[b, c]|, |[b, d]|))) c= K by XBOOLE_1: 7;



 A4: ( LSeg ( |[b, c]|, |[b, d]|)) c= (( LSeg ( |[a, c]|, |[b, c]|)) \/ ( LSeg ( |[b, c]|, |[b, d]|))) by XBOOLE_1: 7;



 A5: ( SE-corner K) = |[( E-bound K), ( S-bound K)]| by PSCOMP_1:def 14;



 A6: ( NE-corner K) = |[b, d]| by A1, A2, Th41;



 A7: ( E-bound K) = b by A1, A2, Th38;



 A8: ( S-bound K) = c by A1, A2, Th39;



 thus ( E-most K) = (( LSeg (( SE-corner K),( NE-corner K))) /\ K) by PSCOMP_1:def 17

 .= ( LSeg ( |[b, c]|, |[b, d]|)) by A3, A4, A5, A6, A7, A8, XBOOLE_1: 1, XBOOLE_1: 28;

 end;

 theorem :: JGRAPH_6:46



 Th46: for a,b,c,d be Real st a <= b & c <= d holds ( W-min ( rectangle (a,b,c,d))) = |[a, c]| & ( E-max ( rectangle (a,b,c,d))) = |[b, d]|

 proof

 let a,b,c,d be Real;

 set K = ( rectangle (a,b,c,d));

 assume that

 A1: a <= b and

 A2: c <= d;



 A3: ( lower_bound ( proj2 | ( LSeg ( |[a, c]|, |[a, d]|)))) = c

 proof

 set X = ( LSeg ( |[a, c]|, |[a, d]|));

 reconsider Z = (( proj2 | X) .: the carrier of (( TOP-REAL 2) | X)) as Subset of REAL ;



 A4: X = the carrier of (( TOP-REAL 2) | X) by PRE_TOPC: 8;



 A5: for p be Real st p in Z holds p >= c

 proof

 let p be Real;

 assume p in Z;

 then

 consider p0 be object such that

 A6: p0 in the carrier of (( TOP-REAL 2) | X) and p0 in the carrier of (( TOP-REAL 2) | X) and

 A7: p = (( proj2 | X) . p0) by FUNCT_2: 64;

 reconsider p0 as Point of ( TOP-REAL 2) by A4, A6;



 A8: ( |[a, c]| `2 ) = c by EUCLID: 52;

 ( |[a, d]| `2 ) = d by EUCLID: 52;

 then (p0 `2 ) >= c by A2, A4, A6, A8, TOPREAL1: 4;

 hence thesis by A4, A6, A7, PSCOMP_1: 23;

 end;



 A9: for q be Real st for p be Real st p in Z holds p >= q holds c >= q

 proof

 let q be Real such that

 A10: for p be Real st p in Z holds p >= q;



 A11: |[a, c]| in X by RLTOPSP1: 68;

 (( proj2 | X) . |[a, c]|) = ( |[a, c]| `2 ) by PSCOMP_1: 23, RLTOPSP1: 68

 .= c by EUCLID: 52;

 hence thesis by A4, A10, A11, FUNCT_2: 35;

 end;



 thus ( lower_bound ( proj2 | X)) = ( lower_bound Z) by PSCOMP_1:def 1

 .= c by A5, A9, SEQ_4: 44;

 end;



 A12: ( W-most K) = ( LSeg ( |[a, c]|, |[a, d]|)) by A1, A2, Th44;



 A13: ( W-bound K) = a by A1, A2, Th36;



 A14: ( upper_bound ( proj2 | ( LSeg ( |[b, c]|, |[b, d]|)))) = d

 proof

 set X = ( LSeg ( |[b, c]|, |[b, d]|));

 reconsider Z = (( proj2 | X) .: the carrier of (( TOP-REAL 2) | X)) as Subset of REAL ;



 A15: X = the carrier of (( TOP-REAL 2) | X) by PRE_TOPC: 8;



 A16: for p be Real st p in Z holds p <= d

 proof

 let p be Real;

 assume p in Z;

 then

 consider p0 be object such that

 A17: p0 in the carrier of (( TOP-REAL 2) | X) and p0 in the carrier of (( TOP-REAL 2) | X) and

 A18: p = (( proj2 | X) . p0) by FUNCT_2: 64;

 reconsider p0 as Point of ( TOP-REAL 2) by A15, A17;



 A19: ( |[b, c]| `2 ) = c by EUCLID: 52;

 ( |[b, d]| `2 ) = d by EUCLID: 52;

 then (p0 `2 ) <= d by A2, A15, A17, A19, TOPREAL1: 4;

 hence thesis by A15, A17, A18, PSCOMP_1: 23;

 end;



 A20: for q be Real st for p be Real st p in Z holds p <= q holds d <= q

 proof

 let q be Real such that

 A21: for p be Real st p in Z holds p <= q;



 A22: |[b, d]| in X by RLTOPSP1: 68;

 (( proj2 | X) . |[b, d]|) = ( |[b, d]| `2 ) by PSCOMP_1: 23, RLTOPSP1: 68

 .= d by EUCLID: 52;

 hence thesis by A15, A21, A22, FUNCT_2: 35;

 end;



 thus ( upper_bound ( proj2 | X)) = ( upper_bound Z) by PSCOMP_1:def 2

 .= d by A16, A20, SEQ_4: 46;

 end;



 A23: ( E-most K) = ( LSeg ( |[b, c]|, |[b, d]|)) by A1, A2, Th45;

 ( E-bound K) = b by A1, A2, Th38;

 hence thesis by A3, A12, A13, A14, A23, PSCOMP_1:def 19, PSCOMP_1:def 23;

 end;

 theorem :: JGRAPH_6:47



 Th47: for a,b,c,d be Real st a < b & c < d holds (( LSeg ( |[a, c]|, |[a, d]|)) \/ ( LSeg ( |[a, d]|, |[b, d]|))) is_an_arc_of (( W-min ( rectangle (a,b,c,d))),( E-max ( rectangle (a,b,c,d)))) & (( LSeg ( |[a, c]|, |[b, c]|)) \/ ( LSeg ( |[b, c]|, |[b, d]|))) is_an_arc_of (( E-max ( rectangle (a,b,c,d))),( W-min ( rectangle (a,b,c,d))))

 proof

 let a,b,c,d be Real;

 set K = ( rectangle (a,b,c,d));

 assume that

 A1: a < b and

 A2: c < d;



 A3: ( W-min K) = |[a, c]| by A1, A2, Th46;



 A4: ( E-max K) = |[b, d]| by A1, A2, Th46;

 ( |[a, c]| `2 ) = c by EUCLID: 52;

 then

 A5: |[a, c]| <> |[a, d]| by A2, EUCLID: 52;

 set p1 = |[a, c]|, p2 = |[a, d]|, q1 = |[b, d]|;



 A6: (( LSeg (p1,p2)) /\ ( LSeg (p2,q1))) = {p2} by A1, A2, Th34;

 ( |[a, c]| `1 ) = a by EUCLID: 52;

 then

 A7: |[a, c]| <> |[b, c]| by A1, EUCLID: 52;

 set q2 = |[b, c]|;

 (( LSeg (q1,q2)) /\ ( LSeg (q2,p1))) = {q2} by A1, A2, Th32;

 hence thesis by A3, A4, A5, A6, A7, TOPREAL1: 12;

 end;

 theorem :: JGRAPH_6:48



 Th48: for a,b,c,d be Real, f1,f2 be FinSequence of ( TOP-REAL 2), p0,p1,p01,p10 be Point of ( TOP-REAL 2) st a & f2 = <*p0, p10, p1*> holds f1 is being_S-Seq & ( L~ f1) = (( LSeg (p0,p01)) \/ ( LSeg (p01,p1))) & f2 is being_S-Seq & ( L~ f2) = (( LSeg (p0,p10)) \/ ( LSeg (p10,p1))) & ( rectangle (a,b,c,d)) = (( L~ f1) \/ ( L~ f2)) & (( L~ f1) /\ ( L~ f2)) = {p0, p1} & (f1 /. 1) = p0 & (f1 /. ( len f1)) = p1 & (f2 /. 1) = p0 & (f2 /. ( len f2)) = p1

 proof

 let a,b,c,d be Real, f1,f2 be FinSequence of ( TOP-REAL 2), p0,p1,p01,p10 be Point of ( TOP-REAL 2);

 assume that

 A1: a and

 A8: f2 = <*p0, p10, p1*>;

 set P = ( rectangle (a,b,c,d));

 set L1 = { p : (p `1 ) = a & (p `2 ) <= d & (p `2 ) >= c };

 set L2 = { p : (p `1 ) <= b & (p `1 ) >= a & (p `2 ) = d };

 set L3 = { p : (p `1 ) <= b & (p `1 ) >= a & (p `2 ) = c };

 set L4 = { p : (p `1 ) = b & (p `2 ) <= d & (p `2 ) >= c };



 A9: (p1 `1 ) = b by A4, EUCLID: 52;



 A10: (p1 `2 ) = d by A4, EUCLID: 52;



 A11: (p10 `1 ) = b by A6, EUCLID: 52;



 A12: (p10 `2 ) = c by A6, EUCLID: 52;



 A13: (p0 `1 ) = a by A3, EUCLID: 52;



 A14: (p0 `2 ) = c by A3, EUCLID: 52;



 A15: ( len f1) = (1 + 2) by A7, FINSEQ_1: 45;



 A16: (f1 /. 1) = p0 by A7, FINSEQ_4: 18;



 A17: (f1 /. 2) = p01 by A7, FINSEQ_4: 18;



 A18: (f1 /. 3) = p1 by A7, FINSEQ_4: 18;

 thus f1 is being_S-Seq

 proof



 A19: p0 <> p01 by A2, A5, A14, EUCLID: 52;

 p01 <> p1 by A1, A5, A9, EUCLID: 52;

 hence f1 is one-to-one by A1, A7, A9, A13, A19, FINSEQ_3: 95;

 thus ( len f1) >= 2 by A15;

 thus f1 is unfolded

 proof

 let i be Nat;

 assume that

 A20: 1 <= i and

 A21: (i + 2) <= ( len f1);

 i <= 1 by A15, A21, XREAL_1: 6;

 then

 A22: i = 1 by A20, XXREAL_0: 1;

 reconsider n2 = (1 + 1) as Nat;

 n2 in ( Seg ( len f1)) by A15, FINSEQ_1: 1;

 then

 A23: ( LSeg (f1,1)) = ( LSeg (p0,p01)) by A15, A16, A17, TOPREAL1:def 3;



 A24: ( LSeg (f1,n2)) = ( LSeg (p01,p1)) by A15, A17, A18, TOPREAL1:def 3;

 for x be object holds x in (( LSeg (p0,p01)) /\ ( LSeg (p01,p1))) iff x = p01

 proof

 let x be object;

 thus x in (( LSeg (p0,p01)) /\ ( LSeg (p01,p1))) implies x = p01

 proof

 assume

 A25: x in (( LSeg (p0,p01)) /\ ( LSeg (p01,p1)));

 then

 A26: x in ( LSeg (p0,p01)) by XBOOLE_0:def 4;



 A27: x in ( LSeg (p01,p1)) by A25, XBOOLE_0:def 4;



 A28: x in { p : (p `1 ) = a & (p `2 ) <= d & (p `2 ) >= c } by A2, A3, A5, A26, Th30;



 A29: x in { p2 : (p2 `1 ) <= b & (p2 `1 ) >= a & (p2 `2 ) = d } by A1, A4, A5, A27, Th30;



 A30: ex p st p = x & (p `1 ) = a & (p `2 ) <= d & (p `2 ) >= c by A28;

 ex p2 st p2 = x & (p2 `1 ) <= b & (p2 `1 ) >= a & (p2 `2 ) = d by A29;

 hence thesis by A5, A30, EUCLID: 53;

 end;

 assume

 A31: x = p01;

 then

 A32: x in ( LSeg (p0,p01)) by RLTOPSP1: 68;

 x in ( LSeg (p01,p1)) by A31, RLTOPSP1: 68;

 hence thesis by A32, XBOOLE_0:def 4;

 end;

 hence thesis by A17, A22, A23, A24, TARSKI:def 1;

 end;

 thus f1 is s.n.c.

 proof

 let i,j be Nat such that

 A33: (i + 1) < j;

 now

 per cases ;

 suppose 1 <= i;

 then

 A34: (1 + 1) <= (i + 1) by XREAL_1: 6;

 now

 per cases ;

 case 1 <= j & (j + 1) <= ( len f1);

 then j <= 2 by A15, XREAL_1: 6;

 hence contradiction by A33, A34, XXREAL_0: 2;

 end;

 case not (1 <= j & (j + 1) <= ( len f1));

 then ( LSeg (f1,j)) = {} by TOPREAL1:def 3;

 hence (( LSeg (f1,i)) /\ ( LSeg (f1,j))) = {} ;

 end;

 end;

 hence (( LSeg (f1,i)) /\ ( LSeg (f1,j))) = {} ;

 end;

 suppose not (1 <= i & (i + 1) <= ( len f1));

 then ( LSeg (f1,i)) = {} by TOPREAL1:def 3;

 hence (( LSeg (f1,i)) /\ ( LSeg (f1,j))) = {} ;

 end;

 end;

 hence (( LSeg (f1,i)) /\ ( LSeg (f1,j))) = {} ;

 end;

 let i be Nat;

 assume that

 A35: 1 <= i and

 A36: (i + 1) <= ( len f1);



 A37: i <= (1 + 1) by A15, A36, XREAL_1: 6;

 now

 per cases by A35, A37, NAT_1: 9;

 suppose

 A38: i = 1;



 then ((f1 /. i) `1 ) = (p0 `1 ) by A7, FINSEQ_4: 18

 .= a by A3, EUCLID: 52

 .= ((f1 /. (i + 1)) `1 ) by A5, A17, A38, EUCLID: 52;

 hence thesis;

 end;

 suppose

 A39: i = 2;



 then ((f1 /. i) `2 ) = (p01 `2 ) by A7, FINSEQ_4: 18

 .= d by A5, EUCLID: 52

 .= ((f1 /. (i + 1)) `2 ) by A4, A18, A39, EUCLID: 52;

 hence thesis;

 end;

 end;

 hence thesis;

 end;



 A40: (1 + 1) in ( Seg ( len f1)) by A15, FINSEQ_1: 1;



 A41: (1 + 1) <= ( len f1) by A15;

 ( LSeg (p0,p01)) = ( LSeg (f1,1)) by A15, A16, A17, A40, TOPREAL1:def 3;

 then

 A42: ( LSeg (p0,p01)) in { ( LSeg (f1,i)) : 1 <= i & (i + 1) <= ( len f1) } by A41;

 ( LSeg (p01,p1)) = ( LSeg (f1,2)) by A15, A17, A18, TOPREAL1:def 3;

 then ( LSeg (p01,p1)) in { ( LSeg (f1,i)) : 1 <= i & (i + 1) <= ( len f1) } by A15;

 then

 A43: {( LSeg (p0,p01)), ( LSeg (p01,p1))} c= { ( LSeg (f1,i)) : 1 <= i & (i + 1) <= ( len f1) } by A42, ZFMISC_1: 32;

 { ( LSeg (f1,i)) : 1 <= i & (i + 1) <= ( len f1) } c= {( LSeg (p0,p01)), ( LSeg (p01,p1))}

 proof

 let a be object;

 assume a in { ( LSeg (f1,i)) : 1 <= i & (i + 1) <= ( len f1) };

 then

 consider i such that

 A44: a = ( LSeg (f1,i)) and

 A45: 1 <= i and

 A46: (i + 1) <= ( len f1);

 (i + 1) <= (2 + 1) by A7, A46, FINSEQ_1: 45;

 then i <= (1 + 1) by XREAL_1: 6;

 then i = 1 or i = 2 by A45, NAT_1: 9;

 then a = ( LSeg (p0,p01)) or a = ( LSeg (p01,p1)) by A16, A17, A18, A44, A46, TOPREAL1:def 3;

 hence thesis by TARSKI:def 2;

 end;

 then ( L~ f1) = ( union {( LSeg (p0,p01)), ( LSeg (p01,p1))}) by A43, XBOOLE_0:def 10;

 hence

 A47: ( L~ f1) = (( LSeg (p0,p01)) \/ ( LSeg (p01,p1))) by ZFMISC_1: 75;



 then

 A48: ( L~ f1) = (L1 \/ ( LSeg (p01,p1))) by A2, A3, A5, Th30

 .= (L1 \/ L2) by A1, A4, A5, Th30;



 A49: ( len f2) = (1 + 2) by A8, FINSEQ_1: 45;



 A50: (f2 /. 1) = p0 by A8, FINSEQ_4: 18;



 A51: (f2 /. 2) = p10 by A8, FINSEQ_4: 18;



 A52: (f2 /. 3) = p1 by A8, FINSEQ_4: 18;

 thus f2 is being_S-Seq

 proof

 thus f2 is one-to-one by A1, A2, A8, A9, A10, A11, A12, A13, FINSEQ_3: 95;

 thus ( len f2) >= 2 by A49;

 thus f2 is unfolded

 proof

 let i be Nat;

 assume that

 A53: 1 <= i and

 A54: (i + 2) <= ( len f2);

 i <= 1 by A49, A54, XREAL_1: 6;

 then

 A55: i = 1 by A53, XXREAL_0: 1;

 (1 + 1) in ( Seg ( len f2)) by A49, FINSEQ_1: 1;

 then

 A56: ( LSeg (f2,1)) = ( LSeg (p0,p10)) by A49, A50, A51, TOPREAL1:def 3;



 A57: ( LSeg (f2,(1 + 1))) = ( LSeg (p10,p1)) by A49, A51, A52, TOPREAL1:def 3;

 for x be object holds x in (( LSeg (p0,p10)) /\ ( LSeg (p10,p1))) iff x = p10

 proof

 let x be object;

 thus x in (( LSeg (p0,p10)) /\ ( LSeg (p10,p1))) implies x = p10

 proof

 assume

 A58: x in (( LSeg (p0,p10)) /\ ( LSeg (p10,p1)));

 then

 A59: x in ( LSeg (p0,p10)) by XBOOLE_0:def 4;



 A60: x in ( LSeg (p10,p1)) by A58, XBOOLE_0:def 4;



 A61: x in { p : (p `1 ) <= b & (p `1 ) >= a & (p `2 ) = c } by A1, A3, A6, A59, Th30;



 A62: x in { p2 : (p2 `1 ) = b & (p2 `2 ) <= d & (p2 `2 ) >= c } by A2, A4, A6, A60, Th30;



 A63: ex p st p = x & (p `1 ) <= b & (p `1 ) >= a & (p `2 ) = c by A61;

 ex p2 st p2 = x & (p2 `1 ) = b & (p2 `2 ) <= d & (p2 `2 ) >= c by A62;

 hence thesis by A6, A63, EUCLID: 53;

 end;

 assume

 A64: x = p10;

 then

 A65: x in ( LSeg (p0,p10)) by RLTOPSP1: 68;

 x in ( LSeg (p10,p1)) by A64, RLTOPSP1: 68;

 hence thesis by A65, XBOOLE_0:def 4;

 end;

 hence thesis by A51, A55, A56, A57, TARSKI:def 1;

 end;

 thus f2 is s.n.c.

 proof

 let i,j be Nat such that

 A66: (i + 1) < j;

 now

 per cases ;

 suppose 1 <= i;

 then

 A67: (1 + 1) <= (i + 1) by XREAL_1: 6;

 now

 per cases ;

 case 1 <= j & (j + 1) <= ( len f2);

 then j <= 2 by A49, XREAL_1: 6;

 hence contradiction by A66, A67, XXREAL_0: 2;

 end;

 case not (1 <= j & (j + 1) <= ( len f2));

 then ( LSeg (f2,j)) = {} by TOPREAL1:def 3;

 hence (( LSeg (f2,i)) /\ ( LSeg (f2,j))) = {} ;

 end;

 end;

 hence (( LSeg (f2,i)) /\ ( LSeg (f2,j))) = {} ;

 end;

 suppose not (1 <= i & (i + 1) <= ( len f2));

 then ( LSeg (f2,i)) = {} by TOPREAL1:def 3;

 hence (( LSeg (f2,i)) /\ ( LSeg (f2,j))) = {} ;

 end;

 end;

 hence (( LSeg (f2,i)) /\ ( LSeg (f2,j))) = {} ;

 end;

 let i be Nat;

 assume that

 A68: 1 <= i and

 A69: (i + 1) <= ( len f2);



 A70: i <= (1 + 1) by A49, A69, XREAL_1: 6;

 per cases by A68, A70, NAT_1: 9;

 suppose

 A71: i = 1;



 then ((f2 /. i) `2 ) = (p0 `2 ) by A8, FINSEQ_4: 18

 .= c by A3, EUCLID: 52

 .= ((f2 /. (i + 1)) `2 ) by A6, A51, A71, EUCLID: 52;

 hence thesis;

 end;

 suppose

 A72: i = 2;



 then ((f2 /. i) `1 ) = (p10 `1 ) by A8, FINSEQ_4: 18

 .= b by A6, EUCLID: 52

 .= ((f2 /. (i + 1)) `1 ) by A4, A52, A72, EUCLID: 52;

 hence thesis;

 end;

 end;



 A73: (1 + 1) in ( Seg ( len f2)) by A49, FINSEQ_1: 1;



 A74: (1 + 1) <= ( len f2) by A49;

 ( LSeg (p0,p10)) = ( LSeg (f2,1)) by A49, A50, A51, A73, TOPREAL1:def 3;

 then

 A75: ( LSeg (p0,p10)) in { ( LSeg (f2,i)) : 1 <= i & (i + 1) <= ( len f2) } by A74;

 ( LSeg (p10,p1)) = ( LSeg (f2,2)) by A49, A51, A52, TOPREAL1:def 3;

 then ( LSeg (p10,p1)) in { ( LSeg (f2,i)) : 1 <= i & (i + 1) <= ( len f2) } by A49;

 then

 A76: {( LSeg (p0,p10)), ( LSeg (p10,p1))} c= { ( LSeg (f2,i)) : 1 <= i & (i + 1) <= ( len f2) } by A75, ZFMISC_1: 32;

 { ( LSeg (f2,i)) : 1 <= i & (i + 1) <= ( len f2) } c= {( LSeg (p0,p10)), ( LSeg (p10,p1))}

 proof

 let ax be object;

 assume ax in { ( LSeg (f2,i)) : 1 <= i & (i + 1) <= ( len f2) };

 then

 consider i such that

 A77: ax = ( LSeg (f2,i)) and

 A78: 1 <= i and

 A79: (i + 1) <= ( len f2);

 (i + 1) <= (2 + 1) by A8, A79, FINSEQ_1: 45;

 then i <= (1 + 1) by XREAL_1: 6;

 then i = 1 or i = 2 by A78, NAT_1: 9;

 then ax = ( LSeg (p0,p10)) or ax = ( LSeg (p10,p1)) by A50, A51, A52, A77, A79, TOPREAL1:def 3;

 hence thesis by TARSKI:def 2;

 end;

 then

 A80: ( L~ f2) = ( union {( LSeg (p0,p10)), ( LSeg (p10,p1))}) by A76, XBOOLE_0:def 10;

 hence ( L~ f2) = (( LSeg (p0,p10)) \/ ( LSeg (p10,p1))) by ZFMISC_1: 75;

 ( L~ f2) = (( LSeg (p0,p10)) \/ ( LSeg (p10,p1))) by A80, ZFMISC_1: 75;



 then

 A81: ( L~ f2) = (L3 \/ ( LSeg (p10,p1))) by A1, A3, A6, Th30

 .= (L3 \/ L4) by A2, A4, A6, Th30;

 P = ((( LSeg (p0,p01)) \/ ( LSeg (p01,p1))) \/ (( LSeg (p0,p10)) \/ ( LSeg (p10,p1)))) by A3, A4, A5, A6, SPPOL_2:def 3;

 hence P = (( L~ f1) \/ ( L~ f2)) by A47, A80, ZFMISC_1: 75;

 now

 assume L2 meets L3;

 then

 consider x be object such that

 A82: x in L2 and

 A83: x in L3 by XBOOLE_0: 3;



 A84: ex p st p = x & (p `1 ) <= b & (p `1 ) >= a & (p `2 ) = d by A82;

 ex p2 st p2 = x & (p2 `1 ) <= b & (p2 `1 ) >= a & (p2 `2 ) = c by A83;

 hence contradiction by A2, A84;

 end;

 then

 A85: (L2 /\ L3) = {} ;



 A86: ( LSeg ( |[a, c]|, |[a, d]|)) = { p3 : (p3 `1 ) = a & (p3 `2 ) <= d & (p3 `2 ) >= c } by A2, Th30;



 A87: ( LSeg ( |[a, d]|, |[b, d]|)) = { p2 : (p2 `1 ) <= b & (p2 `1 ) >= a & (p2 `2 ) = d } by A1, Th30;



 A88: ( LSeg ( |[a, c]|, |[b, c]|)) = { q1 : (q1 `1 ) <= b & (q1 `1 ) >= a & (q1 `2 ) = c } by A1, Th30;



 A89: ( LSeg ( |[b, c]|, |[b, d]|)) = { q2 : (q2 `1 ) = b & (q2 `2 ) <= d & (q2 `2 ) >= c } by A2, Th30;



 A90: (( LSeg ( |[a, c]|, |[a, d]|)) /\ ( LSeg ( |[a, c]|, |[b, c]|))) = { |[a, c]|} by A1, A2, Th31;



 A91: (( LSeg ( |[a, d]|, |[b, d]|)) /\ ( LSeg ( |[b, c]|, |[b, d]|))) = { |[b, d]|} by A1, A2, Th33;

 now

 assume L1 meets L4;

 then

 consider x be object such that

 A92: x in L1 and

 A93: x in L4 by XBOOLE_0: 3;



 A94: ex p st p = x & (p `1 ) = a & (p `2 ) <= d & (p `2 ) >= c by A92;

 ex p2 st p2 = x & (p2 `1 ) = b & (p2 `2 ) <= d & (p2 `2 ) >= c by A93;

 hence contradiction by A1, A94;

 end;

 then

 A95: (L1 /\ L4) = {} ;



 thus (( L~ f1) /\ ( L~ f2)) = (((L1 \/ L2) /\ L3) \/ ((L1 \/ L2) /\ L4)) by A48, A81, XBOOLE_1: 23

 .= (((L1 /\ L3) \/ (L2 /\ L3)) \/ ((L1 \/ L2) /\ L4)) by XBOOLE_1: 23

 .= ((L1 /\ L3) \/ ((L1 /\ L4) \/ (L2 /\ L4))) by A85, XBOOLE_1: 23

 .= {p0, p1} by A3, A4, A86, A87, A88, A89, A90, A91, A95, ENUMSET1: 1;

 thus thesis by A7, A8, A15, A49, FINSEQ_4: 18;

 end;

 theorem :: JGRAPH_6:49



 Th49: for P1,P2 be Subset of ( TOP-REAL 2), a,b,c,d be Real, f1,f2 be FinSequence of ( TOP-REAL 2), p1,p2 be Point of ( TOP-REAL 2) st a & f2 = <* |[a, c]|, |[b, c]|, |[b, d]|*> & P1 = ( L~ f1) & P2 = ( L~ f2) holds P1 is_an_arc_of (p1,p2) & P2 is_an_arc_of (p1,p2) & P1 is non empty & P2 is non empty & ( rectangle (a,b,c,d)) = (P1 \/ P2) & (P1 /\ P2) = {p1, p2}

 proof

 let P1,P2 be Subset of ( TOP-REAL 2), a,b,c,d be Real, f1,f2 be FinSequence of ( TOP-REAL 2), p1,p2 be Point of ( TOP-REAL 2);

 assume that

 A1: a and

 A6: f2 = <* |[a, c]|, |[b, c]|, |[b, d]|*> and

 A7: P1 = ( L~ f1) and

 A8: P2 = ( L~ f2);

 ( |[a, c]| `2 ) = c by EUCLID: 52;

 then

 A9: |[a, c]| <> |[a, d]| or |[a, d]| <> |[b, d]| by A2, EUCLID: 52;



 A10: P1 = (( LSeg ( |[a, c]|, |[a, d]|)) \/ ( LSeg ( |[a, d]|, |[b, d]|))) by A1, A2, A5, A6, A7, Th48;



 A11: (( LSeg ( |[a, c]|, |[a, d]|)) /\ ( LSeg ( |[a, d]|, |[b, d]|))) = { |[a, d]|} by A1, A2, Th34;

 ( |[b, c]| `2 ) = c by EUCLID: 52;

 then

 A12: |[a, c]| <> |[b, c]| or |[b, c]| <> |[b, d]| by A2, EUCLID: 52;



 A13: P2 = (( LSeg ( |[a, c]|, |[b, c]|)) \/ ( LSeg ( |[b, c]|, |[b, d]|))) by A1, A2, A5, A6, A8, Th48;

 (( LSeg ( |[a, c]|, |[b, c]|)) /\ ( LSeg ( |[b, c]|, |[b, d]|))) = { |[b, c]|} by A1, A2, Th32;

 hence thesis by A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, A12, A13, Th48, TOPREAL1: 12;

 end;

 theorem :: JGRAPH_6:50



 Th50: for a,b,c,d be Real st a < b & c < d holds ( rectangle (a,b,c,d)) is being_simple_closed_curve

 proof

 let a,b,c,d be Real;

 assume that

 A1: a as FinSequence of ( TOP-REAL 2);

 reconsider f2 = <* |[a, c]|, |[b, c]|, |[b, d]|*> as FinSequence of ( TOP-REAL 2);

 set P1 = ( L~ f1), P2 = ( L~ f2);



 A3: a < b & c < d & P = { p : (p `1 ) = a & c <= (p `2 ) & (p `2 ) <= d or (p `2 ) = d & a <= (p `1 ) & (p `1 ) <= b or (p `1 ) = b & c <= (p `2 ) & (p `2 ) <= d or (p `2 ) = c & a <= (p `1 ) & (p `1 ) <= b } & p1 = |[a, c]| & p2 = |[b, d]| & f1 = <* |[a, c]|, |[a, d]|, |[b, d]|*> & f2 = <* |[a, c]|, |[b, c]|, |[b, d]|*> & P1 = ( L~ f1) & P2 = ( L~ f2) implies P1 is_an_arc_of (p1,p2) & P2 is_an_arc_of (p1,p2) & P1 is non empty & P2 is non empty & P = (P1 \/ P2) & (P1 /\ P2) = {p1, p2} by Th49;

 ( |[a, c]| `1 ) = a by EUCLID: 52;

 then

 A4: p1 <> p2 by A1, EUCLID: 52;

 p1 in (P1 /\ P2) by A1, A2, A3, Lm15, TARSKI:def 2;

 then p1 in P1 by XBOOLE_0:def 4;

 then

 A5: p1 in P by A1, A2, A3, Lm15, XBOOLE_0:def 3;

 p2 in (P1 /\ P2) by A1, A2, A3, Lm15, TARSKI:def 2;

 then p2 in P1 by XBOOLE_0:def 4;

 then p2 in P by A1, A2, A3, Lm15, XBOOLE_0:def 3;

 hence thesis by A1, A2, A3, A4, A5, Lm15, TOPREAL2: 6;

 end;

 theorem :: JGRAPH_6:51



 Th51: for a,b,c,d be Real st a < b & c < d holds ( Upper_Arc ( rectangle (a,b,c,d))) = (( LSeg ( |[a, c]|, |[a, d]|)) \/ ( LSeg ( |[a, d]|, |[b, d]|)))

 proof

 let a,b,c,d be Real;

 set K = ( rectangle (a,b,c,d));

 assume that

 A1: a < b and

 A2: c < d;



 A3: K is being_simple_closed_curve by A1, A2, Th50;

 set P = K;



 A4: ( W-min K) = |[a, c]| by A1, A2, Th46;



 A5: ( E-max K) = |[b, d]| by A1, A2, Th46;

 reconsider U = (( LSeg ( |[a, c]|, |[a, d]|)) \/ ( LSeg ( |[a, d]|, |[b, d]|))) as non empty Subset of ( TOP-REAL 2);



 A6: U is_an_arc_of (( W-min P),( E-max P)) by A1, A2, Th47;

 reconsider P3 = (( LSeg ( |[a, c]|, |[b, c]|)) \/ ( LSeg ( |[b, c]|, |[b, d]|))) as non empty Subset of ( TOP-REAL 2);



 A7: P3 is_an_arc_of (( E-max P),( W-min P)) by A1, A2, Th47;

 reconsider f1 = <* |[a, c]|, |[a, d]|, |[b, d]|*>, f2 = <* |[a, c]|, |[b, c]|, |[b, d]|*> as FinSequence of ( TOP-REAL 2);

 set p0 = |[a, c]|, p01 = |[a, d]|, p10 = |[b, c]|, p1 = |[b, d]|;



 A8: a & f2 = <*p0, p10, p1*> implies f1 is being_S-Seq & ( L~ f1) = (( LSeg (p0,p01)) \/ ( LSeg (p01,p1))) & f2 is being_S-Seq & ( L~ f2) = (( LSeg (p0,p10)) \/ ( LSeg (p10,p1))) & K = (( L~ f1) \/ ( L~ f2)) & (( L~ f1) /\ ( L~ f2)) = {p0, p1} & (f1 /. 1) = p0 & (f1 /. ( len f1)) = p1 & (f2 /. 1) = p0 & (f2 /. ( len f2)) = p1 by Th48;



 A9: ( Vertical_Line ((( W-bound P) + ( E-bound P)) / 2)) = ( Vertical_Line ((a + ( E-bound P)) / 2)) by A1, A2, Th36

 .= ( Vertical_Line ((a + b) / 2)) by A1, A2, Th38;

 set Q = ( Vertical_Line ((( W-bound P) + ( E-bound P)) / 2));

 reconsider a2 = a, b2 = b, c2 = c, d2 = d as Real;



 A10: (U /\ ( Vertical_Line ((( W-bound P) + ( E-bound P)) / 2))) = { |[((a + b) / 2), d]|}

 proof

 thus (U /\ ( Vertical_Line ((( W-bound P) + ( E-bound P)) / 2))) c= { |[((a + b) / 2), d]|}

 proof

 let x be object;

 assume

 A11: x in (U /\ ( Vertical_Line ((( W-bound P) + ( E-bound P)) / 2)));

 then

 A12: x in U by XBOOLE_0:def 4;

 x in ( Vertical_Line ((( W-bound P) + ( E-bound P)) / 2)) by A11, XBOOLE_0:def 4;

 then x in { p where p be Point of ( TOP-REAL 2) : (p `1 ) = ((a + b) / 2) } by A9, JORDAN6:def 6;

 then

 consider p such that

 A13: x = p and

 A14: (p `1 ) = ((a + b) / 2);

 now

 assume p in ( LSeg ( |[a, c]|, |[a, d]|));

 then (p `1 ) = a by TOPREAL3: 11;

 hence contradiction by A1, A14;

 end;

 then p in ( LSeg ( |[a2, d2]|, |[b2, d2]|)) by A12, A13, XBOOLE_0:def 3;

 then (p `2 ) = d by TOPREAL3: 12;

 then x = |[((a + b) / 2), d]| by A13, A14, EUCLID: 53;

 hence thesis by TARSKI:def 1;

 end;

 let x be object;

 assume x in { |[((a + b) / 2), d]|};

 then

 A15: x = |[((a + b) / 2), d]| by TARSKI:def 1;

 ( |[((a + b) / 2), d]| `1 ) = ((a + b) / 2) by EUCLID: 52;

 then x in { p where p be Point of ( TOP-REAL 2) : (p `1 ) = ((a + b) / 2) } by A15;

 then

 A16: x in ( Vertical_Line ((( W-bound P) + ( E-bound P)) / 2)) by A9, JORDAN6:def 6;



 A17: ( |[b, d]| `1 ) = b by EUCLID: 52;



 A18: ( |[b, d]| `2 ) = d by EUCLID: 52;



 A19: ( |[a, d]| `1 ) = a by EUCLID: 52;

 ( |[a, d]| `2 ) = d by EUCLID: 52;

 then x in ( LSeg ( |[b, d]|, |[a, d]|)) by A1, A15, A17, A18, A19, TOPREAL3: 13;

 then x in U by XBOOLE_0:def 3;

 hence thesis by A16, XBOOLE_0:def 4;

 end;

 then |[((a + b) / 2), d]| in (U /\ Q) by TARSKI:def 1;

 then U meets Q;

 then ( First_Point (U,( W-min P),( E-max P),( Vertical_Line ((( W-bound P) + ( E-bound P)) / 2)))) in { |[((a + b) / 2), d]|} by A6, A10, JORDAN5C:def 1;

 then ( First_Point (U,( W-min P),( E-max P),( Vertical_Line ((( W-bound P) + ( E-bound P)) / 2)))) = |[((a + b) / 2), d]| by TARSKI:def 1;

 then

 A20: (( First_Point (U,( W-min P),( E-max P),( Vertical_Line ((( W-bound P) + ( E-bound P)) / 2)))) `2 ) = d by EUCLID: 52;



 A21: (P3 /\ ( Vertical_Line ((( W-bound P) + ( E-bound P)) / 2))) = { |[((a + b) / 2), c]|}

 proof

 thus (P3 /\ ( Vertical_Line ((( W-bound P) + ( E-bound P)) / 2))) c= { |[((a + b) / 2), c]|}

 proof

 let x be object;

 assume

 A22: x in (P3 /\ ( Vertical_Line ((( W-bound P) + ( E-bound P)) / 2)));

 then

 A23: x in P3 by XBOOLE_0:def 4;

 x in ( Vertical_Line ((( W-bound P) + ( E-bound P)) / 2)) by A22, XBOOLE_0:def 4;

 then x in { p where p be Point of ( TOP-REAL 2) : (p `1 ) = ((a + b) / 2) } by A9, JORDAN6:def 6;

 then

 consider p such that

 A24: x = p and

 A25: (p `1 ) = ((a + b) / 2);

 now

 assume p in ( LSeg ( |[b, c]|, |[b, d]|));

 then (p `1 ) = b by TOPREAL3: 11;

 hence contradiction by A1, A25;

 end;

 then p in ( LSeg ( |[a2, c2]|, |[b2, c2]|)) by A23, A24, XBOOLE_0:def 3;

 then (p `2 ) = c by TOPREAL3: 12;

 then x = |[((a + b) / 2), c]| by A24, A25, EUCLID: 53;

 hence thesis by TARSKI:def 1;

 end;

 let x be object;

 assume x in { |[((a + b) / 2), c]|};

 then

 A26: x = |[((a + b) / 2), c]| by TARSKI:def 1;

 ( |[((a + b) / 2), c]| `1 ) = ((a + b) / 2) by EUCLID: 52;

 then x in { p where p be Point of ( TOP-REAL 2) : (p `1 ) = ((a + b) / 2) } by A26;

 then

 A27: x in ( Vertical_Line ((( W-bound P) + ( E-bound P)) / 2)) by A9, JORDAN6:def 6;



 A28: ( |[b, c]| `1 ) = b by EUCLID: 52;



 A29: ( |[b, c]| `2 ) = c by EUCLID: 52;



 A30: ( |[a, c]| `1 ) = a by EUCLID: 52;

 ( |[a, c]| `2 ) = c by EUCLID: 52;

 then |[((b + a) / 2), c]| in ( LSeg ( |[a, c]|, |[b, c]|)) by A1, A28, A29, A30, TOPREAL3: 13;

 then x in P3 by A26, XBOOLE_0:def 3;

 hence thesis by A27, XBOOLE_0:def 4;

 end;

 then |[((a + b) / 2), c]| in (P3 /\ Q) by TARSKI:def 1;

 then P3 meets Q;

 then ( Last_Point (P3,( E-max P),( W-min P),( Vertical_Line ((( W-bound P) + ( E-bound P)) / 2)))) in { |[((a + b) / 2), c]|} by A7, A21, JORDAN5C:def 2;

 then ( Last_Point (P3,( E-max P),( W-min P),( Vertical_Line ((( W-bound P) + ( E-bound P)) / 2)))) = |[((a + b) / 2), c]| by TARSKI:def 1;

 then (( Last_Point (P3,( E-max P),( W-min P),( Vertical_Line ((( W-bound P) + ( E-bound P)) / 2)))) `2 ) = c by EUCLID: 52;

 hence thesis by A1, A2, A3, A4, A5, A6, A7, A8, A20, JORDAN6:def 8;

 end;

 theorem :: JGRAPH_6:52



 Th52: for a,b,c,d be Real st a < b & c < d holds ( Lower_Arc ( rectangle (a,b,c,d))) = (( LSeg ( |[a, c]|, |[b, c]|)) \/ ( LSeg ( |[b, c]|, |[b, d]|)))

 proof

 let a,b,c,d be Real;

 set K = ( rectangle (a,b,c,d));

 assume that

 A1: a < b and

 A2: c < d;



 A3: K is being_simple_closed_curve by A1, A2, Th50;

 set P = K;



 A4: ( W-min K) = |[a, c]| by A1, A2, Th46;



 A5: ( E-max K) = |[b, d]| by A1, A2, Th46;

 reconsider U = (( LSeg ( |[a, c]|, |[a, d]|)) \/ ( LSeg ( |[a, d]|, |[b, d]|))) as non empty Subset of ( TOP-REAL 2);



 A6: U is_an_arc_of (( W-min P),( E-max P)) by A1, A2, Th47;

 reconsider P3 = (( LSeg ( |[a, c]|, |[b, c]|)) \/ ( LSeg ( |[b, c]|, |[b, d]|))) as non empty Subset of ( TOP-REAL 2);



 A7: P3 is_an_arc_of (( E-max P),( W-min P)) by A1, A2, Th47;

 reconsider f1 = <* |[a, c]|, |[a, d]|, |[b, d]|*>, f2 = <* |[a, c]|, |[b, c]|, |[b, d]|*> as FinSequence of ( TOP-REAL 2);

 set p0 = |[a, c]|, p01 = |[a, d]|, p10 = |[b, c]|, p1 = |[b, d]|;



 A8: a & f2 = <*p0, p10, p1*> implies f1 is being_S-Seq & ( L~ f1) = (( LSeg (p0,p01)) \/ ( LSeg (p01,p1))) & f2 is being_S-Seq & ( L~ f2) = (( LSeg (p0,p10)) \/ ( LSeg (p10,p1))) & K = (( L~ f1) \/ ( L~ f2)) & (( L~ f1) /\ ( L~ f2)) = {p0, p1} & (f1 /. 1) = p0 & (f1 /. ( len f1)) = p1 & (f2 /. 1) = p0 & (f2 /. ( len f2)) = p1 by Th48;



 A9: ( Vertical_Line ((( W-bound P) + ( E-bound P)) / 2)) = ( Vertical_Line ((a + ( E-bound P)) / 2)) by A1, A2, Th36

 .= ( Vertical_Line ((a + b) / 2)) by A1, A2, Th38;

 set Q = ( Vertical_Line ((( W-bound P) + ( E-bound P)) / 2));

 reconsider a2 = a, b2 = b, c2 = c, d2 = d as Real;



 A10: (U /\ ( Vertical_Line ((( W-bound P) + ( E-bound P)) / 2))) = { |[((a + b) / 2), d]|}

 proof

 thus (U /\ ( Vertical_Line ((( W-bound P) + ( E-bound P)) / 2))) c= { |[((a + b) / 2), d]|}

 proof

 let x be object;

 assume

 A11: x in (U /\ ( Vertical_Line ((( W-bound P) + ( E-bound P)) / 2)));

 then

 A12: x in U by XBOOLE_0:def 4;

 x in ( Vertical_Line ((( W-bound P) + ( E-bound P)) / 2)) by A11, XBOOLE_0:def 4;

 then x in { p where p be Point of ( TOP-REAL 2) : (p `1 ) = ((a + b) / 2) } by A9, JORDAN6:def 6;

 then

 consider p such that

 A13: x = p and

 A14: (p `1 ) = ((a + b) / 2);

 now

 assume p in ( LSeg ( |[a, c]|, |[a, d]|));

 then (p `1 ) = a by TOPREAL3: 11;

 hence contradiction by A1, A14;

 end;

 then p in ( LSeg ( |[a2, d2]|, |[b2, d2]|)) by A12, A13, XBOOLE_0:def 3;

 then (p `2 ) = d by TOPREAL3: 12;

 then x = |[((a + b) / 2), d]| by A13, A14, EUCLID: 53;

 hence thesis by TARSKI:def 1;

 end;

 let x be object;

 assume x in { |[((a + b) / 2), d]|};

 then

 A15: x = |[((a + b) / 2), d]| by TARSKI:def 1;

 ( |[((a + b) / 2), d]| `1 ) = ((a + b) / 2) by EUCLID: 52;

 then x in { p where p be Point of ( TOP-REAL 2) : (p `1 ) = ((a + b) / 2) } by A15;

 then

 A16: x in ( Vertical_Line ((( W-bound P) + ( E-bound P)) / 2)) by A9, JORDAN6:def 6;



 A17: ( |[b, d]| `1 ) = b by EUCLID: 52;



 A18: ( |[b, d]| `2 ) = d by EUCLID: 52;



 A19: ( |[a, d]| `1 ) = a by EUCLID: 52;

 ( |[a, d]| `2 ) = d by EUCLID: 52;

 then x in ( LSeg ( |[b, d]|, |[a, d]|)) by A1, A15, A17, A18, A19, TOPREAL3: 13;

 then x in U by XBOOLE_0:def 3;

 hence thesis by A16, XBOOLE_0:def 4;

 end;

 then |[((a + b) / 2), d]| in (U /\ Q) by TARSKI:def 1;

 then U meets Q;

 then ( First_Point (U,( W-min P),( E-max P),( Vertical_Line ((( W-bound P) + ( E-bound P)) / 2)))) in { |[((a + b) / 2), d]|} by A6, A10, JORDAN5C:def 1;

 then ( First_Point (U,( W-min P),( E-max P),( Vertical_Line ((( W-bound P) + ( E-bound P)) / 2)))) = |[((a + b) / 2), d]| by TARSKI:def 1;

 then

 A20: (( First_Point (U,( W-min P),( E-max P),( Vertical_Line ((( W-bound P) + ( E-bound P)) / 2)))) `2 ) = d by EUCLID: 52;



 A21: (P3 /\ ( Vertical_Line ((( W-bound P) + ( E-bound P)) / 2))) = { |[((a + b) / 2), c]|}

 proof

 thus (P3 /\ ( Vertical_Line ((( W-bound P) + ( E-bound P)) / 2))) c= { |[((a + b) / 2), c]|}

 proof

 let x be object;

 assume

 A22: x in (P3 /\ ( Vertical_Line ((( W-bound P) + ( E-bound P)) / 2)));

 then

 A23: x in P3 by XBOOLE_0:def 4;

 x in ( Vertical_Line ((( W-bound P) + ( E-bound P)) / 2)) by A22, XBOOLE_0:def 4;

 then x in { p where p be Point of ( TOP-REAL 2) : (p `1 ) = ((a + b) / 2) } by A9, JORDAN6:def 6;

 then

 consider p such that

 A24: x = p and

 A25: (p `1 ) = ((a + b) / 2);

 now

 assume p in ( LSeg ( |[b, c]|, |[b, d]|));

 then (p `1 ) = b by TOPREAL3: 11;

 hence contradiction by A1, A25;

 end;

 then p in ( LSeg ( |[a2, c2]|, |[b2, c2]|)) by A23, A24, XBOOLE_0:def 3;

 then (p `2 ) = c by TOPREAL3: 12;

 then x = |[((a + b) / 2), c]| by A24, A25, EUCLID: 53;

 hence thesis by TARSKI:def 1;

 end;

 let x be object;

 assume x in { |[((a + b) / 2), c]|};

 then

 A26: x = |[((a + b) / 2), c]| by TARSKI:def 1;

 ( |[((a + b) / 2), c]| `1 ) = ((a + b) / 2) by EUCLID: 52;

 then x in { p where p be Point of ( TOP-REAL 2) : (p `1 ) = ((a + b) / 2) } by A26;

 then

 A27: x in ( Vertical_Line ((( W-bound P) + ( E-bound P)) / 2)) by A9, JORDAN6:def 6;



 A28: ( |[b, c]| `1 ) = b by EUCLID: 52;



 A29: ( |[b, c]| `2 ) = c by EUCLID: 52;



 A30: ( |[a, c]| `1 ) = a by EUCLID: 52;

 ( |[a, c]| `2 ) = c by EUCLID: 52;

 then |[((a + b) / 2), c]| in ( LSeg ( |[a, c]|, |[b, c]|)) by A1, A28, A29, A30, TOPREAL3: 13;

 then x in P3 by A26, XBOOLE_0:def 3;

 hence thesis by A27, XBOOLE_0:def 4;

 end;

 then |[((a + b) / 2), c]| in (P3 /\ Q) by TARSKI:def 1;

 then P3 meets Q;

 then ( Last_Point (P3,( E-max P),( W-min P),( Vertical_Line ((( W-bound P) + ( E-bound P)) / 2)))) in { |[((a + b) / 2), c]|} by A7, A21, JORDAN5C:def 2;

 then ( Last_Point (P3,( E-max P),( W-min P),( Vertical_Line ((( W-bound P) + ( E-bound P)) / 2)))) = |[((a + b) / 2), c]| by TARSKI:def 1;

 then

 A31: (( Last_Point (P3,( E-max P),( W-min P),( Vertical_Line ((( W-bound P) + ( E-bound P)) / 2)))) `2 ) = c by EUCLID: 52;



 A32: P3 is_an_arc_of (( E-max P),( W-min P)) by A1, A2, Th47;



 A33: (( Upper_Arc P) /\ P3) = {( W-min P), ( E-max P)} by A1, A2, A4, A5, A8, Th51;



 A34: (( Upper_Arc P) \/ P3) = P by A1, A2, A8, Th51;

 (( First_Point (( Upper_Arc P),( W-min P),( E-max P),( Vertical_Line ((( W-bound P) + ( E-bound P)) / 2)))) `2 ) > (( Last_Point (P3,( E-max P),( W-min P),( Vertical_Line ((( W-bound P) + ( E-bound P)) / 2)))) `2 ) by A1, A2, A20, A31, Th51;

 hence thesis by A3, A32, A33, A34, JORDAN6:def 9;

 end;

 theorem :: JGRAPH_6:53



 Th53: for a,b,c,d be Real st a < b & c < d holds ex f be Function of I[01] , (( TOP-REAL 2) | ( Upper_Arc ( rectangle (a,b,c,d)))) st f is being_homeomorphism & (f . 0 ) = ( W-min ( rectangle (a,b,c,d))) & (f . 1) = ( E-max ( rectangle (a,b,c,d))) & ( rng f) = ( Upper_Arc ( rectangle (a,b,c,d))) & (for r be Real st r in [. 0 , (1 / 2).] holds (f . r) = (((1 - (2 * r)) * |[a, c]|) + ((2 * r) * |[a, d]|))) & (for r be Real st r in [.(1 / 2), 1.] holds (f . r) = (((1 - ((2 * r) - 1)) * |[a, d]|) + (((2 * r) - 1) * |[b, d]|))) & (for p be Point of ( TOP-REAL 2) st p in ( LSeg ( |[a, c]|, |[a, d]|)) holds 0 <= ((((p `2 ) - c) / (d - c)) / 2) & ((((p `2 ) - c) / (d - c)) / 2) <= 1 & (f . ((((p `2 ) - c) / (d - c)) / 2)) = p) & for p be Point of ( TOP-REAL 2) st p in ( LSeg ( |[a, d]|, |[b, d]|)) holds 0 <= (((((p `1 ) - a) / (b - a)) / 2) + (1 / 2)) & (((((p `1 ) - a) / (b - a)) / 2) + (1 / 2)) <= 1 & (f . (((((p `1 ) - a) / (b - a)) / 2) + (1 / 2))) = p

 proof

 let a,b,c,d be Real;

 set K = ( rectangle (a,b,c,d));

 assume that

 A1: a < b and

 A2: c < d;

 defpred P[ object, object] means for r be Real st $1 = r holds (r in [. 0 , (1 / 2).] implies $2 = (((1 - (2 * r)) * |[a, c]|) + ((2 * r) * |[a, d]|))) & (r in [.(1 / 2), 1.] implies $2 = (((1 - ((2 * r) - 1)) * |[a, d]|) + (((2 * r) - 1) * |[b, d]|)));



 A3: [. 0 , 1.] = ( [. 0 , (1 / 2).] \/ [.(1 / 2), 1.]) by XXREAL_1: 165;



 A4: for x be object st x in [. 0 , 1.] holds ex y be object st P[x, y]

 proof

 let x be object;

 assume

 A5: x in [. 0 , 1.];

 now

 per cases by A3, A5, XBOOLE_0:def 3;

 case

 A6: x in [. 0 , (1 / 2).];

 then

 reconsider r = x as Real;



 A7: r <= (1 / 2) by A6, XXREAL_1: 1;

 set y0 = (((1 - (2 * r)) * |[a, c]|) + ((2 * r) * |[a, d]|));

 r in [.(1 / 2), 1.] implies y0 = (((1 - ((2 * r) - 1)) * |[a, d]|) + (((2 * r) - 1) * |[b, d]|))

 proof

 assume r in [.(1 / 2), 1.];

 then (1 / 2) <= r by XXREAL_1: 1;

 then

 A8: r = (1 / 2) by A7, XXREAL_0: 1;



 then

 A9: y0 = (( 0 * |[a, c]|) + |[a, d]|) by RLVECT_1:def 8

 .= (( 0. ( TOP-REAL 2)) + |[a, d]|) by RLVECT_1: 10

 .= |[a, d]| by RLVECT_1: 4;

 (((1 - ((2 * r) - 1)) * |[a, d]|) + (((2 * r) - 1) * |[b, d]|)) = ((1 * |[a, d]|) + ( 0. ( TOP-REAL 2))) by A8, RLVECT_1: 10

 .= ( |[a, d]| + ( 0. ( TOP-REAL 2))) by RLVECT_1:def 8

 .= |[a, d]| by RLVECT_1: 4;

 hence thesis by A9;

 end;

 then for r2 be Real st x = r2 holds (r2 in [. 0 , (1 / 2).] implies y0 = (((1 - (2 * r2)) * |[a, c]|) + ((2 * r2) * |[a, d]|))) & (r2 in [.(1 / 2), 1.] implies y0 = (((1 - ((2 * r2) - 1)) * |[a, d]|) + (((2 * r2) - 1) * |[b, d]|)));

 hence thesis;

 end;

 case

 A10: x in [.(1 / 2), 1.];

 then

 reconsider r = x as Real;



 A11: (1 / 2) <= r by A10, XXREAL_1: 1;

 set y0 = (((1 - ((2 * r) - 1)) * |[a, d]|) + (((2 * r) - 1) * |[b, d]|));

 r in [. 0 , (1 / 2).] implies y0 = (((1 - (2 * r)) * |[a, c]|) + ((2 * r) * |[a, d]|))

 proof

 assume r in [. 0 , (1 / 2).];

 then r <= (1 / 2) by XXREAL_1: 1;

 then

 A12: r = (1 / 2) by A11, XXREAL_0: 1;



 then

 A13: y0 = ( |[a, d]| + ( 0 * |[b, d]|)) by RLVECT_1:def 8

 .= ( |[a, d]| + ( 0. ( TOP-REAL 2))) by RLVECT_1: 10

 .= |[a, d]| by RLVECT_1: 4;

 (((1 - (2 * r)) * |[a, c]|) + ((2 * r) * |[a, d]|)) = (( 0. ( TOP-REAL 2)) + (1 * |[a, d]|)) by A12, RLVECT_1: 10

 .= (( 0. ( TOP-REAL 2)) + |[a, d]|) by RLVECT_1:def 8

 .= |[a, d]| by RLVECT_1: 4;

 hence thesis by A13;

 end;

 then for r2 be Real st x = r2 holds (r2 in [. 0 , (1 / 2).] implies y0 = (((1 - (2 * r2)) * |[a, c]|) + ((2 * r2) * |[a, d]|))) & (r2 in [.(1 / 2), 1.] implies y0 = (((1 - ((2 * r2) - 1)) * |[a, d]|) + (((2 * r2) - 1) * |[b, d]|)));

 hence thesis;

 end;

 end;

 hence thesis;

 end;

 ex f2 be Function st ( dom f2) = [. 0 , 1.] & for x be object st x in [. 0 , 1.] holds P[x, (f2 . x)] from CLASSES1:sch 1( A4);

 then

 consider f2 be Function such that

 A14: ( dom f2) = [. 0 , 1.] and

 A15: for x be object st x in [. 0 , 1.] holds P[x, (f2 . x)];

 ( rng f2) c= the carrier of (( TOP-REAL 2) | ( Upper_Arc K))

 proof

 let y be object;

 assume y in ( rng f2);

 then

 consider x be object such that

 A16: x in ( dom f2) and

 A17: y = (f2 . x) by FUNCT_1:def 3;

 now

 per cases by A3, A14, A16, XBOOLE_0:def 3;

 case

 A18: x in [. 0 , (1 / 2).];

 then

 reconsider r = x as Real;



 A19: 0 <= r by A18, XXREAL_1: 1;

 r <= (1 / 2) by A18, XXREAL_1: 1;

 then

 A20: (r * 2) <= ((1 / 2) * 2) by XREAL_1: 64;

 (f2 . x) = (((1 - (2 * r)) * |[a, c]|) + ((2 * r) * |[a, d]|)) by A14, A15, A16, A18;

 then

 A21: y in { (((1 - lambda) * |[a, c]|) + (lambda * |[a, d]|)) where lambda be Real : 0 <= lambda & lambda <= 1 } by A17, A19, A20;

 ( Upper_Arc K) = (( LSeg ( |[a, c]|, |[a, d]|)) \/ ( LSeg ( |[a, d]|, |[b, d]|))) by A1, A2, Th51;

 then y in ( Upper_Arc K) by A21, XBOOLE_0:def 3;

 hence thesis by PRE_TOPC: 8;

 end;

 case

 A22: x in [.(1 / 2), 1.];

 then

 reconsider r = x as Real;



 A23: (1 / 2) <= r by A22, XXREAL_1: 1;



 A24: r <= 1 by A22, XXREAL_1: 1;

 (r * 2) >= ((1 / 2) * 2) by A23, XREAL_1: 64;

 then

 A25: ((2 * r) - 1) >= 0 by XREAL_1: 48;

 (2 * 1) >= (2 * r) by A24, XREAL_1: 64;

 then

 A26: ((1 + 1) - 1) >= ((2 * r) - 1) by XREAL_1: 9;

 (f2 . x) = (((1 - ((2 * r) - 1)) * |[a, d]|) + (((2 * r) - 1) * |[b, d]|)) by A14, A15, A16, A22;

 then

 A27: y in { (((1 - lambda) * |[a, d]|) + (lambda * |[b, d]|)) where lambda be Real : 0 <= lambda & lambda <= 1 } by A17, A25, A26;

 ( Upper_Arc K) = (( LSeg ( |[a, c]|, |[a, d]|)) \/ ( LSeg ( |[a, d]|, |[b, d]|))) by A1, A2, Th51;

 then y in ( Upper_Arc K) by A27, XBOOLE_0:def 3;

 hence thesis by PRE_TOPC: 8;

 end;

 end;

 hence thesis;

 end;

 then

 reconsider f3 = f2 as Function of I[01] , (( TOP-REAL 2) | ( Upper_Arc K)) by A14, BORSUK_1: 40, FUNCT_2: 2;



 A28: 0 in [. 0 , 1.] by XXREAL_1: 1;

 0 in [. 0 , (1 / 2).] by XXREAL_1: 1;



 then

 A29: (f3 . 0 ) = (((1 - (2 * 0 )) * |[a, c]|) + ((2 * 0 ) * |[a, d]|)) by A15, A28

 .= ((1 * |[a, c]|) + ( 0. ( TOP-REAL 2))) by RLVECT_1: 10

 .= ( |[a, c]| + ( 0. ( TOP-REAL 2))) by RLVECT_1:def 8

 .= |[a, c]| by RLVECT_1: 4

 .= ( W-min K) by A1, A2, Th46;



 A30: 1 in [. 0 , 1.] by XXREAL_1: 1;

 1 in [.(1 / 2), 1.] by XXREAL_1: 1;



 then

 A31: (f3 . 1) = (((1 - ((2 * 1) - 1)) * |[a, d]|) + (((2 * 1) - 1) * |[b, d]|)) by A15, A30

 .= (( 0 * |[a, d]|) + |[b, d]|) by RLVECT_1:def 8

 .= (( 0. ( TOP-REAL 2)) + |[b, d]|) by RLVECT_1: 10

 .= |[b, d]| by RLVECT_1: 4

 .= ( E-max K) by A1, A2, Th46;



 A32: for r be Real st r in [. 0 , (1 / 2).] holds (f3 . r) = (((1 - (2 * r)) * |[a, c]|) + ((2 * r) * |[a, d]|))

 proof

 let r be Real;

 assume

 A33: r in [. 0 , (1 / 2).];

 then

 A34: 0 <= r by XXREAL_1: 1;

 r <= (1 / 2) by A33, XXREAL_1: 1;

 then r <= 1 by XXREAL_0: 2;

 then r in [. 0 , 1.] by A34, XXREAL_1: 1;

 hence thesis by A15, A33;

 end;



 A35: for r be Real st r in [.(1 / 2), 1.] holds (f3 . r) = (((1 - ((2 * r) - 1)) * |[a, d]|) + (((2 * r) - 1) * |[b, d]|))

 proof

 let r be Real;

 assume

 A36: r in [.(1 / 2), 1.];

 then

 A37: (1 / 2) <= r by XXREAL_1: 1;

 r <= 1 by A36, XXREAL_1: 1;

 then r in [. 0 , 1.] by A37, XXREAL_1: 1;

 hence thesis by A15, A36;

 end;



 A38: for p be Point of ( TOP-REAL 2) st p in ( LSeg ( |[a, c]|, |[a, d]|)) holds 0 <= ((((p `2 ) - c) / (d - c)) / 2) & ((((p `2 ) - c) / (d - c)) / 2) <= 1 & (f3 . ((((p `2 ) - c) / (d - c)) / 2)) = p

 proof

 let p be Point of ( TOP-REAL 2);

 assume

 A39: p in ( LSeg ( |[a, c]|, |[a, d]|));



 A40: ( |[a, c]| `2 ) = c by EUCLID: 52;



 A41: ( |[a, d]| `2 ) = d by EUCLID: 52;

 then

 A42: c <= (p `2 ) by A2, A39, A40, TOPREAL1: 4;



 A43: (p `2 ) <= d by A2, A39, A40, A41, TOPREAL1: 4;



 A44: (d - c) > 0 by A2, XREAL_1: 50;



 A45: ((p `2 ) - c) >= 0 by A42, XREAL_1: 48;



 A46: (d - c) > 0 by A2, XREAL_1: 50;

 ((p `2 ) - c) <= (d - c) by A43, XREAL_1: 9;

 then (((p `2 ) - c) / (d - c)) <= ((d - c) / (d - c)) by A46, XREAL_1: 72;

 then (((p `2 ) - c) / (d - c)) <= 1 by A46, XCMPLX_1: 60;

 then

 A47: ((((p `2 ) - c) / (d - c)) / 2) <= (1 / 2) by XREAL_1: 72;

 set r = ((((p `2 ) - c) / (d - c)) / 2);

 r in [. 0 , (1 / 2).] by A44, A45, A47, XXREAL_1: 1;



 then (f3 . ((((p `2 ) - c) / (d - c)) / 2)) = (((1 - (2 * r)) * |[a, c]|) + ((2 * r) * |[a, d]|)) by A32

 .= ( |[((1 - (2 * r)) * a), ((1 - (2 * r)) * c)]| + ((2 * r) * |[a, d]|)) by EUCLID: 58

 .= ( |[((1 - (2 * r)) * a), ((1 - (2 * r)) * c)]| + |[((2 * r) * a), ((2 * r) * d)]|) by EUCLID: 58

 .= |[(((1 * a) - ((2 * r) * a)) + ((2 * r) * a)), (((1 - (2 * r)) * c) + ((2 * r) * d))]| by EUCLID: 56

 .= |[a, ((1 * c) + ((((p `2 ) - c) / (d - c)) * (d - c)))]|

 .= |[a, ((1 * c) + ((p `2 ) - c))]| by A46, XCMPLX_1: 87

 .= |[(p `1 ), (p `2 )]| by A39, TOPREAL3: 11

 .= p by EUCLID: 53;

 hence thesis by A44, A45, A47, XXREAL_0: 2;

 end;



 A48: for p be Point of ( TOP-REAL 2) st p in ( LSeg ( |[a, d]|, |[b, d]|)) holds 0 <= (((((p `1 ) - a) / (b - a)) / 2) + (1 / 2)) & (((((p `1 ) - a) / (b - a)) / 2) + (1 / 2)) <= 1 & (f3 . (((((p `1 ) - a) / (b - a)) / 2) + (1 / 2))) = p

 proof

 let p be Point of ( TOP-REAL 2);

 assume

 A49: p in ( LSeg ( |[a, d]|, |[b, d]|));



 A50: ( |[a, d]| `1 ) = a by EUCLID: 52;



 A51: ( |[b, d]| `1 ) = b by EUCLID: 52;

 then

 A52: a <= (p `1 ) by A1, A49, A50, TOPREAL1: 3;



 A53: (p `1 ) <= b by A1, A49, A50, A51, TOPREAL1: 3;



 A54: (b - a) > 0 by A1, XREAL_1: 50;



 A55: ((p `1 ) - a) >= 0 by A52, XREAL_1: 48;

 then

 A56: (((((p `1 ) - a) / (b - a)) / 2) + (1 / 2)) >= ( 0 + (1 / 2)) by A54, XREAL_1: 7;



 A57: (b - a) > 0 by A1, XREAL_1: 50;

 ((p `1 ) - a) <= (b - a) by A53, XREAL_1: 9;

 then (((p `1 ) - a) / (b - a)) <= ((b - a) / (b - a)) by A57, XREAL_1: 72;

 then (((p `1 ) - a) / (b - a)) <= 1 by A57, XCMPLX_1: 60;

 then ((((p `1 ) - a) / (b - a)) / 2) <= (1 / 2) by XREAL_1: 72;

 then

 A58: (((((p `1 ) - a) / (b - a)) / 2) + (1 / 2)) <= ((1 / 2) + (1 / 2)) by XREAL_1: 7;

 set r = (((((p `1 ) - a) / (b - a)) / 2) + (1 / 2));

 r in [.(1 / 2), 1.] by A56, A58, XXREAL_1: 1;



 then (f3 . (((((p `1 ) - a) / (b - a)) / 2) + (1 / 2))) = (((1 - ((2 * r) - 1)) * |[a, d]|) + (((2 * r) - 1) * |[b, d]|)) by A35

 .= ( |[((1 - ((2 * r) - 1)) * a), ((1 - ((2 * r) - 1)) * d)]| + (((2 * r) - 1) * |[b, d]|)) by EUCLID: 58

 .= ( |[((1 - ((2 * r) - 1)) * a), ((1 - ((2 * r) - 1)) * d)]| + |[(((2 * r) - 1) * b), (((2 * r) - 1) * d)]|) by EUCLID: 58

 .= |[(((1 - ((2 * r) - 1)) * a) + (((2 * r) - 1) * b)), (((1 * d) - (((2 * r) - 1) * d)) + (((2 * r) - 1) * d))]| by EUCLID: 56

 .= |[((1 * a) + ((((p `1 ) - a) / (b - a)) * (b - a))), d]|

 .= |[((1 * a) + ((p `1 ) - a)), d]| by A57, XCMPLX_1: 87

 .= |[(p `1 ), (p `2 )]| by A49, TOPREAL3: 12

 .= p by EUCLID: 53;

 hence thesis by A54, A55, A58;

 end;

 reconsider B00 = [. 0 , 1.] as Subset of R^1 by TOPMETR: 17;

 reconsider B01 = B00 as non empty Subset of R^1 by XXREAL_1: 1;

 I[01] = ( R^1 | B01) by TOPMETR: 19, TOPMETR: 20;

 then

 consider h1 be Function of I[01] , R^1 such that

 A59: for p be Point of I[01] holds (h1 . p) = p and

 A60: h1 is continuous by Th6;

 consider h2 be Function of I[01] , R^1 such that

 A61: for p be Point of I[01] , r1 be Real st (h1 . p) = r1 holds (h2 . p) = (2 * r1) and

 A62: h2 is continuous by A60, JGRAPH_2: 23;

 consider h5 be Function of I[01] , R^1 such that

 A63: for p be Point of I[01] , r1 be Real st (h2 . p) = r1 holds (h5 . p) = (1 - r1) and

 A64: h5 is continuous by A62, Th8;

 consider h3 be Function of I[01] , R^1 such that

 A65: for p be Point of I[01] , r1 be Real st (h2 . p) = r1 holds (h3 . p) = (r1 - 1) and

 A66: h3 is continuous by A62, Th7;

 consider h4 be Function of I[01] , R^1 such that

 A67: for p be Point of I[01] , r1 be Real st (h3 . p) = r1 holds (h4 . p) = (1 - r1) and

 A68: h4 is continuous by A66, Th8;

 consider g1 be Function of I[01] , ( TOP-REAL 2) such that

 A69: for r be Point of I[01] holds (g1 . r) = (((h5 . r) * |[a, c]|) + ((h2 . r) * |[a, d]|)) and

 A70: g1 is continuous by A62, A64, Th13;



 A71: for r be Point of I[01] , s be Real st r = s holds (g1 . r) = (((1 - (2 * s)) * |[a, c]|) + ((2 * s) * |[a, d]|))

 proof

 let r be Point of I[01] , s be Real;

 assume

 A72: r = s;

 (g1 . r) = (((h5 . r) * |[a, c]|) + ((h2 . r) * |[a, d]|)) by A69

 .= (((1 - (2 * (h1 . r))) * |[a, c]|) + ((h2 . r) * |[a, d]|)) by A61, A63

 .= (((1 - (2 * (h1 . r))) * |[a, c]|) + ((2 * (h1 . r)) * |[a, d]|)) by A61

 .= (((1 - (2 * s)) * |[a, c]|) + ((2 * (h1 . r)) * |[a, d]|)) by A59, A72

 .= (((1 - (2 * s)) * |[a, c]|) + ((2 * s) * |[a, d]|)) by A59, A72;

 hence thesis;

 end;

 consider g2 be Function of I[01] , ( TOP-REAL 2) such that

 A73: for r be Point of I[01] holds (g2 . r) = (((h4 . r) * |[a, d]|) + ((h3 . r) * |[b, d]|)) and

 A74: g2 is continuous by A66, A68, Th13;



 A75: for r be Point of I[01] , s be Real st r = s holds (g2 . r) = (((1 - ((2 * s) - 1)) * |[a, d]|) + (((2 * s) - 1) * |[b, d]|))

 proof

 let r be Point of I[01] , s be Real;

 assume

 A76: r = s;

 (g2 . r) = (((h4 . r) * |[a, d]|) + ((h3 . r) * |[b, d]|)) by A73

 .= (((1 - ((h2 . r) - 1)) * |[a, d]|) + ((h3 . r) * |[b, d]|)) by A65, A67

 .= (((1 - ((h2 . r) - 1)) * |[a, d]|) + (((h2 . r) - 1) * |[b, d]|)) by A65

 .= (((1 - ((2 * (h1 . r)) - 1)) * |[a, d]|) + (((h2 . r) - 1) * |[b, d]|)) by A61

 .= (((1 - ((2 * (h1 . r)) - 1)) * |[a, d]|) + (((2 * (h1 . r)) - 1) * |[b, d]|)) by A61

 .= (((1 - ((2 * s) - 1)) * |[a, d]|) + (((2 * (h1 . r)) - 1) * |[b, d]|)) by A59, A76

 .= (((1 - ((2 * s) - 1)) * |[a, d]|) + (((2 * s) - 1) * |[b, d]|)) by A59, A76;

 hence thesis;

 end;

 reconsider B11 = [. 0 , (1 / 2).] as non empty Subset of I[01] by A3, BORSUK_1: 40, XBOOLE_1: 7, XXREAL_1: 1;



 A77: ( dom (g1 | B11)) = (( dom g1) /\ B11) by RELAT_1: 61

 .= (the carrier of I[01] /\ B11) by FUNCT_2:def 1

 .= B11 by XBOOLE_1: 28

 .= the carrier of ( I[01] | B11) by PRE_TOPC: 8;

 ( rng (g1 | B11)) c= the carrier of ( TOP-REAL 2);

 then

 reconsider g11 = (g1 | B11) as Function of ( I[01] | B11), ( TOP-REAL 2) by A77, FUNCT_2: 2;



 A78: ( TOP-REAL 2) is SubSpace of ( TOP-REAL 2) by TSEP_1: 2;

 then

 A79: g11 is continuous by A70, BORSUK_4: 44;

 reconsider B22 = [.(1 / 2), 1.] as non empty Subset of I[01] by A3, BORSUK_1: 40, XBOOLE_1: 7, XXREAL_1: 1;



 A80: ( dom (g2 | B22)) = (( dom g2) /\ B22) by RELAT_1: 61

 .= (the carrier of I[01] /\ B22) by FUNCT_2:def 1

 .= B22 by XBOOLE_1: 28

 .= the carrier of ( I[01] | B22) by PRE_TOPC: 8;

 ( rng (g2 | B22)) c= the carrier of ( TOP-REAL 2);

 then

 reconsider g22 = (g2 | B22) as Function of ( I[01] | B22), ( TOP-REAL 2) by A80, FUNCT_2: 2;



 A81: g22 is continuous by A74, A78, BORSUK_4: 44;



 A82: B11 = ( [#] ( I[01] | B11)) by PRE_TOPC:def 5;



 A83: B22 = ( [#] ( I[01] | B22)) by PRE_TOPC:def 5;



 A84: B11 is closed by Th4;



 A85: B22 is closed by Th4;



 A86: (( [#] ( I[01] | B11)) \/ ( [#] ( I[01] | B22))) = ( [#] I[01] ) by A82, A83, BORSUK_1: 40, XXREAL_1: 165;

 for p be object st p in (( [#] ( I[01] | B11)) /\ ( [#] ( I[01] | B22))) holds (g11 . p) = (g22 . p)

 proof

 let p be object;

 assume

 A87: p in (( [#] ( I[01] | B11)) /\ ( [#] ( I[01] | B22)));

 then

 A88: p in ( [#] ( I[01] | B11)) by XBOOLE_0:def 4;



 A89: p in ( [#] ( I[01] | B22)) by A87;



 A90: p in B11 by A88, PRE_TOPC:def 5;



 A91: p in B22 by A89, PRE_TOPC:def 5;

 reconsider rp = p as Real by A90;



 A92: rp <= (1 / 2) by A90, XXREAL_1: 1;

 rp >= (1 / 2) by A91, XXREAL_1: 1;

 then rp = (1 / 2) by A92, XXREAL_0: 1;

 then

 A93: (2 * rp) = 1;



 thus (g11 . p) = (g1 . p) by A90, FUNCT_1: 49

 .= (((1 - 1) * |[a, c]|) + (1 * |[a, d]|)) by A71, A90, A93

 .= (( 0. ( TOP-REAL 2)) + (1 * |[a, d]|)) by RLVECT_1: 10

 .= (((1 - 0 ) * |[a, d]|) + ((1 - 1) * |[b, d]|)) by RLVECT_1: 10

 .= (g2 . p) by A75, A90, A93

 .= (g22 . p) by A91, FUNCT_1: 49;

 end;

 then

 consider h be Function of I[01] , ( TOP-REAL 2) such that

 A94: h = (g11 +* g22) and

 A95: h is continuous by A79, A81, A82, A83, A84, A85, A86, JGRAPH_2: 1;



 A96: ( dom f3) = ( dom h) by Th5;



 A97: ( dom f3) = the carrier of I[01] by Th5;

 for x be object st x in ( dom f2) holds (f3 . x) = (h . x)

 proof

 let x be object;

 assume

 A98: x in ( dom f2);

 then

 reconsider rx = x as Real by A97;



 A99: 0 <= rx by A96, A98, BORSUK_1: 40, XXREAL_1: 1;



 A100: rx <= 1 by A96, A98, BORSUK_1: 40, XXREAL_1: 1;

 now

 per cases ;

 case

 A101: rx < (1 / 2);

 then

 A102: rx in [. 0 , (1 / 2).] by A99, XXREAL_1: 1;

 not rx in ( dom g22) by A83, A101, XXREAL_1: 1;



 then (h . rx) = (g11 . rx) by A94, FUNCT_4: 11

 .= (g1 . rx) by A102, FUNCT_1: 49

 .= (((1 - (2 * rx)) * |[a, c]|) + ((2 * rx) * |[a, d]|)) by A71, A96, A98

 .= (f3 . rx) by A32, A102;

 hence thesis;

 end;

 case rx >= (1 / 2);

 then

 A103: rx in [.(1 / 2), 1.] by A100, XXREAL_1: 1;

 then rx in ( [#] ( I[01] | B22)) by PRE_TOPC:def 5;



 then (h . rx) = (g22 . rx) by A80, A94, FUNCT_4: 13

 .= (g2 . rx) by A103, FUNCT_1: 49

 .= (((1 - ((2 * rx) - 1)) * |[a, d]|) + (((2 * rx) - 1) * |[b, d]|)) by A75, A96, A98

 .= (f3 . rx) by A35, A103;

 hence thesis;

 end;

 end;

 hence thesis;

 end;

 then

 A104: f2 = h by A96, FUNCT_1: 2;

 for x1,x2 be object st x1 in ( dom f3) & x2 in ( dom f3) & (f3 . x1) = (f3 . x2) holds x1 = x2

 proof

 let x1,x2 be object;

 assume that

 A105: x1 in ( dom f3) and

 A106: x2 in ( dom f3) and

 A107: (f3 . x1) = (f3 . x2);

 reconsider r1 = x1 as Real by A105;

 reconsider r2 = x2 as Real by A106;



 A108: (( LSeg ( |[a, c]|, |[a, d]|)) /\ ( LSeg ( |[a, d]|, |[b, d]|))) = { |[a, d]|} by A1, A2, Th34;

 now

 per cases by A3, A14, A105, A106, XBOOLE_0:def 3;

 case

 A109: x1 in [. 0 , (1 / 2).] & x2 in [. 0 , (1 / 2).];

 then (f3 . r1) = (((1 - (2 * r1)) * |[a, c]|) + ((2 * r1) * |[a, d]|)) by A32;

 then (((1 - (2 * r2)) * |[a, c]|) + ((2 * r2) * |[a, d]|)) = (((1 - (2 * r1)) * |[a, c]|) + ((2 * r1) * |[a, d]|)) by A32, A107, A109;

 then ((((1 - (2 * r2)) * |[a, c]|) + ((2 * r2) * |[a, d]|)) - ((2 * r1) * |[a, d]|)) = ((1 - (2 * r1)) * |[a, c]|) by RLVECT_4: 1;

 then (((1 - (2 * r2)) * |[a, c]|) + (((2 * r2) * |[a, d]|) - ((2 * r1) * |[a, d]|))) = ((1 - (2 * r1)) * |[a, c]|) by RLVECT_1:def 3;

 then (((1 - (2 * r2)) * |[a, c]|) + (((2 * r2) - (2 * r1)) * |[a, d]|)) = ((1 - (2 * r1)) * |[a, c]|) by RLVECT_1: 35;

 then ((((2 * r2) - (2 * r1)) * |[a, d]|) + (((1 - (2 * r2)) * |[a, c]|) - ((1 - (2 * r1)) * |[a, c]|))) = (((1 - (2 * r1)) * |[a, c]|) - ((1 - (2 * r1)) * |[a, c]|)) by RLVECT_1:def 3;

 then ((((2 * r2) - (2 * r1)) * |[a, d]|) + (((1 - (2 * r2)) * |[a, c]|) - ((1 - (2 * r1)) * |[a, c]|))) = ( 0. ( TOP-REAL 2)) by RLVECT_1: 5;

 then ((((2 * r2) - (2 * r1)) * |[a, d]|) + (((1 - (2 * r2)) - (1 - (2 * r1))) * |[a, c]|)) = ( 0. ( TOP-REAL 2)) by RLVECT_1: 35;

 then ((((2 * r2) - (2 * r1)) * |[a, d]|) + (( - ((2 * r2) - (2 * r1))) * |[a, c]|)) = ( 0. ( TOP-REAL 2));

 then ((((2 * r2) - (2 * r1)) * |[a, d]|) + ( - (((2 * r2) - (2 * r1)) * |[a, c]|))) = ( 0. ( TOP-REAL 2)) by RLVECT_1: 79;

 then ((((2 * r2) - (2 * r1)) * |[a, d]|) - (((2 * r2) - (2 * r1)) * |[a, c]|)) = ( 0. ( TOP-REAL 2));

 then (((2 * r2) - (2 * r1)) * ( |[a, d]| - |[a, c]|)) = ( 0. ( TOP-REAL 2)) by RLVECT_1: 34;

 then ((2 * r2) - (2 * r1)) = 0 or ( |[a, d]| - |[a, c]|) = ( 0. ( TOP-REAL 2)) by RLVECT_1: 11;

 then ((2 * r2) - (2 * r1)) = 0 or |[a, d]| = |[a, c]| by RLVECT_1: 21;

 then ((2 * r2) - (2 * r1)) = 0 or d = ( |[a, c]| `2 ) by EUCLID: 52;

 hence thesis by A2, EUCLID: 52;

 end;

 case

 A110: x1 in [. 0 , (1 / 2).] & x2 in [.(1 / 2), 1.];

 then

 A111: (f3 . r1) = (((1 - (2 * r1)) * |[a, c]|) + ((2 * r1) * |[a, d]|)) by A32;



 A112: 0 <= r1 by A110, XXREAL_1: 1;

 r1 <= (1 / 2) by A110, XXREAL_1: 1;

 then (r1 * 2) <= ((1 / 2) * 2) by XREAL_1: 64;

 then

 A113: (f3 . r1) in ( LSeg ( |[a, c]|, |[a, d]|)) by A111, A112;



 A114: (f3 . r2) = (((1 - ((2 * r2) - 1)) * |[a, d]|) + (((2 * r2) - 1) * |[b, d]|)) by A35, A110;



 A115: (1 / 2) <= r2 by A110, XXREAL_1: 1;



 A116: r2 <= 1 by A110, XXREAL_1: 1;

 (r2 * 2) >= ((1 / 2) * 2) by A115, XREAL_1: 64;

 then

 A117: ((2 * r2) - 1) >= 0 by XREAL_1: 48;

 (2 * 1) >= (2 * r2) by A116, XREAL_1: 64;

 then ((1 + 1) - 1) >= ((2 * r2) - 1) by XREAL_1: 9;

 then (f3 . r2) in { (((1 - lambda) * |[a, d]|) + (lambda * |[b, d]|)) where lambda be Real : 0 <= lambda & lambda <= 1 } by A114, A117;

 then (f3 . r1) in (( LSeg ( |[a, c]|, |[a, d]|)) /\ ( LSeg ( |[a, d]|, |[b, d]|))) by A107, A113, XBOOLE_0:def 4;

 then

 A118: (f3 . r1) = |[a, d]| by A108, TARSKI:def 1;

 then ((((1 - (2 * r1)) * |[a, c]|) + ((2 * r1) * |[a, d]|)) + ( - |[a, d]|)) = ( 0. ( TOP-REAL 2)) by A111, RLVECT_1: 5;

 then ((((1 - (2 * r1)) * |[a, c]|) + ((2 * r1) * |[a, d]|)) + (( - 1) * |[a, d]|)) = ( 0. ( TOP-REAL 2)) by RLVECT_1: 16;

 then (((1 - (2 * r1)) * |[a, c]|) + (((2 * r1) * |[a, d]|) + (( - 1) * |[a, d]|))) = ( 0. ( TOP-REAL 2)) by RLVECT_1:def 3;

 then (((1 - (2 * r1)) * |[a, c]|) + (((2 * r1) + ( - 1)) * |[a, d]|)) = ( 0. ( TOP-REAL 2)) by RLVECT_1:def 6;

 then (((1 - (2 * r1)) * |[a, c]|) + (( - (1 - (2 * r1))) * |[a, d]|)) = ( 0. ( TOP-REAL 2));

 then (((1 - (2 * r1)) * |[a, c]|) + ( - ((1 - (2 * r1)) * |[a, d]|))) = ( 0. ( TOP-REAL 2)) by RLVECT_1: 79;

 then (((1 - (2 * r1)) * |[a, c]|) - ((1 - (2 * r1)) * |[a, d]|)) = ( 0. ( TOP-REAL 2));

 then ((1 - (2 * r1)) * ( |[a, c]| - |[a, d]|)) = ( 0. ( TOP-REAL 2)) by RLVECT_1: 34;

 then (1 - (2 * r1)) = 0 or ( |[a, c]| - |[a, d]|) = ( 0. ( TOP-REAL 2)) by RLVECT_1: 11;

 then (1 - (2 * r1)) = 0 or |[a, c]| = |[a, d]| by RLVECT_1: 21;

 then

 A119: (1 - (2 * r1)) = 0 or c = ( |[a, d]| `2 ) by EUCLID: 52;

 ((((1 - ((2 * r2) - 1)) * |[a, d]|) + (((2 * r2) - 1) * |[b, d]|)) + ( - |[a, d]|)) = ( 0. ( TOP-REAL 2)) by A107, A114, A118, RLVECT_1: 5;

 then ((((1 - ((2 * r2) - 1)) * |[a, d]|) + (((2 * r2) - 1) * |[b, d]|)) + (( - 1) * |[a, d]|)) = ( 0. ( TOP-REAL 2)) by RLVECT_1: 16;

 then ((((2 * r2) - 1) * |[b, d]|) + (((1 - ((2 * r2) - 1)) * |[a, d]|) + (( - 1) * |[a, d]|))) = ( 0. ( TOP-REAL 2)) by RLVECT_1:def 3;

 then ((((2 * r2) - 1) * |[b, d]|) + (((1 - ((2 * r2) - 1)) + ( - 1)) * |[a, d]|)) = ( 0. ( TOP-REAL 2)) by RLVECT_1:def 6;

 then ((((2 * r2) - 1) * |[b, d]|) + (( - ((2 * r2) - 1)) * |[a, d]|)) = ( 0. ( TOP-REAL 2));

 then ((((2 * r2) - 1) * |[b, d]|) + ( - (((2 * r2) - 1) * |[a, d]|))) = ( 0. ( TOP-REAL 2)) by RLVECT_1: 79;

 then ((((2 * r2) - 1) * |[b, d]|) - (((2 * r2) - 1) * |[a, d]|)) = ( 0. ( TOP-REAL 2));

 then (((2 * r2) - 1) * ( |[b, d]| - |[a, d]|)) = ( 0. ( TOP-REAL 2)) by RLVECT_1: 34;

 then ((2 * r2) - 1) = 0 or ( |[b, d]| - |[a, d]|) = ( 0. ( TOP-REAL 2)) by RLVECT_1: 11;

 then ((2 * r2) - 1) = 0 or |[b, d]| = |[a, d]| by RLVECT_1: 21;

 then ((2 * r2) - 1) = 0 or b = ( |[a, d]| `1 ) by EUCLID: 52;

 hence thesis by A1, A2, A119, EUCLID: 52;

 end;

 case

 A120: x1 in [.(1 / 2), 1.] & x2 in [. 0 , (1 / 2).];

 then

 A121: (f3 . r2) = (((1 - (2 * r2)) * |[a, c]|) + ((2 * r2) * |[a, d]|)) by A32;



 A122: 0 <= r2 by A120, XXREAL_1: 1;

 r2 <= (1 / 2) by A120, XXREAL_1: 1;

 then (r2 * 2) <= ((1 / 2) * 2) by XREAL_1: 64;

 then

 A123: (f3 . r2) in ( LSeg ( |[a, c]|, |[a, d]|)) by A121, A122;



 A124: (f3 . r1) = (((1 - ((2 * r1) - 1)) * |[a, d]|) + (((2 * r1) - 1) * |[b, d]|)) by A35, A120;



 A125: (1 / 2) <= r1 by A120, XXREAL_1: 1;



 A126: r1 <= 1 by A120, XXREAL_1: 1;

 (r1 * 2) >= ((1 / 2) * 2) by A125, XREAL_1: 64;

 then

 A127: ((2 * r1) - 1) >= 0 by XREAL_1: 48;

 (2 * 1) >= (2 * r1) by A126, XREAL_1: 64;

 then ((1 + 1) - 1) >= ((2 * r1) - 1) by XREAL_1: 9;

 then (f3 . r1) in { (((1 - lambda) * |[a, d]|) + (lambda * |[b, d]|)) where lambda be Real : 0 <= lambda & lambda <= 1 } by A124, A127;

 then (f3 . r2) in (( LSeg ( |[a, c]|, |[a, d]|)) /\ ( LSeg ( |[a, d]|, |[b, d]|))) by A107, A123, XBOOLE_0:def 4;

 then

 A128: (f3 . r2) = |[a, d]| by A108, TARSKI:def 1;

 then ((((1 - (2 * r2)) * |[a, c]|) + ((2 * r2) * |[a, d]|)) + ( - |[a, d]|)) = ( 0. ( TOP-REAL 2)) by A121, RLVECT_1: 5;

 then ((((1 - (2 * r2)) * |[a, c]|) + ((2 * r2) * |[a, d]|)) + (( - 1) * |[a, d]|)) = ( 0. ( TOP-REAL 2)) by RLVECT_1: 16;

 then (((1 - (2 * r2)) * |[a, c]|) + (((2 * r2) * |[a, d]|) + (( - 1) * |[a, d]|))) = ( 0. ( TOP-REAL 2)) by RLVECT_1:def 3;

 then (((1 - (2 * r2)) * |[a, c]|) + (((2 * r2) + ( - 1)) * |[a, d]|)) = ( 0. ( TOP-REAL 2)) by RLVECT_1:def 6;

 then (((1 - (2 * r2)) * |[a, c]|) + (( - (1 - (2 * r2))) * |[a, d]|)) = ( 0. ( TOP-REAL 2));

 then (((1 - (2 * r2)) * |[a, c]|) + ( - ((1 - (2 * r2)) * |[a, d]|))) = ( 0. ( TOP-REAL 2)) by RLVECT_1: 79;

 then (((1 - (2 * r2)) * |[a, c]|) - ((1 - (2 * r2)) * |[a, d]|)) = ( 0. ( TOP-REAL 2));

 then ((1 - (2 * r2)) * ( |[a, c]| - |[a, d]|)) = ( 0. ( TOP-REAL 2)) by RLVECT_1: 34;

 then (1 - (2 * r2)) = 0 or ( |[a, c]| - |[a, d]|) = ( 0. ( TOP-REAL 2)) by RLVECT_1: 11;

 then (1 - (2 * r2)) = 0 or |[a, c]| = |[a, d]| by RLVECT_1: 21;

 then

 A129: (1 - (2 * r2)) = 0 or c = ( |[a, d]| `2 ) by EUCLID: 52;

 ((((1 - ((2 * r1) - 1)) * |[a, d]|) + (((2 * r1) - 1) * |[b, d]|)) + ( - |[a, d]|)) = ( 0. ( TOP-REAL 2)) by A107, A124, A128, RLVECT_1: 5;

 then ((((1 - ((2 * r1) - 1)) * |[a, d]|) + (((2 * r1) - 1) * |[b, d]|)) + (( - 1) * |[a, d]|)) = ( 0. ( TOP-REAL 2)) by RLVECT_1: 16;

 then ((((2 * r1) - 1) * |[b, d]|) + (((1 - ((2 * r1) - 1)) * |[a, d]|) + (( - 1) * |[a, d]|))) = ( 0. ( TOP-REAL 2)) by RLVECT_1:def 3;

 then ((((2 * r1) - 1) * |[b, d]|) + ((( - 1) + (1 - ((2 * r1) - 1))) * |[a, d]|)) = ( 0. ( TOP-REAL 2)) by RLVECT_1:def 6;

 then ((((2 * r1) - 1) * |[b, d]|) + (( - ((2 * r1) - 1)) * |[a, d]|)) = ( 0. ( TOP-REAL 2));

 then ((((2 * r1) - 1) * |[b, d]|) + ( - (((2 * r1) - 1) * |[a, d]|))) = ( 0. ( TOP-REAL 2)) by RLVECT_1: 79;

 then ((((2 * r1) - 1) * |[b, d]|) - (((2 * r1) - 1) * |[a, d]|)) = ( 0. ( TOP-REAL 2));

 then (((2 * r1) - 1) * ( |[b, d]| - |[a, d]|)) = ( 0. ( TOP-REAL 2)) by RLVECT_1: 34;

 then ((2 * r1) - 1) = 0 or ( |[b, d]| - |[a, d]|) = ( 0. ( TOP-REAL 2)) by RLVECT_1: 11;

 then ((2 * r1) - 1) = 0 or |[b, d]| = |[a, d]| by RLVECT_1: 21;

 then ((2 * r1) - 1) = 0 or b = ( |[a, d]| `1 ) by EUCLID: 52;

 hence thesis by A1, A2, A129, EUCLID: 52;

 end;

 case

 A130: x1 in [.(1 / 2), 1.] & x2 in [.(1 / 2), 1.];

 then (f3 . r1) = (((1 - ((2 * r1) - 1)) * |[a, d]|) + (((2 * r1) - 1) * |[b, d]|)) by A35;

 then (((1 - ((2 * r2) - 1)) * |[a, d]|) + (((2 * r2) - 1) * |[b, d]|)) = (((1 - ((2 * r1) - 1)) * |[a, d]|) + (((2 * r1) - 1) * |[b, d]|)) by A35, A107, A130;

 then ((((1 - ((2 * r2) - 1)) * |[a, d]|) + (((2 * r2) - 1) * |[b, d]|)) - (((2 * r1) - 1) * |[b, d]|)) = ((1 - ((2 * r1) - 1)) * |[a, d]|) by RLVECT_4: 1;

 then (((1 - ((2 * r2) - 1)) * |[a, d]|) + ((((2 * r2) - 1) * |[b, d]|) - (((2 * r1) - 1) * |[b, d]|))) = ((1 - ((2 * r1) - 1)) * |[a, d]|) by RLVECT_1:def 3;

 then (((1 - ((2 * r2) - 1)) * |[a, d]|) + ((((2 * r2) - 1) - ((2 * r1) - 1)) * |[b, d]|)) = ((1 - ((2 * r1) - 1)) * |[a, d]|) by RLVECT_1: 35;

 then (((((2 * r2) - 1) - ((2 * r1) - 1)) * |[b, d]|) + (((1 - ((2 * r2) - 1)) * |[a, d]|) - ((1 - ((2 * r1) - 1)) * |[a, d]|))) = (((1 - ((2 * r1) - 1)) * |[a, d]|) - ((1 - ((2 * r1) - 1)) * |[a, d]|)) by RLVECT_1:def 3;

 then (((((2 * r2) - 1) - ((2 * r1) - 1)) * |[b, d]|) + (((1 - ((2 * r2) - 1)) * |[a, d]|) - ((1 - ((2 * r1) - 1)) * |[a, d]|))) = ( 0. ( TOP-REAL 2)) by RLVECT_1: 5;

 then (((((2 * r2) - 1) - ((2 * r1) - 1)) * |[b, d]|) + (((1 - ((2 * r2) - 1)) - (1 - ((2 * r1) - 1))) * |[a, d]|)) = ( 0. ( TOP-REAL 2)) by RLVECT_1: 35;

 then (((((2 * r2) - 1) - ((2 * r1) - 1)) * |[b, d]|) + (( - (((2 * r2) - 1) - ((2 * r1) - 1))) * |[a, d]|)) = ( 0. ( TOP-REAL 2));

 then (((((2 * r2) - 1) - ((2 * r1) - 1)) * |[b, d]|) + ( - ((((2 * r2) - 1) - ((2 * r1) - 1)) * |[a, d]|))) = ( 0. ( TOP-REAL 2)) by RLVECT_1: 79;

 then (((((2 * r2) - 1) - ((2 * r1) - 1)) * |[b, d]|) - ((((2 * r2) - 1) - ((2 * r1) - 1)) * |[a, d]|)) = ( 0. ( TOP-REAL 2));

 then ((((2 * r2) - 1) - ((2 * r1) - 1)) * ( |[b, d]| - |[a, d]|)) = ( 0. ( TOP-REAL 2)) by RLVECT_1: 34;

 then (((2 * r2) - 1) - ((2 * r1) - 1)) = 0 or ( |[b, d]| - |[a, d]|) = ( 0. ( TOP-REAL 2)) by RLVECT_1: 11;

 then (((2 * r2) - 1) - ((2 * r1) - 1)) = 0 or |[b, d]| = |[a, d]| by RLVECT_1: 21;

 then (((2 * r2) - 1) - ((2 * r1) - 1)) = 0 or b = ( |[a, d]| `1 ) by EUCLID: 52;

 hence thesis by A1, EUCLID: 52;

 end;

 end;

 hence thesis;

 end;

 then

 A131: f3 is one-to-one by FUNCT_1:def 4;

 ( [#] (( TOP-REAL 2) | ( Upper_Arc K))) c= ( rng f3)

 proof

 let y be object;

 assume y in ( [#] (( TOP-REAL 2) | ( Upper_Arc K)));

 then

 A132: y in ( Upper_Arc K) by PRE_TOPC:def 5;

 then

 reconsider q = y as Point of ( TOP-REAL 2);



 A133: ( Upper_Arc K) = (( LSeg ( |[a, c]|, |[a, d]|)) \/ ( LSeg ( |[a, d]|, |[b, d]|))) by A1, A2, Th51;

 now

 per cases by A132, A133, XBOOLE_0:def 3;

 case

 A134: q in ( LSeg ( |[a, c]|, |[a, d]|));

 then

 A135: 0 <= ((((q `2 ) - c) / (d - c)) / 2) by A38;



 A136: ((((q `2 ) - c) / (d - c)) / 2) <= 1 by A38, A134;



 A137: (f3 . ((((q `2 ) - c) / (d - c)) / 2)) = q by A38, A134;

 ((((q `2 ) - c) / (d - c)) / 2) in [. 0 , 1.] by A135, A136, XXREAL_1: 1;

 hence thesis by A14, A137, FUNCT_1:def 3;

 end;

 case

 A138: q in ( LSeg ( |[a, d]|, |[b, d]|));

 then

 A139: 0 <= (((((q `1 ) - a) / (b - a)) / 2) + (1 / 2)) by A48;



 A140: (((((q `1 ) - a) / (b - a)) / 2) + (1 / 2)) <= 1 by A48, A138;



 A141: (f3 . (((((q `1 ) - a) / (b - a)) / 2) + (1 / 2))) = q by A48, A138;

 (((((q `1 ) - a) / (b - a)) / 2) + (1 / 2)) in [. 0 , 1.] by A139, A140, XXREAL_1: 1;

 hence thesis by A14, A141, FUNCT_1:def 3;

 end;

 end;

 hence thesis;

 end;

 then

 A142: ( rng f3) = ( [#] (( TOP-REAL 2) | ( Upper_Arc K)));

 I[01] is compact by HEINE: 4, TOPMETR: 20;

 then

 A143: f3 is being_homeomorphism by A95, A104, A131, A142, COMPTS_1: 17, JGRAPH_1: 45;

 ( rng f3) = ( Upper_Arc K) by A142, PRE_TOPC:def 5;

 hence thesis by A29, A31, A32, A35, A38, A48, A143;

 end;

 theorem :: JGRAPH_6:54



 Th54: for a,b,c,d be Real st a < b & c < d holds ex f be Function of I[01] , (( TOP-REAL 2) | ( Lower_Arc ( rectangle (a,b,c,d)))) st f is being_homeomorphism & (f . 0 ) = ( E-max ( rectangle (a,b,c,d))) & (f . 1) = ( W-min ( rectangle (a,b,c,d))) & ( rng f) = ( Lower_Arc ( rectangle (a,b,c,d))) & (for r be Real st r in [. 0 , (1 / 2).] holds (f . r) = (((1 - (2 * r)) * |[b, d]|) + ((2 * r) * |[b, c]|))) & (for r be Real st r in [.(1 / 2), 1.] holds (f . r) = (((1 - ((2 * r) - 1)) * |[b, c]|) + (((2 * r) - 1) * |[a, c]|))) & (for p be Point of ( TOP-REAL 2) st p in ( LSeg ( |[b, d]|, |[b, c]|)) holds 0 <= ((((p `2 ) - d) / (c - d)) / 2) & ((((p `2 ) - d) / (c - d)) / 2) <= 1 & (f . ((((p `2 ) - d) / (c - d)) / 2)) = p) & for p be Point of ( TOP-REAL 2) st p in ( LSeg ( |[b, c]|, |[a, c]|)) holds 0 <= (((((p `1 ) - b) / (a - b)) / 2) + (1 / 2)) & (((((p `1 ) - b) / (a - b)) / 2) + (1 / 2)) <= 1 & (f . (((((p `1 ) - b) / (a - b)) / 2) + (1 / 2))) = p

 proof

 let a,b,c,d be Real;

 set K = ( rectangle (a,b,c,d));

 assume that

 A1: a < b and

 A2: c < d;

 defpred P[ object, object] means for r be Real st $1 = r holds (r in [. 0 , (1 / 2).] implies $2 = (((1 - (2 * r)) * |[b, d]|) + ((2 * r) * |[b, c]|))) & (r in [.(1 / 2), 1.] implies $2 = (((1 - ((2 * r) - 1)) * |[b, c]|) + (((2 * r) - 1) * |[a, c]|)));



 A3: [. 0 , 1.] = ( [. 0 , (1 / 2).] \/ [.(1 / 2), 1.]) by XXREAL_1: 165;



 A4: for x be object st x in [. 0 , 1.] holds ex y be object st P[x, y]

 proof

 let x be object;

 assume

 A5: x in [. 0 , 1.];

 now

 per cases by A3, A5, XBOOLE_0:def 3;

 case

 A6: x in [. 0 , (1 / 2).];

 then

 reconsider r = x as Real;



 A7: r <= (1 / 2) by A6, XXREAL_1: 1;

 set y0 = (((1 - (2 * r)) * |[b, d]|) + ((2 * r) * |[b, c]|));

 r in [.(1 / 2), 1.] implies y0 = (((1 - ((2 * r) - 1)) * |[b, c]|) + (((2 * r) - 1) * |[a, c]|))

 proof

 assume r in [.(1 / 2), 1.];

 then (1 / 2) <= r by XXREAL_1: 1;

 then

 A8: r = (1 / 2) by A7, XXREAL_0: 1;



 then

 A9: y0 = (( 0 * |[b, d]|) + |[b, c]|) by RLVECT_1:def 8

 .= (( 0. ( TOP-REAL 2)) + |[b, c]|) by RLVECT_1: 10

 .= |[b, c]| by RLVECT_1: 4;

 (((1 - ((2 * r) - 1)) * |[b, c]|) + (((2 * r) - 1) * |[a, c]|)) = ((1 * |[b, c]|) + ( 0. ( TOP-REAL 2))) by A8, RLVECT_1: 10

 .= ( |[b, c]| + ( 0. ( TOP-REAL 2))) by RLVECT_1:def 8

 .= |[b, c]| by RLVECT_1: 4;

 hence thesis by A9;

 end;

 then for r2 be Real st x = r2 holds (r2 in [. 0 , (1 / 2).] implies y0 = (((1 - (2 * r2)) * |[b, d]|) + ((2 * r2) * |[b, c]|))) & (r2 in [.(1 / 2), 1.] implies y0 = (((1 - ((2 * r2) - 1)) * |[b, c]|) + (((2 * r2) - 1) * |[a, c]|)));

 hence thesis;

 end;

 case

 A10: x in [.(1 / 2), 1.];

 then

 reconsider r = x as Real;



 A11: (1 / 2) <= r by A10, XXREAL_1: 1;

 set y0 = (((1 - ((2 * r) - 1)) * |[b, c]|) + (((2 * r) - 1) * |[a, c]|));

 r in [. 0 , (1 / 2).] implies y0 = (((1 - (2 * r)) * |[b, d]|) + ((2 * r) * |[b, c]|))

 proof

 assume r in [. 0 , (1 / 2).];

 then r <= (1 / 2) by XXREAL_1: 1;

 then

 A12: r = (1 / 2) by A11, XXREAL_0: 1;



 then

 A13: y0 = ( |[b, c]| + ( 0 * |[a, c]|)) by RLVECT_1:def 8

 .= ( |[b, c]| + ( 0. ( TOP-REAL 2))) by RLVECT_1: 10

 .= |[b, c]| by RLVECT_1: 4;

 (((1 - (2 * r)) * |[b, d]|) + ((2 * r) * |[b, c]|)) = (( 0. ( TOP-REAL 2)) + (1 * |[b, c]|)) by A12, RLVECT_1: 10

 .= (( 0. ( TOP-REAL 2)) + |[b, c]|) by RLVECT_1:def 8

 .= |[b, c]| by RLVECT_1: 4;

 hence thesis by A13;

 end;

 then for r2 be Real st x = r2 holds (r2 in [. 0 , (1 / 2).] implies y0 = (((1 - (2 * r2)) * |[b, d]|) + ((2 * r2) * |[b, c]|))) & (r2 in [.(1 / 2), 1.] implies y0 = (((1 - ((2 * r2) - 1)) * |[b, c]|) + (((2 * r2) - 1) * |[a, c]|)));

 hence thesis;

 end;

 end;

 hence thesis;

 end;

 ex f2 be Function st ( dom f2) = [. 0 , 1.] & for x be object st x in [. 0 , 1.] holds P[x, (f2 . x)] from CLASSES1:sch 1( A4);

 then

 consider f2 be Function such that

 A14: ( dom f2) = [. 0 , 1.] and

 A15: for x be object st x in [. 0 , 1.] holds P[x, (f2 . x)];

 ( rng f2) c= the carrier of (( TOP-REAL 2) | ( Lower_Arc K))

 proof

 let y be object;

 assume y in ( rng f2);

 then

 consider x be object such that

 A16: x in ( dom f2) and

 A17: y = (f2 . x) by FUNCT_1:def 3;

 now

 per cases by A3, A14, A16, XBOOLE_0:def 3;

 case

 A18: x in [. 0 , (1 / 2).];

 then

 reconsider r = x as Real;



 A19: 0 <= r by A18, XXREAL_1: 1;

 r <= (1 / 2) by A18, XXREAL_1: 1;

 then

 A20: (r * 2) <= ((1 / 2) * 2) by XREAL_1: 64;

 (f2 . x) = (((1 - (2 * r)) * |[b, d]|) + ((2 * r) * |[b, c]|)) by A14, A15, A16, A18;

 then

 A21: y in ( LSeg ( |[b, d]|, |[b, c]|)) by A17, A19, A20;

 ( Lower_Arc K) = (( LSeg ( |[a, c]|, |[b, c]|)) \/ ( LSeg ( |[b, c]|, |[b, d]|))) by A1, A2, Th52;

 then y in ( Lower_Arc K) by A21, XBOOLE_0:def 3;

 hence thesis by PRE_TOPC: 8;

 end;

 case

 A22: x in [.(1 / 2), 1.];

 then

 reconsider r = x as Real;



 A23: (1 / 2) <= r by A22, XXREAL_1: 1;



 A24: r <= 1 by A22, XXREAL_1: 1;

 (r * 2) >= ((1 / 2) * 2) by A23, XREAL_1: 64;

 then

 A25: ((2 * r) - 1) >= 0 by XREAL_1: 48;

 (2 * 1) >= (2 * r) by A24, XREAL_1: 64;

 then

 A26: ((1 + 1) - 1) >= ((2 * r) - 1) by XREAL_1: 9;

 (f2 . x) = (((1 - ((2 * r) - 1)) * |[b, c]|) + (((2 * r) - 1) * |[a, c]|)) by A14, A15, A16, A22;

 then

 A27: y in ( LSeg ( |[b, c]|, |[a, c]|)) by A17, A25, A26;

 ( Lower_Arc K) = (( LSeg ( |[a, c]|, |[b, c]|)) \/ ( LSeg ( |[b, c]|, |[b, d]|))) by A1, A2, Th52;

 then y in ( Lower_Arc K) by A27, XBOOLE_0:def 3;

 hence thesis by PRE_TOPC: 8;

 end;

 end;

 hence thesis;

 end;

 then

 reconsider f3 = f2 as Function of I[01] , (( TOP-REAL 2) | ( Lower_Arc K)) by A14, BORSUK_1: 40, FUNCT_2: 2;



 A28: 0 in [. 0 , 1.] by XXREAL_1: 1;

 0 in [. 0 , (1 / 2).] by XXREAL_1: 1;



 then

 A29: (f3 . 0 ) = (((1 - (2 * 0 )) * |[b, d]|) + ((2 * 0 ) * |[b, c]|)) by A15, A28

 .= ((1 * |[b, d]|) + ( 0. ( TOP-REAL 2))) by RLVECT_1: 10

 .= ( |[b, d]| + ( 0. ( TOP-REAL 2))) by RLVECT_1:def 8

 .= |[b, d]| by RLVECT_1: 4

 .= ( E-max K) by A1, A2, Th46;



 A30: 1 in [. 0 , 1.] by XXREAL_1: 1;

 1 in [.(1 / 2), 1.] by XXREAL_1: 1;



 then

 A31: (f3 . 1) = (((1 - ((2 * 1) - 1)) * |[b, c]|) + (((2 * 1) - 1) * |[a, c]|)) by A15, A30

 .= (( 0 * |[b, c]|) + |[a, c]|) by RLVECT_1:def 8

 .= (( 0. ( TOP-REAL 2)) + |[a, c]|) by RLVECT_1: 10

 .= |[a, c]| by RLVECT_1: 4

 .= ( W-min K) by A1, A2, Th46;



 A32: for r be Real st r in [. 0 , (1 / 2).] holds (f3 . r) = (((1 - (2 * r)) * |[b, d]|) + ((2 * r) * |[b, c]|))

 proof

 let r be Real;

 assume

 A33: r in [. 0 , (1 / 2).];

 then

 A34: 0 <= r by XXREAL_1: 1;

 r <= (1 / 2) by A33, XXREAL_1: 1;

 then r <= 1 by XXREAL_0: 2;

 then r in [. 0 , 1.] by A34, XXREAL_1: 1;

 hence thesis by A15, A33;

 end;



 A35: for r be Real st r in [.(1 / 2), 1.] holds (f3 . r) = (((1 - ((2 * r) - 1)) * |[b, c]|) + (((2 * r) - 1) * |[a, c]|))

 proof

 let r be Real;

 assume

 A36: r in [.(1 / 2), 1.];

 then

 A37: (1 / 2) <= r by XXREAL_1: 1;

 r <= 1 by A36, XXREAL_1: 1;

 then r in [. 0 , 1.] by A37, XXREAL_1: 1;

 hence thesis by A15, A36;

 end;



 A38: for p be Point of ( TOP-REAL 2) st p in ( LSeg ( |[b, d]|, |[b, c]|)) holds 0 <= ((((p `2 ) - d) / (c - d)) / 2) & ((((p `2 ) - d) / (c - d)) / 2) <= 1 & (f3 . ((((p `2 ) - d) / (c - d)) / 2)) = p

 proof

 let p be Point of ( TOP-REAL 2);

 assume

 A39: p in ( LSeg ( |[b, d]|, |[b, c]|));



 A40: ( |[b, d]| `2 ) = d by EUCLID: 52;



 A41: ( |[b, c]| `2 ) = c by EUCLID: 52;

 then

 A42: c <= (p `2 ) by A2, A39, A40, TOPREAL1: 4;



 A43: (p `2 ) <= d by A2, A39, A40, A41, TOPREAL1: 4;

 (d - c) > 0 by A2, XREAL_1: 50;

 then

 A44: ( - (d - c)) < ( - 0 ) by XREAL_1: 24;

 (d - (p `2 )) >= 0 by A43, XREAL_1: 48;

 then

 A45: ( - (d - (p `2 ))) <= ( - 0 );

 ((p `2 ) - d) >= (c - d) by A42, XREAL_1: 9;

 then (((p `2 ) - d) / (c - d)) <= ((c - d) / (c - d)) by A44, XREAL_1: 73;

 then (((p `2 ) - d) / (c - d)) <= 1 by A44, XCMPLX_1: 60;

 then

 A46: ((((p `2 ) - d) / (c - d)) / 2) <= (1 / 2) by XREAL_1: 72;

 set r = ((((p `2 ) - d) / (c - d)) / 2);

 r in [. 0 , (1 / 2).] by A44, A45, A46, XXREAL_1: 1;



 then (f3 . ((((p `2 ) - d) / (c - d)) / 2)) = (((1 - (2 * r)) * |[b, d]|) + ((2 * r) * |[b, c]|)) by A32

 .= ( |[((1 - (2 * r)) * b), ((1 - (2 * r)) * d)]| + ((2 * r) * |[b, c]|)) by EUCLID: 58

 .= ( |[((1 - (2 * r)) * b), ((1 - (2 * r)) * d)]| + |[((2 * r) * b), ((2 * r) * c)]|) by EUCLID: 58

 .= |[(((1 * b) - ((2 * r) * b)) + ((2 * r) * b)), (((1 - (2 * r)) * d) + ((2 * r) * c))]| by EUCLID: 56

 .= |[b, ((1 * d) + ((((p `2 ) - d) / (c - d)) * (c - d)))]|

 .= |[b, ((1 * d) + ((p `2 ) - d))]| by A44, XCMPLX_1: 87

 .= |[(p `1 ), (p `2 )]| by A39, TOPREAL3: 11

 .= p by EUCLID: 53;

 hence thesis by A44, A45, A46, XXREAL_0: 2;

 end;



 A47: for p be Point of ( TOP-REAL 2) st p in ( LSeg ( |[b, c]|, |[a, c]|)) holds 0 <= (((((p `1 ) - b) / (a - b)) / 2) + (1 / 2)) & (((((p `1 ) - b) / (a - b)) / 2) + (1 / 2)) <= 1 & (f3 . (((((p `1 ) - b) / (a - b)) / 2) + (1 / 2))) = p

 proof

 let p be Point of ( TOP-REAL 2);

 assume

 A48: p in ( LSeg ( |[b, c]|, |[a, c]|));



 A49: ( |[b, c]| `1 ) = b by EUCLID: 52;



 A50: ( |[a, c]| `1 ) = a by EUCLID: 52;

 then

 A51: a <= (p `1 ) by A1, A48, A49, TOPREAL1: 3;



 A52: (p `1 ) <= b by A1, A48, A49, A50, TOPREAL1: 3;

 (b - a) > 0 by A1, XREAL_1: 50;

 then

 A53: ( - (b - a)) < ( - 0 ) by XREAL_1: 24;

 (b - (p `1 )) >= 0 by A52, XREAL_1: 48;

 then

 A54: ( - (b - (p `1 ))) <= ( - 0 );

 then

 A55: (((((p `1 ) - b) / (a - b)) / 2) + (1 / 2)) >= ( 0 + (1 / 2)) by A53, XREAL_1: 7;

 ((p `1 ) - b) >= (a - b) by A51, XREAL_1: 9;

 then (((p `1 ) - b) / (a - b)) <= ((a - b) / (a - b)) by A53, XREAL_1: 73;

 then (((p `1 ) - b) / (a - b)) <= 1 by A53, XCMPLX_1: 60;

 then ((((p `1 ) - b) / (a - b)) / 2) <= (1 / 2) by XREAL_1: 72;

 then

 A56: (((((p `1 ) - b) / (a - b)) / 2) + (1 / 2)) <= ((1 / 2) + (1 / 2)) by XREAL_1: 7;

 set r = (((((p `1 ) - b) / (a - b)) / 2) + (1 / 2));

 r in [.(1 / 2), 1.] by A55, A56, XXREAL_1: 1;



 then (f3 . (((((p `1 ) - b) / (a - b)) / 2) + (1 / 2))) = (((1 - ((2 * r) - 1)) * |[b, c]|) + (((2 * r) - 1) * |[a, c]|)) by A35

 .= ( |[((1 - ((2 * r) - 1)) * b), ((1 - ((2 * r) - 1)) * c)]| + (((2 * r) - 1) * |[a, c]|)) by EUCLID: 58

 .= ( |[((1 - ((2 * r) - 1)) * b), ((1 - ((2 * r) - 1)) * c)]| + |[(((2 * r) - 1) * a), (((2 * r) - 1) * c)]|) by EUCLID: 58

 .= |[(((1 - ((2 * r) - 1)) * b) + (((2 * r) - 1) * a)), (((1 * c) - (((2 * r) - 1) * c)) + (((2 * r) - 1) * c))]| by EUCLID: 56

 .= |[((1 * b) + ((((p `1 ) - b) / (a - b)) * (a - b))), c]|

 .= |[((1 * b) + ((p `1 ) - b)), c]| by A53, XCMPLX_1: 87

 .= |[(p `1 ), (p `2 )]| by A48, TOPREAL3: 12

 .= p by EUCLID: 53;

 hence thesis by A53, A54, A56;

 end;

 reconsider B00 = [. 0 , 1.] as Subset of R^1 by TOPMETR: 17;

 reconsider B01 = B00 as non empty Subset of R^1 by XXREAL_1: 1;

 I[01] = ( R^1 | B01) by TOPMETR: 19, TOPMETR: 20;

 then

 consider h1 be Function of I[01] , R^1 such that

 A57: for p be Point of I[01] holds (h1 . p) = p and

 A58: h1 is continuous by Th6;

 consider h2 be Function of I[01] , R^1 such that

 A59: for p be Point of I[01] , r1 be Real st (h1 . p) = r1 holds (h2 . p) = (2 * r1) and

 A60: h2 is continuous by A58, JGRAPH_2: 23;

 consider h5 be Function of I[01] , R^1 such that

 A61: for p be Point of I[01] , r1 be Real st (h2 . p) = r1 holds (h5 . p) = (1 - r1) and

 A62: h5 is continuous by A60, Th8;

 consider h3 be Function of I[01] , R^1 such that

 A63: for p be Point of I[01] , r1 be Real st (h2 . p) = r1 holds (h3 . p) = (r1 - 1) and

 A64: h3 is continuous by A60, Th7;

 consider h4 be Function of I[01] , R^1 such that

 A65: for p be Point of I[01] , r1 be Real st (h3 . p) = r1 holds (h4 . p) = (1 - r1) and

 A66: h4 is continuous by A64, Th8;

 consider g1 be Function of I[01] , ( TOP-REAL 2) such that

 A67: for r be Point of I[01] holds (g1 . r) = (((h5 . r) * |[b, d]|) + ((h2 . r) * |[b, c]|)) and

 A68: g1 is continuous by A60, A62, Th13;



 A69: for r be Point of I[01] , s be Real st r = s holds (g1 . r) = (((1 - (2 * s)) * |[b, d]|) + ((2 * s) * |[b, c]|))

 proof

 let r be Point of I[01] , s be Real;

 assume

 A70: r = s;

 (g1 . r) = (((h5 . r) * |[b, d]|) + ((h2 . r) * |[b, c]|)) by A67

 .= (((1 - (2 * (h1 . r))) * |[b, d]|) + ((h2 . r) * |[b, c]|)) by A59, A61

 .= (((1 - (2 * (h1 . r))) * |[b, d]|) + ((2 * (h1 . r)) * |[b, c]|)) by A59

 .= (((1 - (2 * s)) * |[b, d]|) + ((2 * (h1 . r)) * |[b, c]|)) by A57, A70

 .= (((1 - (2 * s)) * |[b, d]|) + ((2 * s) * |[b, c]|)) by A57, A70;

 hence thesis;

 end;

 consider g2 be Function of I[01] , ( TOP-REAL 2) such that

 A71: for r be Point of I[01] holds (g2 . r) = (((h4 . r) * |[b, c]|) + ((h3 . r) * |[a, c]|)) and

 A72: g2 is continuous by A64, A66, Th13;



 A73: for r be Point of I[01] , s be Real st r = s holds (g2 . r) = (((1 - ((2 * s) - 1)) * |[b, c]|) + (((2 * s) - 1) * |[a, c]|))

 proof

 let r be Point of I[01] , s be Real;

 assume

 A74: r = s;

 (g2 . r) = (((h4 . r) * |[b, c]|) + ((h3 . r) * |[a, c]|)) by A71

 .= (((1 - ((h2 . r) - 1)) * |[b, c]|) + ((h3 . r) * |[a, c]|)) by A63, A65

 .= (((1 - ((h2 . r) - 1)) * |[b, c]|) + (((h2 . r) - 1) * |[a, c]|)) by A63

 .= (((1 - ((2 * (h1 . r)) - 1)) * |[b, c]|) + (((h2 . r) - 1) * |[a, c]|)) by A59

 .= (((1 - ((2 * (h1 . r)) - 1)) * |[b, c]|) + (((2 * (h1 . r)) - 1) * |[a, c]|)) by A59

 .= (((1 - ((2 * s) - 1)) * |[b, c]|) + (((2 * (h1 . r)) - 1) * |[a, c]|)) by A57, A74

 .= (((1 - ((2 * s) - 1)) * |[b, c]|) + (((2 * s) - 1) * |[a, c]|)) by A57, A74;

 hence thesis;

 end;

 reconsider B11 = [. 0 , (1 / 2).] as non empty Subset of I[01] by A3, BORSUK_1: 40, XBOOLE_1: 7, XXREAL_1: 1;



 A75: ( dom (g1 | B11)) = (( dom g1) /\ B11) by RELAT_1: 61

 .= (the carrier of I[01] /\ B11) by FUNCT_2:def 1

 .= B11 by XBOOLE_1: 28

 .= the carrier of ( I[01] | B11) by PRE_TOPC: 8;

 ( rng (g1 | B11)) c= the carrier of ( TOP-REAL 2);

 then

 reconsider g11 = (g1 | B11) as Function of ( I[01] | B11), ( TOP-REAL 2) by A75, FUNCT_2: 2;



 A76: ( TOP-REAL 2) is SubSpace of ( TOP-REAL 2) by TSEP_1: 2;

 then

 A77: g11 is continuous by A68, BORSUK_4: 44;

 reconsider B22 = [.(1 / 2), 1.] as non empty Subset of I[01] by A3, BORSUK_1: 40, XBOOLE_1: 7, XXREAL_1: 1;



 A78: ( dom (g2 | B22)) = (( dom g2) /\ B22) by RELAT_1: 61

 .= (the carrier of I[01] /\ B22) by FUNCT_2:def 1

 .= B22 by XBOOLE_1: 28

 .= the carrier of ( I[01] | B22) by PRE_TOPC: 8;

 ( rng (g2 | B22)) c= the carrier of ( TOP-REAL 2);

 then

 reconsider g22 = (g2 | B22) as Function of ( I[01] | B22), ( TOP-REAL 2) by A78, FUNCT_2: 2;



 A79: g22 is continuous by A72, A76, BORSUK_4: 44;



 A80: B11 = ( [#] ( I[01] | B11)) by PRE_TOPC:def 5;



 A81: B22 = ( [#] ( I[01] | B22)) by PRE_TOPC:def 5;



 A82: B11 is closed by Th4;



 A83: B22 is closed by Th4;



 A84: (( [#] ( I[01] | B11)) \/ ( [#] ( I[01] | B22))) = ( [#] I[01] ) by A80, A81, BORSUK_1: 40, XXREAL_1: 165;

 for p be object st p in (( [#] ( I[01] | B11)) /\ ( [#] ( I[01] | B22))) holds (g11 . p) = (g22 . p)

 proof

 let p be object;

 assume

 A85: p in (( [#] ( I[01] | B11)) /\ ( [#] ( I[01] | B22)));

 then

 A86: p in ( [#] ( I[01] | B11)) by XBOOLE_0:def 4;



 A87: p in ( [#] ( I[01] | B22)) by A85;



 A88: p in B11 by A86, PRE_TOPC:def 5;



 A89: p in B22 by A87, PRE_TOPC:def 5;

 reconsider rp = p as Real by A88;



 A90: rp <= (1 / 2) by A88, XXREAL_1: 1;

 rp >= (1 / 2) by A89, XXREAL_1: 1;

 then rp = (1 / 2) by A90, XXREAL_0: 1;

 then

 A91: (2 * rp) = 1;



 thus (g11 . p) = (g1 . p) by A88, FUNCT_1: 49

 .= (((1 - 1) * |[b, d]|) + (1 * |[b, c]|)) by A69, A88, A91

 .= (( 0. ( TOP-REAL 2)) + (1 * |[b, c]|)) by RLVECT_1: 10

 .= (((1 - 0 ) * |[b, c]|) + ((1 - 1) * |[a, c]|)) by RLVECT_1: 10

 .= (g2 . p) by A73, A88, A91

 .= (g22 . p) by A89, FUNCT_1: 49;

 end;

 then

 consider h be Function of I[01] , ( TOP-REAL 2) such that

 A92: h = (g11 +* g22) and

 A93: h is continuous by A77, A79, A80, A81, A82, A83, A84, JGRAPH_2: 1;



 A94: ( dom f3) = ( dom h) by Th5;



 A95: ( dom f3) = the carrier of I[01] by Th5;

 for x be object st x in ( dom f2) holds (f3 . x) = (h . x)

 proof

 let x be object;

 assume

 A96: x in ( dom f2);

 then

 reconsider rx = x as Real by A95;



 A97: 0 <= rx by A94, A96, BORSUK_1: 40, XXREAL_1: 1;



 A98: rx <= 1 by A94, A96, BORSUK_1: 40, XXREAL_1: 1;

 per cases ;

 suppose

 A99: rx < (1 / 2);

 then

 A100: rx in [. 0 , (1 / 2).] by A97, XXREAL_1: 1;

 not rx in ( dom g22) by A81, A99, XXREAL_1: 1;



 then (h . rx) = (g11 . rx) by A92, FUNCT_4: 11

 .= (g1 . rx) by A100, FUNCT_1: 49

 .= (((1 - (2 * rx)) * |[b, d]|) + ((2 * rx) * |[b, c]|)) by A69, A94, A96

 .= (f3 . rx) by A32, A100;

 hence thesis;

 end;

 suppose rx >= (1 / 2);

 then

 A101: rx in [.(1 / 2), 1.] by A98, XXREAL_1: 1;

 then rx in ( [#] ( I[01] | B22)) by PRE_TOPC:def 5;



 then (h . rx) = (g22 . rx) by A78, A92, FUNCT_4: 13

 .= (g2 . rx) by A101, FUNCT_1: 49

 .= (((1 - ((2 * rx) - 1)) * |[b, c]|) + (((2 * rx) - 1) * |[a, c]|)) by A73, A94, A96

 .= (f3 . rx) by A35, A101;

 hence thesis;

 end;

 end;

 then

 A102: f2 = h by A94, FUNCT_1: 2;

 for x1,x2 be object st x1 in ( dom f3) & x2 in ( dom f3) & (f3 . x1) = (f3 . x2) holds x1 = x2

 proof

 let x1,x2 be object;

 assume that

 A103: x1 in ( dom f3) and

 A104: x2 in ( dom f3) and

 A105: (f3 . x1) = (f3 . x2);

 reconsider r1 = x1 as Real by A103;

 reconsider r2 = x2 as Real by A104;



 A106: (( LSeg ( |[b, d]|, |[b, c]|)) /\ ( LSeg ( |[b, c]|, |[a, c]|))) = { |[b, c]|} by A1, A2, Th32;

 now

 per cases by A3, A14, A103, A104, XBOOLE_0:def 3;

 case

 A107: x1 in [. 0 , (1 / 2).] & x2 in [. 0 , (1 / 2).];

 then (f3 . r1) = (((1 - (2 * r1)) * |[b, d]|) + ((2 * r1) * |[b, c]|)) by A32;

 then (((1 - (2 * r2)) * |[b, d]|) + ((2 * r2) * |[b, c]|)) = (((1 - (2 * r1)) * |[b, d]|) + ((2 * r1) * |[b, c]|)) by A32, A105, A107;

 then ((((1 - (2 * r2)) * |[b, d]|) + ((2 * r2) * |[b, c]|)) - ((2 * r1) * |[b, c]|)) = ((1 - (2 * r1)) * |[b, d]|) by RLVECT_4: 1;

 then (((1 - (2 * r2)) * |[b, d]|) + (((2 * r2) * |[b, c]|) - ((2 * r1) * |[b, c]|))) = ((1 - (2 * r1)) * |[b, d]|) by RLVECT_1:def 3;

 then (((1 - (2 * r2)) * |[b, d]|) + (((2 * r2) - (2 * r1)) * |[b, c]|)) = ((1 - (2 * r1)) * |[b, d]|) by RLVECT_1: 35;

 then ((((2 * r2) - (2 * r1)) * |[b, c]|) + (((1 - (2 * r2)) * |[b, d]|) - ((1 - (2 * r1)) * |[b, d]|))) = (((1 - (2 * r1)) * |[b, d]|) - ((1 - (2 * r1)) * |[b, d]|)) by RLVECT_1:def 3;

 then ((((2 * r2) - (2 * r1)) * |[b, c]|) + (((1 - (2 * r2)) * |[b, d]|) - ((1 - (2 * r1)) * |[b, d]|))) = ( 0. ( TOP-REAL 2)) by RLVECT_1: 5;

 then ((((2 * r2) - (2 * r1)) * |[b, c]|) + (((1 - (2 * r2)) - (1 - (2 * r1))) * |[b, d]|)) = ( 0. ( TOP-REAL 2)) by RLVECT_1: 35;

 then ((((2 * r2) - (2 * r1)) * |[b, c]|) + (( - ((2 * r2) - (2 * r1))) * |[b, d]|)) = ( 0. ( TOP-REAL 2));

 then ((((2 * r2) - (2 * r1)) * |[b, c]|) + ( - (((2 * r2) - (2 * r1)) * |[b, d]|))) = ( 0. ( TOP-REAL 2)) by RLVECT_1: 79;

 then ((((2 * r2) - (2 * r1)) * |[b, c]|) - (((2 * r2) - (2 * r1)) * |[b, d]|)) = ( 0. ( TOP-REAL 2));

 then (((2 * r2) - (2 * r1)) * ( |[b, c]| - |[b, d]|)) = ( 0. ( TOP-REAL 2)) by RLVECT_1: 34;

 then ((2 * r2) - (2 * r1)) = 0 or ( |[b, c]| - |[b, d]|) = ( 0. ( TOP-REAL 2)) by RLVECT_1: 11;

 then ((2 * r2) - (2 * r1)) = 0 or |[b, c]| = |[b, d]| by RLVECT_1: 21;

 then ((2 * r2) - (2 * r1)) = 0 or d = ( |[b, c]| `2 ) by EUCLID: 52;

 hence thesis by A2, EUCLID: 52;

 end;

 case

 A108: x1 in [. 0 , (1 / 2).] & x2 in [.(1 / 2), 1.];

 then

 A109: (f3 . r1) = (((1 - (2 * r1)) * |[b, d]|) + ((2 * r1) * |[b, c]|)) by A32;



 A110: 0 <= r1 by A108, XXREAL_1: 1;

 r1 <= (1 / 2) by A108, XXREAL_1: 1;

 then (r1 * 2) <= ((1 / 2) * 2) by XREAL_1: 64;

 then

 A111: (f3 . r1) in ( LSeg ( |[b, d]|, |[b, c]|)) by A109, A110;



 A112: (f3 . r2) = (((1 - ((2 * r2) - 1)) * |[b, c]|) + (((2 * r2) - 1) * |[a, c]|)) by A35, A108;



 A113: (1 / 2) <= r2 by A108, XXREAL_1: 1;



 A114: r2 <= 1 by A108, XXREAL_1: 1;

 (r2 * 2) >= ((1 / 2) * 2) by A113, XREAL_1: 64;

 then

 A115: ((2 * r2) - 1) >= 0 by XREAL_1: 48;

 (2 * 1) >= (2 * r2) by A114, XREAL_1: 64;

 then ((1 + 1) - 1) >= ((2 * r2) - 1) by XREAL_1: 9;

 then (f3 . r2) in { (((1 - lambda) * |[b, c]|) + (lambda * |[a, c]|)) where lambda be Real : 0 <= lambda & lambda <= 1 } by A112, A115;

 then (f3 . r1) in (( LSeg ( |[b, d]|, |[b, c]|)) /\ ( LSeg ( |[b, c]|, |[a, c]|))) by A105, A111, XBOOLE_0:def 4;

 then

 A116: (f3 . r1) = |[b, c]| by A106, TARSKI:def 1;

 then ((((1 - (2 * r1)) * |[b, d]|) + ((2 * r1) * |[b, c]|)) + ( - |[b, c]|)) = ( 0. ( TOP-REAL 2)) by A109, RLVECT_1: 5;

 then ((((1 - (2 * r1)) * |[b, d]|) + ((2 * r1) * |[b, c]|)) + (( - 1) * |[b, c]|)) = ( 0. ( TOP-REAL 2)) by RLVECT_1: 16;

 then (((1 - (2 * r1)) * |[b, d]|) + (((2 * r1) * |[b, c]|) + (( - 1) * |[b, c]|))) = ( 0. ( TOP-REAL 2)) by RLVECT_1:def 3;

 then (((1 - (2 * r1)) * |[b, d]|) + (((2 * r1) + ( - 1)) * |[b, c]|)) = ( 0. ( TOP-REAL 2)) by RLVECT_1:def 6;

 then (((1 - (2 * r1)) * |[b, d]|) + (( - (1 - (2 * r1))) * |[b, c]|)) = ( 0. ( TOP-REAL 2));

 then (((1 - (2 * r1)) * |[b, d]|) + ( - ((1 - (2 * r1)) * |[b, c]|))) = ( 0. ( TOP-REAL 2)) by RLVECT_1: 79;

 then (((1 - (2 * r1)) * |[b, d]|) - ((1 - (2 * r1)) * |[b, c]|)) = ( 0. ( TOP-REAL 2));

 then ((1 - (2 * r1)) * ( |[b, d]| - |[b, c]|)) = ( 0. ( TOP-REAL 2)) by RLVECT_1: 34;

 then (1 - (2 * r1)) = 0 or ( |[b, d]| - |[b, c]|) = ( 0. ( TOP-REAL 2)) by RLVECT_1: 11;

 then (1 - (2 * r1)) = 0 or |[b, d]| = |[b, c]| by RLVECT_1: 21;

 then

 A117: (1 - (2 * r1)) = 0 or d = ( |[b, c]| `2 ) by EUCLID: 52;

 ((((1 - ((2 * r2) - 1)) * |[b, c]|) + (((2 * r2) - 1) * |[a, c]|)) + ( - |[b, c]|)) = ( 0. ( TOP-REAL 2)) by A105, A112, A116, RLVECT_1: 5;

 then ((((1 - ((2 * r2) - 1)) * |[b, c]|) + (((2 * r2) - 1) * |[a, c]|)) + (( - 1) * |[b, c]|)) = ( 0. ( TOP-REAL 2)) by RLVECT_1: 16;

 then ((((2 * r2) - 1) * |[a, c]|) + (((1 - ((2 * r2) - 1)) * |[b, c]|) + (( - 1) * |[b, c]|))) = ( 0. ( TOP-REAL 2)) by RLVECT_1:def 3;

 then ((((2 * r2) - 1) * |[a, c]|) + (((1 - ((2 * r2) - 1)) + ( - 1)) * |[b, c]|)) = ( 0. ( TOP-REAL 2)) by RLVECT_1:def 6;

 then ((((2 * r2) - 1) * |[a, c]|) + (( - ((2 * r2) - 1)) * |[b, c]|)) = ( 0. ( TOP-REAL 2));

 then ((((2 * r2) - 1) * |[a, c]|) + ( - (((2 * r2) - 1) * |[b, c]|))) = ( 0. ( TOP-REAL 2)) by RLVECT_1: 79;

 then ((((2 * r2) - 1) * |[a, c]|) - (((2 * r2) - 1) * |[b, c]|)) = ( 0. ( TOP-REAL 2));

 then (((2 * r2) - 1) * ( |[a, c]| - |[b, c]|)) = ( 0. ( TOP-REAL 2)) by RLVECT_1: 34;

 then ((2 * r2) - 1) = 0 or ( |[a, c]| - |[b, c]|) = ( 0. ( TOP-REAL 2)) by RLVECT_1: 11;

 then ((2 * r2) - 1) = 0 or |[a, c]| = |[b, c]| by RLVECT_1: 21;

 then ((2 * r2) - 1) = 0 or a = ( |[b, c]| `1 ) by EUCLID: 52;

 hence thesis by A1, A2, A117, EUCLID: 52;

 end;

 case

 A118: x1 in [.(1 / 2), 1.] & x2 in [. 0 , (1 / 2).];

 then

 A119: (f3 . r2) = (((1 - (2 * r2)) * |[b, d]|) + ((2 * r2) * |[b, c]|)) by A32;



 A120: 0 <= r2 by A118, XXREAL_1: 1;

 r2 <= (1 / 2) by A118, XXREAL_1: 1;

 then (r2 * 2) <= ((1 / 2) * 2) by XREAL_1: 64;

 then

 A121: (f3 . r2) in ( LSeg ( |[b, d]|, |[b, c]|)) by A119, A120;



 A122: (f3 . r1) = (((1 - ((2 * r1) - 1)) * |[b, c]|) + (((2 * r1) - 1) * |[a, c]|)) by A35, A118;



 A123: (1 / 2) <= r1 by A118, XXREAL_1: 1;



 A124: r1 <= 1 by A118, XXREAL_1: 1;

 (r1 * 2) >= ((1 / 2) * 2) by A123, XREAL_1: 64;

 then

 A125: ((2 * r1) - 1) >= 0 by XREAL_1: 48;

 (2 * 1) >= (2 * r1) by A124, XREAL_1: 64;

 then ((1 + 1) - 1) >= ((2 * r1) - 1) by XREAL_1: 9;

 then (f3 . r1) in { (((1 - lambda) * |[b, c]|) + (lambda * |[a, c]|)) where lambda be Real : 0 <= lambda & lambda <= 1 } by A122, A125;

 then (f3 . r2) in (( LSeg ( |[b, d]|, |[b, c]|)) /\ ( LSeg ( |[b, c]|, |[a, c]|))) by A105, A121, XBOOLE_0:def 4;

 then

 A126: (f3 . r2) = |[b, c]| by A106, TARSKI:def 1;

 then ((((1 - (2 * r2)) * |[b, d]|) + ((2 * r2) * |[b, c]|)) + ( - |[b, c]|)) = ( 0. ( TOP-REAL 2)) by A119, RLVECT_1: 5;

 then ((((1 - (2 * r2)) * |[b, d]|) + ((2 * r2) * |[b, c]|)) + (( - 1) * |[b, c]|)) = ( 0. ( TOP-REAL 2)) by RLVECT_1: 16;

 then (((1 - (2 * r2)) * |[b, d]|) + (((2 * r2) * |[b, c]|) + (( - 1) * |[b, c]|))) = ( 0. ( TOP-REAL 2)) by RLVECT_1:def 3;

 then (((1 - (2 * r2)) * |[b, d]|) + (((2 * r2) + ( - 1)) * |[b, c]|)) = ( 0. ( TOP-REAL 2)) by RLVECT_1:def 6;

 then (((1 - (2 * r2)) * |[b, d]|) + (( - (1 - (2 * r2))) * |[b, c]|)) = ( 0. ( TOP-REAL 2));

 then (((1 - (2 * r2)) * |[b, d]|) + ( - ((1 - (2 * r2)) * |[b, c]|))) = ( 0. ( TOP-REAL 2)) by RLVECT_1: 79;

 then (((1 - (2 * r2)) * |[b, d]|) - ((1 - (2 * r2)) * |[b, c]|)) = ( 0. ( TOP-REAL 2));

 then ((1 - (2 * r2)) * ( |[b, d]| - |[b, c]|)) = ( 0. ( TOP-REAL 2)) by RLVECT_1: 34;

 then (1 - (2 * r2)) = 0 or ( |[b, d]| - |[b, c]|) = ( 0. ( TOP-REAL 2)) by RLVECT_1: 11;

 then (1 - (2 * r2)) = 0 or |[b, d]| = |[b, c]| by RLVECT_1: 21;

 then

 A127: (1 - (2 * r2)) = 0 or d = ( |[b, c]| `2 ) by EUCLID: 52;

 ((((1 - ((2 * r1) - 1)) * |[b, c]|) + (((2 * r1) - 1) * |[a, c]|)) + ( - |[b, c]|)) = ( 0. ( TOP-REAL 2)) by A105, A122, A126, RLVECT_1: 5;

 then ((((1 - ((2 * r1) - 1)) * |[b, c]|) + (((2 * r1) - 1) * |[a, c]|)) + (( - 1) * |[b, c]|)) = ( 0. ( TOP-REAL 2)) by RLVECT_1: 16;

 then ((((2 * r1) - 1) * |[a, c]|) + (((1 - ((2 * r1) - 1)) * |[b, c]|) + (( - 1) * |[b, c]|))) = ( 0. ( TOP-REAL 2)) by RLVECT_1:def 3;

 then ((((2 * r1) - 1) * |[a, c]|) + (((1 - ((2 * r1) - 1)) + ( - 1)) * |[b, c]|)) = ( 0. ( TOP-REAL 2)) by RLVECT_1:def 6;

 then ((((2 * r1) - 1) * |[a, c]|) + (( - ((2 * r1) - 1)) * |[b, c]|)) = ( 0. ( TOP-REAL 2));

 then ((((2 * r1) - 1) * |[a, c]|) + ( - (((2 * r1) - 1) * |[b, c]|))) = ( 0. ( TOP-REAL 2)) by RLVECT_1: 79;

 then ((((2 * r1) - 1) * |[a, c]|) - (((2 * r1) - 1) * |[b, c]|)) = ( 0. ( TOP-REAL 2));

 then (((2 * r1) - 1) * ( |[a, c]| - |[b, c]|)) = ( 0. ( TOP-REAL 2)) by RLVECT_1: 34;

 then ((2 * r1) - 1) = 0 or ( |[a, c]| - |[b, c]|) = ( 0. ( TOP-REAL 2)) by RLVECT_1: 11;

 then ((2 * r1) - 1) = 0 or |[a, c]| = |[b, c]| by RLVECT_1: 21;

 then ((2 * r1) - 1) = 0 or a = ( |[b, c]| `1 ) by EUCLID: 52;

 hence thesis by A1, A2, A127, EUCLID: 52;

 end;

 case

 A128: x1 in [.(1 / 2), 1.] & x2 in [.(1 / 2), 1.];

 then (f3 . r1) = (((1 - ((2 * r1) - 1)) * |[b, c]|) + (((2 * r1) - 1) * |[a, c]|)) by A35;

 then (((1 - ((2 * r2) - 1)) * |[b, c]|) + (((2 * r2) - 1) * |[a, c]|)) = (((1 - ((2 * r1) - 1)) * |[b, c]|) + (((2 * r1) - 1) * |[a, c]|)) by A35, A105, A128;

 then ((((1 - ((2 * r2) - 1)) * |[b, c]|) + (((2 * r2) - 1) * |[a, c]|)) - (((2 * r1) - 1) * |[a, c]|)) = ((1 - ((2 * r1) - 1)) * |[b, c]|) by RLVECT_4: 1;

 then (((1 - ((2 * r2) - 1)) * |[b, c]|) + ((((2 * r2) - 1) * |[a, c]|) - (((2 * r1) - 1) * |[a, c]|))) = ((1 - ((2 * r1) - 1)) * |[b, c]|) by RLVECT_1:def 3;

 then (((1 - ((2 * r2) - 1)) * |[b, c]|) + ((((2 * r2) - 1) - ((2 * r1) - 1)) * |[a, c]|)) = ((1 - ((2 * r1) - 1)) * |[b, c]|) by RLVECT_1: 35;

 then (((((2 * r2) - 1) - ((2 * r1) - 1)) * |[a, c]|) + (((1 - ((2 * r2) - 1)) * |[b, c]|) - ((1 - ((2 * r1) - 1)) * |[b, c]|))) = (((1 - ((2 * r1) - 1)) * |[b, c]|) - ((1 - ((2 * r1) - 1)) * |[b, c]|)) by RLVECT_1:def 3;

 then (((((2 * r2) - 1) - ((2 * r1) - 1)) * |[a, c]|) + (((1 - ((2 * r2) - 1)) * |[b, c]|) - ((1 - ((2 * r1) - 1)) * |[b, c]|))) = ( 0. ( TOP-REAL 2)) by RLVECT_1: 5;

 then (((((2 * r2) - 1) - ((2 * r1) - 1)) * |[a, c]|) + (((1 - ((2 * r2) - 1)) - (1 - ((2 * r1) - 1))) * |[b, c]|)) = ( 0. ( TOP-REAL 2)) by RLVECT_1: 35;

 then (((((2 * r2) - 1) - ((2 * r1) - 1)) * |[a, c]|) + (( - (((2 * r2) - 1) - ((2 * r1) - 1))) * |[b, c]|)) = ( 0. ( TOP-REAL 2));

 then (((((2 * r2) - 1) - ((2 * r1) - 1)) * |[a, c]|) + ( - ((((2 * r2) - 1) - ((2 * r1) - 1)) * |[b, c]|))) = ( 0. ( TOP-REAL 2)) by RLVECT_1: 79;

 then (((((2 * r2) - 1) - ((2 * r1) - 1)) * |[a, c]|) - ((((2 * r2) - 1) - ((2 * r1) - 1)) * |[b, c]|)) = ( 0. ( TOP-REAL 2));

 then ((((2 * r2) - 1) - ((2 * r1) - 1)) * ( |[a, c]| - |[b, c]|)) = ( 0. ( TOP-REAL 2)) by RLVECT_1: 34;

 then (((2 * r2) - 1) - ((2 * r1) - 1)) = 0 or ( |[a, c]| - |[b, c]|) = ( 0. ( TOP-REAL 2)) by RLVECT_1: 11;

 then (((2 * r2) - 1) - ((2 * r1) - 1)) = 0 or |[a, c]| = |[b, c]| by RLVECT_1: 21;

 then (((2 * r2) - 1) - ((2 * r1) - 1)) = 0 or a = ( |[b, c]| `1 ) by EUCLID: 52;

 hence thesis by A1, EUCLID: 52;

 end;

 end;

 hence thesis;

 end;

 then

 A129: f3 is one-to-one by FUNCT_1:def 4;

 ( [#] (( TOP-REAL 2) | ( Lower_Arc K))) c= ( rng f3)

 proof

 let y be object;

 assume y in ( [#] (( TOP-REAL 2) | ( Lower_Arc K)));

 then

 A130: y in ( Lower_Arc K) by PRE_TOPC:def 5;

 then

 reconsider q = y as Point of ( TOP-REAL 2);



 A131: ( Lower_Arc K) = (( LSeg ( |[b, d]|, |[b, c]|)) \/ ( LSeg ( |[b, c]|, |[a, c]|))) by A1, A2, Th52;

 now

 per cases by A130, A131, XBOOLE_0:def 3;

 case

 A132: q in ( LSeg ( |[b, d]|, |[b, c]|));

 then

 A133: 0 <= ((((q `2 ) - d) / (c - d)) / 2) by A38;



 A134: ((((q `2 ) - d) / (c - d)) / 2) <= 1 by A38, A132;



 A135: (f3 . ((((q `2 ) - d) / (c - d)) / 2)) = q by A38, A132;

 ((((q `2 ) - d) / (c - d)) / 2) in [. 0 , 1.] by A133, A134, XXREAL_1: 1;

 hence thesis by A14, A135, FUNCT_1:def 3;

 end;

 case

 A136: q in ( LSeg ( |[b, c]|, |[a, c]|));

 then

 A137: 0 <= (((((q `1 ) - b) / (a - b)) / 2) + (1 / 2)) by A47;



 A138: (((((q `1 ) - b) / (a - b)) / 2) + (1 / 2)) <= 1 by A47, A136;



 A139: (f3 . (((((q `1 ) - b) / (a - b)) / 2) + (1 / 2))) = q by A47, A136;

 (((((q `1 ) - b) / (a - b)) / 2) + (1 / 2)) in [. 0 , 1.] by A137, A138, XXREAL_1: 1;

 hence thesis by A14, A139, FUNCT_1:def 3;

 end;

 end;

 hence thesis;

 end;

 then

 A140: ( rng f3) = ( [#] (( TOP-REAL 2) | ( Lower_Arc K)));

 I[01] is compact by HEINE: 4, TOPMETR: 20;

 then

 A141: f3 is being_homeomorphism by A93, A102, A129, A140, COMPTS_1: 17, JGRAPH_1: 45;

 ( rng f3) = ( Lower_Arc K) by A140, PRE_TOPC:def 5;

 hence thesis by A29, A31, A32, A35, A38, A47, A141;

 end;

 theorem :: JGRAPH_6:55



 Th55: for a,b,c,d be Real, p1,p2 be Point of ( TOP-REAL 2) st a < b & c < d & p1 in ( LSeg ( |[a, c]|, |[a, d]|)) & p2 in ( LSeg ( |[a, c]|, |[a, d]|)) holds LE (p1,p2,( rectangle (a,b,c,d))) iff (p1 `2 ) <= (p2 `2 )

 proof

 let a,b,c,d be Real, p1,p2 be Point of ( TOP-REAL 2);

 set K = ( rectangle (a,b,c,d));

 assume that

 A1: a < b and

 A2: c < d and

 A3: p1 in ( LSeg ( |[a, c]|, |[a, d]|)) and

 A4: p2 in ( LSeg ( |[a, c]|, |[a, d]|));



 A5: K is being_simple_closed_curve by A1, A2, Th50;



 A6: (p1 `1 ) = a by A2, A3, Th1;



 A7: c <= (p1 `2 ) by A2, A3, Th1;



 A8: (p2 `1 ) = a by A2, A4, Th1;



 A9: ( E-max K) = |[b, d]| by A1, A2, Th46;



 A10: ( Upper_Arc K) = (( LSeg ( |[a, c]|, |[a, d]|)) \/ ( LSeg ( |[a, d]|, |[b, d]|))) by A1, A2, Th51;

 then

 A11: ( LSeg ( |[a, c]|, |[a, d]|)) c= ( Upper_Arc K) by XBOOLE_1: 7;



 A12: (( Upper_Arc K) /\ ( Lower_Arc K)) = {( W-min K), ( E-max K)} by A5, JORDAN6:def 9;

 A13:

 now

 assume p2 in ( Lower_Arc K);

 then

 A14: p2 in (( Upper_Arc K) /\ ( Lower_Arc K)) by A4, A11, XBOOLE_0:def 4;

 now

 assume p2 = ( E-max K);

 then (p2 `1 ) = b by A9, EUCLID: 52;

 hence contradiction by A1, A4, TOPREAL3: 11;

 end;

 hence p2 = ( W-min K) by A12, A14, TARSKI:def 2;

 end;

 thus LE (p1,p2,K) implies (p1 `2 ) <= (p2 `2 )

 proof

 assume LE (p1,p2,K);

 then

 A15: p1 in ( Upper_Arc K) & p2 in ( Lower_Arc K) & not p2 = ( W-min K) or p1 in ( Upper_Arc K) & p2 in ( Upper_Arc K) & LE (p1,p2,( Upper_Arc K),( W-min K),( E-max K)) or p1 in ( Lower_Arc K) & p2 in ( Lower_Arc K) & not p2 = ( W-min K) & LE (p1,p2,( Lower_Arc K),( E-max K),( W-min K)) by JORDAN6:def 10;

 consider f be Function of I[01] , (( TOP-REAL 2) | ( Upper_Arc K)) such that

 A16: f is being_homeomorphism and

 A17: (f . 0 ) = ( W-min K) and

 A18: (f . 1) = ( E-max K) and ( rng f) = ( Upper_Arc K) and for r be Real st r in [. 0 , (1 / 2).] holds (f . r) = (((1 - (2 * r)) * |[a, c]|) + ((2 * r) * |[a, d]|)) and for r be Real st r in [.(1 / 2), 1.] holds (f . r) = (((1 - ((2 * r) - 1)) * |[a, d]|) + (((2 * r) - 1) * |[b, d]|)) and

 A19: for p be Point of ( TOP-REAL 2) st p in ( LSeg ( |[a, c]|, |[a, d]|)) holds 0 <= ((((p `2 ) - c) / (d - c)) / 2) & ((((p `2 ) - c) / (d - c)) / 2) <= 1 & (f . ((((p `2 ) - c) / (d - c)) / 2)) = p and for p be Point of ( TOP-REAL 2) st p in ( LSeg ( |[a, d]|, |[b, d]|)) holds 0 <= (((((p `1 ) - a) / (b - a)) / 2) + (1 / 2)) & (((((p `1 ) - a) / (b - a)) / 2) + (1 / 2)) <= 1 & (f . (((((p `1 ) - a) / (b - a)) / 2) + (1 / 2))) = p by A1, A2, Th53;

 reconsider s1 = ((((p1 `2 ) - c) / (d - c)) / 2), s2 = ((((p2 `2 ) - c) / (d - c)) / 2) as Real;



 A20: (f . s1) = p1 by A3, A19;



 A21: (f . s2) = p2 by A4, A19;



 A22: (d - c) > 0 by A2, XREAL_1: 50;



 A23: s1 <= 1 by A3, A19;



 A24: 0 <= s2 by A4, A19;

 s2 <= 1 by A4, A19;

 then s1 <= s2 by A13, A15, A16, A17, A18, A20, A21, A23, A24, JORDAN5C:def 3;

 then (((((p1 `2 ) - c) / (d - c)) / 2) * 2) <= (((((p2 `2 ) - c) / (d - c)) / 2) * 2) by XREAL_1: 64;

 then ((((p1 `2 ) - c) / (d - c)) * (d - c)) <= ((((p2 `2 ) - c) / (d - c)) * (d - c)) by A22, XREAL_1: 64;

 then ((p1 `2 ) - c) <= ((((p2 `2 ) - c) / (d - c)) * (d - c)) by A22, XCMPLX_1: 87;

 then ((p1 `2 ) - c) <= ((p2 `2 ) - c) by A22, XCMPLX_1: 87;

 then (((p1 `2 ) - c) + c) <= (((p2 `2 ) - c) + c) by XREAL_1: 7;

 hence thesis;

 end;

 thus (p1 `2 ) <= (p2 `2 ) implies LE (p1,p2,K)

 proof

 assume

 A25: (p1 `2 ) <= (p2 `2 );

 for g be Function of I[01] , (( TOP-REAL 2) | ( Upper_Arc K)), s1,s2 be Real st g is being_homeomorphism & (g . 0 ) = ( W-min K) & (g . 1) = ( E-max K) & (g . s1) = p1 & 0 <= s1 & s1 <= 1 & (g . s2) = p2 & 0 <= s2 & s2 <= 1 holds s1 <= s2

 proof

 let g be Function of I[01] , (( TOP-REAL 2) | ( Upper_Arc K)), s1,s2 be Real;

 assume that

 A26: g is being_homeomorphism and

 A27: (g . 0 ) = ( W-min K) and (g . 1) = ( E-max K) and

 A28: (g . s1) = p1 and

 A29: 0 <= s1 and

 A30: s1 <= 1 and

 A31: (g . s2) = p2 and

 A32: 0 <= s2 and

 A33: s2 <= 1;



 A34: ( dom g) = the carrier of I[01] by FUNCT_2:def 1;



 A35: g is one-to-one by A26, TOPS_2:def 5;



 A36: the carrier of (( TOP-REAL 2) | ( Upper_Arc K)) = ( Upper_Arc K) by PRE_TOPC: 8;

 then

 reconsider g1 = g as Function of I[01] , ( TOP-REAL 2) by FUNCT_2: 7;

 g is continuous by A26, TOPS_2:def 5;

 then

 A37: g1 is continuous by PRE_TOPC: 26;

 reconsider h1 = proj1 as Function of ( TOP-REAL 2), R^1 by TOPMETR: 17;

 reconsider h2 = proj2 as Function of ( TOP-REAL 2), R^1 by TOPMETR: 17;

 reconsider hh1 = h1 as Function of the TopStruct of ( TOP-REAL 2), R^1 ;

 reconsider hh2 = h2 as Function of the TopStruct of ( TOP-REAL 2), R^1 ;



 A38: the TopStruct of ( TOP-REAL 2) = ( the TopStruct of ( TOP-REAL 2) | ( [#] the TopStruct of ( TOP-REAL 2))) by TSEP_1: 3

 .= the TopStruct of (( TOP-REAL 2) | ( [#] ( TOP-REAL 2))) by PRE_TOPC: 36

 .= (( TOP-REAL 2) | ( [#] ( TOP-REAL 2)));

 then (for p be Point of (( TOP-REAL 2) | ( [#] ( TOP-REAL 2))) holds (hh1 . p) = ( proj1 . p)) implies hh1 is continuous by JGRAPH_2: 29;

 then

 A39: (for p be Point of (( TOP-REAL 2) | ( [#] ( TOP-REAL 2))) holds (hh1 . p) = ( proj1 . p)) implies h1 is continuous by PRE_TOPC: 32;

 (for p be Point of (( TOP-REAL 2) | ( [#] ( TOP-REAL 2))) holds (hh2 . p) = ( proj2 . p)) implies hh2 is continuous by A38, JGRAPH_2: 30;

 then (for p be Point of (( TOP-REAL 2) | ( [#] ( TOP-REAL 2))) holds (hh2 . p) = ( proj2 . p)) implies h2 is continuous by PRE_TOPC: 32;

 then

 consider h be Function of ( TOP-REAL 2), R^1 such that

 A40: for p be Point of ( TOP-REAL 2), r1,r2 be Real st (hh1 . p) = r1 & (hh2 . p) = r2 holds (h . p) = (r1 + r2) and

 A41: h is continuous by A39, JGRAPH_2: 19;

 reconsider k = (h * g1) as Function of I[01] , R^1 ;



 A42: ( W-min K) = |[a, c]| by A1, A2, Th46;

 now

 assume

 A43: s1 > s2;



 A44: ( dom g) = [. 0 , 1.] by BORSUK_1: 40, FUNCT_2:def 1;

 0 in [. 0 , 1.] by XXREAL_1: 1;



 then

 A45: (k . 0 ) = (h . ( W-min K)) by A27, A44, FUNCT_1: 13

 .= ((h1 . ( W-min K)) + (h2 . ( W-min K))) by A40

 .= ((( W-min K) `1 ) + ( proj2 . ( W-min K))) by PSCOMP_1:def 5

 .= ((( W-min K) `1 ) + (( W-min K) `2 )) by PSCOMP_1:def 6

 .= (a + (( W-min K) `2 )) by A42, EUCLID: 52

 .= (a + c) by A42, EUCLID: 52;

 s1 in [. 0 , 1.] by A29, A30, XXREAL_1: 1;



 then

 A46: (k . s1) = (h . p1) by A28, A44, FUNCT_1: 13

 .= ((h1 . p1) + (h2 . p1)) by A40

 .= ((p1 `1 ) + ( proj2 . p1)) by PSCOMP_1:def 5

 .= (a + (p1 `2 )) by A6, PSCOMP_1:def 6;



 A47: s2 in [. 0 , 1.] by A32, A33, XXREAL_1: 1;



 then

 A48: (k . s2) = (h . p2) by A31, A44, FUNCT_1: 13

 .= ((h1 . p2) + (h2 . p2)) by A40

 .= ((p2 `1 ) + ( proj2 . p2)) by PSCOMP_1:def 5

 .= (a + (p2 `2 )) by A8, PSCOMP_1:def 6;



 A49: (k . 0 ) <= (k . s1) by A7, A45, A46, XREAL_1: 7;



 A50: (k . s1) <= (k . s2) by A25, A46, A48, XREAL_1: 7;



 A51: 0 in [. 0 , 1.] by XXREAL_1: 1;

 then

 A52: [. 0 , s2.] c= [. 0 , 1.] by A47, XXREAL_2:def 12;

 reconsider B = [. 0 , s2.] as Subset of I[01] by A47, A51, BORSUK_1: 40, XXREAL_2:def 12;



 A53: B is connected by A32, A47, A51, BORSUK_1: 40, BORSUK_4: 24;



 A54: 0 in B by A32, XXREAL_1: 1;



 A55: s2 in B by A32, XXREAL_1: 1;

 consider xc be Point of I[01] such that

 A56: xc in B and

 A57: (k . xc) = (k . s1) by A37, A41, A49, A50, A53, A54, A55, TOPREAL5: 5;

 reconsider rxc = xc as Real;



 A58: for x1,x2 be set st x1 in ( dom k) & x2 in ( dom k) & (k . x1) = (k . x2) holds x1 = x2

 proof

 let x1,x2 be set;

 assume that

 A59: x1 in ( dom k) and

 A60: x2 in ( dom k) and

 A61: (k . x1) = (k . x2);

 reconsider r1 = x1 as Point of I[01] by A59;

 reconsider r2 = x2 as Point of I[01] by A60;



 A62: (k . x1) = (h . (g1 . x1)) by A59, FUNCT_1: 12

 .= ((h1 . (g1 . r1)) + (h2 . (g1 . r1))) by A40

 .= (((g1 . r1) `1 ) + ( proj2 . (g1 . r1))) by PSCOMP_1:def 5

 .= (((g1 . r1) `1 ) + ((g1 . r1) `2 )) by PSCOMP_1:def 6;



 A63: (k . x2) = (h . (g1 . x2)) by A60, FUNCT_1: 12

 .= ((h1 . (g1 . r2)) + (h2 . (g1 . r2))) by A40

 .= (((g1 . r2) `1 ) + ( proj2 . (g1 . r2))) by PSCOMP_1:def 5

 .= (((g1 . r2) `1 ) + ((g1 . r2) `2 )) by PSCOMP_1:def 6;



 A64: (g . r1) in ( Upper_Arc K) by A36;



 A65: (g . r2) in ( Upper_Arc K) by A36;

 reconsider gr1 = (g . r1) as Point of ( TOP-REAL 2) by A64;

 reconsider gr2 = (g . r2) as Point of ( TOP-REAL 2) by A65;

 now

 per cases by A10, A36, XBOOLE_0:def 3;

 case

 A66: (g . r1) in ( LSeg ( |[a, c]|, |[a, d]|)) & (g . r2) in ( LSeg ( |[a, c]|, |[a, d]|));

 then

 A67: (gr1 `1 ) = a by A2, Th1;

 (gr2 `1 ) = a by A2, A66, Th1;

 then |[(gr1 `1 ), (gr1 `2 )]| = (g . r2) by A61, A62, A63, A67, EUCLID: 53;

 then (g . r1) = (g . r2) by EUCLID: 53;

 hence thesis by A34, A35, FUNCT_1:def 4;

 end;

 case

 A68: (g . r1) in ( LSeg ( |[a, c]|, |[a, d]|)) & (g . r2) in ( LSeg ( |[a, d]|, |[b, d]|));

 then

 A69: (gr1 `1 ) = a by A2, Th1;



 A70: (gr1 `2 ) <= d by A2, A68, Th1;



 A71: (gr2 `2 ) = d by A1, A68, Th3;



 A72: a <= (gr2 `1 ) by A1, A68, Th3;



 A73: (a + (gr1 `2 )) = ((gr2 `1 ) + d) by A1, A61, A62, A63, A68, A69, Th3;

 A74:

 now

 assume a <> (gr2 `1 );

 then a < (gr2 `1 ) by A72, XXREAL_0: 1;

 hence contradiction by A70, A73, XREAL_1: 8;

 end;

 now

 assume (gr1 `2 ) <> d;

 then d > (gr1 `2 ) by A70, XXREAL_0: 1;

 hence contradiction by A61, A62, A63, A69, A71, A72, XREAL_1: 8;

 end;

 then |[(gr1 `1 ), (gr1 `2 )]| = (g . r2) by A69, A71, A74, EUCLID: 53;

 then (g . r1) = (g . r2) by EUCLID: 53;

 hence thesis by A34, A35, FUNCT_1:def 4;

 end;

 case

 A75: (g . r1) in ( LSeg ( |[a, d]|, |[b, d]|)) & (g . r2) in ( LSeg ( |[a, c]|, |[a, d]|));

 then

 A76: (gr2 `1 ) = a by A2, Th1;



 A77: (gr2 `2 ) <= d by A2, A75, Th1;



 A78: (gr1 `2 ) = d by A1, A75, Th3;



 A79: a <= (gr1 `1 ) by A1, A75, Th3;



 A80: (a + (gr2 `2 )) = ((gr1 `1 ) + d) by A1, A61, A62, A63, A75, A76, Th3;

 A81:

 now

 assume a <> (gr1 `1 );

 then a < (gr1 `1 ) by A79, XXREAL_0: 1;

 hence contradiction by A77, A80, XREAL_1: 8;

 end;

 now

 assume (gr2 `2 ) <> d;

 then d > (gr2 `2 ) by A77, XXREAL_0: 1;

 hence contradiction by A61, A62, A63, A76, A78, A79, XREAL_1: 8;

 end;

 then |[(gr2 `1 ), (gr2 `2 )]| = (g . r1) by A76, A78, A81, EUCLID: 53;

 then (g . r1) = (g . r2) by EUCLID: 53;

 hence thesis by A34, A35, FUNCT_1:def 4;

 end;

 case

 A82: (g . r1) in ( LSeg ( |[a, d]|, |[b, d]|)) & (g . r2) in ( LSeg ( |[a, d]|, |[b, d]|));

 then

 A83: (gr1 `2 ) = d by A1, Th3;

 (gr2 `2 ) = d by A1, A82, Th3;

 then |[(gr1 `1 ), (gr1 `2 )]| = (g . r2) by A61, A62, A63, A83, EUCLID: 53;

 then (g . r1) = (g . r2) by EUCLID: 53;

 hence thesis by A34, A35, FUNCT_1:def 4;

 end;

 end;

 hence thesis;

 end;



 A84: ( dom k) = [. 0 , 1.] by BORSUK_1: 40, FUNCT_2:def 1;

 then s1 in ( dom k) by A29, A30, XXREAL_1: 1;

 then rxc = s1 by A52, A56, A57, A58, A84;

 hence contradiction by A43, A56, XXREAL_1: 1;

 end;

 hence thesis;

 end;

 then LE (p1,p2,( Upper_Arc K),( W-min K),( E-max K)) by A3, A4, A11, JORDAN5C:def 3;

 hence thesis by A3, A4, A11, JORDAN6:def 10;

 end;

 end;

 theorem :: JGRAPH_6:56



 Th56: for a,b,c,d be Real, p1,p2 be Point of ( TOP-REAL 2) st a < b & c < d & p1 in ( LSeg ( |[a, d]|, |[b, d]|)) & p2 in ( LSeg ( |[a, d]|, |[b, d]|)) holds LE (p1,p2,( rectangle (a,b,c,d))) iff (p1 `1 ) <= (p2 `1 )

 proof

 let a,b,c,d be Real, p1,p2 be Point of ( TOP-REAL 2);

 set K = ( rectangle (a,b,c,d));

 assume that

 A1: a < b and

 A2: c < d and

 A3: p1 in ( LSeg ( |[a, d]|, |[b, d]|)) and

 A4: p2 in ( LSeg ( |[a, d]|, |[b, d]|));



 A5: K is being_simple_closed_curve by A1, A2, Th50;



 A6: (p1 `2 ) = d by A1, A3, Th3;



 A7: a <= (p1 `1 ) by A1, A3, Th3;



 A8: (p1 `1 ) <= b by A1, A3, Th3;



 A9: (p2 `2 ) = d by A1, A4, Th3;



 A10: ( W-min K) = |[a, c]| by A1, A2, Th46;



 A11: ( E-max K) = |[b, d]| by A1, A2, Th46;



 A12: ( Upper_Arc K) = (( LSeg ( |[a, c]|, |[a, d]|)) \/ ( LSeg ( |[a, d]|, |[b, d]|))) by A1, A2, Th51;

 then

 A13: ( LSeg ( |[a, d]|, |[b, d]|)) c= ( Upper_Arc K) by XBOOLE_1: 7;



 A14: (( Upper_Arc K) /\ ( Lower_Arc K)) = {( W-min K), ( E-max K)} by A5, JORDAN6:def 9;

 A15:

 now

 assume p2 in ( Lower_Arc K);

 then

 A16: p2 in (( Upper_Arc K) /\ ( Lower_Arc K)) by A4, A13, XBOOLE_0:def 4;

 now

 assume p2 = ( W-min K);

 then (p2 `2 ) = c by A10, EUCLID: 52;

 hence contradiction by A2, A4, TOPREAL3: 12;

 end;

 hence p2 = ( E-max K) by A14, A16, TARSKI:def 2;

 end;

 thus LE (p1,p2,K) implies (p1 `1 ) <= (p2 `1 )

 proof

 assume LE (p1,p2,K);

 then

 A17: p1 in ( Upper_Arc K) & p2 in ( Lower_Arc K) & not p2 = ( W-min K) or p1 in ( Upper_Arc K) & p2 in ( Upper_Arc K) & LE (p1,p2,( Upper_Arc K),( W-min K),( E-max K)) or p1 in ( Lower_Arc K) & p2 in ( Lower_Arc K) & not p2 = ( W-min K) & LE (p1,p2,( Lower_Arc K),( E-max K),( W-min K)) by JORDAN6:def 10;

 now

 per cases ;

 case p2 = ( E-max K);

 hence thesis by A8, A11, EUCLID: 52;

 end;

 case

 A18: p2 <> ( E-max K);

 consider f be Function of I[01] , (( TOP-REAL 2) | ( Upper_Arc K)) such that

 A19: f is being_homeomorphism and

 A20: (f . 0 ) = ( W-min K) and

 A21: (f . 1) = ( E-max K) and ( rng f) = ( Upper_Arc K) and for r be Real st r in [. 0 , (1 / 2).] holds (f . r) = (((1 - (2 * r)) * |[a, c]|) + ((2 * r) * |[a, d]|)) and for r be Real st r in [.(1 / 2), 1.] holds (f . r) = (((1 - ((2 * r) - 1)) * |[a, d]|) + (((2 * r) - 1) * |[b, d]|)) and for p be Point of ( TOP-REAL 2) st p in ( LSeg ( |[a, c]|, |[a, d]|)) holds 0 <= ((((p `2 ) - c) / (d - c)) / 2) & ((((p `2 ) - c) / (d - c)) / 2) <= 1 & (f . ((((p `2 ) - c) / (d - c)) / 2)) = p and

 A22: for p be Point of ( TOP-REAL 2) st p in ( LSeg ( |[a, d]|, |[b, d]|)) holds 0 <= (((((p `1 ) - a) / (b - a)) / 2) + (1 / 2)) & (((((p `1 ) - a) / (b - a)) / 2) + (1 / 2)) <= 1 & (f . (((((p `1 ) - a) / (b - a)) / 2) + (1 / 2))) = p by A1, A2, Th53;

 reconsider s1 = (((((p1 `1 ) - a) / (b - a)) / 2) + (1 / 2)), s2 = (((((p2 `1 ) - a) / (b - a)) / 2) + (1 / 2)) as Real;



 A23: (f . s1) = p1 by A3, A22;



 A24: (f . s2) = p2 by A4, A22;



 A25: (b - a) > 0 by A1, XREAL_1: 50;



 A26: s1 <= 1 by A3, A22;



 A27: 0 <= s2 by A4, A22;

 s2 <= 1 by A4, A22;

 then s1 <= s2 by A15, A17, A18, A19, A20, A21, A23, A24, A26, A27, JORDAN5C:def 3;

 then ((((p1 `1 ) - a) / (b - a)) / 2) <= ((((p2 `1 ) - a) / (b - a)) / 2) by XREAL_1: 6;

 then (((((p1 `1 ) - a) / (b - a)) / 2) * 2) <= (((((p2 `1 ) - a) / (b - a)) / 2) * 2) by XREAL_1: 64;

 then ((((p1 `1 ) - a) / (b - a)) * (b - a)) <= ((((p2 `1 ) - a) / (b - a)) * (b - a)) by A25, XREAL_1: 64;

 then ((p1 `1 ) - a) <= ((((p2 `1 ) - a) / (b - a)) * (b - a)) by A25, XCMPLX_1: 87;

 then ((p1 `1 ) - a) <= ((p2 `1 ) - a) by A25, XCMPLX_1: 87;

 then (((p1 `1 ) - a) + a) <= (((p2 `1 ) - a) + a) by XREAL_1: 7;

 hence thesis;

 end;

 end;

 hence thesis;

 end;

 thus (p1 `1 ) <= (p2 `1 ) implies LE (p1,p2,K)

 proof

 assume

 A28: (p1 `1 ) <= (p2 `1 );

 for g be Function of I[01] , (( TOP-REAL 2) | ( Upper_Arc K)), s1,s2 be Real st g is being_homeomorphism & (g . 0 ) = ( W-min K) & (g . 1) = ( E-max K) & (g . s1) = p1 & 0 <= s1 & s1 <= 1 & (g . s2) = p2 & 0 <= s2 & s2 <= 1 holds s1 <= s2

 proof

 let g be Function of I[01] , (( TOP-REAL 2) | ( Upper_Arc K)), s1,s2 be Real;

 assume that

 A29: g is being_homeomorphism and

 A30: (g . 0 ) = ( W-min K) and (g . 1) = ( E-max K) and

 A31: (g . s1) = p1 and

 A32: 0 <= s1 and

 A33: s1 <= 1 and

 A34: (g . s2) = p2 and

 A35: 0 <= s2 and

 A36: s2 <= 1;



 A37: ( dom g) = the carrier of I[01] by FUNCT_2:def 1;



 A38: g is one-to-one by A29, TOPS_2:def 5;



 A39: the carrier of (( TOP-REAL 2) | ( Upper_Arc K)) = ( Upper_Arc K) by PRE_TOPC: 8;

 then

 reconsider g1 = g as Function of I[01] , ( TOP-REAL 2) by FUNCT_2: 7;

 g is continuous by A29, TOPS_2:def 5;

 then

 A40: g1 is continuous by PRE_TOPC: 26;

 reconsider h1 = proj1 as Function of ( TOP-REAL 2), R^1 by TOPMETR: 17;

 reconsider h2 = proj2 as Function of ( TOP-REAL 2), R^1 by TOPMETR: 17;

 reconsider hh1 = h1 as Function of the TopStruct of ( TOP-REAL 2), R^1 ;

 reconsider hh2 = h2 as Function of the TopStruct of ( TOP-REAL 2), R^1 ;



 A41: the TopStruct of ( TOP-REAL 2) = ( the TopStruct of ( TOP-REAL 2) | ( [#] the TopStruct of ( TOP-REAL 2))) by TSEP_1: 3

 .= the TopStruct of (( TOP-REAL 2) | ( [#] ( TOP-REAL 2))) by PRE_TOPC: 36

 .= (( TOP-REAL 2) | ( [#] ( TOP-REAL 2)));

 then (for p be Point of (( TOP-REAL 2) | ( [#] ( TOP-REAL 2))) holds (hh1 . p) = ( proj1 . p)) implies hh1 is continuous by JGRAPH_2: 29;

 then

 A42: (for p be Point of (( TOP-REAL 2) | ( [#] ( TOP-REAL 2))) holds (hh1 . p) = ( proj1 . p)) implies h1 is continuous by PRE_TOPC: 32;

 (for p be Point of (( TOP-REAL 2) | ( [#] ( TOP-REAL 2))) holds (hh2 . p) = ( proj2 . p)) implies hh2 is continuous by A41, JGRAPH_2: 30;

 then (for p be Point of (( TOP-REAL 2) | ( [#] ( TOP-REAL 2))) holds (hh2 . p) = ( proj2 . p)) implies h2 is continuous by PRE_TOPC: 32;

 then

 consider h be Function of ( TOP-REAL 2), R^1 such that

 A43: for p be Point of ( TOP-REAL 2), r1,r2 be Real st (h1 . p) = r1 & (h2 . p) = r2 holds (h . p) = (r1 + r2) and

 A44: h is continuous by A42, JGRAPH_2: 19;

 reconsider k = (h * g1) as Function of I[01] , R^1 ;



 A45: ( W-min K) = |[a, c]| by A1, A2, Th46;

 now

 assume

 A46: s1 > s2;



 A47: ( dom g) = [. 0 , 1.] by BORSUK_1: 40, FUNCT_2:def 1;

 0 in [. 0 , 1.] by XXREAL_1: 1;



 then

 A48: (k . 0 ) = (h . ( W-min K)) by A30, A47, FUNCT_1: 13

 .= ((h1 . ( W-min K)) + (h2 . ( W-min K))) by A43

 .= ((( W-min K) `1 ) + ( proj2 . ( W-min K))) by PSCOMP_1:def 5

 .= ((( W-min K) `1 ) + (( W-min K) `2 )) by PSCOMP_1:def 6

 .= ((( W-min K) `1 ) + c) by A45, EUCLID: 52

 .= (a + c) by A45, EUCLID: 52;

 s1 in [. 0 , 1.] by A32, A33, XXREAL_1: 1;



 then

 A49: (k . s1) = (h . p1) by A31, A47, FUNCT_1: 13

 .= ((h1 . p1) + (h2 . p1)) by A43

 .= ((p1 `1 ) + ( proj2 . p1)) by PSCOMP_1:def 5

 .= ((p1 `1 ) + d) by A6, PSCOMP_1:def 6;



 A50: s2 in [. 0 , 1.] by A35, A36, XXREAL_1: 1;



 then

 A51: (k . s2) = (h . p2) by A34, A47, FUNCT_1: 13

 .= ((h1 . p2) + (h2 . p2)) by A43

 .= ((p2 `1 ) + ( proj2 . p2)) by PSCOMP_1:def 5

 .= ((p2 `1 ) + d) by A9, PSCOMP_1:def 6;



 A52: (k . 0 ) <= (k . s1) by A2, A7, A48, A49, XREAL_1: 7;



 A53: (k . s1) <= (k . s2) by A28, A49, A51, XREAL_1: 7;



 A54: 0 in [. 0 , 1.] by XXREAL_1: 1;

 then

 A55: [. 0 , s2.] c= [. 0 , 1.] by A50, XXREAL_2:def 12;

 reconsider B = [. 0 , s2.] as Subset of I[01] by A50, A54, BORSUK_1: 40, XXREAL_2:def 12;



 A56: B is connected by A35, A50, A54, BORSUK_1: 40, BORSUK_4: 24;



 A57: 0 in B by A35, XXREAL_1: 1;



 A58: s2 in B by A35, XXREAL_1: 1;

 consider xc be Point of I[01] such that

 A59: xc in B and

 A60: (k . xc) = (k . s1) by A40, A44, A52, A53, A56, A57, A58, TOPREAL5: 5;

 reconsider rxc = xc as Real;



 A61: for x1,x2 be set st x1 in ( dom k) & x2 in ( dom k) & (k . x1) = (k . x2) holds x1 = x2

 proof

 let x1,x2 be set;

 assume that

 A62: x1 in ( dom k) and

 A63: x2 in ( dom k) and

 A64: (k . x1) = (k . x2);

 reconsider r1 = x1 as Point of I[01] by A62;

 reconsider r2 = x2 as Point of I[01] by A63;



 A65: (k . x1) = (h . (g1 . x1)) by A62, FUNCT_1: 12

 .= ((h1 . (g1 . r1)) + (h2 . (g1 . r1))) by A43

 .= (((g1 . r1) `1 ) + ( proj2 . (g1 . r1))) by PSCOMP_1:def 5

 .= (((g1 . r1) `1 ) + ((g1 . r1) `2 )) by PSCOMP_1:def 6;



 A66: (k . x2) = (h . (g1 . x2)) by A63, FUNCT_1: 12

 .= ((h1 . (g1 . r2)) + (h2 . (g1 . r2))) by A43

 .= (((g1 . r2) `1 ) + ( proj2 . (g1 . r2))) by PSCOMP_1:def 5

 .= (((g1 . r2) `1 ) + ((g1 . r2) `2 )) by PSCOMP_1:def 6;



 A67: (g . r1) in ( Upper_Arc K) by A39;



 A68: (g . r2) in ( Upper_Arc K) by A39;

 reconsider gr1 = (g . r1) as Point of ( TOP-REAL 2) by A67;

 reconsider gr2 = (g . r2) as Point of ( TOP-REAL 2) by A68;

 now

 per cases by A12, A39, XBOOLE_0:def 3;

 case

 A69: (g . r1) in ( LSeg ( |[a, c]|, |[a, d]|)) & (g . r2) in ( LSeg ( |[a, c]|, |[a, d]|));

 then

 A70: (gr1 `1 ) = a by A2, Th1;

 (gr2 `1 ) = a by A2, A69, Th1;

 then |[(gr1 `1 ), (gr1 `2 )]| = (g . r2) by A64, A65, A66, A70, EUCLID: 53;

 then (g . r1) = (g . r2) by EUCLID: 53;

 hence thesis by A37, A38, FUNCT_1:def 4;

 end;

 case

 A71: (g . r1) in ( LSeg ( |[a, c]|, |[a, d]|)) & (g . r2) in ( LSeg ( |[a, d]|, |[b, d]|));

 then

 A72: (gr1 `1 ) = a by A2, Th1;



 A73: (gr1 `2 ) <= d by A2, A71, Th1;



 A74: (gr2 `2 ) = d by A1, A71, Th3;



 A75: a <= (gr2 `1 ) by A1, A71, Th3;



 A76: (a + (gr1 `2 )) = ((gr2 `1 ) + d) by A1, A64, A65, A66, A71, A72, Th3;

 A77:

 now

 assume a <> (gr2 `1 );

 then a < (gr2 `1 ) by A75, XXREAL_0: 1;

 hence contradiction by A73, A76, XREAL_1: 8;

 end;

 now

 assume (gr1 `2 ) <> d;

 then d > (gr1 `2 ) by A73, XXREAL_0: 1;

 hence contradiction by A64, A65, A66, A72, A74, A75, XREAL_1: 8;

 end;

 then |[(gr1 `1 ), (gr1 `2 )]| = (g . r2) by A72, A74, A77, EUCLID: 53;

 then (g . r1) = (g . r2) by EUCLID: 53;

 hence thesis by A37, A38, FUNCT_1:def 4;

 end;

 case

 A78: (g . r1) in ( LSeg ( |[a, d]|, |[b, d]|)) & (g . r2) in ( LSeg ( |[a, c]|, |[a, d]|));

 then

 A79: (gr2 `1 ) = a by A2, Th1;



 A80: (gr2 `2 ) <= d by A2, A78, Th1;



 A81: (gr1 `2 ) = d by A1, A78, Th3;



 A82: a <= (gr1 `1 ) by A1, A78, Th3;



 A83: (a + (gr2 `2 )) = ((gr1 `1 ) + d) by A1, A64, A65, A66, A78, A79, Th3;

 A84:

 now

 assume a <> (gr1 `1 );

 then a < (gr1 `1 ) by A82, XXREAL_0: 1;

 hence contradiction by A80, A83, XREAL_1: 8;

 end;

 now

 assume (gr2 `2 ) <> d;

 then d > (gr2 `2 ) by A80, XXREAL_0: 1;

 hence contradiction by A64, A65, A66, A79, A81, A82, XREAL_1: 8;

 end;

 then |[(gr2 `1 ), (gr2 `2 )]| = (g . r1) by A79, A81, A84, EUCLID: 53;

 then (g . r1) = (g . r2) by EUCLID: 53;

 hence thesis by A37, A38, FUNCT_1:def 4;

 end;

 case

 A85: (g . r1) in ( LSeg ( |[a, d]|, |[b, d]|)) & (g . r2) in ( LSeg ( |[a, d]|, |[b, d]|));

 then

 A86: (gr1 `2 ) = d by A1, Th3;

 (gr2 `2 ) = d by A1, A85, Th3;

 then |[(gr1 `1 ), (gr1 `2 )]| = (g . r2) by A64, A65, A66, A86, EUCLID: 53;

 then (g . r1) = (g . r2) by EUCLID: 53;

 hence thesis by A37, A38, FUNCT_1:def 4;

 end;

 end;

 hence thesis;

 end;



 A87: ( dom k) = [. 0 , 1.] by BORSUK_1: 40, FUNCT_2:def 1;

 then s1 in ( dom k) by A32, A33, XXREAL_1: 1;

 then rxc = s1 by A55, A59, A60, A61, A87;

 hence contradiction by A46, A59, XXREAL_1: 1;

 end;

 hence thesis;

 end;

 then LE (p1,p2,( Upper_Arc K),( W-min K),( E-max K)) by A3, A4, A13, JORDAN5C:def 3;

 hence thesis by A3, A4, A13, JORDAN6:def 10;

 end;

 end;

 theorem :: JGRAPH_6:57



 Th57: for a,b,c,d be Real, p1,p2 be Point of ( TOP-REAL 2) st a = (p2 `2 )

 proof

 let a,b,c,d be Real, p1,p2 be Point of ( TOP-REAL 2);

 set K = ( rectangle (a,b,c,d));

 assume that

 A1: a < b and

 A2: c < d and

 A3: p1 in ( LSeg ( |[b, c]|, |[b, d]|)) and

 A4: p2 in ( LSeg ( |[b, c]|, |[b, d]|));



 A5: K is being_simple_closed_curve by A1, A2, Th50;



 A6: (p1 `1 ) = b by A2, A3, Th1;



 A7: (p1 `2 ) <= d by A2, A3, Th1;



 A8: (p2 `1 ) = b by A2, A4, Th1;



 A9: (p2 `2 ) <= d by A2, A4, Th1;



 A10: ( W-min K) = |[a, c]| by A1, A2, Th46;



 A11: ( E-max K) = |[b, d]| by A1, A2, Th46;



 A12: ( Lower_Arc K) = (( LSeg ( |[b, d]|, |[b, c]|)) \/ ( LSeg ( |[b, c]|, |[a, c]|))) by A1, A2, Th52;

 then

 A13: ( LSeg ( |[b, d]|, |[b, c]|)) c= ( Lower_Arc K) by XBOOLE_1: 7;



 A14: (( Upper_Arc K) /\ ( Lower_Arc K)) = {( W-min K), ( E-max K)} by A5, JORDAN6:def 9;

 A15:

 now

 assume p1 in ( Upper_Arc K);

 then

 A16: p1 in (( Upper_Arc K) /\ ( Lower_Arc K)) by A3, A13, XBOOLE_0:def 4;

 now

 assume p1 = ( W-min K);

 then (p1 `1 ) = a by A10, EUCLID: 52;

 hence contradiction by A1, A3, TOPREAL3: 11;

 end;

 hence p1 = ( E-max K) by A14, A16, TARSKI:def 2;

 end;

 thus LE (p1,p2,K) implies (p1 `2 ) >= (p2 `2 )

 proof

 assume LE (p1,p2,K);

 then

 A17: p1 in ( Upper_Arc K) & p2 in ( Lower_Arc K) & not p2 = ( W-min K) or p1 in ( Upper_Arc K) & p2 in ( Upper_Arc K) & LE (p1,p2,( Upper_Arc K),( W-min K),( E-max K)) or p1 in ( Lower_Arc K) & p2 in ( Lower_Arc K) & not p2 = ( W-min K) & LE (p1,p2,( Lower_Arc K),( E-max K),( W-min K)) by JORDAN6:def 10;

 now

 per cases ;

 case p1 = ( E-max K);

 hence thesis by A9, A11, EUCLID: 52;

 end;

 case

 A18: p1 <> ( E-max K);

 consider f be Function of I[01] , (( TOP-REAL 2) | ( Lower_Arc K)) such that

 A19: f is being_homeomorphism and

 A20: (f . 0 ) = ( E-max K) and

 A21: (f . 1) = ( W-min K) and ( rng f) = ( Lower_Arc K) and for r be Real st r in [. 0 , (1 / 2).] holds (f . r) = (((1 - (2 * r)) * |[b, d]|) + ((2 * r) * |[b, c]|)) and for r be Real st r in [.(1 / 2), 1.] holds (f . r) = (((1 - ((2 * r) - 1)) * |[b, c]|) + (((2 * r) - 1) * |[a, c]|)) and

 A22: for p be Point of ( TOP-REAL 2) st p in ( LSeg ( |[b, d]|, |[b, c]|)) holds 0 <= ((((p `2 ) - d) / (c - d)) / 2) & ((((p `2 ) - d) / (c - d)) / 2) <= 1 & (f . ((((p `2 ) - d) / (c - d)) / 2)) = p and for p be Point of ( TOP-REAL 2) st p in ( LSeg ( |[b, c]|, |[a, c]|)) holds 0 <= (((((p `1 ) - b) / (a - b)) / 2) + (1 / 2)) & (((((p `1 ) - b) / (a - b)) / 2) + (1 / 2)) <= 1 & (f . (((((p `1 ) - b) / (a - b)) / 2) + (1 / 2))) = p by A1, A2, Th54;

 reconsider s1 = ((((p1 `2 ) - d) / (c - d)) / 2), s2 = ((((p2 `2 ) - d) / (c - d)) / 2) as Real;



 A23: (f . s1) = p1 by A3, A22;



 A24: (f . s2) = p2 by A4, A22;

 (d - c) > 0 by A2, XREAL_1: 50;

 then

 A25: ( - (d - c)) < ( - 0 ) by XREAL_1: 24;



 A26: s1 <= 1 by A3, A22;



 A27: 0 <= s2 by A4, A22;

 s2 <= 1 by A4, A22;

 then s1 <= s2 by A15, A17, A18, A19, A20, A21, A23, A24, A26, A27, JORDAN5C:def 3;

 then (((((p1 `2 ) - d) / (c - d)) / 2) * 2) <= (((((p2 `2 ) - d) / (c - d)) / 2) * 2) by XREAL_1: 64;

 then ((((p1 `2 ) - d) / (c - d)) * (c - d)) >= ((((p2 `2 ) - d) / (c - d)) * (c - d)) by A25, XREAL_1: 65;

 then ((p1 `2 ) - d) >= ((((p2 `2 ) - d) / (c - d)) * (c - d)) by A25, XCMPLX_1: 87;

 then ((p1 `2 ) - d) >= ((p2 `2 ) - d) by A25, XCMPLX_1: 87;

 then (((p1 `2 ) - d) + d) >= (((p2 `2 ) - d) + d) by XREAL_1: 7;

 hence thesis;

 end;

 end;

 hence thesis;

 end;

 thus (p1 `2 ) >= (p2 `2 ) implies LE (p1,p2,K)

 proof

 assume

 A28: (p1 `2 ) >= (p2 `2 );

 now

 per cases ;

 case p2 = ( W-min K);

 then p2 = |[a, c]| by A1, A2, Th46;

 hence contradiction by A1, A8, EUCLID: 52;

 end;

 case

 A29: p2 <> ( W-min K);

 for g be Function of I[01] , (( TOP-REAL 2) | ( Lower_Arc K)), s1,s2 be Real st g is being_homeomorphism & (g . 0 ) = ( E-max K) & (g . 1) = ( W-min K) & (g . s1) = p1 & 0 <= s1 & s1 <= 1 & (g . s2) = p2 & 0 <= s2 & s2 <= 1 holds s1 <= s2

 proof

 let g be Function of I[01] , (( TOP-REAL 2) | ( Lower_Arc K)), s1,s2 be Real;

 assume that

 A30: g is being_homeomorphism and

 A31: (g . 0 ) = ( E-max K) and (g . 1) = ( W-min K) and

 A32: (g . s1) = p1 and

 A33: 0 <= s1 and

 A34: s1 <= 1 and

 A35: (g . s2) = p2 and

 A36: 0 <= s2 and

 A37: s2 <= 1;



 A38: ( dom g) = the carrier of I[01] by FUNCT_2:def 1;



 A39: g is one-to-one by A30, TOPS_2:def 5;



 A40: the carrier of (( TOP-REAL 2) | ( Lower_Arc K)) = ( Lower_Arc K) by PRE_TOPC: 8;

 then

 reconsider g1 = g as Function of I[01] , ( TOP-REAL 2) by FUNCT_2: 7;

 g is continuous by A30, TOPS_2:def 5;

 then

 A41: g1 is continuous by PRE_TOPC: 26;

 reconsider h1 = proj1 as Function of ( TOP-REAL 2), R^1 by TOPMETR: 17;

 reconsider h2 = proj2 as Function of ( TOP-REAL 2), R^1 by TOPMETR: 17;

 reconsider hh1 = h1 as Function of the TopStruct of ( TOP-REAL 2), R^1 ;

 reconsider hh2 = h2 as Function of the TopStruct of ( TOP-REAL 2), R^1 ;



 A42: the TopStruct of ( TOP-REAL 2) = ( the TopStruct of ( TOP-REAL 2) | ( [#] the TopStruct of ( TOP-REAL 2))) by TSEP_1: 3

 .= the TopStruct of (( TOP-REAL 2) | ( [#] ( TOP-REAL 2))) by PRE_TOPC: 36

 .= (( TOP-REAL 2) | ( [#] ( TOP-REAL 2)));

 then (for p be Point of (( TOP-REAL 2) | ( [#] ( TOP-REAL 2))) holds (hh1 . p) = ( proj1 . p)) implies hh1 is continuous by JGRAPH_2: 29;

 then

 A43: (for p be Point of (( TOP-REAL 2) | ( [#] ( TOP-REAL 2))) holds (hh1 . p) = ( proj1 . p)) implies h1 is continuous by PRE_TOPC: 32;

 (for p be Point of (( TOP-REAL 2) | ( [#] ( TOP-REAL 2))) holds (hh2 . p) = ( proj2 . p)) implies hh2 is continuous by A42, JGRAPH_2: 30;

 then (for p be Point of (( TOP-REAL 2) | ( [#] ( TOP-REAL 2))) holds (hh2 . p) = ( proj2 . p)) implies h2 is continuous by PRE_TOPC: 32;

 then

 consider h be Function of ( TOP-REAL 2), R^1 such that

 A44: for p be Point of ( TOP-REAL 2), r1,r2 be Real st (h1 . p) = r1 & (h2 . p) = r2 holds (h . p) = (r1 + r2) and

 A45: h is continuous by A43, JGRAPH_2: 19;

 reconsider k = (h * g1) as Function of I[01] , R^1 ;



 A46: ( E-max K) = |[b, d]| by A1, A2, Th46;

 now

 assume

 A47: s1 > s2;



 A48: ( dom g) = [. 0 , 1.] by BORSUK_1: 40, FUNCT_2:def 1;

 0 in [. 0 , 1.] by XXREAL_1: 1;



 then

 A49: (k . 0 ) = (h . ( E-max K)) by A31, A48, FUNCT_1: 13

 .= ((h1 . ( E-max K)) + (h2 . ( E-max K))) by A44

 .= ((( E-max K) `1 ) + ( proj2 . ( E-max K))) by PSCOMP_1:def 5

 .= ((( E-max K) `1 ) + (( E-max K) `2 )) by PSCOMP_1:def 6

 .= (b + (( E-max K) `2 )) by A46, EUCLID: 52

 .= (b + d) by A46, EUCLID: 52;

 s1 in [. 0 , 1.] by A33, A34, XXREAL_1: 1;



 then

 A50: (k . s1) = (h . p1) by A32, A48, FUNCT_1: 13

 .= ((h1 . p1) + (h2 . p1)) by A44

 .= ((p1 `1 ) + ( proj2 . p1)) by PSCOMP_1:def 5

 .= (b + (p1 `2 )) by A6, PSCOMP_1:def 6;



 A51: s2 in [. 0 , 1.] by A36, A37, XXREAL_1: 1;



 then

 A52: (k . s2) = (h . p2) by A35, A48, FUNCT_1: 13

 .= (( proj1 . p2) + ( proj2 . p2)) by A44

 .= ((p2 `1 ) + ( proj2 . p2)) by PSCOMP_1:def 5

 .= (b + (p2 `2 )) by A8, PSCOMP_1:def 6;



 A53: (k . 0 ) >= (k . s1) by A7, A49, A50, XREAL_1: 7;



 A54: (k . s1) >= (k . s2) by A28, A50, A52, XREAL_1: 7;



 A55: 0 in [. 0 , 1.] by XXREAL_1: 1;

 then

 A56: [. 0 , s2.] c= [. 0 , 1.] by A51, XXREAL_2:def 12;

 reconsider B = [. 0 , s2.] as Subset of I[01] by A51, A55, BORSUK_1: 40, XXREAL_2:def 12;



 A57: B is connected by A36, A51, A55, BORSUK_1: 40, BORSUK_4: 24;



 A58: 0 in B by A36, XXREAL_1: 1;



 A59: s2 in B by A36, XXREAL_1: 1;

 consider xc be Point of I[01] such that

 A60: xc in B and

 A61: (k . xc) = (k . s1) by A41, A45, A53, A54, A57, A58, A59, TOPREAL5: 5;

 reconsider rxc = xc as Real;



 A62: for x1,x2 be set st x1 in ( dom k) & x2 in ( dom k) & (k . x1) = (k . x2) holds x1 = x2

 proof

 let x1,x2 be set;

 assume that

 A63: x1 in ( dom k) and

 A64: x2 in ( dom k) and

 A65: (k . x1) = (k . x2);

 reconsider r1 = x1 as Point of I[01] by A63;

 reconsider r2 = x2 as Point of I[01] by A64;



 A66: (k . x1) = (h . (g1 . x1)) by A63, FUNCT_1: 12

 .= ((h1 . (g1 . r1)) + (h2 . (g1 . r1))) by A44

 .= (((g1 . r1) `1 ) + ( proj2 . (g1 . r1))) by PSCOMP_1:def 5

 .= (((g1 . r1) `1 ) + ((g1 . r1) `2 )) by PSCOMP_1:def 6;



 A67: (k . x2) = (h . (g1 . x2)) by A64, FUNCT_1: 12

 .= ((h1 . (g1 . r2)) + (h2 . (g1 . r2))) by A44

 .= (((g1 . r2) `1 ) + ( proj2 . (g1 . r2))) by PSCOMP_1:def 5

 .= (((g1 . r2) `1 ) + ((g1 . r2) `2 )) by PSCOMP_1:def 6;



 A68: (g . r1) in ( Lower_Arc K) by A40;



 A69: (g . r2) in ( Lower_Arc K) by A40;

 reconsider gr1 = (g . r1) as Point of ( TOP-REAL 2) by A68;

 reconsider gr2 = (g . r2) as Point of ( TOP-REAL 2) by A69;

 now

 per cases by A12, A40, XBOOLE_0:def 3;

 case

 A70: (g . r1) in ( LSeg ( |[b, c]|, |[b, d]|)) & (g . r2) in ( LSeg ( |[b, c]|, |[b, d]|));

 then

 A71: (gr1 `1 ) = b by A2, Th1;

 (gr2 `1 ) = b by A2, A70, Th1;

 then |[(gr1 `1 ), (gr1 `2 )]| = (g . r2) by A65, A66, A67, A71, EUCLID: 53;

 then (g . r1) = (g . r2) by EUCLID: 53;

 hence thesis by A38, A39, FUNCT_1:def 4;

 end;

 case

 A72: (g . r1) in ( LSeg ( |[b, c]|, |[b, d]|)) & (g . r2) in ( LSeg ( |[b, c]|, |[a, c]|));

 then

 A73: (gr1 `1 ) = b by A2, Th1;



 A74: c <= (gr1 `2 ) by A2, A72, Th1;



 A75: (gr2 `2 ) = c by A1, A72, Th3;



 A76: (gr2 `1 ) <= b by A1, A72, Th3;



 A77: (b + (gr1 `2 )) = ((gr2 `1 ) + c) by A1, A65, A66, A67, A72, A73, Th3;

 A78:

 now

 assume b <> (gr2 `1 );

 then b > (gr2 `1 ) by A76, XXREAL_0: 1;

 hence contradiction by A74, A77, XREAL_1: 8;

 end;

 now

 assume (gr1 `2 ) <> c;

 then c < (gr1 `2 ) by A74, XXREAL_0: 1;

 hence contradiction by A65, A66, A67, A73, A75, A76, XREAL_1: 8;

 end;

 then |[(gr1 `1 ), (gr1 `2 )]| = (g . r2) by A73, A75, A78, EUCLID: 53;

 then (g . r1) = (g . r2) by EUCLID: 53;

 hence thesis by A38, A39, FUNCT_1:def 4;

 end;

 case

 A79: (g . r1) in ( LSeg ( |[b, c]|, |[a, c]|)) & (g . r2) in ( LSeg ( |[b, c]|, |[b, d]|));

 then

 A80: (gr2 `1 ) = b by A2, Th1;



 A81: c <= (gr2 `2 ) by A2, A79, Th1;



 A82: (gr1 `2 ) = c by A1, A79, Th3;



 A83: (gr1 `1 ) <= b by A1, A79, Th3;



 A84: (b + (gr2 `2 )) = ((gr1 `1 ) + c) by A1, A65, A66, A67, A79, A80, Th3;

 A85:

 now

 assume b <> (gr1 `1 );

 then b > (gr1 `1 ) by A83, XXREAL_0: 1;

 hence contradiction by A81, A84, XREAL_1: 8;

 end;

 now

 assume (gr2 `2 ) <> c;

 then c < (gr2 `2 ) by A81, XXREAL_0: 1;

 hence contradiction by A65, A66, A67, A80, A82, A83, XREAL_1: 8;

 end;

 then |[(gr2 `1 ), (gr2 `2 )]| = (g . r1) by A80, A82, A85, EUCLID: 53;

 then (g . r1) = (g . r2) by EUCLID: 53;

 hence thesis by A38, A39, FUNCT_1:def 4;

 end;

 case

 A86: (g . r1) in ( LSeg ( |[b, c]|, |[a, c]|)) & (g . r2) in ( LSeg ( |[b, c]|, |[a, c]|));

 then

 A87: (gr1 `2 ) = c by A1, Th3;

 (gr2 `2 ) = c by A1, A86, Th3;

 then |[(gr1 `1 ), (gr1 `2 )]| = (g . r2) by A65, A66, A67, A87, EUCLID: 53;

 then (g . r1) = (g . r2) by EUCLID: 53;

 hence thesis by A38, A39, FUNCT_1:def 4;

 end;

 end;

 hence thesis;

 end;



 A88: ( dom k) = [. 0 , 1.] by BORSUK_1: 40, FUNCT_2:def 1;

 then s1 in ( dom k) by A33, A34, XXREAL_1: 1;

 then rxc = s1 by A56, A60, A61, A62, A88;

 hence contradiction by A47, A60, XXREAL_1: 1;

 end;

 hence thesis;

 end;

 then LE (p1,p2,( Lower_Arc K),( E-max K),( W-min K)) by A3, A4, A13, JORDAN5C:def 3;

 hence thesis by A3, A4, A13, A29, JORDAN6:def 10;

 end;

 end;

 hence thesis;

 end;

 end;

 theorem :: JGRAPH_6:58



 Th58: for a,b,c,d be Real, p1,p2 be Point of ( TOP-REAL 2) st a ( W-min ( rectangle (a,b,c,d))) iff (p1 `1 ) >= (p2 `1 ) & p2 <> ( W-min ( rectangle (a,b,c,d)))

 proof

 let a,b,c,d be Real, p1,p2 be Point of ( TOP-REAL 2);

 set K = ( rectangle (a,b,c,d));

 assume that

 A1: a < b and

 A2: c < d and

 A3: p1 in ( LSeg ( |[a, c]|, |[b, c]|)) and

 A4: p2 in ( LSeg ( |[a, c]|, |[b, c]|));



 A5: K is being_simple_closed_curve by A1, A2, Th50;



 A6: (p1 `2 ) = c by A1, A3, Th3;



 A7: (p1 `1 ) <= b by A1, A3, Th3;



 A8: (p2 `2 ) = c by A1, A4, Th3;



 A9: a <= (p2 `1 ) by A1, A4, Th3;



 A10: ( W-min K) = |[a, c]| by A1, A2, Th46;



 A11: ( E-max K) = |[b, d]| by A1, A2, Th46;



 A12: ( Lower_Arc K) = (( LSeg ( |[b, d]|, |[b, c]|)) \/ ( LSeg ( |[b, c]|, |[a, c]|))) by A1, A2, Th52;

 then

 A13: ( LSeg ( |[b, c]|, |[a, c]|)) c= ( Lower_Arc K) by XBOOLE_1: 7;

 then

 A14: p1 in ( Lower_Arc K) by A3;



 A15: ( Lower_Arc K) c= K by A5, JORDAN6: 61;



 A16: (( Upper_Arc K) /\ ( Lower_Arc K)) = {( W-min K), ( E-max K)} by A5, JORDAN6:def 9;

 A17:

 now

 assume p1 in ( Upper_Arc K);

 then p1 in (( Upper_Arc K) /\ ( Lower_Arc K)) by A3, A13, XBOOLE_0:def 4;

 then p1 = ( W-min K) or p1 = ( E-max K) by A16, TARSKI:def 2;

 hence p1 = ( W-min K) by A2, A6, A11, EUCLID: 52;

 end;

 thus LE (p1,p2,K) & p1 <> ( W-min K) implies (p1 `1 ) >= (p2 `1 ) & p2 <> ( W-min K)

 proof

 assume that

 A18: LE (p1,p2,K) and

 A19: p1 <> ( W-min K);



 A20: p1 in ( Upper_Arc K) & p2 in ( Lower_Arc K) & not p2 = ( W-min K) or p1 in ( Upper_Arc K) & p2 in ( Upper_Arc K) & LE (p1,p2,( Upper_Arc K),( W-min K),( E-max K)) or p1 in ( Lower_Arc K) & p2 in ( Lower_Arc K) & not p2 = ( W-min K) & LE (p1,p2,( Lower_Arc K),( E-max K),( W-min K)) by A18, JORDAN6:def 10;

 consider f be Function of I[01] , (( TOP-REAL 2) | ( Lower_Arc K)) such that

 A21: f is being_homeomorphism and

 A22: (f . 0 ) = ( E-max K) and

 A23: (f . 1) = ( W-min K) and ( rng f) = ( Lower_Arc K) and for r be Real st r in [. 0 , (1 / 2).] holds (f . r) = (((1 - (2 * r)) * |[b, d]|) + ((2 * r) * |[b, c]|)) and for r be Real st r in [.(1 / 2), 1.] holds (f . r) = (((1 - ((2 * r) - 1)) * |[b, c]|) + (((2 * r) - 1) * |[a, c]|)) and for p be Point of ( TOP-REAL 2) st p in ( LSeg ( |[b, d]|, |[b, c]|)) holds 0 <= ((((p `2 ) - d) / (c - d)) / 2) & ((((p `2 ) - d) / (c - d)) / 2) <= 1 & (f . ((((p `2 ) - d) / (c - d)) / 2)) = p and

 A24: for p be Point of ( TOP-REAL 2) st p in ( LSeg ( |[b, c]|, |[a, c]|)) holds 0 <= (((((p `1 ) - b) / (a - b)) / 2) + (1 / 2)) & (((((p `1 ) - b) / (a - b)) / 2) + (1 / 2)) <= 1 & (f . (((((p `1 ) - b) / (a - b)) / 2) + (1 / 2))) = p by A1, A2, Th54;

 reconsider s1 = (((((p1 `1 ) - b) / (a - b)) / 2) + (1 / 2)), s2 = (((((p2 `1 ) - b) / (a - b)) / 2) + (1 / 2)) as Real;



 A25: (f . s1) = p1 by A3, A24;



 A26: (f . s2) = p2 by A4, A24;

 (b - a) > 0 by A1, XREAL_1: 50;

 then

 A27: ( - (b - a)) < ( - 0 ) by XREAL_1: 24;



 A28: s1 <= 1 by A3, A24;



 A29: 0 <= s2 by A4, A24;

 s2 <= 1 by A4, A24;

 then s1 <= s2 by A17, A19, A20, A21, A22, A23, A25, A26, A28, A29, JORDAN5C:def 3;

 then ((((p1 `1 ) - b) / (a - b)) / 2) <= ((((p2 `1 ) - b) / (a - b)) / 2) by XREAL_1: 6;

 then (((((p1 `1 ) - b) / (a - b)) / 2) * 2) <= (((((p2 `1 ) - b) / (a - b)) / 2) * 2) by XREAL_1: 64;

 then ((((p1 `1 ) - b) / (a - b)) * (a - b)) >= ((((p2 `1 ) - b) / (a - b)) * (a - b)) by A27, XREAL_1: 65;

 then ((p1 `1 ) - b) >= ((((p2 `1 ) - b) / (a - b)) * (a - b)) by A27, XCMPLX_1: 87;

 then ((p1 `1 ) - b) >= ((p2 `1 ) - b) by A27, XCMPLX_1: 87;

 then (((p1 `1 ) - b) + b) >= (((p2 `1 ) - b) + b) by XREAL_1: 7;

 hence (p1 `1 ) >= (p2 `1 );

 now

 assume

 A30: p2 = ( W-min K);

 then LE (p2,p1,K) by A5, A14, A15, JORDAN7: 3;

 hence contradiction by A1, A2, A18, A19, A30, Th50, JORDAN6: 57;

 end;

 hence thesis;

 end;

 thus (p1 `1 ) >= (p2 `1 ) & p2 <> ( W-min K) implies LE (p1,p2,K) & p1 <> ( W-min K)

 proof

 assume that

 A31: (p1 `1 ) >= (p2 `1 ) and

 A32: p2 <> ( W-min K);



 A33: for g be Function of I[01] , (( TOP-REAL 2) | ( Lower_Arc K)), s1,s2 be Real st g is being_homeomorphism & (g . 0 ) = ( E-max K) & (g . 1) = ( W-min K) & (g . s1) = p1 & 0 <= s1 & s1 <= 1 & (g . s2) = p2 & 0 <= s2 & s2 <= 1 holds s1 <= s2

 proof

 let g be Function of I[01] , (( TOP-REAL 2) | ( Lower_Arc K)), s1,s2 be Real;

 assume that

 A34: g is being_homeomorphism and

 A35: (g . 0 ) = ( E-max K) and (g . 1) = ( W-min K) and

 A36: (g . s1) = p1 and

 A37: 0 <= s1 and

 A38: s1 <= 1 and

 A39: (g . s2) = p2 and

 A40: 0 <= s2 and

 A41: s2 <= 1;



 A42: ( dom g) = the carrier of I[01] by FUNCT_2:def 1;



 A43: g is one-to-one by A34, TOPS_2:def 5;



 A44: the carrier of (( TOP-REAL 2) | ( Lower_Arc K)) = ( Lower_Arc K) by PRE_TOPC: 8;

 then

 reconsider g1 = g as Function of I[01] , ( TOP-REAL 2) by FUNCT_2: 7;

 g is continuous by A34, TOPS_2:def 5;

 then

 A45: g1 is continuous by PRE_TOPC: 26;

 reconsider h1 = proj1 as Function of ( TOP-REAL 2), R^1 by TOPMETR: 17;

 reconsider h2 = proj2 as Function of ( TOP-REAL 2), R^1 by TOPMETR: 17;

 reconsider hh1 = h1 as Function of the TopStruct of ( TOP-REAL 2), R^1 ;

 reconsider hh2 = h2 as Function of the TopStruct of ( TOP-REAL 2), R^1 ;



 A46: the TopStruct of ( TOP-REAL 2) = ( the TopStruct of ( TOP-REAL 2) | ( [#] the TopStruct of ( TOP-REAL 2))) by TSEP_1: 3

 .= the TopStruct of (( TOP-REAL 2) | ( [#] ( TOP-REAL 2))) by PRE_TOPC: 36

 .= (( TOP-REAL 2) | ( [#] ( TOP-REAL 2)));

 then (for p be Point of (( TOP-REAL 2) | ( [#] ( TOP-REAL 2))) holds (hh1 . p) = ( proj1 . p)) implies hh1 is continuous by JGRAPH_2: 29;

 then

 A47: (for p be Point of (( TOP-REAL 2) | ( [#] ( TOP-REAL 2))) holds (hh1 . p) = ( proj1 . p)) implies h1 is continuous by PRE_TOPC: 32;

 (for p be Point of (( TOP-REAL 2) | ( [#] ( TOP-REAL 2))) holds (hh2 . p) = ( proj2 . p)) implies hh2 is continuous by A46, JGRAPH_2: 30;

 then (for p be Point of (( TOP-REAL 2) | ( [#] ( TOP-REAL 2))) holds (hh2 . p) = ( proj2 . p)) implies h2 is continuous by PRE_TOPC: 32;

 then

 consider h be Function of ( TOP-REAL 2), R^1 such that

 A48: for p be Point of ( TOP-REAL 2), r1,r2 be Real st (h1 . p) = r1 & (h2 . p) = r2 holds (h . p) = (r1 + r2) and

 A49: h is continuous by A47, JGRAPH_2: 19;

 reconsider k = (h * g1) as Function of I[01] , R^1 ;



 A50: ( E-max K) = |[b, d]| by A1, A2, Th46;

 now

 assume

 A51: s1 > s2;



 A52: ( dom g) = [. 0 , 1.] by BORSUK_1: 40, FUNCT_2:def 1;

 0 in [. 0 , 1.] by XXREAL_1: 1;



 then

 A53: (k . 0 ) = (h . ( E-max K)) by A35, A52, FUNCT_1: 13

 .= ((h1 . ( E-max K)) + (h2 . ( E-max K))) by A48

 .= ((( E-max K) `1 ) + ( proj2 . ( E-max K))) by PSCOMP_1:def 5

 .= ((( E-max K) `1 ) + (( E-max K) `2 )) by PSCOMP_1:def 6

 .= ((( E-max K) `1 ) + d) by A50, EUCLID: 52

 .= (b + d) by A50, EUCLID: 52;

 s1 in [. 0 , 1.] by A37, A38, XXREAL_1: 1;



 then

 A54: (k . s1) = (h . p1) by A36, A52, FUNCT_1: 13

 .= (( proj1 . p1) + ( proj2 . p1)) by A48

 .= ((p1 `1 ) + ( proj2 . p1)) by PSCOMP_1:def 5

 .= ((p1 `1 ) + c) by A6, PSCOMP_1:def 6;



 A55: s2 in [. 0 , 1.] by A40, A41, XXREAL_1: 1;



 then

 A56: (k . s2) = (h . p2) by A39, A52, FUNCT_1: 13

 .= ((h1 . p2) + (h2 . p2)) by A48

 .= ((p2 `1 ) + ( proj2 . p2)) by PSCOMP_1:def 5

 .= ((p2 `1 ) + c) by A8, PSCOMP_1:def 6;



 A57: (k . 0 ) >= (k . s1) by A2, A7, A53, A54, XREAL_1: 7;



 A58: (k . s1) >= (k . s2) by A31, A54, A56, XREAL_1: 7;



 A59: 0 in [. 0 , 1.] by XXREAL_1: 1;

 then

 A60: [. 0 , s2.] c= [. 0 , 1.] by A55, XXREAL_2:def 12;

 reconsider B = [. 0 , s2.] as Subset of I[01] by A55, A59, BORSUK_1: 40, XXREAL_2:def 12;



 A61: B is connected by A40, A55, A59, BORSUK_1: 40, BORSUK_4: 24;



 A62: 0 in B by A40, XXREAL_1: 1;



 A63: s2 in B by A40, XXREAL_1: 1;

 consider xc be Point of I[01] such that

 A64: xc in B and

 A65: (k . xc) = (k . s1) by A45, A49, A57, A58, A61, A62, A63, TOPREAL5: 5;

 reconsider rxc = xc as Real;



 A66: for x1,x2 be set st x1 in ( dom k) & x2 in ( dom k) & (k . x1) = (k . x2) holds x1 = x2

 proof

 let x1,x2 be set;

 assume that

 A67: x1 in ( dom k) and

 A68: x2 in ( dom k) and

 A69: (k . x1) = (k . x2);

 reconsider r1 = x1 as Point of I[01] by A67;

 reconsider r2 = x2 as Point of I[01] by A68;



 A70: (k . x1) = (h . (g1 . x1)) by A67, FUNCT_1: 12

 .= ((h1 . (g1 . r1)) + (h2 . (g1 . r1))) by A48

 .= (((g1 . r1) `1 ) + ( proj2 . (g1 . r1))) by PSCOMP_1:def 5

 .= (((g1 . r1) `1 ) + ((g1 . r1) `2 )) by PSCOMP_1:def 6;



 A71: (k . x2) = (h . (g1 . x2)) by A68, FUNCT_1: 12

 .= ((h1 . (g1 . r2)) + (h2 . (g1 . r2))) by A48

 .= (((g1 . r2) `1 ) + ( proj2 . (g1 . r2))) by PSCOMP_1:def 5

 .= (((g1 . r2) `1 ) + ((g1 . r2) `2 )) by PSCOMP_1:def 6;



 A72: (g . r1) in ( Lower_Arc K) by A44;



 A73: (g . r2) in ( Lower_Arc K) by A44;

 reconsider gr1 = (g . r1) as Point of ( TOP-REAL 2) by A72;

 reconsider gr2 = (g . r2) as Point of ( TOP-REAL 2) by A73;

 now

 per cases by A12, A44, XBOOLE_0:def 3;

 case

 A74: (g . r1) in ( LSeg ( |[b, d]|, |[b, c]|)) & (g . r2) in ( LSeg ( |[b, d]|, |[b, c]|));

 then

 A75: (gr1 `1 ) = b by A2, Th1;

 (gr2 `1 ) = b by A2, A74, Th1;

 then |[(gr1 `1 ), (gr1 `2 )]| = (g . r2) by A69, A70, A71, A75, EUCLID: 53;

 then (g . r1) = (g . r2) by EUCLID: 53;

 hence thesis by A42, A43, FUNCT_1:def 4;

 end;

 case

 A76: (g . r1) in ( LSeg ( |[b, d]|, |[b, c]|)) & (g . r2) in ( LSeg ( |[b, c]|, |[a, c]|));

 then

 A77: (gr1 `1 ) = b by A2, Th1;



 A78: c <= (gr1 `2 ) by A2, A76, Th1;



 A79: (gr2 `2 ) = c by A1, A76, Th3;



 A80: (gr2 `1 ) <= b by A1, A76, Th3;



 A81: (b + (gr1 `2 )) = ((gr2 `1 ) + c) by A1, A69, A70, A71, A76, A77, Th3;

 A82:

 now

 assume b <> (gr2 `1 );

 then b > (gr2 `1 ) by A80, XXREAL_0: 1;

 hence contradiction by A78, A81, XREAL_1: 8;

 end;

 now

 assume (gr1 `2 ) <> c;

 then c < (gr1 `2 ) by A78, XXREAL_0: 1;

 hence contradiction by A69, A70, A71, A77, A79, A80, XREAL_1: 8;

 end;

 then |[(gr1 `1 ), (gr1 `2 )]| = (g . r2) by A77, A79, A82, EUCLID: 53;

 then (g . r1) = (g . r2) by EUCLID: 53;

 hence thesis by A42, A43, FUNCT_1:def 4;

 end;

 case

 A83: (g . r1) in ( LSeg ( |[b, c]|, |[a, c]|)) & (g . r2) in ( LSeg ( |[b, d]|, |[b, c]|));

 then

 A84: (gr2 `1 ) = b by A2, Th1;



 A85: c <= (gr2 `2 ) by A2, A83, Th1;



 A86: (gr1 `2 ) = c by A1, A83, Th3;



 A87: (gr1 `1 ) <= b by A1, A83, Th3;



 A88: (b + (gr2 `2 )) = ((gr1 `1 ) + c) by A1, A69, A70, A71, A83, A84, Th3;

 A89:

 now

 assume b <> (gr1 `1 );

 then b > (gr1 `1 ) by A87, XXREAL_0: 1;

 hence contradiction by A85, A88, XREAL_1: 8;

 end;

 now

 assume (gr2 `2 ) <> c;

 then c < (gr2 `2 ) by A85, XXREAL_0: 1;

 hence contradiction by A69, A70, A71, A84, A86, A87, XREAL_1: 8;

 end;

 then |[(gr2 `1 ), (gr2 `2 )]| = (g . r1) by A84, A86, A89, EUCLID: 53;

 then (g . r1) = (g . r2) by EUCLID: 53;

 hence thesis by A42, A43, FUNCT_1:def 4;

 end;

 case

 A90: (g . r1) in ( LSeg ( |[b, c]|, |[a, c]|)) & (g . r2) in ( LSeg ( |[b, c]|, |[a, c]|));

 then

 A91: (gr1 `2 ) = c by A1, Th3;

 (gr2 `2 ) = c by A1, A90, Th3;

 then |[(gr1 `1 ), (gr1 `2 )]| = (g . r2) by A69, A70, A71, A91, EUCLID: 53;

 then (g . r1) = (g . r2) by EUCLID: 53;

 hence thesis by A42, A43, FUNCT_1:def 4;

 end;

 end;

 hence thesis;

 end;



 A92: ( dom k) = [. 0 , 1.] by BORSUK_1: 40, FUNCT_2:def 1;

 then s1 in ( dom k) by A37, A38, XXREAL_1: 1;

 then rxc = s1 by A60, A64, A65, A66, A92;

 hence contradiction by A51, A64, XXREAL_1: 1;

 end;

 hence thesis;

 end;

 A93:

 now

 assume

 A94: p1 = ( W-min K);

 then (p1 `1 ) = a by A10, EUCLID: 52;

 then (p1 `1 ) = (p2 `1 ) by A9, A31, XXREAL_0: 1;

 then |[(p1 `1 ), (p1 `2 )]| = p2 by A6, A8, EUCLID: 53;

 hence contradiction by A32, A94, EUCLID: 53;

 end;

 LE (p1,p2,( Lower_Arc K),( E-max K),( W-min K)) by A3, A4, A13, A33, JORDAN5C:def 3;

 hence LE (p1,p2,K) by A3, A4, A13, A32, JORDAN6:def 10;

 thus thesis by A93;

 end;

 end;

 theorem :: JGRAPH_6:59



 Th59: for a,b,c,d be Real, p1,p2 be Point of ( TOP-REAL 2) st a < b & c < d & p1 in ( LSeg ( |[a, c]|, |[a, d]|)) holds LE (p1,p2,( rectangle (a,b,c,d))) iff p2 in ( LSeg ( |[a, c]|, |[a, d]|)) & (p1 `2 ) <= (p2 `2 ) or p2 in ( LSeg ( |[a, d]|, |[b, d]|)) or p2 in ( LSeg ( |[b, d]|, |[b, c]|)) or p2 in ( LSeg ( |[b, c]|, |[a, c]|)) & p2 <> ( W-min ( rectangle (a,b,c,d)))

 proof

 let a,b,c,d be Real, p1,p2 be Point of ( TOP-REAL 2);

 set K = ( rectangle (a,b,c,d));

 assume that

 A1: a < b and

 A2: c < d and

 A3: p1 in ( LSeg ( |[a, c]|, |[a, d]|));



 A4: K is being_simple_closed_curve by A1, A2, Th50;

 ( Upper_Arc K) = (( LSeg ( |[a, c]|, |[a, d]|)) \/ ( LSeg ( |[a, d]|, |[b, d]|))) by A1, A2, Th51;

 then

 A5: ( LSeg ( |[a, c]|, |[a, d]|)) c= ( Upper_Arc K) by XBOOLE_1: 7;



 A6: (p1 `1 ) = a by A2, A3, Th1;



 A7: c <= (p1 `2 ) by A2, A3, Th1;



 A8: (p1 `2 ) <= d by A2, A3, Th1;

 thus LE (p1,p2,K) implies p2 in ( LSeg ( |[a, c]|, |[a, d]|)) & (p1 `2 ) <= (p2 `2 ) or p2 in ( LSeg ( |[a, d]|, |[b, d]|)) or p2 in ( LSeg ( |[b, d]|, |[b, c]|)) or p2 in ( LSeg ( |[b, c]|, |[a, c]|)) & p2 <> ( W-min K)

 proof

 assume

 A9: LE (p1,p2,K);

 then

 A10: p1 in K by A4, JORDAN7: 5;



 A11: p2 in K by A4, A9, JORDAN7: 5;

 K = ((( LSeg ( |[a, c]|, |[a, d]|)) \/ ( LSeg ( |[a, d]|, |[b, d]|))) \/ (( LSeg ( |[a, c]|, |[b, c]|)) \/ ( LSeg ( |[b, c]|, |[b, d]|)))) by SPPOL_2:def 3

 .= (((( LSeg ( |[a, c]|, |[a, d]|)) \/ ( LSeg ( |[a, d]|, |[b, d]|))) \/ ( LSeg ( |[b, d]|, |[b, c]|))) \/ ( LSeg ( |[b, c]|, |[a, c]|))) by XBOOLE_1: 4;

 then p2 in ((( LSeg ( |[a, c]|, |[a, d]|)) \/ ( LSeg ( |[a, d]|, |[b, d]|))) \/ ( LSeg ( |[b, d]|, |[b, c]|))) or p2 in ( LSeg ( |[b, c]|, |[a, c]|)) by A11, XBOOLE_0:def 3;

 then

 A12: p2 in (( LSeg ( |[a, c]|, |[a, d]|)) \/ ( LSeg ( |[a, d]|, |[b, d]|))) or p2 in ( LSeg ( |[b, d]|, |[b, c]|)) or p2 in ( LSeg ( |[b, c]|, |[a, c]|)) by XBOOLE_0:def 3;

 now

 per cases by A12, XBOOLE_0:def 3;

 case p2 in ( LSeg ( |[a, c]|, |[a, d]|));

 hence thesis by A1, A2, A3, A9, Th55;

 end;

 case p2 in ( LSeg ( |[a, d]|, |[b, d]|));

 hence thesis;

 end;

 case p2 in ( LSeg ( |[b, d]|, |[b, c]|));

 hence thesis;

 end;

 case

 A13: p2 in ( LSeg ( |[b, c]|, |[a, c]|));

 now

 per cases ;

 case p2 = ( W-min K);

 then LE (p2,p1,K) by A4, A10, JORDAN7: 3;

 hence thesis by A1, A2, A3, A9, Th50, JORDAN6: 57;

 end;

 case p2 <> ( W-min K);

 hence thesis by A13;

 end;

 end;

 hence thesis;

 end;

 end;

 hence thesis;

 end;



 A14: ( W-min K) = |[a, c]| by A1, A2, Th46;

 thus p2 in ( LSeg ( |[a, c]|, |[a, d]|)) & (p1 `2 ) <= (p2 `2 ) or p2 in ( LSeg ( |[a, d]|, |[b, d]|)) or p2 in ( LSeg ( |[b, d]|, |[b, c]|)) or p2 in ( LSeg ( |[b, c]|, |[a, c]|)) & p2 <> ( W-min K) implies LE (p1,p2,K)

 proof

 assume that

 A15: p2 in ( LSeg ( |[a, c]|, |[a, d]|)) & (p1 `2 ) <= (p2 `2 ) or p2 in ( LSeg ( |[a, d]|, |[b, d]|)) or p2 in ( LSeg ( |[b, d]|, |[b, c]|)) or p2 in ( LSeg ( |[b, c]|, |[a, c]|)) & p2 <> ( W-min K);

 now

 per cases by A15;

 case p2 in ( LSeg ( |[a, c]|, |[a, d]|)) & (p1 `2 ) <= (p2 `2 );

 hence thesis by A1, A2, A3, Th55;

 end;

 case

 A16: p2 in ( LSeg ( |[a, d]|, |[b, d]|));

 then

 A17: (p2 `2 ) = d by A1, Th3;



 A18: a <= (p2 `1 ) by A1, A16, Th3;



 A19: ( Upper_Arc K) = (( LSeg ( |[a, c]|, |[a, d]|)) \/ ( LSeg ( |[a, d]|, |[b, d]|))) by A1, A2, Th51;

 then

 A20: p2 in ( Upper_Arc K) by A16, XBOOLE_0:def 3;



 A21: p1 in ( Upper_Arc K) by A3, A19, XBOOLE_0:def 3;

 for g be Function of I[01] , (( TOP-REAL 2) | ( Upper_Arc K)), s1,s2 be Real st g is being_homeomorphism & (g . 0 ) = ( W-min K) & (g . 1) = ( E-max K) & (g . s1) = p1 & 0 <= s1 & s1 <= 1 & (g . s2) = p2 & 0 <= s2 & s2 <= 1 holds s1 <= s2

 proof

 let g be Function of I[01] , (( TOP-REAL 2) | ( Upper_Arc K)), s1,s2 be Real;

 assume that

 A22: g is being_homeomorphism and

 A23: (g . 0 ) = ( W-min K) and (g . 1) = ( E-max K) and

 A24: (g . s1) = p1 and

 A25: 0 <= s1 and

 A26: s1 <= 1 and

 A27: (g . s2) = p2 and

 A28: 0 <= s2 and

 A29: s2 <= 1;



 A30: ( dom g) = the carrier of I[01] by FUNCT_2:def 1;



 A31: g is one-to-one by A22, TOPS_2:def 5;



 A32: the carrier of (( TOP-REAL 2) | ( Upper_Arc K)) = ( Upper_Arc K) by PRE_TOPC: 8;

 then

 reconsider g1 = g as Function of I[01] , ( TOP-REAL 2) by FUNCT_2: 7;

 g is continuous by A22, TOPS_2:def 5;

 then

 A33: g1 is continuous by PRE_TOPC: 26;

 reconsider h1 = proj1 as Function of ( TOP-REAL 2), R^1 by TOPMETR: 17;

 reconsider h2 = proj2 as Function of ( TOP-REAL 2), R^1 by TOPMETR: 17;

 reconsider hh1 = h1 as Function of the TopStruct of ( TOP-REAL 2), R^1 ;

 reconsider hh2 = h2 as Function of the TopStruct of ( TOP-REAL 2), R^1 ;



 A34: the TopStruct of ( TOP-REAL 2) = ( the TopStruct of ( TOP-REAL 2) | ( [#] the TopStruct of ( TOP-REAL 2))) by TSEP_1: 3

 .= the TopStruct of (( TOP-REAL 2) | ( [#] ( TOP-REAL 2))) by PRE_TOPC: 36

 .= (( TOP-REAL 2) | ( [#] ( TOP-REAL 2)));

 then (for p be Point of (( TOP-REAL 2) | ( [#] ( TOP-REAL 2))) holds (hh1 . p) = ( proj1 . p)) implies hh1 is continuous by JGRAPH_2: 29;

 then

 A35: h1 is continuous by PRE_TOPC: 32;

 (for p be Point of (( TOP-REAL 2) | ( [#] ( TOP-REAL 2))) holds (hh2 . p) = ( proj2 . p)) implies hh2 is continuous by A34, JGRAPH_2: 30;

 then (for p be Point of (( TOP-REAL 2) | ( [#] ( TOP-REAL 2))) holds (hh2 . p) = ( proj2 . p)) implies h2 is continuous by PRE_TOPC: 32;

 then

 consider h be Function of ( TOP-REAL 2), R^1 such that

 A36: for p be Point of ( TOP-REAL 2), r1,r2 be Real st (h1 . p) = r1 & (h2 . p) = r2 holds (h . p) = (r1 + r2) and

 A37: h is continuous by A35, JGRAPH_2: 19;

 reconsider k = (h * g1) as Function of I[01] , R^1 ;



 A38: ( W-min K) = |[a, c]| by A1, A2, Th46;

 now

 assume

 A39: s1 > s2;



 A40: ( dom g) = [. 0 , 1.] by BORSUK_1: 40, FUNCT_2:def 1;

 0 in [. 0 , 1.] by XXREAL_1: 1;



 then

 A41: (k . 0 ) = (h . ( W-min K)) by A23, A40, FUNCT_1: 13

 .= ((h1 . ( W-min K)) + (h2 . ( W-min K))) by A36

 .= ((( W-min K) `1 ) + ( proj2 . ( W-min K))) by PSCOMP_1:def 5

 .= ((( W-min K) `1 ) + (( W-min K) `2 )) by PSCOMP_1:def 6

 .= ((( W-min K) `1 ) + c) by A38, EUCLID: 52

 .= (a + c) by A38, EUCLID: 52;

 s1 in [. 0 , 1.] by A25, A26, XXREAL_1: 1;



 then

 A42: (k . s1) = (h . p1) by A24, A40, FUNCT_1: 13

 .= (( proj1 . p1) + ( proj2 . p1)) by A36

 .= ((p1 `1 ) + ( proj2 . p1)) by PSCOMP_1:def 5

 .= (a + (p1 `2 )) by A6, PSCOMP_1:def 6;



 A43: s2 in [. 0 , 1.] by A28, A29, XXREAL_1: 1;



 then

 A44: (k . s2) = (h . p2) by A27, A40, FUNCT_1: 13

 .= (( proj1 . p2) + ( proj2 . p2)) by A36

 .= ((p2 `1 ) + ( proj2 . p2)) by PSCOMP_1:def 5

 .= ((p2 `1 ) + d) by A17, PSCOMP_1:def 6;



 A45: (k . 0 ) <= (k . s1) by A7, A41, A42, XREAL_1: 7;



 A46: (k . s1) <= (k . s2) by A8, A18, A42, A44, XREAL_1: 7;



 A47: 0 in [. 0 , 1.] by XXREAL_1: 1;

 then

 A48: [. 0 , s2.] c= [. 0 , 1.] by A43, XXREAL_2:def 12;

 reconsider B = [. 0 , s2.] as Subset of I[01] by A43, A47, BORSUK_1: 40, XXREAL_2:def 12;



 A49: B is connected by A28, A43, A47, BORSUK_1: 40, BORSUK_4: 24;



 A50: 0 in B by A28, XXREAL_1: 1;



 A51: s2 in B by A28, XXREAL_1: 1;

 consider xc be Point of I[01] such that

 A52: xc in B and

 A53: (k . xc) = (k . s1) by A33, A37, A45, A46, A49, A50, A51, TOPREAL5: 5;

 reconsider rxc = xc as Real;



 A54: for x1,x2 be set st x1 in ( dom k) & x2 in ( dom k) & (k . x1) = (k . x2) holds x1 = x2

 proof

 let x1,x2 be set;

 assume that

 A55: x1 in ( dom k) and

 A56: x2 in ( dom k) and

 A57: (k . x1) = (k . x2);

 reconsider r1 = x1 as Point of I[01] by A55;

 reconsider r2 = x2 as Point of I[01] by A56;



 A58: (k . x1) = (h . (g1 . x1)) by A55, FUNCT_1: 12

 .= ((h1 . (g1 . r1)) + (h2 . (g1 . r1))) by A36

 .= (((g1 . r1) `1 ) + ( proj2 . (g1 . r1))) by PSCOMP_1:def 5

 .= (((g1 . r1) `1 ) + ((g1 . r1) `2 )) by PSCOMP_1:def 6;



 A59: (k . x2) = (h . (g1 . x2)) by A56, FUNCT_1: 12

 .= ((h1 . (g1 . r2)) + (h2 . (g1 . r2))) by A36

 .= (((g1 . r2) `1 ) + ( proj2 . (g1 . r2))) by PSCOMP_1:def 5

 .= (((g1 . r2) `1 ) + ((g1 . r2) `2 )) by PSCOMP_1:def 6;



 A60: (g . r1) in ( Upper_Arc K) by A32;



 A61: (g . r2) in ( Upper_Arc K) by A32;

 reconsider gr1 = (g . r1) as Point of ( TOP-REAL 2) by A60;

 reconsider gr2 = (g . r2) as Point of ( TOP-REAL 2) by A61;

 now

 per cases by A19, A32, XBOOLE_0:def 3;

 case

 A62: (g . r1) in ( LSeg ( |[a, c]|, |[a, d]|)) & (g . r2) in ( LSeg ( |[a, c]|, |[a, d]|));

 then

 A63: (gr1 `1 ) = a by A2, Th1;

 (gr2 `1 ) = a by A2, A62, Th1;

 then |[(gr1 `1 ), (gr1 `2 )]| = (g . r2) by A57, A58, A59, A63, EUCLID: 53;

 then (g . r1) = (g . r2) by EUCLID: 53;

 hence thesis by A30, A31, FUNCT_1:def 4;

 end;

 case

 A64: (g . r1) in ( LSeg ( |[a, c]|, |[a, d]|)) & (g . r2) in ( LSeg ( |[a, d]|, |[b, d]|));

 then

 A65: (gr1 `1 ) = a by A2, Th1;



 A66: (gr1 `2 ) <= d by A2, A64, Th1;



 A67: (gr2 `2 ) = d by A1, A64, Th3;



 A68: a <= (gr2 `1 ) by A1, A64, Th3;



 A69: (a + (gr1 `2 )) = ((gr2 `1 ) + d) by A1, A57, A58, A59, A64, A65, Th3;

 A70:

 now

 assume a <> (gr2 `1 );

 then a < (gr2 `1 ) by A68, XXREAL_0: 1;

 hence contradiction by A66, A69, XREAL_1: 8;

 end;

 now

 assume (gr1 `2 ) <> d;

 then d > (gr1 `2 ) by A66, XXREAL_0: 1;

 hence contradiction by A57, A58, A59, A65, A67, A68, XREAL_1: 8;

 end;

 then |[(gr1 `1 ), (gr1 `2 )]| = (g . r2) by A65, A67, A70, EUCLID: 53;

 then (g . r1) = (g . r2) by EUCLID: 53;

 hence thesis by A30, A31, FUNCT_1:def 4;

 end;

 case

 A71: (g . r1) in ( LSeg ( |[a, d]|, |[b, d]|)) & (g . r2) in ( LSeg ( |[a, c]|, |[a, d]|));

 then

 A72: (gr2 `1 ) = a by A2, Th1;



 A73: (gr2 `2 ) <= d by A2, A71, Th1;



 A74: (gr1 `2 ) = d by A1, A71, Th3;



 A75: a <= (gr1 `1 ) by A1, A71, Th3;



 A76: (a + (gr2 `2 )) = ((gr1 `1 ) + d) by A1, A57, A58, A59, A71, A72, Th3;

 A77:

 now

 assume a <> (gr1 `1 );

 then a < (gr1 `1 ) by A75, XXREAL_0: 1;

 hence contradiction by A73, A76, XREAL_1: 8;

 end;

 now

 assume (gr2 `2 ) <> d;

 then d > (gr2 `2 ) by A73, XXREAL_0: 1;

 hence contradiction by A57, A58, A59, A72, A74, A75, XREAL_1: 8;

 end;

 then |[(gr2 `1 ), (gr2 `2 )]| = (g . r1) by A72, A74, A77, EUCLID: 53;

 then (g . r1) = (g . r2) by EUCLID: 53;

 hence thesis by A30, A31, FUNCT_1:def 4;

 end;

 case

 A78: (g . r1) in ( LSeg ( |[a, d]|, |[b, d]|)) & (g . r2) in ( LSeg ( |[a, d]|, |[b, d]|));

 then

 A79: (gr1 `2 ) = d by A1, Th3;

 (gr2 `2 ) = d by A1, A78, Th3;

 then |[(gr1 `1 ), (gr1 `2 )]| = (g . r2) by A57, A58, A59, A79, EUCLID: 53;

 then (g . r1) = (g . r2) by EUCLID: 53;

 hence thesis by A30, A31, FUNCT_1:def 4;

 end;

 end;

 hence thesis;

 end;



 A80: ( dom k) = [. 0 , 1.] by BORSUK_1: 40, FUNCT_2:def 1;

 then s1 in ( dom k) by A25, A26, XXREAL_1: 1;

 then rxc = s1 by A48, A52, A53, A54, A80;

 hence contradiction by A39, A52, XXREAL_1: 1;

 end;

 hence thesis;

 end;

 then LE (p1,p2,( Upper_Arc K),( W-min K),( E-max K)) by A20, A21, JORDAN5C:def 3;

 hence thesis by A20, A21, JORDAN6:def 10;

 end;

 case

 A81: p2 in ( LSeg ( |[b, d]|, |[b, c]|));

 then

 A82: (p2 `1 ) = b by TOPREAL3: 11;

 ( Lower_Arc K) = (( LSeg ( |[b, d]|, |[b, c]|)) \/ ( LSeg ( |[b, c]|, |[a, c]|))) by A1, A2, Th52;

 then

 A83: ( LSeg ( |[b, d]|, |[b, c]|)) c= ( Lower_Arc K) by XBOOLE_1: 7;

 p2 <> ( W-min K) by A1, A14, A82, EUCLID: 52;

 hence thesis by A3, A5, A81, A83, JORDAN6:def 10;

 end;

 case

 A84: p2 in ( LSeg ( |[b, c]|, |[a, c]|)) & p2 <> ( W-min K);

 ( Lower_Arc K) = (( LSeg ( |[b, d]|, |[b, c]|)) \/ ( LSeg ( |[b, c]|, |[a, c]|))) by A1, A2, Th52;

 then ( LSeg ( |[b, c]|, |[a, c]|)) c= ( Lower_Arc K) by XBOOLE_1: 7;

 hence thesis by A3, A5, A84, JORDAN6:def 10;

 end;

 end;

 hence thesis;

 end;

 end;

 theorem :: JGRAPH_6:60



 Th60: for a,b,c,d be Real, p1,p2 be Point of ( TOP-REAL 2) st a < b & c < d & p1 in ( LSeg ( |[a, d]|, |[b, d]|)) holds LE (p1,p2,( rectangle (a,b,c,d))) iff p2 in ( LSeg ( |[a, d]|, |[b, d]|)) & (p1 `1 ) <= (p2 `1 ) or p2 in ( LSeg ( |[b, d]|, |[b, c]|)) or p2 in ( LSeg ( |[b, c]|, |[a, c]|)) & p2 <> ( W-min ( rectangle (a,b,c,d)))

 proof

 let a,b,c,d be Real, p1,p2 be Point of ( TOP-REAL 2);

 set K = ( rectangle (a,b,c,d));

 assume that

 A1: a < b and

 A2: c < d and

 A3: p1 in ( LSeg ( |[a, d]|, |[b, d]|));



 A4: K is being_simple_closed_curve by A1, A2, Th50;

 ( Upper_Arc K) = (( LSeg ( |[a, c]|, |[a, d]|)) \/ ( LSeg ( |[a, d]|, |[b, d]|))) by A1, A2, Th51;

 then

 A5: ( LSeg ( |[a, d]|, |[b, d]|)) c= ( Upper_Arc K) by XBOOLE_1: 7;



 A6: (p1 `2 ) = d by A1, A3, Th3;



 A7: a <= (p1 `1 ) by A1, A3, Th3;

 thus LE (p1,p2,K) implies p2 in ( LSeg ( |[a, d]|, |[b, d]|)) & (p1 `1 ) <= (p2 `1 ) or p2 in ( LSeg ( |[b, d]|, |[b, c]|)) or p2 in ( LSeg ( |[b, c]|, |[a, c]|)) & p2 <> ( W-min K)

 proof

 assume

 A8: LE (p1,p2,K);

 then

 A9: p1 in K by A4, JORDAN7: 5;



 A10: p2 in K by A4, A8, JORDAN7: 5;

 K = ((( LSeg ( |[a, c]|, |[a, d]|)) \/ ( LSeg ( |[a, d]|, |[b, d]|))) \/ (( LSeg ( |[a, c]|, |[b, c]|)) \/ ( LSeg ( |[b, c]|, |[b, d]|)))) by SPPOL_2:def 3

 .= (((( LSeg ( |[a, c]|, |[a, d]|)) \/ ( LSeg ( |[a, d]|, |[b, d]|))) \/ ( LSeg ( |[b, d]|, |[b, c]|))) \/ ( LSeg ( |[b, c]|, |[a, c]|))) by XBOOLE_1: 4;

 then p2 in ((( LSeg ( |[a, c]|, |[a, d]|)) \/ ( LSeg ( |[a, d]|, |[b, d]|))) \/ ( LSeg ( |[b, d]|, |[b, c]|))) or p2 in ( LSeg ( |[b, c]|, |[a, c]|)) by A10, XBOOLE_0:def 3;

 then

 A11: p2 in (( LSeg ( |[a, c]|, |[a, d]|)) \/ ( LSeg ( |[a, d]|, |[b, d]|))) or p2 in ( LSeg ( |[b, d]|, |[b, c]|)) or p2 in ( LSeg ( |[b, c]|, |[a, c]|)) by XBOOLE_0:def 3;

 now

 per cases by A11, XBOOLE_0:def 3;

 case p2 in ( LSeg ( |[a, c]|, |[a, d]|));

 then LE (p2,p1,K) by A1, A2, A3, Th59;

 hence thesis by A1, A2, A3, A8, Th50, JORDAN6: 57;

 end;

 case p2 in ( LSeg ( |[a, d]|, |[b, d]|));

 hence thesis by A1, A2, A3, A8, Th56;

 end;

 case p2 in ( LSeg ( |[b, d]|, |[b, c]|));

 hence thesis;

 end;

 case

 A12: p2 in ( LSeg ( |[b, c]|, |[a, c]|));

 now

 per cases ;

 case p2 = ( W-min K);

 then LE (p2,p1,K) by A4, A9, JORDAN7: 3;

 hence thesis by A1, A2, A3, A8, Th50, JORDAN6: 57;

 end;

 case p2 <> ( W-min K);

 hence thesis by A12;

 end;

 end;

 hence thesis;

 end;

 end;

 hence thesis;

 end;



 A13: ( W-min K) = |[a, c]| by A1, A2, Th46;

 thus p2 in ( LSeg ( |[a, d]|, |[b, d]|)) & (p1 `1 ) <= (p2 `1 ) or p2 in ( LSeg ( |[b, d]|, |[b, c]|)) or p2 in ( LSeg ( |[b, c]|, |[a, c]|)) & p2 <> ( W-min K) implies LE (p1,p2,K)

 proof

 assume that

 A14: p2 in ( LSeg ( |[a, d]|, |[b, d]|)) & (p1 `1 ) <= (p2 `1 ) or p2 in ( LSeg ( |[b, d]|, |[b, c]|)) or p2 in ( LSeg ( |[b, c]|, |[a, c]|)) & p2 <> ( W-min K);

 now

 per cases by A14;

 case

 A15: p2 in ( LSeg ( |[a, d]|, |[b, d]|)) & (p1 `1 ) <= (p2 `1 );

 then

 A16: (p2 `2 ) = d by A1, Th3;



 A17: ( Upper_Arc K) = (( LSeg ( |[a, c]|, |[a, d]|)) \/ ( LSeg ( |[a, d]|, |[b, d]|))) by A1, A2, Th51;

 then

 A18: p2 in ( Upper_Arc K) by A15, XBOOLE_0:def 3;



 A19: p1 in ( Upper_Arc K) by A3, A17, XBOOLE_0:def 3;

 for g be Function of I[01] , (( TOP-REAL 2) | ( Upper_Arc K)), s1,s2 be Real st g is being_homeomorphism & (g . 0 ) = ( W-min K) & (g . 1) = ( E-max K) & (g . s1) = p1 & 0 <= s1 & s1 <= 1 & (g . s2) = p2 & 0 <= s2 & s2 <= 1 holds s1 <= s2

 proof

 let g be Function of I[01] , (( TOP-REAL 2) | ( Upper_Arc K)), s1,s2 be Real;

 assume that

 A20: g is being_homeomorphism and

 A21: (g . 0 ) = ( W-min K) and (g . 1) = ( E-max K) and

 A22: (g . s1) = p1 and

 A23: 0 <= s1 and

 A24: s1 <= 1 and

 A25: (g . s2) = p2 and

 A26: 0 <= s2 and

 A27: s2 <= 1;



 A28: ( dom g) = the carrier of I[01] by FUNCT_2:def 1;



 A29: g is one-to-one by A20, TOPS_2:def 5;



 A30: the carrier of (( TOP-REAL 2) | ( Upper_Arc K)) = ( Upper_Arc K) by PRE_TOPC: 8;

 then

 reconsider g1 = g as Function of I[01] , ( TOP-REAL 2) by FUNCT_2: 7;

 g is continuous by A20, TOPS_2:def 5;

 then

 A31: g1 is continuous by PRE_TOPC: 26;

 reconsider h1 = proj1 as Function of ( TOP-REAL 2), R^1 by TOPMETR: 17;

 reconsider h2 = proj2 as Function of ( TOP-REAL 2), R^1 by TOPMETR: 17;

 reconsider hh1 = h1 as Function of the TopStruct of ( TOP-REAL 2), R^1 ;

 reconsider hh2 = h2 as Function of the TopStruct of ( TOP-REAL 2), R^1 ;



 A32: the TopStruct of ( TOP-REAL 2) = ( the TopStruct of ( TOP-REAL 2) | ( [#] the TopStruct of ( TOP-REAL 2))) by TSEP_1: 3

 .= the TopStruct of (( TOP-REAL 2) | ( [#] ( TOP-REAL 2))) by PRE_TOPC: 36

 .= (( TOP-REAL 2) | ( [#] ( TOP-REAL 2)));

 then (for p be Point of (( TOP-REAL 2) | ( [#] ( TOP-REAL 2))) holds (hh1 . p) = ( proj1 . p)) implies hh1 is continuous by JGRAPH_2: 29;

 then

 A33: (for p be Point of (( TOP-REAL 2) | ( [#] ( TOP-REAL 2))) holds (hh1 . p) = ( proj1 . p)) implies h1 is continuous by PRE_TOPC: 32;

 (for p be Point of (( TOP-REAL 2) | ( [#] ( TOP-REAL 2))) holds (hh2 . p) = ( proj2 . p)) implies hh2 is continuous by A32, JGRAPH_2: 30;

 then (for p be Point of (( TOP-REAL 2) | ( [#] ( TOP-REAL 2))) holds (hh2 . p) = ( proj2 . p)) implies h2 is continuous by PRE_TOPC: 32;

 then

 consider h be Function of ( TOP-REAL 2), R^1 such that

 A34: for p be Point of ( TOP-REAL 2), r1,r2 be Real st (h1 . p) = r1 & (h2 . p) = r2 holds (h . p) = (r1 + r2) and

 A35: h is continuous by A33, JGRAPH_2: 19;

 reconsider k = (h * g1) as Function of I[01] , R^1 ;



 A36: ( W-min K) = |[a, c]| by A1, A2, Th46;

 now

 assume

 A37: s1 > s2;



 A38: ( dom g) = [. 0 , 1.] by BORSUK_1: 40, FUNCT_2:def 1;

 0 in [. 0 , 1.] by XXREAL_1: 1;



 then

 A39: (k . 0 ) = (h . ( W-min K)) by A21, A38, FUNCT_1: 13

 .= ((h1 . ( W-min K)) + (h2 . ( W-min K))) by A34

 .= ((( W-min K) `1 ) + ( proj2 . ( W-min K))) by PSCOMP_1:def 5

 .= ((( W-min K) `1 ) + (( W-min K) `2 )) by PSCOMP_1:def 6

 .= ((( W-min K) `1 ) + c) by A36, EUCLID: 52

 .= (a + c) by A36, EUCLID: 52;

 s1 in [. 0 , 1.] by A23, A24, XXREAL_1: 1;



 then

 A40: (k . s1) = (h . p1) by A22, A38, FUNCT_1: 13

 .= (( proj1 . p1) + ( proj2 . p1)) by A34

 .= ((p1 `1 ) + ( proj2 . p1)) by PSCOMP_1:def 5

 .= ((p1 `1 ) + d) by A6, PSCOMP_1:def 6;



 A41: s2 in [. 0 , 1.] by A26, A27, XXREAL_1: 1;



 then

 A42: (k . s2) = (h . p2) by A25, A38, FUNCT_1: 13

 .= (( proj1 . p2) + ( proj2 . p2)) by A34

 .= ((p2 `1 ) + ( proj2 . p2)) by PSCOMP_1:def 5

 .= ((p2 `1 ) + d) by A16, PSCOMP_1:def 6;



 A43: (k . 0 ) <= (k . s1) by A2, A7, A39, A40, XREAL_1: 7;



 A44: (k . s1) <= (k . s2) by A15, A40, A42, XREAL_1: 7;



 A45: 0 in [. 0 , 1.] by XXREAL_1: 1;

 then

 A46: [. 0 , s2.] c= [. 0 , 1.] by A41, XXREAL_2:def 12;

 reconsider B = [. 0 , s2.] as Subset of I[01] by A41, A45, BORSUK_1: 40, XXREAL_2:def 12;



 A47: B is connected by A26, A41, A45, BORSUK_1: 40, BORSUK_4: 24;



 A48: 0 in B by A26, XXREAL_1: 1;



 A49: s2 in B by A26, XXREAL_1: 1;

 consider xc be Point of I[01] such that

 A50: xc in B and

 A51: (k . xc) = (k . s1) by A31, A35, A43, A44, A47, A48, A49, TOPREAL5: 5;

 reconsider rxc = xc as Real;



 A52: for x1,x2 be set st x1 in ( dom k) & x2 in ( dom k) & (k . x1) = (k . x2) holds x1 = x2

 proof

 let x1,x2 be set;

 assume that

 A53: x1 in ( dom k) and

 A54: x2 in ( dom k) and

 A55: (k . x1) = (k . x2);

 reconsider r1 = x1 as Point of I[01] by A53;

 reconsider r2 = x2 as Point of I[01] by A54;



 A56: (k . x1) = (h . (g1 . x1)) by A53, FUNCT_1: 12

 .= ((h1 . (g1 . r1)) + (h2 . (g1 . r1))) by A34

 .= (((g1 . r1) `1 ) + ( proj2 . (g1 . r1))) by PSCOMP_1:def 5

 .= (((g1 . r1) `1 ) + ((g1 . r1) `2 )) by PSCOMP_1:def 6;



 A57: (k . x2) = (h . (g1 . x2)) by A54, FUNCT_1: 12

 .= ((h1 . (g1 . r2)) + (h2 . (g1 . r2))) by A34

 .= (((g1 . r2) `1 ) + ( proj2 . (g1 . r2))) by PSCOMP_1:def 5

 .= (((g1 . r2) `1 ) + ((g1 . r2) `2 )) by PSCOMP_1:def 6;



 A58: (g . r1) in ( Upper_Arc K) by A30;



 A59: (g . r2) in ( Upper_Arc K) by A30;

 reconsider gr1 = (g . r1) as Point of ( TOP-REAL 2) by A58;

 reconsider gr2 = (g . r2) as Point of ( TOP-REAL 2) by A59;

 now

 per cases by A17, A30, XBOOLE_0:def 3;

 case

 A60: (g . r1) in ( LSeg ( |[a, c]|, |[a, d]|)) & (g . r2) in ( LSeg ( |[a, c]|, |[a, d]|));

 then

 A61: (gr1 `1 ) = a by A2, Th1;

 (gr2 `1 ) = a by A2, A60, Th1;

 then |[(gr1 `1 ), (gr1 `2 )]| = (g . r2) by A55, A56, A57, A61, EUCLID: 53;

 then (g . r1) = (g . r2) by EUCLID: 53;

 hence thesis by A28, A29, FUNCT_1:def 4;

 end;

 case

 A62: (g . r1) in ( LSeg ( |[a, c]|, |[a, d]|)) & (g . r2) in ( LSeg ( |[a, d]|, |[b, d]|));

 then

 A63: (gr1 `1 ) = a by A2, Th1;



 A64: (gr1 `2 ) <= d by A2, A62, Th1;



 A65: (gr2 `2 ) = d by A1, A62, Th3;



 A66: a <= (gr2 `1 ) by A1, A62, Th3;



 A67: (a + (gr1 `2 )) = ((gr2 `1 ) + d) by A1, A55, A56, A57, A62, A63, Th3;

 A68:

 now

 assume a <> (gr2 `1 );

 then a < (gr2 `1 ) by A66, XXREAL_0: 1;

 hence contradiction by A64, A67, XREAL_1: 8;

 end;

 now

 assume (gr1 `2 ) <> d;

 then d > (gr1 `2 ) by A64, XXREAL_0: 1;

 hence contradiction by A55, A56, A57, A63, A65, A66, XREAL_1: 8;

 end;

 then |[(gr1 `1 ), (gr1 `2 )]| = (g . r2) by A63, A65, A68, EUCLID: 53;

 then (g . r1) = (g . r2) by EUCLID: 53;

 hence thesis by A28, A29, FUNCT_1:def 4;

 end;

 case

 A69: (g . r1) in ( LSeg ( |[a, d]|, |[b, d]|)) & (g . r2) in ( LSeg ( |[a, c]|, |[a, d]|));

 then

 A70: (gr2 `1 ) = a by A2, Th1;



 A71: (gr2 `2 ) <= d by A2, A69, Th1;



 A72: (gr1 `2 ) = d by A1, A69, Th3;



 A73: a <= (gr1 `1 ) by A1, A69, Th3;



 A74: (a + (gr2 `2 )) = ((gr1 `1 ) + d) by A1, A55, A56, A57, A69, A70, Th3;

 A75:

 now

 assume a <> (gr1 `1 );

 then a < (gr1 `1 ) by A73, XXREAL_0: 1;

 hence contradiction by A71, A74, XREAL_1: 8;

 end;

 now

 assume (gr2 `2 ) <> d;

 then d > (gr2 `2 ) by A71, XXREAL_0: 1;

 hence contradiction by A55, A56, A57, A70, A72, A73, XREAL_1: 8;

 end;

 then |[(gr2 `1 ), (gr2 `2 )]| = (g . r1) by A70, A72, A75, EUCLID: 53;

 then (g . r1) = (g . r2) by EUCLID: 53;

 hence thesis by A28, A29, FUNCT_1:def 4;

 end;

 case

 A76: (g . r1) in ( LSeg ( |[a, d]|, |[b, d]|)) & (g . r2) in ( LSeg ( |[a, d]|, |[b, d]|));

 then

 A77: (gr1 `2 ) = d by A1, Th3;

 (gr2 `2 ) = d by A1, A76, Th3;

 then |[(gr1 `1 ), (gr1 `2 )]| = (g . r2) by A55, A56, A57, A77, EUCLID: 53;

 then (g . r1) = (g . r2) by EUCLID: 53;

 hence thesis by A28, A29, FUNCT_1:def 4;

 end;

 end;

 hence thesis;

 end;



 A78: ( dom k) = [. 0 , 1.] by BORSUK_1: 40, FUNCT_2:def 1;

 then s1 in ( dom k) by A23, A24, XXREAL_1: 1;

 then rxc = s1 by A46, A50, A51, A52, A78;

 hence contradiction by A37, A50, XXREAL_1: 1;

 end;

 hence thesis;

 end;

 then LE (p1,p2,( Upper_Arc K),( W-min K),( E-max K)) by A18, A19, JORDAN5C:def 3;

 hence thesis by A18, A19, JORDAN6:def 10;

 end;

 case

 A79: p2 in ( LSeg ( |[b, d]|, |[b, c]|));

 then

 A80: (p2 `1 ) = b by TOPREAL3: 11;

 ( Lower_Arc K) = (( LSeg ( |[b, d]|, |[b, c]|)) \/ ( LSeg ( |[b, c]|, |[a, c]|))) by A1, A2, Th52;

 then

 A81: ( LSeg ( |[b, d]|, |[b, c]|)) c= ( Lower_Arc K) by XBOOLE_1: 7;

 p2 <> ( W-min K) by A1, A13, A80, EUCLID: 52;

 hence thesis by A3, A5, A79, A81, JORDAN6:def 10;

 end;

 case

 A82: p2 in ( LSeg ( |[b, c]|, |[a, c]|)) & p2 <> ( W-min K);

 ( Lower_Arc K) = (( LSeg ( |[b, d]|, |[b, c]|)) \/ ( LSeg ( |[b, c]|, |[a, c]|))) by A1, A2, Th52;

 then ( LSeg ( |[b, c]|, |[a, c]|)) c= ( Lower_Arc K) by XBOOLE_1: 7;

 hence thesis by A3, A5, A82, JORDAN6:def 10;

 end;

 end;

 hence thesis;

 end;

 end;

 theorem :: JGRAPH_6:61



 Th61: for a,b,c,d be Real, p1,p2 be Point of ( TOP-REAL 2) st a = (p2 `2 ) or p2 in ( LSeg ( |[b, c]|, |[a, c]|)) & p2 <> ( W-min ( rectangle (a,b,c,d)))

 proof

 let a,b,c,d be Real, p1,p2 be Point of ( TOP-REAL 2);

 set K = ( rectangle (a,b,c,d));

 assume that

 A1: a < b and

 A2: c < d and

 A3: p1 in ( LSeg ( |[b, d]|, |[b, c]|));



 A4: K is being_simple_closed_curve by A1, A2, Th50;



 A5: (p1 `1 ) = b by A2, A3, Th1;



 A6: c <= (p1 `2 ) by A2, A3, Th1;



 A7: (p1 `2 ) <= d by A2, A3, Th1;

 thus LE (p1,p2,K) implies p2 in ( LSeg ( |[b, d]|, |[b, c]|)) & (p1 `2 ) >= (p2 `2 ) or p2 in ( LSeg ( |[b, c]|, |[a, c]|)) & p2 <> ( W-min K)

 proof

 assume

 A8: LE (p1,p2,K);

 then

 A9: p1 in K by A4, JORDAN7: 5;



 A10: p2 in K by A4, A8, JORDAN7: 5;

 K = ((( LSeg ( |[a, c]|, |[a, d]|)) \/ ( LSeg ( |[a, d]|, |[b, d]|))) \/ (( LSeg ( |[a, c]|, |[b, c]|)) \/ ( LSeg ( |[b, c]|, |[b, d]|)))) by SPPOL_2:def 3

 .= (((( LSeg ( |[a, c]|, |[a, d]|)) \/ ( LSeg ( |[a, d]|, |[b, d]|))) \/ ( LSeg ( |[b, d]|, |[b, c]|))) \/ ( LSeg ( |[b, c]|, |[a, c]|))) by XBOOLE_1: 4;

 then p2 in ((( LSeg ( |[a, c]|, |[a, d]|)) \/ ( LSeg ( |[a, d]|, |[b, d]|))) \/ ( LSeg ( |[b, d]|, |[b, c]|))) or p2 in ( LSeg ( |[b, c]|, |[a, c]|)) by A10, XBOOLE_0:def 3;

 then

 A11: p2 in (( LSeg ( |[a, c]|, |[a, d]|)) \/ ( LSeg ( |[a, d]|, |[b, d]|))) or p2 in ( LSeg ( |[b, d]|, |[b, c]|)) or p2 in ( LSeg ( |[b, c]|, |[a, c]|)) by XBOOLE_0:def 3;

 now

 per cases by A11, XBOOLE_0:def 3;

 case p2 in ( LSeg ( |[a, c]|, |[a, d]|));

 then LE (p2,p1,K) by A1, A2, A3, Th59;

 hence thesis by A1, A2, A3, A8, Th50, JORDAN6: 57;

 end;

 case p2 in ( LSeg ( |[a, d]|, |[b, d]|));

 then LE (p2,p1,K) by A1, A2, A3, Th60;

 hence thesis by A1, A2, A3, A8, Th50, JORDAN6: 57;

 end;

 case p2 in ( LSeg ( |[b, d]|, |[b, c]|));

 hence thesis by A1, A2, A3, A8, Th57;

 end;

 case

 A12: p2 in ( LSeg ( |[b, c]|, |[a, c]|));

 now

 per cases ;

 case p2 = ( W-min K);

 then LE (p2,p1,K) by A4, A9, JORDAN7: 3;

 hence thesis by A1, A2, A3, A8, Th50, JORDAN6: 57;

 end;

 case p2 <> ( W-min K);

 hence thesis by A12;

 end;

 end;

 hence thesis;

 end;

 end;

 hence thesis;

 end;

 thus p2 in ( LSeg ( |[b, d]|, |[b, c]|)) & (p1 `2 ) >= (p2 `2 ) or p2 in ( LSeg ( |[b, c]|, |[a, c]|)) & p2 <> ( W-min K) implies LE (p1,p2,K)

 proof

 assume that

 A13: p2 in ( LSeg ( |[b, d]|, |[b, c]|)) & (p1 `2 ) >= (p2 `2 ) or p2 in ( LSeg ( |[b, c]|, |[a, c]|)) & p2 <> ( W-min K);

 now

 per cases by A13;

 case

 A14: p2 in ( LSeg ( |[b, d]|, |[b, c]|)) & (p1 `2 ) >= (p2 `2 );

 then

 A15: (p2 `1 ) = b by A2, Th1;

 ( W-min K) = |[a, c]| by A1, A2, Th46;

 then

 A16: p2 <> ( W-min K) by A1, A15, EUCLID: 52;



 A17: ( Lower_Arc K) = (( LSeg ( |[b, d]|, |[b, c]|)) \/ ( LSeg ( |[b, c]|, |[a, c]|))) by A1, A2, Th52;

 then

 A18: p2 in ( Lower_Arc K) by A14, XBOOLE_0:def 3;



 A19: p1 in ( Lower_Arc K) by A3, A17, XBOOLE_0:def 3;

 for g be Function of I[01] , (( TOP-REAL 2) | ( Lower_Arc K)), s1,s2 be Real st g is being_homeomorphism & (g . 0 ) = ( E-max K) & (g . 1) = ( W-min K) & (g . s1) = p1 & 0 <= s1 & s1 <= 1 & (g . s2) = p2 & 0 <= s2 & s2 <= 1 holds s1 <= s2

 proof

 let g be Function of I[01] , (( TOP-REAL 2) | ( Lower_Arc K)), s1,s2 be Real;

 assume that

 A20: g is being_homeomorphism and

 A21: (g . 0 ) = ( E-max K) and (g . 1) = ( W-min K) and

 A22: (g . s1) = p1 and

 A23: 0 <= s1 and

 A24: s1 <= 1 and

 A25: (g . s2) = p2 and

 A26: 0 <= s2 and

 A27: s2 <= 1;



 A28: ( dom g) = the carrier of I[01] by FUNCT_2:def 1;



 A29: g is one-to-one by A20, TOPS_2:def 5;



 A30: the carrier of (( TOP-REAL 2) | ( Lower_Arc K)) = ( Lower_Arc K) by PRE_TOPC: 8;

 then

 reconsider g1 = g as Function of I[01] , ( TOP-REAL 2) by FUNCT_2: 7;

 g is continuous by A20, TOPS_2:def 5;

 then

 A31: g1 is continuous by PRE_TOPC: 26;

 reconsider h1 = proj1 as Function of ( TOP-REAL 2), R^1 by TOPMETR: 17;

 reconsider h2 = proj2 as Function of ( TOP-REAL 2), R^1 by TOPMETR: 17;

 reconsider hh1 = h1 as Function of the TopStruct of ( TOP-REAL 2), R^1 ;

 reconsider hh2 = h2 as Function of the TopStruct of ( TOP-REAL 2), R^1 ;



 A32: the TopStruct of ( TOP-REAL 2) = ( the TopStruct of ( TOP-REAL 2) | ( [#] the TopStruct of ( TOP-REAL 2))) by TSEP_1: 3

 .= the TopStruct of (( TOP-REAL 2) | ( [#] ( TOP-REAL 2))) by PRE_TOPC: 36

 .= (( TOP-REAL 2) | ( [#] ( TOP-REAL 2)));

 then (for p be Point of (( TOP-REAL 2) | ( [#] ( TOP-REAL 2))) holds (hh1 . p) = ( proj1 . p)) implies hh1 is continuous by JGRAPH_2: 29;

 then

 A33: (for p be Point of (( TOP-REAL 2) | ( [#] ( TOP-REAL 2))) holds (hh1 . p) = ( proj1 . p)) implies h1 is continuous by PRE_TOPC: 32;

 (for p be Point of (( TOP-REAL 2) | ( [#] ( TOP-REAL 2))) holds (hh2 . p) = ( proj2 . p)) implies hh2 is continuous by A32, JGRAPH_2: 30;

 then (for p be Point of (( TOP-REAL 2) | ( [#] ( TOP-REAL 2))) holds (hh2 . p) = ( proj2 . p)) implies h2 is continuous by PRE_TOPC: 32;

 then

 consider h be Function of ( TOP-REAL 2), R^1 such that

 A34: for p be Point of ( TOP-REAL 2), r1,r2 be Real st (h1 . p) = r1 & (h2 . p) = r2 holds (h . p) = (r1 + r2) and

 A35: h is continuous by A33, JGRAPH_2: 19;

 reconsider k = (h * g1) as Function of I[01] , R^1 ;



 A36: ( E-max K) = |[b, d]| by A1, A2, Th46;

 now

 assume

 A37: s1 > s2;



 A38: ( dom g) = [. 0 , 1.] by BORSUK_1: 40, FUNCT_2:def 1;

 0 in [. 0 , 1.] by XXREAL_1: 1;



 then

 A39: (k . 0 ) = (h . ( E-max K)) by A21, A38, FUNCT_1: 13

 .= ((h1 . ( E-max K)) + (h2 . ( E-max K))) by A34

 .= ((( E-max K) `1 ) + ( proj2 . ( E-max K))) by PSCOMP_1:def 5

 .= ((( E-max K) `1 ) + (( E-max K) `2 )) by PSCOMP_1:def 6

 .= ((( E-max K) `1 ) + d) by A36, EUCLID: 52

 .= (b + d) by A36, EUCLID: 52;

 s1 in [. 0 , 1.] by A23, A24, XXREAL_1: 1;



 then

 A40: (k . s1) = (h . p1) by A22, A38, FUNCT_1: 13

 .= (( proj1 . p1) + ( proj2 . p1)) by A34

 .= ((p1 `1 ) + ( proj2 . p1)) by PSCOMP_1:def 5

 .= (b + (p1 `2 )) by A5, PSCOMP_1:def 6;



 A41: s2 in [. 0 , 1.] by A26, A27, XXREAL_1: 1;



 then

 A42: (k . s2) = (h . p2) by A25, A38, FUNCT_1: 13

 .= (( proj1 . p2) + ( proj2 . p2)) by A34

 .= ((p2 `1 ) + ( proj2 . p2)) by PSCOMP_1:def 5

 .= (b + (p2 `2 )) by A15, PSCOMP_1:def 6;



 A43: (k . 0 ) >= (k . s1) by A7, A39, A40, XREAL_1: 7;



 A44: (k . s1) >= (k . s2) by A14, A40, A42, XREAL_1: 7;



 A45: 0 in [. 0 , 1.] by XXREAL_1: 1;

 then

 A46: [. 0 , s2.] c= [. 0 , 1.] by A41, XXREAL_2:def 12;

 reconsider B = [. 0 , s2.] as Subset of I[01] by A41, A45, BORSUK_1: 40, XXREAL_2:def 12;



 A47: B is connected by A26, A41, A45, BORSUK_1: 40, BORSUK_4: 24;



 A48: 0 in B by A26, XXREAL_1: 1;



 A49: s2 in B by A26, XXREAL_1: 1;

 consider xc be Point of I[01] such that

 A50: xc in B and

 A51: (k . xc) = (k . s1) by A31, A35, A43, A44, A47, A48, A49, TOPREAL5: 5;

 reconsider rxc = xc as Real;



 A52: for x1,x2 be set st x1 in ( dom k) & x2 in ( dom k) & (k . x1) = (k . x2) holds x1 = x2

 proof

 let x1,x2 be set;

 assume that

 A53: x1 in ( dom k) and

 A54: x2 in ( dom k) and

 A55: (k . x1) = (k . x2);

 reconsider r1 = x1 as Point of I[01] by A53;

 reconsider r2 = x2 as Point of I[01] by A54;



 A56: (k . x1) = (h . (g1 . x1)) by A53, FUNCT_1: 12

 .= ((h1 . (g1 . r1)) + (h2 . (g1 . r1))) by A34

 .= (((g1 . r1) `1 ) + ( proj2 . (g1 . r1))) by PSCOMP_1:def 5

 .= (((g1 . r1) `1 ) + ((g1 . r1) `2 )) by PSCOMP_1:def 6;



 A57: (k . x2) = (h . (g1 . x2)) by A54, FUNCT_1: 12

 .= ((h1 . (g1 . r2)) + (h2 . (g1 . r2))) by A34

 .= (((g1 . r2) `1 ) + ( proj2 . (g1 . r2))) by PSCOMP_1:def 5

 .= (((g1 . r2) `1 ) + ((g1 . r2) `2 )) by PSCOMP_1:def 6;



 A58: (g . r1) in ( Lower_Arc K) by A30;



 A59: (g . r2) in ( Lower_Arc K) by A30;

 reconsider gr1 = (g . r1) as Point of ( TOP-REAL 2) by A58;

 reconsider gr2 = (g . r2) as Point of ( TOP-REAL 2) by A59;

 now

 per cases by A17, A30, XBOOLE_0:def 3;

 case

 A60: (g . r1) in ( LSeg ( |[b, d]|, |[b, c]|)) & (g . r2) in ( LSeg ( |[b, d]|, |[b, c]|));

 then

 A61: (gr1 `1 ) = b by A2, Th1;

 (gr2 `1 ) = b by A2, A60, Th1;

 then |[(gr1 `1 ), (gr1 `2 )]| = (g . r2) by A55, A56, A57, A61, EUCLID: 53;

 then (g . r1) = (g . r2) by EUCLID: 53;

 hence thesis by A28, A29, FUNCT_1:def 4;

 end;

 case

 A62: (g . r1) in ( LSeg ( |[b, d]|, |[b, c]|)) & (g . r2) in ( LSeg ( |[b, c]|, |[a, c]|));

 then

 A63: (gr1 `1 ) = b by A2, Th1;



 A64: c <= (gr1 `2 ) by A2, A62, Th1;



 A65: (gr2 `2 ) = c by A1, A62, Th3;



 A66: (gr2 `1 ) <= b by A1, A62, Th3;



 A67: (b + (gr1 `2 )) = ((gr2 `1 ) + c) by A2, A55, A56, A57, A62, A65, Th1;

 A68:

 now

 assume b <> (gr2 `1 );

 then b > (gr2 `1 ) by A66, XXREAL_0: 1;

 hence contradiction by A55, A56, A57, A63, A64, A65, XREAL_1: 8;

 end;

 now

 assume (gr1 `2 ) <> c;

 then c < (gr1 `2 ) by A64, XXREAL_0: 1;

 hence contradiction by A66, A67, XREAL_1: 8;

 end;

 then |[(gr1 `1 ), (gr1 `2 )]| = (g . r2) by A63, A65, A68, EUCLID: 53;

 then (g . r1) = (g . r2) by EUCLID: 53;

 hence thesis by A28, A29, FUNCT_1:def 4;

 end;

 case

 A69: (g . r1) in ( LSeg ( |[b, c]|, |[a, c]|)) & (g . r2) in ( LSeg ( |[b, d]|, |[b, c]|));

 then

 A70: (gr2 `1 ) = b by A2, Th1;



 A71: c <= (gr2 `2 ) by A2, A69, Th1;



 A72: (gr1 `2 ) = c by A1, A69, Th3;



 A73: (gr1 `1 ) <= b by A1, A69, Th3;



 A74: (b + (gr2 `2 )) = ((gr1 `1 ) + c) by A1, A55, A56, A57, A69, A70, Th3;

 A75:

 now

 assume b <> (gr1 `1 );

 then b > (gr1 `1 ) by A73, XXREAL_0: 1;

 hence contradiction by A71, A74, XREAL_1: 8;

 end;

 now

 assume (gr2 `2 ) <> c;

 then c < (gr2 `2 ) by A71, XXREAL_0: 1;

 hence contradiction by A55, A56, A57, A70, A72, A73, XREAL_1: 8;

 end;

 then |[(gr2 `1 ), (gr2 `2 )]| = (g . r1) by A70, A72, A75, EUCLID: 53;

 then (g . r1) = (g . r2) by EUCLID: 53;

 hence thesis by A28, A29, FUNCT_1:def 4;

 end;

 case

 A76: (g . r1) in ( LSeg ( |[b, c]|, |[a, c]|)) & (g . r2) in ( LSeg ( |[b, c]|, |[a, c]|));

 then

 A77: (gr1 `2 ) = c by A1, Th3;

 (gr2 `2 ) = c by A1, A76, Th3;

 then |[(gr1 `1 ), (gr1 `2 )]| = (g . r2) by A55, A56, A57, A77, EUCLID: 53;

 then (g . r1) = (g . r2) by EUCLID: 53;

 hence thesis by A28, A29, FUNCT_1:def 4;

 end;

 end;

 hence thesis;

 end;



 A78: ( dom k) = [. 0 , 1.] by BORSUK_1: 40, FUNCT_2:def 1;

 then s1 in ( dom k) by A23, A24, XXREAL_1: 1;

 then rxc = s1 by A46, A50, A51, A52, A78;

 hence contradiction by A37, A50, XXREAL_1: 1;

 end;

 hence thesis;

 end;

 then LE (p1,p2,( Lower_Arc K),( E-max K),( W-min K)) by A18, A19, JORDAN5C:def 3;

 hence thesis by A16, A18, A19, JORDAN6:def 10;

 end;

 case

 A79: p2 in ( LSeg ( |[b, c]|, |[a, c]|)) & p2 <> ( W-min K);

 then

 A80: (p2 `2 ) = c by A1, Th3;



 A81: (p2 `1 ) <= b by A1, A79, Th3;



 A82: ( Lower_Arc K) = (( LSeg ( |[b, d]|, |[b, c]|)) \/ ( LSeg ( |[b, c]|, |[a, c]|))) by A1, A2, Th52;

 then

 A83: p2 in ( Lower_Arc K) by A79, XBOOLE_0:def 3;



 A84: p1 in ( Lower_Arc K) by A3, A82, XBOOLE_0:def 3;

 for g be Function of I[01] , (( TOP-REAL 2) | ( Lower_Arc K)), s1,s2 be Real st g is being_homeomorphism & (g . 0 ) = ( E-max K) & (g . 1) = ( W-min K) & (g . s1) = p1 & 0 <= s1 & s1 <= 1 & (g . s2) = p2 & 0 <= s2 & s2 <= 1 holds s1 <= s2

 proof

 let g be Function of I[01] , (( TOP-REAL 2) | ( Lower_Arc K)), s1,s2 be Real;

 assume that

 A85: g is being_homeomorphism and

 A86: (g . 0 ) = ( E-max K) and (g . 1) = ( W-min K) and

 A87: (g . s1) = p1 and

 A88: 0 <= s1 and

 A89: s1 <= 1 and

 A90: (g . s2) = p2 and

 A91: 0 <= s2 and

 A92: s2 <= 1;



 A93: ( dom g) = the carrier of I[01] by FUNCT_2:def 1;



 A94: g is one-to-one by A85, TOPS_2:def 5;



 A95: the carrier of (( TOP-REAL 2) | ( Lower_Arc K)) = ( Lower_Arc K) by PRE_TOPC: 8;

 then

 reconsider g1 = g as Function of I[01] , ( TOP-REAL 2) by FUNCT_2: 7;

 g is continuous by A85, TOPS_2:def 5;

 then

 A96: g1 is continuous by PRE_TOPC: 26;

 reconsider h1 = proj1 as Function of ( TOP-REAL 2), R^1 by TOPMETR: 17;

 reconsider h2 = proj2 as Function of ( TOP-REAL 2), R^1 by TOPMETR: 17;

 reconsider hh1 = h1 as Function of the TopStruct of ( TOP-REAL 2), R^1 ;

 reconsider hh2 = h2 as Function of the TopStruct of ( TOP-REAL 2), R^1 ;



 A97: the TopStruct of ( TOP-REAL 2) = ( the TopStruct of ( TOP-REAL 2) | ( [#] the TopStruct of ( TOP-REAL 2))) by TSEP_1: 3

 .= the TopStruct of (( TOP-REAL 2) | ( [#] ( TOP-REAL 2))) by PRE_TOPC: 36

 .= (( TOP-REAL 2) | ( [#] ( TOP-REAL 2)));

 then (for p be Point of (( TOP-REAL 2) | ( [#] ( TOP-REAL 2))) holds (hh1 . p) = ( proj1 . p)) implies hh1 is continuous by JGRAPH_2: 29;

 then

 A98: (for p be Point of (( TOP-REAL 2) | ( [#] ( TOP-REAL 2))) holds (hh1 . p) = ( proj1 . p)) implies h1 is continuous by PRE_TOPC: 32;

 (for p be Point of (( TOP-REAL 2) | ( [#] ( TOP-REAL 2))) holds (hh2 . p) = ( proj2 . p)) implies hh2 is continuous by A97, JGRAPH_2: 30;

 then (for p be Point of (( TOP-REAL 2) | ( [#] ( TOP-REAL 2))) holds (hh2 . p) = ( proj2 . p)) implies h2 is continuous by PRE_TOPC: 32;

 then

 consider h be Function of ( TOP-REAL 2), R^1 such that

 A99: for p be Point of ( TOP-REAL 2), r1,r2 be Real st (h1 . p) = r1 & (h2 . p) = r2 holds (h . p) = (r1 + r2) and

 A100: h is continuous by A98, JGRAPH_2: 19;

 reconsider k = (h * g1) as Function of I[01] , R^1 ;



 A101: ( E-max K) = |[b, d]| by A1, A2, Th46;

 now

 assume

 A102: s1 > s2;



 A103: ( dom g) = [. 0 , 1.] by BORSUK_1: 40, FUNCT_2:def 1;

 0 in [. 0 , 1.] by XXREAL_1: 1;



 then

 A104: (k . 0 ) = (h . ( E-max K)) by A86, A103, FUNCT_1: 13

 .= ((h1 . ( E-max K)) + (h2 . ( E-max K))) by A99

 .= ((( E-max K) `1 ) + ( proj2 . ( E-max K))) by PSCOMP_1:def 5

 .= ((( E-max K) `1 ) + (( E-max K) `2 )) by PSCOMP_1:def 6

 .= ((( E-max K) `1 ) + d) by A101, EUCLID: 52

 .= (b + d) by A101, EUCLID: 52;

 s1 in [. 0 , 1.] by A88, A89, XXREAL_1: 1;



 then

 A105: (k . s1) = (h . p1) by A87, A103, FUNCT_1: 13

 .= (( proj1 . p1) + ( proj2 . p1)) by A99

 .= ((p1 `1 ) + ( proj2 . p1)) by PSCOMP_1:def 5

 .= (b + (p1 `2 )) by A5, PSCOMP_1:def 6;



 A106: s2 in [. 0 , 1.] by A91, A92, XXREAL_1: 1;



 then

 A107: (k . s2) = (h . p2) by A90, A103, FUNCT_1: 13

 .= (( proj1 . p2) + ( proj2 . p2)) by A99

 .= ((p2 `1 ) + ( proj2 . p2)) by PSCOMP_1:def 5

 .= ((p2 `1 ) + c) by A80, PSCOMP_1:def 6;



 A108: (k . 0 ) >= (k . s1) by A7, A104, A105, XREAL_1: 7;



 A109: (k . s1) >= (k . s2) by A6, A81, A105, A107, XREAL_1: 7;



 A110: 0 in [. 0 , 1.] by XXREAL_1: 1;

 then

 A111: [. 0 , s2.] c= [. 0 , 1.] by A106, XXREAL_2:def 12;

 reconsider B = [. 0 , s2.] as Subset of I[01] by A106, A110, BORSUK_1: 40, XXREAL_2:def 12;



 A112: B is connected by A91, A106, A110, BORSUK_1: 40, BORSUK_4: 24;



 A113: 0 in B by A91, XXREAL_1: 1;



 A114: s2 in B by A91, XXREAL_1: 1;

 consider xc be Point of I[01] such that

 A115: xc in B and

 A116: (k . xc) = (k . s1) by A96, A100, A108, A109, A112, A113, A114, TOPREAL5: 5;

 reconsider rxc = xc as Real;



 A117: for x1,x2 be set st x1 in ( dom k) & x2 in ( dom k) & (k . x1) = (k . x2) holds x1 = x2

 proof

 let x1,x2 be set;

 assume that

 A118: x1 in ( dom k) and

 A119: x2 in ( dom k) and

 A120: (k . x1) = (k . x2);

 reconsider r1 = x1 as Point of I[01] by A118;

 reconsider r2 = x2 as Point of I[01] by A119;



 A121: (k . x1) = (h . (g1 . x1)) by A118, FUNCT_1: 12

 .= ((h1 . (g1 . r1)) + (h2 . (g1 . r1))) by A99

 .= (((g1 . r1) `1 ) + ( proj2 . (g1 . r1))) by PSCOMP_1:def 5

 .= (((g1 . r1) `1 ) + ((g1 . r1) `2 )) by PSCOMP_1:def 6;



 A122: (k . x2) = (h . (g1 . x2)) by A119, FUNCT_1: 12

 .= ((h1 . (g1 . r2)) + (h2 . (g1 . r2))) by A99

 .= (((g1 . r2) `1 ) + ( proj2 . (g1 . r2))) by PSCOMP_1:def 5

 .= (((g1 . r2) `1 ) + ((g1 . r2) `2 )) by PSCOMP_1:def 6;



 A123: (g . r1) in ( Lower_Arc K) by A95;



 A124: (g . r2) in ( Lower_Arc K) by A95;

 reconsider gr1 = (g . r1) as Point of ( TOP-REAL 2) by A123;

 reconsider gr2 = (g . r2) as Point of ( TOP-REAL 2) by A124;

 now

 per cases by A82, A95, XBOOLE_0:def 3;

 case

 A125: (g . r1) in ( LSeg ( |[b, d]|, |[b, c]|)) & (g . r2) in ( LSeg ( |[b, d]|, |[b, c]|));

 then

 A126: (gr1 `1 ) = b by A2, Th1;

 (gr2 `1 ) = b by A2, A125, Th1;

 then |[(gr1 `1 ), (gr1 `2 )]| = (g . r2) by A120, A121, A122, A126, EUCLID: 53;

 then (g . r1) = (g . r2) by EUCLID: 53;

 hence thesis by A93, A94, FUNCT_1:def 4;

 end;

 case

 A127: (g . r1) in ( LSeg ( |[b, d]|, |[b, c]|)) & (g . r2) in ( LSeg ( |[b, c]|, |[a, c]|));

 then

 A128: (gr1 `1 ) = b by A2, Th1;



 A129: c <= (gr1 `2 ) by A2, A127, Th1;



 A130: (gr2 `2 ) = c by A1, A127, Th3;



 A131: (gr2 `1 ) <= b by A1, A127, Th3;



 A132: (b + (gr1 `2 )) = ((gr2 `1 ) + c) by A2, A120, A121, A122, A127, A130, Th1;

 A133:

 now

 assume b <> (gr2 `1 );

 then b > (gr2 `1 ) by A131, XXREAL_0: 1;

 hence contradiction by A120, A121, A122, A128, A129, A130, XREAL_1: 8;

 end;

 now

 assume (gr1 `2 ) <> c;

 then c < (gr1 `2 ) by A129, XXREAL_0: 1;

 hence contradiction by A131, A132, XREAL_1: 8;

 end;

 then |[(gr1 `1 ), (gr1 `2 )]| = (g . r2) by A128, A130, A133, EUCLID: 53;

 then (g . r1) = (g . r2) by EUCLID: 53;

 hence thesis by A93, A94, FUNCT_1:def 4;

 end;

 case

 A134: (g . r1) in ( LSeg ( |[b, c]|, |[a, c]|)) & (g . r2) in ( LSeg ( |[b, d]|, |[b, c]|));

 then

 A135: (gr2 `1 ) = b by A2, Th1;



 A136: c <= (gr2 `2 ) by A2, A134, Th1;



 A137: (gr1 `2 ) = c by A1, A134, Th3;



 A138: (gr1 `1 ) <= b by A1, A134, Th3;



 A139: (b + (gr2 `2 )) = ((gr1 `1 ) + c) by A1, A120, A121, A122, A134, A135, Th3;

 A140:

 now

 assume b <> (gr1 `1 );

 then b > (gr1 `1 ) by A138, XXREAL_0: 1;

 hence contradiction by A136, A139, XREAL_1: 8;

 end;

 now

 assume (gr2 `2 ) <> c;

 then c < (gr2 `2 ) by A136, XXREAL_0: 1;

 hence contradiction by A120, A121, A122, A135, A137, A138, XREAL_1: 8;

 end;

 then |[(gr2 `1 ), (gr2 `2 )]| = (g . r1) by A135, A137, A140, EUCLID: 53;

 then (g . r1) = (g . r2) by EUCLID: 53;

 hence thesis by A93, A94, FUNCT_1:def 4;

 end;

 case

 A141: (g . r1) in ( LSeg ( |[b, c]|, |[a, c]|)) & (g . r2) in ( LSeg ( |[b, c]|, |[a, c]|));

 then

 A142: (gr1 `2 ) = c by A1, Th3;

 (gr2 `2 ) = c by A1, A141, Th3;

 then |[(gr1 `1 ), (gr1 `2 )]| = (g . r2) by A120, A121, A122, A142, EUCLID: 53;

 then (g . r1) = (g . r2) by EUCLID: 53;

 hence thesis by A93, A94, FUNCT_1:def 4;

 end;

 end;

 hence thesis;

 end;



 A143: ( dom k) = [. 0 , 1.] by BORSUK_1: 40, FUNCT_2:def 1;

 then s1 in ( dom k) by A88, A89, XXREAL_1: 1;

 then rxc = s1 by A111, A115, A116, A117, A143;

 hence contradiction by A102, A115, XXREAL_1: 1;

 end;

 hence thesis;

 end;

 then LE (p1,p2,( Lower_Arc K),( E-max K),( W-min K)) by A83, A84, JORDAN5C:def 3;

 hence thesis by A79, A83, A84, JORDAN6:def 10;

 end;

 end;

 hence thesis;

 end;

 end;

 theorem :: JGRAPH_6:62



 Th62: for a,b,c,d be Real, p1,p2 be Point of ( TOP-REAL 2) st a ( W-min ( rectangle (a,b,c,d))) holds LE (p1,p2,( rectangle (a,b,c,d))) iff p2 in ( LSeg ( |[b, c]|, |[a, c]|)) & (p1 `1 ) >= (p2 `1 ) & p2 <> ( W-min ( rectangle (a,b,c,d)))

 proof

 let a,b,c,d be Real, p1,p2 be Point of ( TOP-REAL 2);

 set K = ( rectangle (a,b,c,d));

 assume that

 A1: a ( W-min K);



 A5: K is being_simple_closed_curve by A1, A2, Th50;



 A6: (p1 `2 ) = c by A1, A3, Th3;



 A7: (p1 `1 ) <= b by A1, A3, Th3;

 thus LE (p1,p2,K) implies p2 in ( LSeg ( |[b, c]|, |[a, c]|)) & (p1 `1 ) >= (p2 `1 ) & p2 <> ( W-min K)

 proof

 assume

 A8: LE (p1,p2,K);

 then

 A9: p2 in K by A5, JORDAN7: 5;

 K = ((( LSeg ( |[a, c]|, |[a, d]|)) \/ ( LSeg ( |[a, d]|, |[b, d]|))) \/ (( LSeg ( |[a, c]|, |[b, c]|)) \/ ( LSeg ( |[b, c]|, |[b, d]|)))) by SPPOL_2:def 3

 .= (((( LSeg ( |[a, c]|, |[a, d]|)) \/ ( LSeg ( |[a, d]|, |[b, d]|))) \/ ( LSeg ( |[b, d]|, |[b, c]|))) \/ ( LSeg ( |[b, c]|, |[a, c]|))) by XBOOLE_1: 4;

 then p2 in ((( LSeg ( |[a, c]|, |[a, d]|)) \/ ( LSeg ( |[a, d]|, |[b, d]|))) \/ ( LSeg ( |[b, d]|, |[b, c]|))) or p2 in ( LSeg ( |[b, c]|, |[a, c]|)) by A9, XBOOLE_0:def 3;

 then

 A10: p2 in (( LSeg ( |[a, c]|, |[a, d]|)) \/ ( LSeg ( |[a, d]|, |[b, d]|))) or p2 in ( LSeg ( |[b, d]|, |[b, c]|)) or p2 in ( LSeg ( |[b, c]|, |[a, c]|)) by XBOOLE_0:def 3;

 now

 per cases by A10, XBOOLE_0:def 3;

 case p2 in ( LSeg ( |[a, c]|, |[a, d]|));

 then LE (p2,p1,K) by A1, A2, A3, A4, Th59;

 hence thesis by A1, A2, A3, A4, A8, Th50, JORDAN6: 57;

 end;

 case p2 in ( LSeg ( |[a, d]|, |[b, d]|));

 then LE (p2,p1,K) by A1, A2, A3, A4, Th60;

 hence thesis by A1, A2, A3, A4, A8, Th50, JORDAN6: 57;

 end;

 case p2 in ( LSeg ( |[b, d]|, |[b, c]|));

 then LE (p2,p1,K) by A1, A2, A3, A4, Th61;

 hence thesis by A1, A2, A3, A4, A8, Th50, JORDAN6: 57;

 end;

 case p2 in ( LSeg ( |[b, c]|, |[a, c]|));

 hence thesis by A1, A2, A3, A4, A8, Th58;

 end;

 end;

 hence thesis;

 end;

 thus p2 in ( LSeg ( |[b, c]|, |[a, c]|)) & (p1 `1 ) >= (p2 `1 ) & p2 <> ( W-min K) implies LE (p1,p2,K)

 proof

 assume that

 A11: p2 in ( LSeg ( |[b, c]|, |[a, c]|)) and

 A12: (p1 `1 ) >= (p2 `1 ) and

 A13: p2 <> ( W-min K);

 now

 per cases by A11, A12;

 case

 A14: p2 in ( LSeg ( |[b, c]|, |[a, c]|)) & (p1 `1 ) >= (p2 `1 );

 then

 A15: (p2 `2 ) = c by A1, Th3;



 A16: ( Lower_Arc K) = (( LSeg ( |[b, d]|, |[b, c]|)) \/ ( LSeg ( |[b, c]|, |[a, c]|))) by A1, A2, Th52;

 then

 A17: p2 in ( Lower_Arc K) by A14, XBOOLE_0:def 3;



 A18: p1 in ( Lower_Arc K) by A3, A16, XBOOLE_0:def 3;

 for g be Function of I[01] , (( TOP-REAL 2) | ( Lower_Arc K)), s1,s2 be Real st g is being_homeomorphism & (g . 0 ) = ( E-max K) & (g . 1) = ( W-min K) & (g . s1) = p1 & 0 <= s1 & s1 <= 1 & (g . s2) = p2 & 0 <= s2 & s2 <= 1 holds s1 <= s2

 proof

 let g be Function of I[01] , (( TOP-REAL 2) | ( Lower_Arc K)), s1,s2 be Real;

 assume that

 A19: g is being_homeomorphism and

 A20: (g . 0 ) = ( E-max K) and (g . 1) = ( W-min K) and

 A21: (g . s1) = p1 and

 A22: 0 <= s1 and

 A23: s1 <= 1 and

 A24: (g . s2) = p2 and

 A25: 0 <= s2 and

 A26: s2 <= 1;



 A27: ( dom g) = the carrier of I[01] by FUNCT_2:def 1;



 A28: g is one-to-one by A19, TOPS_2:def 5;



 A29: the carrier of (( TOP-REAL 2) | ( Lower_Arc K)) = ( Lower_Arc K) by PRE_TOPC: 8;

 then

 reconsider g1 = g as Function of I[01] , ( TOP-REAL 2) by FUNCT_2: 7;

 g is continuous by A19, TOPS_2:def 5;

 then

 A30: g1 is continuous by PRE_TOPC: 26;

 reconsider h1 = proj1 as Function of ( TOP-REAL 2), R^1 by TOPMETR: 17;

 reconsider h2 = proj2 as Function of ( TOP-REAL 2), R^1 by TOPMETR: 17;

 reconsider hh1 = h1 as Function of the TopStruct of ( TOP-REAL 2), R^1 ;

 reconsider hh2 = h2 as Function of the TopStruct of ( TOP-REAL 2), R^1 ;



 A31: the TopStruct of ( TOP-REAL 2) = ( the TopStruct of ( TOP-REAL 2) | ( [#] the TopStruct of ( TOP-REAL 2))) by TSEP_1: 3

 .= the TopStruct of (( TOP-REAL 2) | ( [#] ( TOP-REAL 2))) by PRE_TOPC: 36

 .= (( TOP-REAL 2) | ( [#] ( TOP-REAL 2)));

 then (for p be Point of (( TOP-REAL 2) | ( [#] ( TOP-REAL 2))) holds (hh1 . p) = ( proj1 . p)) implies hh1 is continuous by JGRAPH_2: 29;

 then

 A32: (for p be Point of (( TOP-REAL 2) | ( [#] ( TOP-REAL 2))) holds (hh1 . p) = ( proj1 . p)) implies h1 is continuous by PRE_TOPC: 32;

 (for p be Point of (( TOP-REAL 2) | ( [#] ( TOP-REAL 2))) holds (hh2 . p) = ( proj2 . p)) implies hh2 is continuous by A31, JGRAPH_2: 30;

 then (for p be Point of (( TOP-REAL 2) | ( [#] ( TOP-REAL 2))) holds (hh2 . p) = ( proj2 . p)) implies h2 is continuous by PRE_TOPC: 32;

 then

 consider h be Function of ( TOP-REAL 2), R^1 such that

 A33: for p be Point of ( TOP-REAL 2), r1,r2 be Real st (h1 . p) = r1 & (h2 . p) = r2 holds (h . p) = (r1 + r2) and

 A34: h is continuous by A32, JGRAPH_2: 19;

 reconsider k = (h * g1) as Function of I[01] , R^1 ;



 A35: ( E-max K) = |[b, d]| by A1, A2, Th46;

 now

 assume

 A36: s1 > s2;



 A37: ( dom g) = [. 0 , 1.] by BORSUK_1: 40, FUNCT_2:def 1;

 0 in [. 0 , 1.] by XXREAL_1: 1;



 then

 A38: (k . 0 ) = (h . ( E-max K)) by A20, A37, FUNCT_1: 13

 .= ((h1 . ( E-max K)) + (h2 . ( E-max K))) by A33

 .= ((( E-max K) `1 ) + ( proj2 . ( E-max K))) by PSCOMP_1:def 5

 .= ((( E-max K) `1 ) + (( E-max K) `2 )) by PSCOMP_1:def 6

 .= ((( E-max K) `1 ) + d) by A35, EUCLID: 52

 .= (b + d) by A35, EUCLID: 52;

 s1 in [. 0 , 1.] by A22, A23, XXREAL_1: 1;



 then

 A39: (k . s1) = (h . p1) by A21, A37, FUNCT_1: 13

 .= (( proj1 . p1) + ( proj2 . p1)) by A33

 .= ((p1 `1 ) + ( proj2 . p1)) by PSCOMP_1:def 5

 .= ((p1 `1 ) + c) by A6, PSCOMP_1:def 6;



 A40: s2 in [. 0 , 1.] by A25, A26, XXREAL_1: 1;



 then

 A41: (k . s2) = (h . p2) by A24, A37, FUNCT_1: 13

 .= (( proj1 . p2) + ( proj2 . p2)) by A33

 .= ((p2 `1 ) + ( proj2 . p2)) by PSCOMP_1:def 5

 .= ((p2 `1 ) + c) by A15, PSCOMP_1:def 6;



 A42: (k . 0 ) >= (k . s1) by A2, A7, A38, A39, XREAL_1: 7;



 A43: (k . s1) >= (k . s2) by A14, A39, A41, XREAL_1: 7;



 A44: 0 in [. 0 , 1.] by XXREAL_1: 1;

 then

 A45: [. 0 , s2.] c= [. 0 , 1.] by A40, XXREAL_2:def 12;

 reconsider B = [. 0 , s2.] as Subset of I[01] by A40, A44, BORSUK_1: 40, XXREAL_2:def 12;



 A46: B is connected by A25, A40, A44, BORSUK_1: 40, BORSUK_4: 24;



 A47: 0 in B by A25, XXREAL_1: 1;



 A48: s2 in B by A25, XXREAL_1: 1;

 consider xc be Point of I[01] such that

 A49: xc in B and

 A50: (k . xc) = (k . s1) by A30, A34, A42, A43, A46, A47, A48, TOPREAL5: 5;

 reconsider rxc = xc as Real;



 A51: for x1,x2 be set st x1 in ( dom k) & x2 in ( dom k) & (k . x1) = (k . x2) holds x1 = x2

 proof

 let x1,x2 be set;

 assume that

 A52: x1 in ( dom k) and

 A53: x2 in ( dom k) and

 A54: (k . x1) = (k . x2);

 reconsider r1 = x1 as Point of I[01] by A52;

 reconsider r2 = x2 as Point of I[01] by A53;



 A55: (k . x1) = (h . (g1 . x1)) by A52, FUNCT_1: 12

 .= ((h1 . (g1 . r1)) + (h2 . (g1 . r1))) by A33

 .= (((g1 . r1) `1 ) + ( proj2 . (g1 . r1))) by PSCOMP_1:def 5

 .= (((g1 . r1) `1 ) + ((g1 . r1) `2 )) by PSCOMP_1:def 6;



 A56: (k . x2) = (h . (g1 . x2)) by A53, FUNCT_1: 12

 .= ((h1 . (g1 . r2)) + (h2 . (g1 . r2))) by A33

 .= (((g1 . r2) `1 ) + ( proj2 . (g1 . r2))) by PSCOMP_1:def 5

 .= (((g1 . r2) `1 ) + ((g1 . r2) `2 )) by PSCOMP_1:def 6;



 A57: (g . r1) in ( Lower_Arc K) by A29;



 A58: (g . r2) in ( Lower_Arc K) by A29;

 reconsider gr1 = (g . r1) as Point of ( TOP-REAL 2) by A57;

 reconsider gr2 = (g . r2) as Point of ( TOP-REAL 2) by A58;

 now

 per cases by A16, A29, XBOOLE_0:def 3;

 case

 A59: (g . r1) in ( LSeg ( |[b, d]|, |[b, c]|)) & (g . r2) in ( LSeg ( |[b, d]|, |[b, c]|));

 then

 A60: (gr1 `1 ) = b by A2, Th1;

 (gr2 `1 ) = b by A2, A59, Th1;

 then |[(gr1 `1 ), (gr1 `2 )]| = (g . r2) by A54, A55, A56, A60, EUCLID: 53;

 then (g . r1) = (g . r2) by EUCLID: 53;

 hence thesis by A27, A28, FUNCT_1:def 4;

 end;

 case

 A61: (g . r1) in ( LSeg ( |[b, d]|, |[b, c]|)) & (g . r2) in ( LSeg ( |[b, c]|, |[a, c]|));

 then

 A62: (gr1 `1 ) = b by A2, Th1;



 A63: c <= (gr1 `2 ) by A2, A61, Th1;



 A64: (gr2 `2 ) = c by A1, A61, Th3;



 A65: (gr2 `1 ) <= b by A1, A61, Th3;



 A66: (b + (gr1 `2 )) = ((gr2 `1 ) + c) by A2, A54, A55, A56, A61, A64, Th1;

 A67:

 now

 assume b <> (gr2 `1 );

 then b > (gr2 `1 ) by A65, XXREAL_0: 1;

 hence contradiction by A54, A55, A56, A62, A63, A64, XREAL_1: 8;

 end;

 now

 assume (gr1 `2 ) <> c;

 then c < (gr1 `2 ) by A63, XXREAL_0: 1;

 hence contradiction by A65, A66, XREAL_1: 8;

 end;

 then |[(gr1 `1 ), (gr1 `2 )]| = (g . r2) by A62, A64, A67, EUCLID: 53;

 then (g . r1) = (g . r2) by EUCLID: 53;

 hence thesis by A27, A28, FUNCT_1:def 4;

 end;

 case

 A68: (g . r1) in ( LSeg ( |[b, c]|, |[a, c]|)) & (g . r2) in ( LSeg ( |[b, d]|, |[b, c]|));

 then

 A69: (gr2 `1 ) = b by A2, Th1;



 A70: c <= (gr2 `2 ) by A2, A68, Th1;



 A71: (gr1 `2 ) = c by A1, A68, Th3;



 A72: (gr1 `1 ) <= b by A1, A68, Th3;



 A73: (b + (gr2 `2 )) = ((gr1 `1 ) + c) by A1, A54, A55, A56, A68, A69, Th3;

 A74:

 now

 assume b <> (gr1 `1 );

 then b > (gr1 `1 ) by A72, XXREAL_0: 1;

 hence contradiction by A70, A73, XREAL_1: 8;

 end;

 now

 assume (gr2 `2 ) <> c;

 then c < (gr2 `2 ) by A70, XXREAL_0: 1;

 hence contradiction by A54, A55, A56, A69, A71, A72, XREAL_1: 8;

 end;

 then |[(gr2 `1 ), (gr2 `2 )]| = (g . r1) by A69, A71, A74, EUCLID: 53;

 then (g . r1) = (g . r2) by EUCLID: 53;

 hence thesis by A27, A28, FUNCT_1:def 4;

 end;

 case

 A75: (g . r1) in ( LSeg ( |[b, c]|, |[a, c]|)) & (g . r2) in ( LSeg ( |[b, c]|, |[a, c]|));

 then

 A76: (gr1 `2 ) = c by A1, Th3;

 (gr2 `2 ) = c by A1, A75, Th3;

 then |[(gr1 `1 ), (gr1 `2 )]| = (g . r2) by A54, A55, A56, A76, EUCLID: 53;

 then (g . r1) = (g . r2) by EUCLID: 53;

 hence thesis by A27, A28, FUNCT_1:def 4;

 end;

 end;

 hence thesis;

 end;



 A77: ( dom k) = [. 0 , 1.] by BORSUK_1: 40, FUNCT_2:def 1;

 then s1 in ( dom k) by A22, A23, XXREAL_1: 1;

 then rxc = s1 by A45, A49, A50, A51, A77;

 hence contradiction by A36, A49, XXREAL_1: 1;

 end;

 hence thesis;

 end;

 then LE (p1,p2,( Lower_Arc K),( E-max K),( W-min K)) by A17, A18, JORDAN5C:def 3;

 hence thesis by A13, A17, A18, JORDAN6:def 10;

 end;

 case

 A78: p2 in ( LSeg ( |[b, c]|, |[a, c]|)) & p2 <> ( W-min K);

 then

 A79: (p2 `2 ) = c by A1, Th3;



 A80: ( Lower_Arc K) = (( LSeg ( |[b, d]|, |[b, c]|)) \/ ( LSeg ( |[b, c]|, |[a, c]|))) by A1, A2, Th52;

 then

 A81: p2 in ( Lower_Arc K) by A78, XBOOLE_0:def 3;



 A82: p1 in ( Lower_Arc K) by A3, A80, XBOOLE_0:def 3;

 for g be Function of I[01] , (( TOP-REAL 2) | ( Lower_Arc K)), s1,s2 be Real st g is being_homeomorphism & (g . 0 ) = ( E-max K) & (g . 1) = ( W-min K) & (g . s1) = p1 & 0 <= s1 & s1 <= 1 & (g . s2) = p2 & 0 <= s2 & s2 <= 1 holds s1 <= s2

 proof

 let g be Function of I[01] , (( TOP-REAL 2) | ( Lower_Arc K)), s1,s2 be Real;

 assume that

 A83: g is being_homeomorphism and

 A84: (g . 0 ) = ( E-max K) and (g . 1) = ( W-min K) and

 A85: (g . s1) = p1 and

 A86: 0 <= s1 and

 A87: s1 <= 1 and

 A88: (g . s2) = p2 and

 A89: 0 <= s2 and

 A90: s2 <= 1;



 A91: ( dom g) = the carrier of I[01] by FUNCT_2:def 1;



 A92: g is one-to-one by A83, TOPS_2:def 5;



 A93: the carrier of (( TOP-REAL 2) | ( Lower_Arc K)) = ( Lower_Arc K) by PRE_TOPC: 8;

 then

 reconsider g1 = g as Function of I[01] , ( TOP-REAL 2) by FUNCT_2: 7;

 g is continuous by A83, TOPS_2:def 5;

 then

 A94: g1 is continuous by PRE_TOPC: 26;

 reconsider h1 = proj1 as Function of ( TOP-REAL 2), R^1 by TOPMETR: 17;

 reconsider h2 = proj2 as Function of ( TOP-REAL 2), R^1 by TOPMETR: 17;

 reconsider hh1 = h1 as Function of the TopStruct of ( TOP-REAL 2), R^1 ;

 reconsider hh2 = h2 as Function of the TopStruct of ( TOP-REAL 2), R^1 ;



 A95: the TopStruct of ( TOP-REAL 2) = ( the TopStruct of ( TOP-REAL 2) | ( [#] the TopStruct of ( TOP-REAL 2))) by TSEP_1: 3

 .= the TopStruct of (( TOP-REAL 2) | ( [#] ( TOP-REAL 2))) by PRE_TOPC: 36

 .= (( TOP-REAL 2) | ( [#] ( TOP-REAL 2)));

 then (for p be Point of (( TOP-REAL 2) | ( [#] ( TOP-REAL 2))) holds (hh1 . p) = ( proj1 . p)) implies hh1 is continuous by JGRAPH_2: 29;

 then

 A96: (for p be Point of (( TOP-REAL 2) | ( [#] ( TOP-REAL 2))) holds (hh1 . p) = ( proj1 . p)) implies h1 is continuous by PRE_TOPC: 32;

 (for p be Point of (( TOP-REAL 2) | ( [#] ( TOP-REAL 2))) holds (hh2 . p) = ( proj2 . p)) implies hh2 is continuous by A95, JGRAPH_2: 30;

 then (for p be Point of (( TOP-REAL 2) | ( [#] ( TOP-REAL 2))) holds (hh2 . p) = ( proj2 . p)) implies h2 is continuous by PRE_TOPC: 32;

 then

 consider h be Function of ( TOP-REAL 2), R^1 such that

 A97: for p be Point of ( TOP-REAL 2), r1,r2 be Real st (h1 . p) = r1 & (h2 . p) = r2 holds (h . p) = (r1 + r2) and

 A98: h is continuous by A96, JGRAPH_2: 19;

 reconsider k = (h * g1) as Function of I[01] , R^1 ;



 A99: ( E-max K) = |[b, d]| by A1, A2, Th46;

 now

 assume

 A100: s1 > s2;



 A101: ( dom g) = [. 0 , 1.] by BORSUK_1: 40, FUNCT_2:def 1;

 0 in [. 0 , 1.] by XXREAL_1: 1;



 then

 A102: (k . 0 ) = (h . ( E-max K)) by A84, A101, FUNCT_1: 13

 .= ((h1 . ( E-max K)) + (h2 . ( E-max K))) by A97

 .= ((( E-max K) `1 ) + ( proj2 . ( E-max K))) by PSCOMP_1:def 5

 .= ((( E-max K) `1 ) + (( E-max K) `2 )) by PSCOMP_1:def 6

 .= ((( E-max K) `1 ) + d) by A99, EUCLID: 52

 .= (b + d) by A99, EUCLID: 52;

 s1 in [. 0 , 1.] by A86, A87, XXREAL_1: 1;



 then

 A103: (k . s1) = (h . p1) by A85, A101, FUNCT_1: 13

 .= (( proj1 . p1) + ( proj2 . p1)) by A97

 .= ((p1 `1 ) + ( proj2 . p1)) by PSCOMP_1:def 5

 .= ((p1 `1 ) + c) by A6, PSCOMP_1:def 6;



 A104: s2 in [. 0 , 1.] by A89, A90, XXREAL_1: 1;



 then

 A105: (k . s2) = (h . p2) by A88, A101, FUNCT_1: 13

 .= (( proj1 . p2) + ( proj2 . p2)) by A97

 .= ((p2 `1 ) + ( proj2 . p2)) by PSCOMP_1:def 5

 .= ((p2 `1 ) + c) by A79, PSCOMP_1:def 6;



 A106: (k . 0 ) >= (k . s1) by A2, A7, A102, A103, XREAL_1: 7;



 A107: (k . s1) >= (k . s2) by A12, A103, A105, XREAL_1: 7;



 A108: 0 in [. 0 , 1.] by XXREAL_1: 1;

 then

 A109: [. 0 , s2.] c= [. 0 , 1.] by A104, XXREAL_2:def 12;

 reconsider B = [. 0 , s2.] as Subset of I[01] by A104, A108, BORSUK_1: 40, XXREAL_2:def 12;



 A110: B is connected by A89, A104, A108, BORSUK_1: 40, BORSUK_4: 24;



 A111: 0 in B by A89, XXREAL_1: 1;



 A112: s2 in B by A89, XXREAL_1: 1;

 consider xc be Point of I[01] such that

 A113: xc in B and

 A114: (k . xc) = (k . s1) by A94, A98, A106, A107, A110, A111, A112, TOPREAL5: 5;

 reconsider rxc = xc as Real;



 A115: for x1,x2 be set st x1 in ( dom k) & x2 in ( dom k) & (k . x1) = (k . x2) holds x1 = x2

 proof

 let x1,x2 be set;

 assume that

 A116: x1 in ( dom k) and

 A117: x2 in ( dom k) and

 A118: (k . x1) = (k . x2);

 reconsider r1 = x1 as Point of I[01] by A116;

 reconsider r2 = x2 as Point of I[01] by A117;



 A119: (k . x1) = (h . (g1 . x1)) by A116, FUNCT_1: 12

 .= ((h1 . (g1 . r1)) + (h2 . (g1 . r1))) by A97

 .= (((g1 . r1) `1 ) + ( proj2 . (g1 . r1))) by PSCOMP_1:def 5

 .= (((g1 . r1) `1 ) + ((g1 . r1) `2 )) by PSCOMP_1:def 6;



 A120: (k . x2) = (h . (g1 . x2)) by A117, FUNCT_1: 12

 .= ((h1 . (g1 . r2)) + (h2 . (g1 . r2))) by A97

 .= (((g1 . r2) `1 ) + ( proj2 . (g1 . r2))) by PSCOMP_1:def 5

 .= (((g1 . r2) `1 ) + ((g1 . r2) `2 )) by PSCOMP_1:def 6;



 A121: (g . r1) in ( Lower_Arc K) by A93;



 A122: (g . r2) in ( Lower_Arc K) by A93;

 reconsider gr1 = (g . r1) as Point of ( TOP-REAL 2) by A121;

 reconsider gr2 = (g . r2) as Point of ( TOP-REAL 2) by A122;

 now

 per cases by A80, A93, XBOOLE_0:def 3;

 case

 A123: (g . r1) in ( LSeg ( |[b, d]|, |[b, c]|)) & (g . r2) in ( LSeg ( |[b, d]|, |[b, c]|));

 then

 A124: (gr1 `1 ) = b by A2, Th1;

 (gr2 `1 ) = b by A2, A123, Th1;

 then |[(gr1 `1 ), (gr1 `2 )]| = (g . r2) by A118, A119, A120, A124, EUCLID: 53;

 then (g . r1) = (g . r2) by EUCLID: 53;

 hence thesis by A91, A92, FUNCT_1:def 4;

 end;

 case

 A125: (g . r1) in ( LSeg ( |[b, d]|, |[b, c]|)) & (g . r2) in ( LSeg ( |[b, c]|, |[a, c]|));

 then

 A126: (gr1 `1 ) = b by A2, Th1;



 A127: c <= (gr1 `2 ) by A2, A125, Th1;



 A128: (gr2 `2 ) = c by A1, A125, Th3;



 A129: (gr2 `1 ) <= b by A1, A125, Th3;



 A130: (b + (gr1 `2 )) = ((gr2 `1 ) + c) by A2, A118, A119, A120, A125, A128, Th1;

 A131:

 now

 assume b <> (gr2 `1 );

 then b > (gr2 `1 ) by A129, XXREAL_0: 1;

 hence contradiction by A118, A119, A120, A126, A127, A128, XREAL_1: 8;

 end;

 now

 assume (gr1 `2 ) <> c;

 then c < (gr1 `2 ) by A127, XXREAL_0: 1;

 hence contradiction by A129, A130, XREAL_1: 8;

 end;

 then |[(gr1 `1 ), (gr1 `2 )]| = (g . r2) by A126, A128, A131, EUCLID: 53;

 then (g . r1) = (g . r2) by EUCLID: 53;

 hence thesis by A91, A92, FUNCT_1:def 4;

 end;

 case

 A132: (g . r1) in ( LSeg ( |[b, c]|, |[a, c]|)) & (g . r2) in ( LSeg ( |[b, d]|, |[b, c]|));

 then

 A133: (gr2 `1 ) = b by A2, Th1;



 A134: c <= (gr2 `2 ) by A2, A132, Th1;



 A135: (gr1 `2 ) = c by A1, A132, Th3;



 A136: (gr1 `1 ) <= b by A1, A132, Th3;



 A137: (b + (gr2 `2 )) = ((gr1 `1 ) + c) by A1, A118, A119, A120, A132, A133, Th3;

 A138:

 now

 assume b <> (gr1 `1 );

 then b > (gr1 `1 ) by A136, XXREAL_0: 1;

 hence contradiction by A134, A137, XREAL_1: 8;

 end;

 now

 assume (gr2 `2 ) <> c;

 then c < (gr2 `2 ) by A134, XXREAL_0: 1;

 hence contradiction by A118, A119, A120, A133, A135, A136, XREAL_1: 8;

 end;

 then |[(gr2 `1 ), (gr2 `2 )]| = (g . r1) by A133, A135, A138, EUCLID: 53;

 then (g . r1) = (g . r2) by EUCLID: 53;

 hence thesis by A91, A92, FUNCT_1:def 4;

 end;

 case

 A139: (g . r1) in ( LSeg ( |[b, c]|, |[a, c]|)) & (g . r2) in ( LSeg ( |[b, c]|, |[a, c]|));

 then

 A140: (gr1 `2 ) = c by A1, Th3;

 (gr2 `2 ) = c by A1, A139, Th3;

 then |[(gr1 `1 ), (gr1 `2 )]| = (g . r2) by A118, A119, A120, A140, EUCLID: 53;

 then (g . r1) = (g . r2) by EUCLID: 53;

 hence thesis by A91, A92, FUNCT_1:def 4;

 end;

 end;

 hence thesis;

 end;



 A141: ( dom k) = [. 0 , 1.] by BORSUK_1: 40, FUNCT_2:def 1;

 then s1 in ( dom k) by A86, A87, XXREAL_1: 1;

 then rxc = s1 by A109, A113, A114, A115, A141;

 hence contradiction by A100, A113, XXREAL_1: 1;

 end;

 hence thesis;

 end;

 then LE (p1,p2,( Lower_Arc K),( E-max K),( W-min K)) by A81, A82, JORDAN5C:def 3;

 hence thesis by A78, A81, A82, JORDAN6:def 10;

 end;

 end;

 hence thesis;

 end;

 end;

 theorem :: JGRAPH_6:63



 Th63: for x be set, a,b,c,d be Real st x in ( rectangle (a,b,c,d)) & a < b & c < d holds x in ( LSeg ( |[a, c]|, |[a, d]|)) or x in ( LSeg ( |[a, d]|, |[b, d]|)) or x in ( LSeg ( |[b, d]|, |[b, c]|)) or x in ( LSeg ( |[b, c]|, |[a, c]|))

 proof

 let x be set, a,b,c,d be Real;

 assume that

 A1: x in ( rectangle (a,b,c,d)) and

 A2: a = c or (q `1 ) <= b & (q `1 ) >= a & (q `2 ) = d or (q `1 ) <= b & (q `1 ) >= a & (q `2 ) = c or (q `1 ) = b & (q `2 ) <= d & (q `2 ) >= c } by A1, A2, A3, SPPOL_2: 54;

 then

 consider p such that

 A4: p = x and

 A5: (p `1 ) = a & (p `2 ) <= d & (p `2 ) >= c or (p `1 ) <= b & (p `1 ) >= a & (p `2 ) = d or (p `1 ) <= b & (p `1 ) >= a & (p `2 ) = c or (p `1 ) = b & (p `2 ) <= d & (p `2 ) >= c;

 now

 per cases by A5;

 case

 A6: (p `1 ) = a & c <= (p `2 ) & (p `2 ) <= d;



 A7: (d - c) > 0 by A3, XREAL_1: 50;



 A8: ((p `2 ) - c) >= 0 by A6, XREAL_1: 48;



 A9: (d - (p `2 )) >= 0 by A6, XREAL_1: 48;

 reconsider r = (((p `2 ) - c) / (d - c)) as Real;



 A10: (1 - r) = (((d - c) / (d - c)) - (((p `2 ) - c) / (d - c))) by A7, XCMPLX_1: 60

 .= (((d - c) - ((p `2 ) - c)) / (d - c)) by XCMPLX_1: 120

 .= ((d - (p `2 )) / (d - c));

 then

 A11: ((1 - r) + r) >= ( 0 + r) by A7, A9, XREAL_1: 7;



 A12: ((((1 - r) * |[a, c]|) + (r * |[a, d]|)) `1 ) = ((((1 - r) * |[a, c]|) `1 ) + ((r * |[a, d]|) `1 )) by TOPREAL3: 2

 .= (((1 - r) * ( |[a, c]| `1 )) + ((r * |[a, d]|) `1 )) by TOPREAL3: 4

 .= (((1 - r) * a) + ((r * |[a, d]|) `1 )) by EUCLID: 52

 .= (((1 - r) * a) + (r * ( |[a, d]| `1 ))) by TOPREAL3: 4

 .= (((1 - r) * a) + (r * a)) by EUCLID: 52

 .= (p `1 ) by A6;

 ((((1 - r) * |[a, c]|) + (r * |[a, d]|)) `2 ) = ((((1 - r) * |[a, c]|) `2 ) + ((r * |[a, d]|) `2 )) by TOPREAL3: 2

 .= (((1 - r) * ( |[a, c]| `2 )) + ((r * |[a, d]|) `2 )) by TOPREAL3: 4

 .= (((1 - r) * c) + ((r * |[a, d]|) `2 )) by EUCLID: 52

 .= (((1 - r) * c) + (r * ( |[a, d]| `2 ))) by TOPREAL3: 4

 .= ((((d - (p `2 )) / (d - c)) * c) + ((((p `2 ) - c) / (d - c)) * d)) by A10, EUCLID: 52

 .= ((((d - (p `2 )) * ((d - c) " )) * c) + ((((p `2 ) - c) / (d - c)) * d)) by XCMPLX_0:def 9

 .= ((((d - c) " ) * ((d - (p `2 )) * c)) + ((((d - c) " ) * ((p `2 ) - c)) * d)) by XCMPLX_0:def 9

 .= ((((d - c) " ) * (d - c)) * (p `2 ))

 .= (1 * (p `2 )) by A7, XCMPLX_0:def 7

 .= (p `2 );



 then p = |[((((1 - r) * |[a, c]|) + (r * |[a, d]|)) `1 ), ((((1 - r) * |[a, c]|) + (r * |[a, d]|)) `2 )]| by A12, EUCLID: 53

 .= (((1 - r) * |[a, c]|) + (r * |[a, d]|)) by EUCLID: 53;

 hence thesis by A4, A7, A8, A11;

 end;

 case

 A13: (p `2 ) = d & a <= (p `1 ) & (p `1 ) <= b;



 A14: (b - a) > 0 by A2, XREAL_1: 50;



 A15: ((p `1 ) - a) >= 0 by A13, XREAL_1: 48;



 A16: (b - (p `1 )) >= 0 by A13, XREAL_1: 48;

 reconsider r = (((p `1 ) - a) / (b - a)) as Real;



 A17: (1 - r) = (((b - a) / (b - a)) - (((p `1 ) - a) / (b - a))) by A14, XCMPLX_1: 60

 .= (((b - a) - ((p `1 ) - a)) / (b - a)) by XCMPLX_1: 120

 .= ((b - (p `1 )) / (b - a));

 then

 A18: ((1 - r) + r) >= ( 0 + r) by A14, A16, XREAL_1: 7;



 A19: ((((1 - r) * |[a, d]|) + (r * |[b, d]|)) `1 ) = ((((1 - r) * |[a, d]|) `1 ) + ((r * |[b, d]|) `1 )) by TOPREAL3: 2

 .= (((1 - r) * ( |[a, d]| `1 )) + ((r * |[b, d]|) `1 )) by TOPREAL3: 4

 .= (((1 - r) * a) + ((r * |[b, d]|) `1 )) by EUCLID: 52

 .= (((1 - r) * a) + (r * ( |[b, d]| `1 ))) by TOPREAL3: 4

 .= ((((b - (p `1 )) / (b - a)) * a) + ((((p `1 ) - a) / (b - a)) * b)) by A17, EUCLID: 52

 .= ((((b - (p `1 )) * ((b - a) " )) * a) + ((((p `1 ) - a) / (b - a)) * b)) by XCMPLX_0:def 9

 .= ((((b - a) " ) * ((b - (p `1 )) * a)) + ((((b - a) " ) * ((p `1 ) - a)) * b)) by XCMPLX_0:def 9

 .= ((((b - a) " ) * (b - a)) * (p `1 ))

 .= (1 * (p `1 )) by A14, XCMPLX_0:def 7

 .= (p `1 );

 ((((1 - r) * |[a, d]|) + (r * |[b, d]|)) `2 ) = ((((1 - r) * |[a, d]|) `2 ) + ((r * |[b, d]|) `2 )) by TOPREAL3: 2

 .= (((1 - r) * ( |[a, d]| `2 )) + ((r * |[b, d]|) `2 )) by TOPREAL3: 4

 .= (((1 - r) * d) + ((r * |[b, d]|) `2 )) by EUCLID: 52

 .= (((1 - r) * d) + (r * ( |[b, d]| `2 ))) by TOPREAL3: 4

 .= (((1 - r) * d) + (r * d)) by EUCLID: 52

 .= (p `2 ) by A13;



 then p = |[((((1 - r) * |[a, d]|) + (r * |[b, d]|)) `1 ), ((((1 - r) * |[a, d]|) + (r * |[b, d]|)) `2 )]| by A19, EUCLID: 53

 .= (((1 - r) * |[a, d]|) + (r * |[b, d]|)) by EUCLID: 53;

 hence thesis by A4, A14, A15, A18;

 end;

 case

 A20: (p `1 ) = b & c <= (p `2 ) & (p `2 ) <= d;



 A21: (d - c) > 0 by A3, XREAL_1: 50;



 A22: ((p `2 ) - c) >= 0 by A20, XREAL_1: 48;



 A23: (d - (p `2 )) >= 0 by A20, XREAL_1: 48;

 reconsider r = ((d - (p `2 )) / (d - c)) as Real;



 A24: (1 - r) = (((d - c) / (d - c)) - ((d - (p `2 )) / (d - c))) by A21, XCMPLX_1: 60

 .= (((d - c) - (d - (p `2 ))) / (d - c)) by XCMPLX_1: 120

 .= (((p `2 ) - c) / (d - c));

 then

 A25: ((1 - r) + r) >= ( 0 + r) by A21, A22, XREAL_1: 7;



 A26: ((((1 - r) * |[b, d]|) + (r * |[b, c]|)) `1 ) = ((((1 - r) * |[b, d]|) `1 ) + ((r * |[b, c]|) `1 )) by TOPREAL3: 2

 .= (((1 - r) * ( |[b, d]| `1 )) + ((r * |[b, c]|) `1 )) by TOPREAL3: 4

 .= (((1 - r) * b) + ((r * |[b, c]|) `1 )) by EUCLID: 52

 .= (((1 - r) * b) + (r * ( |[b, c]| `1 ))) by TOPREAL3: 4

 .= (((1 - r) * b) + (r * b)) by EUCLID: 52

 .= (p `1 ) by A20;

 ((((1 - r) * |[b, d]|) + (r * |[b, c]|)) `2 ) = ((((1 - r) * |[b, d]|) `2 ) + ((r * |[b, c]|) `2 )) by TOPREAL3: 2

 .= (((1 - r) * ( |[b, d]| `2 )) + ((r * |[b, c]|) `2 )) by TOPREAL3: 4

 .= (((1 - r) * d) + ((r * |[b, c]|) `2 )) by EUCLID: 52

 .= (((1 - r) * d) + (r * ( |[b, c]| `2 ))) by TOPREAL3: 4

 .= (((((p `2 ) - c) / (d - c)) * d) + (((d - (p `2 )) / (d - c)) * c)) by A24, EUCLID: 52

 .= (((((p `2 ) - c) * ((d - c) " )) * d) + (((d - (p `2 )) / (d - c)) * c)) by XCMPLX_0:def 9

 .= ((((d - c) " ) * (((p `2 ) - c) * d)) + ((((d - c) " ) * (d - (p `2 ))) * c)) by XCMPLX_0:def 9

 .= ((((d - c) " ) * (d - c)) * (p `2 ))

 .= (1 * (p `2 )) by A21, XCMPLX_0:def 7

 .= (p `2 );



 then p = |[((((1 - r) * |[b, d]|) + (r * |[b, c]|)) `1 ), ((((1 - r) * |[b, d]|) + (r * |[b, c]|)) `2 )]| by A26, EUCLID: 53

 .= (((1 - r) * |[b, d]|) + (r * |[b, c]|)) by EUCLID: 53;

 hence thesis by A4, A21, A23, A25;

 end;

 case

 A27: (p `2 ) = c & a <= (p `1 ) & (p `1 ) <= b;



 A28: (b - a) > 0 by A2, XREAL_1: 50;



 A29: ((p `1 ) - a) >= 0 by A27, XREAL_1: 48;



 A30: (b - (p `1 )) >= 0 by A27, XREAL_1: 48;

 reconsider r = ((b - (p `1 )) / (b - a)) as Real;



 A31: (1 - r) = (((b - a) / (b - a)) - ((b - (p `1 )) / (b - a))) by A28, XCMPLX_1: 60

 .= (((b - a) - (b - (p `1 ))) / (b - a)) by XCMPLX_1: 120

 .= (((p `1 ) - a) / (b - a));

 then

 A32: ((1 - r) + r) >= ( 0 + r) by A28, A29, XREAL_1: 7;



 A33: ((((1 - r) * |[b, c]|) + (r * |[a, c]|)) `1 ) = ((((1 - r) * |[b, c]|) `1 ) + ((r * |[a, c]|) `1 )) by TOPREAL3: 2

 .= (((1 - r) * ( |[b, c]| `1 )) + ((r * |[a, c]|) `1 )) by TOPREAL3: 4

 .= (((1 - r) * b) + ((r * |[a, c]|) `1 )) by EUCLID: 52

 .= (((1 - r) * b) + (r * ( |[a, c]| `1 ))) by TOPREAL3: 4

 .= (((((p `1 ) - a) / (b - a)) * b) + (((b - (p `1 )) / (b - a)) * a)) by A31, EUCLID: 52

 .= (((((p `1 ) - a) * ((b - a) " )) * b) + (((b - (p `1 )) / (b - a)) * a)) by XCMPLX_0:def 9

 .= ((((b - a) " ) * (((p `1 ) - a) * b)) + ((((b - a) " ) * (b - (p `1 ))) * a)) by XCMPLX_0:def 9

 .= ((((b - a) " ) * (b - a)) * (p `1 ))

 .= (1 * (p `1 )) by A28, XCMPLX_0:def 7

 .= (p `1 );

 ((((1 - r) * |[b, c]|) + (r * |[a, c]|)) `2 ) = ((((1 - r) * |[b, c]|) `2 ) + ((r * |[a, c]|) `2 )) by TOPREAL3: 2

 .= (((1 - r) * ( |[b, c]| `2 )) + ((r * |[a, c]|) `2 )) by TOPREAL3: 4

 .= (((1 - r) * c) + ((r * |[a, c]|) `2 )) by EUCLID: 52

 .= (((1 - r) * c) + (r * ( |[a, c]| `2 ))) by TOPREAL3: 4

 .= (((1 - r) * c) + (r * c)) by EUCLID: 52

 .= (p `2 ) by A27;



 then p = |[((((1 - r) * |[b, c]|) + (r * |[a, c]|)) `1 ), ((((1 - r) * |[b, c]|) + (r * |[a, c]|)) `2 )]| by A33, EUCLID: 53

 .= (((1 - r) * |[b, c]|) + (r * |[a, c]|)) by EUCLID: 53;

 hence thesis by A4, A28, A30, A32;

 end;

 end;

 hence thesis;

 end;

 begin

 theorem :: JGRAPH_6:64



 Th64: for p1,p2 be Point of ( TOP-REAL 2) st LE (p1,p2,( rectangle (( - 1),1,( - 1),1))) & p1 in ( LSeg ( |[( - 1), ( - 1)]|, |[( - 1), 1]|)) holds p2 in ( LSeg ( |[( - 1), ( - 1)]|, |[( - 1), 1]|)) & (p2 `2 ) >= (p1 `2 ) or p2 in ( LSeg ( |[( - 1), 1]|, |[1, 1]|)) or p2 in ( LSeg ( |[1, 1]|, |[1, ( - 1)]|)) or p2 in ( LSeg ( |[1, ( - 1)]|, |[( - 1), ( - 1)]|)) & p2 <> |[( - 1), ( - 1)]|

 proof

 let p1,p2 be Point of ( TOP-REAL 2);

 set K = ( rectangle (( - 1),1,( - 1),1));

 assume that

 A1: LE (p1,p2,K) and

 A2: p1 in ( LSeg ( |[( - 1), ( - 1)]|, |[( - 1), 1]|));

 p2 in ( LSeg ( |[( - 1), ( - 1)]|, |[( - 1), 1]|)) & (p1 `2 ) <= (p2 `2 ) or p2 in ( LSeg ( |[( - 1), 1]|, |[1, 1]|)) or p2 in ( LSeg ( |[1, 1]|, |[1, ( - 1)]|)) or p2 in ( LSeg ( |[1, ( - 1)]|, |[( - 1), ( - 1)]|)) & p2 <> ( W-min K) by A1, A2, Th59;

 hence thesis by Th46;

 end;

 theorem :: JGRAPH_6:65



 Th65: for p1,p2 be Point of ( TOP-REAL 2), P be non empty compact Subset of ( TOP-REAL 2), f be Function of ( TOP-REAL 2), ( TOP-REAL 2) st P = ( circle ( 0 , 0 ,1)) & f = Sq_Circ & p1 in ( LSeg ( |[( - 1), ( - 1)]|, |[( - 1), 1]|)) & (p1 `2 ) >= 0 & LE (p1,p2,( rectangle (( - 1),1,( - 1),1))) holds LE ((f . p1),(f . p2),P)

 proof

 let p1,p2 be Point of ( TOP-REAL 2), P be non empty compact Subset of ( TOP-REAL 2), f be Function of ( TOP-REAL 2), ( TOP-REAL 2);

 set K = ( rectangle (( - 1),1,( - 1),1));

 assume that

 A1: P = ( circle ( 0 , 0 ,1)) and

 A2: f = Sq_Circ and

 A3: p1 in ( LSeg ( |[( - 1), ( - 1)]|, |[( - 1), 1]|)) and

 A4: (p1 `2 ) >= 0 and

 A5: LE (p1,p2,K);



 A6: K is being_simple_closed_curve by Th50;



 A7: P = { p : |.p.| = 1 } by A1, Th24;



 A8: (p1 `1 ) = ( - 1) by A3, Th1;



 A9: (p1 `2 ) <= 1 by A3, Th1;



 A10: p1 in K by A5, A6, JORDAN7: 5;



 A11: p2 in K by A5, A6, JORDAN7: 5;



 A12: (f .: K) = P by A2, A7, Lm15, Th35, JGRAPH_3: 23;



 A13: ( dom f) = the carrier of ( TOP-REAL 2) by FUNCT_2:def 1;

 then

 A14: (f . p1) in P by A10, A12, FUNCT_1:def 6;



 A15: (f . p2) in P by A11, A12, A13, FUNCT_1:def 6;



 A16: (p1 `1 ) = ( - 1) by A3, Th1;



 A17: ((p1 `2 ) ^2 ) >= 0 by XREAL_1: 63;

 then

 A18: ( sqrt (1 + ((p1 `2 ) ^2 ))) > 0 by SQUARE_1: 25;



 A19: (p1 `2 ) <= ( - (p1 `1 )) by A3, A8, Th1;

 p1 <> ( 0. ( TOP-REAL 2)) by A8, EUCLID: 52, EUCLID: 54;

 then

 A20: (f . p1) = |[((p1 `1 ) / ( sqrt (1 + (((p1 `2 ) / (p1 `1 )) ^2 )))), ((p1 `2 ) / ( sqrt (1 + (((p1 `2 ) / (p1 `1 )) ^2 ))))]| by A2, A4, A16, A19, JGRAPH_3:def 1;



 then

 A21: ((f . p1) `1 ) = ((p1 `1 ) / ( sqrt (1 + (((p1 `2 ) / ( - 1)) ^2 )))) by A16, EUCLID: 52

 .= ((p1 `1 ) / ( sqrt (1 + ((p1 `2 ) ^2 ))));



 A22: ((f . p1) `2 ) = ((p1 `2 ) / ( sqrt (1 + (((p1 `2 ) / ( - 1)) ^2 )))) by A16, A20, EUCLID: 52

 .= ((p1 `2 ) / ( sqrt (1 + ((p1 `2 ) ^2 ))));



 A23: ((f . p1) `1 ) < 0 by A16, A17, A21, SQUARE_1: 25, XREAL_1: 141;



 A24: ((f . p1) `2 ) >= 0 by A4, A18, A22;

 (f . p1) in { p9 where p9 be Point of ( TOP-REAL 2) : p9 in P & (p9 `2 ) >= 0 } by A4, A14, A18, A22;

 then

 A25: (f . p1) in ( Upper_Arc P) by A7, JGRAPH_5: 34;

 now

 per cases by A3, A5, Th64;

 case

 A26: p2 in ( LSeg ( |[( - 1), ( - 1)]|, |[( - 1), 1]|)) & (p2 `2 ) >= (p1 `2 );



 A27: ((p2 `2 ) ^2 ) >= 0 by XREAL_1: 63;

 then

 A28: ( sqrt (1 + ((p2 `2 ) ^2 ))) > 0 by SQUARE_1: 25;



 A29: (p2 `1 ) = ( - 1) by A26, Th1;



 A30: ( - 1) <= (p2 `2 ) by A26, Th1;



 A31: (p2 `2 ) <= ( - (p2 `1 )) by A26, A29, Th1;

 p2 <> ( 0. ( TOP-REAL 2)) by A29, EUCLID: 52, EUCLID: 54;

 then

 A32: (f . p2) = |[((p2 `1 ) / ( sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))), ((p2 `2 ) / ( sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))))]| by A2, A29, A30, A31, JGRAPH_3:def 1;



 then

 A33: ((f . p2) `1 ) = ((p2 `1 ) / ( sqrt (1 + (((p2 `2 ) / ( - 1)) ^2 )))) by A29, EUCLID: 52

 .= ((p2 `1 ) / ( sqrt (1 + ((p2 `2 ) ^2 ))));



 A34: ((f . p2) `2 ) = ((p2 `2 ) / ( sqrt (1 + (((p2 `2 ) / ( - 1)) ^2 )))) by A29, A32, EUCLID: 52

 .= ((p2 `2 ) / ( sqrt (1 + ((p2 `2 ) ^2 ))));



 A35: ((f . p2) `1 ) < 0 by A27, A29, A33, SQUARE_1: 25, XREAL_1: 141;

 ((p1 `2 ) * ( sqrt (1 + ((p2 `2 ) ^2 )))) <= ((p2 `2 ) * ( sqrt (1 + ((p1 `2 ) ^2 )))) by A4, A26, Lm3;

 then (((p1 `2 ) * ( sqrt (1 + ((p2 `2 ) ^2 )))) / ( sqrt (1 + ((p2 `2 ) ^2 )))) <= (((p2 `2 ) * ( sqrt (1 + ((p1 `2 ) ^2 )))) / ( sqrt (1 + ((p2 `2 ) ^2 )))) by A28, XREAL_1: 72;

 then (p1 `2 ) <= (((p2 `2 ) * ( sqrt (1 + ((p1 `2 ) ^2 )))) / ( sqrt (1 + ((p2 `2 ) ^2 )))) by A28, XCMPLX_1: 89;

 then ((p1 `2 ) / ( sqrt (1 + ((p1 `2 ) ^2 )))) <= ((((p2 `2 ) * ( sqrt (1 + ((p1 `2 ) ^2 )))) / ( sqrt (1 + ((p2 `2 ) ^2 )))) / ( sqrt (1 + ((p1 `2 ) ^2 )))) by A18, XREAL_1: 72;

 then ((p1 `2 ) / ( sqrt (1 + ((p1 `2 ) ^2 )))) <= ((((p2 `2 ) * ( sqrt (1 + ((p1 `2 ) ^2 )))) / ( sqrt (1 + ((p1 `2 ) ^2 )))) / ( sqrt (1 + ((p2 `2 ) ^2 )))) by XCMPLX_1: 48;

 then ((f . p1) `2 ) <= ((f . p2) `2 ) by A18, A22, A34, XCMPLX_1: 89;

 hence thesis by A7, A14, A15, A23, A24, A35, JGRAPH_5: 53;

 end;

 case

 A36: p2 in ( LSeg ( |[( - 1), 1]|, |[1, 1]|));

 then

 A37: (p2 `2 ) = 1 by Th3;



 A38: ( - 1) <= (p2 `1 ) by A36, Th3;



 A39: (p2 `1 ) <= 1 by A36, Th3;

 ((p2 `1 ) ^2 ) >= 0 by XREAL_1: 63;

 then

 A40: ( sqrt (1 + ((p2 `1 ) ^2 ))) > 0 by SQUARE_1: 25;

 p2 <> ( 0. ( TOP-REAL 2)) by A37, EUCLID: 52, EUCLID: 54;

 then

 A41: (f . p2) = |[((p2 `1 ) / ( sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))), ((p2 `2 ) / ( sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))))]| by A2, A37, A38, A39, JGRAPH_3: 4;

 then

 A42: ((f . p2) `1 ) = ((p2 `1 ) / ( sqrt (1 + ((p2 `1 ) ^2 )))) by A37, EUCLID: 52;



 A43: ((f . p2) `2 ) >= 0 by A37, A40, A41, EUCLID: 52;

 ( - ( sqrt (1 + ((p2 `1 ) ^2 )))) <= ((p2 `1 ) * ( sqrt (1 + ((p1 `2 ) ^2 )))) by A4, A9, A38, A39, SQUARE_1: 55;

 then (((p1 `1 ) * ( sqrt (1 + ((p2 `1 ) ^2 )))) / ( sqrt (1 + ((p2 `1 ) ^2 )))) <= (((p2 `1 ) * ( sqrt (1 + ((p1 `2 ) ^2 )))) / ( sqrt (1 + ((p2 `1 ) ^2 )))) by A8, A40, XREAL_1: 72;

 then (p1 `1 ) <= (((p2 `1 ) * ( sqrt (1 + ((p1 `2 ) ^2 )))) / ( sqrt (1 + ((p2 `1 ) ^2 )))) by A40, XCMPLX_1: 89;

 then ((p1 `1 ) / ( sqrt (1 + ((p1 `2 ) ^2 )))) <= ((((p2 `1 ) * ( sqrt (1 + ((p1 `2 ) ^2 )))) / ( sqrt (1 + ((p2 `1 ) ^2 )))) / ( sqrt (1 + ((p1 `2 ) ^2 )))) by A18, XREAL_1: 72;

 then ((p1 `1 ) / ( sqrt (1 + ((p1 `2 ) ^2 )))) <= ((((p2 `1 ) * ( sqrt (1 + ((p1 `2 ) ^2 )))) / ( sqrt (1 + ((p1 `2 ) ^2 )))) / ( sqrt (1 + ((p2 `1 ) ^2 )))) by XCMPLX_1: 48;

 then ((f . p1) `1 ) <= ((f . p2) `1 ) by A18, A21, A42, XCMPLX_1: 89;

 hence thesis by A4, A7, A14, A15, A18, A22, A43, JGRAPH_5: 54;

 end;

 case

 A44: p2 in ( LSeg ( |[1, 1]|, |[1, ( - 1)]|));

 then

 A45: (p2 `1 ) = 1 by Th1;



 A46: ( - 1) <= (p2 `2 ) by A44, Th1;



 A47: (p2 `2 ) <= 1 by A44, Th1;

 ((p2 `2 ) ^2 ) >= 0 by XREAL_1: 63;

 then

 A48: ( sqrt (1 + ((p2 `2 ) ^2 ))) > 0 by SQUARE_1: 25;

 p2 <> ( 0. ( TOP-REAL 2)) by A45, EUCLID: 52, EUCLID: 54;

 then (f . p2) = |[((p2 `1 ) / ( sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))), ((p2 `2 ) / ( sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))))]| by A2, A45, A46, A47, JGRAPH_3:def 1;

 then

 A49: ((f . p2) `1 ) = ((p2 `1 ) / ( sqrt (1 + ((p2 `2 ) ^2 )))) by A45, EUCLID: 52;

 ((p1 `1 ) / ( sqrt (1 + ((p1 `2 ) ^2 )))) <= ((((p2 `1 ) * ( sqrt (1 + ((p1 `2 ) ^2 )))) / ( sqrt (1 + ((p2 `2 ) ^2 )))) / ( sqrt (1 + ((p1 `2 ) ^2 )))) by A8, A18, A45, A48, XREAL_1: 72;

 then ((p1 `1 ) / ( sqrt (1 + ((p1 `2 ) ^2 )))) <= ((((p2 `1 ) * ( sqrt (1 + ((p1 `2 ) ^2 )))) / ( sqrt (1 + ((p1 `2 ) ^2 )))) / ( sqrt (1 + ((p2 `2 ) ^2 )))) by XCMPLX_1: 48;

 then

 A50: ((f . p1) `1 ) <= ((f . p2) `1 ) by A18, A21, A49, XCMPLX_1: 89;

 now

 per cases ;

 case ((f . p2) `2 ) >= 0 ;

 hence thesis by A4, A7, A14, A15, A18, A22, A50, JGRAPH_5: 54;

 end;

 case

 A51: ((f . p2) `2 ) < 0 ;

 then (f . p2) in { p9 where p9 be Point of ( TOP-REAL 2) : p9 in P & (p9 `2 ) <= 0 } by A15;

 then

 A52: (f . p2) in ( Lower_Arc P) by A7, JGRAPH_5: 35;

 ( W-min P) = |[( - 1), 0 ]| by A7, JGRAPH_5: 29;

 then (f . p2) <> ( W-min P) by A51, EUCLID: 52;

 hence thesis by A25, A52, JORDAN6:def 10;

 end;

 end;

 hence thesis;

 end;

 case

 A53: p2 in ( LSeg ( |[1, ( - 1)]|, |[( - 1), ( - 1)]|)) & p2 <> |[( - 1), ( - 1)]|;

 then

 A54: (p2 `2 ) = ( - 1) by Th3;



 A55: ( - 1) <= (p2 `1 ) by A53, Th3;



 A56: ((p2 `1 ) ^2 ) >= 0 by XREAL_1: 63;



 A57: (p2 `1 ) <= ( - (p2 `2 )) by A53, A54, Th3;

 p2 <> ( 0. ( TOP-REAL 2)) by A54, EUCLID: 52, EUCLID: 54;

 then (f . p2) = |[((p2 `1 ) / ( sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))), ((p2 `2 ) / ( sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))))]| by A2, A54, A55, A57, JGRAPH_3: 4;



 then ((f . p2) `2 ) = ((p2 `2 ) / ( sqrt (1 + (((p2 `1 ) / ( - 1)) ^2 )))) by A54, EUCLID: 52

 .= ((p2 `2 ) / ( sqrt (1 + ((p2 `1 ) ^2 ))));

 then

 A58: ((f . p2) `2 ) < 0 by A54, A56, SQUARE_1: 25, XREAL_1: 141;

 then (f . p2) in { p9 where p9 be Point of ( TOP-REAL 2) : p9 in P & (p9 `2 ) <= 0 } by A15;

 then

 A59: (f . p2) in ( Lower_Arc P) by A7, JGRAPH_5: 35;

 ( W-min P) = |[( - 1), 0 ]| by A7, JGRAPH_5: 29;

 then (f . p2) <> ( W-min P) by A58, EUCLID: 52;

 hence thesis by A25, A59, JORDAN6:def 10;

 end;

 end;

 hence thesis;

 end;

 theorem :: JGRAPH_6:66



 Th66: for p1,p2,p3 be Point of ( TOP-REAL 2), P be non empty compact Subset of ( TOP-REAL 2), f be Function of ( TOP-REAL 2), ( TOP-REAL 2) st P = ( circle ( 0 , 0 ,1)) & f = Sq_Circ & p1 in ( LSeg ( |[( - 1), ( - 1)]|, |[( - 1), 1]|)) & (p1 `2 ) >= 0 & LE (p1,p2,( rectangle (( - 1),1,( - 1),1))) & LE (p2,p3,( rectangle (( - 1),1,( - 1),1))) holds LE ((f . p2),(f . p3),P)

 proof

 let p1,p2,p3 be Point of ( TOP-REAL 2), P be non empty compact Subset of ( TOP-REAL 2), f be Function of ( TOP-REAL 2), ( TOP-REAL 2);

 set K = ( rectangle (( - 1),1,( - 1),1));

 assume that

 A1: P = ( circle ( 0 , 0 ,1)) and

 A2: f = Sq_Circ and

 A3: p1 in ( LSeg ( |[( - 1), ( - 1)]|, |[( - 1), 1]|)) and

 A4: (p1 `2 ) >= 0 and

 A5: LE (p1,p2,K) and

 A6: LE (p2,p3,K);



 A7: K is being_simple_closed_curve by Th50;



 A8: P = { p : |.p.| = 1 } by A1, Th24;



 A9: p3 in K by A6, A7, JORDAN7: 5;



 A10: p2 in K by A5, A7, JORDAN7: 5;



 A11: (f .: K) = P by A2, A8, Lm15, Th35, JGRAPH_3: 23;



 A12: ( dom f) = the carrier of ( TOP-REAL 2) by FUNCT_2:def 1;

 then

 A13: (f . p2) in P by A10, A11, FUNCT_1:def 6;



 A14: (f . p3) in P by A9, A11, A12, FUNCT_1:def 6;

 now

 per cases by A3, A5, Th64;

 case p2 in ( LSeg ( |[( - 1), ( - 1)]|, |[( - 1), 1]|)) & (p2 `2 ) >= (p1 `2 );

 hence thesis by A1, A2, A4, A6, Th65;

 end;

 case

 A15: p2 in ( LSeg ( |[( - 1), 1]|, |[1, 1]|));

 then

 A16: (p2 `2 ) = 1 by Th3;



 A17: ( - 1) <= (p2 `1 ) by A15, Th3;



 A18: (p2 `1 ) <= 1 by A15, Th3;

 ((p2 `1 ) ^2 ) >= 0 by XREAL_1: 63;

 then

 A19: ( sqrt (1 + ((p2 `1 ) ^2 ))) > 0 by SQUARE_1: 25;

 p2 <> ( 0. ( TOP-REAL 2)) by A16, EUCLID: 52, EUCLID: 54;

 then

 A20: (f . p2) = |[((p2 `1 ) / ( sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))), ((p2 `2 ) / ( sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))))]| by A2, A16, A17, A18, JGRAPH_3: 4;

 then

 A21: ((f . p2) `1 ) = ((p2 `1 ) / ( sqrt (1 + ((p2 `1 ) ^2 )))) by A16, EUCLID: 52;



 A22: ((f . p2) `2 ) >= 0 by A16, A19, A20, EUCLID: 52;

 then (f . p2) in { p9 where p9 be Point of ( TOP-REAL 2) : p9 in P & (p9 `2 ) >= 0 } by A13;

 then

 A23: (f . p2) in ( Upper_Arc P) by A8, JGRAPH_5: 34;

 now

 per cases by A6, A15, Th60;

 case

 A24: p3 in ( LSeg ( |[( - 1), 1]|, |[1, 1]|)) & (p2 `1 ) <= (p3 `1 );

 then

 A25: (p3 `2 ) = 1 by Th3;



 A26: ( - 1) <= (p3 `1 ) by A24, Th3;



 A27: (p3 `1 ) <= 1 by A24, Th3;

 ((p3 `1 ) ^2 ) >= 0 by XREAL_1: 63;

 then

 A28: ( sqrt (1 + ((p3 `1 ) ^2 ))) > 0 by SQUARE_1: 25;

 p3 <> ( 0. ( TOP-REAL 2)) by A25, EUCLID: 52, EUCLID: 54;

 then

 A29: (f . p3) = |[((p3 `1 ) / ( sqrt (1 + (((p3 `1 ) / (p3 `2 )) ^2 )))), ((p3 `2 ) / ( sqrt (1 + (((p3 `1 ) / (p3 `2 )) ^2 ))))]| by A2, A25, A26, A27, JGRAPH_3: 4;

 then

 A30: ((f . p3) `1 ) = ((p3 `1 ) / ( sqrt (1 + ((p3 `1 ) ^2 )))) by A25, EUCLID: 52;



 A31: ((f . p3) `2 ) >= 0 by A25, A28, A29, EUCLID: 52;

 ((p2 `1 ) * ( sqrt (1 + ((p3 `1 ) ^2 )))) <= ((p3 `1 ) * ( sqrt (1 + ((p2 `1 ) ^2 )))) by A24, SQUARE_1: 57;

 then (((p2 `1 ) * ( sqrt (1 + ((p3 `1 ) ^2 )))) / ( sqrt (1 + ((p3 `1 ) ^2 )))) <= (((p3 `1 ) * ( sqrt (1 + ((p2 `1 ) ^2 )))) / ( sqrt (1 + ((p3 `1 ) ^2 )))) by A28, XREAL_1: 72;

 then (p2 `1 ) <= (((p3 `1 ) * ( sqrt (1 + ((p2 `1 ) ^2 )))) / ( sqrt (1 + ((p3 `1 ) ^2 )))) by A28, XCMPLX_1: 89;

 then ((p2 `1 ) / ( sqrt (1 + ((p2 `1 ) ^2 )))) <= ((((p3 `1 ) * ( sqrt (1 + ((p2 `1 ) ^2 )))) / ( sqrt (1 + ((p3 `1 ) ^2 )))) / ( sqrt (1 + ((p2 `1 ) ^2 )))) by A19, XREAL_1: 72;

 then ((p2 `1 ) / ( sqrt (1 + ((p2 `1 ) ^2 )))) <= ((((p3 `1 ) * ( sqrt (1 + ((p2 `1 ) ^2 )))) / ( sqrt (1 + ((p2 `1 ) ^2 )))) / ( sqrt (1 + ((p3 `1 ) ^2 )))) by XCMPLX_1: 48;

 then ((f . p2) `1 ) <= ((f . p3) `1 ) by A19, A21, A30, XCMPLX_1: 89;

 hence thesis by A8, A13, A14, A22, A31, JGRAPH_5: 54;

 end;

 case

 A32: p3 in ( LSeg ( |[1, 1]|, |[1, ( - 1)]|));

 then

 A33: (p3 `1 ) = 1 by Th1;



 A34: ( - 1) <= (p3 `2 ) by A32, Th1;



 A35: (p3 `2 ) <= 1 by A32, Th1;

 ((p3 `2 ) ^2 ) >= 0 by XREAL_1: 63;

 then

 A36: ( sqrt (1 + ((p3 `2 ) ^2 ))) > 0 by SQUARE_1: 25;

 p3 <> ( 0. ( TOP-REAL 2)) by A33, EUCLID: 52, EUCLID: 54;

 then (f . p3) = |[((p3 `1 ) / ( sqrt (1 + (((p3 `2 ) / (p3 `1 )) ^2 )))), ((p3 `2 ) / ( sqrt (1 + (((p3 `2 ) / (p3 `1 )) ^2 ))))]| by A2, A33, A34, A35, JGRAPH_3:def 1;

 then

 A37: ((f . p3) `1 ) = ((p3 `1 ) / ( sqrt (1 + ((p3 `2 ) ^2 )))) by A33, EUCLID: 52;



 A38: ( - 1) <= ( - (p2 `1 )) by A18, XREAL_1: 24;



 A39: ( - ( - 1)) >= ( - (p2 `1 )) by A17, XREAL_1: 24;

 ((p2 `1 ) ^2 ) = (( - (p2 `1 )) ^2 );

 then (( - ( - (p2 `1 ))) * ( sqrt (1 + ((p3 `2 ) ^2 )))) <= ( sqrt (1 + ((p2 `1 ) ^2 ))) by A34, A35, A38, A39, SQUARE_1: 55;

 then (((p2 `1 ) * ( sqrt (1 + ((p3 `2 ) ^2 )))) / ( sqrt (1 + ((p3 `2 ) ^2 )))) <= (((p3 `1 ) * ( sqrt (1 + ((p2 `1 ) ^2 )))) / ( sqrt (1 + ((p3 `2 ) ^2 )))) by A33, A36, XREAL_1: 72;

 then (p2 `1 ) <= (((p3 `1 ) * ( sqrt (1 + ((p2 `1 ) ^2 )))) / ( sqrt (1 + ((p3 `2 ) ^2 )))) by A36, XCMPLX_1: 89;

 then ((p2 `1 ) / ( sqrt (1 + ((p2 `1 ) ^2 )))) <= ((((p3 `1 ) * ( sqrt (1 + ((p2 `1 ) ^2 )))) / ( sqrt (1 + ((p3 `2 ) ^2 )))) / ( sqrt (1 + ((p2 `1 ) ^2 )))) by A19, XREAL_1: 72;

 then ((p2 `1 ) / ( sqrt (1 + ((p2 `1 ) ^2 )))) <= ((((p3 `1 ) * ( sqrt (1 + ((p2 `1 ) ^2 )))) / ( sqrt (1 + ((p2 `1 ) ^2 )))) / ( sqrt (1 + ((p3 `2 ) ^2 )))) by XCMPLX_1: 48;

 then

 A40: ((f . p2) `1 ) <= ((f . p3) `1 ) by A19, A21, A37, XCMPLX_1: 89;

 now

 per cases ;

 case ((f . p3) `2 ) >= 0 ;

 hence thesis by A8, A13, A14, A22, A40, JGRAPH_5: 54;

 end;

 case

 A41: ((f . p3) `2 ) < 0 ;

 then (f . p3) in { p9 where p9 be Point of ( TOP-REAL 2) : p9 in P & (p9 `2 ) <= 0 } by A14;

 then

 A42: (f . p3) in ( Lower_Arc P) by A8, JGRAPH_5: 35;

 ( W-min P) = |[( - 1), 0 ]| by A8, JGRAPH_5: 29;

 then (f . p3) <> ( W-min P) by A41, EUCLID: 52;

 hence thesis by A23, A42, JORDAN6:def 10;

 end;

 end;

 hence thesis;

 end;

 case

 A43: p3 in ( LSeg ( |[1, ( - 1)]|, |[( - 1), ( - 1)]|)) & p3 <> ( W-min K);

 then

 A44: (p3 `2 ) = ( - 1) by Th3;



 A45: ( - 1) <= (p3 `1 ) by A43, Th3;



 A46: ((p3 `1 ) ^2 ) >= 0 by XREAL_1: 63;



 A47: ( - (p3 `2 )) >= (p3 `1 ) by A43, A44, Th3;

 p3 <> ( 0. ( TOP-REAL 2)) by A44, EUCLID: 52, EUCLID: 54;

 then (f . p3) = |[((p3 `1 ) / ( sqrt (1 + (((p3 `1 ) / (p3 `2 )) ^2 )))), ((p3 `2 ) / ( sqrt (1 + (((p3 `1 ) / (p3 `2 )) ^2 ))))]| by A2, A44, A45, A47, JGRAPH_3: 4;



 then ((f . p3) `2 ) = ((p3 `2 ) / ( sqrt (1 + (((p3 `1 ) / ( - 1)) ^2 )))) by A44, EUCLID: 52

 .= ((p3 `2 ) / ( sqrt (1 + ((p3 `1 ) ^2 ))));

 then

 A48: ((f . p3) `2 ) < 0 by A44, A46, SQUARE_1: 25, XREAL_1: 141;

 then (f . p3) in { p9 where p9 be Point of ( TOP-REAL 2) : p9 in P & (p9 `2 ) <= 0 } by A14;

 then

 A49: (f . p3) in ( Lower_Arc P) by A8, JGRAPH_5: 35;

 ( W-min P) = |[( - 1), 0 ]| by A8, JGRAPH_5: 29;

 then (f . p3) <> ( W-min P) by A48, EUCLID: 52;

 hence thesis by A23, A49, JORDAN6:def 10;

 end;

 end;

 hence thesis;

 end;

 case

 A50: p2 in ( LSeg ( |[1, 1]|, |[1, ( - 1)]|));

 then

 A51: (p2 `1 ) = 1 by Th1;



 A52: ( - 1) <= (p2 `2 ) by A50, Th1;



 A53: (p2 `2 ) <= 1 by A50, Th1;

 ((p2 `2 ) ^2 ) >= 0 by XREAL_1: 63;

 then

 A54: ( sqrt (1 + ((p2 `2 ) ^2 ))) > 0 by SQUARE_1: 25;

 p2 <> ( 0. ( TOP-REAL 2)) by A51, EUCLID: 52, EUCLID: 54;

 then

 A55: (f . p2) = |[((p2 `1 ) / ( sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))), ((p2 `2 ) / ( sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))))]| by A2, A51, A52, A53, JGRAPH_3:def 1;

 then

 A56: ((f . p2) `1 ) = ((p2 `1 ) / ( sqrt (1 + ((p2 `2 ) ^2 )))) by A51, EUCLID: 52;



 A57: ((f . p2) `2 ) = ((p2 `2 ) / ( sqrt (1 + ((p2 `2 ) ^2 )))) by A51, A55, EUCLID: 52;

 now

 per cases by A6, A50, Th61;

 case

 A58: p3 in ( LSeg ( |[1, 1]|, |[1, ( - 1)]|)) & (p2 `2 ) >= (p3 `2 );

 then

 A59: (p3 `1 ) = 1 by Th1;



 A60: ( - 1) <= (p3 `2 ) by A58, Th1;



 A61: (p3 `2 ) <= 1 by A58, Th1;

 ((p3 `2 ) ^2 ) >= 0 by XREAL_1: 63;

 then

 A62: ( sqrt (1 + ((p3 `2 ) ^2 ))) > 0 by SQUARE_1: 25;

 p3 <> ( 0. ( TOP-REAL 2)) by A59, EUCLID: 52, EUCLID: 54;

 then

 A63: (f . p3) = |[((p3 `1 ) / ( sqrt (1 + (((p3 `2 ) / (p3 `1 )) ^2 )))), ((p3 `2 ) / ( sqrt (1 + (((p3 `2 ) / (p3 `1 )) ^2 ))))]| by A2, A59, A60, A61, JGRAPH_3:def 1;

 then

 A64: ((f . p3) `2 ) = ((p3 `2 ) / ( sqrt (1 + ((p3 `2 ) ^2 )))) by A59, EUCLID: 52;



 A65: ((f . p3) `1 ) >= 0 by A59, A62, A63, EUCLID: 52;

 ((p2 `2 ) * ( sqrt (1 + ((p3 `2 ) ^2 )))) >= ((p3 `2 ) * ( sqrt (1 + ((p2 `2 ) ^2 )))) by A58, SQUARE_1: 57;

 then (((p2 `2 ) * ( sqrt (1 + ((p3 `2 ) ^2 )))) / ( sqrt (1 + ((p3 `2 ) ^2 )))) >= (((p3 `2 ) * ( sqrt (1 + ((p2 `2 ) ^2 )))) / ( sqrt (1 + ((p3 `2 ) ^2 )))) by A62, XREAL_1: 72;

 then (p2 `2 ) >= (((p3 `2 ) * ( sqrt (1 + ((p2 `2 ) ^2 )))) / ( sqrt (1 + ((p3 `2 ) ^2 )))) by A62, XCMPLX_1: 89;

 then ((p2 `2 ) / ( sqrt (1 + ((p2 `2 ) ^2 )))) >= ((((p3 `2 ) * ( sqrt (1 + ((p2 `2 ) ^2 )))) / ( sqrt (1 + ((p3 `2 ) ^2 )))) / ( sqrt (1 + ((p2 `2 ) ^2 )))) by A54, XREAL_1: 72;

 then ((p2 `2 ) / ( sqrt (1 + ((p2 `2 ) ^2 )))) >= ((((p3 `2 ) * ( sqrt (1 + ((p2 `2 ) ^2 )))) / ( sqrt (1 + ((p2 `2 ) ^2 )))) / ( sqrt (1 + ((p3 `2 ) ^2 )))) by XCMPLX_1: 48;

 then ((p2 `2 ) / ( sqrt (1 + ((p2 `2 ) ^2 )))) >= ((p3 `2 ) / ( sqrt (1 + ((p3 `2 ) ^2 )))) by A54, XCMPLX_1: 89;

 hence thesis by A8, A13, A14, A51, A54, A56, A57, A64, A65, JGRAPH_5: 55;

 end;

 case

 A66: p3 in ( LSeg ( |[1, ( - 1)]|, |[( - 1), ( - 1)]|)) & p3 <> ( W-min K);

 then

 A67: (p3 `2 ) = ( - 1) by Th3;



 A68: ( - 1) <= (p3 `1 ) by A66, Th3;



 A69: (p3 `1 ) <= 1 by A66, Th3;



 A70: ((p3 `1 ) ^2 ) >= 0 by XREAL_1: 63;

 then

 A71: ( sqrt (1 + ((p3 `1 ) ^2 ))) > 0 by SQUARE_1: 25;



 A72: ( - (p3 `2 )) >= (p3 `1 ) by A66, A67, Th3;

 p3 <> ( 0. ( TOP-REAL 2)) by A67, EUCLID: 52, EUCLID: 54;

 then

 A73: (f . p3) = |[((p3 `1 ) / ( sqrt (1 + (((p3 `1 ) / (p3 `2 )) ^2 )))), ((p3 `2 ) / ( sqrt (1 + (((p3 `1 ) / (p3 `2 )) ^2 ))))]| by A2, A67, A68, A72, JGRAPH_3: 4;



 then

 A74: ((f . p3) `1 ) = ((p3 `1 ) / ( sqrt (1 + (((p3 `1 ) / ( - 1)) ^2 )))) by A67, EUCLID: 52

 .= ((p3 `1 ) / ( sqrt (1 + ((p3 `1 ) ^2 ))));



 A75: ((f . p3) `2 ) = ((p3 `2 ) / ( sqrt (1 + (((p3 `1 ) / ( - 1)) ^2 )))) by A67, A73, EUCLID: 52

 .= ((p3 `2 ) / ( sqrt (1 + ((p3 `1 ) ^2 ))));

 then

 A76: ((f . p3) `2 ) < 0 by A67, A70, SQUARE_1: 25, XREAL_1: 141;

 (f . p3) in { p9 where p9 be Point of ( TOP-REAL 2) : p9 in P & (p9 `2 ) <= 0 } by A14, A67, A71, A75;

 then

 A77: (f . p3) in ( Lower_Arc P) by A8, JGRAPH_5: 35;

 ( W-min P) = |[( - 1), 0 ]| by A8, JGRAPH_5: 29;

 then

 A78: (f . p3) <> ( W-min P) by A76, EUCLID: 52;

 now

 per cases ;

 case ((f . p2) `2 ) >= 0 ;

 then (f . p2) in { p9 where p9 be Point of ( TOP-REAL 2) : p9 in P & (p9 `2 ) >= 0 } by A13;

 then (f . p2) in ( Upper_Arc P) by A8, JGRAPH_5: 34;

 hence thesis by A77, A78, JORDAN6:def 10;

 end;

 case

 A79: ((f . p2) `2 ) < 0 ;

 (((p2 `1 ) * ( sqrt (1 + ((p3 `1 ) ^2 )))) / ( sqrt (1 + ((p3 `1 ) ^2 )))) >= (((p3 `1 ) * ( sqrt (1 + ((p2 `2 ) ^2 )))) / ( sqrt (1 + ((p3 `1 ) ^2 )))) by A51, A52, A53, A68, A69, A71, SQUARE_1: 56, XREAL_1: 72;

 then (p2 `1 ) >= (((p3 `1 ) * ( sqrt (1 + ((p2 `2 ) ^2 )))) / ( sqrt (1 + ((p3 `1 ) ^2 )))) by A71, XCMPLX_1: 89;

 then ((p2 `1 ) / ( sqrt (1 + ((p2 `2 ) ^2 )))) >= ((((p3 `1 ) * ( sqrt (1 + ((p2 `2 ) ^2 )))) / ( sqrt (1 + ((p3 `1 ) ^2 )))) / ( sqrt (1 + ((p2 `2 ) ^2 )))) by A54, XREAL_1: 72;

 then ((p2 `1 ) / ( sqrt (1 + ((p2 `2 ) ^2 )))) >= ((((p3 `1 ) * ( sqrt (1 + ((p2 `2 ) ^2 )))) / ( sqrt (1 + ((p2 `2 ) ^2 )))) / ( sqrt (1 + ((p3 `1 ) ^2 )))) by XCMPLX_1: 48;

 then ((p2 `1 ) / ( sqrt (1 + ((p2 `2 ) ^2 )))) >= ((p3 `1 ) / ( sqrt (1 + ((p3 `1 ) ^2 )))) by A54, XCMPLX_1: 89;

 hence thesis by A8, A13, A14, A56, A67, A71, A74, A75, A78, A79, JGRAPH_5: 56;

 end;

 end;

 hence thesis;

 end;

 end;

 hence thesis;

 end;

 case

 A80: p2 in ( LSeg ( |[1, ( - 1)]|, |[( - 1), ( - 1)]|)) & p2 <> |[( - 1), ( - 1)]|;

 then

 A81: (p2 `2 ) = ( - 1) by Th3;



 A82: ( - 1) <= (p2 `1 ) by A80, Th3;

 ((p2 `1 ) ^2 ) >= 0 by XREAL_1: 63;

 then

 A83: ( sqrt (1 + ((p2 `1 ) ^2 ))) > 0 by SQUARE_1: 25;



 A84: ( - (p2 `2 )) >= (p2 `1 ) by A80, A81, Th3;

 p2 <> ( 0. ( TOP-REAL 2)) by A81, EUCLID: 52, EUCLID: 54;

 then

 A85: (f . p2) = |[((p2 `1 ) / ( sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))), ((p2 `2 ) / ( sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))))]| by A2, A81, A82, A84, JGRAPH_3: 4;



 then

 A86: ((f . p2) `1 ) = ((p2 `1 ) / ( sqrt (1 + (((p2 `1 ) / ( - 1)) ^2 )))) by A81, EUCLID: 52

 .= ((p2 `1 ) / ( sqrt (1 + ((p2 `1 ) ^2 ))));



 A87: ((f . p2) `2 ) = ((p2 `2 ) / ( sqrt (1 + (((p2 `1 ) / ( - 1)) ^2 )))) by A81, A85, EUCLID: 52

 .= ((p2 `2 ) / ( sqrt (1 + ((p2 `1 ) ^2 ))));



 A88: ( W-min K) = |[( - 1), ( - 1)]| by Th46;

 then

 A89: p3 in ( LSeg ( |[1, ( - 1)]|, |[( - 1), ( - 1)]|)) by A6, A80, Th62;



 A90: (p2 `1 ) >= (p3 `1 ) by A6, A80, A88, Th62;



 A91: (p3 `2 ) = ( - 1) by A89, Th3;



 A92: ( - 1) <= (p3 `1 ) by A89, Th3;



 A93: ((p3 `1 ) ^2 ) >= 0 by XREAL_1: 63;

 then

 A94: ( sqrt (1 + ((p3 `1 ) ^2 ))) > 0 by SQUARE_1: 25;



 A95: ( - (p3 `2 )) >= (p3 `1 ) by A89, A91, Th3;

 p3 <> ( 0. ( TOP-REAL 2)) by A91, EUCLID: 52, EUCLID: 54;

 then

 A96: (f . p3) = |[((p3 `1 ) / ( sqrt (1 + (((p3 `1 ) / (p3 `2 )) ^2 )))), ((p3 `2 ) / ( sqrt (1 + (((p3 `1 ) / (p3 `2 )) ^2 ))))]| by A2, A91, A92, A95, JGRAPH_3: 4;



 then

 A97: ((f . p3) `1 ) = ((p3 `1 ) / ( sqrt (1 + (((p3 `1 ) / ( - 1)) ^2 )))) by A91, EUCLID: 52

 .= ((p3 `1 ) / ( sqrt (1 + ((p3 `1 ) ^2 ))));

 ((f . p3) `2 ) = ((p3 `2 ) / ( sqrt (1 + (((p3 `1 ) / ( - 1)) ^2 )))) by A91, A96, EUCLID: 52

 .= ((p3 `2 ) / ( sqrt (1 + ((p3 `1 ) ^2 ))));

 then

 A98: ((f . p3) `2 ) < 0 by A91, A93, SQUARE_1: 25, XREAL_1: 141;

 ( W-min P) = |[( - 1), 0 ]| by A8, JGRAPH_5: 29;

 then

 A99: (f . p3) <> ( W-min P) by A98, EUCLID: 52;

 ((p2 `1 ) * ( sqrt (1 + ((p3 `1 ) ^2 )))) >= ((p3 `1 ) * ( sqrt (1 + ((p2 `1 ) ^2 )))) by A90, SQUARE_1: 57;

 then (((p2 `1 ) * ( sqrt (1 + ((p3 `1 ) ^2 )))) / ( sqrt (1 + ((p3 `1 ) ^2 )))) >= (((p3 `1 ) * ( sqrt (1 + ((p2 `1 ) ^2 )))) / ( sqrt (1 + ((p3 `1 ) ^2 )))) by A94, XREAL_1: 72;

 then (p2 `1 ) >= (((p3 `1 ) * ( sqrt (1 + ((p2 `1 ) ^2 )))) / ( sqrt (1 + ((p3 `1 ) ^2 )))) by A94, XCMPLX_1: 89;

 then ((p2 `1 ) / ( sqrt (1 + ((p2 `1 ) ^2 )))) >= ((((p3 `1 ) * ( sqrt (1 + ((p2 `1 ) ^2 )))) / ( sqrt (1 + ((p3 `1 ) ^2 )))) / ( sqrt (1 + ((p2 `1 ) ^2 )))) by A83, XREAL_1: 72;

 then ((p2 `1 ) / ( sqrt (1 + ((p2 `1 ) ^2 )))) >= ((((p3 `1 ) * ( sqrt (1 + ((p2 `1 ) ^2 )))) / ( sqrt (1 + ((p2 `1 ) ^2 )))) / ( sqrt (1 + ((p3 `1 ) ^2 )))) by XCMPLX_1: 48;

 then ((p2 `1 ) / ( sqrt (1 + ((p2 `1 ) ^2 )))) >= ((p3 `1 ) / ( sqrt (1 + ((p3 `1 ) ^2 )))) by A83, XCMPLX_1: 89;

 hence thesis by A8, A13, A14, A81, A83, A86, A87, A97, A98, A99, JGRAPH_5: 56;

 end;

 end;

 hence thesis;

 end;

 theorem :: JGRAPH_6:67



 Th67: for p be Point of ( TOP-REAL 2), f be Function of ( TOP-REAL 2), ( TOP-REAL 2) st f = Sq_Circ & (p `1 ) = ( - 1) & (p `2 ) < 0 holds ((f . p) `1 ) < 0 & ((f . p) `2 ) < 0

 proof

 let p be Point of ( TOP-REAL 2), f be Function of ( TOP-REAL 2), ( TOP-REAL 2);

 assume that

 A1: f = Sq_Circ and

 A2: (p `1 ) = ( - 1) and

 A3: (p `2 ) < 0 ;

 now

 per cases ;

 case p = ( 0. ( TOP-REAL 2));

 hence contradiction by A2, EUCLID: 52, EUCLID: 54;

 end;

 case

 A4: p <> ( 0. ( TOP-REAL 2));

 now

 per cases ;

 case (p `2 ) <= (p `1 ) & ( - (p `1 )) <= (p `2 ) or (p `2 ) >= (p `1 ) & (p `2 ) <= ( - (p `1 ));

 then

 A5: (f . p) = |[((p `1 ) / ( sqrt (1 + (((p `2 ) / (p `1 )) ^2 )))), ((p `2 ) / ( sqrt (1 + (((p `2 ) / (p `1 )) ^2 ))))]| by A1, A4, JGRAPH_3:def 1;

 then

 A6: ((f . p) `1 ) = ((p `1 ) / ( sqrt (1 + (((p `2 ) / (p `1 )) ^2 )))) by EUCLID: 52;

 ((f . p) `2 ) = ((p `2 ) / ( sqrt (1 + (((p `2 ) / (p `1 )) ^2 )))) by A5, EUCLID: 52;

 hence thesis by A2, A3, A6, SQUARE_1: 25, XREAL_1: 141;

 end;

 case not ((p `2 ) <= (p `1 ) & ( - (p `1 )) <= (p `2 ) or (p `2 ) >= (p `1 ) & (p `2 ) <= ( - (p `1 )));

 then

 A7: (f . p) = |[((p `1 ) / ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 )))), ((p `2 ) / ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 ))))]| by A1, A4, JGRAPH_3:def 1;

 then

 A8: ((f . p) `1 ) = ((p `1 ) / ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 )))) by EUCLID: 52;

 ((f . p) `2 ) = ((p `2 ) / ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 )))) by A7, EUCLID: 52;

 hence thesis by A2, A3, A8, SQUARE_1: 25, XREAL_1: 141;

 end;

 end;

 hence thesis;

 end;

 end;

 hence thesis;

 end;

 theorem :: JGRAPH_6:68



 Th68: for p be Point of ( TOP-REAL 2), P be non empty compact Subset of ( TOP-REAL 2), f be Function of ( TOP-REAL 2), ( TOP-REAL 2) st f = Sq_Circ holds ((f . p) `1 ) >= 0 iff (p `1 ) >= 0

 proof

 let p be Point of ( TOP-REAL 2), P be non empty compact Subset of ( TOP-REAL 2), f be Function of ( TOP-REAL 2), ( TOP-REAL 2);

 assume that

 A1: f = Sq_Circ ;

 thus ((f . p) `1 ) >= 0 implies (p `1 ) >= 0

 proof

 assume

 A2: ((f . p) `1 ) >= 0 ;

 reconsider g = ( Sq_Circ " ) as Function of ( TOP-REAL 2), ( TOP-REAL 2) by JGRAPH_3: 29;



 A3: ( dom f) = the carrier of ( TOP-REAL 2) by FUNCT_2:def 1;

 set q = (f . p);

 now

 per cases ;

 case q = ( 0. ( TOP-REAL 2));

 hence ((g . q) `1 ) >= 0 by A2, JGRAPH_3: 28;

 end;

 case

 A4: q <> ( 0. ( TOP-REAL 2));

 now

 per cases ;

 case (q `2 ) <= (q `1 ) & ( - (q `1 )) <= (q `2 ) or (q `2 ) >= (q `1 ) & (q `2 ) <= ( - (q `1 ));

 then

 A5: (g . q) = |[((q `1 ) * ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))), ((q `2 ) * ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 ))))]| by A4, JGRAPH_3: 28;

 (((q `2 ) / (q `1 )) ^2 ) >= 0 by XREAL_1: 63;

 then ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 ))) > 0 by SQUARE_1: 25;

 hence ((g . q) `1 ) >= 0 by A2, A5, EUCLID: 52;

 end;

 case not ((q `2 ) <= (q `1 ) & ( - (q `1 )) <= (q `2 ) or (q `2 ) >= (q `1 ) & (q `2 ) <= ( - (q `1 )));

 then

 A6: (g . q) = |[((q `1 ) * ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))), ((q `2 ) * ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 ))))]| by JGRAPH_3: 28;

 (((q `1 ) / (q `2 )) ^2 ) >= 0 by XREAL_1: 63;

 then ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 ))) > 0 by SQUARE_1: 25;

 hence ((g . q) `1 ) >= 0 by A2, A6, EUCLID: 52;

 end;

 end;

 hence ((g . q) `1 ) >= 0 ;

 end;

 end;

 hence thesis by A1, A3, FUNCT_1: 34;

 end;

 thus (p `1 ) >= 0 implies ((f . p) `1 ) >= 0

 proof

 assume

 A7: (p `1 ) >= 0 ;

 now

 per cases ;

 case p = ( 0. ( TOP-REAL 2));

 hence thesis by A1, A7, JGRAPH_3:def 1;

 end;

 case

 A8: p <> ( 0. ( TOP-REAL 2));

 now

 per cases ;

 case (p `2 ) <= (p `1 ) & ( - (p `1 )) <= (p `2 ) or (p `2 ) >= (p `1 ) & (p `2 ) <= ( - (p `1 ));

 then

 A9: (f . p) = |[((p `1 ) / ( sqrt (1 + (((p `2 ) / (p `1 )) ^2 )))), ((p `2 ) / ( sqrt (1 + (((p `2 ) / (p `1 )) ^2 ))))]| by A1, A8, JGRAPH_3:def 1;

 (((p `2 ) / (p `1 )) ^2 ) >= 0 by XREAL_1: 63;

 then ( sqrt (1 + (((p `2 ) / (p `1 )) ^2 ))) > 0 by SQUARE_1: 25;

 hence thesis by A7, A9, EUCLID: 52;

 end;

 case not ((p `2 ) <= (p `1 ) & ( - (p `1 )) <= (p `2 ) or (p `2 ) >= (p `1 ) & (p `2 ) <= ( - (p `1 )));

 then

 A10: (f . p) = |[((p `1 ) / ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 )))), ((p `2 ) / ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 ))))]| by A1, A8, JGRAPH_3:def 1;

 (((p `1 ) / (p `2 )) ^2 ) >= 0 by XREAL_1: 63;

 then ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 ))) > 0 by SQUARE_1: 25;

 hence thesis by A7, A10, EUCLID: 52;

 end;

 end;

 hence thesis;

 end;

 end;

 hence thesis;

 end;

 end;

 theorem :: JGRAPH_6:69



 Th69: for p be Point of ( TOP-REAL 2), P be non empty compact Subset of ( TOP-REAL 2), f be Function of ( TOP-REAL 2), ( TOP-REAL 2) st f = Sq_Circ holds ((f . p) `2 ) >= 0 iff (p `2 ) >= 0

 proof

 let p be Point of ( TOP-REAL 2), P be non empty compact Subset of ( TOP-REAL 2), f be Function of ( TOP-REAL 2), ( TOP-REAL 2);

 assume

 A1: f = Sq_Circ ;

 thus ((f . p) `2 ) >= 0 implies (p `2 ) >= 0

 proof

 assume

 A2: ((f . p) `2 ) >= 0 ;

 reconsider g = ( Sq_Circ " ) as Function of ( TOP-REAL 2), ( TOP-REAL 2) by JGRAPH_3: 29;



 A3: ( dom f) = the carrier of ( TOP-REAL 2) by FUNCT_2:def 1;

 set q = (f . p);

 now

 per cases ;

 case q = ( 0. ( TOP-REAL 2));

 hence ((g . q) `2 ) >= 0 by A2, JGRAPH_3: 28;

 end;

 case

 A4: q <> ( 0. ( TOP-REAL 2));

 now

 per cases ;

 case (q `2 ) <= (q `1 ) & ( - (q `1 )) <= (q `2 ) or (q `2 ) >= (q `1 ) & (q `2 ) <= ( - (q `1 ));

 then

 A5: (g . q) = |[((q `1 ) * ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))), ((q `2 ) * ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 ))))]| by A4, JGRAPH_3: 28;

 (((q `2 ) / (q `1 )) ^2 ) >= 0 by XREAL_1: 63;

 then ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 ))) > 0 by SQUARE_1: 25;

 hence ((g . q) `2 ) >= 0 by A2, A5, EUCLID: 52;

 end;

 case not ((q `2 ) <= (q `1 ) & ( - (q `1 )) <= (q `2 ) or (q `2 ) >= (q `1 ) & (q `2 ) <= ( - (q `1 )));

 then

 A6: (g . q) = |[((q `1 ) * ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))), ((q `2 ) * ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 ))))]| by JGRAPH_3: 28;

 (((q `1 ) / (q `2 )) ^2 ) >= 0 by XREAL_1: 63;

 then ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 ))) > 0 by SQUARE_1: 25;

 hence ((g . q) `2 ) >= 0 by A2, A6, EUCLID: 52;

 end;

 end;

 hence ((g . q) `2 ) >= 0 ;

 end;

 end;

 hence thesis by A1, A3, FUNCT_1: 34;

 end;

 thus (p `2 ) >= 0 implies ((f . p) `2 ) >= 0

 proof

 assume

 A7: (p `2 ) >= 0 ;

 now

 per cases ;

 case p = ( 0. ( TOP-REAL 2));

 hence thesis by A1, A7, JGRAPH_3:def 1;

 end;

 case

 A8: p <> ( 0. ( TOP-REAL 2));

 now

 per cases ;

 case (p `2 ) <= (p `1 ) & ( - (p `1 )) <= (p `2 ) or (p `2 ) >= (p `1 ) & (p `2 ) <= ( - (p `1 ));

 then

 A9: (f . p) = |[((p `1 ) / ( sqrt (1 + (((p `2 ) / (p `1 )) ^2 )))), ((p `2 ) / ( sqrt (1 + (((p `2 ) / (p `1 )) ^2 ))))]| by A1, A8, JGRAPH_3:def 1;

 (((p `2 ) / (p `1 )) ^2 ) >= 0 by XREAL_1: 63;

 then ( sqrt (1 + (((p `2 ) / (p `1 )) ^2 ))) > 0 by SQUARE_1: 25;

 hence thesis by A7, A9, EUCLID: 52;

 end;

 case not ((p `2 ) <= (p `1 ) & ( - (p `1 )) <= (p `2 ) or (p `2 ) >= (p `1 ) & (p `2 ) <= ( - (p `1 )));

 then

 A10: (f . p) = |[((p `1 ) / ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 )))), ((p `2 ) / ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 ))))]| by A1, A8, JGRAPH_3:def 1;

 (((p `1 ) / (p `2 )) ^2 ) >= 0 by XREAL_1: 63;

 then ( sqrt (1 + (((p `1 ) / (p `2 )) ^2 ))) > 0 by SQUARE_1: 25;

 hence thesis by A7, A10, EUCLID: 52;

 end;

 end;

 hence thesis;

 end;

 end;

 hence thesis;

 end;

 end;

 theorem :: JGRAPH_6:70



 Th70: for p,q be Point of ( TOP-REAL 2), f be Function of ( TOP-REAL 2), ( TOP-REAL 2) st f = Sq_Circ & p in ( LSeg ( |[( - 1), ( - 1)]|, |[( - 1), 1]|)) & q in ( LSeg ( |[1, ( - 1)]|, |[( - 1), ( - 1)]|)) holds ((f . p) `1 ) <= ((f . q) `1 )

 proof

 let p,q be Point of ( TOP-REAL 2), f be Function of ( TOP-REAL 2), ( TOP-REAL 2);

 assume that

 A1: f = Sq_Circ and

 A2: p in ( LSeg ( |[( - 1), ( - 1)]|, |[( - 1), 1]|)) and

 A3: q in ( LSeg ( |[1, ( - 1)]|, |[( - 1), ( - 1)]|));



 A4: (p `1 ) = ( - 1) by A2, Th1;



 A5: ( - 1) <= (p `2 ) by A2, Th1;



 A6: (p `2 ) <= 1 by A2, Th1;



 A7: (q `2 ) = ( - 1) by A3, Th3;



 A8: ( - 1) <= (q `1 ) by A3, Th3;



 A9: (q `1 ) <= 1 by A3, Th3;



 A10: p <> ( 0. ( TOP-REAL 2)) by A4, EUCLID: 52, EUCLID: 54;



 A11: q <> ( 0. ( TOP-REAL 2)) by A7, EUCLID: 52, EUCLID: 54;

 (p `2 ) <= ( - (p `1 )) by A2, A4, Th1;

 then (f . p) = |[((p `1 ) / ( sqrt (1 + (((p `2 ) / (p `1 )) ^2 )))), ((p `2 ) / ( sqrt (1 + (((p `2 ) / (p `1 )) ^2 ))))]| by A1, A4, A5, A10, JGRAPH_3:def 1;



 then

 A12: ((f . p) `1 ) = (( - 1) / ( sqrt (1 + (((p `2 ) / ( - 1)) ^2 )))) by A4, EUCLID: 52

 .= (( - 1) / ( sqrt (1 + ((p `2 ) ^2 ))));

 ((p `2 ) ^2 ) >= 0 by XREAL_1: 63;

 then

 A13: ( sqrt (1 + ((p `2 ) ^2 ))) > 0 by SQUARE_1: 25;

 ((q `1 ) ^2 ) >= 0 by XREAL_1: 63;

 then

 A14: ( sqrt (1 + ((q `1 ) ^2 ))) > 0 by SQUARE_1: 25;

 (q `1 ) <= ( - (q `2 )) by A3, A7, Th3;

 then (f . q) = |[((q `1 ) / ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))), ((q `2 ) / ( sqrt (1 + (((q `1 ) / (q `2 )) ^2 ))))]| by A1, A7, A8, A11, JGRAPH_3: 4;



 then

 A15: ((f . q) `1 ) = ((q `1 ) / ( sqrt (1 + (((q `1 ) / ( - 1)) ^2 )))) by A7, EUCLID: 52

 .= ((q `1 ) / ( sqrt (1 + ((q `1 ) ^2 ))));

 ( - ( sqrt (1 + ((q `1 ) ^2 )))) <= ((q `1 ) * ( sqrt (1 + ((p `2 ) ^2 )))) by A5, A6, A8, A9, SQUARE_1: 55;

 then ((( - 1) * ( sqrt (1 + ((q `1 ) ^2 )))) / ( sqrt (1 + ((q `1 ) ^2 )))) <= (((q `1 ) * ( sqrt (1 + ((p `2 ) ^2 )))) / ( sqrt (1 + ((q `1 ) ^2 )))) by A14, XREAL_1: 72;

 then ( - 1) <= (((q `1 ) * ( sqrt (1 + ((p `2 ) ^2 )))) / ( sqrt (1 + ((q `1 ) ^2 )))) by A14, XCMPLX_1: 89;

 then ( - 1) <= (((q `1 ) / ( sqrt (1 + ((q `1 ) ^2 )))) * ( sqrt (1 + ((p `2 ) ^2 )))) by XCMPLX_1: 74;

 then (( - 1) / ( sqrt (1 + ((p `2 ) ^2 )))) <= ((((q `1 ) / ( sqrt (1 + ((q `1 ) ^2 )))) * ( sqrt (1 + ((p `2 ) ^2 )))) / ( sqrt (1 + ((p `2 ) ^2 )))) by A13, XREAL_1: 72;

 hence thesis by A12, A13, A15, XCMPLX_1: 89;

 end;

 theorem :: JGRAPH_6:71



 Th71: for p,q be Point of ( TOP-REAL 2), f be Function of ( TOP-REAL 2), ( TOP-REAL 2) st f = Sq_Circ & p in ( LSeg ( |[( - 1), ( - 1)]|, |[( - 1), 1]|)) & q in ( LSeg ( |[( - 1), ( - 1)]|, |[( - 1), 1]|)) & (p `2 ) >= (q `2 ) & (p `2 ) < 0 holds ((f . p) `2 ) >= ((f . q) `2 )

 proof

 let p,q be Point of ( TOP-REAL 2), f be Function of ( TOP-REAL 2), ( TOP-REAL 2);

 assume that

 A1: f = Sq_Circ and

 A2: p in ( LSeg ( |[( - 1), ( - 1)]|, |[( - 1), 1]|)) and

 A3: q in ( LSeg ( |[( - 1), ( - 1)]|, |[( - 1), 1]|)) and

 A4: (p `2 ) >= (q `2 ) and

 A5: (p `2 ) < 0 ;



 A6: (p `1 ) = ( - 1) by A2, Th1;



 A7: ( - 1) <= (p `2 ) by A2, Th1;

 ((p `2 ) ^2 ) >= 0 by XREAL_1: 63;

 then

 A8: ( sqrt (1 + ((p `2 ) ^2 ))) > 0 by SQUARE_1: 25;

 ((q `2 ) ^2 ) >= 0 by XREAL_1: 63;

 then

 A9: ( sqrt (1 + ((q `2 ) ^2 ))) > 0 by SQUARE_1: 25;



 A10: (p `2 ) <= ( - (p `1 )) by A5, A6;

 p <> ( 0. ( TOP-REAL 2)) by A5, EUCLID: 52, EUCLID: 54;

 then (f . p) = |[((p `1 ) / ( sqrt (1 + (((p `2 ) / (p `1 )) ^2 )))), ((p `2 ) / ( sqrt (1 + (((p `2 ) / (p `1 )) ^2 ))))]| by A1, A6, A7, A10, JGRAPH_3:def 1;



 then

 A11: ((f . p) `2 ) = ((p `2 ) / ( sqrt (1 + (((p `2 ) / ( - 1)) ^2 )))) by A6, EUCLID: 52

 .= ((p `2 ) / ( sqrt (1 + ((p `2 ) ^2 ))));



 A12: (q `1 ) = ( - 1) by A3, Th1;



 A13: ( - 1) <= (q `2 ) by A3, Th1;



 A14: (q `2 ) <= ( - (q `1 )) by A4, A5, A12;

 q <> ( 0. ( TOP-REAL 2)) by A4, A5, EUCLID: 52, EUCLID: 54;

 then (f . q) = |[((q `1 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))), ((q `2 ) / ( sqrt (1 + (((q `2 ) / (q `1 )) ^2 ))))]| by A1, A12, A13, A14, JGRAPH_3:def 1;



 then

 A15: ((f . q) `2 ) = ((q `2 ) / ( sqrt (1 + (((q `2 ) / ( - 1)) ^2 )))) by A12, EUCLID: 52

 .= ((q `2 ) / ( sqrt (1 + ((q `2 ) ^2 ))));

 ((p `2 ) * ( sqrt (1 + ((q `2 ) ^2 )))) >= ((q `2 ) * ( sqrt (1 + ((p `2 ) ^2 )))) by A4, A5, Lm2;

 then (((p `2 ) * ( sqrt (1 + ((q `2 ) ^2 )))) / ( sqrt (1 + ((q `2 ) ^2 )))) >= (((q `2 ) * ( sqrt (1 + ((p `2 ) ^2 )))) / ( sqrt (1 + ((q `2 ) ^2 )))) by A9, XREAL_1: 72;

 then (p `2 ) >= (((q `2 ) * ( sqrt (1 + ((p `2 ) ^2 )))) / ( sqrt (1 + ((q `2 ) ^2 )))) by A9, XCMPLX_1: 89;

 then ((p `2 ) / ( sqrt (1 + ((p `2 ) ^2 )))) >= ((((q `2 ) * ( sqrt (1 + ((p `2 ) ^2 )))) / ( sqrt (1 + ((q `2 ) ^2 )))) / ( sqrt (1 + ((p `2 ) ^2 )))) by A8, XREAL_1: 72;

 then ((p `2 ) / ( sqrt (1 + ((p `2 ) ^2 )))) >= ((((q `2 ) * ( sqrt (1 + ((p `2 ) ^2 )))) / ( sqrt (1 + ((p `2 ) ^2 )))) / ( sqrt (1 + ((q `2 ) ^2 )))) by XCMPLX_1: 48;

 hence thesis by A8, A11, A15, XCMPLX_1: 89;

 end;

 theorem :: JGRAPH_6:72



 Th72: for p1,p2,p3,p4 be Point of ( TOP-REAL 2), P be non empty compact Subset of ( TOP-REAL 2), f be Function of ( TOP-REAL 2), ( TOP-REAL 2) st P = ( circle ( 0 , 0 ,1)) & f = Sq_Circ holds LE (p1,p2,( rectangle (( - 1),1,( - 1),1))) & LE (p2,p3,( rectangle (( - 1),1,( - 1),1))) & LE (p3,p4,( rectangle (( - 1),1,( - 1),1))) implies ((f . p1),(f . p2),(f . p3),(f . p4)) are_in_this_order_on P

 proof

 let p1,p2,p3,p4 be Point of ( TOP-REAL 2), P be non empty compact Subset of ( TOP-REAL 2), f be Function of ( TOP-REAL 2), ( TOP-REAL 2);

 set K = ( rectangle (( - 1),1,( - 1),1));

 assume that

 A1: P = ( circle ( 0 , 0 ,1)) and

 A2: f = Sq_Circ ;



 A3: K is being_simple_closed_curve by Th50;



 A4: K = { p : (p `1 ) = ( - 1) & ( - 1) <= (p `2 ) & (p `2 ) <= 1 or (p `2 ) = 1 & ( - 1) <= (p `1 ) & (p `1 ) <= 1 or (p `1 ) = 1 & ( - 1) <= (p `2 ) & (p `2 ) <= 1 or (p `2 ) = ( - 1) & ( - 1) <= (p `1 ) & (p `1 ) <= 1 } by Lm15;



 A5: P = { p : |.p.| = 1 } by A1, Th24;

 thus LE (p1,p2,K) & LE (p2,p3,K) & LE (p3,p4,K) implies ((f . p1),(f . p2),(f . p3),(f . p4)) are_in_this_order_on P

 proof

 assume that

 A6: LE (p1,p2,K) and

 A7: LE (p2,p3,K) and

 A8: LE (p3,p4,K);



 A9: p1 in K by A3, A6, JORDAN7: 5;



 A10: p2 in K by A3, A6, JORDAN7: 5;



 A11: p3 in K by A3, A7, JORDAN7: 5;



 A12: p4 in K by A3, A8, JORDAN7: 5;

 then

 A13: ex q8 be Point of ( TOP-REAL 2) st (q8 = p4) & ((q8 `1 ) = ( - 1) & ( - 1) <= (q8 `2 ) & (q8 `2 ) <= 1 or (q8 `2 ) = 1 & ( - 1) <= (q8 `1 ) & (q8 `1 ) <= 1 or (q8 `1 ) = 1 & ( - 1) <= (q8 `2 ) & (q8 `2 ) <= 1 or (q8 `2 ) = ( - 1) & ( - 1) <= (q8 `1 ) & (q8 `1 ) <= 1) by A4;



 A14: LE (p1,p3,K) by A6, A7, Th50, JORDAN6: 58;



 A15: LE (p2,p4,K) by A7, A8, Th50, JORDAN6: 58;



 A16: ( W-min K) = |[( - 1), ( - 1)]| by Th46;



 A17: ( |[( - 1), 0 ]| `2 ) = 0 by EUCLID: 52;



 A18: ((1 / 2) * ( |[( - 1), ( - 1)]| + |[( - 1), 1]|)) = (((1 / 2) * |[( - 1), ( - 1)]|) + ((1 / 2) * |[( - 1), 1]|)) by RLVECT_1:def 5

 .= ( |[((1 / 2) * ( - 1)), ((1 / 2) * ( - 1))]| + ((1 / 2) * |[( - 1), 1]|)) by EUCLID: 58

 .= ( |[((1 / 2) * ( - 1)), ((1 / 2) * ( - 1))]| + |[((1 / 2) * ( - 1)), ((1 / 2) * 1)]|) by EUCLID: 58

 .= |[(((1 / 2) * ( - 1)) + ((1 / 2) * ( - 1))), (((1 / 2) * ( - 1)) + ((1 / 2) * 1))]| by EUCLID: 56

 .= |[( - 1), 0 ]|;

 then

 A19: |[( - 1), 0 ]| in ( LSeg ( |[( - 1), ( - 1)]|, |[( - 1), 1]|)) by RLTOPSP1: 69;

 now

 per cases by A9, A16, Th63, RLTOPSP1: 68;

 case

 A20: p1 in ( LSeg ( |[( - 1), ( - 1)]|, |[( - 1), 1]|));

 then

 A21: (p1 `1 ) = ( - 1) by Th1;

 then

 A22: ((f . p1) `1 ) < 0 by A2, Th68;



 A23: (f .: K) = P by A2, A5, Lm15, Th35, JGRAPH_3: 23;



 A24: ( dom f) = the carrier of ( TOP-REAL 2) by FUNCT_2:def 1;

 then

 A25: (f . p1) in P by A9, A23, FUNCT_1:def 6;

 now

 per cases ;

 case

 A26: (p1 `2 ) >= 0 ;

 then

 A27: LE ((f . p1),(f . p2),P) by A1, A2, A6, A20, Th65;



 A28: LE ((f . p2),(f . p3),P) by A1, A2, A6, A7, A20, A26, Th66;

 LE ((f . p3),(f . p4),P) by A1, A2, A8, A14, A20, A26, Th66;

 hence thesis by A27, A28, JORDAN17:def 1;

 end;

 case

 A29: (p1 `2 ) < 0 ;

 now

 per cases ;

 case

 A30: (p2 `2 ) < 0 & p2 in ( LSeg ( |[( - 1), ( - 1)]|, |[( - 1), 1]|));

 then

 A31: (p2 `1 ) = ( - 1) by Th1;



 A32: (f . p2) in P by A10, A23, A24, FUNCT_1:def 6;



 A33: (p1 `2 ) <= (p2 `2 ) by A6, A20, A30, Th55;

 now

 per cases ;

 case

 A34: (p3 `2 ) < 0 & p3 in ( LSeg ( |[( - 1), ( - 1)]|, |[( - 1), 1]|));

 then

 A35: (p3 `1 ) = ( - 1) by Th1;



 A36: (f . p3) in P by A11, A23, A24, FUNCT_1:def 6;



 A37: (p2 `2 ) <= (p3 `2 ) by A7, A30, A34, Th55;

 now

 per cases ;

 case

 A38: (p4 `2 ) < 0 & p4 in ( LSeg ( |[( - 1), ( - 1)]|, |[( - 1), 1]|));

 then

 A39: (p4 `1 ) = ( - 1) by Th1;



 A40: ((f . p2) `1 ) < 0 by A2, A30, A31, Th67;



 A41: ((f . p2) `2 ) < 0 by A2, A30, A31, Th67;



 A42: ((f . p3) `1 ) < 0 by A2, A34, A35, Th67;



 A43: ((f . p3) `2 ) < 0 by A2, A34, A35, Th67;



 A44: ((f . p4) `1 ) < 0 by A2, A38, A39, Th67;



 A45: ((f . p4) `2 ) < 0 by A2, A38, A39, Th67;

 ((f . p1) `2 ) <= ((f . p2) `2 ) by A2, A20, A30, A33, Th71;

 then

 A46: LE ((f . p1),(f . p2),P) by A5, A22, A25, A32, A40, A41, JGRAPH_5: 51;

 ((f . p2) `2 ) <= ((f . p3) `2 ) by A2, A30, A34, A37, Th71;

 then

 A47: LE ((f . p2),(f . p3),P) by A5, A32, A36, A40, A42, A43, JGRAPH_5: 51;



 A48: (f . p4) in P by A12, A23, A24, FUNCT_1:def 6;

 (p3 `2 ) <= (p4 `2 ) by A8, A34, A38, Th55;

 then ((f . p3) `2 ) <= ((f . p4) `2 ) by A2, A34, A38, Th71;

 then LE ((f . p3),(f . p4),P) by A5, A36, A42, A44, A45, A48, JGRAPH_5: 51;

 hence thesis by A46, A47, JORDAN17:def 1;

 end;

 case

 A49: not ((p4 `2 ) < 0 & p4 in ( LSeg ( |[( - 1), ( - 1)]|, |[( - 1), 1]|)));

 A50:

 now

 per cases by A12, Th63;

 case p4 in ( LSeg ( |[( - 1), ( - 1)]|, |[( - 1), 1]|));

 hence p4 in ( LSeg ( |[( - 1), ( - 1)]|, |[( - 1), 1]|)) & ( |[( - 1), 0 ]| `2 ) <= (p4 `2 ) or p4 in ( LSeg ( |[( - 1), 1]|, |[1, 1]|)) or p4 in ( LSeg ( |[1, 1]|, |[1, ( - 1)]|)) or p4 in ( LSeg ( |[1, ( - 1)]|, |[( - 1), ( - 1)]|)) & p4 <> ( W-min K) by A49, EUCLID: 52;

 end;

 case p4 in ( LSeg ( |[( - 1), 1]|, |[1, 1]|));

 hence p4 in ( LSeg ( |[( - 1), ( - 1)]|, |[( - 1), 1]|)) & ( |[( - 1), 0 ]| `2 ) <= (p4 `2 ) or p4 in ( LSeg ( |[( - 1), 1]|, |[1, 1]|)) or p4 in ( LSeg ( |[1, 1]|, |[1, ( - 1)]|)) or p4 in ( LSeg ( |[1, ( - 1)]|, |[( - 1), ( - 1)]|)) & p4 <> ( W-min K);

 end;

 case p4 in ( LSeg ( |[1, 1]|, |[1, ( - 1)]|));

 hence p4 in ( LSeg ( |[( - 1), ( - 1)]|, |[( - 1), 1]|)) & ( |[( - 1), 0 ]| `2 ) <= (p4 `2 ) or p4 in ( LSeg ( |[( - 1), 1]|, |[1, 1]|)) or p4 in ( LSeg ( |[1, 1]|, |[1, ( - 1)]|)) or p4 in ( LSeg ( |[1, ( - 1)]|, |[( - 1), ( - 1)]|)) & p4 <> ( W-min K);

 end;

 case

 A51: p4 in ( LSeg ( |[1, ( - 1)]|, |[( - 1), ( - 1)]|));



 A52: ( W-min K) = |[( - 1), ( - 1)]| by Th46;

 now

 assume

 A53: p4 = ( W-min K);

 then (p4 `2 ) = ( - 1) by A52, EUCLID: 52;

 hence contradiction by A49, A52, A53, RLTOPSP1: 68;

 end;

 hence p4 in ( LSeg ( |[( - 1), ( - 1)]|, |[( - 1), 1]|)) & ( |[( - 1), 0 ]| `2 ) <= (p4 `2 ) or p4 in ( LSeg ( |[( - 1), 1]|, |[1, 1]|)) or p4 in ( LSeg ( |[1, 1]|, |[1, ( - 1)]|)) or p4 in ( LSeg ( |[1, ( - 1)]|, |[( - 1), ( - 1)]|)) & p4 <> ( W-min K) by A51;

 end;

 end;



 A54: ((f . p2) `1 ) < 0 by A2, A30, A31, Th67;



 A55: ((f . p2) `2 ) < 0 by A2, A30, A31, Th67;



 A56: ((f . p3) `1 ) < 0 by A2, A34, A35, Th67;



 A57: ((f . p3) `2 ) < 0 by A2, A34, A35, Th67;

 ((f . p1) `2 ) <= ((f . p2) `2 ) by A2, A20, A30, A33, Th71;

 then

 A58: LE ((f . p1),(f . p2),P) by A5, A22, A25, A32, A54, A55, JGRAPH_5: 51;

 ((f . p2) `2 ) <= ((f . p3) `2 ) by A2, A30, A34, A37, Th71;

 then

 A59: LE ((f . p2),(f . p3),P) by A5, A32, A36, A54, A56, A57, JGRAPH_5: 51;

 A60:

 now

 per cases ;

 case

 A61: (p4 `1 ) = ( - 1) & (p4 `2 ) < 0 & (p1 `2 ) <= (p4 `2 );

 now

 per cases by A50;

 case p4 in ( LSeg ( |[( - 1), ( - 1)]|, |[( - 1), 1]|)) & ( |[( - 1), 0 ]| `2 ) <= (p4 `2 );

 hence contradiction by A61, EUCLID: 52;

 end;

 case p4 in ( LSeg ( |[( - 1), 1]|, |[1, 1]|));

 hence contradiction by A61, Th3;

 end;

 case p4 in ( LSeg ( |[1, 1]|, |[1, ( - 1)]|));

 hence contradiction by A61, Th1;

 end;

 case

 A62: p4 in ( LSeg ( |[1, ( - 1)]|, |[( - 1), ( - 1)]|)) & p4 <> ( W-min K);

 then

 A63: (p4 `2 ) = ( - 1) by Th3;



 A64: ( W-min K) = |[( - 1), ( - 1)]| by Th46;

 then

 A65: (( W-min K) `1 ) = ( - 1) by EUCLID: 52;

 (( W-min K) `2 ) = ( - 1) by A64, EUCLID: 52;

 hence contradiction by A61, A62, A63, A65, TOPREAL3: 6;

 end;

 end;

 hence contradiction;

 end;

 case

 A66: not ((p4 `1 ) = ( - 1) & (p4 `2 ) < 0 & (p1 `2 ) <= (p4 `2 ));



 A67: p4 in ( LSeg ( |[( - 1), ( - 1)]|, |[( - 1), 1]|)) or p4 in ( LSeg ( |[( - 1), 1]|, |[1, 1]|)) or p4 in ( LSeg ( |[1, 1]|, |[1, ( - 1)]|)) or p4 in ( LSeg ( |[1, ( - 1)]|, |[( - 1), ( - 1)]|)) by A12, Th63;

 now

 per cases by A66;

 case

 A68: (p4 `1 ) <> ( - 1);



 A69: (f .: K) = P by A2, A5, Lm15, Th35, JGRAPH_3: 23;



 A70: ( dom f) = the carrier of ( TOP-REAL 2) by FUNCT_2:def 1;



 A71: ((f . p1) `2 ) <= 0 by A2, A21, A29, Th67;



 A72: ( Upper_Arc P) = { p where p be Point of ( TOP-REAL 2) : p in P & (p `2 ) >= 0 } by A5, JGRAPH_5: 34;



 A73: (f . p1) in P by A9, A69, A70, FUNCT_1:def 6;

 ( Lower_Arc P) = { p where p be Point of ( TOP-REAL 2) : p in P & (p `2 ) <= 0 } by A5, JGRAPH_5: 35;

 then

 A74: (f . p1) in ( Lower_Arc P) by A71, A73;



 A75: (f . |[( - 1), 0 ]|) = ( W-min P) by A2, A5, Th10, JGRAPH_5: 29;

 A76:

 now

 assume (f . p1) = ( W-min P);

 then p1 = |[( - 1), 0 ]| by A2, A70, A75, FUNCT_1:def 4;

 hence contradiction by A29, EUCLID: 52;

 end;

 now

 per cases by A67, A68, Th1;

 case p4 in ( LSeg ( |[( - 1), 1]|, |[1, 1]|));

 then

 A77: (p4 `2 ) = 1 by Th3;



 A78: (f . p4) in P by A12, A69, A70, FUNCT_1:def 6;

 ((f . p4) `2 ) >= 0 by A2, A77, Th69;

 then (f . p4) in ( Upper_Arc P) by A72, A78;

 hence LE ((f . p4),(f . p1),P) by A74, A76, JORDAN6:def 10;

 end;

 case p4 in ( LSeg ( |[1, 1]|, |[1, ( - 1)]|));

 then

 A79: (p4 `1 ) = 1 by Th1;



 A80: (f .: K) = P by A2, A5, Lm15, Th35, JGRAPH_3: 23;



 A81: ( dom f) = the carrier of ( TOP-REAL 2) by FUNCT_2:def 1;

 then

 A82: (f . p4) in P by A12, A80, FUNCT_1:def 6;



 A83: (f . p1) in P by A9, A80, A81, FUNCT_1:def 6;



 A84: ((f . p1) `1 ) < 0 by A2, A21, A29, Th67;



 A85: ((f . p1) `2 ) <= 0 by A2, A21, A29, Th67;



 A86: (f . |[( - 1), 0 ]|) = ( W-min P) by A2, A5, Th10, JGRAPH_5: 29;

 A87:

 now

 assume (f . p1) = ( W-min P);

 then p1 = |[( - 1), 0 ]| by A2, A81, A86, FUNCT_1:def 4;

 hence contradiction by A29, EUCLID: 52;

 end;



 A88: ((f . p4) `1 ) >= 0 by A2, A79, Th68;

 now

 per cases ;

 case

 A89: ((f . p4) `2 ) >= 0 ;



 A90: ((f . p1) `2 ) <= 0 by A2, A21, A29, Th67;



 A91: ( Upper_Arc P) = { p where p be Point of ( TOP-REAL 2) : p in P & (p `2 ) >= 0 } by A5, JGRAPH_5: 34;



 A92: (f . |[( - 1), 0 ]|) = ( W-min P) by A2, A5, Th10, JGRAPH_5: 29;

 A93:

 now

 assume (f . p1) = ( W-min P);

 then p1 = |[( - 1), 0 ]| by A2, A81, A92, FUNCT_1:def 4;

 hence contradiction by A29, EUCLID: 52;

 end;



 A94: (f . p4) in ( Upper_Arc P) by A82, A89, A91;

 ( Lower_Arc P) = { p where p be Point of ( TOP-REAL 2) : p in P & (p `2 ) <= 0 } by A5, JGRAPH_5: 35;

 then (f . p1) in ( Lower_Arc P) by A83, A90;

 hence LE ((f . p4),(f . p1),P) by A93, A94, JORDAN6:def 10;

 end;

 case ((f . p4) `2 ) < 0 ;

 hence LE ((f . p4),(f . p1),P) by A5, A82, A83, A84, A85, A87, A88, JGRAPH_5: 56;

 end;

 end;

 hence LE ((f . p4),(f . p1),P);

 end;

 case

 A95: p4 in ( LSeg ( |[1, ( - 1)]|, |[( - 1), ( - 1)]|));

 then (p4 `2 ) = ( - 1) by Th3;

 then

 A96: ((f . p4) `2 ) < 0 by A2, Th69;



 A97: (f . p4) in P by A12, A69, A70, FUNCT_1:def 6;



 A98: ((f . p1) `2 ) <= 0 by A2, A21, A29, Th67;

 ((f . p4) `1 ) >= ((f . p1) `1 ) by A2, A20, A95, Th70;

 hence LE ((f . p4),(f . p1),P) by A5, A73, A76, A96, A97, A98, JGRAPH_5: 56;

 end;

 end;

 hence LE ((f . p4),(f . p1),P);

 end;

 case

 A99: (p4 `1 ) = ( - 1) & (p4 `2 ) >= 0 ;



 A100: (f .: K) = P by A2, A5, Lm15, Th35, JGRAPH_3: 23;



 A101: ( dom f) = the carrier of ( TOP-REAL 2) by FUNCT_2:def 1;

 then

 A102: (f . p4) in P by A12, A100, FUNCT_1:def 6;



 A103: (f . p1) in P by A9, A100, A101, FUNCT_1:def 6;



 A104: ((f . p4) `2 ) >= 0 by A2, A99, Th69;



 A105: ((f . p1) `2 ) <= 0 by A2, A21, A29, Th67;



 A106: ( Upper_Arc P) = { p where p be Point of ( TOP-REAL 2) : p in P & (p `2 ) >= 0 } by A5, JGRAPH_5: 34;



 A107: (f . |[( - 1), 0 ]|) = ( W-min P) by A2, A5, Th10, JGRAPH_5: 29;

 A108:

 now

 assume (f . p1) = ( W-min P);

 then p1 = |[( - 1), 0 ]| by A2, A101, A107, FUNCT_1:def 4;

 hence contradiction by A29, EUCLID: 52;

 end;



 A109: (f . p4) in ( Upper_Arc P) by A102, A104, A106;

 ( Lower_Arc P) = { p where p be Point of ( TOP-REAL 2) : p in P & (p `2 ) <= 0 } by A5, JGRAPH_5: 35;

 then (f . p1) in ( Lower_Arc P) by A103, A105;

 hence LE ((f . p4),(f . p1),P) by A108, A109, JORDAN6:def 10;

 end;

 case

 A110: (p4 `1 ) = ( - 1) & (p4 `2 ) < 0 & (p1 `2 ) > (p4 `2 );

 then

 A111: p4 in ( LSeg ( |[( - 1), ( - 1)]|, |[( - 1), 1]|)) by A13, Th2;



 A112: (f .: K) = P by A2, A5, Lm15, Th35, JGRAPH_3: 23;



 A113: ( dom f) = the carrier of ( TOP-REAL 2) by FUNCT_2:def 1;

 then

 A114: (f . p4) in P by A12, A112, FUNCT_1:def 6;



 A115: (f . p1) in P by A9, A112, A113, FUNCT_1:def 6;



 A116: ((f . p1) `1 ) < 0 by A2, A21, A29, Th67;



 A117: ((f . p1) `2 ) < 0 by A2, A21, A29, Th67;



 A118: ((f . p4) `2 ) <= ((f . p1) `2 ) by A2, A20, A29, A110, A111, Th71;

 ((f . p4) `1 ) < 0 by A2, A110, Th68;

 hence LE ((f . p4),(f . p1),P) by A5, A114, A115, A116, A117, A118, JGRAPH_5: 51;

 end;

 end;

 hence LE ((f . p4),(f . p1),P);

 end;

 end;



 A119: K = { p : (p `1 ) = ( - 1) & ( - 1) <= (p `2 ) & (p `2 ) <= 1 or (p `2 ) = 1 & ( - 1) <= (p `1 ) & (p `1 ) <= 1 or (p `1 ) = 1 & ( - 1) <= (p `2 ) & (p `2 ) <= 1 or (p `2 ) = ( - 1) & ( - 1) <= (p `1 ) & (p `1 ) <= 1 } by Lm15;

 thus K = { p : (p `1 ) = ( - 1) & ( - 1) <= (p `2 ) & (p `2 ) <= 1 or (p `1 ) = 1 & ( - 1) <= (p `2 ) & (p `2 ) <= 1 or ( - 1) = (p `2 ) & ( - 1) <= (p `1 ) & (p `1 ) <= 1 or 1 = (p `2 ) & ( - 1) <= (p `1 ) & (p `1 ) <= 1 }

 proof

 thus K c= { p : (p `1 ) = ( - 1) & ( - 1) <= (p `2 ) & (p `2 ) <= 1 or (p `1 ) = 1 & ( - 1) <= (p `2 ) & (p `2 ) <= 1 or ( - 1) = (p `2 ) & ( - 1) <= (p `1 ) & (p `1 ) <= 1 or 1 = (p `2 ) & ( - 1) <= (p `1 ) & (p `1 ) <= 1 }

 proof

 let x be object;

 assume x in K;

 then ex p st (p = x) & ((p `1 ) = ( - 1) & ( - 1) <= (p `2 ) & (p `2 ) <= 1 or (p `2 ) = 1 & ( - 1) <= (p `1 ) & (p `1 ) <= 1 or (p `1 ) = 1 & ( - 1) <= (p `2 ) & (p `2 ) <= 1 or (p `2 ) = ( - 1) & ( - 1) <= (p `1 ) & (p `1 ) <= 1) by A119;

 hence thesis;

 end;

 thus { p : (p `1 ) = ( - 1) & ( - 1) <= (p `2 ) & (p `2 ) <= 1 or (p `1 ) = 1 & ( - 1) <= (p `2 ) & (p `2 ) <= 1 or ( - 1) = (p `2 ) & ( - 1) <= (p `1 ) & (p `1 ) <= 1 or 1 = (p `2 ) & ( - 1) <= (p `1 ) & (p `1 ) <= 1 } c= K

 proof

 let x be object;

 assume x in { p : (p `1 ) = ( - 1) & ( - 1) <= (p `2 ) & (p `2 ) <= 1 or (p `1 ) = 1 & ( - 1) <= (p `2 ) & (p `2 ) <= 1 or ( - 1) = (p `2 ) & ( - 1) <= (p `1 ) & (p `1 ) <= 1 or 1 = (p `2 ) & ( - 1) <= (p `1 ) & (p `1 ) <= 1 };

 then ex p st (p = x) & ((p `1 ) = ( - 1) & ( - 1) <= (p `2 ) & (p `2 ) <= 1 or (p `1 ) = 1 & ( - 1) <= (p `2 ) & (p `2 ) <= 1 or ( - 1) = (p `2 ) & ( - 1) <= (p `1 ) & (p `1 ) <= 1 or 1 = (p `2 ) & ( - 1) <= (p `1 ) & (p `1 ) <= 1);

 hence thesis by A119;

 end;

 end;

 thus thesis by A58, A59, A60, JORDAN17:def 1;

 end;

 end;

 hence thesis;

 end;

 case

 A120: not ((p3 `2 ) < 0 & p3 in ( LSeg ( |[( - 1), ( - 1)]|, |[( - 1), 1]|)));

 A121:

 now

 per cases by A11, Th63;

 case p3 in ( LSeg ( |[( - 1), ( - 1)]|, |[( - 1), 1]|));

 hence p3 in ( LSeg ( |[( - 1), ( - 1)]|, |[( - 1), 1]|)) & ( |[( - 1), 0 ]| `2 ) <= (p3 `2 ) or p3 in ( LSeg ( |[( - 1), 1]|, |[1, 1]|)) or p3 in ( LSeg ( |[1, 1]|, |[1, ( - 1)]|)) or p3 in ( LSeg ( |[1, ( - 1)]|, |[( - 1), ( - 1)]|)) & p3 <> ( W-min K) by A120, EUCLID: 52;

 end;

 case p3 in ( LSeg ( |[( - 1), 1]|, |[1, 1]|));

 hence p3 in ( LSeg ( |[( - 1), ( - 1)]|, |[( - 1), 1]|)) & ( |[( - 1), 0 ]| `2 ) <= (p3 `2 ) or p3 in ( LSeg ( |[( - 1), 1]|, |[1, 1]|)) or p3 in ( LSeg ( |[1, 1]|, |[1, ( - 1)]|)) or p3 in ( LSeg ( |[1, ( - 1)]|, |[( - 1), ( - 1)]|)) & p3 <> ( W-min K);

 end;

 case p3 in ( LSeg ( |[1, 1]|, |[1, ( - 1)]|));

 hence p3 in ( LSeg ( |[( - 1), ( - 1)]|, |[( - 1), 1]|)) & ( |[( - 1), 0 ]| `2 ) <= (p3 `2 ) or p3 in ( LSeg ( |[( - 1), 1]|, |[1, 1]|)) or p3 in ( LSeg ( |[1, 1]|, |[1, ( - 1)]|)) or p3 in ( LSeg ( |[1, ( - 1)]|, |[( - 1), ( - 1)]|)) & p3 <> ( W-min K);

 end;

 case

 A122: p3 in ( LSeg ( |[1, ( - 1)]|, |[( - 1), ( - 1)]|));



 A123: ( W-min K) = |[( - 1), ( - 1)]| by Th46;

 now

 assume

 A124: p3 = ( W-min K);

 then (p3 `2 ) = ( - 1) by A123, EUCLID: 52;

 hence contradiction by A120, A123, A124, RLTOPSP1: 68;

 end;

 hence p3 in ( LSeg ( |[( - 1), ( - 1)]|, |[( - 1), 1]|)) & ( |[( - 1), 0 ]| `2 ) <= (p3 `2 ) or p3 in ( LSeg ( |[( - 1), 1]|, |[1, 1]|)) or p3 in ( LSeg ( |[1, 1]|, |[1, ( - 1)]|)) or p3 in ( LSeg ( |[1, ( - 1)]|, |[( - 1), ( - 1)]|)) & p3 <> ( W-min K) by A122;

 end;

 end;

 then

 A125: LE ( |[( - 1), 0 ]|,p3,K) by A19, Th59;



 A126: ((f . p2) `1 ) < 0 by A2, A30, A31, Th67;



 A127: ((f . p2) `2 ) < 0 by A2, A30, A31, Th67;

 ((f . p1) `2 ) <= ((f . p2) `2 ) by A2, A20, A30, A33, Th71;

 then

 A128: LE ((f . p1),(f . p2),P) by A5, A22, A25, A32, A126, A127, JGRAPH_5: 51;



 A129: LE ((f . p3),(f . p4),P) by A1, A2, A8, A17, A18, A125, Th66, RLTOPSP1: 69;

 A130:

 now

 per cases ;

 case

 A131: (p4 `1 ) = ( - 1) & (p4 `2 ) < 0 & (p1 `2 ) <= (p4 `2 );



 A132: ( |[( - 1), ( - 1)]| `1 ) = ( - 1) by EUCLID: 52;



 A133: ( |[( - 1), ( - 1)]| `2 ) = ( - 1) by EUCLID: 52;



 A134: ( |[( - 1), 1]| `1 ) = ( - 1) by EUCLID: 52;



 A135: ( |[( - 1), 1]| `2 ) = 1 by EUCLID: 52;

 ( - 1) <= (p4 `2 ) by A12, Th19;

 then

 A136: p4 in ( LSeg ( |[( - 1), ( - 1)]|, |[( - 1), 1]|)) by A131, A132, A133, A134, A135, GOBOARD7: 7;

 now

 per cases by A121;

 case

 A137: p3 in ( LSeg ( |[( - 1), ( - 1)]|, |[( - 1), 1]|)) & ( |[( - 1), 0 ]| `2 ) <= (p3 `2 );

 then 0 <= (p3 `2 ) by EUCLID: 52;

 hence contradiction by A8, A131, A136, A137, Th55;

 end;

 case

 A138: p3 in ( LSeg ( |[( - 1), 1]|, |[1, 1]|));

 then LE (p4,p3,K) by A136, Th59;

 then p3 = p4 by A8, Th50, JORDAN6: 57;

 hence contradiction by A131, A138, Th3;

 end;

 case

 A139: p3 in ( LSeg ( |[1, 1]|, |[1, ( - 1)]|));

 then LE (p4,p3,K) by A136, Th59;

 then p3 = p4 by A8, Th50, JORDAN6: 57;

 hence contradiction by A131, A139, Th1;

 end;

 case

 A140: p3 in ( LSeg ( |[1, ( - 1)]|, |[( - 1), ( - 1)]|)) & p3 <> ( W-min K);

 then LE (p4,p3,K) by A136, Th59;

 then

 A141: p3 = p4 by A8, Th50, JORDAN6: 57;



 A142: (p3 `2 ) = ( - 1) by A140, Th3;



 A143: ( W-min K) = |[( - 1), ( - 1)]| by Th46;

 then

 A144: (( W-min K) `1 ) = ( - 1) by EUCLID: 52;

 (( W-min K) `2 ) = ( - 1) by A143, EUCLID: 52;

 hence contradiction by A131, A140, A141, A142, A144, TOPREAL3: 6;

 end;

 end;

 hence contradiction;

 end;

 case

 A145: not ((p4 `1 ) = ( - 1) & (p4 `2 ) < 0 & (p1 `2 ) <= (p4 `2 ));



 A146: p4 in ( LSeg ( |[( - 1), ( - 1)]|, |[( - 1), 1]|)) or p4 in ( LSeg ( |[( - 1), 1]|, |[1, 1]|)) or p4 in ( LSeg ( |[1, 1]|, |[1, ( - 1)]|)) or p4 in ( LSeg ( |[1, ( - 1)]|, |[( - 1), ( - 1)]|)) by A12, Th63;

 now

 per cases by A145;

 case

 A147: (p4 `1 ) <> ( - 1);



 A148: (f .: K) = P by A2, A5, Lm15, Th35, JGRAPH_3: 23;



 A149: ( dom f) = the carrier of ( TOP-REAL 2) by FUNCT_2:def 1;



 A150: ((f . p1) `2 ) <= 0 by A2, A21, A29, Th67;



 A151: ( Upper_Arc P) = { p where p be Point of ( TOP-REAL 2) : p in P & (p `2 ) >= 0 } by A5, JGRAPH_5: 34;



 A152: (f . p1) in P by A9, A148, A149, FUNCT_1:def 6;

 ( Lower_Arc P) = { p where p be Point of ( TOP-REAL 2) : p in P & (p `2 ) <= 0 } by A5, JGRAPH_5: 35;

 then

 A153: (f . p1) in ( Lower_Arc P) by A150, A152;



 A154: (f . |[( - 1), 0 ]|) = ( W-min P) by A2, A5, Th10, JGRAPH_5: 29;

 A155:

 now

 assume (f . p1) = ( W-min P);

 then p1 = |[( - 1), 0 ]| by A2, A149, A154, FUNCT_1:def 4;

 hence contradiction by A29, EUCLID: 52;

 end;

 now

 per cases by A146, A147, Th1;

 case p4 in ( LSeg ( |[( - 1), 1]|, |[1, 1]|));

 then

 A156: (p4 `2 ) = 1 by Th3;



 A157: (f . p4) in P by A12, A148, A149, FUNCT_1:def 6;

 ((f . p4) `2 ) >= 0 by A2, A156, Th69;

 then (f . p4) in ( Upper_Arc P) by A151, A157;

 hence LE ((f . p4),(f . p1),P) by A153, A155, JORDAN6:def 10;

 end;

 case p4 in ( LSeg ( |[1, 1]|, |[1, ( - 1)]|));

 then

 A158: (p4 `1 ) = 1 by Th1;



 A159: (f .: K) = P by A2, A5, Lm15, Th35, JGRAPH_3: 23;



 A160: ( dom f) = the carrier of ( TOP-REAL 2) by FUNCT_2:def 1;

 then

 A161: (f . p4) in P by A12, A159, FUNCT_1:def 6;



 A162: (f . p1) in P by A9, A159, A160, FUNCT_1:def 6;



 A163: ((f . p1) `1 ) < 0 by A2, A21, A29, Th67;



 A164: ((f . p1) `2 ) <= 0 by A2, A21, A29, Th67;



 A165: (f . |[( - 1), 0 ]|) = ( W-min P) by A2, A5, Th10, JGRAPH_5: 29;

 A166:

 now

 assume (f . p1) = ( W-min P);

 then p1 = |[( - 1), 0 ]| by A2, A160, A165, FUNCT_1:def 4;

 hence contradiction by A29, EUCLID: 52;

 end;



 A167: ((f . p4) `1 ) >= 0 by A2, A158, Th68;

 now

 per cases ;

 case

 A168: ((f . p4) `2 ) >= 0 ;



 A169: ((f . p1) `2 ) <= 0 by A2, A21, A29, Th67;



 A170: ( Upper_Arc P) = { p where p be Point of ( TOP-REAL 2) : p in P & (p `2 ) >= 0 } by A5, JGRAPH_5: 34;



 A171: (f . |[( - 1), 0 ]|) = ( W-min P) by A2, A5, Th10, JGRAPH_5: 29;

 A172:

 now

 assume (f . p1) = ( W-min P);

 then p1 = |[( - 1), 0 ]| by A2, A160, A171, FUNCT_1:def 4;

 hence contradiction by A29, EUCLID: 52;

 end;



 A173: (f . p4) in ( Upper_Arc P) by A161, A168, A170;

 ( Lower_Arc P) = { p where p be Point of ( TOP-REAL 2) : p in P & (p `2 ) <= 0 } by A5, JGRAPH_5: 35;

 then (f . p1) in ( Lower_Arc P) by A162, A169;

 hence LE ((f . p4),(f . p1),P) by A172, A173, JORDAN6:def 10;

 end;

 case ((f . p4) `2 ) < 0 ;

 hence LE ((f . p4),(f . p1),P) by A5, A161, A162, A163, A164, A166, A167, JGRAPH_5: 56;

 end;

 end;

 hence LE ((f . p4),(f . p1),P);

 end;

 case

 A174: p4 in ( LSeg ( |[1, ( - 1)]|, |[( - 1), ( - 1)]|));

 then (p4 `2 ) = ( - 1) by Th3;

 then

 A175: ((f . p4) `2 ) < 0 by A2, Th69;



 A176: (f . p4) in P by A12, A148, A149, FUNCT_1:def 6;



 A177: ((f . p1) `2 ) <= 0 by A2, A21, A29, Th67;

 ((f . p4) `1 ) >= ((f . p1) `1 ) by A2, A20, A174, Th70;

 hence LE ((f . p4),(f . p1),P) by A5, A152, A155, A175, A176, A177, JGRAPH_5: 56;

 end;

 end;

 hence LE ((f . p4),(f . p1),P);

 end;

 case

 A178: (p4 `1 ) = ( - 1) & (p4 `2 ) >= 0 ;



 A179: (f .: K) = P by A2, A5, Lm15, Th35, JGRAPH_3: 23;



 A180: ( dom f) = the carrier of ( TOP-REAL 2) by FUNCT_2:def 1;

 then

 A181: (f . p4) in P by A12, A179, FUNCT_1:def 6;



 A182: (f . p1) in P by A9, A179, A180, FUNCT_1:def 6;



 A183: ((f . p4) `2 ) >= 0 by A2, A178, Th69;



 A184: ((f . p1) `2 ) <= 0 by A2, A21, A29, Th67;



 A185: ( Upper_Arc P) = { p where p be Point of ( TOP-REAL 2) : p in P & (p `2 ) >= 0 } by A5, JGRAPH_5: 34;



 A186: (f . |[( - 1), 0 ]|) = ( W-min P) by A2, A5, Th10, JGRAPH_5: 29;

 A187:

 now

 assume (f . p1) = ( W-min P);

 then p1 = |[( - 1), 0 ]| by A2, A180, A186, FUNCT_1:def 4;

 hence contradiction by A29, EUCLID: 52;

 end;



 A188: (f . p4) in ( Upper_Arc P) by A181, A183, A185;

 ( Lower_Arc P) = { p where p be Point of ( TOP-REAL 2) : p in P & (p `2 ) <= 0 } by A5, JGRAPH_5: 35;

 then (f . p1) in ( Lower_Arc P) by A182, A184;

 hence LE ((f . p4),(f . p1),P) by A187, A188, JORDAN6:def 10;

 end;

 case

 A189: (p4 `1 ) = ( - 1) & (p4 `2 ) < 0 & (p1 `2 ) > (p4 `2 );

 then

 A190: p4 in ( LSeg ( |[( - 1), ( - 1)]|, |[( - 1), 1]|)) by A13, Th2;



 A191: (f .: K) = P by A2, A5, Lm15, Th35, JGRAPH_3: 23;



 A192: ( dom f) = the carrier of ( TOP-REAL 2) by FUNCT_2:def 1;

 then

 A193: (f . p4) in P by A12, A191, FUNCT_1:def 6;



 A194: (f . p1) in P by A9, A191, A192, FUNCT_1:def 6;



 A195: ((f . p1) `1 ) < 0 by A2, A21, A29, Th67;



 A196: ((f . p1) `2 ) < 0 by A2, A21, A29, Th67;



 A197: ((f . p4) `2 ) <= ((f . p1) `2 ) by A2, A20, A29, A189, A190, Th71;

 ((f . p4) `1 ) < 0 by A2, A189, Th68;

 hence LE ((f . p4),(f . p1),P) by A5, A193, A194, A195, A196, A197, JGRAPH_5: 51;

 end;

 end;

 hence LE ((f . p4),(f . p1),P);

 end;

 end;



 A198: K = { p : (p `1 ) = ( - 1) & ( - 1) <= (p `2 ) & (p `2 ) <= 1 or (p `2 ) = 1 & ( - 1) <= (p `1 ) & (p `1 ) <= 1 or (p `1 ) = 1 & ( - 1) <= (p `2 ) & (p `2 ) <= 1 or (p `2 ) = ( - 1) & ( - 1) <= (p `1 ) & (p `1 ) <= 1 } by Lm15;

 thus K = { p : (p `1 ) = ( - 1) & ( - 1) <= (p `2 ) & (p `2 ) <= 1 or (p `1 ) = 1 & ( - 1) <= (p `2 ) & (p `2 ) <= 1 or ( - 1) = (p `2 ) & ( - 1) <= (p `1 ) & (p `1 ) <= 1 or 1 = (p `2 ) & ( - 1) <= (p `1 ) & (p `1 ) <= 1 }

 proof

 thus K c= { p : (p `1 ) = ( - 1) & ( - 1) <= (p `2 ) & (p `2 ) <= 1 or (p `1 ) = 1 & ( - 1) <= (p `2 ) & (p `2 ) <= 1 or ( - 1) = (p `2 ) & ( - 1) <= (p `1 ) & (p `1 ) <= 1 or 1 = (p `2 ) & ( - 1) <= (p `1 ) & (p `1 ) <= 1 }

 proof

 let x be object;

 assume x in K;

 then ex p st (p = x) & ((p `1 ) = ( - 1) & ( - 1) <= (p `2 ) & (p `2 ) <= 1 or (p `2 ) = 1 & ( - 1) <= (p `1 ) & (p `1 ) <= 1 or (p `1 ) = 1 & ( - 1) <= (p `2 ) & (p `2 ) <= 1 or (p `2 ) = ( - 1) & ( - 1) <= (p `1 ) & (p `1 ) <= 1) by A198;

 hence thesis;

 end;

 thus { p : (p `1 ) = ( - 1) & ( - 1) <= (p `2 ) & (p `2 ) <= 1 or (p `1 ) = 1 & ( - 1) <= (p `2 ) & (p `2 ) <= 1 or ( - 1) = (p `2 ) & ( - 1) <= (p `1 ) & (p `1 ) <= 1 or 1 = (p `2 ) & ( - 1) <= (p `1 ) & (p `1 ) <= 1 } c= K

 proof

 let x be object;

 assume x in { p : (p `1 ) = ( - 1) & ( - 1) <= (p `2 ) & (p `2 ) <= 1 or (p `1 ) = 1 & ( - 1) <= (p `2 ) & (p `2 ) <= 1 or ( - 1) = (p `2 ) & ( - 1) <= (p `1 ) & (p `1 ) <= 1 or 1 = (p `2 ) & ( - 1) <= (p `1 ) & (p `1 ) <= 1 };

 then ex p st (p = x) & ((p `1 ) = ( - 1) & ( - 1) <= (p `2 ) & (p `2 ) <= 1 or (p `1 ) = 1 & ( - 1) <= (p `2 ) & (p `2 ) <= 1 or ( - 1) = (p `2 ) & ( - 1) <= (p `1 ) & (p `1 ) <= 1 or 1 = (p `2 ) & ( - 1) <= (p `1 ) & (p `1 ) <= 1);

 hence thesis by A198;

 end;

 end;

 thus thesis by A128, A129, A130, JORDAN17:def 1;

 end;

 end;

 hence thesis;

 end;

 case

 A199: not ((p2 `2 ) < 0 & p2 in ( LSeg ( |[( - 1), ( - 1)]|, |[( - 1), 1]|)));

 A200:

 now

 per cases by A10, Th63;

 case p2 in ( LSeg ( |[( - 1), ( - 1)]|, |[( - 1), 1]|));

 hence p2 in ( LSeg ( |[( - 1), ( - 1)]|, |[( - 1), 1]|)) & ( |[( - 1), 0 ]| `2 ) <= (p2 `2 ) or p2 in ( LSeg ( |[( - 1), 1]|, |[1, 1]|)) or p2 in ( LSeg ( |[1, 1]|, |[1, ( - 1)]|)) or p2 in ( LSeg ( |[1, ( - 1)]|, |[( - 1), ( - 1)]|)) & p2 <> ( W-min K) by A199, EUCLID: 52;

 end;

 case p2 in ( LSeg ( |[( - 1), 1]|, |[1, 1]|));

 hence p2 in ( LSeg ( |[( - 1), ( - 1)]|, |[( - 1), 1]|)) & ( |[( - 1), 0 ]| `2 ) <= (p2 `2 ) or p2 in ( LSeg ( |[( - 1), 1]|, |[1, 1]|)) or p2 in ( LSeg ( |[1, 1]|, |[1, ( - 1)]|)) or p2 in ( LSeg ( |[1, ( - 1)]|, |[( - 1), ( - 1)]|)) & p2 <> ( W-min K);

 end;

 case p2 in ( LSeg ( |[1, 1]|, |[1, ( - 1)]|));

 hence p2 in ( LSeg ( |[( - 1), ( - 1)]|, |[( - 1), 1]|)) & ( |[( - 1), 0 ]| `2 ) <= (p2 `2 ) or p2 in ( LSeg ( |[( - 1), 1]|, |[1, 1]|)) or p2 in ( LSeg ( |[1, 1]|, |[1, ( - 1)]|)) or p2 in ( LSeg ( |[1, ( - 1)]|, |[( - 1), ( - 1)]|)) & p2 <> ( W-min K);

 end;

 case

 A201: p2 in ( LSeg ( |[1, ( - 1)]|, |[( - 1), ( - 1)]|));



 A202: ( W-min K) = |[( - 1), ( - 1)]| by Th46;

 now

 assume

 A203: p2 = ( W-min K);

 then (p2 `2 ) = ( - 1) by A202, EUCLID: 52;

 hence contradiction by A199, A202, A203, RLTOPSP1: 68;

 end;

 hence p2 in ( LSeg ( |[( - 1), ( - 1)]|, |[( - 1), 1]|)) & ( |[( - 1), 0 ]| `2 ) <= (p2 `2 ) or p2 in ( LSeg ( |[( - 1), 1]|, |[1, 1]|)) or p2 in ( LSeg ( |[1, 1]|, |[1, ( - 1)]|)) or p2 in ( LSeg ( |[1, ( - 1)]|, |[( - 1), ( - 1)]|)) & p2 <> ( W-min K) by A201;

 end;

 end;

 then

 A204: LE ( |[( - 1), 0 ]|,p2,K) by A19, Th59;

 then

 A205: LE ((f . p2),(f . p3),P) by A1, A2, A7, A17, A18, Th66, RLTOPSP1: 69;

 LE ( |[( - 1), 0 ]|,p3,K) by A7, A204, Th50, JORDAN6: 58;

 then

 A206: LE ((f . p3),(f . p4),P) by A1, A2, A8, A17, A18, Th66, RLTOPSP1: 69;

 A207:

 now

 per cases ;

 case

 A208: (p4 `1 ) = ( - 1) & (p4 `2 ) < 0 & (p1 `2 ) <= (p4 `2 );



 A209: ( |[( - 1), ( - 1)]| `1 ) = ( - 1) by EUCLID: 52;



 A210: ( |[( - 1), ( - 1)]| `2 ) = ( - 1) by EUCLID: 52;



 A211: ( |[( - 1), 1]| `1 ) = ( - 1) by EUCLID: 52;



 A212: ( |[( - 1), 1]| `2 ) = 1 by EUCLID: 52;

 ( - 1) <= (p4 `2 ) by A12, Th19;

 then

 A213: p4 in ( LSeg ( |[( - 1), ( - 1)]|, |[( - 1), 1]|)) by A208, A209, A210, A211, A212, GOBOARD7: 7;

 now

 per cases by A200;

 case

 A214: p2 in ( LSeg ( |[( - 1), ( - 1)]|, |[( - 1), 1]|)) & ( |[( - 1), 0 ]| `2 ) <= (p2 `2 );

 then 0 <= (p2 `2 ) by EUCLID: 52;

 hence contradiction by A15, A208, A213, A214, Th55;

 end;

 case

 A215: p2 in ( LSeg ( |[( - 1), 1]|, |[1, 1]|));

 then LE (p4,p2,K) by A213, Th59;

 then p2 = p4 by A15, Th50, JORDAN6: 57;

 hence contradiction by A208, A215, Th3;

 end;

 case

 A216: p2 in ( LSeg ( |[1, 1]|, |[1, ( - 1)]|));

 then LE (p4,p2,K) by A213, Th59;

 then p2 = p4 by A15, Th50, JORDAN6: 57;

 hence contradiction by A208, A216, Th1;

 end;

 case

 A217: p2 in ( LSeg ( |[1, ( - 1)]|, |[( - 1), ( - 1)]|)) & p2 <> ( W-min K);

 then LE (p4,p2,K) by A213, Th59;

 then

 A218: p2 = p4 by A15, Th50, JORDAN6: 57;



 A219: (p2 `2 ) = ( - 1) by A217, Th3;



 A220: ( W-min K) = |[( - 1), ( - 1)]| by Th46;

 then

 A221: (( W-min K) `1 ) = ( - 1) by EUCLID: 52;

 (( W-min K) `2 ) = ( - 1) by A220, EUCLID: 52;

 hence contradiction by A208, A217, A218, A219, A221, TOPREAL3: 6;

 end;

 end;

 hence contradiction;

 end;

 case

 A222: not ((p4 `1 ) = ( - 1) & (p4 `2 ) < 0 & (p1 `2 ) <= (p4 `2 ));



 A223: p4 in ( LSeg ( |[( - 1), ( - 1)]|, |[( - 1), 1]|)) or p4 in ( LSeg ( |[( - 1), 1]|, |[1, 1]|)) or p4 in ( LSeg ( |[1, 1]|, |[1, ( - 1)]|)) or p4 in ( LSeg ( |[1, ( - 1)]|, |[( - 1), ( - 1)]|)) by A12, Th63;

 now

 per cases by A222;

 case

 A224: (p4 `1 ) <> ( - 1);



 A225: (f .: K) = P by A2, A5, Lm15, Th35, JGRAPH_3: 23;



 A226: ( dom f) = the carrier of ( TOP-REAL 2) by FUNCT_2:def 1;



 A227: ((f . p1) `2 ) <= 0 by A2, A21, A29, Th67;



 A228: ( Upper_Arc P) = { p where p be Point of ( TOP-REAL 2) : p in P & (p `2 ) >= 0 } by A5, JGRAPH_5: 34;



 A229: (f . p1) in P by A9, A225, A226, FUNCT_1:def 6;

 ( Lower_Arc P) = { p where p be Point of ( TOP-REAL 2) : p in P & (p `2 ) <= 0 } by A5, JGRAPH_5: 35;

 then

 A230: (f . p1) in ( Lower_Arc P) by A227, A229;



 A231: (f . |[( - 1), 0 ]|) = ( W-min P) by A2, A5, Th10, JGRAPH_5: 29;

 A232:

 now

 assume (f . p1) = ( W-min P);

 then p1 = |[( - 1), 0 ]| by A2, A226, A231, FUNCT_1:def 4;

 hence contradiction by A29, EUCLID: 52;

 end;

 now

 per cases by A223, A224, Th1;

 case p4 in ( LSeg ( |[( - 1), 1]|, |[1, 1]|));

 then

 A233: (p4 `2 ) = 1 by Th3;



 A234: (f . p4) in P by A12, A225, A226, FUNCT_1:def 6;

 ((f . p4) `2 ) >= 0 by A2, A233, Th69;

 then (f . p4) in ( Upper_Arc P) by A228, A234;

 hence LE ((f . p4),(f . p1),P) by A230, A232, JORDAN6:def 10;

 end;

 case p4 in ( LSeg ( |[1, 1]|, |[1, ( - 1)]|));

 then

 A235: (p4 `1 ) = 1 by Th1;



 A236: (f .: K) = P by A2, A5, Lm15, Th35, JGRAPH_3: 23;



 A237: ( dom f) = the carrier of ( TOP-REAL 2) by FUNCT_2:def 1;

 then

 A238: (f . p4) in P by A12, A236, FUNCT_1:def 6;



 A239: (f . p1) in P by A9, A236, A237, FUNCT_1:def 6;



 A240: ((f . p1) `1 ) < 0 by A2, A21, A29, Th67;



 A241: ((f . p1) `2 ) <= 0 by A2, A21, A29, Th67;



 A242: (f . |[( - 1), 0 ]|) = ( W-min P) by A2, A5, Th10, JGRAPH_5: 29;

 A243:

 now

 assume (f . p1) = ( W-min P);

 then p1 = |[( - 1), 0 ]| by A2, A237, A242, FUNCT_1:def 4;

 hence contradiction by A29, EUCLID: 52;

 end;



 A244: ((f . p4) `1 ) >= 0 by A2, A235, Th68;

 now

 per cases ;

 case

 A245: ((f . p4) `2 ) >= 0 ;



 A246: ((f . p1) `2 ) <= 0 by A2, A21, A29, Th67;



 A247: ( Upper_Arc P) = { p where p be Point of ( TOP-REAL 2) : p in P & (p `2 ) >= 0 } by A5, JGRAPH_5: 34;



 A248: (f . |[( - 1), 0 ]|) = ( W-min P) by A2, A5, Th10, JGRAPH_5: 29;

 A249:

 now

 assume (f . p1) = ( W-min P);

 then p1 = |[( - 1), 0 ]| by A2, A237, A248, FUNCT_1:def 4;

 hence contradiction by A29, EUCLID: 52;

 end;



 A250: (f . p4) in ( Upper_Arc P) by A238, A245, A247;

 ( Lower_Arc P) = { p where p be Point of ( TOP-REAL 2) : p in P & (p `2 ) <= 0 } by A5, JGRAPH_5: 35;

 then (f . p1) in ( Lower_Arc P) by A239, A246;

 hence LE ((f . p4),(f . p1),P) by A249, A250, JORDAN6:def 10;

 end;

 case ((f . p4) `2 ) < 0 ;

 hence LE ((f . p4),(f . p1),P) by A5, A238, A239, A240, A241, A243, A244, JGRAPH_5: 56;

 end;

 end;

 hence LE ((f . p4),(f . p1),P);

 end;

 case

 A251: p4 in ( LSeg ( |[1, ( - 1)]|, |[( - 1), ( - 1)]|));

 then (p4 `2 ) = ( - 1) by Th3;

 then

 A252: ((f . p4) `2 ) < 0 by A2, Th69;



 A253: (f . p4) in P by A12, A225, A226, FUNCT_1:def 6;



 A254: ((f . p1) `2 ) <= 0 by A2, A21, A29, Th67;

 ((f . p4) `1 ) >= ((f . p1) `1 ) by A2, A20, A251, Th70;

 hence LE ((f . p4),(f . p1),P) by A5, A229, A232, A252, A253, A254, JGRAPH_5: 56;

 end;

 end;

 hence LE ((f . p4),(f . p1),P);

 end;

 case

 A255: (p4 `1 ) = ( - 1) & (p4 `2 ) >= 0 ;



 A256: (f .: K) = P by A2, A5, Lm15, Th35, JGRAPH_3: 23;



 A257: ( dom f) = the carrier of ( TOP-REAL 2) by FUNCT_2:def 1;

 then

 A258: (f . p4) in P by A12, A256, FUNCT_1:def 6;



 A259: (f . p1) in P by A9, A256, A257, FUNCT_1:def 6;



 A260: ((f . p4) `2 ) >= 0 by A2, A255, Th69;



 A261: ((f . p1) `2 ) <= 0 by A2, A21, A29, Th67;



 A262: ( Upper_Arc P) = { p where p be Point of ( TOP-REAL 2) : p in P & (p `2 ) >= 0 } by A5, JGRAPH_5: 34;



 A263: (f . |[( - 1), 0 ]|) = ( W-min P) by A2, A5, Th10, JGRAPH_5: 29;

 A264:

 now

 assume (f . p1) = ( W-min P);

 then p1 = |[( - 1), 0 ]| by A2, A257, A263, FUNCT_1:def 4;

 hence contradiction by A29, EUCLID: 52;

 end;



 A265: (f . p4) in ( Upper_Arc P) by A258, A260, A262;

 ( Lower_Arc P) = { p where p be Point of ( TOP-REAL 2) : p in P & (p `2 ) <= 0 } by A5, JGRAPH_5: 35;

 then (f . p1) in ( Lower_Arc P) by A259, A261;

 hence LE ((f . p4),(f . p1),P) by A264, A265, JORDAN6:def 10;

 end;

 case

 A266: (p4 `1 ) = ( - 1) & (p4 `2 ) < 0 & (p1 `2 ) > (p4 `2 );

 then

 A267: p4 in ( LSeg ( |[( - 1), ( - 1)]|, |[( - 1), 1]|)) by A13, Th2;



 A268: (f .: K) = P by A2, A5, Lm15, Th35, JGRAPH_3: 23;



 A269: ( dom f) = the carrier of ( TOP-REAL 2) by FUNCT_2:def 1;

 then

 A270: (f . p4) in P by A12, A268, FUNCT_1:def 6;



 A271: (f . p1) in P by A9, A268, A269, FUNCT_1:def 6;



 A272: ((f . p1) `1 ) < 0 by A2, A21, A29, Th67;



 A273: ((f . p1) `2 ) < 0 by A2, A21, A29, Th67;



 A274: ((f . p4) `2 ) <= ((f . p1) `2 ) by A2, A20, A29, A266, A267, Th71;

 ((f . p4) `1 ) < 0 by A2, A266, Th68;

 hence LE ((f . p4),(f . p1),P) by A5, A270, A271, A272, A273, A274, JGRAPH_5: 51;

 end;

 end;

 hence LE ((f . p4),(f . p1),P);

 end;

 end;



 A275: K = { p : (p `1 ) = ( - 1) & ( - 1) <= (p `2 ) & (p `2 ) <= 1 or (p `2 ) = 1 & ( - 1) <= (p `1 ) & (p `1 ) <= 1 or (p `1 ) = 1 & ( - 1) <= (p `2 ) & (p `2 ) <= 1 or (p `2 ) = ( - 1) & ( - 1) <= (p `1 ) & (p `1 ) <= 1 } by Lm15;

 thus K = { p : (p `1 ) = ( - 1) & ( - 1) <= (p `2 ) & (p `2 ) <= 1 or (p `1 ) = 1 & ( - 1) <= (p `2 ) & (p `2 ) <= 1 or ( - 1) = (p `2 ) & ( - 1) <= (p `1 ) & (p `1 ) <= 1 or 1 = (p `2 ) & ( - 1) <= (p `1 ) & (p `1 ) <= 1 }

 proof

 thus K c= { p : (p `1 ) = ( - 1) & ( - 1) <= (p `2 ) & (p `2 ) <= 1 or (p `1 ) = 1 & ( - 1) <= (p `2 ) & (p `2 ) <= 1 or ( - 1) = (p `2 ) & ( - 1) <= (p `1 ) & (p `1 ) <= 1 or 1 = (p `2 ) & ( - 1) <= (p `1 ) & (p `1 ) <= 1 }

 proof

 let x be object;

 assume x in K;

 then ex p st (p = x) & ((p `1 ) = ( - 1) & ( - 1) <= (p `2 ) & (p `2 ) <= 1 or (p `2 ) = 1 & ( - 1) <= (p `1 ) & (p `1 ) <= 1 or (p `1 ) = 1 & ( - 1) <= (p `2 ) & (p `2 ) <= 1 or (p `2 ) = ( - 1) & ( - 1) <= (p `1 ) & (p `1 ) <= 1) by A275;

 hence thesis;

 end;

 thus { p : (p `1 ) = ( - 1) & ( - 1) <= (p `2 ) & (p `2 ) <= 1 or (p `1 ) = 1 & ( - 1) <= (p `2 ) & (p `2 ) <= 1 or ( - 1) = (p `2 ) & ( - 1) <= (p `1 ) & (p `1 ) <= 1 or 1 = (p `2 ) & ( - 1) <= (p `1 ) & (p `1 ) <= 1 } c= K

 proof

 let x be object;

 assume x in { p : (p `1 ) = ( - 1) & ( - 1) <= (p `2 ) & (p `2 ) <= 1 or (p `1 ) = 1 & ( - 1) <= (p `2 ) & (p `2 ) <= 1 or ( - 1) = (p `2 ) & ( - 1) <= (p `1 ) & (p `1 ) <= 1 or 1 = (p `2 ) & ( - 1) <= (p `1 ) & (p `1 ) <= 1 };

 then ex p st (p = x) & ((p `1 ) = ( - 1) & ( - 1) <= (p `2 ) & (p `2 ) <= 1 or (p `1 ) = 1 & ( - 1) <= (p `2 ) & (p `2 ) <= 1 or ( - 1) = (p `2 ) & ( - 1) <= (p `1 ) & (p `1 ) <= 1 or 1 = (p `2 ) & ( - 1) <= (p `1 ) & (p `1 ) <= 1);

 hence thesis by A275;

 end;

 end;

 thus thesis by A205, A206, A207, JORDAN17:def 1;

 end;

 end;

 hence thesis;

 end;

 end;

 hence thesis;

 end;

 case

 A276: p1 in ( LSeg ( |[( - 1), 1]|, |[1, 1]|));



 A277: |[( - 1), 1]| in ( LSeg ( |[( - 1), 1]|, |[1, 1]|)) by RLTOPSP1: 68;



 A278: ( |[( - 1), 1]| `1 ) = ( - 1) by EUCLID: 52;



 A279: ( |[( - 1), 1]| `2 ) = 1 by EUCLID: 52;

 ( - 1) <= (p1 `1 ) by A276, Th3;

 then

 A280: LE ( |[( - 1), 1]|,p1,K) by A276, A277, A278, Th60;

 then

 A281: LE ((f . p1),(f . p2),P) by A1, A2, A6, A279, Th66, RLTOPSP1: 68;



 A282: LE ( |[( - 1), 1]|,p2,K) by A6, A280, Th50, JORDAN6: 58;

 then

 A283: LE ((f . p2),(f . p3),P) by A1, A2, A7, A279, Th66, RLTOPSP1: 68;

 LE ( |[( - 1), 1]|,p3,K) by A7, A282, Th50, JORDAN6: 58;

 then LE ((f . p3),(f . p4),P) by A1, A2, A8, A279, Th66, RLTOPSP1: 68;

 hence thesis by A281, A283, JORDAN17:def 1;

 end;

 case

 A284: p1 in ( LSeg ( |[1, 1]|, |[1, ( - 1)]|));



 A285: |[( - 1), 1]| in ( LSeg ( |[( - 1), 1]|, |[1, 1]|)) by RLTOPSP1: 68;



 A286: ( |[( - 1), 1]| `1 ) = ( - 1) by EUCLID: 52;



 A287: ( |[( - 1), 1]| `2 ) = 1 by EUCLID: 52;



 A288: |[1, 1]| in ( LSeg ( |[1, 1]|, |[1, ( - 1)]|)) by RLTOPSP1: 68;



 A289: |[1, 1]| in ( LSeg ( |[( - 1), 1]|, |[1, 1]|)) by RLTOPSP1: 68;



 A290: ( |[1, 1]| `1 ) = 1 by EUCLID: 52;



 A291: ( |[1, 1]| `2 ) = 1 by EUCLID: 52;



 A292: LE ( |[( - 1), 1]|, |[1, 1]|,K) by A285, A286, A289, A290, Th60;

 (p1 `2 ) <= 1 by A284, Th1;

 then LE ( |[1, 1]|,p1,K) by A284, A288, A291, Th61;

 then

 A293: LE ( |[( - 1), 1]|,p1,K) by A292, Th50, JORDAN6: 58;

 then

 A294: LE ((f . p1),(f . p2),P) by A1, A2, A6, A287, Th66, RLTOPSP1: 68;



 A295: LE ( |[( - 1), 1]|,p2,K) by A6, A293, Th50, JORDAN6: 58;

 then

 A296: LE ((f . p2),(f . p3),P) by A1, A2, A7, A287, Th66, RLTOPSP1: 68;

 LE ( |[( - 1), 1]|,p3,K) by A7, A295, Th50, JORDAN6: 58;

 then LE ((f . p3),(f . p4),P) by A1, A2, A8, A287, Th66, RLTOPSP1: 68;

 hence thesis by A294, A296, JORDAN17:def 1;

 end;

 case

 A297: p1 in ( LSeg ( |[1, ( - 1)]|, |[( - 1), ( - 1)]|)) & p1 <> ( W-min K);



 A298: |[( - 1), 1]| in ( LSeg ( |[( - 1), 1]|, |[1, 1]|)) by RLTOPSP1: 68;



 A299: ( |[( - 1), 1]| `1 ) = ( - 1) by EUCLID: 52;



 A300: ( |[( - 1), 1]| `2 ) = 1 by EUCLID: 52;



 A301: |[1, 1]| in ( LSeg ( |[( - 1), 1]|, |[1, 1]|)) by RLTOPSP1: 68;

 ( |[1, 1]| `1 ) = 1 by EUCLID: 52;

 then

 A302: LE ( |[( - 1), 1]|, |[1, 1]|,K) by A298, A299, A301, Th60;



 A303: |[1, ( - 1)]| in ( LSeg ( |[1, 1]|, |[1, ( - 1)]|)) by RLTOPSP1: 68;



 A304: |[1, ( - 1)]| in ( LSeg ( |[1, ( - 1)]|, |[( - 1), ( - 1)]|)) by RLTOPSP1: 68;



 A305: ( |[1, ( - 1)]| `1 ) = 1 by EUCLID: 52;

 LE ( |[1, 1]|, |[1, ( - 1)]|,K) by A301, A303, Th60;

 then

 A306: LE ( |[( - 1), 1]|, |[1, ( - 1)]|,K) by A302, Th50, JORDAN6: 58;

 ( W-min K) = |[( - 1), ( - 1)]| by Th46;

 then

 A307: (( W-min K) `1 ) = ( - 1) by EUCLID: 52;

 (p1 `1 ) <= 1 by A297, Th3;

 then LE ( |[1, ( - 1)]|,p1,K) by A297, A304, A305, A307, Th62;

 then

 A308: LE ( |[( - 1), 1]|,p1,K) by A306, Th50, JORDAN6: 58;

 then

 A309: LE ((f . p1),(f . p2),P) by A1, A2, A6, A300, Th66, RLTOPSP1: 68;



 A310: LE ( |[( - 1), 1]|,p2,K) by A6, A308, Th50, JORDAN6: 58;

 then

 A311: LE ((f . p2),(f . p3),P) by A1, A2, A7, A300, Th66, RLTOPSP1: 68;

 LE ( |[( - 1), 1]|,p3,K) by A7, A310, Th50, JORDAN6: 58;

 then LE ((f . p3),(f . p4),P) by A1, A2, A8, A300, Th66, RLTOPSP1: 68;

 hence thesis by A309, A311, JORDAN17:def 1;

 end;

 end;

 hence thesis;

 end;

 end;

 theorem :: JGRAPH_6:73



 Th73: for p1,p2 be Point of ( TOP-REAL 2), P be non empty compact Subset of ( TOP-REAL 2) st P is being_simple_closed_curve & p1 in P & p2 in P & not LE (p1,p2,P) holds LE (p2,p1,P)

 proof

 let p1,p2 be Point of ( TOP-REAL 2), P be non empty compact Subset of ( TOP-REAL 2);

 assume that

 A1: P is being_simple_closed_curve and

 A2: p1 in P and

 A3: p2 in P and

 A4: not LE (p1,p2,P);



 A5: P = (( Upper_Arc P) \/ ( Lower_Arc P)) by A1, JORDAN6:def 9;



 A6: not p1 = ( W-min P) by A1, A3, A4, JORDAN7: 3;

 per cases by A2, A3, A5, XBOOLE_0:def 3;

 suppose

 A7: p1 in ( Upper_Arc P) & p2 in ( Upper_Arc P);



 A8: ( Upper_Arc P) is_an_arc_of (( W-min P),( E-max P)) by A1, JORDAN6:def 8;

 set q1 = ( W-min P), q2 = ( E-max P);

 set Q = ( Upper_Arc P);

 now

 per cases ;

 case

 A9: p1 <> p2;

 now

 per cases by A7, A8, A9, JORDAN5C: 14;

 case LE (p1,p2,Q,q1,q2) & not LE (p2,p1,Q,q1,q2);

 hence contradiction by A4, A7, JORDAN6:def 10;

 end;

 case LE (p2,p1,Q,q1,q2) & not LE (p1,p2,Q,q1,q2);

 hence thesis by A7, JORDAN6:def 10;

 end;

 end;

 hence thesis;

 end;

 case p1 = p2;

 hence thesis by A1, A2, JORDAN6: 56;

 end;

 end;

 hence thesis;

 end;

 suppose

 A10: p1 in ( Upper_Arc P) & p2 in ( Lower_Arc P);

 now

 per cases ;

 case p2 = ( W-min P);

 hence thesis by A1, A2, JORDAN7: 3;

 end;

 case p2 <> ( W-min P);

 hence contradiction by A4, A10, JORDAN6:def 10;

 end;

 end;

 hence thesis;

 end;

 suppose p1 in ( Lower_Arc P) & p2 in ( Upper_Arc P);

 hence thesis by A6, JORDAN6:def 10;

 end;

 suppose

 A11: p1 in ( Lower_Arc P) & p2 in ( Lower_Arc P);



 A12: ( Lower_Arc P) is_an_arc_of (( E-max P),( W-min P)) by A1, JORDAN6: 50;

 set q2 = ( W-min P), q1 = ( E-max P);

 set Q = ( Lower_Arc P);

 now

 per cases ;

 case

 A13: p1 <> p2;

 now

 per cases by A11, A12, A13, JORDAN5C: 14;

 case

 A14: LE (p1,p2,Q,q1,q2) & not LE (p2,p1,Q,q1,q2);

 now

 per cases ;

 case p2 = ( W-min P);

 hence thesis by A1, A2, JORDAN7: 3;

 end;

 case p2 <> ( W-min P);

 hence contradiction by A4, A11, A14, JORDAN6:def 10;

 end;

 end;

 hence thesis;

 end;

 case LE (p2,p1,Q,q1,q2) & not LE (p1,p2,Q,q1,q2);

 hence thesis by A6, A11, JORDAN6:def 10;

 end;

 end;

 hence thesis;

 end;

 case p1 = p2;

 hence thesis by A1, A2, JORDAN6: 56;

 end;

 end;

 hence thesis;

 end;

 end;

 theorem :: JGRAPH_6:74

 for p1,p2,p3 be Point of ( TOP-REAL 2), P be non empty compact Subset of ( TOP-REAL 2) st P is being_simple_closed_curve & p1 in P & p2 in P & p3 in P holds LE (p1,p2,P) & LE (p2,p3,P) or LE (p1,p3,P) & LE (p3,p2,P) or LE (p2,p1,P) & LE (p1,p3,P) or LE (p2,p3,P) & LE (p3,p1,P) or LE (p3,p1,P) & LE (p1,p2,P) or LE (p3,p2,P) & LE (p2,p1,P) by Th73;

 theorem :: JGRAPH_6:75

 for p1,p2,p3 be Point of ( TOP-REAL 2), P be non empty compact Subset of ( TOP-REAL 2) st P is being_simple_closed_curve & p1 in P & p2 in P & p3 in P & LE (p2,p3,P) holds LE (p1,p2,P) or LE (p2,p1,P) & LE (p1,p3,P) or LE (p3,p1,P) by Th73;

 theorem :: JGRAPH_6:76

 for p1,p2,p3,p4 be Point of ( TOP-REAL 2), P be non empty compact Subset of ( TOP-REAL 2) st P is being_simple_closed_curve & p1 in P & p2 in P & p3 in P & p4 in P & LE (p2,p3,P) & LE (p3,p4,P) holds LE (p1,p2,P) or LE (p2,p1,P) & LE (p1,p3,P) or LE (p3,p1,P) & LE (p1,p4,P) or LE (p4,p1,P) by Th73;

 theorem :: JGRAPH_6:77



 Th77: for p1,p2,p3,p4 be Point of ( TOP-REAL 2), P be non empty compact Subset of ( TOP-REAL 2), f be Function of ( TOP-REAL 2), ( TOP-REAL 2) st P = ( circle ( 0 , 0 ,1)) & f = Sq_Circ & LE ((f . p1),(f . p2),P) & LE ((f . p2),(f . p3),P) & LE ((f . p3),(f . p4),P) holds (p1,p2,p3,p4) are_in_this_order_on ( rectangle (( - 1),1,( - 1),1))

 proof

 let p1,p2,p3,p4 be Point of ( TOP-REAL 2), P be non empty compact Subset of ( TOP-REAL 2), f be Function of ( TOP-REAL 2), ( TOP-REAL 2);

 set K = ( rectangle (( - 1),1,( - 1),1));

 assume that

 A1: P = ( circle ( 0 , 0 ,1)) and

 A2: f = Sq_Circ and

 A3: LE ((f . p1),(f . p2),P) and

 A4: LE ((f . p2),(f . p3),P) and

 A5: LE ((f . p3),(f . p4),P);



 A6: K is being_simple_closed_curve by Th50;



 A7: P = { p : |.p.| = 1 } by A1, Th24;

 then

 A8: LE ((f . p1),(f . p3),P) by A3, A4, JGRAPH_3: 26, JORDAN6: 58;



 A9: LE ((f . p2),(f . p4),P) by A4, A5, A7, JGRAPH_3: 26, JORDAN6: 58;



 A10: (f .: K) = P by A2, A7, Lm15, Th35, JGRAPH_3: 23;



 A11: ( dom f) = the carrier of ( TOP-REAL 2) by FUNCT_2:def 1;



 A12: P = { p : |.p.| = 1 } by A1, Th24;

 then

 A13: P is being_simple_closed_curve by JGRAPH_3: 26;

 then (f . p1) in P by A3, JORDAN7: 5;

 then ex x1 be object st (x1 in ( dom f)) & (x1 in K) & ((f . p1) = (f . x1)) by A10, FUNCT_1:def 6;

 then

 A14: p1 in K by A2, A11, FUNCT_1:def 4;

 (f . p2) in P by A3, A13, JORDAN7: 5;

 then ex x2 be object st (x2 in ( dom f)) & (x2 in K) & ((f . p2) = (f . x2)) by A10, FUNCT_1:def 6;

 then

 A15: p2 in K by A2, A11, FUNCT_1:def 4;

 (f . p3) in P by A4, A13, JORDAN7: 5;

 then ex x3 be object st (x3 in ( dom f)) & (x3 in K) & ((f . p3) = (f . x3)) by A10, FUNCT_1:def 6;

 then

 A16: p3 in K by A2, A11, FUNCT_1:def 4;

 (f . p4) in P by A5, A13, JORDAN7: 5;

 then ex x4 be object st (x4 in ( dom f)) & (x4 in K) & ((f . p4) = (f . x4)) by A10, FUNCT_1:def 6;

 then

 A17: p4 in K by A2, A11, FUNCT_1:def 4;

 now

 assume

 A18: not (p1,p2,p3,p4) are_in_this_order_on K;

 A19:

 now

 assume

 A20: (p1,p2,p4,p3) are_in_this_order_on K;

 now

 per cases by A20, JORDAN17:def 1;

 case

 A21: LE (p1,p2,K) & LE (p2,p4,K) & LE (p4,p3,K);

 then ((f . p1),(f . p2),(f . p4),(f . p3)) are_in_this_order_on P by A1, A2, Th72;

 then LE ((f . p1),(f . p2),P) & LE ((f . p2),(f . p4),P) & LE ((f . p4),(f . p3),P) or LE ((f . p2),(f . p4),P) & LE ((f . p4),(f . p3),P) & LE ((f . p3),(f . p1),P) or LE ((f . p4),(f . p3),P) & LE ((f . p3),(f . p1),P) & LE ((f . p1),(f . p2),P) or LE ((f . p3),(f . p1),P) & LE ((f . p1),(f . p2),P) & LE ((f . p2),(f . p4),P) by JORDAN17:def 1;

 then (f . p3) = (f . p4) or (f . p3) = (f . p1) by A5, A8, A12, JGRAPH_3: 26, JORDAN6: 57;

 then

 A22: p3 = p4 or p3 = p1 by A2, A11, FUNCT_1:def 4;

 LE (p1,p4,K) by A21, Th50, JORDAN6: 58;

 then p1 = p4 by A18, A20, A21, A22, Th50, JORDAN6: 57;

 hence contradiction by A18, A20, A22;

 end;

 case

 A23: LE (p2,p4,K) & LE (p4,p3,K) & LE (p3,p1,K);

 then ((f . p2),(f . p4),(f . p3),(f . p1)) are_in_this_order_on P by A1, A2, Th72;

 then LE ((f . p2),(f . p4),P) & LE ((f . p4),(f . p3),P) & LE ((f . p3),(f . p1),P) or LE ((f . p4),(f . p3),P) & LE ((f . p3),(f . p1),P) & LE ((f . p1),(f . p2),P) or LE ((f . p3),(f . p1),P) & LE ((f . p1),(f . p2),P) & LE ((f . p2),(f . p4),P) or LE ((f . p1),(f . p2),P) & LE ((f . p2),(f . p4),P) & LE ((f . p4),(f . p3),P) by JORDAN17:def 1;

 then (f . p3) = (f . p4) or LE ((f . p3),(f . p2),P) by A5, A13, JORDAN6: 57, JORDAN6: 58;

 then (f . p3) = (f . p4) or (f . p3) = (f . p2) by A4, A12, JGRAPH_3: 26, JORDAN6: 57;

 then

 A24: p3 = p4 or p3 = p2 by A2, A11, FUNCT_1:def 4;

 then p4 = p2 by A18, A20, A23, Th50, JORDAN6: 57;

 hence contradiction by A18, A20, A24;

 end;

 case

 A25: LE (p4,p3,K) & LE (p3,p1,K) & LE (p1,p2,K);

 then ((f . p4),(f . p3),(f . p1),(f . p2)) are_in_this_order_on P by A1, A2, Th72;

 then LE ((f . p4),(f . p3),P) & LE ((f . p3),(f . p1),P) & LE ((f . p1),(f . p2),P) or LE ((f . p3),(f . p1),P) & LE ((f . p1),(f . p2),P) & LE ((f . p2),(f . p4),P) or LE ((f . p1),(f . p2),P) & LE ((f . p2),(f . p4),P) & LE ((f . p4),(f . p3),P) or LE ((f . p2),(f . p4),P) & LE ((f . p4),(f . p3),P) & LE ((f . p3),(f . p1),P) by JORDAN17:def 1;

 then (f . p3) = (f . p4) or LE ((f . p3),(f . p2),P) by A5, A13, JORDAN6: 57, JORDAN6: 58;

 then (f . p3) = (f . p4) or (f . p3) = (f . p2) by A4, A12, JGRAPH_3: 26, JORDAN6: 57;

 then

 A26: p3 = p4 or p3 = p2 by A2, A11, FUNCT_1:def 4;

 then p3 = p1 by A18, A20, A25, Th50, JORDAN6: 57;

 hence contradiction by A6, A18, A20, A26, JORDAN17: 12;

 end;

 case

 A27: LE (p3,p1,K) & LE (p1,p2,K) & LE (p2,p4,K);

 then ((f . p3),(f . p1),(f . p2),(f . p4)) are_in_this_order_on P by A1, A2, Th72;

 then LE ((f . p4),(f . p3),P) & LE ((f . p3),(f . p1),P) & LE ((f . p1),(f . p2),P) or LE ((f . p3),(f . p1),P) & LE ((f . p1),(f . p2),P) & LE ((f . p2),(f . p4),P) or LE ((f . p1),(f . p2),P) & LE ((f . p2),(f . p4),P) & LE ((f . p4),(f . p3),P) or LE ((f . p2),(f . p4),P) & LE ((f . p4),(f . p3),P) & LE ((f . p3),(f . p1),P) by JORDAN17:def 1;

 then (f . p3) = (f . p4) or LE ((f . p3),(f . p2),P) by A5, A13, JORDAN6: 57, JORDAN6: 58;

 then (f . p3) = (f . p4) or (f . p3) = (f . p2) by A4, A12, JGRAPH_3: 26, JORDAN6: 57;

 then

 A28: p3 = p4 or p3 = p2 by A2, A11, FUNCT_1:def 4;

 then p3 = p1 by A18, A20, A27, Th50, JORDAN6: 57;

 hence contradiction by A6, A18, A20, A28, JORDAN17: 12;

 end;

 end;

 hence contradiction;

 end;

 A29:

 now

 assume

 A30: (p1,p3,p4,p2) are_in_this_order_on K;

 now

 per cases by A30, JORDAN17:def 1;

 case LE (p1,p3,K) & LE (p3,p4,K) & LE (p4,p2,K);

 then ((f . p1),(f . p3),(f . p4),(f . p2)) are_in_this_order_on P by A1, A2, Th72;

 then LE ((f . p1),(f . p3),P) & LE ((f . p3),(f . p4),P) & LE ((f . p4),(f . p2),P) or LE ((f . p3),(f . p4),P) & LE ((f . p4),(f . p2),P) & LE ((f . p2),(f . p1),P) or LE ((f . p4),(f . p2),P) & LE ((f . p2),(f . p1),P) & LE ((f . p1),(f . p3),P) or LE ((f . p2),(f . p1),P) & LE ((f . p1),(f . p3),P) & LE ((f . p3),(f . p4),P) by JORDAN17:def 1;

 then (f . p4) = (f . p2) or (f . p2) = (f . p1) by A3, A9, A12, JGRAPH_3: 26, JORDAN6: 57;

 then

 A31: p4 = p2 or p2 = p1 by A2, A11, FUNCT_1:def 4;

 then (f . p3) = (f . p2) or (f . p4) = (f . p1) by A4, A5, A6, A12, A18, A30, JGRAPH_3: 26, JORDAN17: 12, JORDAN6: 57;

 then p3 = p2 or p4 = p1 by A2, A11, FUNCT_1:def 4;

 hence contradiction by A6, A18, A30, A31, JORDAN17: 12;

 end;

 case LE (p3,p4,K) & LE (p4,p2,K) & LE (p2,p1,K);

 then ((f . p3),(f . p4),(f . p2),(f . p1)) are_in_this_order_on P by A1, A2, Th72;

 then

 A32: LE ((f . p1),(f . p3),P) & LE ((f . p3),(f . p4),P) & LE ((f . p4),(f . p2),P) or LE ((f . p3),(f . p4),P) & LE ((f . p4),(f . p2),P) & LE ((f . p2),(f . p1),P) or LE ((f . p4),(f . p2),P) & LE ((f . p2),(f . p1),P) & LE ((f . p1),(f . p3),P) or LE ((f . p2),(f . p1),P) & LE ((f . p1),(f . p3),P) & LE ((f . p3),(f . p4),P) by JORDAN17:def 1;

 then (f . p4) = (f . p2) or (f . p2) = (f . p1) by A3, A9, A12, JGRAPH_3: 26, JORDAN6: 57;

 then

 A33: p4 = p2 or p2 = p1 by A2, A11, FUNCT_1:def 4;

 (f . p2) = (f . p1) or LE ((f . p3),(f . p2),P) by A3, A13, A32, JORDAN6: 57, JORDAN6: 58;

 then (f . p2) = (f . p1) or (f . p3) = (f . p2) by A4, A12, JGRAPH_3: 26, JORDAN6: 57;

 then p2 = p1 or p3 = p2 by A2, A11, FUNCT_1:def 4;

 hence contradiction by A6, A18, A30, A33, JORDAN17: 12;

 end;

 case LE (p4,p2,K) & LE (p2,p1,K) & LE (p1,p3,K);

 then ((f . p4),(f . p2),(f . p1),(f . p3)) are_in_this_order_on P by A1, A2, Th72;

 then

 A34: LE ((f . p1),(f . p3),P) & LE ((f . p3),(f . p4),P) & LE ((f . p4),(f . p2),P) or LE ((f . p3),(f . p4),P) & LE ((f . p4),(f . p2),P) & LE ((f . p2),(f . p1),P) or LE ((f . p4),(f . p2),P) & LE ((f . p2),(f . p1),P) & LE ((f . p1),(f . p3),P) or LE ((f . p2),(f . p1),P) & LE ((f . p1),(f . p3),P) & LE ((f . p3),(f . p4),P) by JORDAN17:def 1;

 then (f . p4) = (f . p2) or (f . p2) = (f . p1) by A3, A9, A12, JGRAPH_3: 26, JORDAN6: 57;

 then

 A35: p4 = p2 or p2 = p1 by A2, A11, FUNCT_1:def 4;

 (f . p2) = (f . p1) or LE ((f . p3),(f . p2),P) by A3, A13, A34, JORDAN6: 57, JORDAN6: 58;

 then (f . p2) = (f . p1) or (f . p3) = (f . p2) by A4, A12, JGRAPH_3: 26, JORDAN6: 57;

 then p2 = p1 or p3 = p2 by A2, A11, FUNCT_1:def 4;

 hence contradiction by A6, A18, A30, A35, JORDAN17: 12;

 end;

 case

 A36: LE (p2,p1,K) & LE (p1,p3,K) & LE (p3,p4,K);

 then ((f . p2),(f . p1),(f . p3),(f . p4)) are_in_this_order_on P by A1, A2, Th72;

 then LE ((f . p1),(f . p3),P) & LE ((f . p3),(f . p4),P) & LE ((f . p4),(f . p2),P) or LE ((f . p3),(f . p4),P) & LE ((f . p4),(f . p2),P) & LE ((f . p2),(f . p1),P) or LE ((f . p4),(f . p2),P) & LE ((f . p2),(f . p1),P) & LE ((f . p1),(f . p3),P) or LE ((f . p2),(f . p1),P) & LE ((f . p1),(f . p3),P) & LE ((f . p3),(f . p4),P) by JORDAN17:def 1;

 then (f . p4) = (f . p2) or (f . p2) = (f . p1) by A3, A9, A12, JGRAPH_3: 26, JORDAN6: 57;

 then

 A37: p4 = p2 or p2 = p1 by A2, A11, FUNCT_1:def 4;

 LE (p2,p3,K) by A36, Th50, JORDAN6: 58;

 then p2 = p3 by A6, A18, A30, A36, A37, JORDAN17: 12, JORDAN6: 57;

 hence contradiction by A6, A18, A30, A37, JORDAN17: 12;

 end;

 end;

 hence contradiction;

 end;

 now

 per cases by A6, A14, A15, A16, A17, A18, JORDAN17: 27;

 case (p1,p2,p4,p3) are_in_this_order_on K;

 hence contradiction by A19;

 end;

 case

 A38: (p1,p3,p2,p4) are_in_this_order_on K;

 now

 per cases by A38, JORDAN17:def 1;

 case

 A39: LE (p1,p3,K) & LE (p3,p2,K) & LE (p2,p4,K);

 then ((f . p1),(f . p3),(f . p2),(f . p4)) are_in_this_order_on P by A1, A2, Th72;

 then LE ((f . p1),(f . p3),P) & LE ((f . p3),(f . p2),P) & LE ((f . p2),(f . p4),P) or LE ((f . p3),(f . p2),P) & LE ((f . p2),(f . p4),P) & LE ((f . p4),(f . p1),P) or LE ((f . p2),(f . p4),P) & LE ((f . p4),(f . p1),P) & LE ((f . p1),(f . p3),P) or LE ((f . p4),(f . p1),P) & LE ((f . p1),(f . p3),P) & LE ((f . p3),(f . p2),P) by JORDAN17:def 1;

 then (f . p3) = (f . p2) or LE ((f . p2),(f . p1),P) by A4, A13, JORDAN6: 57, JORDAN6: 58;

 then (f . p3) = (f . p2) or (f . p2) = (f . p1) by A3, A12, JGRAPH_3: 26, JORDAN6: 57;

 then

 A40: p3 = p2 or p2 = p1 by A2, A11, FUNCT_1:def 4;

 then p3 = p1 by A18, A38, A39, Th50, JORDAN6: 57;

 hence contradiction by A18, A38, A40;

 end;

 case

 A41: LE (p3,p2,K) & LE (p2,p4,K) & LE (p4,p1,K);

 then ((f . p3),(f . p2),(f . p4),(f . p1)) are_in_this_order_on P by A1, A2, Th72;

 then LE ((f . p1),(f . p3),P) & LE ((f . p3),(f . p2),P) & LE ((f . p2),(f . p4),P) or LE ((f . p3),(f . p2),P) & LE ((f . p2),(f . p4),P) & LE ((f . p4),(f . p1),P) or LE ((f . p2),(f . p4),P) & LE ((f . p4),(f . p1),P) & LE ((f . p1),(f . p3),P) or LE ((f . p4),(f . p1),P) & LE ((f . p1),(f . p3),P) & LE ((f . p3),(f . p2),P) by JORDAN17:def 1;

 then (f . p3) = (f . p2) or LE ((f . p2),(f . p1),P) by A4, A13, JORDAN6: 57, JORDAN6: 58;

 then (f . p3) = (f . p2) or (f . p2) = (f . p1) by A3, A12, JGRAPH_3: 26, JORDAN6: 57;

 then

 A42: p3 = p2 or p2 = p1 by A2, A11, FUNCT_1:def 4;

 then p4 = p1 by A18, A38, A41, Th50, JORDAN6: 57;

 hence contradiction by A6, A18, A38, A42, JORDAN17: 12;

 end;

 case

 A43: LE (p2,p4,K) & LE (p4,p1,K) & LE (p1,p3,K);

 then ((f . p2),(f . p4),(f . p1),(f . p3)) are_in_this_order_on P by A1, A2, Th72;

 then LE ((f . p1),(f . p3),P) & LE ((f . p3),(f . p2),P) & LE ((f . p2),(f . p4),P) or LE ((f . p3),(f . p2),P) & LE ((f . p2),(f . p4),P) & LE ((f . p4),(f . p1),P) or LE ((f . p2),(f . p4),P) & LE ((f . p4),(f . p1),P) & LE ((f . p1),(f . p3),P) or LE ((f . p4),(f . p1),P) & LE ((f . p1),(f . p3),P) & LE ((f . p3),(f . p2),P) by JORDAN17:def 1;

 then (f . p3) = (f . p2) or LE ((f . p2),(f . p1),P) by A4, A13, JORDAN6: 57, JORDAN6: 58;

 then (f . p3) = (f . p2) or (f . p2) = (f . p1) by A3, A12, JGRAPH_3: 26, JORDAN6: 57;

 then

 A44: p3 = p2 or p2 = p1 by A2, A11, FUNCT_1:def 4;

 then p4 = p1 by A18, A38, A43, Th50, JORDAN6: 57;

 hence contradiction by A6, A18, A38, A44, JORDAN17: 12;

 end;

 case

 A45: LE (p4,p1,K) & LE (p1,p3,K) & LE (p3,p2,K);

 then ((f . p4),(f . p1),(f . p3),(f . p2)) are_in_this_order_on P by A1, A2, Th72;

 then LE ((f . p1),(f . p3),P) & LE ((f . p3),(f . p2),P) & LE ((f . p2),(f . p4),P) or LE ((f . p3),(f . p2),P) & LE ((f . p2),(f . p4),P) & LE ((f . p4),(f . p1),P) or LE ((f . p2),(f . p4),P) & LE ((f . p4),(f . p1),P) & LE ((f . p1),(f . p3),P) or LE ((f . p4),(f . p1),P) & LE ((f . p1),(f . p3),P) & LE ((f . p3),(f . p2),P) by JORDAN17:def 1;

 then (f . p3) = (f . p2) or LE ((f . p2),(f . p1),P) by A4, A13, JORDAN6: 57, JORDAN6: 58;

 then (f . p3) = (f . p2) or (f . p2) = (f . p1) by A3, A12, JGRAPH_3: 26, JORDAN6: 57;

 then

 A46: p3 = p2 or p2 = p1 by A2, A11, FUNCT_1:def 4;

 then p3 = p1 by A18, A38, A45, Th50, JORDAN6: 57;

 hence contradiction by A18, A38, A46;

 end;

 end;

 hence contradiction;

 end;

 case (p1,p3,p4,p2) are_in_this_order_on K;

 hence contradiction by A29;

 end;

 case

 A47: (p1,p4,p2,p3) are_in_this_order_on K;

 now

 per cases by A47, JORDAN17:def 1;

 case

 A48: LE (p1,p4,K) & LE (p4,p2,K) & LE (p2,p3,K);

 then ((f . p1),(f . p4),(f . p2),(f . p3)) are_in_this_order_on P by A1, A2, Th72;

 then

 A49: LE ((f . p1),(f . p4),P) & LE ((f . p4),(f . p2),P) & LE ((f . p2),(f . p3),P) or LE ((f . p4),(f . p2),P) & LE ((f . p2),(f . p3),P) & LE ((f . p3),(f . p1),P) or LE ((f . p2),(f . p3),P) & LE ((f . p3),(f . p1),P) & LE ((f . p1),(f . p4),P) or LE ((f . p3),(f . p1),P) & LE ((f . p1),(f . p4),P) & LE ((f . p4),(f . p2),P) by JORDAN17:def 1;

 then (f . p4) = (f . p2) or LE ((f . p2),(f . p1),P) by A9, A13, JORDAN6: 57, JORDAN6: 58;

 then (f . p4) = (f . p2) or (f . p2) = (f . p1) by A3, A12, JGRAPH_3: 26, JORDAN6: 57;

 then

 A50: p4 = p2 or p2 = p1 by A2, A11, FUNCT_1:def 4;

 then

 A51: p4 = p2 by A48, Th50, JORDAN6: 57;

 (f . p3) = (f . p1) or LE ((f . p4),(f . p3),P) by A8, A13, A49, JORDAN6: 57, JORDAN6: 58;

 then (f . p3) = (f . p1) or (f . p4) = (f . p3) by A5, A12, JGRAPH_3: 26, JORDAN6: 57;

 then

 A52: p3 = p1 or p4 = p3 by A2, A11, FUNCT_1:def 4;

 then p1 = p2 by A18, A47, A48, A50, Th50, JORDAN6: 57;

 hence contradiction by A18, A47, A51, A52;

 end;

 case

 A53: LE (p4,p2,K) & LE (p2,p3,K) & LE (p3,p1,K);

 then ((f . p4),(f . p2),(f . p3),(f . p1)) are_in_this_order_on P by A1, A2, Th72;

 then

 A54: LE ((f . p1),(f . p4),P) & LE ((f . p4),(f . p2),P) & LE ((f . p2),(f . p3),P) or LE ((f . p4),(f . p2),P) & LE ((f . p2),(f . p3),P) & LE ((f . p3),(f . p1),P) or LE ((f . p2),(f . p3),P) & LE ((f . p3),(f . p1),P) & LE ((f . p1),(f . p4),P) or LE ((f . p3),(f . p1),P) & LE ((f . p1),(f . p4),P) & LE ((f . p4),(f . p2),P) by JORDAN17:def 1;

 then (f . p4) = (f . p2) or LE ((f . p2),(f . p1),P) by A9, A13, JORDAN6: 57, JORDAN6: 58;

 then (f . p4) = (f . p2) or (f . p2) = (f . p1) by A3, A12, JGRAPH_3: 26, JORDAN6: 57;

 then p4 = p2 or p2 = p1 by A2, A11, FUNCT_1:def 4;

 then

 A55: p4 = p2 or p2 = p1 & p3 = p1 by A53, Th50, JORDAN6: 57;

 (f . p3) = (f . p1) or LE ((f . p4),(f . p3),P) by A8, A13, A54, JORDAN6: 57, JORDAN6: 58;

 then (f . p3) = (f . p1) or (f . p4) = (f . p3) by A5, A12, JGRAPH_3: 26, JORDAN6: 57;

 then p3 = p1 or p4 = p3 by A2, A11, FUNCT_1:def 4;

 hence contradiction by A6, A29, A47, A55, JORDAN17: 12;

 end;

 case

 A56: LE (p2,p3,K) & LE (p3,p1,K) & LE (p1,p4,K);

 then ((f . p2),(f . p3),(f . p1),(f . p4)) are_in_this_order_on P by A1, A2, Th72;

 then

 A57: LE ((f . p1),(f . p4),P) & LE ((f . p4),(f . p2),P) & LE ((f . p2),(f . p3),P) or LE ((f . p4),(f . p2),P) & LE ((f . p2),(f . p3),P) & LE ((f . p3),(f . p1),P) or LE ((f . p2),(f . p3),P) & LE ((f . p3),(f . p1),P) & LE ((f . p1),(f . p4),P) or LE ((f . p3),(f . p1),P) & LE ((f . p1),(f . p4),P) & LE ((f . p4),(f . p2),P) by JORDAN17:def 1;

 then (f . p4) = (f . p2) or LE ((f . p2),(f . p1),P) by A9, A13, JORDAN6: 57, JORDAN6: 58;

 then (f . p4) = (f . p2) or (f . p2) = (f . p1) by A3, A12, JGRAPH_3: 26, JORDAN6: 57;

 then p4 = p2 or p2 = p1 by A2, A11, FUNCT_1:def 4;

 then

 A58: p4 = p2 or p2 = p1 & p3 = p1 by A56, Th50, JORDAN6: 57;

 (f . p3) = (f . p1) or LE ((f . p4),(f . p3),P) by A8, A13, A57, JORDAN6: 57, JORDAN6: 58;

 then (f . p3) = (f . p1) or (f . p4) = (f . p3) by A5, A12, JGRAPH_3: 26, JORDAN6: 57;

 then p3 = p1 or p4 = p3 by A2, A11, FUNCT_1:def 4;

 hence contradiction by A6, A29, A47, A58, JORDAN17: 12;

 end;

 case

 A59: LE (p3,p1,K) & LE (p1,p4,K) & LE (p4,p2,K);

 then ((f . p3),(f . p1),(f . p4),(f . p2)) are_in_this_order_on P by A1, A2, Th72;

 then

 A60: LE ((f . p1),(f . p4),P) & LE ((f . p4),(f . p2),P) & LE ((f . p2),(f . p3),P) or LE ((f . p4),(f . p2),P) & LE ((f . p2),(f . p3),P) & LE ((f . p3),(f . p1),P) or LE ((f . p2),(f . p3),P) & LE ((f . p3),(f . p1),P) & LE ((f . p1),(f . p4),P) or LE ((f . p3),(f . p1),P) & LE ((f . p1),(f . p4),P) & LE ((f . p4),(f . p2),P) by JORDAN17:def 1;

 then (f . p4) = (f . p2) or LE ((f . p2),(f . p1),P) by A9, A13, JORDAN6: 57, JORDAN6: 58;

 then (f . p4) = (f . p2) or (f . p2) = (f . p1) by A3, A12, JGRAPH_3: 26, JORDAN6: 57;

 then p4 = p2 or p2 = p1 by A2, A11, FUNCT_1:def 4;

 then

 A61: p4 = p2 by A59, Th50, JORDAN6: 57;

 (f . p3) = (f . p1) or LE ((f . p4),(f . p3),P) by A8, A13, A60, JORDAN6: 57, JORDAN6: 58;

 then (f . p3) = (f . p1) or (f . p4) = (f . p3) by A5, A12, JGRAPH_3: 26, JORDAN6: 57;

 then p3 = p1 or p4 = p3 by A2, A11, FUNCT_1:def 4;

 hence contradiction by A6, A29, A47, A61, JORDAN17: 12;

 end;

 end;

 hence contradiction;

 end;

 case

 A62: (p1,p4,p3,p2) are_in_this_order_on K;

 now

 per cases by A62, JORDAN17:def 1;

 case

 A63: LE (p1,p4,K) & LE (p4,p3,K) & LE (p3,p2,K);

 then ((f . p1),(f . p4),(f . p3),(f . p2)) are_in_this_order_on P by A1, A2, Th72;

 then

 A64: LE ((f . p1),(f . p4),P) & LE ((f . p4),(f . p3),P) & LE ((f . p3),(f . p2),P) or LE ((f . p4),(f . p3),P) & LE ((f . p3),(f . p2),P) & LE ((f . p2),(f . p1),P) or LE ((f . p3),(f . p2),P) & LE ((f . p2),(f . p1),P) & LE ((f . p1),(f . p4),P) or LE ((f . p2),(f . p1),P) & LE ((f . p1),(f . p4),P) & LE ((f . p4),(f . p3),P) by JORDAN17:def 1;

 then (f . p3) = (f . p2) or (f . p2) = (f . p1) by A3, A4, A12, JGRAPH_3: 26, JORDAN6: 57;

 then

 A65: p3 = p2 or p2 = p1 by A2, A11, FUNCT_1:def 4;

 LE (p1,p3,K) by A63, Th50, JORDAN6: 58;

 then

 A66: p3 = p2 by A63, A65, Th50, JORDAN6: 57;

 (f . p4) = (f . p3) or (f . p2) = (f . p1) by A3, A5, A12, A64, JGRAPH_3: 26, JORDAN6: 57;

 then p4 = p3 or p2 = p1 by A2, A11, FUNCT_1:def 4;

 then p4 = p3 or (p2,p3,p4,p1) are_in_this_order_on K by A6, A62, A66, JORDAN17: 12;

 hence contradiction by A6, A18, A62, A65, JORDAN17: 12;

 end;

 case LE (p4,p3,K) & LE (p3,p2,K) & LE (p2,p1,K);

 then ((f . p4),(f . p3),(f . p2),(f . p1)) are_in_this_order_on P by A1, A2, Th72;

 then

 A67: LE ((f . p1),(f . p4),P) & LE ((f . p4),(f . p3),P) & LE ((f . p3),(f . p2),P) or LE ((f . p4),(f . p3),P) & LE ((f . p3),(f . p2),P) & LE ((f . p2),(f . p1),P) or LE ((f . p3),(f . p2),P) & LE ((f . p2),(f . p1),P) & LE ((f . p1),(f . p4),P) or LE ((f . p2),(f . p1),P) & LE ((f . p1),(f . p4),P) & LE ((f . p4),(f . p3),P) by JORDAN17:def 1;

 then (f . p3) = (f . p2) or (f . p2) = (f . p1) by A3, A4, A12, JGRAPH_3: 26, JORDAN6: 57;

 then

 A68: p3 = p2 or p2 = p1 by A2, A11, FUNCT_1:def 4;

 (f . p4) = (f . p3) or (f . p2) = (f . p1) by A3, A5, A12, A67, JGRAPH_3: 26, JORDAN6: 57;

 then p4 = p3 or p2 = p1 by A2, A11, FUNCT_1:def 4;

 hence contradiction by A6, A19, A62, A68, JORDAN17: 12;

 end;

 case LE (p3,p2,K) & LE (p2,p1,K) & LE (p1,p4,K);

 then ((f . p3),(f . p2),(f . p1),(f . p4)) are_in_this_order_on P by A1, A2, Th72;

 then LE ((f . p1),(f . p4),P) & LE ((f . p4),(f . p3),P) & LE ((f . p3),(f . p2),P) or LE ((f . p4),(f . p3),P) & LE ((f . p3),(f . p2),P) & LE ((f . p2),(f . p1),P) or LE ((f . p3),(f . p2),P) & LE ((f . p2),(f . p1),P) & LE ((f . p1),(f . p4),P) or LE ((f . p2),(f . p1),P) & LE ((f . p1),(f . p4),P) & LE ((f . p4),(f . p3),P) by JORDAN17:def 1;

 then (f . p4) = (f . p3) or (f . p2) = (f . p1) by A3, A5, A12, JGRAPH_3: 26, JORDAN6: 57;

 then p4 = p3 or p2 = p1 by A2, A11, FUNCT_1:def 4;

 hence contradiction by A6, A19, A29, A62, JORDAN17: 12;

 end;

 case

 A69: LE (p2,p1,K) & LE (p1,p4,K) & LE (p4,p3,K);

 then ((f . p2),(f . p1),(f . p4),(f . p3)) are_in_this_order_on P by A1, A2, Th72;

 then

 A70: LE ((f . p1),(f . p4),P) & LE ((f . p4),(f . p3),P) & LE ((f . p3),(f . p2),P) or LE ((f . p4),(f . p3),P) & LE ((f . p3),(f . p2),P) & LE ((f . p2),(f . p1),P) or LE ((f . p3),(f . p2),P) & LE ((f . p2),(f . p1),P) & LE ((f . p1),(f . p4),P) or LE ((f . p2),(f . p1),P) & LE ((f . p1),(f . p4),P) & LE ((f . p4),(f . p3),P) by JORDAN17:def 1;

 then (f . p3) = (f . p2) or (f . p2) = (f . p1) by A3, A4, A12, JGRAPH_3: 26, JORDAN6: 57;

 then

 A71: p3 = p2 or p2 = p1 by A2, A11, FUNCT_1:def 4;

 LE (p1,p3,K) by A69, Th50, JORDAN6: 58;

 then

 A72: p1 = p2 by A69, A71, Th50, JORDAN6: 57;

 (f . p4) = (f . p3) or (f . p2) = (f . p3) by A4, A5, A12, A70, JGRAPH_3: 26, JORDAN6: 57;

 then p4 = p3 or p2 = p3 by A2, A11, FUNCT_1:def 4;

 hence contradiction by A6, A29, A62, A72, JORDAN17: 12;

 end;

 end;

 hence contradiction;

 end;

 end;

 hence contradiction;

 end;

 hence thesis;

 end;

 theorem :: JGRAPH_6:78



 Th78: for p1,p2,p3,p4 be Point of ( TOP-REAL 2), P be non empty compact Subset of ( TOP-REAL 2), f be Function of ( TOP-REAL 2), ( TOP-REAL 2) st P = ( circle ( 0 , 0 ,1)) & f = Sq_Circ holds (p1,p2,p3,p4) are_in_this_order_on ( rectangle (( - 1),1,( - 1),1)) iff ((f . p1),(f . p2),(f . p3),(f . p4)) are_in_this_order_on P

 proof

 let p1,p2,p3,p4 be Point of ( TOP-REAL 2), P be non empty compact Subset of ( TOP-REAL 2), f be Function of ( TOP-REAL 2), ( TOP-REAL 2);

 set K = ( rectangle (( - 1),1,( - 1),1));

 assume that

 A1: P = ( circle ( 0 , 0 ,1)) and

 A2: f = Sq_Circ ;



 A3: K is being_simple_closed_curve by Th50;

 ( circle ( 0 , 0 ,1)) = { p5 where p5 be Point of ( TOP-REAL 2) : |.p5.| = 1 } by Th24;

 then

 A4: P is being_simple_closed_curve by A1, JGRAPH_3: 26;

 thus (p1,p2,p3,p4) are_in_this_order_on K implies ((f . p1),(f . p2),(f . p3),(f . p4)) are_in_this_order_on P

 proof

 assume

 A5: (p1,p2,p3,p4) are_in_this_order_on K;

 now

 per cases by A5, JORDAN17:def 1;

 case LE (p1,p2,K) & LE (p2,p3,K) & LE (p3,p4,K);

 hence thesis by A1, A2, Th72;

 end;

 case LE (p2,p3,K) & LE (p3,p4,K) & LE (p4,p1,K);

 hence thesis by A1, A2, A4, Th72, JORDAN17: 12;

 end;

 case LE (p3,p4,K) & LE (p4,p1,K) & LE (p1,p2,K);

 hence thesis by A1, A2, A4, Th72, JORDAN17: 11;

 end;

 case LE (p4,p1,K) & LE (p1,p2,K) & LE (p2,p3,K);

 hence thesis by A1, A2, A4, Th72, JORDAN17: 10;

 end;

 end;

 hence thesis;

 end;

 thus ((f . p1),(f . p2),(f . p3),(f . p4)) are_in_this_order_on P implies (p1,p2,p3,p4) are_in_this_order_on K

 proof

 assume

 A6: ((f . p1),(f . p2),(f . p3),(f . p4)) are_in_this_order_on P;

 now

 per cases by A6, JORDAN17:def 1;

 case LE ((f . p1),(f . p2),P) & LE ((f . p2),(f . p3),P) & LE ((f . p3),(f . p4),P);

 hence thesis by A1, A2, Th77;

 end;

 case LE ((f . p2),(f . p3),P) & LE ((f . p3),(f . p4),P) & LE ((f . p4),(f . p1),P);

 then (p2,p3,p4,p1) are_in_this_order_on K by A1, A2, Th77;

 hence thesis by A3, JORDAN17: 12;

 end;

 case LE ((f . p3),(f . p4),P) & LE ((f . p4),(f . p1),P) & LE ((f . p1),(f . p2),P);

 then (p3,p4,p1,p2) are_in_this_order_on K by A1, A2, Th77;

 hence thesis by A3, JORDAN17: 11;

 end;

 case LE ((f . p4),(f . p1),P) & LE ((f . p1),(f . p2),P) & LE ((f . p2),(f . p3),P);

 then (p4,p1,p2,p3) are_in_this_order_on K by A1, A2, Th77;

 hence thesis by A3, JORDAN17: 10;

 end;

 end;

 hence thesis;

 end;

 end;

 theorem :: JGRAPH_6:79

 for p1,p2,p3,p4 be Point of ( TOP-REAL 2) st (p1,p2,p3,p4) are_in_this_order_on ( rectangle (( - 1),1,( - 1),1)) holds for f,g be Function of I[01] , ( TOP-REAL 2) st f is continuous one-to-one & g is continuous one-to-one & (f . 0 ) = p1 & (f . 1) = p3 & (g . 0 ) = p2 & (g . 1) = p4 & ( rng f) c= ( closed_inside_of_rectangle (( - 1),1,( - 1),1)) & ( rng g) c= ( closed_inside_of_rectangle (( - 1),1,( - 1),1)) holds ( rng f) meets ( rng g)

 proof

 let p1,p2,p3,p4 be Point of ( TOP-REAL 2);

 set K = ( rectangle (( - 1),1,( - 1),1)), K0 = ( closed_inside_of_rectangle (( - 1),1,( - 1),1));

 assume

 A1: (p1,p2,p3,p4) are_in_this_order_on K;

 reconsider j = 1 as non negative Real;

 reconsider P = ( circle ( 0 , 0 ,j)) as non empty compact Subset of ( TOP-REAL 2);



 A2: P = { p6 where p6 be Point of ( TOP-REAL 2) : |.p6.| = 1 } by Th24;

 thus for f,g be Function of I[01] , ( TOP-REAL 2) st f is continuous one-to-one & g is continuous one-to-one & (f . 0 ) = p1 & (f . 1) = p3 & (g . 0 ) = p2 & (g . 1) = p4 & ( rng f) c= K0 & ( rng g) c= K0 holds ( rng f) meets ( rng g)

 proof

 let f,g be Function of I[01] , ( TOP-REAL 2);

 assume that

 A3: f is continuous one-to-one and

 A4: g is continuous one-to-one and

 A5: (f . 0 ) = p1 and

 A6: (f . 1) = p3 and

 A7: (g . 0 ) = p2 and

 A8: (g . 1) = p4 and

 A9: ( rng f) c= K0 and

 A10: ( rng g) c= K0;

 reconsider s = Sq_Circ as Function of ( TOP-REAL 2), ( TOP-REAL 2);



 A11: ( dom s) = the carrier of ( TOP-REAL 2) by FUNCT_2:def 1;

 reconsider f1 = (s * f) as Function of I[01] , ( TOP-REAL 2);

 reconsider g1 = (s * g) as Function of I[01] , ( TOP-REAL 2);

 s is being_homeomorphism by JGRAPH_3: 43;

 then

 A12: s is continuous by TOPS_2:def 5;



 A13: ( dom f) = [. 0 , 1.] by BORSUK_1: 40, FUNCT_2:def 1;

 then 0 in ( dom f) by XXREAL_1: 1;

 then

 A14: (f1 . 0 ) = ( Sq_Circ . p1) by A5, FUNCT_1: 13;

 1 in ( dom f) by A13, XXREAL_1: 1;

 then

 A15: (f1 . 1) = ( Sq_Circ . p3) by A6, FUNCT_1: 13;



 A16: ( dom g) = [. 0 , 1.] by BORSUK_1: 40, FUNCT_2:def 1;

 then 0 in ( dom g) by XXREAL_1: 1;

 then

 A17: (g1 . 0 ) = ( Sq_Circ . p2) by A7, FUNCT_1: 13;

 1 in ( dom g) by A16, XXREAL_1: 1;

 then

 A18: (g1 . 1) = ( Sq_Circ . p4) by A8, FUNCT_1: 13;

 defpred P[ Point of ( TOP-REAL 2)] means |.$1.| <= 1;

 { p8 where p8 be Point of ( TOP-REAL 2) : P[p8] } is Subset of ( TOP-REAL 2) from DOMAIN_1:sch 7;

 then

 reconsider C0 = { p8 where p8 be Point of ( TOP-REAL 2) : |.p8.| <= 1 } as Subset of ( TOP-REAL 2);



 A19: (s .: K0) = C0 by Th27;



 A20: ( rng f1) c= C0

 proof

 let y be object;

 assume y in ( rng f1);

 then

 consider x be object such that

 A21: x in ( dom f1) and

 A22: y = (f1 . x) by FUNCT_1:def 3;



 A23: x in ( dom f) by A21, FUNCT_1: 11;



 A24: (f . x) in ( dom s) by A21, FUNCT_1: 11;

 (f . x) in ( rng f) by A23, FUNCT_1: 3;

 then (s . (f . x)) in (s .: K0) by A9, A24, FUNCT_1:def 6;

 hence thesis by A19, A21, A22, FUNCT_1: 12;

 end;



 A25: ( rng g1) c= C0

 proof

 let y be object;

 assume y in ( rng g1);

 then

 consider x be object such that

 A26: x in ( dom g1) and

 A27: y = (g1 . x) by FUNCT_1:def 3;



 A28: x in ( dom g) by A26, FUNCT_1: 11;



 A29: (g . x) in ( dom s) by A26, FUNCT_1: 11;

 (g . x) in ( rng g) by A28, FUNCT_1: 3;

 then (s . (g . x)) in (s .: K0) by A10, A29, FUNCT_1:def 6;

 hence thesis by A19, A26, A27, FUNCT_1: 12;

 end;

 reconsider q1 = (s . p1), q2 = (s . p2), q3 = (s . p3), q4 = (s . p4) as Point of ( TOP-REAL 2);

 (q1,q2,q3,q4) are_in_this_order_on P by A1, Th78;

 then ( rng f1) meets ( rng g1) by A2, A3, A4, A12, A14, A15, A17, A18, A20, A25, Th18;

 then

 consider y be object such that

 A30: y in ( rng f1) and

 A31: y in ( rng g1) by XBOOLE_0: 3;

 consider x1 be object such that

 A32: x1 in ( dom f1) and

 A33: y = (f1 . x1) by A30, FUNCT_1:def 3;

 consider x2 be object such that

 A34: x2 in ( dom g1) and

 A35: y = (g1 . x2) by A31, FUNCT_1:def 3;

 ( dom f1) c= ( dom f) by RELAT_1: 25;

 then

 A36: (f . x1) in ( rng f) by A32, FUNCT_1: 3;

 ( dom g1) c= ( dom g) by RELAT_1: 25;

 then

 A37: (g . x2) in ( rng g) by A34, FUNCT_1: 3;

 y = ( Sq_Circ . (f . x1)) by A32, A33, FUNCT_1: 12;

 then

 A38: (( Sq_Circ " ) . y) = (f . x1) by A11, A36, FUNCT_1: 32;

 x1 in ( dom f) by A32, FUNCT_1: 11;

 then

 A39: (f . x1) in ( rng f) by FUNCT_1:def 3;

 y = ( Sq_Circ . (g . x2)) by A34, A35, FUNCT_1: 12;

 then

 A40: (( Sq_Circ " ) . y) = (g . x2) by A11, A37, FUNCT_1: 32;

 x2 in ( dom g) by A34, FUNCT_1: 11;

 then (g . x2) in ( rng g) by FUNCT_1:def 3;

 hence thesis by A38, A39, A40, XBOOLE_0: 3;

 end;

 end;