jgraph_7.miz



    begin

    theorem :: JGRAPH_7:1

    

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

    proof

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

      assume that

       A1: a < b and

       A2: (p `2 ) = d and

       A3: a <= (p `1 ) and

       A4: (p `1 ) <= b;

      reconsider w = (((p `1 ) - a) / (b - a)) as Real;

      

       A5: (b - a) > 0 by A1, XREAL_1: 50;

      ((p `1 ) - a) <= (b - a) by A4, XREAL_1: 9;

      then w <= ((b - a) / (b - a)) by A5, XREAL_1: 72;

      then

       A6: w <= 1 by A5, XCMPLX_1: 60;

      ((p `1 ) - a) >= 0 by A3, XREAL_1: 48;

      then

       A7: 0 <= w by A5, XREAL_1: 136;

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

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

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

      .= |[(a + (w * (b - a))), d]|

      .= |[(a + ((p `1 ) - a)), d]| by A5, XCMPLX_1: 87

      .= p by A2, EUCLID: 53;

      hence thesis by A7, A6;

    end;

    theorem :: JGRAPH_7:2

    

     Th2: for n be Element of NAT , P be Subset of ( TOP-REAL n), p1,p2 be Point of ( TOP-REAL n) st P is_an_arc_of (p1,p2) holds ex f be Function of I[01] , ( TOP-REAL n) st f is continuous one-to-one & ( rng f) = P & (f . 0 ) = p1 & (f . 1) = p2

    proof

      let n be Element of NAT , P be Subset of ( TOP-REAL n), p1,p2 be Point of ( TOP-REAL n);

      assume

       A1: P is_an_arc_of (p1,p2);

      then

      consider f2 be Function of I[01] , (( TOP-REAL n) | P) such that

       A2: f2 is being_homeomorphism and

       A3: (f2 . 0 ) = p1 and

       A4: (f2 . 1) = p2 by TOPREAL1:def 1;

      p1 in P by A1, TOPREAL1: 1;

      then

      consider g be Function of I[01] , ( TOP-REAL n) such that

       A5: f2 = g and

       A6: g is continuous and

       A7: g is one-to-one by A2, JORDAN7: 15;

      ( rng g) = ( [#] (( TOP-REAL n) | P)) by A2, A5, TOPS_2:def 5

      .= P by PRE_TOPC:def 5;

      hence thesis by A3, A4, A5, A6, A7;

    end;

    theorem :: JGRAPH_7:3

    

     Th3: for p1,p2 be Point of ( TOP-REAL 2), b,c,d be Real st (p1 `1 ) < b & (p1 `1 ) = (p2 `1 ) & c <= (p1 `2 ) & (p1 `2 ) < (p2 `2 ) & (p2 `2 ) <= d holds LE (p1,p2,( rectangle ((p1 `1 ),b,c,d)))

    proof

      let p1,p2 be Point of ( TOP-REAL 2), b,c,d be Real;

      set a = (p1 `1 );

      assume that

       A1: (p1 `1 ) < b and

       A2: (p1 `1 ) = (p2 `1 ) and

       A3: c <= (p1 `2 ) and

       A4: (p1 `2 ) < (p2 `2 ) and

       A5: (p2 `2 ) <= d;

      

       A6: (p1 `2 ) < d by A4, A5, XXREAL_0: 2;

      then

       A7: c < d by A3, XXREAL_0: 2;

      then

       A8: p1 in ( LSeg ( |[a, c]|, |[a, d]|)) by A3, A6, JGRAPH_6: 2;

      c <= (p2 `2 ) by A3, A4, XXREAL_0: 2;

      then p2 in ( LSeg ( |[a, c]|, |[a, d]|)) by A2, A5, A7, JGRAPH_6: 2;

      hence thesis by A1, A4, A7, A8, JGRAPH_6: 55;

    end;

    theorem :: JGRAPH_7:4

    

     Th4: for p1,p2 be Point of ( TOP-REAL 2), b,c be Real st (p1 `1 ) < b & c < (p2 `2 ) & c <= (p1 `2 ) & (p1 `2 ) <= (p2 `2 ) & (p1 `1 ) <= (p2 `1 ) & (p2 `1 ) <= b holds LE (p1,p2,( rectangle ((p1 `1 ),b,c,(p2 `2 ))))

    proof

      let p1,p2 be Point of ( TOP-REAL 2), b,c be Real;

      set a = (p1 `1 ), d = (p2 `2 );

      assume that

       A1: a < b and

       A2: c < d and

       A3: c <= (p1 `2 ) and

       A4: (p1 `2 ) <= d and

       A5: a <= (p2 `1 ) and

       A6: (p2 `1 ) <= b;

      

       A7: p1 in ( LSeg ( |[a, c]|, |[a, d]|)) by A2, A3, A4, JGRAPH_6: 2;

      p2 in ( LSeg ( |[a, d]|, |[b, d]|)) by A1, A5, A6, Th1;

      hence thesis by A1, A2, A7, JGRAPH_6: 59;

    end;

    theorem :: JGRAPH_7:5

    

     Th5: for p1,p2 be Point of ( TOP-REAL 2), c,d be Real st (p1 `1 ) < (p2 `1 ) & c < d & c <= (p1 `2 ) & (p1 `2 ) <= d & c <= (p2 `2 ) & (p2 `2 ) <= d holds LE (p1,p2,( rectangle ((p1 `1 ),(p2 `1 ),c,d)))

    proof

      let p1,p2 be Point of ( TOP-REAL 2), c,d be Real;

      set a = (p1 `1 ), b = (p2 `1 );

      assume that

       A1: a < b and

       A2: c < d and

       A3: c <= (p1 `2 ) and

       A4: (p1 `2 ) <= d and

       A5: c <= (p2 `2 ) and

       A6: (p2 `2 ) <= d;

      

       A7: p2 in ( LSeg ( |[b, c]|, |[b, d]|)) by A2, A5, A6, JGRAPH_6: 2;

      p1 in ( LSeg ( |[a, c]|, |[a, d]|)) by A2, A3, A4, JGRAPH_6: 2;

      hence thesis by A1, A2, A7, JGRAPH_6: 59;

    end;

    theorem :: JGRAPH_7:6

    

     Th6: for p1,p2 be Point of ( TOP-REAL 2), b,d be Real st (p2 `2 ) < d & (p2 `2 ) <= (p1 `2 ) & (p1 `2 ) <= d & (p1 `1 ) < (p2 `1 ) & (p2 `1 ) <= b holds LE (p1,p2,( rectangle ((p1 `1 ),b,(p2 `2 ),d)))

    proof

      let p1,p2 be Point of ( TOP-REAL 2), b,d be Real;

      set a = (p1 `1 ), c = (p2 `2 ), K = ( rectangle (a,b,c,d));

      assume that

       A1: c < d and

       A2: c <= (p1 `2 ) and

       A3: (p1 `2 ) <= d and

       A4: a < (p2 `1 ) and

       A5: (p2 `1 ) <= b;

      

       A6: p1 in ( LSeg ( |[a, c]|, |[a, d]|)) by A1, A2, A3, JGRAPH_6: 2;

      

       A7: a < b by A4, A5, XXREAL_0: 2;

      then ( W-min K) = |[a, c]| by A1, JGRAPH_6: 46;

      then

       A8: (( W-min K) `1 ) = a by EUCLID: 52;

      p2 in ( LSeg ( |[b, c]|, |[a, c]|)) by A4, A5, A7, Th1;

      hence thesis by A1, A4, A7, A6, A8, JGRAPH_6: 59;

    end;

    theorem :: JGRAPH_7:7

    

     Th7: for p1,p2 be Point of ( TOP-REAL 2), a,b,c,d be Real st a < b & c < d & (p1 `2 ) = d & (p2 `2 ) = d & a <= (p1 `1 ) & (p1 `1 ) < (p2 `1 ) & (p2 `1 ) <= b holds LE (p1,p2,( rectangle (a,b,c,d)))

    proof

      let p1,p2 be Point of ( TOP-REAL 2), a,b,c,d be Real;

      assume that

       A1: a < b and

       A2: c < d and

       A3: (p1 `2 ) = d and

       A4: (p2 `2 ) = d and

       A5: a <= (p1 `1 ) and

       A6: (p1 `1 ) < (p2 `1 ) and

       A7: (p2 `1 ) <= b;

      a <= (p2 `1 ) by A5, A6, XXREAL_0: 2;

      then

       A8: p2 in ( LSeg ( |[a, d]|, |[b, d]|)) by A1, A4, A7, Th1;

      (p1 `1 ) <= b by A6, A7, XXREAL_0: 2;

      then p1 in ( LSeg ( |[a, d]|, |[b, d]|)) by A1, A3, A5, Th1;

      hence thesis by A1, A2, A6, A8, JGRAPH_6: 60;

    end;

    theorem :: JGRAPH_7:8

    

     Th8: for p1,p2 be Point of ( TOP-REAL 2), a,b,c,d be Real st a < b & c < d & (p1 `2 ) = d & (p2 `1 ) = b & a <= (p1 `1 ) & (p1 `1 ) <= b & c <= (p2 `2 ) & (p2 `2 ) <= d holds LE (p1,p2,( rectangle (a,b,c,d)))

    proof

      let p1,p2 be Point of ( TOP-REAL 2), a,b,c,d be Real;

      assume that

       A1: a < b and

       A2: c < d and

       A3: (p1 `2 ) = d and

       A4: (p2 `1 ) = b and

       A5: a <= (p1 `1 ) and

       A6: (p1 `1 ) <= b and

       A7: c <= (p2 `2 ) and

       A8: (p2 `2 ) <= d;

      

       A9: p2 in ( LSeg ( |[b, d]|, |[b, c]|)) by A2, A4, A7, A8, JGRAPH_6: 2;

      p1 in ( LSeg ( |[a, d]|, |[b, d]|)) by A1, A3, A5, A6, Th1;

      hence thesis by A1, A2, A9, JGRAPH_6: 60;

    end;

    theorem :: JGRAPH_7:9

    

     Th9: for p1,p2 be Point of ( TOP-REAL 2), a,b,c,d be Real st a < b & c < d & (p1 `2 ) = d & (p2 `2 ) = c & a <= (p1 `1 ) & (p1 `1 ) <= b & a < (p2 `1 ) & (p2 `1 ) <= b holds LE (p1,p2,( rectangle (a,b,c,d)))

    proof

      let p1,p2 be Point of ( TOP-REAL 2), a,b,c,d be Real;

      set K = ( rectangle (a,b,c,d));

      assume that

       A1: a < b and

       A2: c < d and

       A3: (p1 `2 ) = d and

       A4: (p2 `2 ) = c and

       A5: a <= (p1 `1 ) and

       A6: (p1 `1 ) <= b and

       A7: a < (p2 `1 ) and

       A8: (p2 `1 ) <= b;

      

       A9: p2 in ( LSeg ( |[b, c]|, |[a, c]|)) by A1, A4, A7, A8, Th1;

      ( W-min K) = |[a, c]| by A1, A2, JGRAPH_6: 46;

      then

       A10: (( W-min K) `1 ) = a by EUCLID: 52;

      p1 in ( LSeg ( |[a, d]|, |[b, d]|)) by A1, A3, A5, A6, Th1;

      hence thesis by A1, A2, A7, A9, A10, JGRAPH_6: 60;

    end;

    theorem :: JGRAPH_7:10

    

     Th10: for p1,p2 be Point of ( TOP-REAL 2), a,b,c,d be Real st a < b & c < d & (p1 `1 ) = b & (p2 `1 ) = b & c <= (p2 `2 ) & (p2 `2 ) < (p1 `2 ) & (p1 `2 ) <= d holds LE (p1,p2,( rectangle (a,b,c,d)))

    proof

      let p1,p2 be Point of ( TOP-REAL 2), a,b,c,d be Real;

      assume that

       A1: a < b and

       A2: c < d and

       A3: (p1 `1 ) = b and

       A4: (p2 `1 ) = b and

       A5: c <= (p2 `2 ) and

       A6: (p2 `2 ) < (p1 `2 ) and

       A7: (p1 `2 ) <= d;

      d >= (p2 `2 ) by A6, A7, XXREAL_0: 2;

      then

       A8: p2 in ( LSeg ( |[b, d]|, |[b, c]|)) by A2, A4, A5, JGRAPH_6: 2;

      (p1 `2 ) >= c by A5, A6, XXREAL_0: 2;

      then p1 in ( LSeg ( |[b, d]|, |[b, c]|)) by A2, A3, A7, JGRAPH_6: 2;

      hence thesis by A1, A2, A6, A8, JGRAPH_6: 61;

    end;

    theorem :: JGRAPH_7:11

    

     Th11: for p1,p2 be Point of ( TOP-REAL 2), a,b,c,d be Real st a < b & c < d & (p1 `1 ) = b & (p2 `2 ) = c & c <= (p1 `2 ) & (p1 `2 ) <= d & a < (p2 `1 ) & (p2 `1 ) <= b holds LE (p1,p2,( rectangle (a,b,c,d)))

    proof

      let p1,p2 be Point of ( TOP-REAL 2), a,b,c,d be Real;

      set K = ( rectangle (a,b,c,d));

      assume that

       A1: a < b and

       A2: c < d and

       A3: (p1 `1 ) = b and

       A4: (p2 `2 ) = c and

       A5: c <= (p1 `2 ) and

       A6: (p1 `2 ) <= d and

       A7: a < (p2 `1 ) and

       A8: (p2 `1 ) <= b;

      

       A9: p1 in ( LSeg ( |[b, d]|, |[b, c]|)) by A2, A3, A5, A6, JGRAPH_6: 2;

      ( W-min K) = |[a, c]| by A1, A2, JGRAPH_6: 46;

      then

       A10: (( W-min K) `1 ) = a by EUCLID: 52;

      p2 in ( LSeg ( |[b, c]|, |[a, c]|)) by A1, A4, A7, A8, Th1;

      hence thesis by A1, A2, A7, A9, A10, JGRAPH_6: 61;

    end;

    theorem :: JGRAPH_7:12

    

     Th12: for p1,p2 be Point of ( TOP-REAL 2), a,b,c,d be Real st a < b & c < d & (p1 `2 ) = c & (p2 `2 ) = c & a < (p2 `1 ) & (p2 `1 ) < (p1 `1 ) & (p1 `1 ) <= b holds LE (p1,p2,( rectangle (a,b,c,d)))

    proof

      let p1,p2 be Point of ( TOP-REAL 2), a,b,c,d be Real;

      set K = ( rectangle (a,b,c,d));

      assume that

       A1: a < b and

       A2: c < d and

       A3: (p1 `2 ) = c and

       A4: (p2 `2 ) = c and

       A5: a < (p2 `1 ) and

       A6: (p2 `1 ) < (p1 `1 ) and

       A7: (p1 `1 ) <= b;

      b > (p2 `1 ) by A6, A7, XXREAL_0: 2;

      then

       A8: p2 in ( LSeg ( |[b, c]|, |[a, c]|)) by A1, A4, A5, Th1;

      ( W-min K) = |[a, c]| by A1, A2, JGRAPH_6: 46;

      then

       A9: (( W-min K) `1 ) = a by EUCLID: 52;

      (p1 `1 ) > a by A5, A6, XXREAL_0: 2;

      then p1 in ( LSeg ( |[b, c]|, |[a, c]|)) by A1, A3, A7, Th1;

      hence thesis by A1, A2, A5, A6, A8, A9, JGRAPH_6: 62;

    end;

    theorem :: JGRAPH_7:13

    

     Th13: for p1,p2 be Point of ( TOP-REAL 2), a,b,c,d be Real st a < b & c < d & (p1 `2 ) = d & (p2 `1 ) = b & a <= (p1 `1 ) & (p1 `1 ) <= b & c <= (p2 `2 ) & (p2 `2 ) <= d holds LE (p1,p2,( rectangle (a,b,c,d)))

    proof

      let p1,p2 be Point of ( TOP-REAL 2), a,b,c,d be Real;

      assume that

       A1: a < b and

       A2: c < d and

       A3: (p1 `2 ) = d and

       A4: (p2 `1 ) = b and

       A5: a <= (p1 `1 ) and

       A6: (p1 `1 ) <= b and

       A7: c <= (p2 `2 ) and

       A8: (p2 `2 ) <= d;

      

       A9: p2 in ( LSeg ( |[b, d]|, |[b, c]|)) by A2, A4, A7, A8, JGRAPH_6: 2;

      p1 in ( LSeg ( |[a, d]|, |[b, d]|)) by A1, A3, A5, A6, Th1;

      hence thesis by A1, A2, A9, JGRAPH_6: 60;

    end;

    theorem :: JGRAPH_7:14

    

     Th14: for p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real st a < b & (p1 `1 ) = a & (p2 `1 ) = a & (p3 `1 ) = a & (p4 `1 ) = a & c <= (p1 `2 ) & (p1 `2 ) < (p2 `2 ) & (p2 `2 ) < (p3 `2 ) & (p3 `2 ) < (p4 `2 ) & (p4 `2 ) <= d holds (p1,p2,p3,p4) are_in_this_order_on ( rectangle (a,b,c,d))

    proof

      let p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real;

      set K = ( rectangle (a,b,c,d));

      assume that

       A1: a < b and

       A2: (p1 `1 ) = a and

       A3: (p2 `1 ) = a and

       A4: (p3 `1 ) = a and

       A5: (p4 `1 ) = a and

       A6: c <= (p1 `2 ) and

       A7: (p1 `2 ) < (p2 `2 ) and

       A8: (p2 `2 ) < (p3 `2 ) and

       A9: (p3 `2 ) < (p4 `2 ) and

       A10: (p4 `2 ) <= d;

      

       A11: (p3 `2 ) < d by A9, A10, XXREAL_0: 2;

      (p2 `2 ) < (p4 `2 ) by A8, A9, XXREAL_0: 2;

      then

       A12: (p2 `2 ) < d by A10, XXREAL_0: 2;

      

       A13: c < (p2 `2 ) by A6, A7, XXREAL_0: 2;

      then c < (p3 `2 ) by A8, XXREAL_0: 2;

      then LE (p1,p2,K) & LE (p2,p3,K) & LE (p3,p4,K) or LE (p2,p3,K) & LE (p3,p4,K) & LE (p4,p1,K) or LE (p3,p4,K) & LE (p4,p1,K) & LE (p1,p2,K) or LE (p4,p1,K) & LE (p1,p2,K) & LE (p2,p3,K) by A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A12, A13, A11, Th3;

      hence thesis by JORDAN17:def 1;

    end;

    theorem :: JGRAPH_7:15

    

     Th15: for p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real st a < b & c < d & (p1 `1 ) = a & (p2 `1 ) = a & (p3 `1 ) = a & (p4 `2 ) = d & c <= (p1 `2 ) & (p1 `2 ) < (p2 `2 ) & (p2 `2 ) < (p3 `2 ) & (p3 `2 ) <= d & a <= (p4 `1 ) & (p4 `1 ) <= b holds (p1,p2,p3,p4) are_in_this_order_on ( rectangle (a,b,c,d))

    proof

      let p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real;

      set K = ( rectangle (a,b,c,d));

      assume that

       A1: a < b and

       A2: c < d and

       A3: (p1 `1 ) = a and

       A4: (p2 `1 ) = a and

       A5: (p3 `1 ) = a and

       A6: (p4 `2 ) = d and

       A7: c <= (p1 `2 ) and

       A8: (p1 `2 ) < (p2 `2 ) and

       A9: (p2 `2 ) < (p3 `2 ) and

       A10: (p3 `2 ) <= d and

       A11: a <= (p4 `1 ) and

       A12: (p4 `1 ) <= b;

      

       A13: (p2 `2 ) < d by A9, A10, XXREAL_0: 2;

      

       A14: c < (p2 `2 ) by A7, A8, XXREAL_0: 2;

      then c < (p3 `2 ) by A9, XXREAL_0: 2;

      then LE (p1,p2,K) & LE (p2,p3,K) & LE (p3,p4,K) or LE (p2,p3,K) & LE (p3,p4,K) & LE (p4,p1,K) or LE (p3,p4,K) & LE (p4,p1,K) & LE (p1,p2,K) or LE (p4,p1,K) & LE (p1,p2,K) & LE (p2,p3,K) by A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, A12, A13, A14, Th3, Th4;

      hence thesis by JORDAN17:def 1;

    end;

    theorem :: JGRAPH_7:16

    

     Th16: for p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real st a < b & c < d & (p1 `1 ) = a & (p2 `1 ) = a & (p3 `1 ) = a & (p4 `1 ) = b & c <= (p1 `2 ) & (p1 `2 ) < (p2 `2 ) & (p2 `2 ) < (p3 `2 ) & (p3 `2 ) <= d & c <= (p4 `2 ) & (p4 `2 ) <= d holds (p1,p2,p3,p4) are_in_this_order_on ( rectangle (a,b,c,d))

    proof

      let p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real;

      set K = ( rectangle (a,b,c,d));

      assume that

       A1: a < b and

       A2: c < d and

       A3: (p1 `1 ) = a and

       A4: (p2 `1 ) = a and

       A5: (p3 `1 ) = a and

       A6: (p4 `1 ) = b and

       A7: c <= (p1 `2 ) and

       A8: (p1 `2 ) < (p2 `2 ) and

       A9: (p2 `2 ) < (p3 `2 ) and

       A10: (p3 `2 ) <= d and

       A11: c <= (p4 `2 ) and

       A12: (p4 `2 ) <= d;

      

       A13: (p2 `2 ) <= d by A9, A10, XXREAL_0: 2;

      

       A14: c < (p2 `2 ) by A7, A8, XXREAL_0: 2;

      then c < (p3 `2 ) by A9, XXREAL_0: 2;

      then LE (p1,p2,K) & LE (p2,p3,K) & LE (p3,p4,K) or LE (p2,p3,K) & LE (p3,p4,K) & LE (p4,p1,K) or LE (p3,p4,K) & LE (p4,p1,K) & LE (p1,p2,K) or LE (p4,p1,K) & LE (p1,p2,K) & LE (p2,p3,K) by A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, A12, A13, A14, Th3, Th5;

      hence thesis by JORDAN17:def 1;

    end;

    theorem :: JGRAPH_7:17

    

     Th17: for p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real st a < b & c < d & (p1 `1 ) = a & (p2 `1 ) = a & (p3 `1 ) = a & (p4 `2 ) = c & c <= (p1 `2 ) & (p1 `2 ) < (p2 `2 ) & (p2 `2 ) < (p3 `2 ) & (p3 `2 ) <= d & a < (p4 `1 ) & (p4 `1 ) <= b holds (p1,p2,p3,p4) are_in_this_order_on ( rectangle (a,b,c,d))

    proof

      let p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real;

      set K = ( rectangle (a,b,c,d));

      assume that

       A1: a < b and

       A2: c < d and

       A3: (p1 `1 ) = a and

       A4: (p2 `1 ) = a and

       A5: (p3 `1 ) = a and

       A6: (p4 `2 ) = c and

       A7: c <= (p1 `2 ) and

       A8: (p1 `2 ) < (p2 `2 ) and

       A9: (p2 `2 ) < (p3 `2 ) and

       A10: (p3 `2 ) <= d and

       A11: a < (p4 `1 ) and

       A12: (p4 `1 ) <= b;

      

       A13: (p2 `2 ) < d by A9, A10, XXREAL_0: 2;

      

       A14: c < (p2 `2 ) by A7, A8, XXREAL_0: 2;

      then c < (p3 `2 ) by A9, XXREAL_0: 2;

      then LE (p1,p2,K) & LE (p2,p3,K) & LE (p3,p4,K) or LE (p2,p3,K) & LE (p3,p4,K) & LE (p4,p1,K) or LE (p3,p4,K) & LE (p4,p1,K) & LE (p1,p2,K) or LE (p4,p1,K) & LE (p1,p2,K) & LE (p2,p3,K) by A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, A12, A13, A14, Th3, Th6;

      hence thesis by JORDAN17:def 1;

    end;

    theorem :: JGRAPH_7:18

    

     Th18: for p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real st a < b & c < d & (p1 `1 ) = a & (p2 `1 ) = a & (p3 `2 ) = d & (p4 `2 ) = d & c <= (p1 `2 ) & (p1 `2 ) < (p2 `2 ) & (p2 `2 ) <= d & a <= (p3 `1 ) & (p3 `1 ) < (p4 `1 ) & (p4 `1 ) <= b holds (p1,p2,p3,p4) are_in_this_order_on ( rectangle (a,b,c,d))

    proof

      let p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real;

      set K = ( rectangle (a,b,c,d));

      assume that

       A1: a < b and

       A2: c < d and

       A3: (p1 `1 ) = a and

       A4: (p2 `1 ) = a and

       A5: (p3 `2 ) = d and

       A6: (p4 `2 ) = d and

       A7: c <= (p1 `2 ) and

       A8: (p1 `2 ) < (p2 `2 ) and

       A9: (p2 `2 ) <= d and

       A10: a <= (p3 `1 ) and

       A11: (p3 `1 ) < (p4 `1 ) and

       A12: (p4 `1 ) <= b;

      

       A13: (p3 `1 ) < b by A11, A12, XXREAL_0: 2;

      c < (p2 `2 ) by A7, A8, XXREAL_0: 2;

      then LE (p1,p2,K) & LE (p2,p3,K) & LE (p3,p4,K) or LE (p2,p3,K) & LE (p3,p4,K) & LE (p4,p1,K) or LE (p3,p4,K) & LE (p4,p1,K) & LE (p1,p2,K) or LE (p4,p1,K) & LE (p1,p2,K) & LE (p2,p3,K) by A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, A12, A13, Th3, Th4, Th7;

      hence thesis by JORDAN17:def 1;

    end;

    theorem :: JGRAPH_7:19

    

     Th19: for p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real st a < b & c < d & (p1 `1 ) = a & (p2 `1 ) = a & (p3 `2 ) = d & (p4 `1 ) = b & c <= (p1 `2 ) & (p1 `2 ) < (p2 `2 ) & (p2 `2 ) <= d & a <= (p3 `1 ) & (p3 `1 ) <= b & c <= (p4 `2 ) & (p4 `2 ) <= d holds (p1,p2,p3,p4) are_in_this_order_on ( rectangle (a,b,c,d))

    proof

      let p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real;

      set K = ( rectangle (a,b,c,d));

      assume that

       A1: a < b and

       A2: c < d and

       A3: (p1 `1 ) = a and

       A4: (p2 `1 ) = a and

       A5: (p3 `2 ) = d and

       A6: (p4 `1 ) = b and

       A7: c <= (p1 `2 ) and

       A8: (p1 `2 ) < (p2 `2 ) and

       A9: (p2 `2 ) <= d and

       A10: a <= (p3 `1 ) and

       A11: (p3 `1 ) <= b and

       A12: c <= (p4 `2 ) and

       A13: (p4 `2 ) <= d;

      c < (p2 `2 ) by A7, A8, XXREAL_0: 2;

      then LE (p1,p2,K) & LE (p2,p3,K) & LE (p3,p4,K) or LE (p2,p3,K) & LE (p3,p4,K) & LE (p4,p1,K) or LE (p3,p4,K) & LE (p4,p1,K) & LE (p1,p2,K) or LE (p4,p1,K) & LE (p1,p2,K) & LE (p2,p3,K) by A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, A12, A13, Th3, Th4, Th8;

      hence thesis by JORDAN17:def 1;

    end;

    theorem :: JGRAPH_7:20

    

     Th20: for p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real st a < b & c < d & (p1 `1 ) = a & (p2 `1 ) = a & (p3 `2 ) = d & (p4 `2 ) = c & c <= (p1 `2 ) & (p1 `2 ) < (p2 `2 ) & (p2 `2 ) <= d & a <= (p3 `1 ) & (p3 `1 ) <= b & a < (p4 `1 ) & (p4 `1 ) <= b holds (p1,p2,p3,p4) are_in_this_order_on ( rectangle (a,b,c,d))

    proof

      let p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real;

      set K = ( rectangle (a,b,c,d));

      assume that

       A1: a < b and

       A2: c < d and

       A3: (p1 `1 ) = a and

       A4: (p2 `1 ) = a and

       A5: (p3 `2 ) = d and

       A6: (p4 `2 ) = c and

       A7: c <= (p1 `2 ) and

       A8: (p1 `2 ) < (p2 `2 ) and

       A9: (p2 `2 ) <= d and

       A10: a <= (p3 `1 ) and

       A11: (p3 `1 ) <= b and

       A12: a < (p4 `1 ) and

       A13: (p4 `1 ) <= b;

      c < (p2 `2 ) by A7, A8, XXREAL_0: 2;

      then LE (p1,p2,K) & LE (p2,p3,K) & LE (p3,p4,K) or LE (p2,p3,K) & LE (p3,p4,K) & LE (p4,p1,K) or LE (p3,p4,K) & LE (p4,p1,K) & LE (p1,p2,K) or LE (p4,p1,K) & LE (p1,p2,K) & LE (p2,p3,K) by A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, A12, A13, Th3, Th4, Th9;

      hence thesis by JORDAN17:def 1;

    end;

    theorem :: JGRAPH_7:21

    

     Th21: for p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real st a < b & c < d & (p1 `1 ) = a & (p2 `1 ) = a & (p3 `1 ) = b & (p4 `1 ) = b & c <= (p1 `2 ) & (p1 `2 ) < (p2 `2 ) & (p2 `2 ) <= d & c <= (p4 `2 ) & (p4 `2 ) < (p3 `2 ) & (p3 `2 ) <= d holds (p1,p2,p3,p4) are_in_this_order_on ( rectangle (a,b,c,d))

    proof

      let p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real;

      set K = ( rectangle (a,b,c,d));

      assume that

       A1: a < b and

       A2: c < d and

       A3: (p1 `1 ) = a and

       A4: (p2 `1 ) = a and

       A5: (p3 `1 ) = b and

       A6: (p4 `1 ) = b and

       A7: c <= (p1 `2 ) and

       A8: (p1 `2 ) < (p2 `2 ) and

       A9: (p2 `2 ) <= d and

       A10: c <= (p4 `2 ) and

       A11: (p4 `2 ) < (p3 `2 ) and

       A12: (p3 `2 ) <= d;

      

       A13: (p3 `2 ) > c by A10, A11, XXREAL_0: 2;

      c < (p2 `2 ) by A7, A8, XXREAL_0: 2;

      then LE (p1,p2,K) & LE (p2,p3,K) & LE (p3,p4,K) or LE (p2,p3,K) & LE (p3,p4,K) & LE (p4,p1,K) or LE (p3,p4,K) & LE (p4,p1,K) & LE (p1,p2,K) or LE (p4,p1,K) & LE (p1,p2,K) & LE (p2,p3,K) by A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, A12, A13, Th3, Th5, Th10;

      hence thesis by JORDAN17:def 1;

    end;

    theorem :: JGRAPH_7:22

    

     Th22: for p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real st a < b & c < d & (p1 `1 ) = a & (p2 `1 ) = a & (p3 `1 ) = b & (p4 `2 ) = c & c <= (p1 `2 ) & (p1 `2 ) < (p2 `2 ) & (p2 `2 ) <= d & c <= (p3 `2 ) & (p3 `2 ) <= d & a < (p4 `1 ) & (p4 `1 ) <= b holds (p1,p2,p3,p4) are_in_this_order_on ( rectangle (a,b,c,d))

    proof

      let p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real;

      set K = ( rectangle (a,b,c,d));

      assume that

       A1: a < b and

       A2: c < d and

       A3: (p1 `1 ) = a and

       A4: (p2 `1 ) = a and

       A5: (p3 `1 ) = b and

       A6: (p4 `2 ) = c and

       A7: c <= (p1 `2 ) and

       A8: (p1 `2 ) < (p2 `2 ) and

       A9: (p2 `2 ) <= d and

       A10: c <= (p3 `2 ) and

       A11: (p3 `2 ) <= d and

       A12: a < (p4 `1 ) and

       A13: (p4 `1 ) <= b;

      c < (p2 `2 ) by A7, A8, XXREAL_0: 2;

      then LE (p1,p2,K) & LE (p2,p3,K) & LE (p3,p4,K) or LE (p2,p3,K) & LE (p3,p4,K) & LE (p4,p1,K) or LE (p3,p4,K) & LE (p4,p1,K) & LE (p1,p2,K) or LE (p4,p1,K) & LE (p1,p2,K) & LE (p2,p3,K) by A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, A12, A13, Th3, Th5, Th11;

      hence thesis by JORDAN17:def 1;

    end;

    theorem :: JGRAPH_7:23

    

     Th23: for p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real st a < b & c < d & (p1 `1 ) = a & (p2 `1 ) = a & (p3 `2 ) = c & (p4 `2 ) = c & c <= (p1 `2 ) & (p1 `2 ) < (p2 `2 ) & (p2 `2 ) <= d & a < (p4 `1 ) & (p4 `1 ) < (p3 `1 ) & (p3 `1 ) <= b holds (p1,p2,p3,p4) are_in_this_order_on ( rectangle (a,b,c,d))

    proof

      let p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real;

      set K = ( rectangle (a,b,c,d));

      assume that

       A1: a < b and

       A2: c < d and

       A3: (p1 `1 ) = a and

       A4: (p2 `1 ) = a and

       A5: (p3 `2 ) = c and

       A6: (p4 `2 ) = c and

       A7: c <= (p1 `2 ) and

       A8: (p1 `2 ) < (p2 `2 ) and

       A9: (p2 `2 ) <= d and

       A10: a < (p4 `1 ) and

       A11: (p4 `1 ) < (p3 `1 ) and

       A12: (p3 `1 ) <= b;

      

       A13: a < (p3 `1 ) by A10, A11, XXREAL_0: 2;

      c < (p2 `2 ) by A7, A8, XXREAL_0: 2;

      then LE (p1,p2,K) & LE (p2,p3,K) & LE (p3,p4,K) or LE (p2,p3,K) & LE (p3,p4,K) & LE (p4,p1,K) or LE (p3,p4,K) & LE (p4,p1,K) & LE (p1,p2,K) or LE (p4,p1,K) & LE (p1,p2,K) & LE (p2,p3,K) by A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, A12, A13, Th3, Th6, Th12;

      hence thesis by JORDAN17:def 1;

    end;

    theorem :: JGRAPH_7:24

    

     Th24: for p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real st a < b & c < d & (p1 `1 ) = a & (p2 `2 ) = d & (p3 `2 ) = d & (p4 `2 ) = d & c <= (p1 `2 ) & (p1 `2 ) <= d & a <= (p2 `1 ) & (p2 `1 ) < (p3 `1 ) & (p3 `1 ) < (p4 `1 ) & (p4 `1 ) <= b holds (p1,p2,p3,p4) are_in_this_order_on ( rectangle (a,b,c,d))

    proof

      let p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real;

      set K = ( rectangle (a,b,c,d));

      assume that

       A1: a < b and

       A2: c < d and

       A3: (p1 `1 ) = a and

       A4: (p2 `2 ) = d and

       A5: (p3 `2 ) = d and

       A6: (p4 `2 ) = d and

       A7: c <= (p1 `2 ) and

       A8: (p1 `2 ) <= d and

       A9: a <= (p2 `1 ) and

       A10: (p2 `1 ) < (p3 `1 ) and

       A11: (p3 `1 ) < (p4 `1 ) and

       A12: (p4 `1 ) <= b;

      

       A13: (p3 `1 ) < b by A11, A12, XXREAL_0: 2;

      (p2 `1 ) < (p4 `1 ) by A10, A11, XXREAL_0: 2;

      then

       A14: (p2 `1 ) < b by A12, XXREAL_0: 2;

      a < (p3 `1 ) by A9, A10, XXREAL_0: 2;

      then LE (p1,p2,K) & LE (p2,p3,K) & LE (p3,p4,K) or LE (p2,p3,K) & LE (p3,p4,K) & LE (p4,p1,K) or LE (p3,p4,K) & LE (p4,p1,K) & LE (p1,p2,K) or LE (p4,p1,K) & LE (p1,p2,K) & LE (p2,p3,K) by A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, A12, A14, A13, Th4, Th7;

      hence thesis by JORDAN17:def 1;

    end;

    theorem :: JGRAPH_7:25

    

     Th25: for p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real st a < b & c < d & (p1 `1 ) = a & (p2 `2 ) = d & (p3 `2 ) = d & (p4 `1 ) = b & c <= (p1 `2 ) & (p1 `2 ) <= d & a <= (p2 `1 ) & (p2 `1 ) < (p3 `1 ) & (p3 `1 ) <= b & c <= (p4 `2 ) & (p4 `2 ) <= d holds (p1,p2,p3,p4) are_in_this_order_on ( rectangle (a,b,c,d))

    proof

      let p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real;

      set K = ( rectangle (a,b,c,d));

      assume that

       A1: a < b and

       A2: c < d and

       A3: (p1 `1 ) = a and

       A4: (p2 `2 ) = d and

       A5: (p3 `2 ) = d and

       A6: (p4 `1 ) = b and

       A7: c <= (p1 `2 ) and

       A8: (p1 `2 ) <= d and

       A9: a <= (p2 `1 ) and

       A10: (p2 `1 ) < (p3 `1 ) and

       A11: (p3 `1 ) <= b and

       A12: c <= (p4 `2 ) and

       A13: (p4 `2 ) <= d;

      

       A14: a < (p3 `1 ) by A9, A10, XXREAL_0: 2;

      (p2 `1 ) < b by A10, A11, XXREAL_0: 2;

      then LE (p1,p2,K) & LE (p2,p3,K) & LE (p3,p4,K) or LE (p2,p3,K) & LE (p3,p4,K) & LE (p4,p1,K) or LE (p3,p4,K) & LE (p4,p1,K) & LE (p1,p2,K) or LE (p4,p1,K) & LE (p1,p2,K) & LE (p2,p3,K) by A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, A12, A13, A14, Th4, Th7, Th13;

      hence thesis by JORDAN17:def 1;

    end;

    theorem :: JGRAPH_7:26

    

     Th26: for p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real st a < b & c < d & (p1 `1 ) = a & (p2 `2 ) = d & (p3 `2 ) = d & (p4 `2 ) = c & c <= (p1 `2 ) & (p1 `2 ) <= d & a <= (p2 `1 ) & (p2 `1 ) < (p3 `1 ) & (p3 `1 ) <= b & a < (p4 `1 ) & (p4 `1 ) <= b holds (p1,p2,p3,p4) are_in_this_order_on ( rectangle (a,b,c,d))

    proof

      let p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real;

      set K = ( rectangle (a,b,c,d));

      assume that

       A1: a < b and

       A2: c < d and

       A3: (p1 `1 ) = a and

       A4: (p2 `2 ) = d and

       A5: (p3 `2 ) = d and

       A6: (p4 `2 ) = c and

       A7: c <= (p1 `2 ) and

       A8: (p1 `2 ) <= d and

       A9: a <= (p2 `1 ) and

       A10: (p2 `1 ) < (p3 `1 ) and

       A11: (p3 `1 ) <= b and

       A12: a < (p4 `1 ) and

       A13: (p4 `1 ) <= b;

      

       A14: a < (p3 `1 ) by A9, A10, XXREAL_0: 2;

      (p2 `1 ) < b by A10, A11, XXREAL_0: 2;

      then LE (p1,p2,K) & LE (p2,p3,K) & LE (p3,p4,K) or LE (p2,p3,K) & LE (p3,p4,K) & LE (p4,p1,K) or LE (p3,p4,K) & LE (p4,p1,K) & LE (p1,p2,K) or LE (p4,p1,K) & LE (p1,p2,K) & LE (p2,p3,K) by A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, A12, A13, A14, Th4, Th7, Th9;

      hence thesis by JORDAN17:def 1;

    end;

    theorem :: JGRAPH_7:27

    

     Th27: for p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real st a < b & c < d & (p1 `1 ) = a & (p2 `2 ) = d & (p3 `1 ) = b & (p4 `1 ) = b & c <= (p1 `2 ) & (p1 `2 ) <= d & a <= (p2 `1 ) & (p2 `1 ) <= b & c <= (p4 `2 ) & (p4 `2 ) < (p3 `2 ) & (p3 `2 ) <= d holds (p1,p2,p3,p4) are_in_this_order_on ( rectangle (a,b,c,d))

    proof

      let p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real;

      set K = ( rectangle (a,b,c,d));

      assume that

       A1: a < b and

       A2: c < d and

       A3: (p1 `1 ) = a and

       A4: (p2 `2 ) = d and

       A5: (p3 `1 ) = b and

       A6: (p4 `1 ) = b and

       A7: c <= (p1 `2 ) and

       A8: (p1 `2 ) <= d and

       A9: a <= (p2 `1 ) and

       A10: (p2 `1 ) <= b and

       A11: c <= (p4 `2 ) and

       A12: (p4 `2 ) < (p3 `2 ) and

       A13: (p3 `2 ) <= d;

      c < (p3 `2 ) by A11, A12, XXREAL_0: 2;

      then LE (p1,p2,K) & LE (p2,p3,K) & LE (p3,p4,K) or LE (p2,p3,K) & LE (p3,p4,K) & LE (p4,p1,K) or LE (p3,p4,K) & LE (p4,p1,K) & LE (p1,p2,K) or LE (p4,p1,K) & LE (p1,p2,K) & LE (p2,p3,K) by A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, A12, A13, Th4, Th8, Th10;

      hence thesis by JORDAN17:def 1;

    end;

    theorem :: JGRAPH_7:28

    

     Th28: for p1,p2,p3,p4 be Point of ( TOP-REAL 2) st (p1 `1 ) <> (p3 `1 ) & (p4 `2 ) <> (p2 `2 ) & (p4 `2 ) <= (p1 `2 ) & (p1 `2 ) <= (p2 `2 ) & (p1 `1 ) <= (p2 `1 ) & (p2 `1 ) <= (p3 `1 ) & (p4 `2 ) <= (p3 `2 ) & (p3 `2 ) <= (p2 `2 ) & (p1 `1 ) < (p4 `1 ) & (p4 `1 ) <= (p3 `1 ) holds (p1,p2,p3,p4) are_in_this_order_on ( rectangle ((p1 `1 ),(p3 `1 ),(p4 `2 ),(p2 `2 )))

    proof

      let p1,p2,p3,p4 be Point of ( TOP-REAL 2);

      set K = ( rectangle ((p1 `1 ),(p3 `1 ),(p4 `2 ),(p2 `2 )));

      assume that

       A1: (p1 `1 ) <> (p3 `1 ) and

       A2: (p4 `2 ) <> (p2 `2 ) and

       A3: (p4 `2 ) <= (p1 `2 ) and

       A4: (p1 `2 ) <= (p2 `2 ) and

       A5: (p1 `1 ) <= (p2 `1 ) and

       A6: (p2 `1 ) <= (p3 `1 ) and

       A7: (p4 `2 ) <= (p3 `2 ) and

       A8: (p3 `2 ) <= (p2 `2 ) and

       A9: (p1 `1 ) < (p4 `1 ) and

       A10: (p4 `1 ) <= (p3 `1 );

      (p4 `2 ) <= (p2 `2 ) by A3, A4, XXREAL_0: 2;

      then

       A11: (p4 `2 ) < (p2 `2 ) by A2, XXREAL_0: 1;

      (p1 `1 ) <= (p3 `1 ) by A5, A6, XXREAL_0: 2;

      then (p1 `1 ) < (p3 `1 ) by A1, XXREAL_0: 1;

      then LE (p1,p2,K) & LE (p2,p3,K) & LE (p3,p4,K) or LE (p2,p3,K) & LE (p3,p4,K) & LE (p4,p1,K) or LE (p3,p4,K) & LE (p4,p1,K) & LE (p1,p2,K) or LE (p4,p1,K) & LE (p1,p2,K) & LE (p2,p3,K) by A3, A4, A5, A6, A7, A8, A9, A10, A11, Th4, Th8, Th11;

      hence thesis by JORDAN17:def 1;

    end;

    theorem :: JGRAPH_7:29

    

     Th29: for p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real st a < b & c < d & (p1 `1 ) = a & (p2 `2 ) = d & (p3 `2 ) = c & (p4 `2 ) = c & c <= (p1 `2 ) & (p1 `2 ) <= d & a <= (p2 `1 ) & (p2 `1 ) <= b & a < (p4 `1 ) & (p4 `1 ) < (p3 `1 ) & (p3 `1 ) <= b holds (p1,p2,p3,p4) are_in_this_order_on ( rectangle (a,b,c,d))

    proof

      let p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real;

      set K = ( rectangle (a,b,c,d));

      assume that

       A1: a < b and

       A2: c < d and

       A3: (p1 `1 ) = a and

       A4: (p2 `2 ) = d and

       A5: (p3 `2 ) = c and

       A6: (p4 `2 ) = c and

       A7: c <= (p1 `2 ) and

       A8: (p1 `2 ) <= d and

       A9: a <= (p2 `1 ) and

       A10: (p2 `1 ) <= b and

       A11: a < (p4 `1 ) and

       A12: (p4 `1 ) < (p3 `1 ) and

       A13: (p3 `1 ) <= b;

      a < (p3 `1 ) by A11, A12, XXREAL_0: 2;

      then LE (p1,p2,K) & LE (p2,p3,K) & LE (p3,p4,K) or LE (p2,p3,K) & LE (p3,p4,K) & LE (p4,p1,K) or LE (p3,p4,K) & LE (p4,p1,K) & LE (p1,p2,K) or LE (p4,p1,K) & LE (p1,p2,K) & LE (p2,p3,K) by A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, A12, A13, Th4, Th9, Th12;

      hence thesis by JORDAN17:def 1;

    end;

    theorem :: JGRAPH_7:30

    

     Th30: for p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real st a < b & c < d & (p1 `1 ) = a & (p2 `1 ) = b & (p3 `1 ) = b & (p4 `1 ) = b & c <= (p1 `2 ) & (p1 `2 ) <= d & d >= (p2 `2 ) & (p2 `2 ) > (p3 `2 ) & (p3 `2 ) > (p4 `2 ) & (p4 `2 ) >= c holds (p1,p2,p3,p4) are_in_this_order_on ( rectangle (a,b,c,d))

    proof

      let p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real;

      set K = ( rectangle (a,b,c,d));

      assume that

       A1: a < b and

       A2: c < d and

       A3: (p1 `1 ) = a and

       A4: (p2 `1 ) = b and

       A5: (p3 `1 ) = b and

       A6: (p4 `1 ) = b and

       A7: c <= (p1 `2 ) and

       A8: (p1 `2 ) <= d and

       A9: d >= (p2 `2 ) and

       A10: (p2 `2 ) > (p3 `2 ) and

       A11: (p3 `2 ) > (p4 `2 ) and

       A12: (p4 `2 ) >= c;

      

       A13: (p3 `2 ) < d by A9, A10, XXREAL_0: 2;

      (p2 `2 ) > (p4 `2 ) by A10, A11, XXREAL_0: 2;

      then

       A14: (p2 `2 ) > c by A12, XXREAL_0: 2;

      c < (p3 `2 ) by A11, A12, XXREAL_0: 2;

      then LE (p1,p2,K) & LE (p2,p3,K) & LE (p3,p4,K) or LE (p2,p3,K) & LE (p3,p4,K) & LE (p4,p1,K) or LE (p3,p4,K) & LE (p4,p1,K) & LE (p1,p2,K) or LE (p4,p1,K) & LE (p1,p2,K) & LE (p2,p3,K) by A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, A12, A14, A13, Th5, Th10;

      hence thesis by JORDAN17:def 1;

    end;

    theorem :: JGRAPH_7:31

    

     Th31: for p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real st a < b & c < d & (p1 `1 ) = a & (p2 `1 ) = b & (p3 `1 ) = b & (p4 `2 ) = c & c <= (p1 `2 ) & (p1 `2 ) <= d & d >= (p2 `2 ) & (p2 `2 ) > (p3 `2 ) & (p3 `2 ) >= c & a < (p4 `1 ) & (p4 `1 ) <= b holds (p1,p2,p3,p4) are_in_this_order_on ( rectangle (a,b,c,d))

    proof

      let p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real;

      set K = ( rectangle (a,b,c,d));

      assume that

       A1: a < b and

       A2: c < d and

       A3: (p1 `1 ) = a and

       A4: (p2 `1 ) = b and

       A5: (p3 `1 ) = b and

       A6: (p4 `2 ) = c and

       A7: c <= (p1 `2 ) and

       A8: (p1 `2 ) <= d and

       A9: d >= (p2 `2 ) and

       A10: (p2 `2 ) > (p3 `2 ) and

       A11: (p3 `2 ) >= c and

       A12: a < (p4 `1 ) and

       A13: (p4 `1 ) <= b;

      

       A14: (p3 `2 ) < d by A9, A10, XXREAL_0: 2;

      (p2 `2 ) > c by A10, A11, XXREAL_0: 2;

      then LE (p1,p2,K) & LE (p2,p3,K) & LE (p3,p4,K) or LE (p2,p3,K) & LE (p3,p4,K) & LE (p4,p1,K) or LE (p3,p4,K) & LE (p4,p1,K) & LE (p1,p2,K) or LE (p4,p1,K) & LE (p1,p2,K) & LE (p2,p3,K) by A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, A12, A13, A14, Th5, Th10, Th11;

      hence thesis by JORDAN17:def 1;

    end;

    theorem :: JGRAPH_7:32

    

     Th32: for p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real st a < b & c < d & (p1 `1 ) = a & (p2 `1 ) = b & (p3 `2 ) = c & (p4 `2 ) = c & c <= (p1 `2 ) & (p1 `2 ) <= d & c <= (p2 `2 ) & (p2 `2 ) <= d & a < (p4 `1 ) & (p4 `1 ) < (p3 `1 ) & (p3 `1 ) <= b holds (p1,p2,p3,p4) are_in_this_order_on ( rectangle (a,b,c,d))

    proof

      let p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real;

      set K = ( rectangle (a,b,c,d));

      assume that

       A1: a < b and

       A2: c < d and

       A3: (p1 `1 ) = a and

       A4: (p2 `1 ) = b and

       A5: (p3 `2 ) = c and

       A6: (p4 `2 ) = c and

       A7: c <= (p1 `2 ) and

       A8: (p1 `2 ) <= d and

       A9: c <= (p2 `2 ) and

       A10: (p2 `2 ) <= d and

       A11: a < (p4 `1 ) and

       A12: (p4 `1 ) < (p3 `1 ) and

       A13: (p3 `1 ) <= b;

      (p3 `1 ) > a by A11, A12, XXREAL_0: 2;

      then LE (p1,p2,K) & LE (p2,p3,K) & LE (p3,p4,K) or LE (p2,p3,K) & LE (p3,p4,K) & LE (p4,p1,K) or LE (p3,p4,K) & LE (p4,p1,K) & LE (p1,p2,K) or LE (p4,p1,K) & LE (p1,p2,K) & LE (p2,p3,K) by A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, A12, A13, Th5, Th11, Th12;

      hence thesis by JORDAN17:def 1;

    end;

    theorem :: JGRAPH_7:33

    

     Th33: for p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real st a < b & c < d & (p1 `1 ) = a & (p2 `2 ) = c & (p3 `2 ) = c & (p4 `2 ) = c & c <= (p1 `2 ) & (p1 `2 ) <= d & a < (p4 `1 ) & (p4 `1 ) < (p3 `1 ) & (p3 `1 ) < (p2 `1 ) & (p2 `1 ) <= b holds (p1,p2,p3,p4) are_in_this_order_on ( rectangle (a,b,c,d))

    proof

      let p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real;

      set K = ( rectangle (a,b,c,d));

      assume that

       A1: a < b and

       A2: c < d and

       A3: (p1 `1 ) = a and

       A4: (p2 `2 ) = c and

       A5: (p3 `2 ) = c and

       A6: (p4 `2 ) = c and

       A7: c <= (p1 `2 ) and

       A8: (p1 `2 ) <= d and

       A9: a < (p4 `1 ) and

       A10: (p4 `1 ) < (p3 `1 ) and

       A11: (p3 `1 ) < (p2 `1 ) and

       A12: (p2 `1 ) <= b;

      

       A13: (p3 `1 ) < b by A11, A12, XXREAL_0: 2;

      (p2 `1 ) > (p4 `1 ) by A10, A11, XXREAL_0: 2;

      then

       A14: (p2 `1 ) > a by A9, XXREAL_0: 2;

      a < (p3 `1 ) by A9, A10, XXREAL_0: 2;

      then LE (p1,p2,K) & LE (p2,p3,K) & LE (p3,p4,K) or LE (p2,p3,K) & LE (p3,p4,K) & LE (p4,p1,K) or LE (p3,p4,K) & LE (p4,p1,K) & LE (p1,p2,K) or LE (p4,p1,K) & LE (p1,p2,K) & LE (p2,p3,K) by A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, A12, A14, A13, Th6, Th12;

      hence thesis by JORDAN17:def 1;

    end;

    theorem :: JGRAPH_7:34

    

     Th34: for p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real st a < b & c < d & (p1 `2 ) = d & (p2 `2 ) = d & (p3 `2 ) = d & (p4 `2 ) = d & a <= (p1 `1 ) & (p1 `1 ) < (p2 `1 ) & (p2 `1 ) < (p3 `1 ) & (p3 `1 ) < (p4 `1 ) & (p4 `1 ) <= b holds (p1,p2,p3,p4) are_in_this_order_on ( rectangle (a,b,c,d))

    proof

      let p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real;

      set K = ( rectangle (a,b,c,d));

      assume that

       A1: a < b and

       A2: c < d and

       A3: (p1 `2 ) = d and

       A4: (p2 `2 ) = d and

       A5: (p3 `2 ) = d and

       A6: (p4 `2 ) = d and

       A7: a <= (p1 `1 ) and

       A8: (p1 `1 ) < (p2 `1 ) and

       A9: (p2 `1 ) < (p3 `1 ) and

       A10: (p3 `1 ) < (p4 `1 ) and

       A11: (p4 `1 ) <= b;

      

       A12: (p3 `1 ) < b by A10, A11, XXREAL_0: 2;

      (p2 `1 ) < (p4 `1 ) by A9, A10, XXREAL_0: 2;

      then

       A13: (p2 `1 ) < b by A11, XXREAL_0: 2;

      

       A14: a < (p2 `1 ) by A7, A8, XXREAL_0: 2;

      then a < (p3 `1 ) by A9, XXREAL_0: 2;

      then LE (p1,p2,K) & LE (p2,p3,K) & LE (p3,p4,K) or LE (p2,p3,K) & LE (p3,p4,K) & LE (p4,p1,K) or LE (p3,p4,K) & LE (p4,p1,K) & LE (p1,p2,K) or LE (p4,p1,K) & LE (p1,p2,K) & LE (p2,p3,K) by A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, A13, A14, A12, Th7;

      hence thesis by JORDAN17:def 1;

    end;

    theorem :: JGRAPH_7:35

    

     Th35: for p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real st a < b & c < d & (p1 `2 ) = d & (p2 `2 ) = d & (p3 `2 ) = d & (p4 `1 ) = b & a <= (p1 `1 ) & (p1 `1 ) < (p2 `1 ) & (p2 `1 ) < (p3 `1 ) & (p3 `1 ) <= b & c <= (p4 `2 ) & (p4 `2 ) <= d holds (p1,p2,p3,p4) are_in_this_order_on ( rectangle (a,b,c,d))

    proof

      let p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real;

      set K = ( rectangle (a,b,c,d));

      assume that

       A1: a < b and

       A2: c < d and

       A3: (p1 `2 ) = d and

       A4: (p2 `2 ) = d and

       A5: (p3 `2 ) = d and

       A6: (p4 `1 ) = b and

       A7: a <= (p1 `1 ) and

       A8: (p1 `1 ) < (p2 `1 ) and

       A9: (p2 `1 ) < (p3 `1 ) and

       A10: (p3 `1 ) <= b and

       A11: c <= (p4 `2 ) and

       A12: (p4 `2 ) <= d;

      

       A13: (p2 `1 ) < b by A9, A10, XXREAL_0: 2;

      

       A14: a < (p2 `1 ) by A7, A8, XXREAL_0: 2;

      then a < (p3 `1 ) by A9, XXREAL_0: 2;

      then LE (p1,p2,K) & LE (p2,p3,K) & LE (p3,p4,K) or LE (p2,p3,K) & LE (p3,p4,K) & LE (p4,p1,K) or LE (p3,p4,K) & LE (p4,p1,K) & LE (p1,p2,K) or LE (p4,p1,K) & LE (p1,p2,K) & LE (p2,p3,K) by A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, A12, A13, A14, Th7, Th8;

      hence thesis by JORDAN17:def 1;

    end;

    theorem :: JGRAPH_7:36

    

     Th36: for p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real st a < b & c < d & (p1 `2 ) = d & (p2 `2 ) = d & (p3 `2 ) = d & (p4 `2 ) = c & a <= (p1 `1 ) & (p1 `1 ) < (p2 `1 ) & (p2 `1 ) < (p3 `1 ) & (p3 `1 ) <= b & a < (p4 `1 ) & (p4 `1 ) <= b holds (p1,p2,p3,p4) are_in_this_order_on ( rectangle (a,b,c,d))

    proof

      let p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real;

      set K = ( rectangle (a,b,c,d));

      assume that

       A1: a < b and

       A2: c < d and

       A3: (p1 `2 ) = d and

       A4: (p2 `2 ) = d and

       A5: (p3 `2 ) = d and

       A6: (p4 `2 ) = c and

       A7: a <= (p1 `1 ) and

       A8: (p1 `1 ) < (p2 `1 ) and

       A9: (p2 `1 ) < (p3 `1 ) and

       A10: (p3 `1 ) <= b and

       A11: a < (p4 `1 ) and

       A12: (p4 `1 ) <= b;

      

       A13: a < (p2 `1 ) by A7, A8, XXREAL_0: 2;

      (p3 `1 ) > (p1 `1 ) by A8, A9, XXREAL_0: 2;

      then

       A14: (p3 `1 ) > a by A7, XXREAL_0: 2;

      (p2 `1 ) <= b by A9, A10, XXREAL_0: 2;

      then LE (p1,p2,K) & LE (p2,p3,K) & LE (p3,p4,K) or LE (p2,p3,K) & LE (p3,p4,K) & LE (p4,p1,K) or LE (p3,p4,K) & LE (p4,p1,K) & LE (p1,p2,K) or LE (p4,p1,K) & LE (p1,p2,K) & LE (p2,p3,K) by A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, A12, A13, A14, Th7, Th9;

      hence thesis by JORDAN17:def 1;

    end;

    theorem :: JGRAPH_7:37

    

     Th37: for p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real st a < b & c < d & (p1 `2 ) = d & (p2 `2 ) = d & (p3 `1 ) = b & (p4 `1 ) = b & a <= (p1 `1 ) & (p1 `1 ) < (p2 `1 ) & (p2 `1 ) <= b & c <= (p4 `2 ) & (p4 `2 ) < (p3 `2 ) & (p3 `2 ) <= d holds (p1,p2,p3,p4) are_in_this_order_on ( rectangle (a,b,c,d))

    proof

      let p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real;

      set K = ( rectangle (a,b,c,d));

      assume that

       A1: a < b and

       A2: c < d and

       A3: (p1 `2 ) = d and

       A4: (p2 `2 ) = d and

       A5: (p3 `1 ) = b and

       A6: (p4 `1 ) = b and

       A7: a <= (p1 `1 ) and

       A8: (p1 `1 ) < (p2 `1 ) and

       A9: (p2 `1 ) <= b and

       A10: c <= (p4 `2 ) and

       A11: (p4 `2 ) < (p3 `2 ) and

       A12: (p3 `2 ) <= d;

      

       A13: (p3 `2 ) > c by A10, A11, XXREAL_0: 2;

      a < (p2 `1 ) by A7, A8, XXREAL_0: 2;

      then LE (p1,p2,K) & LE (p2,p3,K) & LE (p3,p4,K) or LE (p2,p3,K) & LE (p3,p4,K) & LE (p4,p1,K) or LE (p3,p4,K) & LE (p4,p1,K) & LE (p1,p2,K) or LE (p4,p1,K) & LE (p1,p2,K) & LE (p2,p3,K) by A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, A12, A13, Th7, Th8, Th10;

      hence thesis by JORDAN17:def 1;

    end;

    theorem :: JGRAPH_7:38

    

     Th38: for p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real st a < b & c < d & (p1 `2 ) = d & (p2 `2 ) = d & (p3 `1 ) = b & (p4 `2 ) = c & a <= (p1 `1 ) & (p1 `1 ) < (p2 `1 ) & (p2 `1 ) <= b & c <= (p3 `2 ) & (p3 `2 ) <= d & a < (p4 `1 ) & (p4 `1 ) <= b holds (p1,p2,p3,p4) are_in_this_order_on ( rectangle (a,b,c,d))

    proof

      let p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real;

      set K = ( rectangle (a,b,c,d));

      assume that

       A1: a < b and

       A2: c < d and

       A3: (p1 `2 ) = d and

       A4: (p2 `2 ) = d and

       A5: (p3 `1 ) = b and

       A6: (p4 `2 ) = c and

       A7: a <= (p1 `1 ) and

       A8: (p1 `1 ) < (p2 `1 ) and

       A9: (p2 `1 ) <= b and

       A10: c <= (p3 `2 ) and

       A11: (p3 `2 ) <= d and

       A12: a < (p4 `1 ) and

       A13: (p4 `1 ) <= b;

      a < (p2 `1 ) by A7, A8, XXREAL_0: 2;

      then LE (p1,p2,K) & LE (p2,p3,K) & LE (p3,p4,K) or LE (p2,p3,K) & LE (p3,p4,K) & LE (p4,p1,K) or LE (p3,p4,K) & LE (p4,p1,K) & LE (p1,p2,K) or LE (p4,p1,K) & LE (p1,p2,K) & LE (p2,p3,K) by A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, A12, A13, Th7, Th8, Th11;

      hence thesis by JORDAN17:def 1;

    end;

    theorem :: JGRAPH_7:39

    

     Th39: for p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real st a < b & c < d & (p1 `2 ) = d & (p2 `2 ) = d & (p3 `2 ) = c & (p4 `2 ) = c & a <= (p1 `1 ) & (p1 `1 ) < (p2 `1 ) & (p2 `1 ) <= b & a < (p4 `1 ) & (p4 `1 ) < (p3 `1 ) & (p3 `1 ) <= b holds (p1,p2,p3,p4) are_in_this_order_on ( rectangle (a,b,c,d))

    proof

      let p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real;

      set K = ( rectangle (a,b,c,d));

      assume that

       A1: a < b and

       A2: c < d and

       A3: (p1 `2 ) = d and

       A4: (p2 `2 ) = d and

       A5: (p3 `2 ) = c and

       A6: (p4 `2 ) = c and

       A7: a <= (p1 `1 ) and

       A8: (p1 `1 ) < (p2 `1 ) and

       A9: (p2 `1 ) <= b and

       A10: a < (p4 `1 ) and

       A11: (p4 `1 ) < (p3 `1 ) and

       A12: (p3 `1 ) <= b;

      

       A13: (p3 `1 ) > a by A10, A11, XXREAL_0: 2;

      a < (p2 `1 ) by A7, A8, XXREAL_0: 2;

      then LE (p1,p2,K) & LE (p2,p3,K) & LE (p3,p4,K) or LE (p2,p3,K) & LE (p3,p4,K) & LE (p4,p1,K) or LE (p3,p4,K) & LE (p4,p1,K) & LE (p1,p2,K) or LE (p4,p1,K) & LE (p1,p2,K) & LE (p2,p3,K) by A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, A12, A13, Th7, Th9, Th12;

      hence thesis by JORDAN17:def 1;

    end;

    theorem :: JGRAPH_7:40

    

     Th40: for p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real st a < b & c < d & (p1 `2 ) = d & (p2 `1 ) = b & (p3 `1 ) = b & (p4 `1 ) = b & a <= (p1 `1 ) & (p1 `1 ) <= b & d >= (p2 `2 ) & (p2 `2 ) > (p3 `2 ) & (p3 `2 ) > (p4 `2 ) & (p4 `2 ) >= c holds (p1,p2,p3,p4) are_in_this_order_on ( rectangle (a,b,c,d))

    proof

      let p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real;

      set K = ( rectangle (a,b,c,d));

      assume that

       A1: a < b and

       A2: c < d and

       A3: (p1 `2 ) = d and

       A4: (p2 `1 ) = b and

       A5: (p3 `1 ) = b and

       A6: (p4 `1 ) = b and

       A7: a <= (p1 `1 ) and

       A8: (p1 `1 ) <= b and

       A9: d >= (p2 `2 ) and

       A10: (p2 `2 ) > (p3 `2 ) and

       A11: (p3 `2 ) > (p4 `2 ) and

       A12: (p4 `2 ) >= c;

      

       A13: (p3 `2 ) > c by A11, A12, XXREAL_0: 2;

      (p2 `2 ) > (p4 `2 ) by A10, A11, XXREAL_0: 2;

      then

       A14: (p2 `2 ) > c by A12, XXREAL_0: 2;

      d > (p3 `2 ) by A9, A10, XXREAL_0: 2;

      then LE (p1,p2,K) & LE (p2,p3,K) & LE (p3,p4,K) or LE (p2,p3,K) & LE (p3,p4,K) & LE (p4,p1,K) or LE (p3,p4,K) & LE (p4,p1,K) & LE (p1,p2,K) or LE (p4,p1,K) & LE (p1,p2,K) & LE (p2,p3,K) by A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, A12, A14, A13, Th8, Th10;

      hence thesis by JORDAN17:def 1;

    end;

    theorem :: JGRAPH_7:41

    

     Th41: for p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real st a < b & c < d & (p1 `2 ) = d & (p2 `1 ) = b & (p3 `1 ) = b & (p4 `2 ) = c & a <= (p1 `1 ) & (p1 `1 ) <= b & d >= (p2 `2 ) & (p2 `2 ) > (p3 `2 ) & (p3 `2 ) >= c & a < (p4 `1 ) & (p4 `1 ) <= b holds (p1,p2,p3,p4) are_in_this_order_on ( rectangle (a,b,c,d))

    proof

      let p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real;

      set K = ( rectangle (a,b,c,d));

      assume that

       A1: a < b and

       A2: c < d and

       A3: (p1 `2 ) = d and

       A4: (p2 `1 ) = b and

       A5: (p3 `1 ) = b and

       A6: (p4 `2 ) = c and

       A7: a <= (p1 `1 ) and

       A8: (p1 `1 ) <= b and

       A9: d >= (p2 `2 ) and

       A10: (p2 `2 ) > (p3 `2 ) and

       A11: (p3 `2 ) >= c and

       A12: a < (p4 `1 ) and

       A13: (p4 `1 ) <= b;

      

       A14: d > (p3 `2 ) by A9, A10, XXREAL_0: 2;

      (p2 `2 ) > c by A10, A11, XXREAL_0: 2;

      then LE (p1,p2,K) & LE (p2,p3,K) & LE (p3,p4,K) or LE (p2,p3,K) & LE (p3,p4,K) & LE (p4,p1,K) or LE (p3,p4,K) & LE (p4,p1,K) & LE (p1,p2,K) or LE (p4,p1,K) & LE (p1,p2,K) & LE (p2,p3,K) by A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, A12, A13, A14, Th8, Th10, Th11;

      hence thesis by JORDAN17:def 1;

    end;

    theorem :: JGRAPH_7:42

    

     Th42: for p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real st a < b & c < d & (p1 `2 ) = d & (p2 `1 ) = b & (p3 `2 ) = c & (p4 `2 ) = c & a <= (p1 `1 ) & (p1 `1 ) <= b & c <= (p2 `2 ) & (p2 `2 ) <= d & a < (p4 `1 ) & (p4 `1 ) < (p3 `1 ) & (p3 `1 ) <= b holds (p1,p2,p3,p4) are_in_this_order_on ( rectangle (a,b,c,d))

    proof

      let p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real;

      set K = ( rectangle (a,b,c,d));

      assume that

       A1: a < b and

       A2: c < d and

       A3: (p1 `2 ) = d and

       A4: (p2 `1 ) = b and

       A5: (p3 `2 ) = c and

       A6: (p4 `2 ) = c and

       A7: a <= (p1 `1 ) and

       A8: (p1 `1 ) <= b and

       A9: c <= (p2 `2 ) and

       A10: (p2 `2 ) <= d and

       A11: a < (p4 `1 ) and

       A12: (p4 `1 ) < (p3 `1 ) and

       A13: (p3 `1 ) <= b;

      a < (p3 `1 ) by A11, A12, XXREAL_0: 2;

      then LE (p1,p2,K) & LE (p2,p3,K) & LE (p3,p4,K) or LE (p2,p3,K) & LE (p3,p4,K) & LE (p4,p1,K) or LE (p3,p4,K) & LE (p4,p1,K) & LE (p1,p2,K) or LE (p4,p1,K) & LE (p1,p2,K) & LE (p2,p3,K) by A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, A12, A13, Th8, Th11, Th12;

      hence thesis by JORDAN17:def 1;

    end;

    theorem :: JGRAPH_7:43

    

     Th43: for p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real st a < b & c < d & (p1 `2 ) = d & (p2 `2 ) = c & (p3 `2 ) = c & (p4 `2 ) = c & a <= (p1 `1 ) & (p1 `1 ) <= b & a < (p4 `1 ) & (p4 `1 ) < (p3 `1 ) & (p3 `1 ) < (p2 `1 ) & (p2 `1 ) <= b holds (p1,p2,p3,p4) are_in_this_order_on ( rectangle (a,b,c,d))

    proof

      let p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real;

      set K = ( rectangle (a,b,c,d));

      assume that

       A1: a < b and

       A2: c < d and

       A3: (p1 `2 ) = d and

       A4: (p2 `2 ) = c and

       A5: (p3 `2 ) = c and

       A6: (p4 `2 ) = c and

       A7: a <= (p1 `1 ) and

       A8: (p1 `1 ) <= b and

       A9: a < (p4 `1 ) and

       A10: (p4 `1 ) < (p3 `1 ) and

       A11: (p3 `1 ) < (p2 `1 ) and

       A12: (p2 `1 ) <= b;

      

       A13: (p3 `1 ) < b by A11, A12, XXREAL_0: 2;

      (p2 `1 ) > (p4 `1 ) by A10, A11, XXREAL_0: 2;

      then

       A14: (p2 `1 ) > a by A9, XXREAL_0: 2;

      a < (p3 `1 ) by A9, A10, XXREAL_0: 2;

      then LE (p1,p2,K) & LE (p2,p3,K) & LE (p3,p4,K) or LE (p2,p3,K) & LE (p3,p4,K) & LE (p4,p1,K) or LE (p3,p4,K) & LE (p4,p1,K) & LE (p1,p2,K) or LE (p4,p1,K) & LE (p1,p2,K) & LE (p2,p3,K) by A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, A12, A14, A13, Th9, Th12;

      hence thesis by JORDAN17:def 1;

    end;

    theorem :: JGRAPH_7:44

    

     Th44: for p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real st a < b & c < d & (p1 `1 ) = b & (p2 `1 ) = b & (p3 `1 ) = b & (p4 `1 ) = b & d >= (p1 `2 ) & (p1 `2 ) > (p2 `2 ) & (p2 `2 ) > (p3 `2 ) & (p3 `2 ) > (p4 `2 ) & (p4 `2 ) >= c holds (p1,p2,p3,p4) are_in_this_order_on ( rectangle (a,b,c,d))

    proof

      let p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real;

      set K = ( rectangle (a,b,c,d));

      assume that

       A1: a < b and

       A2: c < d and

       A3: (p1 `1 ) = b and

       A4: (p2 `1 ) = b and

       A5: (p3 `1 ) = b and

       A6: (p4 `1 ) = b and

       A7: d >= (p1 `2 ) and

       A8: (p1 `2 ) > (p2 `2 ) and

       A9: (p2 `2 ) > (p3 `2 ) and

       A10: (p3 `2 ) > (p4 `2 ) and

       A11: (p4 `2 ) >= c;

      

       A12: (p3 `2 ) > c by A10, A11, XXREAL_0: 2;

      (p2 `2 ) > (p4 `2 ) by A9, A10, XXREAL_0: 2;

      then

       A13: (p2 `2 ) > c by A11, XXREAL_0: 2;

      

       A14: d > (p2 `2 ) by A7, A8, XXREAL_0: 2;

      then d > (p3 `2 ) by A9, XXREAL_0: 2;

      then LE (p1,p2,K) & LE (p2,p3,K) & LE (p3,p4,K) or LE (p2,p3,K) & LE (p3,p4,K) & LE (p4,p1,K) or LE (p3,p4,K) & LE (p4,p1,K) & LE (p1,p2,K) or LE (p4,p1,K) & LE (p1,p2,K) & LE (p2,p3,K) by A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, A13, A14, A12, Th10;

      hence thesis by JORDAN17:def 1;

    end;

    theorem :: JGRAPH_7:45

    

     Th45: for p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real st a < b & c < d & (p1 `1 ) = b & (p2 `1 ) = b & (p3 `1 ) = b & (p4 `2 ) = c & d >= (p1 `2 ) & (p1 `2 ) > (p2 `2 ) & (p2 `2 ) > (p3 `2 ) & (p3 `2 ) >= c & a < (p4 `1 ) & (p4 `1 ) <= b holds (p1,p2,p3,p4) are_in_this_order_on ( rectangle (a,b,c,d))

    proof

      let p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real;

      set K = ( rectangle (a,b,c,d));

      assume that

       A1: a < b and

       A2: c < d and

       A3: (p1 `1 ) = b and

       A4: (p2 `1 ) = b and

       A5: (p3 `1 ) = b and

       A6: (p4 `2 ) = c and

       A7: d >= (p1 `2 ) and

       A8: (p1 `2 ) > (p2 `2 ) and

       A9: (p2 `2 ) > (p3 `2 ) and

       A10: (p3 `2 ) >= c and

       A11: a < (p4 `1 ) and

       A12: (p4 `1 ) <= b;

      

       A13: (p2 `2 ) > c by A9, A10, XXREAL_0: 2;

      

       A14: d > (p2 `2 ) by A7, A8, XXREAL_0: 2;

      then d > (p3 `2 ) by A9, XXREAL_0: 2;

      then LE (p1,p2,K) & LE (p2,p3,K) & LE (p3,p4,K) or LE (p2,p3,K) & LE (p3,p4,K) & LE (p4,p1,K) or LE (p3,p4,K) & LE (p4,p1,K) & LE (p1,p2,K) or LE (p4,p1,K) & LE (p1,p2,K) & LE (p2,p3,K) by A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, A12, A13, A14, Th10, Th11;

      hence thesis by JORDAN17:def 1;

    end;

    theorem :: JGRAPH_7:46

    

     Th46: for p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real st a < b & c < d & (p1 `1 ) = b & (p2 `1 ) = b & (p3 `2 ) = c & (p4 `2 ) = c & d >= (p1 `2 ) & (p1 `2 ) > (p2 `2 ) & (p2 `2 ) >= c & b >= (p3 `1 ) & (p3 `1 ) > (p4 `1 ) & (p4 `1 ) > a holds (p1,p2,p3,p4) are_in_this_order_on ( rectangle (a,b,c,d))

    proof

      let p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real;

      set K = ( rectangle (a,b,c,d));

      assume that

       A1: a < b and

       A2: c < d and

       A3: (p1 `1 ) = b and

       A4: (p2 `1 ) = b and

       A5: (p3 `2 ) = c and

       A6: (p4 `2 ) = c and

       A7: d >= (p1 `2 ) and

       A8: (p1 `2 ) > (p2 `2 ) and

       A9: (p2 `2 ) >= c and

       A10: b >= (p3 `1 ) and

       A11: (p3 `1 ) > (p4 `1 ) and

       A12: (p4 `1 ) > a;

      

       A13: (p3 `1 ) > a by A11, A12, XXREAL_0: 2;

      d > (p2 `2 ) by A7, A8, XXREAL_0: 2;

      then LE (p1,p2,K) & LE (p2,p3,K) & LE (p3,p4,K) or LE (p2,p3,K) & LE (p3,p4,K) & LE (p4,p1,K) or LE (p3,p4,K) & LE (p4,p1,K) & LE (p1,p2,K) or LE (p4,p1,K) & LE (p1,p2,K) & LE (p2,p3,K) by A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, A12, A13, Th10, Th11, Th12;

      hence thesis by JORDAN17:def 1;

    end;

    theorem :: JGRAPH_7:47

    

     Th47: for p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real st a < b & c < d & (p1 `1 ) = b & (p2 `2 ) = c & (p3 `2 ) = c & (p4 `2 ) = c & c <= (p1 `2 ) & (p1 `2 ) <= d & b >= (p2 `1 ) & (p2 `1 ) > (p3 `1 ) & (p3 `1 ) > (p4 `1 ) & (p4 `1 ) > a holds (p1,p2,p3,p4) are_in_this_order_on ( rectangle (a,b,c,d))

    proof

      let p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real;

      set K = ( rectangle (a,b,c,d));

      assume that

       A1: a < b and

       A2: c < d and

       A3: (p1 `1 ) = b and

       A4: (p2 `2 ) = c and

       A5: (p3 `2 ) = c and

       A6: (p4 `2 ) = c and

       A7: c <= (p1 `2 ) and

       A8: (p1 `2 ) <= d and

       A9: b >= (p2 `1 ) and

       A10: (p2 `1 ) > (p3 `1 ) and

       A11: (p3 `1 ) > (p4 `1 ) and

       A12: (p4 `1 ) > a;

      

       A13: (p3 `1 ) > a by A11, A12, XXREAL_0: 2;

      (p2 `1 ) > (p4 `1 ) by A10, A11, XXREAL_0: 2;

      then

       A14: (p2 `1 ) > a by A12, XXREAL_0: 2;

      b > (p3 `1 ) by A9, A10, XXREAL_0: 2;

      then LE (p1,p2,K) & LE (p2,p3,K) & LE (p3,p4,K) or LE (p2,p3,K) & LE (p3,p4,K) & LE (p4,p1,K) or LE (p3,p4,K) & LE (p4,p1,K) & LE (p1,p2,K) or LE (p4,p1,K) & LE (p1,p2,K) & LE (p2,p3,K) by A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, A12, A14, A13, Th11, Th12;

      hence thesis by JORDAN17:def 1;

    end;

    theorem :: JGRAPH_7:48

    

     Th48: for p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real st a < b & c < d & (p1 `2 ) = c & (p2 `2 ) = c & (p3 `2 ) = c & (p4 `2 ) = c & b >= (p1 `1 ) & (p1 `1 ) > (p2 `1 ) & (p2 `1 ) > (p3 `1 ) & (p3 `1 ) > (p4 `1 ) & (p4 `1 ) > a holds (p1,p2,p3,p4) are_in_this_order_on ( rectangle (a,b,c,d))

    proof

      let p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real;

      set K = ( rectangle (a,b,c,d));

      assume that

       A1: a < b and

       A2: c < d and

       A3: (p1 `2 ) = c and

       A4: (p2 `2 ) = c and

       A5: (p3 `2 ) = c and

       A6: (p4 `2 ) = c and

       A7: b >= (p1 `1 ) and

       A8: (p1 `1 ) > (p2 `1 ) and

       A9: (p2 `1 ) > (p3 `1 ) and

       A10: (p3 `1 ) > (p4 `1 ) and

       A11: (p4 `1 ) > a;

      

       A12: (p3 `1 ) > a by A10, A11, XXREAL_0: 2;

      (p2 `1 ) > (p4 `1 ) by A9, A10, XXREAL_0: 2;

      then

       A13: (p2 `1 ) > a by A11, XXREAL_0: 2;

      

       A14: b > (p2 `1 ) by A7, A8, XXREAL_0: 2;

      then b > (p3 `1 ) by A9, XXREAL_0: 2;

      then LE (p1,p2,K) & LE (p2,p3,K) & LE (p3,p4,K) or LE (p2,p3,K) & LE (p3,p4,K) & LE (p4,p1,K) or LE (p3,p4,K) & LE (p4,p1,K) & LE (p1,p2,K) or LE (p4,p1,K) & LE (p1,p2,K) & LE (p2,p3,K) by A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, A13, A14, A12, Th12;

      hence thesis by JORDAN17:def 1;

    end;

    theorem :: JGRAPH_7:49

    

     Th49: for A,B,C,D be Real, h,g be Function of ( TOP-REAL 2), ( TOP-REAL 2) st A > 0 & C > 0 & h = ( AffineMap (A,B,C,D)) & g = ( AffineMap ((1 / A),( - (B / A)),(1 / C),( - (D / C)))) holds g = (h " ) & h = (g " )

    proof

      let A,B,C,D be Real, h,g be Function of ( TOP-REAL 2), ( TOP-REAL 2);

      assume that

       A1: A > 0 and

       A2: C > 0 and

       A3: h = ( AffineMap (A,B,C,D)) and

       A4: g = ( AffineMap ((1 / A),( - (B / A)),(1 / C),( - (D / C))));

      

       A5: h is one-to-one by A1, A2, A3, JGRAPH_2: 44;

      

       A6: for x,y be object st x in ( dom h) & y in ( dom g) holds (h . x) = y iff (g . y) = x

      proof

        let x,y be object;

        assume that

         A7: x in ( dom h) and

         A8: y in ( dom g);

        reconsider py = y as Point of ( TOP-REAL 2) by A8;

        reconsider px = x as Point of ( TOP-REAL 2) by A7;

        

         A9: (h . x) = y implies (g . y) = x

        proof

          assume

           A10: (h . x) = y;

          

           A11: (h . px) = |[((A * (px `1 )) + B), ((C * (px `2 )) + D)]| by A3, JGRAPH_2:def 2;

          then (py `1 ) = ((A * (px `1 )) + B) by A10, EUCLID: 52;

          

          then

           A12: (((1 / A) * (py `1 )) + ( - (B / A))) = (((((1 / A) * A) * (px `1 )) + ((1 / A) * B)) + ( - (B / A)))

          .= (((1 * (px `1 )) + ((1 / A) * B)) + ( - (B / A))) by A1, XCMPLX_1: 106

          .= (((px `1 ) + (B / A)) + ( - (B / A))) by XCMPLX_1: 99

          .= (px `1 );

          (py `2 ) = ((C * (px `2 )) + D) by A10, A11, EUCLID: 52;

          

          then

           A13: (((1 / C) * (py `2 )) + ( - (D / C))) = (((((1 / C) * C) * (px `2 )) + ((1 / C) * D)) + ( - (D / C)))

          .= (((1 * (px `2 )) + ((1 / C) * D)) + ( - (D / C))) by A2, XCMPLX_1: 106

          .= (((px `2 ) + (D / C)) + ( - (D / C))) by XCMPLX_1: 99

          .= (px `2 );

          (g . py) = |[(((1 / A) * (py `1 )) + ( - (B / A))), (((1 / C) * (py `2 )) + ( - (D / C)))]| by A4, JGRAPH_2:def 2;

          hence thesis by A12, A13, EUCLID: 53;

        end;

        (g . y) = x implies (h . x) = y

        proof

          assume

           A14: (g . y) = x;

          

           A15: (g . py) = |[(((1 / A) * (py `1 )) + ( - (B / A))), (((1 / C) * (py `2 )) + ( - (D / C)))]| by A4, JGRAPH_2:def 2;

          then (px `1 ) = (((1 / A) * (py `1 )) + ( - (B / A))) by A14, EUCLID: 52;

          

          then

           A16: ((A * (px `1 )) + B) = ((((A * (1 / A)) * (py `1 )) + (A * ( - (B / A)))) + B)

          .= (((1 * (py `1 )) + (A * ( - (B / A)))) + B) by A1, XCMPLX_1: 106

          .= (((py `1 ) + (A * (( - B) / A))) + B) by XCMPLX_1: 187

          .= (((py `1 ) + ( - B)) + B) by A1, XCMPLX_1: 87

          .= (py `1 );

          (px `2 ) = (((1 / C) * (py `2 )) + ( - (D / C))) by A14, A15, EUCLID: 52;

          

          then

           A17: ((C * (px `2 )) + D) = ((((C * (1 / C)) * (py `2 )) + (C * ( - (D / C)))) + D)

          .= (((1 * (py `2 )) + (C * ( - (D / C)))) + D) by A2, XCMPLX_1: 106

          .= (((py `2 ) + (C * (( - D) / C))) + D) by XCMPLX_1: 187

          .= (((py `2 ) + ( - D)) + D) by A2, XCMPLX_1: 87

          .= (py `2 );

          (h . px) = |[((A * (px `1 )) + B), ((C * (px `2 )) + D)]| by A3, JGRAPH_2:def 2;

          hence thesis by A16, A17, EUCLID: 53;

        end;

        hence thesis by A9;

      end;

      

       A18: ( dom g) = the carrier of ( TOP-REAL 2) by FUNCT_2:def 1;

      reconsider RD = D as Real;

      reconsider RC = C as Real;

      reconsider RB = B as Real;

      reconsider RA = A as Real;

      

       A19: g = ( AffineMap ((1 / RA),( - (RB / RA)),(1 / RC),( - (RD / RC)))) by A4;

      h = ( AffineMap (RA,RB,RC,RD)) by A3;

      then h is onto by A1, A2, JORDAN1K: 36;

      then

       A20: ( rng h) = the carrier of ( TOP-REAL 2) by FUNCT_2:def 3;

      

       A21: (1 / C) > 0 by A2, XREAL_1: 139;

      (1 / A) > 0 by A1, XREAL_1: 139;

      then g is onto by A21, A19, JORDAN1K: 36;

      then

       A22: ( rng g) = the carrier of ( TOP-REAL 2) by FUNCT_2:def 3;

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

      then g = (h " ) by A5, A18, A20, A22, A6, FUNCT_1: 38;

      hence thesis by A5, FUNCT_1: 43;

    end;

    theorem :: JGRAPH_7:50

    

     Th50: for A,B,C,D be Real, h be Function of ( TOP-REAL 2), ( TOP-REAL 2) st A > 0 & C > 0 & h = ( AffineMap (A,B,C,D)) holds h is being_homeomorphism & for p1,p2 be Point of ( TOP-REAL 2) st (p1 `1 ) < (p2 `1 ) holds ((h . p1) `1 ) < ((h . p2) `1 )

    proof

      let A,B,C,D be Real, h be Function of ( TOP-REAL 2), ( TOP-REAL 2);

      assume that

       A1: A > 0 and

       A2: C > 0 and

       A3: h = ( AffineMap (A,B,C,D));

      

       A4: h is one-to-one by A1, A2, A3, JGRAPH_2: 44;

      set g = ( AffineMap ((1 / A),( - (B / A)),(1 / C),( - (D / C))));

      

       A5: g = (h " ) by A1, A2, A3, Th49;

      

       A6: for p1,p2 be Point of ( TOP-REAL 2) st (p1 `1 ) < (p2 `1 ) holds ((h . p1) `1 ) < ((h . p2) `1 )

      proof

        let p1,p2 be Point of ( TOP-REAL 2);

        (h . p1) = |[((A * (p1 `1 )) + B), ((C * (p1 `2 )) + D)]| by A3, JGRAPH_2:def 2;

        then

         A7: ((h . p1) `1 ) = ((A * (p1 `1 )) + B) by EUCLID: 52;

        (h . p2) = |[((A * (p2 `1 )) + B), ((C * (p2 `2 )) + D)]| by A3, JGRAPH_2:def 2;

        then

         A8: ((h . p2) `1 ) = ((A * (p2 `1 )) + B) by EUCLID: 52;

        assume (p1 `1 ) < (p2 `1 );

        then (A * (p1 `1 )) < (A * (p2 `1 )) by A1, XREAL_1: 68;

        hence thesis by A7, A8, XREAL_1: 8;

      end;

      

       A9: ( dom h) = ( [#] ( TOP-REAL 2)) by FUNCT_2:def 1;

      ( dom g) = ( [#] ( TOP-REAL 2)) by FUNCT_2:def 1;

      then

       A10: ( rng h) = ( [#] ( TOP-REAL 2)) by A4, A5, FUNCT_1: 32;

      then h is onto one-to-one by A1, A2, A3, FUNCT_2:def 3, JGRAPH_2: 44;

      then (h /" ) is continuous by A5, TOPS_2:def 4;

      hence thesis by A3, A4, A9, A10, A6, TOPS_2:def 5;

    end;

    theorem :: JGRAPH_7:51

    

     Th51: for A,B,C,D be Real, h be Function of ( TOP-REAL 2), ( TOP-REAL 2) st A > 0 & C > 0 & h = ( AffineMap (A,B,C,D)) holds h is being_homeomorphism & for p1,p2 be Point of ( TOP-REAL 2) st (p1 `2 ) < (p2 `2 ) holds ((h . p1) `2 ) < ((h . p2) `2 )

    proof

      let A,B,C,D be Real, h be Function of ( TOP-REAL 2), ( TOP-REAL 2);

      assume that

       A1: A > 0 and

       A2: C > 0 and

       A3: h = ( AffineMap (A,B,C,D));

      

       A4: h is one-to-one by A1, A2, A3, JGRAPH_2: 44;

      set g = ( AffineMap ((1 / A),( - (B / A)),(1 / C),( - (D / C))));

      

       A5: g = (h " ) by A1, A2, A3, Th49;

      

       A6: for p1,p2 be Point of ( TOP-REAL 2) st (p1 `2 ) < (p2 `2 ) holds ((h . p1) `2 ) < ((h . p2) `2 )

      proof

        let p1,p2 be Point of ( TOP-REAL 2);

        (h . p1) = |[((A * (p1 `1 )) + B), ((C * (p1 `2 )) + D)]| by A3, JGRAPH_2:def 2;

        then

         A7: ((h . p1) `2 ) = ((C * (p1 `2 )) + D) by EUCLID: 52;

        (h . p2) = |[((A * (p2 `1 )) + B), ((C * (p2 `2 )) + D)]| by A3, JGRAPH_2:def 2;

        then

         A8: ((h . p2) `2 ) = ((C * (p2 `2 )) + D) by EUCLID: 52;

        assume (p1 `2 ) < (p2 `2 );

        then (C * (p1 `2 )) < (C * (p2 `2 )) by A2, XREAL_1: 68;

        hence thesis by A7, A8, XREAL_1: 8;

      end;

      

       A9: ( dom h) = ( [#] ( TOP-REAL 2)) by FUNCT_2:def 1;

      ( dom g) = ( [#] ( TOP-REAL 2)) by FUNCT_2:def 1;

      then

       A10: ( rng h) = ( [#] ( TOP-REAL 2)) by A4, A5, FUNCT_1: 32;

      then h is onto one-to-one by A1, A2, A3, FUNCT_2:def 3, JGRAPH_2: 44;

      then (h /" ) is continuous by A5, TOPS_2:def 4;

      hence thesis by A3, A4, A9, A10, A6, TOPS_2:def 5;

    end;

    theorem :: JGRAPH_7:52

    

     Th52: for a,b,c,d be Real, h be Function of ( TOP-REAL 2), ( TOP-REAL 2), f be Function of I[01] , ( TOP-REAL 2) st a < b & c < d & h = ( AffineMap ((2 / (b - a)),( - ((b + a) / (b - a))),(2 / (d - c)),( - ((d + c) / (d - c))))) & ( rng f) c= ( closed_inside_of_rectangle (a,b,c,d)) holds ( rng (h * f)) c= ( closed_inside_of_rectangle (( - 1),1,( - 1),1))

    proof

      let a,b,c,d be Real, h be Function of ( TOP-REAL 2), ( TOP-REAL 2), f be Function of I[01] , ( TOP-REAL 2);

      set A = (2 / (b - a)), B = ( - ((b + a) / (b - a))), C = (2 / (d - c)), D = ( - ((d + c) / (d - c)));

      assume that

       A1: a < b and

       A2: c < d and

       A3: h = ( AffineMap (A,B,C,D)) and

       A4: ( rng f) c= ( closed_inside_of_rectangle (a,b,c,d));

      let x be object;

      assume x in ( rng (h * f));

      then

      consider y be object such that

       A5: y in ( dom (h * f)) and

       A6: x = ((h * f) . y) by FUNCT_1:def 3;

      reconsider t0 = y as Point of I[01] by A5;

      

       A7: ((h * f) . t0) = (h . (f . t0)) by A5, FUNCT_1: 12;

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

      then (f . t0) in ( rng f) by FUNCT_1:def 3;

      then (f . t0) in ( closed_inside_of_rectangle (a,b,c,d)) by A4;

      then (f . t0) in { p where p be Point of ( TOP-REAL 2) : a <= (p `1 ) & (p `1 ) <= b & c <= (p `2 ) & (p `2 ) <= d } by JGRAPH_6:def 2;

      then

       A8: ex p be Point of ( TOP-REAL 2) st (f . t0) = p & a <= (p `1 ) & (p `1 ) <= b & c <= (p `2 ) & (p `2 ) <= d;

      reconsider p0 = x as Point of ( TOP-REAL 2) by A5, A6, FUNCT_2: 5;

      

       A9: (h . (f . t0)) = |[((A * ((f . t0) `1 )) + B), ((C * ((f . t0) `2 )) + D)]| by A3, JGRAPH_2:def 2;

      

       A10: (b - a) > 0 by A1, XREAL_1: 50;

      then

       A11: A > 0 by XREAL_1: 139;

      ((( - 1) - B) / A) = ((( - 1) + ((b + a) / (b - a))) / (2 / (b - a)))

      .= ((((( - 1) * (b - a)) + (b + a)) / (b - a)) / (2 / (b - a))) by A10, XCMPLX_1: 113

      .= ((((a + a) / (b - a)) / 2) * (b - a)) by XCMPLX_1: 82

      .= (((b - a) * ((a + a) / (b - a))) / 2)

      .= ((a + a) / 2) by A10, XCMPLX_1: 87

      .= a;

      then (A * ((( - 1) - B) / A)) <= (A * ((f . t0) `1 )) by A11, A8, XREAL_1: 64;

      then (( - 1) - B) <= (A * ((f . t0) `1 )) by A11, XCMPLX_1: 87;

      then ((( - 1) - B) + B) <= ((A * ((f . t0) `1 )) + B) by XREAL_1: 6;

      then

       A12: ( - 1) <= (p0 `1 ) by A6, A9, A7, EUCLID: 52;

      

       A13: (d - c) > 0 by A2, XREAL_1: 50;

      then

       A14: C > 0 by XREAL_1: 139;

      ((1 - B) / A) = ((1 + ((b + a) / (b - a))) / (2 / (b - a)))

      .= ((((1 * (b - a)) + (b + a)) / (b - a)) / (2 / (b - a))) by A10, XCMPLX_1: 113

      .= ((((b + b) / (b - a)) / 2) * (b - a)) by XCMPLX_1: 82

      .= (((b - a) * ((b + b) / (b - a))) / 2)

      .= ((b + b) / 2) by A10, XCMPLX_1: 87

      .= b;

      then (A * ((1 - B) / A)) >= (A * ((f . t0) `1 )) by A11, A8, XREAL_1: 64;

      then (1 - B) >= (A * ((f . t0) `1 )) by A11, XCMPLX_1: 87;

      then ((1 - B) + B) >= ((A * ((f . t0) `1 )) + B) by XREAL_1: 6;

      then

       A15: (p0 `1 ) <= 1 by A6, A9, A7, EUCLID: 52;

      ((1 - D) / C) = ((1 + ((d + c) / (d - c))) / (2 / (d - c)))

      .= ((((1 * (d - c)) + (d + c)) / (d - c)) / (2 / (d - c))) by A13, XCMPLX_1: 113

      .= ((((d + d) / (d - c)) / 2) * (d - c)) by XCMPLX_1: 82

      .= (((d - c) * ((d + d) / (d - c))) / 2)

      .= ((d + d) / 2) by A13, XCMPLX_1: 87

      .= d;

      then (C * ((1 - D) / C)) >= (C * ((f . t0) `2 )) by A14, A8, XREAL_1: 64;

      then (1 - D) >= (C * ((f . t0) `2 )) by A14, XCMPLX_1: 87;

      then ((1 - D) + D) >= ((C * ((f . t0) `2 )) + D) by XREAL_1: 6;

      then

       A16: (p0 `2 ) <= 1 by A6, A9, A7, EUCLID: 52;

      ((( - 1) - D) / C) = ((( - 1) + ((d + c) / (d - c))) / (2 / (d - c)))

      .= ((((( - 1) * (d - c)) + (d + c)) / (d - c)) / (2 / (d - c))) by A13, XCMPLX_1: 113

      .= ((((c + c) / (d - c)) / 2) * (d - c)) by XCMPLX_1: 82

      .= (((d - c) * ((c + c) / (d - c))) / 2)

      .= ((c + c) / 2) by A13, XCMPLX_1: 87

      .= c;

      then (C * ((( - 1) - D) / C)) <= (C * ((f . t0) `2 )) by A14, A8, XREAL_1: 64;

      then (( - 1) - D) <= (C * ((f . t0) `2 )) by A14, XCMPLX_1: 87;

      then ((( - 1) - D) + D) <= ((C * ((f . t0) `2 )) + D) by XREAL_1: 6;

      then ( - 1) <= (p0 `2 ) by A6, A9, A7, EUCLID: 52;

      then x in { p2 where p2 be Point of ( TOP-REAL 2) : ( - 1) <= (p2 `1 ) & (p2 `1 ) <= 1 & ( - 1) <= (p2 `2 ) & (p2 `2 ) <= 1 } by A16, A12, A15;

      hence thesis by JGRAPH_6:def 2;

    end;

    theorem :: JGRAPH_7:53

    

     Th53: for a,b,c,d be Real, h be Function of ( TOP-REAL 2), ( TOP-REAL 2), f be Function of I[01] , ( TOP-REAL 2) st a < b & c < d & h = ( AffineMap ((2 / (b - a)),( - ((b + a) / (b - a))),(2 / (d - c)),( - ((d + c) / (d - c))))) & f is continuous one-to-one holds (h * f) is continuous one-to-one

    proof

      let a,b,c,d be Real, h be Function of ( TOP-REAL 2), ( TOP-REAL 2), f be Function of I[01] , ( TOP-REAL 2);

      set A = (2 / (b - a)), B = ( - ((b + a) / (b - a))), C = (2 / (d - c)), D = ( - ((d + c) / (d - c)));

      assume that

       A1: a < b and

       A2: c < d and

       A3: h = ( AffineMap (A,B,C,D)) and

       A4: f is continuous one-to-one;

      (d - c) > 0 by A2, XREAL_1: 50;

      then

       A5: C > 0 by XREAL_1: 139;

      (b - a) > 0 by A1, XREAL_1: 50;

      then A > 0 by XREAL_1: 139;

      then h is being_homeomorphism by A3, A5, Th51;

      then h is one-to-one by TOPS_2:def 5;

      hence thesis by A3, A4, FUNCT_1: 24;

    end;

    theorem :: JGRAPH_7:54

    

     Th54: for a,b,c,d be Real, h be Function of ( TOP-REAL 2), ( TOP-REAL 2), f be Function of I[01] , ( TOP-REAL 2), O be Point of I[01] st a < b & h = ( AffineMap ((2 / (b - a)),( - ((b + a) / (b - a))),(2 / (d - c)),( - ((d + c) / (d - c))))) & ((f . O) `1 ) = a holds (((h * f) . O) `1 ) = ( - 1)

    proof

      let a,b,c,d be Real, h be Function of ( TOP-REAL 2), ( TOP-REAL 2), f be Function of I[01] , ( TOP-REAL 2), O be Point of I[01] ;

      set A = (2 / (b - a)), B = ( - ((b + a) / (b - a))), C = (2 / (d - c)), D = ( - ((d + c) / (d - c)));

      assume that

       A1: a < b and

       A2: h = ( AffineMap (A,B,C,D)) and

       A3: ((f . O) `1 ) = a;

      

       A4: (b - a) > 0 by A1, XREAL_1: 50;

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

      then

       A5: ((h * f) . O) = (h . (f . O)) by FUNCT_1: 13;

      

       A6: (h . (f . O)) = |[((A * ((f . O) `1 )) + B), ((C * ((f . O) `2 )) + D)]| by A2, JGRAPH_2:def 2;

      ((A * ((f . O) `1 )) + B) = (((2 * a) / (b - a)) + ( - ((b + a) / (b - a)))) by A3, XCMPLX_1: 74

      .= (((2 * a) / (b - a)) + (( - (b + a)) / (b - a))) by XCMPLX_1: 187

      .= (((2 * a) + ( - (b + a))) / (b - a)) by XCMPLX_1: 62

      .= (( - (b - a)) / (b - a))

      .= ( - 1) by A4, XCMPLX_1: 197;

      hence thesis by A5, A6, EUCLID: 52;

    end;

    theorem :: JGRAPH_7:55

    

     Th55: for a,b,c,d be Real, h be Function of ( TOP-REAL 2), ( TOP-REAL 2), f be Function of I[01] , ( TOP-REAL 2), I be Point of I[01] st c < d & h = ( AffineMap ((2 / (b - a)),( - ((b + a) / (b - a))),(2 / (d - c)),( - ((d + c) / (d - c))))) & ((f . I) `2 ) = d holds (((h * f) . I) `2 ) = 1

    proof

      let a,b,c,d be Real, h be Function of ( TOP-REAL 2), ( TOP-REAL 2), f be Function of I[01] , ( TOP-REAL 2), I be Point of I[01] ;

      set A = (2 / (b - a)), B = ( - ((b + a) / (b - a))), C = (2 / (d - c)), D = ( - ((d + c) / (d - c)));

      assume that

       A1: c < d and

       A2: h = ( AffineMap (A,B,C,D)) and

       A3: ((f . I) `2 ) = d;

      

       A4: (d - c) > 0 by A1, XREAL_1: 50;

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

      then

       A5: ((h * f) . I) = (h . (f . I)) by FUNCT_1: 13;

      

       A6: (h . (f . I)) = |[((A * ((f . I) `1 )) + B), ((C * ((f . I) `2 )) + D)]| by A2, JGRAPH_2:def 2;

      ((C * ((f . I) `2 )) + D) = (((2 * d) / (d - c)) + ( - ((d + c) / (d - c)))) by A3, XCMPLX_1: 74

      .= (((2 * d) / (d - c)) + (( - (d + c)) / (d - c))) by XCMPLX_1: 187

      .= (((2 * d) + ( - (d + c))) / (d - c)) by XCMPLX_1: 62

      .= 1 by A4, XCMPLX_1: 60;

      hence thesis by A5, A6, EUCLID: 52;

    end;

    theorem :: JGRAPH_7:56

    

     Th56: for a,b,c,d be Real, h be Function of ( TOP-REAL 2), ( TOP-REAL 2), f be Function of I[01] , ( TOP-REAL 2), I be Point of I[01] st a < b & h = ( AffineMap ((2 / (b - a)),( - ((b + a) / (b - a))),(2 / (d - c)),( - ((d + c) / (d - c))))) & ((f . I) `1 ) = b holds (((h * f) . I) `1 ) = 1

    proof

      let a,b,c,d be Real, h be Function of ( TOP-REAL 2), ( TOP-REAL 2), f be Function of I[01] , ( TOP-REAL 2), I be Point of I[01] ;

      set A = (2 / (b - a)), B = ( - ((b + a) / (b - a))), C = (2 / (d - c)), D = ( - ((d + c) / (d - c)));

      assume that

       A1: a < b and

       A2: h = ( AffineMap (A,B,C,D)) and

       A3: ((f . I) `1 ) = b;

      

       A4: (b - a) > 0 by A1, XREAL_1: 50;

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

      then

       A5: ((h * f) . I) = (h . (f . I)) by FUNCT_1: 13;

      

       A6: (h . (f . I)) = |[((A * ((f . I) `1 )) + B), ((C * ((f . I) `2 )) + D)]| by A2, JGRAPH_2:def 2;

      ((A * ((f . I) `1 )) + B) = (((2 * b) / (b - a)) + ( - ((b + a) / (b - a)))) by A3, XCMPLX_1: 74

      .= (((2 * b) / (b - a)) + (( - (b + a)) / (b - a))) by XCMPLX_1: 187

      .= (((b + b) + ( - (b + a))) / (b - a)) by XCMPLX_1: 62

      .= 1 by A4, XCMPLX_1: 60;

      hence thesis by A5, A6, EUCLID: 52;

    end;

    theorem :: JGRAPH_7:57

    

     Th57: for a,b,c,d be Real, h be Function of ( TOP-REAL 2), ( TOP-REAL 2), f be Function of I[01] , ( TOP-REAL 2), I be Point of I[01] st c < d & h = ( AffineMap ((2 / (b - a)),( - ((b + a) / (b - a))),(2 / (d - c)),( - ((d + c) / (d - c))))) & ((f . I) `2 ) = c holds (((h * f) . I) `2 ) = ( - 1)

    proof

      let a,b,c,d be Real, h be Function of ( TOP-REAL 2), ( TOP-REAL 2), f be Function of I[01] , ( TOP-REAL 2), I be Point of I[01] ;

      set A = (2 / (b - a)), B = ( - ((b + a) / (b - a))), C = (2 / (d - c)), D = ( - ((d + c) / (d - c)));

      assume that

       A1: c < d and

       A2: h = ( AffineMap (A,B,C,D)) and

       A3: ((f . I) `2 ) = c;

      

       A4: (d - c) > 0 by A1, XREAL_1: 50;

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

      then

       A5: ((h * f) . I) = (h . (f . I)) by FUNCT_1: 13;

      

       A6: (h . (f . I)) = |[((A * ((f . I) `1 )) + B), ((C * ((f . I) `2 )) + D)]| by A2, JGRAPH_2:def 2;

      ((C * ((f . I) `2 )) + D) = (((2 * c) / (d - c)) + ( - ((d + c) / (d - c)))) by A3, XCMPLX_1: 74

      .= (((2 * c) / (d - c)) + (( - (d + c)) / (d - c))) by XCMPLX_1: 187

      .= (((c + c) + ( - (d + c))) / (d - c)) by XCMPLX_1: 62

      .= (( - (d - c)) / (d - c))

      .= ( - 1) by A4, XCMPLX_1: 197;

      hence thesis by A5, A6, EUCLID: 52;

    end;

    theorem :: JGRAPH_7:58

    

     Th58: for a,b,c,d be Real, h be Function of ( TOP-REAL 2), ( TOP-REAL 2), f be Function of I[01] , ( TOP-REAL 2), O,I be Point of I[01] st c < d & h = ( AffineMap ((2 / (b - a)),( - ((b + a) / (b - a))),(2 / (d - c)),( - ((d + c) / (d - c))))) & c <= ((f . O) `2 ) & ((f . O) `2 ) < ((f . I) `2 ) & ((f . I) `2 ) <= d holds ( - 1) <= (((h * f) . O) `2 ) & (((h * f) . O) `2 ) < (((h * f) . I) `2 ) & (((h * f) . I) `2 ) <= 1

    proof

      let a,b,c,d be Real, h be Function of ( TOP-REAL 2), ( TOP-REAL 2), f be Function of I[01] , ( TOP-REAL 2), O,I be Point of I[01] ;

      set A = (2 / (b - a)), B = ( - ((b + a) / (b - a))), C = (2 / (d - c)), D = ( - ((d + c) / (d - c)));

      assume that

       A1: c < d and

       A2: h = ( AffineMap (A,B,C,D)) and

       A3: c <= ((f . O) `2 ) and

       A4: ((f . O) `2 ) < ((f . I) `2 ) and

       A5: ((f . I) `2 ) <= d;

      

       A6: (h . (f . O)) = |[((A * ((f . O) `1 )) + B), ((C * ((f . O) `2 )) + D)]| by A2, JGRAPH_2:def 2;

      

       A7: (d - c) > 0 by A1, XREAL_1: 50;

      then

       A8: C > 0 by XREAL_1: 139;

      ((( - 1) - D) / C) = ((( - 1) + ((d + c) / (d - c))) / (2 / (d - c)))

      .= ((((( - 1) * (d - c)) + (d + c)) / (d - c)) / (2 / (d - c))) by A7, XCMPLX_1: 113

      .= ((((c + c) / (d - c)) / 2) * (d - c)) by XCMPLX_1: 82

      .= (((d - c) * ((c + c) / (d - c))) / 2)

      .= ((c + c) / 2) by A7, XCMPLX_1: 87

      .= c;

      then (C * ((( - 1) - D) / C)) <= (C * ((f . O) `2 )) by A3, A8, XREAL_1: 64;

      then (( - 1) - D) <= (C * ((f . O) `2 )) by A8, XCMPLX_1: 87;

      then

       A9: ((( - 1) - D) + D) <= ((C * ((f . O) `2 )) + D) by XREAL_1: 6;

      

       A10: ( dom f) = the carrier of I[01] by FUNCT_2:def 1;

      then

       A11: ((h * f) . O) = (h . (f . O)) by FUNCT_1: 13;

      

       A12: ((h * f) . I) = (h . (f . I)) by A10, FUNCT_1: 13;

      ((1 - D) / C) = ((1 + ((d + c) / (d - c))) / (2 / (d - c)))

      .= ((((1 * (d - c)) + (d + c)) / (d - c)) / (2 / (d - c))) by A7, XCMPLX_1: 113

      .= ((((d + d) / (d - c)) / 2) * (d - c)) by XCMPLX_1: 82

      .= (((d - c) * ((d + d) / (d - c))) / 2)

      .= ((d + d) / 2) by A7, XCMPLX_1: 87

      .= d;

      then (C * ((1 - D) / C)) >= (C * ((f . I) `2 )) by A5, A8, XREAL_1: 64;

      then (1 - D) >= (C * ((f . I) `2 )) by A8, XCMPLX_1: 87;

      then

       A13: ((1 - D) + D) >= ((C * ((f . I) `2 )) + D) by XREAL_1: 6;

      

       A14: (h . (f . I)) = |[((A * ((f . I) `1 )) + B), ((C * ((f . I) `2 )) + D)]| by A2, JGRAPH_2:def 2;

      (C * ((f . O) `2 )) < (C * ((f . I) `2 )) by A4, A8, XREAL_1: 68;

      then ((C * ((f . O) `2 )) + D) < ((C * ((f . I) `2 )) + D) by XREAL_1: 8;

      then ((C * ((f . O) `2 )) + D) < (((h * f) . I) `2 ) by A12, A14, EUCLID: 52;

      hence thesis by A11, A12, A6, A14, A9, A13, EUCLID: 52;

    end;

    theorem :: JGRAPH_7:59

    

     Th59: for a,b,c,d be Real, h be Function of ( TOP-REAL 2), ( TOP-REAL 2), f be Function of I[01] , ( TOP-REAL 2), O,I be Point of I[01] st a < b & c < d & h = ( AffineMap ((2 / (b - a)),( - ((b + a) / (b - a))),(2 / (d - c)),( - ((d + c) / (d - c))))) & c <= ((f . O) `2 ) & ((f . O) `2 ) <= d & a <= ((f . I) `1 ) & ((f . I) `1 ) <= b holds ( - 1) <= (((h * f) . O) `2 ) & (((h * f) . O) `2 ) <= 1 & ( - 1) <= (((h * f) . I) `1 ) & (((h * f) . I) `1 ) <= 1

    proof

      let a,b,c,d be Real, h be Function of ( TOP-REAL 2), ( TOP-REAL 2), f be Function of I[01] , ( TOP-REAL 2), O,I be Point of I[01] ;

      set A = (2 / (b - a)), B = ( - ((b + a) / (b - a))), C = (2 / (d - c)), D = ( - ((d + c) / (d - c)));

      assume that

       A1: a < b and

       A2: c < d and

       A3: h = ( AffineMap (A,B,C,D)) and

       A4: c <= ((f . O) `2 ) and

       A5: ((f . O) `2 ) <= d and

       A6: a <= ((f . I) `1 ) and

       A7: ((f . I) `1 ) <= b;

      

       A8: (h . (f . O)) = |[((A * ((f . O) `1 )) + B), ((C * ((f . O) `2 )) + D)]| by A3, JGRAPH_2:def 2;

      

       A9: (d - c) > 0 by A2, XREAL_1: 50;

      then

       A10: C > 0 by XREAL_1: 139;

      then (C * d) >= (C * ((f . O) `2 )) by A5, XREAL_1: 64;

      then

       A11: ((C * d) + D) >= ((C * ((f . O) `2 )) + D) by XREAL_1: 6;

      ((( - 1) - D) / C) = ((( - 1) + ((d + c) / (d - c))) / (2 / (d - c)))

      .= ((((( - 1) * (d - c)) + (d + c)) / (d - c)) / (2 / (d - c))) by A9, XCMPLX_1: 113

      .= ((((c + c) / (d - c)) / 2) * (d - c)) by XCMPLX_1: 82

      .= (((d - c) * ((c + c) / (d - c))) / 2)

      .= ((c + c) / 2) by A9, XCMPLX_1: 87

      .= c;

      then (C * ((( - 1) - D) / C)) <= (C * ((f . O) `2 )) by A4, A10, XREAL_1: 64;

      then (( - 1) - D) <= (C * ((f . O) `2 )) by A10, XCMPLX_1: 87;

      then

       A12: ((( - 1) - D) + D) <= ((C * ((f . O) `2 )) + D) by XREAL_1: 6;

      

       A13: ((C * d) + D) = (((2 * d) / (d - c)) + ( - ((d + c) / (d - c)))) by XCMPLX_1: 74

      .= (((2 * d) / (d - c)) + (( - (d + c)) / (d - c))) by XCMPLX_1: 187

      .= (((2 * d) + ( - (d + c))) / (d - c)) by XCMPLX_1: 62

      .= 1 by A9, XCMPLX_1: 60;

      

       A14: (h . (f . I)) = |[((A * ((f . I) `1 )) + B), ((C * ((f . I) `2 )) + D)]| by A3, JGRAPH_2:def 2;

      

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

      then

       A16: ((h * f) . I) = (h . (f . I)) by FUNCT_1: 13;

      

       A17: (b - a) > 0 by A1, XREAL_1: 50;

      then

       A18: A > 0 by XREAL_1: 139;

      then (A * b) >= (A * ((f . I) `1 )) by A7, XREAL_1: 64;

      then

       A19: ((A * b) + B) >= ((A * ((f . I) `1 )) + B) by XREAL_1: 6;

      (A * a) <= (A * ((f . I) `1 )) by A6, A18, XREAL_1: 64;

      then

       A20: ((A * a) + B) <= ((A * ((f . I) `1 )) + B) by XREAL_1: 7;

      

       A21: ((A * b) + B) = (((2 * b) / (b - a)) + ( - ((b + a) / (b - a)))) by XCMPLX_1: 74

      .= (((2 * b) / (b - a)) + (( - (b + a)) / (b - a))) by XCMPLX_1: 187

      .= (((2 * b) + ( - (b + a))) / (b - a)) by XCMPLX_1: 62

      .= 1 by A17, XCMPLX_1: 60;

      

       A22: ((A * a) + B) = (((2 * a) / (b - a)) + ( - ((b + a) / (b - a)))) by XCMPLX_1: 74

      .= (((2 * a) / (b - a)) + (( - (b + a)) / (b - a))) by XCMPLX_1: 187

      .= (((2 * a) + ( - (b + a))) / (b - a)) by XCMPLX_1: 62

      .= (( - (b - a)) / (b - a))

      .= ( - 1) by A17, XCMPLX_1: 197;

      ((h * f) . O) = (h . (f . O)) by A15, FUNCT_1: 13;

      hence thesis by A16, A8, A14, A22, A21, A13, A12, A11, A19, A20, EUCLID: 52;

    end;

    theorem :: JGRAPH_7:60

    

     Th60: for a,b,c,d be Real, h be Function of ( TOP-REAL 2), ( TOP-REAL 2), f be Function of I[01] , ( TOP-REAL 2), O,I be Point of I[01] st c < d & h = ( AffineMap ((2 / (b - a)),( - ((b + a) / (b - a))),(2 / (d - c)),( - ((d + c) / (d - c))))) & c <= ((f . O) `2 ) & ((f . O) `2 ) <= d & c <= ((f . I) `2 ) & ((f . I) `2 ) <= d holds ( - 1) <= (((h * f) . O) `2 ) & (((h * f) . O) `2 ) <= 1 & ( - 1) <= (((h * f) . I) `2 ) & (((h * f) . I) `2 ) <= 1

    proof

      let a,b,c,d be Real, h be Function of ( TOP-REAL 2), ( TOP-REAL 2), f be Function of I[01] , ( TOP-REAL 2), O,I be Point of I[01] ;

      set A = (2 / (b - a)), B = ( - ((b + a) / (b - a))), C = (2 / (d - c)), D = ( - ((d + c) / (d - c)));

      assume that

       A1: c < d and

       A2: h = ( AffineMap (A,B,C,D)) and

       A3: c <= ((f . O) `2 ) and

       A4: ((f . O) `2 ) <= d and

       A5: c <= ((f . I) `2 ) and

       A6: ((f . I) `2 ) <= d;

      

       A7: (h . (f . O)) = |[((A * ((f . O) `1 )) + B), ((C * ((f . O) `2 )) + D)]| by A2, JGRAPH_2:def 2;

      

       A8: (d - c) > 0 by A1, XREAL_1: 50;

      then

       A9: C > 0 by XREAL_1: 139;

      then (C * d) >= (C * ((f . O) `2 )) by A4, XREAL_1: 64;

      then

       A10: ((C * d) + D) >= ((C * ((f . O) `2 )) + D) by XREAL_1: 6;

      ((( - 1) - D) / C) = ((( - 1) + ((d + c) / (d - c))) / (2 / (d - c)))

      .= ((((( - 1) * (d - c)) + (d + c)) / (d - c)) / (2 / (d - c))) by A8, XCMPLX_1: 113

      .= ((d - c) * (((c + c) / (d - c)) / 2)) by XCMPLX_1: 82

      .= (((d - c) * ((c + c) / (d - c))) / 2)

      .= ((c + c) / 2) by A8, XCMPLX_1: 87

      .= c;

      then (C * ((( - 1) - D) / C)) <= (C * ((f . O) `2 )) by A3, A9, XREAL_1: 64;

      then (( - 1) - D) <= (C * ((f . O) `2 )) by A9, XCMPLX_1: 87;

      then

       A11: ((( - 1) - D) + D) <= ((C * ((f . O) `2 )) + D) by XREAL_1: 6;

      (C * c) <= (C * ((f . I) `2 )) by A5, A9, XREAL_1: 64;

      then

       A12: ((C * c) + D) <= ((C * ((f . I) `2 )) + D) by XREAL_1: 7;

      

       A13: (h . (f . I)) = |[((A * ((f . I) `1 )) + B), ((C * ((f . I) `2 )) + D)]| by A2, JGRAPH_2:def 2;

      

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

      then

       A15: ((h * f) . I) = (h . (f . I)) by FUNCT_1: 13;

      (C * d) >= (C * ((f . I) `2 )) by A6, A9, XREAL_1: 64;

      then

       A16: ((C * d) + D) >= ((C * ((f . I) `2 )) + D) by XREAL_1: 6;

      

       A17: ((C * d) + D) = (((2 * d) / (d - c)) + ( - ((d + c) / (d - c)))) by XCMPLX_1: 74

      .= (((2 * d) / (d - c)) + (( - (d + c)) / (d - c))) by XCMPLX_1: 187

      .= (((d + d) + ( - (d + c))) / (d - c)) by XCMPLX_1: 62

      .= 1 by A8, XCMPLX_1: 60;

      

       A18: ((C * c) + D) = (((2 * c) / (d - c)) + ( - ((d + c) / (d - c)))) by XCMPLX_1: 74

      .= (((2 * c) / (d - c)) + (( - (d + c)) / (d - c))) by XCMPLX_1: 187

      .= (((c + c) + ( - (d + c))) / (d - c)) by XCMPLX_1: 62

      .= (( - (d - c)) / (d - c))

      .= ( - 1) by A8, XCMPLX_1: 197;

      ((h * f) . O) = (h . (f . O)) by A14, FUNCT_1: 13;

      hence thesis by A15, A7, A13, A18, A17, A11, A10, A16, A12, EUCLID: 52;

    end;

    theorem :: JGRAPH_7:61

    

     Th61: for a,b,c,d be Real, h be Function of ( TOP-REAL 2), ( TOP-REAL 2), f be Function of I[01] , ( TOP-REAL 2), O,I be Point of I[01] st a < b & c < d & h = ( AffineMap ((2 / (b - a)),( - ((b + a) / (b - a))),(2 / (d - c)),( - ((d + c) / (d - c))))) & c <= ((f . O) `2 ) & ((f . O) `2 ) <= d & a < ((f . I) `1 ) & ((f . I) `1 ) <= b holds ( - 1) <= (((h * f) . O) `2 ) & (((h * f) . O) `2 ) <= 1 & ( - 1) < (((h * f) . I) `1 ) & (((h * f) . I) `1 ) <= 1

    proof

      let a,b,c,d be Real, h be Function of ( TOP-REAL 2), ( TOP-REAL 2), f be Function of I[01] , ( TOP-REAL 2), O,I be Point of I[01] ;

      set A = (2 / (b - a)), B = ( - ((b + a) / (b - a))), C = (2 / (d - c)), D = ( - ((d + c) / (d - c)));

      assume that

       A1: a < b and

       A2: c < d and

       A3: h = ( AffineMap (A,B,C,D)) and

       A4: c <= ((f . O) `2 ) and

       A5: ((f . O) `2 ) <= d and

       A6: a < ((f . I) `1 ) and

       A7: ((f . I) `1 ) <= b;

      

       A8: (h . (f . O)) = |[((A * ((f . O) `1 )) + B), ((C * ((f . O) `2 )) + D)]| by A3, JGRAPH_2:def 2;

      

       A9: (d - c) > 0 by A2, XREAL_1: 50;

      then

       A10: C > 0 by XREAL_1: 139;

      then (C * d) >= (C * ((f . O) `2 )) by A5, XREAL_1: 64;

      then

       A11: ((C * d) + D) >= ((C * ((f . O) `2 )) + D) by XREAL_1: 6;

      ((( - 1) - D) / C) = ((( - 1) + ((d + c) / (d - c))) / (2 / (d - c)))

      .= ((((( - 1) * (d - c)) + (d + c)) / (d - c)) / (2 / (d - c))) by A9, XCMPLX_1: 113

      .= ((d - c) * (((c + c) / (d - c)) / 2)) by XCMPLX_1: 82

      .= (((d - c) * ((c + c) / (d - c))) / 2)

      .= ((c + c) / 2) by A9, XCMPLX_1: 87

      .= c;

      then (C * ((( - 1) - D) / C)) <= (C * ((f . O) `2 )) by A4, A10, XREAL_1: 64;

      then (( - 1) - D) <= (C * ((f . O) `2 )) by A10, XCMPLX_1: 87;

      then

       A12: ((( - 1) - D) + D) <= ((C * ((f . O) `2 )) + D) by XREAL_1: 6;

      

       A13: ((C * d) + D) = (((2 * d) / (d - c)) + ( - ((d + c) / (d - c)))) by XCMPLX_1: 74

      .= (((2 * d) / (d - c)) + (( - (d + c)) / (d - c))) by XCMPLX_1: 187

      .= (((d + d) + ( - (d + c))) / (d - c)) by XCMPLX_1: 62

      .= 1 by A9, XCMPLX_1: 60;

      

       A14: (h . (f . I)) = |[((A * ((f . I) `1 )) + B), ((C * ((f . I) `2 )) + D)]| by A3, JGRAPH_2:def 2;

      

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

      then

       A16: ((h * f) . I) = (h . (f . I)) by FUNCT_1: 13;

      

       A17: (b - a) > 0 by A1, XREAL_1: 50;

      then

       A18: A > 0 by XREAL_1: 139;

      then (A * b) >= (A * ((f . I) `1 )) by A7, XREAL_1: 64;

      then

       A19: ((A * b) + B) >= ((A * ((f . I) `1 )) + B) by XREAL_1: 6;

      (A * a) < (A * ((f . I) `1 )) by A6, A18, XREAL_1: 68;

      then

       A20: ((A * a) + B) < ((A * ((f . I) `1 )) + B) by XREAL_1: 8;

      

       A21: ((A * b) + B) = (((2 * b) / (b - a)) + ( - ((b + a) / (b - a)))) by XCMPLX_1: 74

      .= (((2 * b) / (b - a)) + (( - (b + a)) / (b - a))) by XCMPLX_1: 187

      .= (((b + b) + ( - (b + a))) / (b - a)) by XCMPLX_1: 62

      .= 1 by A17, XCMPLX_1: 60;

      

       A22: ((A * a) + B) = (((2 * a) / (b - a)) + ( - ((b + a) / (b - a)))) by XCMPLX_1: 74

      .= (((2 * a) / (b - a)) + (( - (b + a)) / (b - a))) by XCMPLX_1: 187

      .= (((a + a) + ( - (b + a))) / (b - a)) by XCMPLX_1: 62

      .= (( - (b - a)) / (b - a))

      .= ( - 1) by A17, XCMPLX_1: 197;

      ((h * f) . O) = (h . (f . O)) by A15, FUNCT_1: 13;

      hence thesis by A16, A8, A14, A22, A13, A21, A12, A11, A19, A20, EUCLID: 52;

    end;

    theorem :: JGRAPH_7:62

    

     Th62: for a,b,c,d be Real, h be Function of ( TOP-REAL 2), ( TOP-REAL 2), f be Function of I[01] , ( TOP-REAL 2), O,I be Point of I[01] st a < b & h = ( AffineMap ((2 / (b - a)),( - ((b + a) / (b - a))),(2 / (d - c)),( - ((d + c) / (d - c))))) & a <= ((f . O) `1 ) & ((f . O) `1 ) < ((f . I) `1 ) & ((f . I) `1 ) <= b holds ( - 1) <= (((h * f) . O) `1 ) & (((h * f) . O) `1 ) < (((h * f) . I) `1 ) & (((h * f) . I) `1 ) <= 1

    proof

      let a,b,c,d be Real, h be Function of ( TOP-REAL 2), ( TOP-REAL 2), f be Function of I[01] , ( TOP-REAL 2), O,I be Point of I[01] ;

      set A = (2 / (b - a)), B = ( - ((b + a) / (b - a))), C = (2 / (d - c)), D = ( - ((d + c) / (d - c)));

      assume that

       A1: a < b and

       A2: h = ( AffineMap (A,B,C,D)) and

       A3: a <= ((f . O) `1 ) and

       A4: ((f . O) `1 ) < ((f . I) `1 ) and

       A5: ((f . I) `1 ) <= b;

      

       A6: (h . (f . O)) = |[((A * ((f . O) `1 )) + B), ((C * ((f . O) `2 )) + D)]| by A2, JGRAPH_2:def 2;

      

       A7: (b - a) > 0 by A1, XREAL_1: 50;

      then

       A8: A > 0 by XREAL_1: 139;

      ((1 - B) / A) = ((1 + ((b + a) / (b - a))) / (2 / (b - a)))

      .= ((((1 * (b - a)) + (b + a)) / (b - a)) / (2 / (b - a))) by A7, XCMPLX_1: 113

      .= ((b - a) * (((b + b) / (b - a)) / 2)) by XCMPLX_1: 82

      .= (((b - a) * ((b + b) / (b - a))) / 2)

      .= ((b + b) / 2) by A7, XCMPLX_1: 87

      .= b;

      then (A * ((1 - B) / A)) >= (A * ((f . I) `1 )) by A5, A8, XREAL_1: 64;

      then (1 - B) >= (A * ((f . I) `1 )) by A8, XCMPLX_1: 87;

      then

       A9: ((1 - B) + B) >= ((A * ((f . I) `1 )) + B) by XREAL_1: 6;

      

       A10: ( dom f) = the carrier of I[01] by FUNCT_2:def 1;

      then

       A11: ((h * f) . O) = (h . (f . O)) by FUNCT_1: 13;

      

       A12: ((h * f) . I) = (h . (f . I)) by A10, FUNCT_1: 13;

      ((( - 1) - B) / A) = ((( - 1) + ((b + a) / (b - a))) / (2 / (b - a)))

      .= ((((( - 1) * (b - a)) + (b + a)) / (b - a)) / (2 / (b - a))) by A7, XCMPLX_1: 113

      .= ((b - a) * (((a + a) / (b - a)) / 2)) by XCMPLX_1: 82

      .= (((b - a) * ((a + a) / (b - a))) / 2)

      .= ((a + a) / 2) by A7, XCMPLX_1: 87

      .= a;

      then (A * ((( - 1) - B) / A)) <= (A * ((f . O) `1 )) by A3, A8, XREAL_1: 64;

      then (( - 1) - B) <= (A * ((f . O) `1 )) by A8, XCMPLX_1: 87;

      then

       A13: ((( - 1) - B) + B) <= ((A * ((f . O) `1 )) + B) by XREAL_1: 6;

      

       A14: (h . (f . I)) = |[((A * ((f . I) `1 )) + B), ((C * ((f . I) `2 )) + D)]| by A2, JGRAPH_2:def 2;

      (A * ((f . O) `1 )) < (A * ((f . I) `1 )) by A4, A8, XREAL_1: 68;

      then ((A * ((f . O) `1 )) + B) < ((A * ((f . I) `1 )) + B) by XREAL_1: 8;

      then ((A * ((f . O) `1 )) + B) < (((h * f) . I) `1 ) by A12, A14, EUCLID: 52;

      hence thesis by A11, A12, A6, A14, A13, A9, EUCLID: 52;

    end;

    theorem :: JGRAPH_7:63

    

     Th63: for a,b,c,d be Real, h be Function of ( TOP-REAL 2), ( TOP-REAL 2), f be Function of I[01] , ( TOP-REAL 2), O,I be Point of I[01] st a < b & c < d & h = ( AffineMap ((2 / (b - a)),( - ((b + a) / (b - a))),(2 / (d - c)),( - ((d + c) / (d - c))))) & a <= ((f . O) `1 ) & ((f . O) `1 ) <= b & c <= ((f . I) `2 ) & ((f . I) `2 ) <= d holds ( - 1) <= (((h * f) . O) `1 ) & (((h * f) . O) `1 ) <= 1 & ( - 1) <= (((h * f) . I) `2 ) & (((h * f) . I) `2 ) <= 1

    proof

      let a,b,c,d be Real, h be Function of ( TOP-REAL 2), ( TOP-REAL 2), f be Function of I[01] , ( TOP-REAL 2), O,I be Point of I[01] ;

      set A = (2 / (b - a)), B = ( - ((b + a) / (b - a))), C = (2 / (d - c)), D = ( - ((d + c) / (d - c)));

      assume that

       A1: a < b and

       A2: c < d and

       A3: h = ( AffineMap (A,B,C,D)) and

       A4: a <= ((f . O) `1 ) and

       A5: ((f . O) `1 ) <= b and

       A6: c <= ((f . I) `2 ) and

       A7: ((f . I) `2 ) <= d;

      

       A8: (h . (f . O)) = |[((A * ((f . O) `1 )) + B), ((C * ((f . O) `2 )) + D)]| by A3, JGRAPH_2:def 2;

      

       A9: (b - a) > 0 by A1, XREAL_1: 50;

      then

       A10: A > 0 by XREAL_1: 139;

      ((( - 1) - B) / A) = ((( - 1) + ((b + a) / (b - a))) / (2 / (b - a)))

      .= ((((( - 1) * (b - a)) + (b + a)) / (b - a)) / (2 / (b - a))) by A9, XCMPLX_1: 113

      .= ((b - a) * (((a + a) / (b - a)) / 2)) by XCMPLX_1: 82

      .= (((b - a) * ((a + a) / (b - a))) / 2)

      .= ((a + a) / 2) by A9, XCMPLX_1: 87

      .= a;

      then (A * ((( - 1) - B) / A)) <= (A * ((f . O) `1 )) by A4, A10, XREAL_1: 64;

      then (( - 1) - B) <= (A * ((f . O) `1 )) by A10, XCMPLX_1: 87;

      then

       A11: ((( - 1) - B) + B) <= ((A * ((f . O) `1 )) + B) by XREAL_1: 6;

      

       A12: (d - c) > 0 by A2, XREAL_1: 50;

      then

       A13: C > 0 by XREAL_1: 139;

      ((( - 1) - D) / C) = ((( - 1) + ((d + c) / (d - c))) / (2 / (d - c)))

      .= ((((( - 1) * (d - c)) + (d + c)) / (d - c)) / (2 / (d - c))) by A12, XCMPLX_1: 113

      .= ((d - c) * (((c + c) / (d - c)) / 2)) by XCMPLX_1: 82

      .= (((d - c) * ((c + c) / (d - c))) / 2)

      .= ((c + c) / 2) by A12, XCMPLX_1: 87

      .= c;

      then (C * ((( - 1) - D) / C)) <= (C * ((f . I) `2 )) by A6, A13, XREAL_1: 64;

      then (( - 1) - D) <= (C * ((f . I) `2 )) by A13, XCMPLX_1: 87;

      then

       A14: ((( - 1) - D) + D) <= ((C * ((f . I) `2 )) + D) by XREAL_1: 6;

      

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

      then

       A16: ((h * f) . I) = (h . (f . I)) by FUNCT_1: 13;

      ((1 - B) / A) = ((1 + ((b + a) / (b - a))) / (2 / (b - a)))

      .= ((((1 * (b - a)) + (b + a)) / (b - a)) / (2 / (b - a))) by A9, XCMPLX_1: 113

      .= ((b - a) * (((b + b) / (b - a)) / 2)) by XCMPLX_1: 82

      .= (((b - a) * ((b + b) / (b - a))) / 2)

      .= ((b + b) / 2) by A9, XCMPLX_1: 87

      .= b;

      then (A * ((1 - B) / A)) >= (A * ((f . O) `1 )) by A5, A10, XREAL_1: 64;

      then (1 - B) >= (A * ((f . O) `1 )) by A10, XCMPLX_1: 87;

      then

       A17: ((1 - B) + B) >= ((A * ((f . O) `1 )) + B) by XREAL_1: 6;

      

       A18: (h . (f . I)) = |[((A * ((f . I) `1 )) + B), ((C * ((f . I) `2 )) + D)]| by A3, JGRAPH_2:def 2;

      ((1 - D) / C) = ((1 + ((d + c) / (d - c))) / (2 / (d - c)))

      .= ((((1 * (d - c)) + (d + c)) / (d - c)) / (2 / (d - c))) by A12, XCMPLX_1: 113

      .= ((d - c) * (((d + d) / (d - c)) / 2)) by XCMPLX_1: 82

      .= (((d - c) * ((d + d) / (d - c))) / 2)

      .= ((d + d) / 2) by A12, XCMPLX_1: 87

      .= d;

      then (C * ((1 - D) / C)) >= (C * ((f . I) `2 )) by A7, A13, XREAL_1: 64;

      then (1 - D) >= (C * ((f . I) `2 )) by A13, XCMPLX_1: 87;

      then

       A19: ((1 - D) + D) >= ((C * ((f . I) `2 )) + D) by XREAL_1: 6;

      ((h * f) . O) = (h . (f . O)) by A15, FUNCT_1: 13;

      hence thesis by A16, A8, A18, A11, A17, A19, A14, EUCLID: 52;

    end;

    theorem :: JGRAPH_7:64

    

     Th64: for a,b,c,d be Real, h be Function of ( TOP-REAL 2), ( TOP-REAL 2), f be Function of I[01] , ( TOP-REAL 2), O,I be Point of I[01] st a < b & h = ( AffineMap ((2 / (b - a)),( - ((b + a) / (b - a))),(2 / (d - c)),( - ((d + c) / (d - c))))) & a <= ((f . O) `1 ) & ((f . O) `1 ) <= b & a < ((f . I) `1 ) & ((f . I) `1 ) <= b holds ( - 1) <= (((h * f) . O) `1 ) & (((h * f) . O) `1 ) <= 1 & ( - 1) < (((h * f) . I) `1 ) & (((h * f) . I) `1 ) <= 1

    proof

      let a,b,c,d be Real, h be Function of ( TOP-REAL 2), ( TOP-REAL 2), f be Function of I[01] , ( TOP-REAL 2), O,I be Point of I[01] ;

      set A = (2 / (b - a)), B = ( - ((b + a) / (b - a))), C = (2 / (d - c)), D = ( - ((d + c) / (d - c)));

      assume that

       A1: a < b and

       A2: h = ( AffineMap (A,B,C,D)) and

       A3: a <= ((f . O) `1 ) and

       A4: ((f . O) `1 ) <= b and

       A5: a < ((f . I) `1 ) and

       A6: ((f . I) `1 ) <= b;

      

       A7: (b - a) > 0 by A1, XREAL_1: 50;

      then

       A8: A > 0 by XREAL_1: 139;

      

       A9: ((1 - B) / A) = ((1 + ((b + a) / (b - a))) / (2 / (b - a)))

      .= ((((1 * (b - a)) + (b + a)) / (b - a)) / (2 / (b - a))) by A7, XCMPLX_1: 113

      .= ((b - a) * (((b + b) / (b - a)) / 2)) by XCMPLX_1: 82

      .= (((b - a) * ((b + b) / (b - a))) / 2)

      .= ((b + b) / 2) by A7, XCMPLX_1: 87

      .= b;

      then (A * ((1 - B) / A)) >= (A * ((f . O) `1 )) by A4, A8, XREAL_1: 64;

      then (1 - B) >= (A * ((f . O) `1 )) by A8, XCMPLX_1: 87;

      then

       A10: ((1 - B) + B) >= ((A * ((f . O) `1 )) + B) by XREAL_1: 6;

      (A * ((1 - B) / A)) >= (A * ((f . I) `1 )) by A6, A8, A9, XREAL_1: 64;

      then (1 - B) >= (A * ((f . I) `1 )) by A8, XCMPLX_1: 87;

      then

       A11: ((1 - B) + B) >= ((A * ((f . I) `1 )) + B) by XREAL_1: 6;

      

       A12: (h . (f . I)) = |[((A * ((f . I) `1 )) + B), ((C * ((f . I) `2 )) + D)]| by A2, JGRAPH_2:def 2;

      

       A13: (h . (f . O)) = |[((A * ((f . O) `1 )) + B), ((C * ((f . O) `2 )) + D)]| by A2, JGRAPH_2:def 2;

      

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

      then

       A15: ((h * f) . I) = (h . (f . I)) by FUNCT_1: 13;

      

       A16: ((( - 1) - B) / A) = ((( - 1) + ((b + a) / (b - a))) / (2 / (b - a)))

      .= ((((( - 1) * (b - a)) + (b + a)) / (b - a)) / (2 / (b - a))) by A7, XCMPLX_1: 113

      .= ((b - a) * (((a + a) / (b - a)) / 2)) by XCMPLX_1: 82

      .= (((b - a) * ((a + a) / (b - a))) / 2)

      .= ((a + a) / 2) by A7, XCMPLX_1: 87

      .= a;

      then (A * ((( - 1) - B) / A)) <= (A * ((f . O) `1 )) by A3, A8, XREAL_1: 64;

      then (( - 1) - B) <= (A * ((f . O) `1 )) by A8, XCMPLX_1: 87;

      then

       A17: ((( - 1) - B) + B) <= ((A * ((f . O) `1 )) + B) by XREAL_1: 6;

      (A * ((( - 1) - B) / A)) < (A * ((f . I) `1 )) by A5, A8, A16, XREAL_1: 68;

      then (( - 1) - B) < (A * ((f . I) `1 )) by A8, XCMPLX_1: 87;

      then

       A18: ((( - 1) - B) + B) < ((A * ((f . I) `1 )) + B) by XREAL_1: 8;

      ((h * f) . O) = (h . (f . O)) by A14, FUNCT_1: 13;

      hence thesis by A15, A13, A12, A17, A10, A11, A18, EUCLID: 52;

    end;

    theorem :: JGRAPH_7:65

    

     Th65: for a,b,c,d be Real, h be Function of ( TOP-REAL 2), ( TOP-REAL 2), f be Function of I[01] , ( TOP-REAL 2), O,I be Point of I[01] st c < d & h = ( AffineMap ((2 / (b - a)),( - ((b + a) / (b - a))),(2 / (d - c)),( - ((d + c) / (d - c))))) & d >= ((f . O) `2 ) & ((f . O) `2 ) > ((f . I) `2 ) & ((f . I) `2 ) >= c holds 1 >= (((h * f) . O) `2 ) & (((h * f) . O) `2 ) > (((h * f) . I) `2 ) & (((h * f) . I) `2 ) >= ( - 1)

    proof

      let a,b,c,d be Real, h be Function of ( TOP-REAL 2), ( TOP-REAL 2), f be Function of I[01] , ( TOP-REAL 2), O,I be Point of I[01] ;

      set A = (2 / (b - a)), B = ( - ((b + a) / (b - a))), C = (2 / (d - c)), D = ( - ((d + c) / (d - c)));

      assume that

       A1: c < d and

       A2: h = ( AffineMap (A,B,C,D)) and

       A3: d >= ((f . O) `2 ) and

       A4: ((f . O) `2 ) > ((f . I) `2 ) and

       A5: ((f . I) `2 ) >= c;

      

       A6: (h . (f . O)) = |[((A * ((f . O) `1 )) + B), ((C * ((f . O) `2 )) + D)]| by A2, JGRAPH_2:def 2;

      

       A7: (d - c) > 0 by A1, XREAL_1: 50;

      then

       A8: C > 0 by XREAL_1: 139;

      ((( - 1) - D) / C) = ((( - 1) + ((d + c) / (d - c))) / (2 / (d - c)))

      .= ((((( - 1) * (d - c)) + (d + c)) / (d - c)) / (2 / (d - c))) by A7, XCMPLX_1: 113

      .= ((d - c) * (((c + c) / (d - c)) / 2)) by XCMPLX_1: 82

      .= (((d - c) * ((c + c) / (d - c))) / 2)

      .= ((c + c) / 2) by A7, XCMPLX_1: 87

      .= c;

      then (C * ((( - 1) - D) / C)) <= (C * ((f . I) `2 )) by A5, A8, XREAL_1: 64;

      then (( - 1) - D) <= (C * ((f . I) `2 )) by A8, XCMPLX_1: 87;

      then

       A9: ((( - 1) - D) + D) <= ((C * ((f . I) `2 )) + D) by XREAL_1: 6;

      

       A10: ( dom f) = the carrier of I[01] by FUNCT_2:def 1;

      then

       A11: ((h * f) . O) = (h . (f . O)) by FUNCT_1: 13;

      

       A12: ((h * f) . I) = (h . (f . I)) by A10, FUNCT_1: 13;

      ((1 - D) / C) = ((1 + ((d + c) / (d - c))) / (2 / (d - c)))

      .= ((((1 * (d - c)) + (d + c)) / (d - c)) / (2 / (d - c))) by A7, XCMPLX_1: 113

      .= ((d - c) * (((d + d) / (d - c)) / 2)) by XCMPLX_1: 82

      .= (((d - c) * ((d + d) / (d - c))) / 2)

      .= ((d + d) / 2) by A7, XCMPLX_1: 87

      .= d;

      then (C * ((1 - D) / C)) >= (C * ((f . O) `2 )) by A3, A8, XREAL_1: 64;

      then (1 - D) >= (C * ((f . O) `2 )) by A8, XCMPLX_1: 87;

      then

       A13: ((1 - D) + D) >= ((C * ((f . O) `2 )) + D) by XREAL_1: 6;

      

       A14: (h . (f . I)) = |[((A * ((f . I) `1 )) + B), ((C * ((f . I) `2 )) + D)]| by A2, JGRAPH_2:def 2;

      (C * ((f . O) `2 )) > (C * ((f . I) `2 )) by A4, A8, XREAL_1: 68;

      then ((C * ((f . O) `2 )) + D) > ((C * ((f . I) `2 )) + D) by XREAL_1: 8;

      then ((C * ((f . O) `2 )) + D) > (((h * f) . I) `2 ) by A12, A14, EUCLID: 52;

      hence thesis by A11, A12, A6, A14, A13, A9, EUCLID: 52;

    end;

    theorem :: JGRAPH_7:66

    

     Th66: for a,b,c,d be Real, h be Function of ( TOP-REAL 2), ( TOP-REAL 2), f be Function of I[01] , ( TOP-REAL 2), O,I be Point of I[01] st a < b & c < d & h = ( AffineMap ((2 / (b - a)),( - ((b + a) / (b - a))),(2 / (d - c)),( - ((d + c) / (d - c))))) & c <= ((f . O) `2 ) & ((f . O) `2 ) <= d & a < ((f . I) `1 ) & ((f . I) `1 ) <= b holds ( - 1) <= (((h * f) . O) `2 ) & (((h * f) . O) `2 ) <= 1 & ( - 1) < (((h * f) . I) `1 ) & (((h * f) . I) `1 ) <= 1

    proof

      let a,b,c,d be Real, h be Function of ( TOP-REAL 2), ( TOP-REAL 2), f be Function of I[01] , ( TOP-REAL 2), O,I be Point of I[01] ;

      set A = (2 / (b - a)), B = ( - ((b + a) / (b - a))), C = (2 / (d - c)), D = ( - ((d + c) / (d - c)));

      assume that

       A1: a < b and

       A2: c < d and

       A3: h = ( AffineMap (A,B,C,D)) and

       A4: c <= ((f . O) `2 ) and

       A5: ((f . O) `2 ) <= d and

       A6: a < ((f . I) `1 ) and

       A7: ((f . I) `1 ) <= b;

      

       A8: (h . (f . O)) = |[((A * ((f . O) `1 )) + B), ((C * ((f . O) `2 )) + D)]| by A3, JGRAPH_2:def 2;

      

       A9: (b - a) > 0 by A1, XREAL_1: 50;

      then

       A10: A > 0 by XREAL_1: 139;

      ((( - 1) - B) / A) = ((( - 1) + ((b + a) / (b - a))) / (2 / (b - a)))

      .= ((((( - 1) * (b - a)) + (b + a)) / (b - a)) / (2 / (b - a))) by A9, XCMPLX_1: 113

      .= ((b - a) * (((a + a) / (b - a)) / 2)) by XCMPLX_1: 82

      .= (((b - a) * ((a + a) / (b - a))) / 2)

      .= ((a + a) / 2) by A9, XCMPLX_1: 87

      .= a;

      then (A * ((( - 1) - B) / A)) < (A * ((f . I) `1 )) by A6, A10, XREAL_1: 68;

      then (( - 1) - B) < (A * ((f . I) `1 )) by A10, XCMPLX_1: 87;

      then

       A11: ((( - 1) - B) + B) < ((A * ((f . I) `1 )) + B) by XREAL_1: 8;

      

       A12: (d - c) > 0 by A2, XREAL_1: 50;

      then

       A13: C > 0 by XREAL_1: 139;

      ((( - 1) - D) / C) = ((( - 1) + ((d + c) / (d - c))) / (2 / (d - c)))

      .= ((((( - 1) * (d - c)) + (d + c)) / (d - c)) / (2 / (d - c))) by A12, XCMPLX_1: 113

      .= ((d - c) * (((c + c) / (d - c)) / 2)) by XCMPLX_1: 82

      .= (((d - c) * ((c + c) / (d - c))) / 2)

      .= ((c + c) / 2) by A12, XCMPLX_1: 87

      .= c;

      then (C * ((( - 1) - D) / C)) <= (C * ((f . O) `2 )) by A4, A13, XREAL_1: 64;

      then (( - 1) - D) <= (C * ((f . O) `2 )) by A13, XCMPLX_1: 87;

      then

       A14: ((( - 1) - D) + D) <= ((C * ((f . O) `2 )) + D) by XREAL_1: 6;

      

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

      then

       A16: ((h * f) . I) = (h . (f . I)) by FUNCT_1: 13;

      ((1 - B) / A) = ((1 + ((b + a) / (b - a))) / (2 / (b - a)))

      .= ((((1 * (b - a)) + (b + a)) / (b - a)) / (2 / (b - a))) by A9, XCMPLX_1: 113

      .= ((b - a) * (((b + b) / (b - a)) / 2)) by XCMPLX_1: 82

      .= (((b - a) * ((b + b) / (b - a))) / 2)

      .= ((b + b) / 2) by A9, XCMPLX_1: 87

      .= b;

      then (A * ((1 - B) / A)) >= (A * ((f . I) `1 )) by A7, A10, XREAL_1: 64;

      then (1 - B) >= (A * ((f . I) `1 )) by A10, XCMPLX_1: 87;

      then

       A17: ((1 - B) + B) >= ((A * ((f . I) `1 )) + B) by XREAL_1: 6;

      

       A18: (h . (f . I)) = |[((A * ((f . I) `1 )) + B), ((C * ((f . I) `2 )) + D)]| by A3, JGRAPH_2:def 2;

      ((1 - D) / C) = ((1 + ((d + c) / (d - c))) / (2 / (d - c)))

      .= ((((1 * (d - c)) + (d + c)) / (d - c)) / (2 / (d - c))) by A12, XCMPLX_1: 113

      .= ((d - c) * (((d + d) / (d - c)) / 2)) by XCMPLX_1: 82

      .= (((d - c) * ((d + d) / (d - c))) / 2)

      .= ((d + d) / 2) by A12, XCMPLX_1: 87

      .= d;

      then (C * ((1 - D) / C)) >= (C * ((f . O) `2 )) by A5, A13, XREAL_1: 64;

      then (1 - D) >= (C * ((f . O) `2 )) by A13, XCMPLX_1: 87;

      then

       A19: ((1 - D) + D) >= ((C * ((f . O) `2 )) + D) by XREAL_1: 6;

      ((h * f) . O) = (h . (f . O)) by A15, FUNCT_1: 13;

      hence thesis by A16, A8, A18, A14, A19, A17, A11, EUCLID: 52;

    end;

    theorem :: JGRAPH_7:67

    

     Th67: for a,b,c,d be Real, h be Function of ( TOP-REAL 2), ( TOP-REAL 2), f be Function of I[01] , ( TOP-REAL 2), O,I be Point of I[01] st a < b & h = ( AffineMap ((2 / (b - a)),( - ((b + a) / (b - a))),(2 / (d - c)),( - ((d + c) / (d - c))))) & a < ((f . I) `1 ) & ((f . I) `1 ) < ((f . O) `1 ) & ((f . O) `1 ) <= b holds ( - 1) < (((h * f) . I) `1 ) & (((h * f) . I) `1 ) < (((h * f) . O) `1 ) & (((h * f) . O) `1 ) <= 1

    proof

      let a,b,c,d be Real, h be Function of ( TOP-REAL 2), ( TOP-REAL 2), f be Function of I[01] , ( TOP-REAL 2), O,I be Point of I[01] ;

      set A = (2 / (b - a)), B = ( - ((b + a) / (b - a))), C = (2 / (d - c)), D = ( - ((d + c) / (d - c)));

      assume that

       A1: a < b and

       A2: h = ( AffineMap (A,B,C,D)) and

       A3: a < ((f . I) `1 ) and

       A4: ((f . I) `1 ) < ((f . O) `1 ) and

       A5: ((f . O) `1 ) <= b;

      

       A6: (h . (f . O)) = |[((A * ((f . O) `1 )) + B), ((C * ((f . O) `2 )) + D)]| by A2, JGRAPH_2:def 2;

      

       A7: (b - a) > 0 by A1, XREAL_1: 50;

      then

       A8: A > 0 by XREAL_1: 139;

      ((1 - B) / A) = ((1 + ((b + a) / (b - a))) / (2 / (b - a)))

      .= ((((1 * (b - a)) + (b + a)) / (b - a)) / (2 / (b - a))) by A7, XCMPLX_1: 113

      .= ((b - a) * (((b + b) / (b - a)) / 2)) by XCMPLX_1: 82

      .= (((b - a) * ((b + b) / (b - a))) / 2)

      .= ((b + b) / 2) by A7, XCMPLX_1: 87

      .= b;

      then (A * ((1 - B) / A)) >= (A * ((f . O) `1 )) by A5, A8, XREAL_1: 64;

      then (1 - B) >= (A * ((f . O) `1 )) by A8, XCMPLX_1: 87;

      then

       A9: ((1 - B) + B) >= ((A * ((f . O) `1 )) + B) by XREAL_1: 6;

      

       A10: ( dom f) = the carrier of I[01] by FUNCT_2:def 1;

      then

       A11: ((h * f) . O) = (h . (f . O)) by FUNCT_1: 13;

      

       A12: ((h * f) . I) = (h . (f . I)) by A10, FUNCT_1: 13;

      ((( - 1) - B) / A) = ((( - 1) + ((b + a) / (b - a))) / (2 / (b - a)))

      .= ((((( - 1) * (b - a)) + (b + a)) / (b - a)) / (2 / (b - a))) by A7, XCMPLX_1: 113

      .= ((b - a) * (((a + a) / (b - a)) / 2)) by XCMPLX_1: 82

      .= (((b - a) * ((a + a) / (b - a))) / 2)

      .= ((a + a) / 2) by A7, XCMPLX_1: 87

      .= a;

      then (A * ((( - 1) - B) / A)) < (A * ((f . I) `1 )) by A3, A8, XREAL_1: 68;

      then (( - 1) - B) < (A * ((f . I) `1 )) by A8, XCMPLX_1: 87;

      then

       A13: ((( - 1) - B) + B) < ((A * ((f . I) `1 )) + B) by XREAL_1: 8;

      

       A14: (h . (f . I)) = |[((A * ((f . I) `1 )) + B), ((C * ((f . I) `2 )) + D)]| by A2, JGRAPH_2:def 2;

      (A * ((f . O) `1 )) > (A * ((f . I) `1 )) by A4, A8, XREAL_1: 68;

      then ((A * ((f . O) `1 )) + B) > ((A * ((f . I) `1 )) + B) by XREAL_1: 8;

      then ((A * ((f . O) `1 )) + B) > (((h * f) . I) `1 ) by A12, A14, EUCLID: 52;

      hence thesis by A11, A12, A6, A14, A9, A13, EUCLID: 52;

    end;

    theorem :: JGRAPH_7:68

    

     Th68: for p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real, f,g be Function of I[01] , ( TOP-REAL 2) st a < b & c < d & (p1 `1 ) = a & (p2 `1 ) = a & (p3 `1 ) = a & (p4 `1 ) = a & c <= (p1 `2 ) & (p1 `2 ) < (p2 `2 ) & (p2 `2 ) < (p3 `2 ) & (p3 `2 ) < (p4 `2 ) & (p4 `2 ) <= d & (f . 0 ) = p1 & (f . 1) = p3 & (g . 0 ) = p2 & (g . 1) = p4 & f is continuous one-to-one & g is continuous one-to-one & ( rng f) c= ( closed_inside_of_rectangle (a,b,c,d)) & ( rng g) c= ( closed_inside_of_rectangle (a,b,c,d)) holds ( rng f) meets ( rng g)

    proof

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

      let p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real, f,g be Function of I[01] , ( TOP-REAL 2);

      assume that

       A1: a < b and

       A2: c < d and

       A3: (p1 `1 ) = a and

       A4: (p2 `1 ) = a and

       A5: (p3 `1 ) = a and

       A6: (p4 `1 ) = a and

       A7: c <= (p1 `2 ) and

       A8: (p1 `2 ) < (p2 `2 ) and

       A9: (p2 `2 ) < (p3 `2 ) and

       A10: (p3 `2 ) < (p4 `2 ) and

       A11: (p4 `2 ) <= d and

       A12: (f . 0 ) = p1 and

       A13: (f . 1) = p3 and

       A14: (g . 0 ) = p2 and

       A15: (g . 1) = p4 and

       A16: f is continuous one-to-one and

       A17: g is continuous one-to-one and

       A18: ( rng f) c= ( closed_inside_of_rectangle (a,b,c,d)) and

       A19: ( rng g) c= ( closed_inside_of_rectangle (a,b,c,d));

      set A = (2 / (b - a)), B = ( - ((b + a) / (b - a))), C = (2 / (d - c)), D = ( - ((d + c) / (d - c)));

      set h = ( AffineMap (A,B,C,D));

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

      

       A20: g2 is continuous one-to-one by A1, A2, A17, Th53;

      

       A21: ((g . O) `1 ) = a by A4, A14;

      

       A22: c < (p2 `2 ) by A7, A8, XXREAL_0: 2;

      (p2 `2 ) < (p4 `2 ) by A9, A10, XXREAL_0: 2;

      then

       A23: ((g2 . I) `2 ) <= 1 by A2, A11, A14, A15, A22, A21, Th58;

      

       A24: ( rng g2) c= ( closed_inside_of_rectangle (( - 1),1,( - 1),1)) by A1, A2, A19, Th52;

      

       A25: ((g2 . I) `1 ) = ( - 1) by A1, A6, A15, Th54;

      

       A26: ((g2 . O) `1 ) = ( - 1) by A1, A4, A14, Th54;

      

       A27: ( dom g) = the carrier of I[01] by FUNCT_2:def 1;

      then

       A28: (h . p2) = (g2 . O) by A14, FUNCT_1: 13;

      

       A29: (h . p4) = (g2 . I) by A15, A27, FUNCT_1: 13;

      (d - c) > 0 by A2, XREAL_1: 50;

      then

       A30: C > 0 by XREAL_1: 139;

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

      

       A31: f2 is continuous one-to-one by A1, A2, A16, Th53;

      

       A32: ( rng f2) c= ( closed_inside_of_rectangle (( - 1),1,( - 1),1)) by A1, A2, A18, Th52;

      

       A33: ((f2 . I) `1 ) = ( - 1) by A1, A5, A13, Th54;

      

       A34: ((f . I) `1 ) = a by A5, A13;

      

       A35: (p3 `2 ) < d by A10, A11, XXREAL_0: 2;

      (p1 `2 ) < (p3 `2 ) by A8, A9, XXREAL_0: 2;

      then

       A36: ( - 1) <= ((f2 . O) `2 ) by A2, A7, A12, A13, A35, A34, Th58;

      

       A37: ((f2 . O) `1 ) = ( - 1) by A1, A3, A12, Th54;

      set x = the Element of (( rng f2) /\ ( rng g2));

      

       A38: ( dom f) = the carrier of I[01] by FUNCT_2:def 1;

      then

       A39: (h . p3) = (f2 . I) by A13, FUNCT_1: 13;

      (b - a) > 0 by A1, XREAL_1: 50;

      then

       A40: A > 0 by XREAL_1: 139;

      then

       A41: ((h . p1) `2 ) < ((h . p2) `2 ) by A8, A30, Th51;

      

       A42: ((h . p3) `2 ) < ((h . p4) `2 ) by A10, A40, A30, Th51;

      

       A43: ((h . p2) `2 ) < ((h . p3) `2 ) by A9, A40, A30, Th51;

      (h . p1) = (f2 . O) by A12, A38, FUNCT_1: 13;

      then ((f2 . O),(g2 . O),(f2 . I),(g2 . I)) are_in_this_order_on ( rectangle (( - 1),1,( - 1),1)) by A41, A43, A42, A28, A39, A29, A37, A33, A36, A26, A25, A23, Th14;

      then ( rng f2) meets ( rng g2) by A31, A32, A20, A24, JGRAPH_6: 79;

      then

       A44: (( rng f2) /\ ( rng g2)) <> {} by XBOOLE_0:def 7;

      then x in ( rng g2) by XBOOLE_0:def 4;

      then

      consider z2 be object such that

       A45: z2 in ( dom g2) and

       A46: x = (g2 . z2) by FUNCT_1:def 3;

      

       A47: x = (h . (g . z2)) by A27, A45, A46, FUNCT_1: 13;

      h is being_homeomorphism by A40, A30, Th51;

      then

       A48: h is one-to-one by TOPS_2:def 5;

      x in ( rng f2) by A44, XBOOLE_0:def 4;

      then

      consider z1 be object such that

       A49: z1 in ( dom f2) and

       A50: x = (f2 . z1) by FUNCT_1:def 3;

      

       A51: (f . z1) in ( rng f) by A38, A49, FUNCT_1:def 3;

      

       A52: (g . z2) in ( rng g) by A27, A45, FUNCT_1:def 3;

      (g . z2) in the carrier of ( TOP-REAL 2) by A45, FUNCT_2: 5;

      then

       A53: (g . z2) in ( dom h) by FUNCT_2:def 1;

      (f . z1) in the carrier of ( TOP-REAL 2) by A49, FUNCT_2: 5;

      then

       A54: (f . z1) in ( dom h) by FUNCT_2:def 1;

      x = (h . (f . z1)) by A38, A49, A50, FUNCT_1: 13;

      then (f . z1) = (g . z2) by A47, A54, A53, A48, FUNCT_1:def 4;

      hence thesis by A51, A52, XBOOLE_0: 3;

    end;

    theorem :: JGRAPH_7:69

    for p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real, P,Q be Subset of ( TOP-REAL 2) st a < b & c < d & (p1 `1 ) = a & (p2 `1 ) = a & (p3 `1 ) = a & (p4 `1 ) = a & c <= (p1 `2 ) & (p1 `2 ) < (p2 `2 ) & (p2 `2 ) < (p3 `2 ) & (p3 `2 ) < (p4 `2 ) & (p4 `2 ) <= d & P is_an_arc_of (p1,p3) & Q is_an_arc_of (p2,p4) & P c= ( closed_inside_of_rectangle (a,b,c,d)) & Q c= ( closed_inside_of_rectangle (a,b,c,d)) holds P meets Q

    proof

      let p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real, P,Q be Subset of ( TOP-REAL 2);

      assume that

       A1: a < b and

       A2: c < d and

       A3: (p1 `1 ) = a and

       A4: (p2 `1 ) = a and

       A5: (p3 `1 ) = a and

       A6: (p4 `1 ) = a and

       A7: c <= (p1 `2 ) and

       A8: (p1 `2 ) < (p2 `2 ) and

       A9: (p2 `2 ) < (p3 `2 ) and

       A10: (p3 `2 ) < (p4 `2 ) and

       A11: (p4 `2 ) <= d and

       A12: P is_an_arc_of (p1,p3) and

       A13: Q is_an_arc_of (p2,p4) and

       A14: P c= ( closed_inside_of_rectangle (a,b,c,d)) and

       A15: Q c= ( closed_inside_of_rectangle (a,b,c,d));

      

       A16: ex g be Function of I[01] , ( TOP-REAL 2) st g is continuous one-to-one & ( rng g) = Q & (g . 0 ) = p2 & (g . 1) = p4 by A13, Th2;

      ex f be Function of I[01] , ( TOP-REAL 2) st f is continuous one-to-one & ( rng f) = P & (f . 0 ) = p1 & (f . 1) = p3 by A12, Th2;

      hence thesis by A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, A14, A15, A16, Th68;

    end;

    theorem :: JGRAPH_7:70

    

     Th70: for p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real, f,g be Function of I[01] , ( TOP-REAL 2) st a < b & c < d & (p1 `1 ) = a & (p2 `1 ) = a & (p3 `1 ) = a & (p4 `2 ) = d & c <= (p1 `2 ) & (p1 `2 ) < (p2 `2 ) & (p2 `2 ) < (p3 `2 ) & (p3 `2 ) <= d & a <= (p4 `1 ) & (p4 `1 ) <= b & (f . 0 ) = p1 & (f . 1) = p3 & (g . 0 ) = p2 & (g . 1) = p4 & f is continuous one-to-one & g is continuous one-to-one & ( rng f) c= ( closed_inside_of_rectangle (a,b,c,d)) & ( rng g) c= ( closed_inside_of_rectangle (a,b,c,d)) holds ( rng f) meets ( rng g)

    proof

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

      let p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real, f,g be Function of I[01] , ( TOP-REAL 2);

      assume that

       A1: a < b and

       A2: c < d and

       A3: (p1 `1 ) = a and

       A4: (p2 `1 ) = a and

       A5: (p3 `1 ) = a and

       A6: (p4 `2 ) = d and

       A7: c <= (p1 `2 ) and

       A8: (p1 `2 ) < (p2 `2 ) and

       A9: (p2 `2 ) < (p3 `2 ) and

       A10: (p3 `2 ) <= d and

       A11: a <= (p4 `1 ) and

       A12: (p4 `1 ) <= b and

       A13: (f . 0 ) = p1 and

       A14: (f . 1) = p3 and

       A15: (g . 0 ) = p2 and

       A16: (g . 1) = p4 and

       A17: f is continuous one-to-one and

       A18: g is continuous one-to-one and

       A19: ( rng f) c= ( closed_inside_of_rectangle (a,b,c,d)) and

       A20: ( rng g) c= ( closed_inside_of_rectangle (a,b,c,d));

      set A = (2 / (b - a)), B = ( - ((b + a) / (b - a))), C = (2 / (d - c)), D = ( - ((d + c) / (d - c)));

      set h = ( AffineMap (A,B,C,D));

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

      

       A21: (p1 `2 ) < (p3 `2 ) by A8, A9, XXREAL_0: 2;

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

      

       A22: ( - 1) <= ((g2 . I) `1 ) by A1, A2, A6, A11, A12, A16, Th59;

      

       A23: ((g2 . I) `2 ) = 1 by A2, A6, A16, Th55;

      

       A24: ((g2 . O) `1 ) = ( - 1) by A1, A4, A15, Th54;

      ((f . I) `1 ) = a by A5, A14;

      then

       A25: ( - 1) <= ((f2 . O) `2 ) by A2, A7, A10, A13, A14, A21, Th58;

      

       A26: g2 is continuous one-to-one by A1, A2, A18, Th53;

      set x = the Element of (( rng f2) /\ ( rng g2));

      

       A27: ( dom f) = the carrier of I[01] by FUNCT_2:def 1;

      then

       A28: (h . p3) = (f2 . I) by A14, FUNCT_1: 13;

      (d - c) > 0 by A2, XREAL_1: 50;

      then

       A29: C > 0 by XREAL_1: 139;

      (b - a) > 0 by A1, XREAL_1: 50;

      then

       A30: A > 0 by XREAL_1: 139;

      then

       A31: ((h . p1) `2 ) < ((h . p2) `2 ) by A8, A29, Th51;

      ((f . O) `1 ) = a by A3, A13;

      then

       A32: ((f2 . I) `2 ) <= 1 by A2, A7, A10, A13, A14, A21, Th58;

      h is being_homeomorphism by A30, A29, Th51;

      then

       A33: h is one-to-one by TOPS_2:def 5;

      

       A34: ( rng f2) c= ( closed_inside_of_rectangle (( - 1),1,( - 1),1)) by A1, A2, A19, Th52;

      

       A35: ((f2 . I) `1 ) = ( - 1) by A1, A5, A14, Th54;

      

       A36: ((f2 . O) `1 ) = ( - 1) by A1, A3, A13, Th54;

      

       A37: f2 is continuous one-to-one by A1, A2, A17, Th53;

      

       A38: ( rng g2) c= ( closed_inside_of_rectangle (( - 1),1,( - 1),1)) by A1, A2, A20, Th52;

      

       A39: ((h . p2) `2 ) < ((h . p3) `2 ) by A9, A30, A29, Th51;

      

       A40: ((g2 . I) `1 ) <= 1 by A1, A2, A6, A11, A12, A16, Th59;

      

       A41: ( dom g) = the carrier of I[01] by FUNCT_2:def 1;

      then

       A42: (h . p2) = (g2 . O) by A15, FUNCT_1: 13;

      (h . p1) = (f2 . O) by A13, A27, FUNCT_1: 13;

      then ( rng f2) meets ( rng g2) by A31, A39, A42, A28, A37, A36, A35, A25, A32, A34, A26, A24, A23, A22, A40, A38, Th15, JGRAPH_6: 79;

      then

       A43: (( rng f2) /\ ( rng g2)) <> {} by XBOOLE_0:def 7;

      then x in ( rng g2) by XBOOLE_0:def 4;

      then

      consider z2 be object such that

       A44: z2 in ( dom g2) and

       A45: x = (g2 . z2) by FUNCT_1:def 3;

      

       A46: x = (h . (g . z2)) by A41, A44, A45, FUNCT_1: 13;

      

       A47: (g . z2) in ( rng g) by A41, A44, FUNCT_1:def 3;

      (g . z2) in the carrier of ( TOP-REAL 2) by A44, FUNCT_2: 5;

      then

       A48: (g . z2) in ( dom h) by FUNCT_2:def 1;

      x in ( rng f2) by A43, XBOOLE_0:def 4;

      then

      consider z1 be object such that

       A49: z1 in ( dom f2) and

       A50: x = (f2 . z1) by FUNCT_1:def 3;

      

       A51: (f . z1) in ( rng f) by A27, A49, FUNCT_1:def 3;

      (f . z1) in the carrier of ( TOP-REAL 2) by A49, FUNCT_2: 5;

      then

       A52: (f . z1) in ( dom h) by FUNCT_2:def 1;

      x = (h . (f . z1)) by A27, A49, A50, FUNCT_1: 13;

      then (f . z1) = (g . z2) by A46, A52, A48, A33, FUNCT_1:def 4;

      hence thesis by A51, A47, XBOOLE_0: 3;

    end;

    theorem :: JGRAPH_7:71

    for p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real, P,Q be Subset of ( TOP-REAL 2) st a < b & c < d & (p1 `1 ) = a & (p2 `1 ) = a & (p3 `1 ) = a & (p4 `2 ) = d & c <= (p1 `2 ) & (p1 `2 ) < (p2 `2 ) & (p2 `2 ) < (p3 `2 ) & (p3 `2 ) <= d & a <= (p4 `1 ) & (p4 `1 ) <= b & P is_an_arc_of (p1,p3) & Q is_an_arc_of (p2,p4) & P c= ( closed_inside_of_rectangle (a,b,c,d)) & Q c= ( closed_inside_of_rectangle (a,b,c,d)) holds P meets Q

    proof

      let p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real, P,Q be Subset of ( TOP-REAL 2);

      assume that

       A1: a < b and

       A2: c < d and

       A3: (p1 `1 ) = a and

       A4: (p2 `1 ) = a and

       A5: (p3 `1 ) = a and

       A6: (p4 `2 ) = d and

       A7: c <= (p1 `2 ) and

       A8: (p1 `2 ) < (p2 `2 ) and

       A9: (p2 `2 ) < (p3 `2 ) and

       A10: (p3 `2 ) <= d and

       A11: a <= (p4 `1 ) and

       A12: (p4 `1 ) <= b and

       A13: P is_an_arc_of (p1,p3) and

       A14: Q is_an_arc_of (p2,p4) and

       A15: P c= ( closed_inside_of_rectangle (a,b,c,d)) and

       A16: Q c= ( closed_inside_of_rectangle (a,b,c,d));

      

       A17: ex g be Function of I[01] , ( TOP-REAL 2) st g is continuous one-to-one & ( rng g) = Q & (g . 0 ) = p2 & (g . 1) = p4 by A14, Th2;

      ex f be Function of I[01] , ( TOP-REAL 2) st f is continuous one-to-one & ( rng f) = P & (f . 0 ) = p1 & (f . 1) = p3 by A13, Th2;

      hence thesis by A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, A12, A15, A16, A17, Th70;

    end;

    theorem :: JGRAPH_7:72

    

     Th72: for p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real, f,g be Function of I[01] , ( TOP-REAL 2) st a < b & c < d & (p1 `1 ) = a & (p2 `1 ) = a & (p3 `1 ) = a & (p4 `1 ) = b & c <= (p1 `2 ) & (p1 `2 ) < (p2 `2 ) & (p2 `2 ) < (p3 `2 ) & (p3 `2 ) <= d & c <= (p4 `2 ) & (p4 `2 ) <= d & (f . 0 ) = p1 & (f . 1) = p3 & (g . 0 ) = p2 & (g . 1) = p4 & f is continuous one-to-one & g is continuous one-to-one & ( rng f) c= ( closed_inside_of_rectangle (a,b,c,d)) & ( rng g) c= ( closed_inside_of_rectangle (a,b,c,d)) holds ( rng f) meets ( rng g)

    proof

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

      let p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real, f,g be Function of I[01] , ( TOP-REAL 2);

      assume that

       A1: a < b and

       A2: c < d and

       A3: (p1 `1 ) = a and

       A4: (p2 `1 ) = a and

       A5: (p3 `1 ) = a and

       A6: (p4 `1 ) = b and

       A7: c <= (p1 `2 ) and

       A8: (p1 `2 ) < (p2 `2 ) and

       A9: (p2 `2 ) < (p3 `2 ) and

       A10: (p3 `2 ) <= d and

       A11: c <= (p4 `2 ) and

       A12: (p4 `2 ) <= d and

       A13: (f . 0 ) = p1 and

       A14: (f . 1) = p3 and

       A15: (g . 0 ) = p2 and

       A16: (g . 1) = p4 and

       A17: f is continuous one-to-one and

       A18: g is continuous one-to-one and

       A19: ( rng f) c= ( closed_inside_of_rectangle (a,b,c,d)) and

       A20: ( rng g) c= ( closed_inside_of_rectangle (a,b,c,d));

      set A = (2 / (b - a)), B = ( - ((b + a) / (b - a))), C = (2 / (d - c)), D = ( - ((d + c) / (d - c)));

      set h = ( AffineMap (A,B,C,D));

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

      

       A21: (p1 `2 ) < (p3 `2 ) by A8, A9, XXREAL_0: 2;

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

      

       A22: ( - 1) <= ((g2 . I) `2 ) by A2, A11, A12, A16, Th60;

      

       A23: ((g2 . I) `1 ) = 1 by A1, A6, A16, Th56;

      

       A24: ((g2 . O) `1 ) = ( - 1) by A1, A4, A15, Th54;

      ((f . I) `1 ) = a by A5, A14;

      then

       A25: ( - 1) <= ((f2 . O) `2 ) by A2, A7, A10, A13, A14, A21, Th58;

      

       A26: g2 is continuous one-to-one by A1, A2, A18, Th53;

      set x = the Element of (( rng f2) /\ ( rng g2));

      

       A27: ( dom f) = the carrier of I[01] by FUNCT_2:def 1;

      then

       A28: (h . p3) = (f2 . I) by A14, FUNCT_1: 13;

      (d - c) > 0 by A2, XREAL_1: 50;

      then

       A29: C > 0 by XREAL_1: 139;

      (b - a) > 0 by A1, XREAL_1: 50;

      then

       A30: A > 0 by XREAL_1: 139;

      then

       A31: ((h . p1) `2 ) < ((h . p2) `2 ) by A8, A29, Th51;

      ((f . O) `1 ) = a by A3, A13;

      then

       A32: ((f2 . I) `2 ) <= 1 by A2, A7, A10, A13, A14, A21, Th58;

      h is being_homeomorphism by A30, A29, Th51;

      then

       A33: h is one-to-one by TOPS_2:def 5;

      

       A34: ( rng f2) c= ( closed_inside_of_rectangle (( - 1),1,( - 1),1)) by A1, A2, A19, Th52;

      

       A35: ((f2 . I) `1 ) = ( - 1) by A1, A5, A14, Th54;

      

       A36: ((f2 . O) `1 ) = ( - 1) by A1, A3, A13, Th54;

      

       A37: f2 is continuous one-to-one by A1, A2, A17, Th53;

      

       A38: ( rng g2) c= ( closed_inside_of_rectangle (( - 1),1,( - 1),1)) by A1, A2, A20, Th52;

      

       A39: ((h . p2) `2 ) < ((h . p3) `2 ) by A9, A30, A29, Th51;

      

       A40: ((g2 . I) `2 ) <= 1 by A2, A11, A12, A16, Th60;

      

       A41: ( dom g) = the carrier of I[01] by FUNCT_2:def 1;

      then

       A42: (h . p2) = (g2 . O) by A15, FUNCT_1: 13;

      (h . p1) = (f2 . O) by A13, A27, FUNCT_1: 13;

      then ( rng f2) meets ( rng g2) by A31, A39, A42, A28, A37, A36, A35, A25, A32, A34, A26, A24, A23, A22, A40, A38, Th16, JGRAPH_6: 79;

      then

       A43: (( rng f2) /\ ( rng g2)) <> {} by XBOOLE_0:def 7;

      then x in ( rng g2) by XBOOLE_0:def 4;

      then

      consider z2 be object such that

       A44: z2 in ( dom g2) and

       A45: x = (g2 . z2) by FUNCT_1:def 3;

      

       A46: x = (h . (g . z2)) by A41, A44, A45, FUNCT_1: 13;

      

       A47: (g . z2) in ( rng g) by A41, A44, FUNCT_1:def 3;

      (g . z2) in the carrier of ( TOP-REAL 2) by A44, FUNCT_2: 5;

      then

       A48: (g . z2) in ( dom h) by FUNCT_2:def 1;

      x in ( rng f2) by A43, XBOOLE_0:def 4;

      then

      consider z1 be object such that

       A49: z1 in ( dom f2) and

       A50: x = (f2 . z1) by FUNCT_1:def 3;

      

       A51: (f . z1) in ( rng f) by A27, A49, FUNCT_1:def 3;

      (f . z1) in the carrier of ( TOP-REAL 2) by A49, FUNCT_2: 5;

      then

       A52: (f . z1) in ( dom h) by FUNCT_2:def 1;

      x = (h . (f . z1)) by A27, A49, A50, FUNCT_1: 13;

      then (f . z1) = (g . z2) by A46, A52, A48, A33, FUNCT_1:def 4;

      hence thesis by A51, A47, XBOOLE_0: 3;

    end;

    theorem :: JGRAPH_7:73

    for p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real, P,Q be Subset of ( TOP-REAL 2) st a < b & c < d & (p1 `1 ) = a & (p2 `1 ) = a & (p3 `1 ) = a & (p4 `1 ) = b & c <= (p1 `2 ) & (p1 `2 ) < (p2 `2 ) & (p2 `2 ) < (p3 `2 ) & (p3 `2 ) <= d & c <= (p4 `2 ) & (p4 `2 ) <= d & P is_an_arc_of (p1,p3) & Q is_an_arc_of (p2,p4) & P c= ( closed_inside_of_rectangle (a,b,c,d)) & Q c= ( closed_inside_of_rectangle (a,b,c,d)) holds P meets Q

    proof

      let p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real, P,Q be Subset of ( TOP-REAL 2);

      assume that

       A1: a < b and

       A2: c < d and

       A3: (p1 `1 ) = a and

       A4: (p2 `1 ) = a and

       A5: (p3 `1 ) = a and

       A6: (p4 `1 ) = b and

       A7: c <= (p1 `2 ) and

       A8: (p1 `2 ) < (p2 `2 ) and

       A9: (p2 `2 ) < (p3 `2 ) and

       A10: (p3 `2 ) <= d and

       A11: c <= (p4 `2 ) and

       A12: (p4 `2 ) <= d and

       A13: P is_an_arc_of (p1,p3) and

       A14: Q is_an_arc_of (p2,p4) and

       A15: P c= ( closed_inside_of_rectangle (a,b,c,d)) and

       A16: Q c= ( closed_inside_of_rectangle (a,b,c,d));

      

       A17: ex g be Function of I[01] , ( TOP-REAL 2) st g is continuous one-to-one & ( rng g) = Q & (g . 0 ) = p2 & (g . 1) = p4 by A14, Th2;

      ex f be Function of I[01] , ( TOP-REAL 2) st f is continuous one-to-one & ( rng f) = P & (f . 0 ) = p1 & (f . 1) = p3 by A13, Th2;

      hence thesis by A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, A12, A15, A16, A17, Th72;

    end;

    theorem :: JGRAPH_7:74

    

     Th74: for p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real, f,g be Function of I[01] , ( TOP-REAL 2) st a < b & c < d & (p1 `1 ) = a & (p2 `1 ) = a & (p3 `1 ) = a & (p4 `2 ) = c & c <= (p1 `2 ) & (p1 `2 ) < (p2 `2 ) & (p2 `2 ) < (p3 `2 ) & (p3 `2 ) <= d & a < (p4 `1 ) & (p4 `1 ) <= b & (f . 0 ) = p1 & (f . 1) = p3 & (g . 0 ) = p2 & (g . 1) = p4 & f is continuous one-to-one & g is continuous one-to-one & ( rng f) c= ( closed_inside_of_rectangle (a,b,c,d)) & ( rng g) c= ( closed_inside_of_rectangle (a,b,c,d)) holds ( rng f) meets ( rng g)

    proof

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

      let p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real, f,g be Function of I[01] , ( TOP-REAL 2);

      assume that

       A1: a < b and

       A2: c < d and

       A3: (p1 `1 ) = a and

       A4: (p2 `1 ) = a and

       A5: (p3 `1 ) = a and

       A6: (p4 `2 ) = c and

       A7: c <= (p1 `2 ) and

       A8: (p1 `2 ) < (p2 `2 ) and

       A9: (p2 `2 ) < (p3 `2 ) and

       A10: (p3 `2 ) <= d and

       A11: a < (p4 `1 ) and

       A12: (p4 `1 ) <= b and

       A13: (f . 0 ) = p1 and

       A14: (f . 1) = p3 and

       A15: (g . 0 ) = p2 and

       A16: (g . 1) = p4 and

       A17: f is continuous one-to-one and

       A18: g is continuous one-to-one and

       A19: ( rng f) c= ( closed_inside_of_rectangle (a,b,c,d)) and

       A20: ( rng g) c= ( closed_inside_of_rectangle (a,b,c,d));

      set A = (2 / (b - a)), B = ( - ((b + a) / (b - a))), C = (2 / (d - c)), D = ( - ((d + c) / (d - c)));

      set h = ( AffineMap (A,B,C,D));

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

      

       A21: (p1 `2 ) < (p3 `2 ) by A8, A9, XXREAL_0: 2;

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

      

       A22: ( - 1) < ((g2 . I) `1 ) by A1, A2, A6, A11, A12, A16, Th61;

      

       A23: ((g2 . I) `2 ) = ( - 1) by A2, A6, A16, Th57;

      

       A24: ((g2 . O) `1 ) = ( - 1) by A1, A4, A15, Th54;

      ((f . I) `1 ) = a by A5, A14;

      then

       A25: ( - 1) <= ((f2 . O) `2 ) by A2, A7, A10, A13, A14, A21, Th58;

      

       A26: g2 is continuous one-to-one by A1, A2, A18, Th53;

      set x = the Element of (( rng f2) /\ ( rng g2));

      

       A27: ( dom f) = the carrier of I[01] by FUNCT_2:def 1;

      then

       A28: (h . p3) = (f2 . I) by A14, FUNCT_1: 13;

      (d - c) > 0 by A2, XREAL_1: 50;

      then

       A29: C > 0 by XREAL_1: 139;

      (b - a) > 0 by A1, XREAL_1: 50;

      then

       A30: A > 0 by XREAL_1: 139;

      then

       A31: ((h . p1) `2 ) < ((h . p2) `2 ) by A8, A29, Th51;

      ((f . O) `1 ) = a by A3, A13;

      then

       A32: ((f2 . I) `2 ) <= 1 by A2, A7, A10, A13, A14, A21, Th58;

      h is being_homeomorphism by A30, A29, Th51;

      then

       A33: h is one-to-one by TOPS_2:def 5;

      

       A34: ( rng f2) c= ( closed_inside_of_rectangle (( - 1),1,( - 1),1)) by A1, A2, A19, Th52;

      

       A35: ((f2 . I) `1 ) = ( - 1) by A1, A5, A14, Th54;

      

       A36: ((f2 . O) `1 ) = ( - 1) by A1, A3, A13, Th54;

      

       A37: f2 is continuous one-to-one by A1, A2, A17, Th53;

      

       A38: ( rng g2) c= ( closed_inside_of_rectangle (( - 1),1,( - 1),1)) by A1, A2, A20, Th52;

      

       A39: ((h . p2) `2 ) < ((h . p3) `2 ) by A9, A30, A29, Th51;

      

       A40: ((g2 . I) `1 ) <= 1 by A1, A2, A6, A11, A12, A16, Th61;

      

       A41: ( dom g) = the carrier of I[01] by FUNCT_2:def 1;

      then

       A42: (h . p2) = (g2 . O) by A15, FUNCT_1: 13;

      (h . p1) = (f2 . O) by A13, A27, FUNCT_1: 13;

      then ( rng f2) meets ( rng g2) by A31, A39, A42, A28, A37, A36, A35, A25, A32, A34, A26, A24, A23, A22, A40, A38, Th17, JGRAPH_6: 79;

      then

       A43: (( rng f2) /\ ( rng g2)) <> {} by XBOOLE_0:def 7;

      then x in ( rng g2) by XBOOLE_0:def 4;

      then

      consider z2 be object such that

       A44: z2 in ( dom g2) and

       A45: x = (g2 . z2) by FUNCT_1:def 3;

      

       A46: x = (h . (g . z2)) by A41, A44, A45, FUNCT_1: 13;

      

       A47: (g . z2) in ( rng g) by A41, A44, FUNCT_1:def 3;

      (g . z2) in the carrier of ( TOP-REAL 2) by A44, FUNCT_2: 5;

      then

       A48: (g . z2) in ( dom h) by FUNCT_2:def 1;

      x in ( rng f2) by A43, XBOOLE_0:def 4;

      then

      consider z1 be object such that

       A49: z1 in ( dom f2) and

       A50: x = (f2 . z1) by FUNCT_1:def 3;

      

       A51: (f . z1) in ( rng f) by A27, A49, FUNCT_1:def 3;

      (f . z1) in the carrier of ( TOP-REAL 2) by A49, FUNCT_2: 5;

      then

       A52: (f . z1) in ( dom h) by FUNCT_2:def 1;

      x = (h . (f . z1)) by A27, A49, A50, FUNCT_1: 13;

      then (f . z1) = (g . z2) by A46, A52, A48, A33, FUNCT_1:def 4;

      hence thesis by A51, A47, XBOOLE_0: 3;

    end;

    theorem :: JGRAPH_7:75

    for p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real, P,Q be Subset of ( TOP-REAL 2) st a < b & c < d & (p1 `1 ) = a & (p2 `1 ) = a & (p3 `1 ) = a & (p4 `2 ) = c & c <= (p1 `2 ) & (p1 `2 ) < (p2 `2 ) & (p2 `2 ) < (p3 `2 ) & (p3 `2 ) <= d & a < (p4 `1 ) & (p4 `1 ) <= b & P is_an_arc_of (p1,p3) & Q is_an_arc_of (p2,p4) & P c= ( closed_inside_of_rectangle (a,b,c,d)) & Q c= ( closed_inside_of_rectangle (a,b,c,d)) holds P meets Q

    proof

      let p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real, P,Q be Subset of ( TOP-REAL 2);

      assume that

       A1: a < b and

       A2: c < d and

       A3: (p1 `1 ) = a and

       A4: (p2 `1 ) = a and

       A5: (p3 `1 ) = a and

       A6: (p4 `2 ) = c and

       A7: c <= (p1 `2 ) and

       A8: (p1 `2 ) < (p2 `2 ) and

       A9: (p2 `2 ) < (p3 `2 ) and

       A10: (p3 `2 ) <= d and

       A11: a < (p4 `1 ) and

       A12: (p4 `1 ) <= b and

       A13: P is_an_arc_of (p1,p3) and

       A14: Q is_an_arc_of (p2,p4) and

       A15: P c= ( closed_inside_of_rectangle (a,b,c,d)) and

       A16: Q c= ( closed_inside_of_rectangle (a,b,c,d));

      

       A17: ex g be Function of I[01] , ( TOP-REAL 2) st g is continuous one-to-one & ( rng g) = Q & (g . 0 ) = p2 & (g . 1) = p4 by A14, Th2;

      ex f be Function of I[01] , ( TOP-REAL 2) st f is continuous one-to-one & ( rng f) = P & (f . 0 ) = p1 & (f . 1) = p3 by A13, Th2;

      hence thesis by A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, A12, A15, A16, A17, Th74;

    end;

    theorem :: JGRAPH_7:76

    

     Th76: for p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real, f,g be Function of I[01] , ( TOP-REAL 2) st a < b & c < d & (p1 `1 ) = a & (p2 `1 ) = a & (p3 `2 ) = d & (p4 `2 ) = d & c <= (p1 `2 ) & (p1 `2 ) < (p2 `2 ) & (p2 `2 ) <= d & a <= (p3 `1 ) & (p3 `1 ) < (p4 `1 ) & (p4 `1 ) <= b & (f . 0 ) = p1 & (f . 1) = p3 & (g . 0 ) = p2 & (g . 1) = p4 & f is continuous one-to-one & g is continuous one-to-one & ( rng f) c= ( closed_inside_of_rectangle (a,b,c,d)) & ( rng g) c= ( closed_inside_of_rectangle (a,b,c,d)) holds ( rng f) meets ( rng g)

    proof

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

      let p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real, f,g be Function of I[01] , ( TOP-REAL 2);

      assume that

       A1: a < b and

       A2: c < d and

       A3: (p1 `1 ) = a and

       A4: (p2 `1 ) = a and

       A5: (p3 `2 ) = d and

       A6: (p4 `2 ) = d and

       A7: c <= (p1 `2 ) and

       A8: (p1 `2 ) < (p2 `2 ) and

       A9: (p2 `2 ) <= d and

       A10: a <= (p3 `1 ) and

       A11: (p3 `1 ) < (p4 `1 ) and

       A12: (p4 `1 ) <= b and

       A13: (f . 0 ) = p1 and

       A14: (f . 1) = p3 and

       A15: (g . 0 ) = p2 and

       A16: (g . 1) = p4 and

       A17: f is continuous one-to-one and

       A18: g is continuous one-to-one and

       A19: ( rng f) c= ( closed_inside_of_rectangle (a,b,c,d)) and

       A20: ( rng g) c= ( closed_inside_of_rectangle (a,b,c,d));

      set A = (2 / (b - a)), B = ( - ((b + a) / (b - a))), C = (2 / (d - c)), D = ( - ((d + c) / (d - c)));

      set h = ( AffineMap (A,B,C,D));

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

      

       A21: g2 is continuous one-to-one by A1, A2, A18, Th53;

      c < (p2 `2 ) by A7, A8, XXREAL_0: 2;

      then

       A22: ((g2 . O) `2 ) <= 1 by A1, A2, A4, A9, A15, Th59;

      

       A23: ( rng g2) c= ( closed_inside_of_rectangle (( - 1),1,( - 1),1)) by A1, A2, A20, Th52;

      

       A24: ((g2 . I) `2 ) = 1 by A2, A6, A16, Th55;

      (d - c) > 0 by A2, XREAL_1: 50;

      then

       A25: C > 0 by XREAL_1: 139;

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

      

       A26: f2 is continuous one-to-one by A1, A2, A17, Th53;

      (p3 `1 ) < b by A11, A12, XXREAL_0: 2;

      then

       A27: ( - 1) <= ((f2 . I) `1 ) by A1, A2, A5, A10, A14, Th59;

      

       A28: ( rng f2) c= ( closed_inside_of_rectangle (( - 1),1,( - 1),1)) by A1, A2, A19, Th52;

      

       A29: ((f2 . I) `2 ) = 1 by A2, A5, A14, Th55;

      (p1 `2 ) <= d by A8, A9, XXREAL_0: 2;

      then

       A30: ( - 1) <= ((f2 . O) `2 ) by A1, A2, A3, A7, A13, Th59;

      

       A31: ((f2 . O) `1 ) = ( - 1) by A1, A3, A13, Th54;

      set x = the Element of (( rng f2) /\ ( rng g2));

      

       A32: ( dom f) = the carrier of I[01] by FUNCT_2:def 1;

      then

       A33: (h . p3) = (f2 . I) by A14, FUNCT_1: 13;

      (b - a) > 0 by A1, XREAL_1: 50;

      then

       A34: A > 0 by XREAL_1: 139;

      then

       A35: ((h . p1) `2 ) < ((h . p2) `2 ) by A8, A25, Th51;

      a < (p4 `1 ) by A10, A11, XXREAL_0: 2;

      then

       A36: ((g2 . I) `1 ) <= 1 by A1, A2, A6, A12, A16, Th59;

      h is being_homeomorphism by A34, A25, Th51;

      then

       A37: h is one-to-one by TOPS_2:def 5;

      

       A38: ((g2 . O) `1 ) = ( - 1) by A1, A4, A15, Th54;

      

       A39: ( dom g) = the carrier of I[01] by FUNCT_2:def 1;

      then

       A40: (h . p2) = (g2 . O) by A15, FUNCT_1: 13;

      

       A41: ((h . p3) `1 ) < ((h . p4) `1 ) by A11, A34, A25, Th50;

      

       A42: (h . p4) = (g2 . I) by A16, A39, FUNCT_1: 13;

      (h . p1) = (f2 . O) by A13, A32, FUNCT_1: 13;

      then ((f2 . O),(g2 . O),(f2 . I),(g2 . I)) are_in_this_order_on ( rectangle (( - 1),1,( - 1),1)) by A35, A41, A40, A33, A42, A31, A29, A30, A27, A38, A24, A22, A36, Th18;

      then ( rng f2) meets ( rng g2) by A26, A28, A21, A23, JGRAPH_6: 79;

      then

       A43: (( rng f2) /\ ( rng g2)) <> {} by XBOOLE_0:def 7;

      then x in ( rng g2) by XBOOLE_0:def 4;

      then

      consider z2 be object such that

       A44: z2 in ( dom g2) and

       A45: x = (g2 . z2) by FUNCT_1:def 3;

      

       A46: x = (h . (g . z2)) by A39, A44, A45, FUNCT_1: 13;

      

       A47: (g . z2) in ( rng g) by A39, A44, FUNCT_1:def 3;

      (g . z2) in the carrier of ( TOP-REAL 2) by A44, FUNCT_2: 5;

      then

       A48: (g . z2) in ( dom h) by FUNCT_2:def 1;

      x in ( rng f2) by A43, XBOOLE_0:def 4;

      then

      consider z1 be object such that

       A49: z1 in ( dom f2) and

       A50: x = (f2 . z1) by FUNCT_1:def 3;

      

       A51: (f . z1) in ( rng f) by A32, A49, FUNCT_1:def 3;

      (f . z1) in the carrier of ( TOP-REAL 2) by A49, FUNCT_2: 5;

      then

       A52: (f . z1) in ( dom h) by FUNCT_2:def 1;

      x = (h . (f . z1)) by A32, A49, A50, FUNCT_1: 13;

      then (f . z1) = (g . z2) by A46, A52, A48, A37, FUNCT_1:def 4;

      hence thesis by A51, A47, XBOOLE_0: 3;

    end;

    theorem :: JGRAPH_7:77

    for p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real, P,Q be Subset of ( TOP-REAL 2) st a < b & c < d & (p1 `1 ) = a & (p2 `1 ) = a & (p3 `2 ) = d & (p4 `2 ) = d & c <= (p1 `2 ) & (p1 `2 ) < (p2 `2 ) & (p2 `2 ) <= d & a <= (p3 `1 ) & (p3 `1 ) < (p4 `1 ) & (p4 `1 ) <= b & P is_an_arc_of (p1,p3) & Q is_an_arc_of (p2,p4) & P c= ( closed_inside_of_rectangle (a,b,c,d)) & Q c= ( closed_inside_of_rectangle (a,b,c,d)) holds P meets Q

    proof

      let p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real, P,Q be Subset of ( TOP-REAL 2);

      assume that

       A1: a < b and

       A2: c < d and

       A3: (p1 `1 ) = a and

       A4: (p2 `1 ) = a and

       A5: (p3 `2 ) = d and

       A6: (p4 `2 ) = d and

       A7: c <= (p1 `2 ) and

       A8: (p1 `2 ) < (p2 `2 ) and

       A9: (p2 `2 ) <= d and

       A10: a <= (p3 `1 ) and

       A11: (p3 `1 ) < (p4 `1 ) and

       A12: (p4 `1 ) <= b and

       A13: P is_an_arc_of (p1,p3) and

       A14: Q is_an_arc_of (p2,p4) and

       A15: P c= ( closed_inside_of_rectangle (a,b,c,d)) and

       A16: Q c= ( closed_inside_of_rectangle (a,b,c,d));

      

       A17: ex g be Function of I[01] , ( TOP-REAL 2) st g is continuous one-to-one & ( rng g) = Q & (g . 0 ) = p2 & (g . 1) = p4 by A14, Th2;

      ex f be Function of I[01] , ( TOP-REAL 2) st f is continuous one-to-one & ( rng f) = P & (f . 0 ) = p1 & (f . 1) = p3 by A13, Th2;

      hence thesis by A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, A12, A15, A16, A17, Th76;

    end;

    theorem :: JGRAPH_7:78

    

     Th78: for p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real, f,g be Function of I[01] , ( TOP-REAL 2) st a < b & c < d & (p1 `1 ) = a & (p2 `1 ) = a & (p3 `2 ) = d & (p4 `1 ) = b & c <= (p1 `2 ) & (p1 `2 ) < (p2 `2 ) & (p2 `2 ) <= d & a <= (p3 `1 ) & (p3 `1 ) <= b & c <= (p4 `2 ) & (p4 `2 ) <= d & (f . 0 ) = p1 & (f . 1) = p3 & (g . 0 ) = p2 & (g . 1) = p4 & f is continuous one-to-one & g is continuous one-to-one & ( rng f) c= ( closed_inside_of_rectangle (a,b,c,d)) & ( rng g) c= ( closed_inside_of_rectangle (a,b,c,d)) holds ( rng f) meets ( rng g)

    proof

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

      let p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real, f,g be Function of I[01] , ( TOP-REAL 2);

      assume that

       A1: a < b and

       A2: c < d and

       A3: (p1 `1 ) = a and

       A4: (p2 `1 ) = a and

       A5: (p3 `2 ) = d and

       A6: (p4 `1 ) = b and

       A7: c <= (p1 `2 ) and

       A8: (p1 `2 ) < (p2 `2 ) and

       A9: (p2 `2 ) <= d and

       A10: a <= (p3 `1 ) and

       A11: (p3 `1 ) <= b and

       A12: c <= (p4 `2 ) and

       A13: (p4 `2 ) <= d and

       A14: (f . 0 ) = p1 and

       A15: (f . 1) = p3 and

       A16: (g . 0 ) = p2 and

       A17: (g . 1) = p4 and

       A18: f is continuous one-to-one and

       A19: g is continuous one-to-one and

       A20: ( rng f) c= ( closed_inside_of_rectangle (a,b,c,d)) and

       A21: ( rng g) c= ( closed_inside_of_rectangle (a,b,c,d));

      set A = (2 / (b - a)), B = ( - ((b + a) / (b - a))), C = (2 / (d - c)), D = ( - ((d + c) / (d - c)));

      set h = ( AffineMap (A,B,C,D));

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

      

       A22: ( - 1) <= ((f2 . I) `1 ) by A1, A2, A5, A10, A11, A15, Th59;

      

       A23: ((f2 . I) `1 ) <= 1 by A1, A2, A5, A10, A11, A15, Th59;

      

       A24: f2 is continuous one-to-one by A1, A2, A18, Th53;

      

       A25: ((f2 . I) `2 ) = 1 by A2, A5, A15, Th55;

      

       A26: ((f2 . O) `1 ) = ( - 1) by A1, A3, A14, Th54;

      

       A27: ( rng f2) c= ( closed_inside_of_rectangle (( - 1),1,( - 1),1)) by A1, A2, A20, Th52;

      (d - c) > 0 by A2, XREAL_1: 50;

      then

       A28: C > 0 by XREAL_1: 139;

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

      

       A29: ( - 1) <= ((g2 . I) `2 ) by A2, A12, A13, A17, Th60;

      

       A30: ((g2 . O) `1 ) = ( - 1) by A1, A4, A16, Th54;

      

       A31: g2 is continuous one-to-one by A1, A2, A19, Th53;

      c < (p2 `2 ) by A7, A8, XXREAL_0: 2;

      then

       A32: ((g2 . O) `2 ) <= 1 by A2, A9, A16, Th60;

      

       A33: ((g2 . I) `2 ) <= 1 by A2, A12, A13, A17, Th60;

      set x = the Element of (( rng f2) /\ ( rng g2));

      

       A34: ( dom g) = the carrier of I[01] by FUNCT_2:def 1;

      then

       A35: (h . p2) = (g2 . O) by A16, FUNCT_1: 13;

      (b - a) > 0 by A1, XREAL_1: 50;

      then

       A36: A > 0 by XREAL_1: 139;

      then

       A37: ((h . p1) `2 ) < ((h . p2) `2 ) by A8, A28, Th51;

      (p1 `2 ) <= d by A8, A9, XXREAL_0: 2;

      then

       A38: ( - 1) <= ((f2 . O) `2 ) by A1, A2, A3, A7, A14, Th59;

      h is being_homeomorphism by A36, A28, Th51;

      then

       A39: h is one-to-one by TOPS_2:def 5;

      

       A40: ( rng g2) c= ( closed_inside_of_rectangle (( - 1),1,( - 1),1)) by A1, A2, A21, Th52;

      

       A41: ((g2 . I) `1 ) = 1 by A1, A6, A17, Th56;

      

       A42: ( dom f) = the carrier of I[01] by FUNCT_2:def 1;

      then (h . p1) = (f2 . O) by A14, FUNCT_1: 13;

      then ( rng f2) meets ( rng g2) by A37, A35, A24, A26, A25, A38, A22, A23, A27, A31, A30, A41, A32, A29, A33, A40, Th19, JGRAPH_6: 79;

      then

       A43: (( rng f2) /\ ( rng g2)) <> {} by XBOOLE_0:def 7;

      then x in ( rng g2) by XBOOLE_0:def 4;

      then

      consider z2 be object such that

       A44: z2 in ( dom g2) and

       A45: x = (g2 . z2) by FUNCT_1:def 3;

      

       A46: x = (h . (g . z2)) by A34, A44, A45, FUNCT_1: 13;

      

       A47: (g . z2) in ( rng g) by A34, A44, FUNCT_1:def 3;

      (g . z2) in the carrier of ( TOP-REAL 2) by A44, FUNCT_2: 5;

      then

       A48: (g . z2) in ( dom h) by FUNCT_2:def 1;

      x in ( rng f2) by A43, XBOOLE_0:def 4;

      then

      consider z1 be object such that

       A49: z1 in ( dom f2) and

       A50: x = (f2 . z1) by FUNCT_1:def 3;

      

       A51: (f . z1) in ( rng f) by A42, A49, FUNCT_1:def 3;

      (f . z1) in the carrier of ( TOP-REAL 2) by A49, FUNCT_2: 5;

      then

       A52: (f . z1) in ( dom h) by FUNCT_2:def 1;

      x = (h . (f . z1)) by A42, A49, A50, FUNCT_1: 13;

      then (f . z1) = (g . z2) by A46, A52, A48, A39, FUNCT_1:def 4;

      hence thesis by A51, A47, XBOOLE_0: 3;

    end;

    theorem :: JGRAPH_7:79

    for p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real, P,Q be Subset of ( TOP-REAL 2) st a < b & c < d & (p1 `1 ) = a & (p2 `1 ) = a & (p3 `2 ) = d & (p4 `1 ) = b & c <= (p1 `2 ) & (p1 `2 ) < (p2 `2 ) & (p2 `2 ) <= d & a <= (p3 `1 ) & (p3 `1 ) <= b & c <= (p4 `2 ) & (p4 `2 ) <= d & P is_an_arc_of (p1,p3) & Q is_an_arc_of (p2,p4) & P c= ( closed_inside_of_rectangle (a,b,c,d)) & Q c= ( closed_inside_of_rectangle (a,b,c,d)) holds P meets Q

    proof

      let p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real, P,Q be Subset of ( TOP-REAL 2);

      assume that

       A1: a < b and

       A2: c < d and

       A3: (p1 `1 ) = a and

       A4: (p2 `1 ) = a and

       A5: (p3 `2 ) = d and

       A6: (p4 `1 ) = b and

       A7: c <= (p1 `2 ) and

       A8: (p1 `2 ) < (p2 `2 ) and

       A9: (p2 `2 ) <= d and

       A10: a <= (p3 `1 ) and

       A11: (p3 `1 ) <= b and

       A12: c <= (p4 `2 ) and

       A13: (p4 `2 ) <= d and

       A14: P is_an_arc_of (p1,p3) and

       A15: Q is_an_arc_of (p2,p4) and

       A16: P c= ( closed_inside_of_rectangle (a,b,c,d)) and

       A17: Q c= ( closed_inside_of_rectangle (a,b,c,d));

      

       A18: ex g be Function of I[01] , ( TOP-REAL 2) st g is continuous one-to-one & ( rng g) = Q & (g . 0 ) = p2 & (g . 1) = p4 by A15, Th2;

      ex f be Function of I[01] , ( TOP-REAL 2) st f is continuous one-to-one & ( rng f) = P & (f . 0 ) = p1 & (f . 1) = p3 by A14, Th2;

      hence thesis by A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, A12, A13, A16, A17, A18, Th78;

    end;

    theorem :: JGRAPH_7:80

    

     Th80: for p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real, f,g be Function of I[01] , ( TOP-REAL 2) st a < b & c < d & (p1 `1 ) = a & (p2 `1 ) = a & (p3 `2 ) = d & (p4 `2 ) = c & c <= (p1 `2 ) & (p1 `2 ) < (p2 `2 ) & (p2 `2 ) <= d & a <= (p3 `1 ) & (p3 `1 ) <= b & a < (p4 `1 ) & (p4 `1 ) <= b & (f . 0 ) = p1 & (f . 1) = p3 & (g . 0 ) = p2 & (g . 1) = p4 & f is continuous one-to-one & g is continuous one-to-one & ( rng f) c= ( closed_inside_of_rectangle (a,b,c,d)) & ( rng g) c= ( closed_inside_of_rectangle (a,b,c,d)) holds ( rng f) meets ( rng g)

    proof

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

      let p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real, f,g be Function of I[01] , ( TOP-REAL 2);

      assume that

       A1: a < b and

       A2: c < d and

       A3: (p1 `1 ) = a and

       A4: (p2 `1 ) = a and

       A5: (p3 `2 ) = d and

       A6: (p4 `2 ) = c and

       A7: c <= (p1 `2 ) and

       A8: (p1 `2 ) < (p2 `2 ) and

       A9: (p2 `2 ) <= d and

       A10: a <= (p3 `1 ) and

       A11: (p3 `1 ) <= b and

       A12: a < (p4 `1 ) and

       A13: (p4 `1 ) <= b and

       A14: (f . 0 ) = p1 and

       A15: (f . 1) = p3 and

       A16: (g . 0 ) = p2 and

       A17: (g . 1) = p4 and

       A18: f is continuous one-to-one and

       A19: g is continuous one-to-one and

       A20: ( rng f) c= ( closed_inside_of_rectangle (a,b,c,d)) and

       A21: ( rng g) c= ( closed_inside_of_rectangle (a,b,c,d));

      set A = (2 / (b - a)), B = ( - ((b + a) / (b - a))), C = (2 / (d - c)), D = ( - ((d + c) / (d - c)));

      set h = ( AffineMap (A,B,C,D));

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

      

       A22: ( - 1) < ((g2 . I) `1 ) by A1, A2, A6, A12, A13, A17, Th61;

      

       A23: ((g2 . I) `1 ) <= 1 by A1, A2, A6, A12, A13, A17, Th61;

      

       A24: g2 is continuous one-to-one by A1, A2, A19, Th53;

      

       A25: ((g2 . I) `2 ) = ( - 1) by A2, A6, A17, Th57;

      

       A26: ((g2 . O) `1 ) = ( - 1) by A1, A4, A16, Th54;

      

       A27: ( rng g2) c= ( closed_inside_of_rectangle (( - 1),1,( - 1),1)) by A1, A2, A21, Th52;

      (d - c) > 0 by A2, XREAL_1: 50;

      then

       A28: C > 0 by XREAL_1: 139;

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

      

       A29: ( - 1) <= ((f2 . I) `1 ) by A1, A2, A5, A10, A11, A15, Th59;

      

       A30: ( rng f2) c= ( closed_inside_of_rectangle (( - 1),1,( - 1),1)) by A1, A2, A20, Th52;

      

       A31: ((f2 . I) `2 ) = 1 by A2, A5, A15, Th55;

      (p1 `2 ) <= d by A8, A9, XXREAL_0: 2;

      then

       A32: ( - 1) <= ((f2 . O) `2 ) by A1, A2, A3, A7, A14, Th59;

      

       A33: ((f2 . O) `1 ) = ( - 1) by A1, A3, A14, Th54;

      

       A34: f2 is continuous one-to-one by A1, A2, A18, Th53;

      

       A35: ((f2 . I) `1 ) <= 1 by A1, A2, A5, A10, A11, A15, Th59;

      set x = the Element of (( rng f2) /\ ( rng g2));

      

       A36: ( dom g) = the carrier of I[01] by FUNCT_2:def 1;

      then

       A37: (h . p2) = (g2 . O) by A16, FUNCT_1: 13;

      

       A38: ((g . I) `2 ) = c by A6, A17;

      c < (p2 `2 ) by A7, A8, XXREAL_0: 2;

      then

       A39: ((g2 . O) `2 ) <= 1 by A1, A2, A9, A12, A13, A16, A17, A38, Th61;

      (b - a) > 0 by A1, XREAL_1: 50;

      then

       A40: A > 0 by XREAL_1: 139;

      then

       A41: ((h . p1) `2 ) < ((h . p2) `2 ) by A8, A28, Th51;

      

       A42: ( dom f) = the carrier of I[01] by FUNCT_2:def 1;

      then (h . p1) = (f2 . O) by A14, FUNCT_1: 13;

      then ( rng f2) meets ( rng g2) by A41, A37, A34, A33, A31, A32, A29, A35, A30, A24, A26, A25, A39, A22, A23, A27, Th20, JGRAPH_6: 79;

      then

       A43: (( rng f2) /\ ( rng g2)) <> {} by XBOOLE_0:def 7;

      then x in ( rng g2) by XBOOLE_0:def 4;

      then

      consider z2 be object such that

       A44: z2 in ( dom g2) and

       A45: x = (g2 . z2) by FUNCT_1:def 3;

      

       A46: x = (h . (g . z2)) by A36, A44, A45, FUNCT_1: 13;

      h is being_homeomorphism by A40, A28, Th51;

      then

       A47: h is one-to-one by TOPS_2:def 5;

      x in ( rng f2) by A43, XBOOLE_0:def 4;

      then

      consider z1 be object such that

       A48: z1 in ( dom f2) and

       A49: x = (f2 . z1) by FUNCT_1:def 3;

      

       A50: (f . z1) in ( rng f) by A42, A48, FUNCT_1:def 3;

      

       A51: (g . z2) in ( rng g) by A36, A44, FUNCT_1:def 3;

      (g . z2) in the carrier of ( TOP-REAL 2) by A44, FUNCT_2: 5;

      then

       A52: (g . z2) in ( dom h) by FUNCT_2:def 1;

      (f . z1) in the carrier of ( TOP-REAL 2) by A48, FUNCT_2: 5;

      then

       A53: (f . z1) in ( dom h) by FUNCT_2:def 1;

      x = (h . (f . z1)) by A42, A48, A49, FUNCT_1: 13;

      then (f . z1) = (g . z2) by A46, A53, A52, A47, FUNCT_1:def 4;

      hence thesis by A50, A51, XBOOLE_0: 3;

    end;

    theorem :: JGRAPH_7:81

    for p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real, P,Q be Subset of ( TOP-REAL 2) st a < b & c < d & (p1 `1 ) = a & (p2 `1 ) = a & (p3 `2 ) = d & (p4 `2 ) = c & c <= (p1 `2 ) & (p1 `2 ) < (p2 `2 ) & (p2 `2 ) <= d & a <= (p3 `1 ) & (p3 `1 ) <= b & a < (p4 `1 ) & (p4 `1 ) <= b & P is_an_arc_of (p1,p3) & Q is_an_arc_of (p2,p4) & P c= ( closed_inside_of_rectangle (a,b,c,d)) & Q c= ( closed_inside_of_rectangle (a,b,c,d)) holds P meets Q

    proof

      let p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real, P,Q be Subset of ( TOP-REAL 2);

      assume that

       A1: a < b and

       A2: c < d and

       A3: (p1 `1 ) = a and

       A4: (p2 `1 ) = a and

       A5: (p3 `2 ) = d and

       A6: (p4 `2 ) = c and

       A7: c <= (p1 `2 ) and

       A8: (p1 `2 ) < (p2 `2 ) and

       A9: (p2 `2 ) <= d and

       A10: a <= (p3 `1 ) and

       A11: (p3 `1 ) <= b and

       A12: a < (p4 `1 ) and

       A13: (p4 `1 ) <= b and

       A14: P is_an_arc_of (p1,p3) and

       A15: Q is_an_arc_of (p2,p4) and

       A16: P c= ( closed_inside_of_rectangle (a,b,c,d)) and

       A17: Q c= ( closed_inside_of_rectangle (a,b,c,d));

      

       A18: ex g be Function of I[01] , ( TOP-REAL 2) st g is continuous one-to-one & ( rng g) = Q & (g . 0 ) = p2 & (g . 1) = p4 by A15, Th2;

      ex f be Function of I[01] , ( TOP-REAL 2) st f is continuous one-to-one & ( rng f) = P & (f . 0 ) = p1 & (f . 1) = p3 by A14, Th2;

      hence thesis by A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, A12, A13, A16, A17, A18, Th80;

    end;

    theorem :: JGRAPH_7:82

    

     Th82: for p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real, f,g be Function of I[01] , ( TOP-REAL 2) st a < b & c < d & (p1 `1 ) = a & (p2 `1 ) = a & (p3 `1 ) = b & (p4 `1 ) = b & c <= (p1 `2 ) & (p1 `2 ) < (p2 `2 ) & (p2 `2 ) <= d & c <= (p4 `2 ) & (p4 `2 ) < (p3 `2 ) & (p3 `2 ) <= d & (f . 0 ) = p1 & (f . 1) = p3 & (g . 0 ) = p2 & (g . 1) = p4 & f is continuous one-to-one & g is continuous one-to-one & ( rng f) c= ( closed_inside_of_rectangle (a,b,c,d)) & ( rng g) c= ( closed_inside_of_rectangle (a,b,c,d)) holds ( rng f) meets ( rng g)

    proof

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

      let p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real, f,g be Function of I[01] , ( TOP-REAL 2);

      assume that

       A1: a < b and

       A2: c < d and

       A3: (p1 `1 ) = a and

       A4: (p2 `1 ) = a and

       A5: (p3 `1 ) = b and

       A6: (p4 `1 ) = b and

       A7: c <= (p1 `2 ) and

       A8: (p1 `2 ) < (p2 `2 ) and

       A9: (p2 `2 ) <= d and

       A10: c <= (p4 `2 ) and

       A11: (p4 `2 ) < (p3 `2 ) and

       A12: (p3 `2 ) <= d and

       A13: (f . 0 ) = p1 and

       A14: (f . 1) = p3 and

       A15: (g . 0 ) = p2 and

       A16: (g . 1) = p4 and

       A17: f is continuous one-to-one and

       A18: g is continuous one-to-one and

       A19: ( rng f) c= ( closed_inside_of_rectangle (a,b,c,d)) and

       A20: ( rng g) c= ( closed_inside_of_rectangle (a,b,c,d));

      set A = (2 / (b - a)), B = ( - ((b + a) / (b - a))), C = (2 / (d - c)), D = ( - ((d + c) / (d - c)));

      set h = ( AffineMap (A,B,C,D));

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

      

       A21: g2 is continuous one-to-one by A1, A2, A18, Th53;

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

      

       A22: f2 is continuous one-to-one by A1, A2, A17, Th53;

      

       A23: ( rng f2) c= ( closed_inside_of_rectangle (( - 1),1,( - 1),1)) by A1, A2, A19, Th52;

      

       A24: ((f2 . I) `1 ) = 1 by A1, A5, A14, Th56;

      c < (p2 `2 ) by A7, A8, XXREAL_0: 2;

      then

       A25: ((g2 . O) `2 ) <= 1 by A2, A9, A15, Th60;

      (p3 `2 ) > c by A10, A11, XXREAL_0: 2;

      then

       A26: ((f2 . I) `2 ) <= 1 by A2, A12, A14, Th60;

      (d - c) > 0 by A2, XREAL_1: 50;

      then

       A27: C > 0 by XREAL_1: 139;

      

       A28: ( rng g2) c= ( closed_inside_of_rectangle (( - 1),1,( - 1),1)) by A1, A2, A20, Th52;

      

       A29: ((g2 . I) `1 ) = 1 by A1, A6, A16, Th56;

      

       A30: ((g2 . O) `1 ) = ( - 1) by A1, A4, A15, Th54;

      

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

      then

       A32: (h . p2) = (g2 . O) by A15, FUNCT_1: 13;

      (p1 `2 ) <= d by A8, A9, XXREAL_0: 2;

      then

       A33: ( - 1) <= ((f2 . O) `2 ) by A2, A7, A13, Th60;

      

       A34: ((f2 . O) `1 ) = ( - 1) by A1, A3, A13, Th54;

      set x = the Element of (( rng f2) /\ ( rng g2));

      

       A35: ( dom f) = the carrier of I[01] by FUNCT_2:def 1;

      then

       A36: (h . p3) = (f2 . I) by A14, FUNCT_1: 13;

      (b - a) > 0 by A1, XREAL_1: 50;

      then

       A37: A > 0 by XREAL_1: 139;

      then

       A38: ((h . p1) `2 ) < ((h . p2) `2 ) by A8, A27, Th51;

      d >= (p4 `2 ) by A11, A12, XXREAL_0: 2;

      then

       A39: ( - 1) <= ((g2 . I) `2 ) by A2, A10, A16, Th60;

      h is being_homeomorphism by A37, A27, Th51;

      then

       A40: h is one-to-one by TOPS_2:def 5;

      

       A41: ((h . p3) `2 ) > ((h . p4) `2 ) by A11, A37, A27, Th51;

      

       A42: (h . p4) = (g2 . I) by A16, A31, FUNCT_1: 13;

      (h . p1) = (f2 . O) by A13, A35, FUNCT_1: 13;

      then ((f2 . O),(g2 . O),(f2 . I),(g2 . I)) are_in_this_order_on ( rectangle (( - 1),1,( - 1),1)) by A38, A41, A32, A36, A42, A34, A24, A33, A26, A30, A29, A25, A39, Th21;

      then ( rng f2) meets ( rng g2) by A22, A23, A21, A28, JGRAPH_6: 79;

      then

       A43: (( rng f2) /\ ( rng g2)) <> {} by XBOOLE_0:def 7;

      then x in ( rng g2) by XBOOLE_0:def 4;

      then

      consider z2 be object such that

       A44: z2 in ( dom g2) and

       A45: x = (g2 . z2) by FUNCT_1:def 3;

      

       A46: x = (h . (g . z2)) by A31, A44, A45, FUNCT_1: 13;

      

       A47: (g . z2) in ( rng g) by A31, A44, FUNCT_1:def 3;

      (g . z2) in the carrier of ( TOP-REAL 2) by A44, FUNCT_2: 5;

      then

       A48: (g . z2) in ( dom h) by FUNCT_2:def 1;

      x in ( rng f2) by A43, XBOOLE_0:def 4;

      then

      consider z1 be object such that

       A49: z1 in ( dom f2) and

       A50: x = (f2 . z1) by FUNCT_1:def 3;

      

       A51: (f . z1) in ( rng f) by A35, A49, FUNCT_1:def 3;

      (f . z1) in the carrier of ( TOP-REAL 2) by A49, FUNCT_2: 5;

      then

       A52: (f . z1) in ( dom h) by FUNCT_2:def 1;

      x = (h . (f . z1)) by A35, A49, A50, FUNCT_1: 13;

      then (f . z1) = (g . z2) by A46, A52, A48, A40, FUNCT_1:def 4;

      hence thesis by A51, A47, XBOOLE_0: 3;

    end;

    theorem :: JGRAPH_7:83

    for p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real, P,Q be Subset of ( TOP-REAL 2) st a < b & c < d & (p1 `1 ) = a & (p2 `1 ) = a & (p3 `1 ) = b & (p4 `1 ) = b & c <= (p1 `2 ) & (p1 `2 ) < (p2 `2 ) & (p2 `2 ) <= d & c <= (p4 `2 ) & (p4 `2 ) < (p3 `2 ) & (p3 `2 ) <= d & P is_an_arc_of (p1,p3) & Q is_an_arc_of (p2,p4) & P c= ( closed_inside_of_rectangle (a,b,c,d)) & Q c= ( closed_inside_of_rectangle (a,b,c,d)) holds P meets Q

    proof

      let p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real, P,Q be Subset of ( TOP-REAL 2);

      assume that

       A1: a < b and

       A2: c < d and

       A3: (p1 `1 ) = a and

       A4: (p2 `1 ) = a and

       A5: (p3 `1 ) = b and

       A6: (p4 `1 ) = b and

       A7: c <= (p1 `2 ) and

       A8: (p1 `2 ) < (p2 `2 ) and

       A9: (p2 `2 ) <= d and

       A10: c <= (p4 `2 ) and

       A11: (p4 `2 ) < (p3 `2 ) and

       A12: (p3 `2 ) <= d and

       A13: P is_an_arc_of (p1,p3) and

       A14: Q is_an_arc_of (p2,p4) and

       A15: P c= ( closed_inside_of_rectangle (a,b,c,d)) and

       A16: Q c= ( closed_inside_of_rectangle (a,b,c,d));

      

       A17: ex g be Function of I[01] , ( TOP-REAL 2) st g is continuous one-to-one & ( rng g) = Q & (g . 0 ) = p2 & (g . 1) = p4 by A14, Th2;

      ex f be Function of I[01] , ( TOP-REAL 2) st f is continuous one-to-one & ( rng f) = P & (f . 0 ) = p1 & (f . 1) = p3 by A13, Th2;

      hence thesis by A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, A12, A15, A16, A17, Th82;

    end;

    theorem :: JGRAPH_7:84

    

     Th84: for p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real, f,g be Function of I[01] , ( TOP-REAL 2) st a < b & c < d & (p1 `1 ) = a & (p2 `1 ) = a & (p3 `1 ) = b & (p4 `2 ) = c & c <= (p1 `2 ) & (p1 `2 ) < (p2 `2 ) & (p2 `2 ) <= d & c <= (p3 `2 ) & (p3 `2 ) <= d & a < (p4 `1 ) & (p4 `1 ) <= b & (f . 0 ) = p1 & (f . 1) = p3 & (g . 0 ) = p2 & (g . 1) = p4 & f is continuous one-to-one & g is continuous one-to-one & ( rng f) c= ( closed_inside_of_rectangle (a,b,c,d)) & ( rng g) c= ( closed_inside_of_rectangle (a,b,c,d)) holds ( rng f) meets ( rng g)

    proof

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

      let p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real, f,g be Function of I[01] , ( TOP-REAL 2);

      assume that

       A1: a < b and

       A2: c < d and

       A3: (p1 `1 ) = a and

       A4: (p2 `1 ) = a and

       A5: (p3 `1 ) = b and

       A6: (p4 `2 ) = c and

       A7: c <= (p1 `2 ) and

       A8: (p1 `2 ) < (p2 `2 ) and

       A9: (p2 `2 ) <= d and

       A10: c <= (p3 `2 ) and

       A11: (p3 `2 ) <= d and

       A12: a < (p4 `1 ) and

       A13: (p4 `1 ) <= b and

       A14: (f . 0 ) = p1 and

       A15: (f . 1) = p3 and

       A16: (g . 0 ) = p2 and

       A17: (g . 1) = p4 and

       A18: f is continuous one-to-one and

       A19: g is continuous one-to-one and

       A20: ( rng f) c= ( closed_inside_of_rectangle (a,b,c,d)) and

       A21: ( rng g) c= ( closed_inside_of_rectangle (a,b,c,d));

      set A = (2 / (b - a)), B = ( - ((b + a) / (b - a))), C = (2 / (d - c)), D = ( - ((d + c) / (d - c)));

      set h = ( AffineMap (A,B,C,D));

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

      

       A22: ( - 1) < ((g2 . I) `1 ) by A1, A2, A6, A12, A13, A17, Th61;

      

       A23: ((g2 . I) `1 ) <= 1 by A1, A2, A6, A12, A13, A17, Th61;

      

       A24: g2 is continuous one-to-one by A1, A2, A19, Th53;

      

       A25: ((g2 . I) `2 ) = ( - 1) by A2, A6, A17, Th57;

      

       A26: ((g2 . O) `1 ) = ( - 1) by A1, A4, A16, Th54;

      

       A27: ( rng g2) c= ( closed_inside_of_rectangle (( - 1),1,( - 1),1)) by A1, A2, A21, Th52;

      (d - c) > 0 by A2, XREAL_1: 50;

      then

       A28: C > 0 by XREAL_1: 139;

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

      

       A29: ( - 1) <= ((f2 . I) `2 ) by A2, A10, A11, A15, Th60;

      

       A30: ((f2 . O) `1 ) = ( - 1) by A1, A3, A14, Th54;

      

       A31: f2 is continuous one-to-one by A1, A2, A18, Th53;

      (p1 `2 ) <= d by A8, A9, XXREAL_0: 2;

      then

       A32: ( - 1) <= ((f2 . O) `2 ) by A2, A7, A14, Th60;

      

       A33: ((f2 . I) `2 ) <= 1 by A2, A10, A11, A15, Th60;

      set x = the Element of (( rng f2) /\ ( rng g2));

      

       A34: ( dom g) = the carrier of I[01] by FUNCT_2:def 1;

      then

       A35: (h . p2) = (g2 . O) by A16, FUNCT_1: 13;

      

       A36: ((g . I) `2 ) = c by A6, A17;

      c < (p2 `2 ) by A7, A8, XXREAL_0: 2;

      then

       A37: ((g2 . O) `2 ) <= 1 by A1, A2, A9, A12, A13, A16, A17, A36, Th61;

      (b - a) > 0 by A1, XREAL_1: 50;

      then

       A38: A > 0 by XREAL_1: 139;

      then

       A39: ((h . p1) `2 ) < ((h . p2) `2 ) by A8, A28, Th51;

      

       A40: ( rng f2) c= ( closed_inside_of_rectangle (( - 1),1,( - 1),1)) by A1, A2, A20, Th52;

      

       A41: ((f2 . I) `1 ) = 1 by A1, A5, A15, Th56;

      

       A42: ( dom f) = the carrier of I[01] by FUNCT_2:def 1;

      then (h . p1) = (f2 . O) by A14, FUNCT_1: 13;

      then ( rng f2) meets ( rng g2) by A39, A35, A31, A30, A41, A32, A29, A33, A40, A24, A26, A25, A37, A22, A23, A27, Th22, JGRAPH_6: 79;

      then

       A43: (( rng f2) /\ ( rng g2)) <> {} by XBOOLE_0:def 7;

      then x in ( rng g2) by XBOOLE_0:def 4;

      then

      consider z2 be object such that

       A44: z2 in ( dom g2) and

       A45: x = (g2 . z2) by FUNCT_1:def 3;

      

       A46: x = (h . (g . z2)) by A34, A44, A45, FUNCT_1: 13;

      h is being_homeomorphism by A38, A28, Th51;

      then

       A47: h is one-to-one by TOPS_2:def 5;

      x in ( rng f2) by A43, XBOOLE_0:def 4;

      then

      consider z1 be object such that

       A48: z1 in ( dom f2) and

       A49: x = (f2 . z1) by FUNCT_1:def 3;

      

       A50: (f . z1) in ( rng f) by A42, A48, FUNCT_1:def 3;

      

       A51: (g . z2) in ( rng g) by A34, A44, FUNCT_1:def 3;

      (g . z2) in the carrier of ( TOP-REAL 2) by A44, FUNCT_2: 5;

      then

       A52: (g . z2) in ( dom h) by FUNCT_2:def 1;

      (f . z1) in the carrier of ( TOP-REAL 2) by A48, FUNCT_2: 5;

      then

       A53: (f . z1) in ( dom h) by FUNCT_2:def 1;

      x = (h . (f . z1)) by A42, A48, A49, FUNCT_1: 13;

      then (f . z1) = (g . z2) by A46, A53, A52, A47, FUNCT_1:def 4;

      hence thesis by A50, A51, XBOOLE_0: 3;

    end;

    theorem :: JGRAPH_7:85

    for p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real, P,Q be Subset of ( TOP-REAL 2) st a < b & c < d & (p1 `1 ) = a & (p2 `1 ) = a & (p3 `1 ) = b & (p4 `2 ) = c & c <= (p1 `2 ) & (p1 `2 ) < (p2 `2 ) & (p2 `2 ) <= d & c <= (p3 `2 ) & (p3 `2 ) <= d & a < (p4 `1 ) & (p4 `1 ) <= b & P is_an_arc_of (p1,p3) & Q is_an_arc_of (p2,p4) & P c= ( closed_inside_of_rectangle (a,b,c,d)) & Q c= ( closed_inside_of_rectangle (a,b,c,d)) holds P meets Q

    proof

      let p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real, P,Q be Subset of ( TOP-REAL 2);

      assume that

       A1: a < b and

       A2: c < d and

       A3: (p1 `1 ) = a and

       A4: (p2 `1 ) = a and

       A5: (p3 `1 ) = b and

       A6: (p4 `2 ) = c and

       A7: c <= (p1 `2 ) and

       A8: (p1 `2 ) < (p2 `2 ) and

       A9: (p2 `2 ) <= d and

       A10: c <= (p3 `2 ) and

       A11: (p3 `2 ) <= d and

       A12: a < (p4 `1 ) and

       A13: (p4 `1 ) <= b and

       A14: P is_an_arc_of (p1,p3) and

       A15: Q is_an_arc_of (p2,p4) and

       A16: P c= ( closed_inside_of_rectangle (a,b,c,d)) and

       A17: Q c= ( closed_inside_of_rectangle (a,b,c,d));

      

       A18: ex g be Function of I[01] , ( TOP-REAL 2) st g is continuous one-to-one & ( rng g) = Q & (g . 0 ) = p2 & (g . 1) = p4 by A15, Th2;

      ex f be Function of I[01] , ( TOP-REAL 2) st f is continuous one-to-one & ( rng f) = P & (f . 0 ) = p1 & (f . 1) = p3 by A14, Th2;

      hence thesis by A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, A12, A13, A16, A17, A18, Th84;

    end;

    theorem :: JGRAPH_7:86

    

     Th86: for p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real, f,g be Function of I[01] , ( TOP-REAL 2) st a < b & c < d & (p1 `1 ) = a & (p2 `1 ) = a & (p3 `2 ) = c & (p4 `2 ) = c & c <= (p1 `2 ) & (p1 `2 ) < (p2 `2 ) & (p2 `2 ) <= d & a < (p4 `1 ) & (p4 `1 ) < (p3 `1 ) & (p3 `1 ) <= b & (f . 0 ) = p1 & (f . 1) = p3 & (g . 0 ) = p2 & (g . 1) = p4 & f is continuous one-to-one & g is continuous one-to-one & ( rng f) c= ( closed_inside_of_rectangle (a,b,c,d)) & ( rng g) c= ( closed_inside_of_rectangle (a,b,c,d)) holds ( rng f) meets ( rng g)

    proof

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

      let p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real, f,g be Function of I[01] , ( TOP-REAL 2);

      assume that

       A1: a < b and

       A2: c < d and

       A3: (p1 `1 ) = a and

       A4: (p2 `1 ) = a and

       A5: (p3 `2 ) = c and

       A6: (p4 `2 ) = c and

       A7: c <= (p1 `2 ) and

       A8: (p1 `2 ) < (p2 `2 ) and

       A9: (p2 `2 ) <= d and

       A10: a < (p4 `1 ) and

       A11: (p4 `1 ) < (p3 `1 ) and

       A12: (p3 `1 ) <= b and

       A13: (f . 0 ) = p1 and

       A14: (f . 1) = p3 and

       A15: (g . 0 ) = p2 and

       A16: (g . 1) = p4 and

       A17: f is continuous one-to-one and

       A18: g is continuous one-to-one and

       A19: ( rng f) c= ( closed_inside_of_rectangle (a,b,c,d)) and

       A20: ( rng g) c= ( closed_inside_of_rectangle (a,b,c,d));

      set A = (2 / (b - a)), B = ( - ((b + a) / (b - a))), C = (2 / (d - c)), D = ( - ((d + c) / (d - c)));

      set h = ( AffineMap (A,B,C,D));

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

      

       A21: g2 is continuous one-to-one by A1, A2, A18, Th53;

      

       A22: b >= (p4 `1 ) by A11, A12, XXREAL_0: 2;

      then

       A23: ( - 1) < ((g2 . I) `1 ) by A1, A2, A6, A10, A16, Th61;

      

       A24: ((g . I) `2 ) = c by A6, A16;

      c < (p2 `2 ) by A7, A8, XXREAL_0: 2;

      then

       A25: ((g2 . O) `2 ) <= 1 by A1, A2, A9, A10, A15, A16, A22, A24, Th61;

      (d - c) > 0 by A2, XREAL_1: 50;

      then

       A26: C > 0 by XREAL_1: 139;

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

      

       A27: f2 is continuous one-to-one by A1, A2, A17, Th53;

      

       A28: (p3 `1 ) > a by A10, A11, XXREAL_0: 2;

      then

       A29: ((f2 . I) `1 ) <= 1 by A1, A2, A5, A12, A14, Th61;

      

       A30: ((f . I) `2 ) = c by A5, A14;

      (p1 `2 ) <= d by A8, A9, XXREAL_0: 2;

      then

       A31: ( - 1) <= ((f2 . O) `2 ) by A1, A2, A7, A12, A13, A14, A28, A30, Th61;

      

       A32: ((f2 . O) `1 ) = ( - 1) by A1, A3, A13, Th54;

      set x = the Element of (( rng f2) /\ ( rng g2));

      

       A33: ( dom f) = the carrier of I[01] by FUNCT_2:def 1;

      then

       A34: (h . p3) = (f2 . I) by A14, FUNCT_1: 13;

      

       A35: ( rng g2) c= ( closed_inside_of_rectangle (( - 1),1,( - 1),1)) by A1, A2, A20, Th52;

      

       A36: ((g2 . I) `2 ) = ( - 1) by A2, A6, A16, Th57;

      

       A37: ((g2 . O) `1 ) = ( - 1) by A1, A4, A15, Th54;

      

       A38: ( dom g) = the carrier of I[01] by FUNCT_2:def 1;

      then

       A39: (h . p2) = (g2 . O) by A15, FUNCT_1: 13;

      

       A40: ( rng f2) c= ( closed_inside_of_rectangle (( - 1),1,( - 1),1)) by A1, A2, A19, Th52;

      

       A41: ((f2 . I) `2 ) = ( - 1) by A2, A5, A14, Th57;

      (b - a) > 0 by A1, XREAL_1: 50;

      then

       A42: A > 0 by XREAL_1: 139;

      then

       A43: ((h . p1) `2 ) < ((h . p2) `2 ) by A8, A26, Th51;

      

       A44: ((h . p3) `1 ) > ((h . p4) `1 ) by A11, A42, A26, Th50;

      

       A45: (h . p4) = (g2 . I) by A16, A38, FUNCT_1: 13;

      (h . p1) = (f2 . O) by A13, A33, FUNCT_1: 13;

      then ((f2 . O),(g2 . O),(f2 . I),(g2 . I)) are_in_this_order_on ( rectangle (( - 1),1,( - 1),1)) by A43, A44, A39, A34, A45, A32, A41, A31, A29, A37, A36, A25, A23, Th23;

      then ( rng f2) meets ( rng g2) by A27, A40, A21, A35, JGRAPH_6: 79;

      then

       A46: (( rng f2) /\ ( rng g2)) <> {} by XBOOLE_0:def 7;

      then x in ( rng g2) by XBOOLE_0:def 4;

      then

      consider z2 be object such that

       A47: z2 in ( dom g2) and

       A48: x = (g2 . z2) by FUNCT_1:def 3;

      

       A49: x = (h . (g . z2)) by A38, A47, A48, FUNCT_1: 13;

      h is being_homeomorphism by A42, A26, Th51;

      then

       A50: h is one-to-one by TOPS_2:def 5;

      x in ( rng f2) by A46, XBOOLE_0:def 4;

      then

      consider z1 be object such that

       A51: z1 in ( dom f2) and

       A52: x = (f2 . z1) by FUNCT_1:def 3;

      

       A53: (f . z1) in ( rng f) by A33, A51, FUNCT_1:def 3;

      

       A54: (g . z2) in ( rng g) by A38, A47, FUNCT_1:def 3;

      (g . z2) in the carrier of ( TOP-REAL 2) by A47, FUNCT_2: 5;

      then

       A55: (g . z2) in ( dom h) by FUNCT_2:def 1;

      (f . z1) in the carrier of ( TOP-REAL 2) by A51, FUNCT_2: 5;

      then

       A56: (f . z1) in ( dom h) by FUNCT_2:def 1;

      x = (h . (f . z1)) by A33, A51, A52, FUNCT_1: 13;

      then (f . z1) = (g . z2) by A49, A56, A55, A50, FUNCT_1:def 4;

      hence thesis by A53, A54, XBOOLE_0: 3;

    end;

    theorem :: JGRAPH_7:87

    for p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real, P,Q be Subset of ( TOP-REAL 2) st a < b & c < d & (p1 `1 ) = a & (p2 `1 ) = a & (p3 `2 ) = c & (p4 `2 ) = c & c <= (p1 `2 ) & (p1 `2 ) < (p2 `2 ) & (p2 `2 ) <= d & a < (p4 `1 ) & (p4 `1 ) < (p3 `1 ) & (p3 `1 ) <= b & P is_an_arc_of (p1,p3) & Q is_an_arc_of (p2,p4) & P c= ( closed_inside_of_rectangle (a,b,c,d)) & Q c= ( closed_inside_of_rectangle (a,b,c,d)) holds P meets Q

    proof

      let p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real, P,Q be Subset of ( TOP-REAL 2);

      assume that

       A1: a < b and

       A2: c < d and

       A3: (p1 `1 ) = a and

       A4: (p2 `1 ) = a and

       A5: (p3 `2 ) = c and

       A6: (p4 `2 ) = c and

       A7: c <= (p1 `2 ) and

       A8: (p1 `2 ) < (p2 `2 ) and

       A9: (p2 `2 ) <= d and

       A10: a < (p4 `1 ) and

       A11: (p4 `1 ) < (p3 `1 ) and

       A12: (p3 `1 ) <= b and

       A13: P is_an_arc_of (p1,p3) and

       A14: Q is_an_arc_of (p2,p4) and

       A15: P c= ( closed_inside_of_rectangle (a,b,c,d)) and

       A16: Q c= ( closed_inside_of_rectangle (a,b,c,d));

      

       A17: ex g be Function of I[01] , ( TOP-REAL 2) st g is continuous one-to-one & ( rng g) = Q & (g . 0 ) = p2 & (g . 1) = p4 by A14, Th2;

      ex f be Function of I[01] , ( TOP-REAL 2) st f is continuous one-to-one & ( rng f) = P & (f . 0 ) = p1 & (f . 1) = p3 by A13, Th2;

      hence thesis by A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, A12, A15, A16, A17, Th86;

    end;

    theorem :: JGRAPH_7:88

    

     Th88: for p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real, f,g be Function of I[01] , ( TOP-REAL 2) st a < b & c < d & (p1 `1 ) = a & (p2 `2 ) = d & (p3 `2 ) = d & (p4 `2 ) = d & c <= (p1 `2 ) & (p1 `2 ) <= d & a <= (p2 `1 ) & (p2 `1 ) < (p3 `1 ) & (p3 `1 ) < (p4 `1 ) & (p4 `1 ) <= b & (f . 0 ) = p1 & (f . 1) = p3 & (g . 0 ) = p2 & (g . 1) = p4 & f is continuous one-to-one & g is continuous one-to-one & ( rng f) c= ( closed_inside_of_rectangle (a,b,c,d)) & ( rng g) c= ( closed_inside_of_rectangle (a,b,c,d)) holds ( rng f) meets ( rng g)

    proof

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

      let p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real, f,g be Function of I[01] , ( TOP-REAL 2);

      assume that

       A1: a < b and

       A2: c < d and

       A3: (p1 `1 ) = a and

       A4: (p2 `2 ) = d and

       A5: (p3 `2 ) = d and

       A6: (p4 `2 ) = d and

       A7: c <= (p1 `2 ) and

       A8: (p1 `2 ) <= d and

       A9: a <= (p2 `1 ) and

       A10: (p2 `1 ) < (p3 `1 ) and

       A11: (p3 `1 ) < (p4 `1 ) and

       A12: (p4 `1 ) <= b and

       A13: (f . 0 ) = p1 and

       A14: (f . 1) = p3 and

       A15: (g . 0 ) = p2 and

       A16: (g . 1) = p4 and

       A17: f is continuous one-to-one and

       A18: g is continuous one-to-one and

       A19: ( rng f) c= ( closed_inside_of_rectangle (a,b,c,d)) and

       A20: ( rng g) c= ( closed_inside_of_rectangle (a,b,c,d));

      set A = (2 / (b - a)), B = ( - ((b + a) / (b - a))), C = (2 / (d - c)), D = ( - ((d + c) / (d - c)));

      set h = ( AffineMap (A,B,C,D));

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

      

       A21: (p2 `1 ) < (p4 `1 ) by A10, A11, XXREAL_0: 2;

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

      

       A22: ( - 1) <= ((f2 . O) `2 ) by A1, A2, A3, A7, A8, A13, Th59;

      

       A23: ((f2 . O) `1 ) = ( - 1) by A1, A3, A13, Th54;

      

       A24: f2 is continuous one-to-one by A1, A2, A17, Th53;

      

       A25: ( rng f2) c= ( closed_inside_of_rectangle (( - 1),1,( - 1),1)) by A1, A2, A19, Th52;

      

       A26: ((f2 . I) `2 ) = 1 by A2, A5, A14, Th55;

      

       A27: ((f2 . O) `2 ) <= 1 by A1, A2, A3, A7, A8, A13, Th59;

      set x = the Element of (( rng f2) /\ ( rng g2));

      

       A28: ( dom f) = the carrier of I[01] by FUNCT_2:def 1;

      then

       A29: (h . p3) = (f2 . I) by A14, FUNCT_1: 13;

      (d - c) > 0 by A2, XREAL_1: 50;

      then

       A30: C > 0 by XREAL_1: 139;

      (b - a) > 0 by A1, XREAL_1: 50;

      then

       A31: A > 0 by XREAL_1: 139;

      then

       A32: ((h . p2) `1 ) < ((h . p3) `1 ) by A10, A30, Th50;

      ((g . O) `2 ) = d by A4, A15;

      then

       A33: ((g2 . I) `1 ) <= 1 by A1, A9, A12, A15, A16, A21, Th62;

      h is being_homeomorphism by A31, A30, Th50;

      then

       A34: h is one-to-one by TOPS_2:def 5;

      

       A35: ( rng g2) c= ( closed_inside_of_rectangle (( - 1),1,( - 1),1)) by A1, A2, A20, Th52;

      

       A36: ((h . p3) `1 ) < ((h . p4) `1 ) by A11, A31, A30, Th50;

      

       A37: ((g2 . I) `2 ) = 1 by A2, A6, A16, Th55;

      

       A38: ((g2 . O) `2 ) = 1 by A2, A4, A15, Th55;

      ((g . I) `2 ) = d by A6, A16;

      then

       A39: ( - 1) <= ((g2 . O) `1 ) by A1, A9, A12, A15, A16, A21, Th62;

      

       A40: g2 is continuous one-to-one by A1, A2, A18, Th53;

      

       A41: ( dom g) = the carrier of I[01] by FUNCT_2:def 1;

      then

       A42: (h . p4) = (g2 . I) by A16, FUNCT_1: 13;

      (h . p2) = (g2 . O) by A15, A41, FUNCT_1: 13;

      then ( rng f2) meets ( rng g2) by A32, A36, A29, A42, A24, A23, A26, A22, A27, A25, A40, A38, A37, A39, A33, A35, Th24, JGRAPH_6: 79;

      then

       A43: (( rng f2) /\ ( rng g2)) <> {} by XBOOLE_0:def 7;

      then x in ( rng g2) by XBOOLE_0:def 4;

      then

      consider z2 be object such that

       A44: z2 in ( dom g2) and

       A45: x = (g2 . z2) by FUNCT_1:def 3;

      

       A46: x = (h . (g . z2)) by A41, A44, A45, FUNCT_1: 13;

      

       A47: (g . z2) in ( rng g) by A41, A44, FUNCT_1:def 3;

      (g . z2) in the carrier of ( TOP-REAL 2) by A44, FUNCT_2: 5;

      then

       A48: (g . z2) in ( dom h) by FUNCT_2:def 1;

      x in ( rng f2) by A43, XBOOLE_0:def 4;

      then

      consider z1 be object such that

       A49: z1 in ( dom f2) and

       A50: x = (f2 . z1) by FUNCT_1:def 3;

      

       A51: (f . z1) in ( rng f) by A28, A49, FUNCT_1:def 3;

      (f . z1) in the carrier of ( TOP-REAL 2) by A49, FUNCT_2: 5;

      then

       A52: (f . z1) in ( dom h) by FUNCT_2:def 1;

      x = (h . (f . z1)) by A28, A49, A50, FUNCT_1: 13;

      then (f . z1) = (g . z2) by A46, A52, A48, A34, FUNCT_1:def 4;

      hence thesis by A51, A47, XBOOLE_0: 3;

    end;

    theorem :: JGRAPH_7:89

    for p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real, P,Q be Subset of ( TOP-REAL 2) st a < b & c < d & (p1 `1 ) = a & (p2 `2 ) = d & (p3 `2 ) = d & (p4 `2 ) = d & c <= (p1 `2 ) & (p1 `2 ) <= d & a <= (p2 `1 ) & (p2 `1 ) < (p3 `1 ) & (p3 `1 ) < (p4 `1 ) & (p4 `1 ) <= b & P is_an_arc_of (p1,p3) & Q is_an_arc_of (p2,p4) & P c= ( closed_inside_of_rectangle (a,b,c,d)) & Q c= ( closed_inside_of_rectangle (a,b,c,d)) holds P meets Q

    proof

      let p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real, P,Q be Subset of ( TOP-REAL 2);

      assume that

       A1: a < b and

       A2: c < d and

       A3: (p1 `1 ) = a and

       A4: (p2 `2 ) = d and

       A5: (p3 `2 ) = d and

       A6: (p4 `2 ) = d and

       A7: c <= (p1 `2 ) and

       A8: (p1 `2 ) <= d and

       A9: a <= (p2 `1 ) and

       A10: (p2 `1 ) < (p3 `1 ) and

       A11: (p3 `1 ) < (p4 `1 ) and

       A12: (p4 `1 ) <= b and

       A13: P is_an_arc_of (p1,p3) and

       A14: Q is_an_arc_of (p2,p4) and

       A15: P c= ( closed_inside_of_rectangle (a,b,c,d)) and

       A16: Q c= ( closed_inside_of_rectangle (a,b,c,d));

      

       A17: ex g be Function of I[01] , ( TOP-REAL 2) st g is continuous one-to-one & ( rng g) = Q & (g . 0 ) = p2 & (g . 1) = p4 by A14, Th2;

      ex f be Function of I[01] , ( TOP-REAL 2) st f is continuous one-to-one & ( rng f) = P & (f . 0 ) = p1 & (f . 1) = p3 by A13, Th2;

      hence thesis by A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, A12, A15, A16, A17, Th88;

    end;

    theorem :: JGRAPH_7:90

    

     Th90: for p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real, f,g be Function of I[01] , ( TOP-REAL 2) st a < b & c < d & (p1 `1 ) = a & (p2 `2 ) = d & (p3 `2 ) = d & (p4 `1 ) = b & c <= (p1 `2 ) & (p1 `2 ) <= d & a <= (p2 `1 ) & (p2 `1 ) < (p3 `1 ) & (p3 `1 ) <= b & c <= (p4 `2 ) & (p4 `2 ) <= d & (f . 0 ) = p1 & (f . 1) = p3 & (g . 0 ) = p2 & (g . 1) = p4 & f is continuous one-to-one & g is continuous one-to-one & ( rng f) c= ( closed_inside_of_rectangle (a,b,c,d)) & ( rng g) c= ( closed_inside_of_rectangle (a,b,c,d)) holds ( rng f) meets ( rng g)

    proof

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

      let p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real, f,g be Function of I[01] , ( TOP-REAL 2);

      assume that

       A1: a < b and

       A2: c < d and

       A3: (p1 `1 ) = a and

       A4: (p2 `2 ) = d and

       A5: (p3 `2 ) = d and

       A6: (p4 `1 ) = b and

       A7: c <= (p1 `2 ) and

       A8: (p1 `2 ) <= d and

       A9: a <= (p2 `1 ) and

       A10: (p2 `1 ) < (p3 `1 ) and

       A11: (p3 `1 ) <= b and

       A12: c <= (p4 `2 ) and

       A13: (p4 `2 ) <= d and

       A14: (f . 0 ) = p1 and

       A15: (f . 1) = p3 and

       A16: (g . 0 ) = p2 and

       A17: (g . 1) = p4 and

       A18: f is continuous one-to-one and

       A19: g is continuous one-to-one and

       A20: ( rng f) c= ( closed_inside_of_rectangle (a,b,c,d)) and

       A21: ( rng g) c= ( closed_inside_of_rectangle (a,b,c,d));

      set A = (2 / (b - a)), B = ( - ((b + a) / (b - a))), C = (2 / (d - c)), D = ( - ((d + c) / (d - c)));

      set h = ( AffineMap (A,B,C,D));

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

      

       A22: ( - 1) <= ((g2 . I) `2 ) by A1, A2, A6, A12, A13, A17, Th63;

      

       A23: ((g2 . I) `2 ) <= 1 by A1, A2, A6, A12, A13, A17, Th63;

      

       A24: g2 is continuous one-to-one by A1, A2, A19, Th53;

      

       A25: ((g2 . I) `1 ) = 1 by A1, A6, A17, Th56;

      

       A26: ((g2 . O) `2 ) = 1 by A2, A4, A16, Th55;

      

       A27: ( rng g2) c= ( closed_inside_of_rectangle (( - 1),1,( - 1),1)) by A1, A2, A21, Th52;

      (d - c) > 0 by A2, XREAL_1: 50;

      then

       A28: C > 0 by XREAL_1: 139;

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

      

       A29: ( - 1) <= ((f2 . O) `2 ) by A1, A2, A3, A7, A8, A14, Th59;

      

       A30: ( rng f2) c= ( closed_inside_of_rectangle (( - 1),1,( - 1),1)) by A1, A2, A20, Th52;

      

       A31: ((f2 . I) `2 ) = 1 by A2, A5, A15, Th55;

      a <= (p3 `1 ) by A9, A10, XXREAL_0: 2;

      then

       A32: ((f2 . I) `1 ) <= 1 by A1, A2, A5, A11, A15, Th59;

      

       A33: ((f2 . O) `1 ) = ( - 1) by A1, A3, A14, Th54;

      

       A34: f2 is continuous one-to-one by A1, A2, A18, Th53;

      

       A35: ((f2 . O) `2 ) <= 1 by A1, A2, A3, A7, A8, A14, Th59;

      set x = the Element of (( rng f2) /\ ( rng g2));

      

       A36: ( dom f) = the carrier of I[01] by FUNCT_2:def 1;

      then

       A37: (h . p3) = (f2 . I) by A15, FUNCT_1: 13;

      (b - a) > 0 by A1, XREAL_1: 50;

      then

       A38: A > 0 by XREAL_1: 139;

      then

       A39: ((h . p2) `1 ) < ((h . p3) `1 ) by A10, A28, Th50;

      (p2 `1 ) < (p4 `1 ) by A6, A10, A11, XXREAL_0: 2;

      then

       A40: ( - 1) <= ((g2 . O) `1 ) by A1, A2, A4, A6, A9, A16, Th63;

      h is being_homeomorphism by A38, A28, Th50;

      then

       A41: h is one-to-one by TOPS_2:def 5;

      

       A42: ( dom g) = the carrier of I[01] by FUNCT_2:def 1;

      then (h . p2) = (g2 . O) by A16, FUNCT_1: 13;

      then ( rng f2) meets ( rng g2) by A39, A37, A34, A33, A31, A29, A35, A32, A30, A24, A26, A25, A40, A22, A23, A27, Th25, JGRAPH_6: 79;

      then

       A43: (( rng f2) /\ ( rng g2)) <> {} by XBOOLE_0:def 7;

      then x in ( rng g2) by XBOOLE_0:def 4;

      then

      consider z2 be object such that

       A44: z2 in ( dom g2) and

       A45: x = (g2 . z2) by FUNCT_1:def 3;

      

       A46: x = (h . (g . z2)) by A42, A44, A45, FUNCT_1: 13;

      

       A47: (g . z2) in ( rng g) by A42, A44, FUNCT_1:def 3;

      (g . z2) in the carrier of ( TOP-REAL 2) by A44, FUNCT_2: 5;

      then

       A48: (g . z2) in ( dom h) by FUNCT_2:def 1;

      x in ( rng f2) by A43, XBOOLE_0:def 4;

      then

      consider z1 be object such that

       A49: z1 in ( dom f2) and

       A50: x = (f2 . z1) by FUNCT_1:def 3;

      

       A51: (f . z1) in ( rng f) by A36, A49, FUNCT_1:def 3;

      (f . z1) in the carrier of ( TOP-REAL 2) by A49, FUNCT_2: 5;

      then

       A52: (f . z1) in ( dom h) by FUNCT_2:def 1;

      x = (h . (f . z1)) by A36, A49, A50, FUNCT_1: 13;

      then (f . z1) = (g . z2) by A46, A52, A48, A41, FUNCT_1:def 4;

      hence thesis by A51, A47, XBOOLE_0: 3;

    end;

    theorem :: JGRAPH_7:91

    for p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real, P,Q be Subset of ( TOP-REAL 2) st a < b & c < d & (p1 `1 ) = a & (p2 `2 ) = d & (p3 `2 ) = d & (p4 `1 ) = b & c <= (p1 `2 ) & (p1 `2 ) <= d & a <= (p2 `1 ) & (p2 `1 ) < (p3 `1 ) & (p3 `1 ) <= b & c <= (p4 `2 ) & (p4 `2 ) <= d & P is_an_arc_of (p1,p3) & Q is_an_arc_of (p2,p4) & P c= ( closed_inside_of_rectangle (a,b,c,d)) & Q c= ( closed_inside_of_rectangle (a,b,c,d)) holds P meets Q

    proof

      let p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real, P,Q be Subset of ( TOP-REAL 2);

      assume that

       A1: a < b and

       A2: c < d and

       A3: (p1 `1 ) = a and

       A4: (p2 `2 ) = d and

       A5: (p3 `2 ) = d and

       A6: (p4 `1 ) = b and

       A7: c <= (p1 `2 ) and

       A8: (p1 `2 ) <= d and

       A9: a <= (p2 `1 ) and

       A10: (p2 `1 ) < (p3 `1 ) and

       A11: (p3 `1 ) <= b and

       A12: c <= (p4 `2 ) and

       A13: (p4 `2 ) <= d and

       A14: P is_an_arc_of (p1,p3) and

       A15: Q is_an_arc_of (p2,p4) and

       A16: P c= ( closed_inside_of_rectangle (a,b,c,d)) and

       A17: Q c= ( closed_inside_of_rectangle (a,b,c,d));

      

       A18: ex g be Function of I[01] , ( TOP-REAL 2) st g is continuous one-to-one & ( rng g) = Q & (g . 0 ) = p2 & (g . 1) = p4 by A15, Th2;

      ex f be Function of I[01] , ( TOP-REAL 2) st f is continuous one-to-one & ( rng f) = P & (f . 0 ) = p1 & (f . 1) = p3 by A14, Th2;

      hence thesis by A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, A12, A13, A16, A17, A18, Th90;

    end;

    theorem :: JGRAPH_7:92

    

     Th92: for p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real, f,g be Function of I[01] , ( TOP-REAL 2) st a < b & c < d & (p1 `1 ) = a & (p2 `2 ) = d & (p3 `2 ) = d & (p4 `2 ) = c & c <= (p1 `2 ) & (p1 `2 ) <= d & a <= (p2 `1 ) & (p2 `1 ) < (p3 `1 ) & (p3 `1 ) <= b & a < (p4 `1 ) & (p4 `1 ) <= b & (f . 0 ) = p1 & (f . 1) = p3 & (g . 0 ) = p2 & (g . 1) = p4 & f is continuous one-to-one & g is continuous one-to-one & ( rng f) c= ( closed_inside_of_rectangle (a,b,c,d)) & ( rng g) c= ( closed_inside_of_rectangle (a,b,c,d)) holds ( rng f) meets ( rng g)

    proof

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

      let p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real, f,g be Function of I[01] , ( TOP-REAL 2);

      assume that

       A1: a < b and

       A2: c < d and

       A3: (p1 `1 ) = a and

       A4: (p2 `2 ) = d and

       A5: (p3 `2 ) = d and

       A6: (p4 `2 ) = c and

       A7: c <= (p1 `2 ) and

       A8: (p1 `2 ) <= d and

       A9: a <= (p2 `1 ) and

       A10: (p2 `1 ) < (p3 `1 ) and

       A11: (p3 `1 ) <= b and

       A12: a < (p4 `1 ) and

       A13: (p4 `1 ) <= b and

       A14: (f . 0 ) = p1 and

       A15: (f . 1) = p3 and

       A16: (g . 0 ) = p2 and

       A17: (g . 1) = p4 and

       A18: f is continuous one-to-one and

       A19: g is continuous one-to-one and

       A20: ( rng f) c= ( closed_inside_of_rectangle (a,b,c,d)) and

       A21: ( rng g) c= ( closed_inside_of_rectangle (a,b,c,d));

      set A = (2 / (b - a)), B = ( - ((b + a) / (b - a))), C = (2 / (d - c)), D = ( - ((d + c) / (d - c)));

      set h = ( AffineMap (A,B,C,D));

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

      

       A22: ( - 1) <= ((f2 . O) `2 ) by A1, A2, A3, A7, A8, A14, Th59;

      

       A23: ((f2 . O) `2 ) <= 1 by A1, A2, A3, A7, A8, A14, Th59;

      

       A24: f2 is continuous one-to-one by A1, A2, A18, Th53;

      

       A25: ((f2 . I) `2 ) = 1 by A2, A5, A15, Th55;

      

       A26: ((f2 . O) `1 ) = ( - 1) by A1, A3, A14, Th54;

      

       A27: ( rng f2) c= ( closed_inside_of_rectangle (( - 1),1,( - 1),1)) by A1, A2, A20, Th52;

      (d - c) > 0 by A2, XREAL_1: 50;

      then

       A28: C > 0 by XREAL_1: 139;

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

      

       A29: ( - 1) < ((g2 . I) `1 ) by A1, A12, A13, A17, Th64;

      

       A30: ((g2 . O) `2 ) = 1 by A2, A4, A16, Th55;

      

       A31: g2 is continuous one-to-one by A1, A2, A19, Th53;

      

       A32: ((g . I) `2 ) = c by A6, A17;

      (p2 `1 ) < b by A10, A11, XXREAL_0: 2;

      then

       A33: ( - 1) <= ((g2 . O) `1 ) by A1, A9, A12, A13, A16, A17, A32, Th64;

      

       A34: ((g2 . I) `1 ) <= 1 by A1, A12, A13, A17, Th64;

      set x = the Element of (( rng f2) /\ ( rng g2));

      

       A35: ( dom f) = the carrier of I[01] by FUNCT_2:def 1;

      then

       A36: (h . p3) = (f2 . I) by A15, FUNCT_1: 13;

      (b - a) > 0 by A1, XREAL_1: 50;

      then

       A37: A > 0 by XREAL_1: 139;

      then

       A38: ((h . p2) `1 ) < ((h . p3) `1 ) by A10, A28, Th50;

      a <= (p3 `1 ) by A9, A10, XXREAL_0: 2;

      then

       A39: ((f2 . I) `1 ) <= 1 by A1, A2, A5, A11, A15, Th59;

      h is being_homeomorphism by A37, A28, Th50;

      then

       A40: h is one-to-one by TOPS_2:def 5;

      

       A41: ( rng g2) c= ( closed_inside_of_rectangle (( - 1),1,( - 1),1)) by A1, A2, A21, Th52;

      

       A42: ((g2 . I) `2 ) = ( - 1) by A2, A6, A17, Th57;

      

       A43: ( dom g) = the carrier of I[01] by FUNCT_2:def 1;

      then (h . p2) = (g2 . O) by A16, FUNCT_1: 13;

      then ( rng f2) meets ( rng g2) by A38, A36, A24, A26, A25, A22, A23, A39, A27, A31, A30, A42, A33, A29, A34, A41, Th26, JGRAPH_6: 79;

      then

       A44: (( rng f2) /\ ( rng g2)) <> {} by XBOOLE_0:def 7;

      then x in ( rng g2) by XBOOLE_0:def 4;

      then

      consider z2 be object such that

       A45: z2 in ( dom g2) and

       A46: x = (g2 . z2) by FUNCT_1:def 3;

      

       A47: x = (h . (g . z2)) by A43, A45, A46, FUNCT_1: 13;

      

       A48: (g . z2) in ( rng g) by A43, A45, FUNCT_1:def 3;

      (g . z2) in the carrier of ( TOP-REAL 2) by A45, FUNCT_2: 5;

      then

       A49: (g . z2) in ( dom h) by FUNCT_2:def 1;

      x in ( rng f2) by A44, XBOOLE_0:def 4;

      then

      consider z1 be object such that

       A50: z1 in ( dom f2) and

       A51: x = (f2 . z1) by FUNCT_1:def 3;

      

       A52: (f . z1) in ( rng f) by A35, A50, FUNCT_1:def 3;

      (f . z1) in the carrier of ( TOP-REAL 2) by A50, FUNCT_2: 5;

      then

       A53: (f . z1) in ( dom h) by FUNCT_2:def 1;

      x = (h . (f . z1)) by A35, A50, A51, FUNCT_1: 13;

      then (f . z1) = (g . z2) by A47, A53, A49, A40, FUNCT_1:def 4;

      hence thesis by A52, A48, XBOOLE_0: 3;

    end;

    theorem :: JGRAPH_7:93

    for p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real, P,Q be Subset of ( TOP-REAL 2) st a < b & c < d & (p1 `1 ) = a & (p2 `2 ) = d & (p3 `2 ) = d & (p4 `2 ) = c & c <= (p1 `2 ) & (p1 `2 ) <= d & a <= (p2 `1 ) & (p2 `1 ) < (p3 `1 ) & (p3 `1 ) <= b & a < (p4 `1 ) & (p4 `1 ) <= b & P is_an_arc_of (p1,p3) & Q is_an_arc_of (p2,p4) & P c= ( closed_inside_of_rectangle (a,b,c,d)) & Q c= ( closed_inside_of_rectangle (a,b,c,d)) holds P meets Q

    proof

      let p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real, P,Q be Subset of ( TOP-REAL 2);

      assume that

       A1: a < b and

       A2: c < d and

       A3: (p1 `1 ) = a and

       A4: (p2 `2 ) = d and

       A5: (p3 `2 ) = d and

       A6: (p4 `2 ) = c and

       A7: c <= (p1 `2 ) and

       A8: (p1 `2 ) <= d and

       A9: a <= (p2 `1 ) and

       A10: (p2 `1 ) < (p3 `1 ) and

       A11: (p3 `1 ) <= b and

       A12: a < (p4 `1 ) and

       A13: (p4 `1 ) <= b and

       A14: P is_an_arc_of (p1,p3) and

       A15: Q is_an_arc_of (p2,p4) and

       A16: P c= ( closed_inside_of_rectangle (a,b,c,d)) and

       A17: Q c= ( closed_inside_of_rectangle (a,b,c,d));

      

       A18: ex g be Function of I[01] , ( TOP-REAL 2) st g is continuous one-to-one & ( rng g) = Q & (g . 0 ) = p2 & (g . 1) = p4 by A15, Th2;

      ex f be Function of I[01] , ( TOP-REAL 2) st f is continuous one-to-one & ( rng f) = P & (f . 0 ) = p1 & (f . 1) = p3 by A14, Th2;

      hence thesis by A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, A12, A13, A16, A17, A18, Th92;

    end;

    theorem :: JGRAPH_7:94

    

     Th94: for p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real, f,g be Function of I[01] , ( TOP-REAL 2) st a < b & c < d & (p1 `1 ) = a & (p2 `2 ) = d & (p3 `1 ) = b & (p4 `1 ) = b & c <= (p1 `2 ) & (p1 `2 ) <= d & a <= (p2 `1 ) & (p2 `1 ) <= b & c <= (p4 `2 ) & (p4 `2 ) < (p3 `2 ) & (p3 `2 ) <= d & (f . 0 ) = p1 & (f . 1) = p3 & (g . 0 ) = p2 & (g . 1) = p4 & f is continuous one-to-one & g is continuous one-to-one & ( rng f) c= ( closed_inside_of_rectangle (a,b,c,d)) & ( rng g) c= ( closed_inside_of_rectangle (a,b,c,d)) holds ( rng f) meets ( rng g)

    proof

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

      let p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real, f,g be Function of I[01] , ( TOP-REAL 2);

      assume that

       A1: a < b and

       A2: c < d and

       A3: (p1 `1 ) = a and

       A4: (p2 `2 ) = d and

       A5: (p3 `1 ) = b and

       A6: (p4 `1 ) = b and

       A7: c <= (p1 `2 ) and

       A8: (p1 `2 ) <= d and

       A9: a <= (p2 `1 ) and

       A10: (p2 `1 ) <= b and

       A11: c <= (p4 `2 ) and

       A12: (p4 `2 ) < (p3 `2 ) and

       A13: (p3 `2 ) <= d and

       A14: (f . 0 ) = p1 and

       A15: (f . 1) = p3 and

       A16: (g . 0 ) = p2 and

       A17: (g . 1) = p4 and

       A18: f is continuous one-to-one and

       A19: g is continuous one-to-one and

       A20: ( rng f) c= ( closed_inside_of_rectangle (a,b,c,d)) and

       A21: ( rng g) c= ( closed_inside_of_rectangle (a,b,c,d));

      set A = (2 / (b - a)), B = ( - ((b + a) / (b - a))), C = (2 / (d - c)), D = ( - ((d + c) / (d - c)));

      set h = ( AffineMap (A,B,C,D));

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

      

       A22: ( - 1) <= ((g2 . O) `1 ) by A1, A2, A4, A9, A10, A16, Th63;

      

       A23: ((g2 . O) `1 ) <= 1 by A1, A2, A4, A9, A10, A16, Th63;

      

       A24: g2 is continuous one-to-one by A1, A2, A19, Th53;

      

       A25: ((g2 . I) `1 ) = 1 by A1, A6, A17, Th56;

      

       A26: ((g2 . O) `2 ) = 1 by A2, A4, A16, Th55;

      

       A27: ( rng g2) c= ( closed_inside_of_rectangle (( - 1),1,( - 1),1)) by A1, A2, A21, Th52;

      (d - c) > 0 by A2, XREAL_1: 50;

      then

       A28: C > 0 by XREAL_1: 139;

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

      

       A29: ( - 1) <= ((f2 . O) `2 ) by A2, A7, A8, A14, Th60;

      

       A30: ((f2 . O) `1 ) = ( - 1) by A1, A3, A14, Th54;

      

       A31: f2 is continuous one-to-one by A1, A2, A18, Th53;

      (p3 `2 ) > c by A11, A12, XXREAL_0: 2;

      then

       A32: ((f2 . I) `2 ) <= 1 by A2, A13, A15, Th60;

      

       A33: ((f2 . O) `2 ) <= 1 by A2, A7, A8, A14, Th60;

      set x = the Element of (( rng f2) /\ ( rng g2));

      

       A34: ( dom g) = the carrier of I[01] by FUNCT_2:def 1;

      then

       A35: (h . p4) = (g2 . I) by A17, FUNCT_1: 13;

      (b - a) > 0 by A1, XREAL_1: 50;

      then

       A36: A > 0 by XREAL_1: 139;

      then

       A37: ((h . p4) `2 ) < ((h . p3) `2 ) by A12, A28, Th51;

      (p4 `2 ) < d by A12, A13, XXREAL_0: 2;

      then

       A38: ( - 1) <= ((g2 . I) `2 ) by A1, A2, A6, A11, A17, Th63;

      h is being_homeomorphism by A36, A28, Th51;

      then

       A39: h is one-to-one by TOPS_2:def 5;

      

       A40: ( rng f2) c= ( closed_inside_of_rectangle (( - 1),1,( - 1),1)) by A1, A2, A20, Th52;

      

       A41: ((f2 . I) `1 ) = 1 by A1, A5, A15, Th56;

      

       A42: ( dom f) = the carrier of I[01] by FUNCT_2:def 1;

      then (h . p3) = (f2 . I) by A15, FUNCT_1: 13;

      then ( rng f2) meets ( rng g2) by A37, A35, A31, A30, A41, A29, A33, A32, A40, A24, A26, A25, A22, A23, A38, A27, Th27, JGRAPH_6: 79;

      then

       A43: (( rng f2) /\ ( rng g2)) <> {} by XBOOLE_0:def 7;

      then x in ( rng g2) by XBOOLE_0:def 4;

      then

      consider z2 be object such that

       A44: z2 in ( dom g2) and

       A45: x = (g2 . z2) by FUNCT_1:def 3;

      

       A46: x = (h . (g . z2)) by A34, A44, A45, FUNCT_1: 13;

      

       A47: (g . z2) in ( rng g) by A34, A44, FUNCT_1:def 3;

      (g . z2) in the carrier of ( TOP-REAL 2) by A44, FUNCT_2: 5;

      then

       A48: (g . z2) in ( dom h) by FUNCT_2:def 1;

      x in ( rng f2) by A43, XBOOLE_0:def 4;

      then

      consider z1 be object such that

       A49: z1 in ( dom f2) and

       A50: x = (f2 . z1) by FUNCT_1:def 3;

      

       A51: (f . z1) in ( rng f) by A42, A49, FUNCT_1:def 3;

      (f . z1) in the carrier of ( TOP-REAL 2) by A49, FUNCT_2: 5;

      then

       A52: (f . z1) in ( dom h) by FUNCT_2:def 1;

      x = (h . (f . z1)) by A42, A49, A50, FUNCT_1: 13;

      then (f . z1) = (g . z2) by A46, A52, A48, A39, FUNCT_1:def 4;

      hence thesis by A51, A47, XBOOLE_0: 3;

    end;

    theorem :: JGRAPH_7:95

    for p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real, P,Q be Subset of ( TOP-REAL 2) st a < b & c < d & (p1 `1 ) = a & (p2 `2 ) = d & (p3 `1 ) = b & (p4 `1 ) = b & c <= (p1 `2 ) & (p1 `2 ) <= d & a <= (p2 `1 ) & (p2 `1 ) <= b & c <= (p4 `2 ) & (p4 `2 ) < (p3 `2 ) & (p3 `2 ) <= d & P is_an_arc_of (p1,p3) & Q is_an_arc_of (p2,p4) & P c= ( closed_inside_of_rectangle (a,b,c,d)) & Q c= ( closed_inside_of_rectangle (a,b,c,d)) holds P meets Q

    proof

      let p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real, P,Q be Subset of ( TOP-REAL 2);

      assume that

       A1: a < b and

       A2: c < d and

       A3: (p1 `1 ) = a and

       A4: (p2 `2 ) = d and

       A5: (p3 `1 ) = b and

       A6: (p4 `1 ) = b and

       A7: c <= (p1 `2 ) and

       A8: (p1 `2 ) <= d and

       A9: a <= (p2 `1 ) and

       A10: (p2 `1 ) <= b and

       A11: c <= (p4 `2 ) and

       A12: (p4 `2 ) < (p3 `2 ) and

       A13: (p3 `2 ) <= d and

       A14: P is_an_arc_of (p1,p3) and

       A15: Q is_an_arc_of (p2,p4) and

       A16: P c= ( closed_inside_of_rectangle (a,b,c,d)) and

       A17: Q c= ( closed_inside_of_rectangle (a,b,c,d));

      

       A18: ex g be Function of I[01] , ( TOP-REAL 2) st g is continuous one-to-one & ( rng g) = Q & (g . 0 ) = p2 & (g . 1) = p4 by A15, Th2;

      ex f be Function of I[01] , ( TOP-REAL 2) st f is continuous one-to-one & ( rng f) = P & (f . 0 ) = p1 & (f . 1) = p3 by A14, Th2;

      hence thesis by A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, A12, A13, A16, A17, A18, Th94;

    end;

    theorem :: JGRAPH_7:96

    

     Th96: for p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real, f,g be Function of I[01] , ( TOP-REAL 2) st a < b & c < d & (p1 `1 ) = a & (p2 `2 ) = d & (p3 `1 ) = b & (p4 `2 ) = c & c <= (p1 `2 ) & (p1 `2 ) <= d & a <= (p2 `1 ) & (p2 `1 ) <= b & c <= (p3 `2 ) & (p3 `2 ) <= d & a < (p4 `1 ) & (p4 `1 ) <= b & (f . 0 ) = p1 & (f . 1) = p3 & (g . 0 ) = p2 & (g . 1) = p4 & f is continuous one-to-one & g is continuous one-to-one & ( rng f) c= ( closed_inside_of_rectangle (a,b,c,d)) & ( rng g) c= ( closed_inside_of_rectangle (a,b,c,d)) holds ( rng f) meets ( rng g)

    proof

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

      let p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real, f,g be Function of I[01] , ( TOP-REAL 2);

      assume that

       A1: a < b and

       A2: c < d and

       A3: (p1 `1 ) = a and

       A4: (p2 `2 ) = d and

       A5: (p3 `1 ) = b and

       A6: (p4 `2 ) = c and

       A7: c <= (p1 `2 ) and

       A8: (p1 `2 ) <= d and

       A9: a <= (p2 `1 ) and

       A10: (p2 `1 ) <= b and

       A11: c <= (p3 `2 ) and

       A12: (p3 `2 ) <= d and

       A13: a < (p4 `1 ) and

       A14: (p4 `1 ) <= b and

       A15: (f . 0 ) = p1 and

       A16: (f . 1) = p3 and

       A17: (g . 0 ) = p2 and

       A18: (g . 1) = p4 and

       A19: f is continuous one-to-one and

       A20: g is continuous one-to-one and

       A21: ( rng f) c= ( closed_inside_of_rectangle (a,b,c,d)) and

       A22: ( rng g) c= ( closed_inside_of_rectangle (a,b,c,d));

      set A = (2 / (b - a)), B = ( - ((b + a) / (b - a))), C = (2 / (d - c)), D = ( - ((d + c) / (d - c)));

      set h = ( AffineMap (A,B,C,D));

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

      

       A23: ( - 1) < ((g2 . I) `1 ) by A1, A13, A14, A18, Th64;

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

      

       A24: ( - 1) <= ((f2 . O) `2 ) by A2, A7, A8, A15, Th60;

      

       A25: ((f2 . O) `2 ) <= 1 by A2, A7, A8, A15, Th60;

      

       A26: f2 is continuous one-to-one by A1, A2, A19, Th53;

      

       A27: ( - 1) <= ((f2 . I) `2 ) by A2, A11, A12, A16, Th60;

      

       A28: ((f2 . O) `1 ) = ( - 1) by A1, A3, A15, Th54;

      

       A29: ( rng f2) c= ( closed_inside_of_rectangle (( - 1),1,( - 1),1)) by A1, A2, A21, Th52;

      set x = the Element of (( rng f2) /\ ( rng g2));

      (d - c) > 0 by A2, XREAL_1: 50;

      then

       A30: C > 0 by XREAL_1: 139;

      

       A31: ((g . I) `2 ) = c by A6, A18;

      then

       A32: ((g2 . O) `1 ) <= 1 by A1, A9, A10, A13, A14, A17, A18, Th64;

      

       A33: ((g2 . I) `1 ) <= 1 by A1, A13, A14, A18, Th64;

      

       A34: g2 is continuous one-to-one by A1, A2, A20, Th53;

      

       A35: ( rng g2) c= ( closed_inside_of_rectangle (( - 1),1,( - 1),1)) by A1, A2, A22, Th52;

      

       A36: ((g2 . I) `2 ) = ( - 1) by A2, A6, A18, Th57;

      

       A37: ((g2 . O) `2 ) = 1 by A2, A4, A17, Th55;

      

       A38: ((f2 . I) `2 ) <= 1 by A2, A11, A12, A16, Th60;

      

       A39: ((f2 . I) `1 ) = 1 by A1, A5, A16, Th56;

      ( - 1) <= ((g2 . O) `1 ) by A1, A9, A10, A13, A14, A17, A18, A31, Th64;

      then ( rng f2) meets ( rng g2) by A26, A28, A39, A24, A25, A27, A38, A29, A34, A37, A36, A32, A23, A33, A35, Th28, JGRAPH_6: 79;

      then

       A40: (( rng f2) /\ ( rng g2)) <> {} by XBOOLE_0:def 7;

      then x in ( rng f2) by XBOOLE_0:def 4;

      then

      consider z1 be object such that

       A41: z1 in ( dom f2) and

       A42: x = (f2 . z1) by FUNCT_1:def 3;

      

       A43: ( dom f) = the carrier of I[01] by FUNCT_2:def 1;

      then

       A44: (f . z1) in ( rng f) by A41, FUNCT_1:def 3;

      (b - a) > 0 by A1, XREAL_1: 50;

      then A > 0 by XREAL_1: 139;

      then h is being_homeomorphism by A30, Th51;

      then

       A45: h is one-to-one by TOPS_2:def 5;

      (f . z1) in the carrier of ( TOP-REAL 2) by A41, FUNCT_2: 5;

      then

       A46: (f . z1) in ( dom h) by FUNCT_2:def 1;

      x in ( rng g2) by A40, XBOOLE_0:def 4;

      then

      consider z2 be object such that

       A47: z2 in ( dom g2) and

       A48: x = (g2 . z2) by FUNCT_1:def 3;

      

       A49: ( dom g) = the carrier of I[01] by FUNCT_2:def 1;

      then

       A50: (g . z2) in ( rng g) by A47, FUNCT_1:def 3;

      

       A51: x = (h . (g . z2)) by A49, A47, A48, FUNCT_1: 13;

      (g . z2) in the carrier of ( TOP-REAL 2) by A47, FUNCT_2: 5;

      then

       A52: (g . z2) in ( dom h) by FUNCT_2:def 1;

      x = (h . (f . z1)) by A43, A41, A42, FUNCT_1: 13;

      then (f . z1) = (g . z2) by A51, A46, A52, A45, FUNCT_1:def 4;

      hence thesis by A44, A50, XBOOLE_0: 3;

    end;

    theorem :: JGRAPH_7:97

    for p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real, P,Q be Subset of ( TOP-REAL 2) st a < b & c < d & (p1 `1 ) = a & (p2 `2 ) = d & (p3 `1 ) = b & (p4 `2 ) = c & c <= (p1 `2 ) & (p1 `2 ) <= d & a <= (p2 `1 ) & (p2 `1 ) <= b & c <= (p3 `2 ) & (p3 `2 ) <= d & a < (p4 `1 ) & (p4 `1 ) <= b & P is_an_arc_of (p1,p3) & Q is_an_arc_of (p2,p4) & P c= ( closed_inside_of_rectangle (a,b,c,d)) & Q c= ( closed_inside_of_rectangle (a,b,c,d)) holds P meets Q

    proof

      let p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real, P,Q be Subset of ( TOP-REAL 2);

      assume that

       A1: a < b and

       A2: c < d and

       A3: (p1 `1 ) = a and

       A4: (p2 `2 ) = d and

       A5: (p3 `1 ) = b and

       A6: (p4 `2 ) = c and

       A7: c <= (p1 `2 ) and

       A8: (p1 `2 ) <= d and

       A9: a <= (p2 `1 ) and

       A10: (p2 `1 ) <= b and

       A11: c <= (p3 `2 ) and

       A12: (p3 `2 ) <= d and

       A13: a < (p4 `1 ) and

       A14: (p4 `1 ) <= b and

       A15: P is_an_arc_of (p1,p3) and

       A16: Q is_an_arc_of (p2,p4) and

       A17: P c= ( closed_inside_of_rectangle (a,b,c,d)) and

       A18: Q c= ( closed_inside_of_rectangle (a,b,c,d));

      

       A19: ex g be Function of I[01] , ( TOP-REAL 2) st g is continuous one-to-one & ( rng g) = Q & (g . 0 ) = p2 & (g . 1) = p4 by A16, Th2;

      ex f be Function of I[01] , ( TOP-REAL 2) st f is continuous one-to-one & ( rng f) = P & (f . 0 ) = p1 & (f . 1) = p3 by A15, Th2;

      hence thesis by A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, A12, A13, A14, A17, A18, A19, Th96;

    end;

    theorem :: JGRAPH_7:98

    

     Th98: for p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real, f,g be Function of I[01] , ( TOP-REAL 2) st a < b & c < d & (p1 `1 ) = a & (p2 `2 ) = d & (p3 `2 ) = c & (p4 `2 ) = c & c <= (p1 `2 ) & (p1 `2 ) <= d & a <= (p2 `1 ) & (p2 `1 ) <= b & a < (p4 `1 ) & (p4 `1 ) < (p3 `1 ) & (p3 `1 ) <= b & (f . 0 ) = p1 & (f . 1) = p3 & (g . 0 ) = p2 & (g . 1) = p4 & f is continuous one-to-one & g is continuous one-to-one & ( rng f) c= ( closed_inside_of_rectangle (a,b,c,d)) & ( rng g) c= ( closed_inside_of_rectangle (a,b,c,d)) holds ( rng f) meets ( rng g)

    proof

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

      let p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real, f,g be Function of I[01] , ( TOP-REAL 2);

      assume that

       A1: a < b and

       A2: c < d and

       A3: (p1 `1 ) = a and

       A4: (p2 `2 ) = d and

       A5: (p3 `2 ) = c and

       A6: (p4 `2 ) = c and

       A7: c <= (p1 `2 ) and

       A8: (p1 `2 ) <= d and

       A9: a <= (p2 `1 ) and

       A10: (p2 `1 ) <= b and

       A11: a < (p4 `1 ) and

       A12: (p4 `1 ) < (p3 `1 ) and

       A13: (p3 `1 ) <= b and

       A14: (f . 0 ) = p1 and

       A15: (f . 1) = p3 and

       A16: (g . 0 ) = p2 and

       A17: (g . 1) = p4 and

       A18: f is continuous one-to-one and

       A19: g is continuous one-to-one and

       A20: ( rng f) c= ( closed_inside_of_rectangle (a,b,c,d)) and

       A21: ( rng g) c= ( closed_inside_of_rectangle (a,b,c,d));

      set A = (2 / (b - a)), B = ( - ((b + a) / (b - a))), C = (2 / (d - c)), D = ( - ((d + c) / (d - c)));

      set h = ( AffineMap (A,B,C,D));

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

      

       A22: g2 is continuous one-to-one by A1, A2, A19, Th53;

      

       A23: (p4 `1 ) < b by A12, A13, XXREAL_0: 2;

      then

       A24: ( - 1) < ((g2 . I) `1 ) by A1, A11, A17, Th64;

      

       A25: ((g . I) `2 ) = c by A6, A17;

      then

       A26: ( - 1) <= ((g2 . O) `1 ) by A1, A9, A10, A11, A16, A17, A23, Th64;

      

       A27: ((g2 . O) `1 ) <= 1 by A1, A9, A10, A11, A16, A17, A23, A25, Th64;

      (d - c) > 0 by A2, XREAL_1: 50;

      then

       A28: C > 0 by XREAL_1: 139;

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

      

       A29: f2 is continuous one-to-one by A1, A2, A18, Th53;

      

       A30: (p3 `1 ) > a by A11, A12, XXREAL_0: 2;

      then

       A31: ((f2 . I) `1 ) <= 1 by A1, A2, A5, A13, A15, Th61;

      

       A32: ((f . I) `2 ) = c by A5, A15;

      then

       A33: ((f2 . O) `2 ) <= 1 by A1, A2, A7, A8, A13, A14, A15, A30, Th61;

      set x = the Element of (( rng f2) /\ ( rng g2));

      

       A34: ( dom g) = the carrier of I[01] by FUNCT_2:def 1;

      then

       A35: (h . p4) = (g2 . I) by A17, FUNCT_1: 13;

      

       A36: ((g2 . I) `2 ) = ( - 1) by A2, A6, A17, Th57;

      

       A37: ((g2 . O) `2 ) = 1 by A2, A4, A16, Th55;

      (b - a) > 0 by A1, XREAL_1: 50;

      then

       A38: A > 0 by XREAL_1: 139;

      then h is being_homeomorphism by A28, Th50;

      then

       A39: h is one-to-one by TOPS_2:def 5;

      

       A40: ( rng g2) c= ( closed_inside_of_rectangle (( - 1),1,( - 1),1)) by A1, A2, A21, Th52;

      

       A41: ( rng f2) c= ( closed_inside_of_rectangle (( - 1),1,( - 1),1)) by A1, A2, A20, Th52;

      

       A42: ((f2 . I) `2 ) = ( - 1) by A2, A5, A15, Th57;

      

       A43: ((f2 . O) `1 ) = ( - 1) by A1, A3, A14, Th54;

      

       A44: ( dom f) = the carrier of I[01] by FUNCT_2:def 1;

      then (h . p3) = (f2 . I) by A15, FUNCT_1: 13;

      then

       A45: ((g2 . I) `1 ) < ((f2 . I) `1 ) by A12, A38, A28, A35, Th50;

      ( - 1) <= ((f2 . O) `2 ) by A1, A2, A7, A8, A13, A14, A15, A30, A32, Th61;

      then ( rng f2) meets ( rng g2) by A29, A43, A42, A33, A31, A41, A22, A37, A36, A26, A27, A24, A40, A45, Th29, JGRAPH_6: 79;

      then

       A46: (( rng f2) /\ ( rng g2)) <> {} by XBOOLE_0:def 7;

      then x in ( rng g2) by XBOOLE_0:def 4;

      then

      consider z2 be object such that

       A47: z2 in ( dom g2) and

       A48: x = (g2 . z2) by FUNCT_1:def 3;

      

       A49: x = (h . (g . z2)) by A34, A47, A48, FUNCT_1: 13;

      

       A50: (g . z2) in ( rng g) by A34, A47, FUNCT_1:def 3;

      (g . z2) in the carrier of ( TOP-REAL 2) by A47, FUNCT_2: 5;

      then

       A51: (g . z2) in ( dom h) by FUNCT_2:def 1;

      x in ( rng f2) by A46, XBOOLE_0:def 4;

      then

      consider z1 be object such that

       A52: z1 in ( dom f2) and

       A53: x = (f2 . z1) by FUNCT_1:def 3;

      

       A54: (f . z1) in ( rng f) by A44, A52, FUNCT_1:def 3;

      (f . z1) in the carrier of ( TOP-REAL 2) by A52, FUNCT_2: 5;

      then

       A55: (f . z1) in ( dom h) by FUNCT_2:def 1;

      x = (h . (f . z1)) by A44, A52, A53, FUNCT_1: 13;

      then (f . z1) = (g . z2) by A49, A55, A51, A39, FUNCT_1:def 4;

      hence thesis by A54, A50, XBOOLE_0: 3;

    end;

    theorem :: JGRAPH_7:99

    for p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real, P,Q be Subset of ( TOP-REAL 2) st a < b & c < d & (p1 `1 ) = a & (p2 `2 ) = d & (p3 `2 ) = c & (p4 `2 ) = c & c <= (p1 `2 ) & (p1 `2 ) <= d & a <= (p2 `1 ) & (p2 `1 ) <= b & a < (p4 `1 ) & (p4 `1 ) < (p3 `1 ) & (p3 `1 ) <= b & P is_an_arc_of (p1,p3) & Q is_an_arc_of (p2,p4) & P c= ( closed_inside_of_rectangle (a,b,c,d)) & Q c= ( closed_inside_of_rectangle (a,b,c,d)) holds P meets Q

    proof

      let p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real, P,Q be Subset of ( TOP-REAL 2);

      assume that

       A1: a < b and

       A2: c < d and

       A3: (p1 `1 ) = a and

       A4: (p2 `2 ) = d and

       A5: (p3 `2 ) = c and

       A6: (p4 `2 ) = c and

       A7: c <= (p1 `2 ) and

       A8: (p1 `2 ) <= d and

       A9: a <= (p2 `1 ) and

       A10: (p2 `1 ) <= b and

       A11: a < (p4 `1 ) and

       A12: (p4 `1 ) < (p3 `1 ) and

       A13: (p3 `1 ) <= b and

       A14: P is_an_arc_of (p1,p3) and

       A15: Q is_an_arc_of (p2,p4) and

       A16: P c= ( closed_inside_of_rectangle (a,b,c,d)) and

       A17: Q c= ( closed_inside_of_rectangle (a,b,c,d));

      

       A18: ex g be Function of I[01] , ( TOP-REAL 2) st g is continuous one-to-one & ( rng g) = Q & (g . 0 ) = p2 & (g . 1) = p4 by A15, Th2;

      ex f be Function of I[01] , ( TOP-REAL 2) st f is continuous one-to-one & ( rng f) = P & (f . 0 ) = p1 & (f . 1) = p3 by A14, Th2;

      hence thesis by A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, A12, A13, A16, A17, A18, Th98;

    end;

    theorem :: JGRAPH_7:100

    

     Th100: for p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real, f,g be Function of I[01] , ( TOP-REAL 2) st a < b & c < d & (p1 `1 ) = a & (p2 `1 ) = b & (p3 `1 ) = b & (p4 `1 ) = b & c <= (p1 `2 ) & (p1 `2 ) <= d & c <= (p4 `2 ) & (p4 `2 ) < (p3 `2 ) & (p3 `2 ) < (p2 `2 ) & (p2 `2 ) <= d & (f . 0 ) = p1 & (f . 1) = p3 & (g . 0 ) = p2 & (g . 1) = p4 & f is continuous one-to-one & g is continuous one-to-one & ( rng f) c= ( closed_inside_of_rectangle (a,b,c,d)) & ( rng g) c= ( closed_inside_of_rectangle (a,b,c,d)) holds ( rng f) meets ( rng g)

    proof

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

      let p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real, f,g be Function of I[01] , ( TOP-REAL 2);

      assume that

       A1: a < b and

       A2: c < d and

       A3: (p1 `1 ) = a and

       A4: (p2 `1 ) = b and

       A5: (p3 `1 ) = b and

       A6: (p4 `1 ) = b and

       A7: c <= (p1 `2 ) and

       A8: (p1 `2 ) <= d and

       A9: c <= (p4 `2 ) and

       A10: (p4 `2 ) < (p3 `2 ) and

       A11: (p3 `2 ) < (p2 `2 ) and

       A12: (p2 `2 ) <= d and

       A13: (f . 0 ) = p1 and

       A14: (f . 1) = p3 and

       A15: (g . 0 ) = p2 and

       A16: (g . 1) = p4 and

       A17: f is continuous one-to-one and

       A18: g is continuous one-to-one and

       A19: ( rng f) c= ( closed_inside_of_rectangle (a,b,c,d)) and

       A20: ( rng g) c= ( closed_inside_of_rectangle (a,b,c,d));

      set A = (2 / (b - a)), B = ( - ((b + a) / (b - a))), C = (2 / (d - c)), D = ( - ((d + c) / (d - c)));

      set h = ( AffineMap (A,B,C,D));

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

      

       A21: (p2 `2 ) > (p4 `2 ) by A10, A11, XXREAL_0: 2;

      ((g . O) `1 ) = b by A4, A15;

      then

       A22: ((g2 . I) `2 ) >= ( - 1) by A2, A9, A12, A15, A16, A21, Th65;

      (d - c) > 0 by A2, XREAL_1: 50;

      then

       A23: C > 0 by XREAL_1: 139;

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

      

       A24: ((f2 . O) `2 ) <= 1 by A2, A7, A8, A13, Th60;

      

       A25: ((f2 . O) `1 ) = ( - 1) by A1, A3, A13, Th54;

      

       A26: f2 is continuous one-to-one by A1, A2, A17, Th53;

      

       A27: ( rng f2) c= ( closed_inside_of_rectangle (( - 1),1,( - 1),1)) by A1, A2, A19, Th52;

      

       A28: ((f2 . I) `1 ) = 1 by A1, A5, A14, Th56;

      set x = the Element of (( rng f2) /\ ( rng g2));

      

       A29: ( dom f) = the carrier of I[01] by FUNCT_2:def 1;

      then

       A30: (h . p3) = (f2 . I) by A14, FUNCT_1: 13;

      ((g . I) `1 ) = b by A6, A16;

      then

       A31: 1 >= ((g2 . O) `2 ) by A2, A9, A12, A15, A16, A21, Th65;

      

       A32: ((g2 . O) `1 ) = 1 by A1, A4, A15, Th56;

      

       A33: g2 is continuous one-to-one by A1, A2, A18, Th53;

      

       A34: ( rng g2) c= ( closed_inside_of_rectangle (( - 1),1,( - 1),1)) by A1, A2, A20, Th52;

      

       A35: ((g2 . I) `1 ) = 1 by A1, A6, A16, Th56;

      (b - a) > 0 by A1, XREAL_1: 50;

      then

       A36: A > 0 by XREAL_1: 139;

      then h is being_homeomorphism by A23, Th50;

      then

       A37: h is one-to-one by TOPS_2:def 5;

      

       A38: ( dom g) = the carrier of I[01] by FUNCT_2:def 1;

      then (h . p2) = (g2 . O) by A15, FUNCT_1: 13;

      then

       A39: ((g2 . O) `2 ) > ((f2 . I) `2 ) by A11, A36, A23, A30, Th51;

      (h . p4) = (g2 . I) by A16, A38, FUNCT_1: 13;

      then

       A40: ((g2 . I) `2 ) < ((f2 . I) `2 ) by A10, A36, A23, A30, Th51;

      ( - 1) <= ((f2 . O) `2 ) by A2, A7, A8, A13, Th60;

      then ( rng f2) meets ( rng g2) by A26, A25, A28, A24, A27, A33, A32, A35, A31, A22, A34, A39, A40, Th30, JGRAPH_6: 79;

      then

       A41: (( rng f2) /\ ( rng g2)) <> {} by XBOOLE_0:def 7;

      then x in ( rng g2) by XBOOLE_0:def 4;

      then

      consider z2 be object such that

       A42: z2 in ( dom g2) and

       A43: x = (g2 . z2) by FUNCT_1:def 3;

      

       A44: x = (h . (g . z2)) by A38, A42, A43, FUNCT_1: 13;

      

       A45: (g . z2) in ( rng g) by A38, A42, FUNCT_1:def 3;

      (g . z2) in the carrier of ( TOP-REAL 2) by A42, FUNCT_2: 5;

      then

       A46: (g . z2) in ( dom h) by FUNCT_2:def 1;

      x in ( rng f2) by A41, XBOOLE_0:def 4;

      then

      consider z1 be object such that

       A47: z1 in ( dom f2) and

       A48: x = (f2 . z1) by FUNCT_1:def 3;

      

       A49: (f . z1) in ( rng f) by A29, A47, FUNCT_1:def 3;

      (f . z1) in the carrier of ( TOP-REAL 2) by A47, FUNCT_2: 5;

      then

       A50: (f . z1) in ( dom h) by FUNCT_2:def 1;

      x = (h . (f . z1)) by A29, A47, A48, FUNCT_1: 13;

      then (f . z1) = (g . z2) by A44, A50, A46, A37, FUNCT_1:def 4;

      hence thesis by A49, A45, XBOOLE_0: 3;

    end;

    theorem :: JGRAPH_7:101

    for p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real, P,Q be Subset of ( TOP-REAL 2) st a < b & c < d & (p1 `1 ) = a & (p2 `1 ) = b & (p3 `1 ) = b & (p4 `1 ) = b & c <= (p1 `2 ) & (p1 `2 ) <= d & c <= (p4 `2 ) & (p4 `2 ) < (p3 `2 ) & (p3 `2 ) < (p2 `2 ) & (p2 `2 ) <= d & P is_an_arc_of (p1,p3) & Q is_an_arc_of (p2,p4) & P c= ( closed_inside_of_rectangle (a,b,c,d)) & Q c= ( closed_inside_of_rectangle (a,b,c,d)) holds P meets Q

    proof

      let p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real, P,Q be Subset of ( TOP-REAL 2);

      assume that

       A1: a < b and

       A2: c < d and

       A3: (p1 `1 ) = a and

       A4: (p2 `1 ) = b and

       A5: (p3 `1 ) = b and

       A6: (p4 `1 ) = b and

       A7: c <= (p1 `2 ) and

       A8: (p1 `2 ) <= d and

       A9: c <= (p4 `2 ) and

       A10: (p4 `2 ) < (p3 `2 ) and

       A11: (p3 `2 ) < (p2 `2 ) and

       A12: (p2 `2 ) <= d and

       A13: P is_an_arc_of (p1,p3) and

       A14: Q is_an_arc_of (p2,p4) and

       A15: P c= ( closed_inside_of_rectangle (a,b,c,d)) and

       A16: Q c= ( closed_inside_of_rectangle (a,b,c,d));

      

       A17: ex g be Function of I[01] , ( TOP-REAL 2) st g is continuous one-to-one & ( rng g) = Q & (g . 0 ) = p2 & (g . 1) = p4 by A14, Th2;

      ex f be Function of I[01] , ( TOP-REAL 2) st f is continuous one-to-one & ( rng f) = P & (f . 0 ) = p1 & (f . 1) = p3 by A13, Th2;

      hence thesis by A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, A12, A15, A16, A17, Th100;

    end;

    theorem :: JGRAPH_7:102

    

     Th102: for p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real, f,g be Function of I[01] , ( TOP-REAL 2) st a < b & c < d & (p1 `1 ) = a & (p2 `1 ) = b & (p3 `1 ) = b & (p4 `2 ) = c & c <= (p1 `2 ) & (p1 `2 ) <= d & c <= (p3 `2 ) & (p3 `2 ) < (p2 `2 ) & (p2 `2 ) <= d & a < (p4 `1 ) & (p4 `1 ) <= b & (f . 0 ) = p1 & (f . 1) = p3 & (g . 0 ) = p2 & (g . 1) = p4 & f is continuous one-to-one & g is continuous one-to-one & ( rng f) c= ( closed_inside_of_rectangle (a,b,c,d)) & ( rng g) c= ( closed_inside_of_rectangle (a,b,c,d)) holds ( rng f) meets ( rng g)

    proof

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

      let p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real, f,g be Function of I[01] , ( TOP-REAL 2);

      assume that

       A1: a < b and

       A2: c < d and

       A3: (p1 `1 ) = a and

       A4: (p2 `1 ) = b and

       A5: (p3 `1 ) = b and

       A6: (p4 `2 ) = c and

       A7: c <= (p1 `2 ) and

       A8: (p1 `2 ) <= d and

       A9: c <= (p3 `2 ) and

       A10: (p3 `2 ) < (p2 `2 ) and

       A11: (p2 `2 ) <= d and

       A12: a < (p4 `1 ) and

       A13: (p4 `1 ) <= b and

       A14: (f . 0 ) = p1 and

       A15: (f . 1) = p3 and

       A16: (g . 0 ) = p2 and

       A17: (g . 1) = p4 and

       A18: f is continuous one-to-one and

       A19: g is continuous one-to-one and

       A20: ( rng f) c= ( closed_inside_of_rectangle (a,b,c,d)) and

       A21: ( rng g) c= ( closed_inside_of_rectangle (a,b,c,d));

      set A = (2 / (b - a)), B = ( - ((b + a) / (b - a))), C = (2 / (d - c)), D = ( - ((d + c) / (d - c)));

      set h = ( AffineMap (A,B,C,D));

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

      

       A22: ( - 1) <= ((f2 . O) `2 ) by A2, A7, A8, A14, Th60;

      

       A23: ( rng f2) c= ( closed_inside_of_rectangle (( - 1),1,( - 1),1)) by A1, A2, A20, Th52;

      (d - c) > 0 by A2, XREAL_1: 50;

      then

       A24: C > 0 by XREAL_1: 139;

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

      

       A25: ((g2 . I) `1 ) > ( - 1) by A1, A2, A6, A12, A13, A17, Th66;

      

       A26: ((g2 . O) `1 ) = 1 by A1, A4, A16, Th56;

      

       A27: g2 is continuous one-to-one by A1, A2, A19, Th53;

      (p2 `2 ) > c by A9, A10, XXREAL_0: 2;

      then

       A28: 1 >= ((g2 . O) `2 ) by A1, A2, A4, A11, A16, Th66;

      set x = the Element of (( rng f2) /\ ( rng g2));

      

       A29: ( dom f) = the carrier of I[01] by FUNCT_2:def 1;

      then

       A30: (h . p3) = (f2 . I) by A15, FUNCT_1: 13;

      

       A31: ((f2 . O) `2 ) <= 1 by A2, A7, A8, A14, Th60;

      

       A32: f2 is continuous one-to-one by A1, A2, A18, Th53;

      (b - a) > 0 by A1, XREAL_1: 50;

      then

       A33: A > 0 by XREAL_1: 139;

      then h is being_homeomorphism by A24, Th50;

      then

       A34: h is one-to-one by TOPS_2:def 5;

      

       A35: ( dom g) = the carrier of I[01] by FUNCT_2:def 1;

      then (h . p2) = (g2 . O) by A16, FUNCT_1: 13;

      then

       A36: ((g2 . O) `2 ) > ((f2 . I) `2 ) by A10, A33, A24, A30, Th51;

      (p3 `2 ) < d by A10, A11, XXREAL_0: 2;

      then

       A37: ( - 1) <= ((f2 . I) `2 ) by A2, A9, A15, Th60;

      

       A38: ((f2 . I) `1 ) = 1 by A1, A5, A15, Th56;

      

       A39: ((f2 . O) `1 ) = ( - 1) by A1, A3, A14, Th54;

      

       A40: ( rng g2) c= ( closed_inside_of_rectangle (( - 1),1,( - 1),1)) by A1, A2, A21, Th52;

      

       A41: ((g2 . I) `2 ) = ( - 1) by A2, A6, A17, Th57;

      1 >= ((g2 . I) `1 ) by A1, A2, A6, A12, A13, A17, Th66;

      then ( rng f2) meets ( rng g2) by A32, A39, A38, A22, A31, A37, A23, A27, A26, A41, A28, A25, A40, A36, Th31, JGRAPH_6: 79;

      then

       A42: (( rng f2) /\ ( rng g2)) <> {} by XBOOLE_0:def 7;

      then x in ( rng g2) by XBOOLE_0:def 4;

      then

      consider z2 be object such that

       A43: z2 in ( dom g2) and

       A44: x = (g2 . z2) by FUNCT_1:def 3;

      

       A45: x = (h . (g . z2)) by A35, A43, A44, FUNCT_1: 13;

      

       A46: (g . z2) in ( rng g) by A35, A43, FUNCT_1:def 3;

      (g . z2) in the carrier of ( TOP-REAL 2) by A43, FUNCT_2: 5;

      then

       A47: (g . z2) in ( dom h) by FUNCT_2:def 1;

      x in ( rng f2) by A42, XBOOLE_0:def 4;

      then

      consider z1 be object such that

       A48: z1 in ( dom f2) and

       A49: x = (f2 . z1) by FUNCT_1:def 3;

      

       A50: (f . z1) in ( rng f) by A29, A48, FUNCT_1:def 3;

      (f . z1) in the carrier of ( TOP-REAL 2) by A48, FUNCT_2: 5;

      then

       A51: (f . z1) in ( dom h) by FUNCT_2:def 1;

      x = (h . (f . z1)) by A29, A48, A49, FUNCT_1: 13;

      then (f . z1) = (g . z2) by A45, A51, A47, A34, FUNCT_1:def 4;

      hence thesis by A50, A46, XBOOLE_0: 3;

    end;

    theorem :: JGRAPH_7:103

    for p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real, P,Q be Subset of ( TOP-REAL 2) st a < b & c < d & (p1 `1 ) = a & (p2 `1 ) = b & (p3 `1 ) = b & (p4 `2 ) = c & c <= (p1 `2 ) & (p1 `2 ) <= d & c <= (p3 `2 ) & (p3 `2 ) < (p2 `2 ) & (p2 `2 ) <= d & a < (p4 `1 ) & (p4 `1 ) <= b & P is_an_arc_of (p1,p3) & Q is_an_arc_of (p2,p4) & P c= ( closed_inside_of_rectangle (a,b,c,d)) & Q c= ( closed_inside_of_rectangle (a,b,c,d)) holds P meets Q

    proof

      let p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real, P,Q be Subset of ( TOP-REAL 2);

      assume that

       A1: a < b and

       A2: c < d and

       A3: (p1 `1 ) = a and

       A4: (p2 `1 ) = b and

       A5: (p3 `1 ) = b and

       A6: (p4 `2 ) = c and

       A7: c <= (p1 `2 ) and

       A8: (p1 `2 ) <= d and

       A9: c <= (p3 `2 ) and

       A10: (p3 `2 ) < (p2 `2 ) and

       A11: (p2 `2 ) <= d and

       A12: a < (p4 `1 ) and

       A13: (p4 `1 ) <= b and

       A14: P is_an_arc_of (p1,p3) and

       A15: Q is_an_arc_of (p2,p4) and

       A16: P c= ( closed_inside_of_rectangle (a,b,c,d)) and

       A17: Q c= ( closed_inside_of_rectangle (a,b,c,d));

      

       A18: ex g be Function of I[01] , ( TOP-REAL 2) st g is continuous one-to-one & ( rng g) = Q & (g . 0 ) = p2 & (g . 1) = p4 by A15, Th2;

      ex f be Function of I[01] , ( TOP-REAL 2) st f is continuous one-to-one & ( rng f) = P & (f . 0 ) = p1 & (f . 1) = p3 by A14, Th2;

      hence thesis by A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, A12, A13, A16, A17, A18, Th102;

    end;

    theorem :: JGRAPH_7:104

    

     Th104: for p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real, f,g be Function of I[01] , ( TOP-REAL 2) st a < b & c < d & (p1 `1 ) = a & (p2 `1 ) = b & (p3 `2 ) = c & (p4 `2 ) = c & c <= (p1 `2 ) & (p1 `2 ) <= d & c <= (p2 `2 ) & (p2 `2 ) <= d & a < (p4 `1 ) & (p4 `1 ) < (p3 `1 ) & (p3 `1 ) <= b & (f . 0 ) = p1 & (f . 1) = p3 & (g . 0 ) = p2 & (g . 1) = p4 & f is continuous one-to-one & g is continuous one-to-one & ( rng f) c= ( closed_inside_of_rectangle (a,b,c,d)) & ( rng g) c= ( closed_inside_of_rectangle (a,b,c,d)) holds ( rng f) meets ( rng g)

    proof

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

      let p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real, f,g be Function of I[01] , ( TOP-REAL 2);

      assume that

       A1: a < b and

       A2: c < d and

       A3: (p1 `1 ) = a and

       A4: (p2 `1 ) = b and

       A5: (p3 `2 ) = c and

       A6: (p4 `2 ) = c and

       A7: c <= (p1 `2 ) and

       A8: (p1 `2 ) <= d and

       A9: c <= (p2 `2 ) and

       A10: (p2 `2 ) <= d and

       A11: a < (p4 `1 ) and

       A12: (p4 `1 ) < (p3 `1 ) and

       A13: (p3 `1 ) <= b and

       A14: (f . 0 ) = p1 and

       A15: (f . 1) = p3 and

       A16: (g . 0 ) = p2 and

       A17: (g . 1) = p4 and

       A18: f is continuous one-to-one and

       A19: g is continuous one-to-one and

       A20: ( rng f) c= ( closed_inside_of_rectangle (a,b,c,d)) and

       A21: ( rng g) c= ( closed_inside_of_rectangle (a,b,c,d));

      set A = (2 / (b - a)), B = ( - ((b + a) / (b - a))), C = (2 / (d - c)), D = ( - ((d + c) / (d - c)));

      set h = ( AffineMap (A,B,C,D));

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

      

       A22: 1 >= ((g2 . O) `2 ) by A1, A2, A4, A9, A10, A16, Th66;

      (p4 `1 ) <= b by A12, A13, XXREAL_0: 2;

      then

       A23: ((g2 . I) `1 ) > ( - 1) by A1, A2, A6, A11, A17, Th66;

      (d - c) > 0 by A2, XREAL_1: 50;

      then

       A24: C > 0 by XREAL_1: 139;

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

      

       A25: f2 is continuous one-to-one by A1, A2, A18, Th53;

      

       A26: (p3 `1 ) > a by A11, A12, XXREAL_0: 2;

      then

       A27: ((f2 . I) `1 ) <= 1 by A1, A2, A5, A13, A15, Th61;

      

       A28: ((f . I) `2 ) = c by A5, A15;

      then

       A29: ((f2 . O) `2 ) <= 1 by A1, A2, A7, A8, A13, A14, A15, A26, Th61;

      set x = the Element of (( rng f2) /\ ( rng g2));

      

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

      then

       A31: (h . p4) = (g2 . I) by A17, FUNCT_1: 13;

      (b - a) > 0 by A1, XREAL_1: 50;

      then

       A32: A > 0 by XREAL_1: 139;

      then h is being_homeomorphism by A24, Th50;

      then

       A33: h is one-to-one by TOPS_2:def 5;

      

       A34: ( dom f) = the carrier of I[01] by FUNCT_2:def 1;

      then (h . p3) = (f2 . I) by A15, FUNCT_1: 13;

      then

       A35: ((g2 . I) `1 ) < ((f2 . I) `1 ) by A12, A32, A24, A31, Th50;

      

       A36: ((g2 . O) `2 ) >= ( - 1) by A1, A2, A4, A9, A10, A16, Th66;

      

       A37: g2 is continuous one-to-one by A1, A2, A19, Th53;

      

       A38: ((g2 . I) `2 ) = ( - 1) by A2, A6, A17, Th57;

      

       A39: ((g2 . O) `1 ) = 1 by A1, A4, A16, Th56;

      

       A40: ( rng g2) c= ( closed_inside_of_rectangle (( - 1),1,( - 1),1)) by A1, A2, A21, Th52;

      

       A41: ( rng f2) c= ( closed_inside_of_rectangle (( - 1),1,( - 1),1)) by A1, A2, A20, Th52;

      

       A42: ((f2 . I) `2 ) = ( - 1) by A2, A5, A15, Th57;

      

       A43: ((f2 . O) `1 ) = ( - 1) by A1, A3, A14, Th54;

      ( - 1) <= ((f2 . O) `2 ) by A1, A2, A7, A8, A13, A14, A15, A26, A28, Th61;

      then ( rng f2) meets ( rng g2) by A25, A43, A42, A29, A27, A41, A37, A39, A38, A22, A36, A23, A40, A35, Th32, JGRAPH_6: 79;

      then

       A44: (( rng f2) /\ ( rng g2)) <> {} by XBOOLE_0:def 7;

      then x in ( rng g2) by XBOOLE_0:def 4;

      then

      consider z2 be object such that

       A45: z2 in ( dom g2) and

       A46: x = (g2 . z2) by FUNCT_1:def 3;

      

       A47: x = (h . (g . z2)) by A30, A45, A46, FUNCT_1: 13;

      

       A48: (g . z2) in ( rng g) by A30, A45, FUNCT_1:def 3;

      (g . z2) in the carrier of ( TOP-REAL 2) by A45, FUNCT_2: 5;

      then

       A49: (g . z2) in ( dom h) by FUNCT_2:def 1;

      x in ( rng f2) by A44, XBOOLE_0:def 4;

      then

      consider z1 be object such that

       A50: z1 in ( dom f2) and

       A51: x = (f2 . z1) by FUNCT_1:def 3;

      

       A52: (f . z1) in ( rng f) by A34, A50, FUNCT_1:def 3;

      (f . z1) in the carrier of ( TOP-REAL 2) by A50, FUNCT_2: 5;

      then

       A53: (f . z1) in ( dom h) by FUNCT_2:def 1;

      x = (h . (f . z1)) by A34, A50, A51, FUNCT_1: 13;

      then (f . z1) = (g . z2) by A47, A53, A49, A33, FUNCT_1:def 4;

      hence thesis by A52, A48, XBOOLE_0: 3;

    end;

    theorem :: JGRAPH_7:105

    for p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real, P,Q be Subset of ( TOP-REAL 2) st a < b & c < d & (p1 `1 ) = a & (p2 `1 ) = b & (p3 `2 ) = c & (p4 `2 ) = c & c <= (p1 `2 ) & (p1 `2 ) <= d & c <= (p2 `2 ) & (p2 `2 ) <= d & a < (p4 `1 ) & (p4 `1 ) < (p3 `1 ) & (p3 `1 ) <= b & P is_an_arc_of (p1,p3) & Q is_an_arc_of (p2,p4) & P c= ( closed_inside_of_rectangle (a,b,c,d)) & Q c= ( closed_inside_of_rectangle (a,b,c,d)) holds P meets Q

    proof

      let p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real, P,Q be Subset of ( TOP-REAL 2);

      assume that

       A1: a < b and

       A2: c < d and

       A3: (p1 `1 ) = a and

       A4: (p2 `1 ) = b and

       A5: (p3 `2 ) = c and

       A6: (p4 `2 ) = c and

       A7: c <= (p1 `2 ) and

       A8: (p1 `2 ) <= d and

       A9: c <= (p2 `2 ) and

       A10: (p2 `2 ) <= d and

       A11: a < (p4 `1 ) and

       A12: (p4 `1 ) < (p3 `1 ) and

       A13: (p3 `1 ) <= b and

       A14: P is_an_arc_of (p1,p3) and

       A15: Q is_an_arc_of (p2,p4) and

       A16: P c= ( closed_inside_of_rectangle (a,b,c,d)) and

       A17: Q c= ( closed_inside_of_rectangle (a,b,c,d));

      

       A18: ex g be Function of I[01] , ( TOP-REAL 2) st g is continuous one-to-one & ( rng g) = Q & (g . 0 ) = p2 & (g . 1) = p4 by A15, Th2;

      ex f be Function of I[01] , ( TOP-REAL 2) st f is continuous one-to-one & ( rng f) = P & (f . 0 ) = p1 & (f . 1) = p3 by A14, Th2;

      hence thesis by A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, A12, A13, A16, A17, A18, Th104;

    end;

    theorem :: JGRAPH_7:106

    

     Th106: for p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real, f,g be Function of I[01] , ( TOP-REAL 2) st a < b & c < d & (p1 `1 ) = a & (p2 `2 ) = c & (p3 `2 ) = c & (p4 `2 ) = c & c <= (p1 `2 ) & (p1 `2 ) <= d & a < (p4 `1 ) & (p4 `1 ) < (p3 `1 ) & (p3 `1 ) < (p2 `1 ) & (p2 `1 ) <= b & (f . 0 ) = p1 & (f . 1) = p3 & (g . 0 ) = p2 & (g . 1) = p4 & f is continuous one-to-one & g is continuous one-to-one & ( rng f) c= ( closed_inside_of_rectangle (a,b,c,d)) & ( rng g) c= ( closed_inside_of_rectangle (a,b,c,d)) holds ( rng f) meets ( rng g)

    proof

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

      let p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real, f,g be Function of I[01] , ( TOP-REAL 2);

      assume that

       A1: a < b and

       A2: c < d and

       A3: (p1 `1 ) = a and

       A4: (p2 `2 ) = c and

       A5: (p3 `2 ) = c and

       A6: (p4 `2 ) = c and

       A7: c <= (p1 `2 ) and

       A8: (p1 `2 ) <= d and

       A9: a < (p4 `1 ) and

       A10: (p4 `1 ) < (p3 `1 ) and

       A11: (p3 `1 ) < (p2 `1 ) and

       A12: (p2 `1 ) <= b and

       A13: (f . 0 ) = p1 and

       A14: (f . 1) = p3 and

       A15: (g . 0 ) = p2 and

       A16: (g . 1) = p4 and

       A17: f is continuous one-to-one and

       A18: g is continuous one-to-one and

       A19: ( rng f) c= ( closed_inside_of_rectangle (a,b,c,d)) and

       A20: ( rng g) c= ( closed_inside_of_rectangle (a,b,c,d));

      set A = (2 / (b - a)), B = ( - ((b + a) / (b - a))), C = (2 / (d - c)), D = ( - ((d + c) / (d - c)));

      set h = ( AffineMap (A,B,C,D));

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

      

       A21: (p2 `1 ) > (p4 `1 ) by A10, A11, XXREAL_0: 2;

      ((g . O) `2 ) = c by A4, A15;

      then

       A22: ((g2 . I) `1 ) > ( - 1) by A1, A9, A12, A15, A16, A21, Th67;

      (d - c) > 0 by A2, XREAL_1: 50;

      then

       A23: C > 0 by XREAL_1: 139;

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

      

       A24: b > (p3 `1 ) by A11, A12, XXREAL_0: 2;

      ((g . I) `2 ) = c by A6, A16;

      then

       A25: 1 >= ((g2 . O) `1 ) by A1, A9, A12, A15, A16, A21, Th67;

      

       A26: ((g2 . O) `2 ) = ( - 1) by A2, A4, A15, Th57;

      

       A27: g2 is continuous one-to-one by A1, A2, A18, Th53;

      

       A28: ((f2 . O) `1 ) = ( - 1) by A1, A3, A13, Th54;

      

       A29: f2 is continuous one-to-one by A1, A2, A17, Th53;

      set x = the Element of (( rng f2) /\ ( rng g2));

      

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

      then

       A31: (h . p3) = (f2 . I) by A14, FUNCT_1: 13;

      

       A32: ( rng g2) c= ( closed_inside_of_rectangle (( - 1),1,( - 1),1)) by A1, A2, A20, Th52;

      

       A33: ((g2 . I) `2 ) = ( - 1) by A2, A6, A16, Th57;

      (b - a) > 0 by A1, XREAL_1: 50;

      then

       A34: A > 0 by XREAL_1: 139;

      then h is being_homeomorphism by A23, Th50;

      then

       A35: h is one-to-one by TOPS_2:def 5;

      

       A36: ((f . I) `2 ) = c by A5, A14;

      

       A37: ( dom g) = the carrier of I[01] by FUNCT_2:def 1;

      then (h . p2) = (g2 . O) by A15, FUNCT_1: 13;

      then

       A38: ((g2 . O) `1 ) > ((f2 . I) `1 ) by A11, A34, A23, A31, Th50;

      

       A39: (p3 `1 ) > a by A9, A10, XXREAL_0: 2;

      then

       A40: ((f2 . O) `2 ) <= 1 by A1, A2, A7, A8, A13, A14, A24, A36, Th61;

      

       A41: ( rng f2) c= ( closed_inside_of_rectangle (( - 1),1,( - 1),1)) by A1, A2, A19, Th52;

      

       A42: ((f2 . I) `2 ) = ( - 1) by A2, A5, A14, Th57;

      (h . p4) = (g2 . I) by A16, A37, FUNCT_1: 13;

      then

       A43: ((g2 . I) `1 ) < ((f2 . I) `1 ) by A10, A34, A23, A31, Th50;

      ( - 1) <= ((f2 . O) `2 ) by A1, A2, A7, A8, A13, A14, A39, A24, A36, Th61;

      then ( rng f2) meets ( rng g2) by A29, A28, A42, A40, A41, A27, A26, A33, A25, A22, A32, A38, A43, Th33, JGRAPH_6: 79;

      then

       A44: (( rng f2) /\ ( rng g2)) <> {} by XBOOLE_0:def 7;

      then x in ( rng g2) by XBOOLE_0:def 4;

      then

      consider z2 be object such that

       A45: z2 in ( dom g2) and

       A46: x = (g2 . z2) by FUNCT_1:def 3;

      

       A47: x = (h . (g . z2)) by A37, A45, A46, FUNCT_1: 13;

      

       A48: (g . z2) in ( rng g) by A37, A45, FUNCT_1:def 3;

      (g . z2) in the carrier of ( TOP-REAL 2) by A45, FUNCT_2: 5;

      then

       A49: (g . z2) in ( dom h) by FUNCT_2:def 1;

      x in ( rng f2) by A44, XBOOLE_0:def 4;

      then

      consider z1 be object such that

       A50: z1 in ( dom f2) and

       A51: x = (f2 . z1) by FUNCT_1:def 3;

      

       A52: (f . z1) in ( rng f) by A30, A50, FUNCT_1:def 3;

      (f . z1) in the carrier of ( TOP-REAL 2) by A50, FUNCT_2: 5;

      then

       A53: (f . z1) in ( dom h) by FUNCT_2:def 1;

      x = (h . (f . z1)) by A30, A50, A51, FUNCT_1: 13;

      then (f . z1) = (g . z2) by A47, A53, A49, A35, FUNCT_1:def 4;

      hence thesis by A52, A48, XBOOLE_0: 3;

    end;

    theorem :: JGRAPH_7:107

    for p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real, P,Q be Subset of ( TOP-REAL 2) st a < b & c < d & (p1 `1 ) = a & (p2 `2 ) = c & (p3 `2 ) = c & (p4 `2 ) = c & c <= (p1 `2 ) & (p1 `2 ) <= d & a < (p4 `1 ) & (p4 `1 ) < (p3 `1 ) & (p3 `1 ) < (p2 `1 ) & (p2 `1 ) <= b & P is_an_arc_of (p1,p3) & Q is_an_arc_of (p2,p4) & P c= ( closed_inside_of_rectangle (a,b,c,d)) & Q c= ( closed_inside_of_rectangle (a,b,c,d)) holds P meets Q

    proof

      let p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real, P,Q be Subset of ( TOP-REAL 2);

      assume that

       A1: a < b and

       A2: c < d and

       A3: (p1 `1 ) = a and

       A4: (p2 `2 ) = c and

       A5: (p3 `2 ) = c and

       A6: (p4 `2 ) = c and

       A7: c <= (p1 `2 ) and

       A8: (p1 `2 ) <= d and

       A9: a < (p4 `1 ) and

       A10: (p4 `1 ) < (p3 `1 ) and

       A11: (p3 `1 ) < (p2 `1 ) and

       A12: (p2 `1 ) <= b and

       A13: P is_an_arc_of (p1,p3) and

       A14: Q is_an_arc_of (p2,p4) and

       A15: P c= ( closed_inside_of_rectangle (a,b,c,d)) and

       A16: Q c= ( closed_inside_of_rectangle (a,b,c,d));

      

       A17: ex g be Function of I[01] , ( TOP-REAL 2) st g is continuous one-to-one & ( rng g) = Q & (g . 0 ) = p2 & (g . 1) = p4 by A14, Th2;

      ex f be Function of I[01] , ( TOP-REAL 2) st f is continuous one-to-one & ( rng f) = P & (f . 0 ) = p1 & (f . 1) = p3 by A13, Th2;

      hence thesis by A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, A12, A15, A16, A17, Th106;

    end;

    theorem :: JGRAPH_7:108

    

     Th108: for p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real, f,g be Function of I[01] , ( TOP-REAL 2) st a < b & c < d & (p1 `2 ) = d & (p2 `2 ) = d & (p3 `2 ) = d & (p4 `2 ) = d & a <= (p1 `1 ) & (p1 `1 ) < (p2 `1 ) & (p2 `1 ) < (p3 `1 ) & (p3 `1 ) < (p4 `1 ) & (p4 `1 ) <= b & (f . 0 ) = p1 & (f . 1) = p3 & (g . 0 ) = p2 & (g . 1) = p4 & f is continuous one-to-one & g is continuous one-to-one & ( rng f) c= ( closed_inside_of_rectangle (a,b,c,d)) & ( rng g) c= ( closed_inside_of_rectangle (a,b,c,d)) holds ( rng f) meets ( rng g)

    proof

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

      let p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real, f,g be Function of I[01] , ( TOP-REAL 2);

      assume that

       A1: a < b and

       A2: c < d and

       A3: (p1 `2 ) = d and

       A4: (p2 `2 ) = d and

       A5: (p3 `2 ) = d and

       A6: (p4 `2 ) = d and

       A7: a <= (p1 `1 ) and

       A8: (p1 `1 ) < (p2 `1 ) and

       A9: (p2 `1 ) < (p3 `1 ) and

       A10: (p3 `1 ) < (p4 `1 ) and

       A11: (p4 `1 ) <= b and

       A12: (f . 0 ) = p1 and

       A13: (f . 1) = p3 and

       A14: (g . 0 ) = p2 and

       A15: (g . 1) = p4 and

       A16: f is continuous one-to-one and

       A17: g is continuous one-to-one and

       A18: ( rng f) c= ( closed_inside_of_rectangle (a,b,c,d)) and

       A19: ( rng g) c= ( closed_inside_of_rectangle (a,b,c,d));

      set A = (2 / (b - a)), B = ( - ((b + a) / (b - a))), C = (2 / (d - c)), D = ( - ((d + c) / (d - c)));

      set h = ( AffineMap (A,B,C,D));

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

      

       A20: g2 is continuous one-to-one by A1, A2, A17, Th53;

      

       A21: ((g . O) `2 ) = d by A4, A14;

      

       A22: a < (p2 `1 ) by A7, A8, XXREAL_0: 2;

      (p2 `1 ) < (p4 `1 ) by A9, A10, XXREAL_0: 2;

      then

       A23: ((g2 . I) `1 ) <= 1 by A1, A11, A14, A15, A22, A21, Th62;

      

       A24: ( rng g2) c= ( closed_inside_of_rectangle (( - 1),1,( - 1),1)) by A1, A2, A19, Th52;

      

       A25: ((g2 . I) `2 ) = 1 by A2, A6, A15, Th55;

      

       A26: ((g2 . O) `2 ) = 1 by A2, A4, A14, Th55;

      

       A27: ( dom g) = the carrier of I[01] by FUNCT_2:def 1;

      then

       A28: (h . p2) = (g2 . O) by A14, FUNCT_1: 13;

      

       A29: (h . p4) = (g2 . I) by A15, A27, FUNCT_1: 13;

      (d - c) > 0 by A2, XREAL_1: 50;

      then

       A30: C > 0 by XREAL_1: 139;

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

      

       A31: f2 is continuous one-to-one by A1, A2, A16, Th53;

      

       A32: ( rng f2) c= ( closed_inside_of_rectangle (( - 1),1,( - 1),1)) by A1, A2, A18, Th52;

      

       A33: ((f2 . I) `2 ) = 1 by A2, A5, A13, Th55;

      

       A34: ((f . I) `2 ) = d by A5, A13;

      

       A35: (p3 `1 ) < b by A10, A11, XXREAL_0: 2;

      (p1 `1 ) < (p3 `1 ) by A8, A9, XXREAL_0: 2;

      then

       A36: ( - 1) <= ((f2 . O) `1 ) by A1, A7, A12, A13, A35, A34, Th62;

      

       A37: ((f2 . O) `2 ) = 1 by A2, A3, A12, Th55;

      set x = the Element of (( rng f2) /\ ( rng g2));

      

       A38: ( dom f) = the carrier of I[01] by FUNCT_2:def 1;

      then

       A39: (h . p3) = (f2 . I) by A13, FUNCT_1: 13;

      (b - a) > 0 by A1, XREAL_1: 50;

      then

       A40: A > 0 by XREAL_1: 139;

      then

       A41: ((h . p1) `1 ) < ((h . p2) `1 ) by A8, A30, Th50;

      

       A42: ((h . p3) `1 ) < ((h . p4) `1 ) by A10, A40, A30, Th50;

      

       A43: ((h . p2) `1 ) < ((h . p3) `1 ) by A9, A40, A30, Th50;

      (h . p1) = (f2 . O) by A12, A38, FUNCT_1: 13;

      then ((f2 . O),(g2 . O),(f2 . I),(g2 . I)) are_in_this_order_on ( rectangle (( - 1),1,( - 1),1)) by A41, A43, A42, A28, A39, A29, A37, A33, A36, A26, A25, A23, Th34;

      then ( rng f2) meets ( rng g2) by A31, A32, A20, A24, JGRAPH_6: 79;

      then

       A44: (( rng f2) /\ ( rng g2)) <> {} by XBOOLE_0:def 7;

      then x in ( rng g2) by XBOOLE_0:def 4;

      then

      consider z2 be object such that

       A45: z2 in ( dom g2) and

       A46: x = (g2 . z2) by FUNCT_1:def 3;

      

       A47: x = (h . (g . z2)) by A27, A45, A46, FUNCT_1: 13;

      h is being_homeomorphism by A40, A30, Th51;

      then

       A48: h is one-to-one by TOPS_2:def 5;

      x in ( rng f2) by A44, XBOOLE_0:def 4;

      then

      consider z1 be object such that

       A49: z1 in ( dom f2) and

       A50: x = (f2 . z1) by FUNCT_1:def 3;

      

       A51: (f . z1) in ( rng f) by A38, A49, FUNCT_1:def 3;

      

       A52: (g . z2) in ( rng g) by A27, A45, FUNCT_1:def 3;

      (g . z2) in the carrier of ( TOP-REAL 2) by A45, FUNCT_2: 5;

      then

       A53: (g . z2) in ( dom h) by FUNCT_2:def 1;

      (f . z1) in the carrier of ( TOP-REAL 2) by A49, FUNCT_2: 5;

      then

       A54: (f . z1) in ( dom h) by FUNCT_2:def 1;

      x = (h . (f . z1)) by A38, A49, A50, FUNCT_1: 13;

      then (f . z1) = (g . z2) by A47, A54, A53, A48, FUNCT_1:def 4;

      hence thesis by A51, A52, XBOOLE_0: 3;

    end;

    theorem :: JGRAPH_7:109

    for p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real, P,Q be Subset of ( TOP-REAL 2) st a < b & c < d & (p1 `2 ) = d & (p2 `2 ) = d & (p3 `2 ) = d & (p4 `2 ) = d & a <= (p1 `1 ) & (p1 `1 ) < (p2 `1 ) & (p2 `1 ) < (p3 `1 ) & (p3 `1 ) < (p4 `1 ) & (p4 `1 ) <= b & P is_an_arc_of (p1,p3) & Q is_an_arc_of (p2,p4) & P c= ( closed_inside_of_rectangle (a,b,c,d)) & Q c= ( closed_inside_of_rectangle (a,b,c,d)) holds P meets Q

    proof

      let p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real, P,Q be Subset of ( TOP-REAL 2);

      assume that

       A1: a < b and

       A2: c < d and

       A3: (p1 `2 ) = d and

       A4: (p2 `2 ) = d and

       A5: (p3 `2 ) = d and

       A6: (p4 `2 ) = d and

       A7: a <= (p1 `1 ) and

       A8: (p1 `1 ) < (p2 `1 ) and

       A9: (p2 `1 ) < (p3 `1 ) and

       A10: (p3 `1 ) < (p4 `1 ) and

       A11: (p4 `1 ) <= b and

       A12: P is_an_arc_of (p1,p3) and

       A13: Q is_an_arc_of (p2,p4) and

       A14: P c= ( closed_inside_of_rectangle (a,b,c,d)) and

       A15: Q c= ( closed_inside_of_rectangle (a,b,c,d));

      

       A16: ex g be Function of I[01] , ( TOP-REAL 2) st g is continuous one-to-one & ( rng g) = Q & (g . 0 ) = p2 & (g . 1) = p4 by A13, Th2;

      ex f be Function of I[01] , ( TOP-REAL 2) st f is continuous one-to-one & ( rng f) = P & (f . 0 ) = p1 & (f . 1) = p3 by A12, Th2;

      hence thesis by A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, A14, A15, A16, Th108;

    end;

    theorem :: JGRAPH_7:110

    

     Th110: for p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real, f,g be Function of I[01] , ( TOP-REAL 2) st a < b & c < d & (p1 `2 ) = d & (p2 `2 ) = d & (p3 `2 ) = d & (p4 `1 ) = b & a <= (p1 `1 ) & (p1 `1 ) < (p2 `1 ) & (p2 `1 ) < (p3 `1 ) & (p3 `1 ) <= b & c <= (p4 `2 ) & (p4 `2 ) <= d & (f . 0 ) = p1 & (f . 1) = p3 & (g . 0 ) = p2 & (g . 1) = p4 & f is continuous one-to-one & g is continuous one-to-one & ( rng f) c= ( closed_inside_of_rectangle (a,b,c,d)) & ( rng g) c= ( closed_inside_of_rectangle (a,b,c,d)) holds ( rng f) meets ( rng g)

    proof

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

      let p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real, f,g be Function of I[01] , ( TOP-REAL 2);

      assume that

       A1: a < b and

       A2: c < d and

       A3: (p1 `2 ) = d and

       A4: (p2 `2 ) = d and

       A5: (p3 `2 ) = d and

       A6: (p4 `1 ) = b and

       A7: a <= (p1 `1 ) and

       A8: (p1 `1 ) < (p2 `1 ) and

       A9: (p2 `1 ) < (p3 `1 ) and

       A10: (p3 `1 ) <= b and

       A11: c <= (p4 `2 ) and

       A12: (p4 `2 ) <= d and

       A13: (f . 0 ) = p1 and

       A14: (f . 1) = p3 and

       A15: (g . 0 ) = p2 and

       A16: (g . 1) = p4 and

       A17: f is continuous one-to-one and

       A18: g is continuous one-to-one and

       A19: ( rng f) c= ( closed_inside_of_rectangle (a,b,c,d)) and

       A20: ( rng g) c= ( closed_inside_of_rectangle (a,b,c,d));

      set A = (2 / (b - a)), B = ( - ((b + a) / (b - a))), C = (2 / (d - c)), D = ( - ((d + c) / (d - c)));

      set h = ( AffineMap (A,B,C,D));

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

      

       A21: (p1 `1 ) < (p3 `1 ) by A8, A9, XXREAL_0: 2;

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

      

       A22: ( - 1) <= ((g2 . I) `2 ) by A1, A2, A6, A11, A12, A16, Th63;

      

       A23: ((g2 . I) `1 ) = 1 by A1, A6, A16, Th56;

      

       A24: ((g2 . O) `2 ) = 1 by A2, A4, A15, Th55;

      ((f . I) `2 ) = d by A5, A14;

      then

       A25: ( - 1) <= ((f2 . O) `1 ) by A1, A7, A10, A13, A14, A21, Th62;

      

       A26: g2 is continuous one-to-one by A1, A2, A18, Th53;

      set x = the Element of (( rng f2) /\ ( rng g2));

      

       A27: ( dom f) = the carrier of I[01] by FUNCT_2:def 1;

      then

       A28: (h . p3) = (f2 . I) by A14, FUNCT_1: 13;

      (d - c) > 0 by A2, XREAL_1: 50;

      then

       A29: C > 0 by XREAL_1: 139;

      (b - a) > 0 by A1, XREAL_1: 50;

      then

       A30: A > 0 by XREAL_1: 139;

      then

       A31: ((h . p1) `1 ) < ((h . p2) `1 ) by A8, A29, Th50;

      ((f . O) `2 ) = d by A3, A13;

      then

       A32: ((f2 . I) `1 ) <= 1 by A1, A7, A10, A13, A14, A21, Th62;

      h is being_homeomorphism by A30, A29, Th50;

      then

       A33: h is one-to-one by TOPS_2:def 5;

      

       A34: ( rng f2) c= ( closed_inside_of_rectangle (( - 1),1,( - 1),1)) by A1, A2, A19, Th52;

      

       A35: ((f2 . I) `2 ) = 1 by A2, A5, A14, Th55;

      

       A36: ((f2 . O) `2 ) = 1 by A2, A3, A13, Th55;

      

       A37: f2 is continuous one-to-one by A1, A2, A17, Th53;

      

       A38: ( rng g2) c= ( closed_inside_of_rectangle (( - 1),1,( - 1),1)) by A1, A2, A20, Th52;

      

       A39: ((h . p2) `1 ) < ((h . p3) `1 ) by A9, A30, A29, Th50;

      

       A40: ((g2 . I) `2 ) <= 1 by A1, A2, A6, A11, A12, A16, Th63;

      

       A41: ( dom g) = the carrier of I[01] by FUNCT_2:def 1;

      then

       A42: (h . p2) = (g2 . O) by A15, FUNCT_1: 13;

      (h . p1) = (f2 . O) by A13, A27, FUNCT_1: 13;

      then ( rng f2) meets ( rng g2) by A31, A39, A42, A28, A37, A36, A35, A25, A32, A34, A26, A24, A23, A22, A40, A38, Th35, JGRAPH_6: 79;

      then

       A43: (( rng f2) /\ ( rng g2)) <> {} by XBOOLE_0:def 7;

      then x in ( rng g2) by XBOOLE_0:def 4;

      then

      consider z2 be object such that

       A44: z2 in ( dom g2) and

       A45: x = (g2 . z2) by FUNCT_1:def 3;

      

       A46: x = (h . (g . z2)) by A41, A44, A45, FUNCT_1: 13;

      

       A47: (g . z2) in ( rng g) by A41, A44, FUNCT_1:def 3;

      (g . z2) in the carrier of ( TOP-REAL 2) by A44, FUNCT_2: 5;

      then

       A48: (g . z2) in ( dom h) by FUNCT_2:def 1;

      x in ( rng f2) by A43, XBOOLE_0:def 4;

      then

      consider z1 be object such that

       A49: z1 in ( dom f2) and

       A50: x = (f2 . z1) by FUNCT_1:def 3;

      

       A51: (f . z1) in ( rng f) by A27, A49, FUNCT_1:def 3;

      (f . z1) in the carrier of ( TOP-REAL 2) by A49, FUNCT_2: 5;

      then

       A52: (f . z1) in ( dom h) by FUNCT_2:def 1;

      x = (h . (f . z1)) by A27, A49, A50, FUNCT_1: 13;

      then (f . z1) = (g . z2) by A46, A52, A48, A33, FUNCT_1:def 4;

      hence thesis by A51, A47, XBOOLE_0: 3;

    end;

    theorem :: JGRAPH_7:111

    for p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real, P,Q be Subset of ( TOP-REAL 2) st a < b & c < d & (p1 `2 ) = d & (p2 `2 ) = d & (p3 `2 ) = d & (p4 `1 ) = b & a <= (p1 `1 ) & (p1 `1 ) < (p2 `1 ) & (p2 `1 ) < (p3 `1 ) & (p3 `1 ) <= b & c <= (p4 `2 ) & (p4 `2 ) <= d & P is_an_arc_of (p1,p3) & Q is_an_arc_of (p2,p4) & P c= ( closed_inside_of_rectangle (a,b,c,d)) & Q c= ( closed_inside_of_rectangle (a,b,c,d)) holds P meets Q

    proof

      let p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real, P,Q be Subset of ( TOP-REAL 2);

      assume that

       A1: a < b and

       A2: c < d and

       A3: (p1 `2 ) = d and

       A4: (p2 `2 ) = d and

       A5: (p3 `2 ) = d and

       A6: (p4 `1 ) = b and

       A7: a <= (p1 `1 ) and

       A8: (p1 `1 ) < (p2 `1 ) and

       A9: (p2 `1 ) < (p3 `1 ) and

       A10: (p3 `1 ) <= b and

       A11: c <= (p4 `2 ) and

       A12: (p4 `2 ) <= d and

       A13: P is_an_arc_of (p1,p3) and

       A14: Q is_an_arc_of (p2,p4) and

       A15: P c= ( closed_inside_of_rectangle (a,b,c,d)) and

       A16: Q c= ( closed_inside_of_rectangle (a,b,c,d));

      

       A17: ex g be Function of I[01] , ( TOP-REAL 2) st g is continuous one-to-one & ( rng g) = Q & (g . 0 ) = p2 & (g . 1) = p4 by A14, Th2;

      ex f be Function of I[01] , ( TOP-REAL 2) st f is continuous one-to-one & ( rng f) = P & (f . 0 ) = p1 & (f . 1) = p3 by A13, Th2;

      hence thesis by A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, A12, A15, A16, A17, Th110;

    end;

    theorem :: JGRAPH_7:112

    

     Th112: for p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real, f,g be Function of I[01] , ( TOP-REAL 2) st a < b & c < d & (p1 `2 ) = d & (p2 `2 ) = d & (p3 `2 ) = d & (p4 `2 ) = c & a <= (p1 `1 ) & (p1 `1 ) < (p2 `1 ) & (p2 `1 ) < (p3 `1 ) & (p3 `1 ) <= b & a < (p4 `1 ) & (p4 `1 ) <= b & (f . 0 ) = p1 & (f . 1) = p3 & (g . 0 ) = p2 & (g . 1) = p4 & f is continuous one-to-one & g is continuous one-to-one & ( rng f) c= ( closed_inside_of_rectangle (a,b,c,d)) & ( rng g) c= ( closed_inside_of_rectangle (a,b,c,d)) holds ( rng f) meets ( rng g)

    proof

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

      let p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real, f,g be Function of I[01] , ( TOP-REAL 2);

      assume that

       A1: a < b and

       A2: c < d and

       A3: (p1 `2 ) = d and

       A4: (p2 `2 ) = d and

       A5: (p3 `2 ) = d and

       A6: (p4 `2 ) = c and

       A7: a <= (p1 `1 ) and

       A8: (p1 `1 ) < (p2 `1 ) and

       A9: (p2 `1 ) < (p3 `1 ) and

       A10: (p3 `1 ) <= b and

       A11: a < (p4 `1 ) and

       A12: (p4 `1 ) <= b and

       A13: (f . 0 ) = p1 and

       A14: (f . 1) = p3 and

       A15: (g . 0 ) = p2 and

       A16: (g . 1) = p4 and

       A17: f is continuous one-to-one and

       A18: g is continuous one-to-one and

       A19: ( rng f) c= ( closed_inside_of_rectangle (a,b,c,d)) and

       A20: ( rng g) c= ( closed_inside_of_rectangle (a,b,c,d));

      set A = (2 / (b - a)), B = ( - ((b + a) / (b - a))), C = (2 / (d - c)), D = ( - ((d + c) / (d - c)));

      set h = ( AffineMap (A,B,C,D));

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

      

       A21: (p1 `1 ) < (p3 `1 ) by A8, A9, XXREAL_0: 2;

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

      

       A22: ( - 1) < ((g2 . I) `1 ) by A1, A11, A12, A16, Th64;

      

       A23: ((g2 . I) `2 ) = ( - 1) by A2, A6, A16, Th57;

      

       A24: ((g2 . O) `2 ) = 1 by A2, A4, A15, Th55;

      ((f . I) `2 ) = d by A5, A14;

      then

       A25: ( - 1) <= ((f2 . O) `1 ) by A1, A7, A10, A13, A14, A21, Th62;

      

       A26: g2 is continuous one-to-one by A1, A2, A18, Th53;

      set x = the Element of (( rng f2) /\ ( rng g2));

      

       A27: ( dom f) = the carrier of I[01] by FUNCT_2:def 1;

      then

       A28: (h . p3) = (f2 . I) by A14, FUNCT_1: 13;

      (d - c) > 0 by A2, XREAL_1: 50;

      then

       A29: C > 0 by XREAL_1: 139;

      (b - a) > 0 by A1, XREAL_1: 50;

      then

       A30: A > 0 by XREAL_1: 139;

      then

       A31: ((h . p1) `1 ) < ((h . p2) `1 ) by A8, A29, Th50;

      ((f . O) `2 ) = d by A3, A13;

      then

       A32: ((f2 . I) `1 ) <= 1 by A1, A7, A10, A13, A14, A21, Th62;

      h is being_homeomorphism by A30, A29, Th50;

      then

       A33: h is one-to-one by TOPS_2:def 5;

      

       A34: ( rng f2) c= ( closed_inside_of_rectangle (( - 1),1,( - 1),1)) by A1, A2, A19, Th52;

      

       A35: ((f2 . I) `2 ) = 1 by A2, A5, A14, Th55;

      

       A36: ((f2 . O) `2 ) = 1 by A2, A3, A13, Th55;

      

       A37: f2 is continuous one-to-one by A1, A2, A17, Th53;

      

       A38: ( rng g2) c= ( closed_inside_of_rectangle (( - 1),1,( - 1),1)) by A1, A2, A20, Th52;

      

       A39: ((h . p2) `1 ) < ((h . p3) `1 ) by A9, A30, A29, Th50;

      

       A40: ((g2 . I) `1 ) <= 1 by A1, A11, A12, A16, Th64;

      

       A41: ( dom g) = the carrier of I[01] by FUNCT_2:def 1;

      then

       A42: (h . p2) = (g2 . O) by A15, FUNCT_1: 13;

      (h . p1) = (f2 . O) by A13, A27, FUNCT_1: 13;

      then ( rng f2) meets ( rng g2) by A31, A39, A42, A28, A37, A36, A35, A25, A32, A34, A26, A24, A23, A22, A40, A38, Th36, JGRAPH_6: 79;

      then

       A43: (( rng f2) /\ ( rng g2)) <> {} by XBOOLE_0:def 7;

      then x in ( rng g2) by XBOOLE_0:def 4;

      then

      consider z2 be object such that

       A44: z2 in ( dom g2) and

       A45: x = (g2 . z2) by FUNCT_1:def 3;

      

       A46: x = (h . (g . z2)) by A41, A44, A45, FUNCT_1: 13;

      

       A47: (g . z2) in ( rng g) by A41, A44, FUNCT_1:def 3;

      (g . z2) in the carrier of ( TOP-REAL 2) by A44, FUNCT_2: 5;

      then

       A48: (g . z2) in ( dom h) by FUNCT_2:def 1;

      x in ( rng f2) by A43, XBOOLE_0:def 4;

      then

      consider z1 be object such that

       A49: z1 in ( dom f2) and

       A50: x = (f2 . z1) by FUNCT_1:def 3;

      

       A51: (f . z1) in ( rng f) by A27, A49, FUNCT_1:def 3;

      (f . z1) in the carrier of ( TOP-REAL 2) by A49, FUNCT_2: 5;

      then

       A52: (f . z1) in ( dom h) by FUNCT_2:def 1;

      x = (h . (f . z1)) by A27, A49, A50, FUNCT_1: 13;

      then (f . z1) = (g . z2) by A46, A52, A48, A33, FUNCT_1:def 4;

      hence thesis by A51, A47, XBOOLE_0: 3;

    end;

    theorem :: JGRAPH_7:113

    for p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real, P,Q be Subset of ( TOP-REAL 2) st a < b & c < d & (p1 `2 ) = d & (p2 `2 ) = d & (p3 `2 ) = d & (p4 `2 ) = c & a <= (p1 `1 ) & (p1 `1 ) < (p2 `1 ) & (p2 `1 ) < (p3 `1 ) & (p3 `1 ) <= b & a < (p4 `1 ) & (p4 `1 ) <= b & P is_an_arc_of (p1,p3) & Q is_an_arc_of (p2,p4) & P c= ( closed_inside_of_rectangle (a,b,c,d)) & Q c= ( closed_inside_of_rectangle (a,b,c,d)) holds P meets Q

    proof

      let p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real, P,Q be Subset of ( TOP-REAL 2);

      assume that

       A1: a < b and

       A2: c < d and

       A3: (p1 `2 ) = d and

       A4: (p2 `2 ) = d and

       A5: (p3 `2 ) = d and

       A6: (p4 `2 ) = c and

       A7: a <= (p1 `1 ) and

       A8: (p1 `1 ) < (p2 `1 ) and

       A9: (p2 `1 ) < (p3 `1 ) and

       A10: (p3 `1 ) <= b and

       A11: a < (p4 `1 ) and

       A12: (p4 `1 ) <= b and

       A13: P is_an_arc_of (p1,p3) and

       A14: Q is_an_arc_of (p2,p4) and

       A15: P c= ( closed_inside_of_rectangle (a,b,c,d)) and

       A16: Q c= ( closed_inside_of_rectangle (a,b,c,d));

      

       A17: ex g be Function of I[01] , ( TOP-REAL 2) st g is continuous one-to-one & ( rng g) = Q & (g . 0 ) = p2 & (g . 1) = p4 by A14, Th2;

      ex f be Function of I[01] , ( TOP-REAL 2) st f is continuous one-to-one & ( rng f) = P & (f . 0 ) = p1 & (f . 1) = p3 by A13, Th2;

      hence thesis by A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, A12, A15, A16, A17, Th112;

    end;

    theorem :: JGRAPH_7:114

    

     Th114: for p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real, f,g be Function of I[01] , ( TOP-REAL 2) st a < b & c < d & (p1 `2 ) = d & (p2 `2 ) = d & (p3 `1 ) = b & (p4 `1 ) = b & a <= (p1 `1 ) & (p1 `1 ) < (p2 `1 ) & (p2 `1 ) <= b & c <= (p4 `2 ) & (p4 `2 ) < (p3 `2 ) & (p3 `2 ) <= d & (f . 0 ) = p1 & (f . 1) = p3 & (g . 0 ) = p2 & (g . 1) = p4 & f is continuous one-to-one & g is continuous one-to-one & ( rng f) c= ( closed_inside_of_rectangle (a,b,c,d)) & ( rng g) c= ( closed_inside_of_rectangle (a,b,c,d)) holds ( rng f) meets ( rng g)

    proof

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

      let p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real, f,g be Function of I[01] , ( TOP-REAL 2);

      assume that

       A1: a < b and

       A2: c < d and

       A3: (p1 `2 ) = d and

       A4: (p2 `2 ) = d and

       A5: (p3 `1 ) = b and

       A6: (p4 `1 ) = b and

       A7: a <= (p1 `1 ) and

       A8: (p1 `1 ) < (p2 `1 ) and

       A9: (p2 `1 ) <= b and

       A10: c <= (p4 `2 ) and

       A11: (p4 `2 ) < (p3 `2 ) and

       A12: (p3 `2 ) <= d and

       A13: (f . 0 ) = p1 and

       A14: (f . 1) = p3 and

       A15: (g . 0 ) = p2 and

       A16: (g . 1) = p4 and

       A17: f is continuous one-to-one and

       A18: g is continuous one-to-one and

       A19: ( rng f) c= ( closed_inside_of_rectangle (a,b,c,d)) and

       A20: ( rng g) c= ( closed_inside_of_rectangle (a,b,c,d));

      set A = (2 / (b - a)), B = ( - ((b + a) / (b - a))), C = (2 / (d - c)), D = ( - ((d + c) / (d - c)));

      set h = ( AffineMap (A,B,C,D));

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

      

       A21: g2 is continuous one-to-one by A1, A2, A18, Th53;

      a < (p2 `1 ) by A7, A8, XXREAL_0: 2;

      then

       A22: ((g2 . O) `1 ) <= 1 by A1, A2, A4, A9, A15, Th63;

      

       A23: ( rng g2) c= ( closed_inside_of_rectangle (( - 1),1,( - 1),1)) by A1, A2, A20, Th52;

      

       A24: ((g2 . I) `1 ) = 1 by A1, A6, A16, Th56;

      (d - c) > 0 by A2, XREAL_1: 50;

      then

       A25: C > 0 by XREAL_1: 139;

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

      

       A26: f2 is continuous one-to-one by A1, A2, A17, Th53;

      (p3 `2 ) > c by A10, A11, XXREAL_0: 2;

      then

       A27: ((f2 . I) `2 ) <= 1 by A1, A2, A5, A12, A14, Th63;

      

       A28: ( rng f2) c= ( closed_inside_of_rectangle (( - 1),1,( - 1),1)) by A1, A2, A19, Th52;

      

       A29: ((f2 . I) `1 ) = 1 by A1, A5, A14, Th56;

      (p1 `1 ) <= b by A8, A9, XXREAL_0: 2;

      then

       A30: ( - 1) <= ((f2 . O) `1 ) by A1, A2, A3, A7, A13, Th63;

      

       A31: ((f2 . O) `2 ) = 1 by A2, A3, A13, Th55;

      set x = the Element of (( rng f2) /\ ( rng g2));

      

       A32: ( dom f) = the carrier of I[01] by FUNCT_2:def 1;

      then

       A33: (h . p3) = (f2 . I) by A14, FUNCT_1: 13;

      (b - a) > 0 by A1, XREAL_1: 50;

      then

       A34: A > 0 by XREAL_1: 139;

      then

       A35: ((h . p1) `1 ) < ((h . p2) `1 ) by A8, A25, Th50;

      d > (p4 `2 ) by A11, A12, XXREAL_0: 2;

      then

       A36: ( - 1) <= ((g2 . I) `2 ) by A1, A2, A6, A10, A16, Th63;

      h is being_homeomorphism by A34, A25, Th51;

      then

       A37: h is one-to-one by TOPS_2:def 5;

      

       A38: ((g2 . O) `2 ) = 1 by A2, A4, A15, Th55;

      

       A39: ( dom g) = the carrier of I[01] by FUNCT_2:def 1;

      then

       A40: (h . p2) = (g2 . O) by A15, FUNCT_1: 13;

      

       A41: ((h . p3) `2 ) > ((h . p4) `2 ) by A11, A34, A25, Th51;

      

       A42: (h . p4) = (g2 . I) by A16, A39, FUNCT_1: 13;

      (h . p1) = (f2 . O) by A13, A32, FUNCT_1: 13;

      then ((f2 . O),(g2 . O),(f2 . I),(g2 . I)) are_in_this_order_on ( rectangle (( - 1),1,( - 1),1)) by A35, A41, A40, A33, A42, A31, A29, A30, A27, A38, A24, A22, A36, Th37;

      then ( rng f2) meets ( rng g2) by A26, A28, A21, A23, JGRAPH_6: 79;

      then

       A43: (( rng f2) /\ ( rng g2)) <> {} by XBOOLE_0:def 7;

      then x in ( rng g2) by XBOOLE_0:def 4;

      then

      consider z2 be object such that

       A44: z2 in ( dom g2) and

       A45: x = (g2 . z2) by FUNCT_1:def 3;

      

       A46: x = (h . (g . z2)) by A39, A44, A45, FUNCT_1: 13;

      

       A47: (g . z2) in ( rng g) by A39, A44, FUNCT_1:def 3;

      (g . z2) in the carrier of ( TOP-REAL 2) by A44, FUNCT_2: 5;

      then

       A48: (g . z2) in ( dom h) by FUNCT_2:def 1;

      x in ( rng f2) by A43, XBOOLE_0:def 4;

      then

      consider z1 be object such that

       A49: z1 in ( dom f2) and

       A50: x = (f2 . z1) by FUNCT_1:def 3;

      

       A51: (f . z1) in ( rng f) by A32, A49, FUNCT_1:def 3;

      (f . z1) in the carrier of ( TOP-REAL 2) by A49, FUNCT_2: 5;

      then

       A52: (f . z1) in ( dom h) by FUNCT_2:def 1;

      x = (h . (f . z1)) by A32, A49, A50, FUNCT_1: 13;

      then (f . z1) = (g . z2) by A46, A52, A48, A37, FUNCT_1:def 4;

      hence thesis by A51, A47, XBOOLE_0: 3;

    end;

    theorem :: JGRAPH_7:115

    for p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real, P,Q be Subset of ( TOP-REAL 2) st a < b & c < d & (p1 `2 ) = d & (p2 `2 ) = d & (p3 `1 ) = b & (p4 `1 ) = b & a <= (p1 `1 ) & (p1 `1 ) < (p2 `1 ) & (p2 `1 ) <= b & c <= (p4 `2 ) & (p4 `2 ) < (p3 `2 ) & (p3 `2 ) <= d & P is_an_arc_of (p1,p3) & Q is_an_arc_of (p2,p4) & P c= ( closed_inside_of_rectangle (a,b,c,d)) & Q c= ( closed_inside_of_rectangle (a,b,c,d)) holds P meets Q

    proof

      let p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real, P,Q be Subset of ( TOP-REAL 2);

      assume that

       A1: a < b and

       A2: c < d and

       A3: (p1 `2 ) = d and

       A4: (p2 `2 ) = d and

       A5: (p3 `1 ) = b and

       A6: (p4 `1 ) = b and

       A7: a <= (p1 `1 ) and

       A8: (p1 `1 ) < (p2 `1 ) and

       A9: (p2 `1 ) <= b and

       A10: c <= (p4 `2 ) and

       A11: (p4 `2 ) < (p3 `2 ) and

       A12: (p3 `2 ) <= d and

       A13: P is_an_arc_of (p1,p3) and

       A14: Q is_an_arc_of (p2,p4) and

       A15: P c= ( closed_inside_of_rectangle (a,b,c,d)) and

       A16: Q c= ( closed_inside_of_rectangle (a,b,c,d));

      

       A17: ex g be Function of I[01] , ( TOP-REAL 2) st g is continuous one-to-one & ( rng g) = Q & (g . 0 ) = p2 & (g . 1) = p4 by A14, Th2;

      ex f be Function of I[01] , ( TOP-REAL 2) st f is continuous one-to-one & ( rng f) = P & (f . 0 ) = p1 & (f . 1) = p3 by A13, Th2;

      hence thesis by A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, A12, A15, A16, A17, Th114;

    end;

    theorem :: JGRAPH_7:116

    

     Th116: for p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real, f,g be Function of I[01] , ( TOP-REAL 2) st a < b & c < d & (p1 `2 ) = d & (p2 `2 ) = d & (p3 `1 ) = b & (p4 `2 ) = c & a <= (p1 `1 ) & (p1 `1 ) < (p2 `1 ) & (p2 `1 ) <= b & c <= (p3 `2 ) & (p3 `2 ) <= d & a < (p4 `1 ) & (p4 `1 ) <= b & (f . 0 ) = p1 & (f . 1) = p3 & (g . 0 ) = p2 & (g . 1) = p4 & f is continuous one-to-one & g is continuous one-to-one & ( rng f) c= ( closed_inside_of_rectangle (a,b,c,d)) & ( rng g) c= ( closed_inside_of_rectangle (a,b,c,d)) holds ( rng f) meets ( rng g)

    proof

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

      let p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real, f,g be Function of I[01] , ( TOP-REAL 2);

      assume that

       A1: a < b and

       A2: c < d and

       A3: (p1 `2 ) = d and

       A4: (p2 `2 ) = d and

       A5: (p3 `1 ) = b and

       A6: (p4 `2 ) = c and

       A7: a <= (p1 `1 ) and

       A8: (p1 `1 ) < (p2 `1 ) and

       A9: (p2 `1 ) <= b and

       A10: c <= (p3 `2 ) and

       A11: (p3 `2 ) <= d and

       A12: a < (p4 `1 ) and

       A13: (p4 `1 ) <= b and

       A14: (f . 0 ) = p1 and

       A15: (f . 1) = p3 and

       A16: (g . 0 ) = p2 and

       A17: (g . 1) = p4 and

       A18: f is continuous one-to-one and

       A19: g is continuous one-to-one and

       A20: ( rng f) c= ( closed_inside_of_rectangle (a,b,c,d)) and

       A21: ( rng g) c= ( closed_inside_of_rectangle (a,b,c,d));

      set A = (2 / (b - a)), B = ( - ((b + a) / (b - a))), C = (2 / (d - c)), D = ( - ((d + c) / (d - c)));

      set h = ( AffineMap (A,B,C,D));

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

      

       A22: ( - 1) <= ((f2 . I) `2 ) by A1, A2, A5, A10, A11, A15, Th63;

      

       A23: ((f2 . I) `2 ) <= 1 by A1, A2, A5, A10, A11, A15, Th63;

      

       A24: f2 is continuous one-to-one by A1, A2, A18, Th53;

      

       A25: ((f2 . I) `1 ) = 1 by A1, A5, A15, Th56;

      

       A26: ((f2 . O) `2 ) = 1 by A2, A3, A14, Th55;

      

       A27: ( rng f2) c= ( closed_inside_of_rectangle (( - 1),1,( - 1),1)) by A1, A2, A20, Th52;

      (d - c) > 0 by A2, XREAL_1: 50;

      then

       A28: C > 0 by XREAL_1: 139;

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

      

       A29: ( - 1) < ((g2 . I) `1 ) by A1, A12, A13, A17, Th64;

      

       A30: ((g2 . O) `2 ) = 1 by A2, A4, A16, Th55;

      

       A31: g2 is continuous one-to-one by A1, A2, A19, Th53;

      a < (p2 `1 ) by A7, A8, XXREAL_0: 2;

      then

       A32: ((g2 . O) `1 ) <= 1 by A1, A9, A16, Th64;

      

       A33: ((g2 . I) `1 ) <= 1 by A1, A12, A13, A17, Th64;

      set x = the Element of (( rng f2) /\ ( rng g2));

      

       A34: ( dom g) = the carrier of I[01] by FUNCT_2:def 1;

      then

       A35: (h . p2) = (g2 . O) by A16, FUNCT_1: 13;

      (b - a) > 0 by A1, XREAL_1: 50;

      then

       A36: A > 0 by XREAL_1: 139;

      then

       A37: ((h . p1) `1 ) < ((h . p2) `1 ) by A8, A28, Th50;

      (p1 `1 ) <= b by A8, A9, XXREAL_0: 2;

      then

       A38: ( - 1) <= ((f2 . O) `1 ) by A1, A2, A3, A7, A14, Th63;

      h is being_homeomorphism by A36, A28, Th51;

      then

       A39: h is one-to-one by TOPS_2:def 5;

      

       A40: ( rng g2) c= ( closed_inside_of_rectangle (( - 1),1,( - 1),1)) by A1, A2, A21, Th52;

      

       A41: ((g2 . I) `2 ) = ( - 1) by A2, A6, A17, Th57;

      

       A42: ( dom f) = the carrier of I[01] by FUNCT_2:def 1;

      then (h . p1) = (f2 . O) by A14, FUNCT_1: 13;

      then ( rng f2) meets ( rng g2) by A37, A35, A24, A26, A25, A38, A22, A23, A27, A31, A30, A41, A32, A29, A33, A40, Th38, JGRAPH_6: 79;

      then

       A43: (( rng f2) /\ ( rng g2)) <> {} by XBOOLE_0:def 7;

      then x in ( rng g2) by XBOOLE_0:def 4;

      then

      consider z2 be object such that

       A44: z2 in ( dom g2) and

       A45: x = (g2 . z2) by FUNCT_1:def 3;

      

       A46: x = (h . (g . z2)) by A34, A44, A45, FUNCT_1: 13;

      

       A47: (g . z2) in ( rng g) by A34, A44, FUNCT_1:def 3;

      (g . z2) in the carrier of ( TOP-REAL 2) by A44, FUNCT_2: 5;

      then

       A48: (g . z2) in ( dom h) by FUNCT_2:def 1;

      x in ( rng f2) by A43, XBOOLE_0:def 4;

      then

      consider z1 be object such that

       A49: z1 in ( dom f2) and

       A50: x = (f2 . z1) by FUNCT_1:def 3;

      

       A51: (f . z1) in ( rng f) by A42, A49, FUNCT_1:def 3;

      (f . z1) in the carrier of ( TOP-REAL 2) by A49, FUNCT_2: 5;

      then

       A52: (f . z1) in ( dom h) by FUNCT_2:def 1;

      x = (h . (f . z1)) by A42, A49, A50, FUNCT_1: 13;

      then (f . z1) = (g . z2) by A46, A52, A48, A39, FUNCT_1:def 4;

      hence thesis by A51, A47, XBOOLE_0: 3;

    end;

    theorem :: JGRAPH_7:117

    for p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real, P,Q be Subset of ( TOP-REAL 2) st a < b & c < d & (p1 `2 ) = d & (p2 `2 ) = d & (p3 `1 ) = b & (p4 `2 ) = c & a <= (p1 `1 ) & (p1 `1 ) < (p2 `1 ) & (p2 `1 ) <= b & c <= (p3 `2 ) & (p3 `2 ) <= d & a < (p4 `1 ) & (p4 `1 ) <= b & P is_an_arc_of (p1,p3) & Q is_an_arc_of (p2,p4) & P c= ( closed_inside_of_rectangle (a,b,c,d)) & Q c= ( closed_inside_of_rectangle (a,b,c,d)) holds P meets Q

    proof

      let p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real, P,Q be Subset of ( TOP-REAL 2);

      assume that

       A1: a < b and

       A2: c < d and

       A3: (p1 `2 ) = d and

       A4: (p2 `2 ) = d and

       A5: (p3 `1 ) = b and

       A6: (p4 `2 ) = c and

       A7: a <= (p1 `1 ) and

       A8: (p1 `1 ) < (p2 `1 ) and

       A9: (p2 `1 ) <= b and

       A10: c <= (p3 `2 ) and

       A11: (p3 `2 ) <= d and

       A12: a < (p4 `1 ) and

       A13: (p4 `1 ) <= b and

       A14: P is_an_arc_of (p1,p3) and

       A15: Q is_an_arc_of (p2,p4) and

       A16: P c= ( closed_inside_of_rectangle (a,b,c,d)) and

       A17: Q c= ( closed_inside_of_rectangle (a,b,c,d));

      

       A18: ex g be Function of I[01] , ( TOP-REAL 2) st g is continuous one-to-one & ( rng g) = Q & (g . 0 ) = p2 & (g . 1) = p4 by A15, Th2;

      ex f be Function of I[01] , ( TOP-REAL 2) st f is continuous one-to-one & ( rng f) = P & (f . 0 ) = p1 & (f . 1) = p3 by A14, Th2;

      hence thesis by A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, A12, A13, A16, A17, A18, Th116;

    end;

    theorem :: JGRAPH_7:118

    

     Th118: for p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real, f,g be Function of I[01] , ( TOP-REAL 2) st a < b & c < d & (p1 `2 ) = d & (p2 `2 ) = d & (p3 `2 ) = c & (p4 `2 ) = c & a <= (p1 `1 ) & (p1 `1 ) < (p2 `1 ) & (p2 `1 ) <= b & a < (p4 `1 ) & (p4 `1 ) < (p3 `1 ) & (p3 `1 ) <= b & (f . 0 ) = p1 & (f . 1) = p3 & (g . 0 ) = p2 & (g . 1) = p4 & f is continuous one-to-one & g is continuous one-to-one & ( rng f) c= ( closed_inside_of_rectangle (a,b,c,d)) & ( rng g) c= ( closed_inside_of_rectangle (a,b,c,d)) holds ( rng f) meets ( rng g)

    proof

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

      let p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real, f,g be Function of I[01] , ( TOP-REAL 2);

      assume that

       A1: a < b and

       A2: c < d and

       A3: (p1 `2 ) = d and

       A4: (p2 `2 ) = d and

       A5: (p3 `2 ) = c and

       A6: (p4 `2 ) = c and

       A7: a <= (p1 `1 ) and

       A8: (p1 `1 ) < (p2 `1 ) and

       A9: (p2 `1 ) <= b and

       A10: a < (p4 `1 ) and

       A11: (p4 `1 ) < (p3 `1 ) and

       A12: (p3 `1 ) <= b and

       A13: (f . 0 ) = p1 and

       A14: (f . 1) = p3 and

       A15: (g . 0 ) = p2 and

       A16: (g . 1) = p4 and

       A17: f is continuous one-to-one and

       A18: g is continuous one-to-one and

       A19: ( rng f) c= ( closed_inside_of_rectangle (a,b,c,d)) and

       A20: ( rng g) c= ( closed_inside_of_rectangle (a,b,c,d));

      set A = (2 / (b - a)), B = ( - ((b + a) / (b - a))), C = (2 / (d - c)), D = ( - ((d + c) / (d - c)));

      set h = ( AffineMap (A,B,C,D));

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

      

       A21: f2 is continuous one-to-one by A1, A2, A17, Th53;

      

       A22: ((f2 . I) `2 ) = ( - 1) by A2, A5, A14, Th57;

      

       A23: ((f2 . O) `2 ) = 1 by A2, A3, A13, Th55;

      (d - c) > 0 by A2, XREAL_1: 50;

      then

       A24: C > 0 by XREAL_1: 139;

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

      

       A25: g2 is continuous one-to-one by A1, A2, A18, Th53;

      b >= (p4 `1 ) by A11, A12, XXREAL_0: 2;

      then

       A26: ( - 1) < ((g2 . I) `1 ) by A1, A10, A16, Th64;

      

       A27: ( rng g2) c= ( closed_inside_of_rectangle (( - 1),1,( - 1),1)) by A1, A2, A20, Th52;

      

       A28: (p3 `1 ) > a by A10, A11, XXREAL_0: 2;

      then

       A29: ((f2 . I) `1 ) <= 1 by A1, A12, A14, Th64;

      

       A30: ((f . I) `2 ) = c by A5, A14;

      (p1 `1 ) <= b by A8, A9, XXREAL_0: 2;

      then

       A31: ( - 1) <= ((f2 . O) `1 ) by A1, A7, A12, A13, A14, A28, A30, Th64;

      

       A32: ((g2 . O) `2 ) = 1 by A2, A4, A15, Th55;

      

       A33: ( dom g) = the carrier of I[01] by FUNCT_2:def 1;

      then

       A34: (h . p2) = (g2 . O) by A15, FUNCT_1: 13;

      a < (p2 `1 ) by A7, A8, XXREAL_0: 2;

      then

       A35: ((g2 . O) `1 ) <= 1 by A1, A9, A15, Th64;

      

       A36: ((g2 . I) `2 ) = ( - 1) by A2, A6, A16, Th57;

      set x = the Element of (( rng f2) /\ ( rng g2));

      

       A37: ( dom f) = the carrier of I[01] by FUNCT_2:def 1;

      then

       A38: (h . p3) = (f2 . I) by A14, FUNCT_1: 13;

      (b - a) > 0 by A1, XREAL_1: 50;

      then

       A39: A > 0 by XREAL_1: 139;

      then

       A40: ((h . p1) `1 ) < ((h . p2) `1 ) by A8, A24, Th50;

      

       A41: ( rng f2) c= ( closed_inside_of_rectangle (( - 1),1,( - 1),1)) by A1, A2, A19, Th52;

      

       A42: ((h . p3) `1 ) > ((h . p4) `1 ) by A11, A39, A24, Th50;

      

       A43: (h . p4) = (g2 . I) by A16, A33, FUNCT_1: 13;

      (h . p1) = (f2 . O) by A13, A37, FUNCT_1: 13;

      then ((f2 . O),(g2 . O),(f2 . I),(g2 . I)) are_in_this_order_on ( rectangle (( - 1),1,( - 1),1)) by A40, A42, A34, A38, A43, A23, A22, A31, A29, A32, A36, A35, A26, Th39;

      then ( rng f2) meets ( rng g2) by A21, A41, A25, A27, JGRAPH_6: 79;

      then

       A44: (( rng f2) /\ ( rng g2)) <> {} by XBOOLE_0:def 7;

      then x in ( rng g2) by XBOOLE_0:def 4;

      then

      consider z2 be object such that

       A45: z2 in ( dom g2) and

       A46: x = (g2 . z2) by FUNCT_1:def 3;

      

       A47: x = (h . (g . z2)) by A33, A45, A46, FUNCT_1: 13;

      h is being_homeomorphism by A39, A24, Th51;

      then

       A48: h is one-to-one by TOPS_2:def 5;

      x in ( rng f2) by A44, XBOOLE_0:def 4;

      then

      consider z1 be object such that

       A49: z1 in ( dom f2) and

       A50: x = (f2 . z1) by FUNCT_1:def 3;

      

       A51: (f . z1) in ( rng f) by A37, A49, FUNCT_1:def 3;

      

       A52: (g . z2) in ( rng g) by A33, A45, FUNCT_1:def 3;

      (g . z2) in the carrier of ( TOP-REAL 2) by A45, FUNCT_2: 5;

      then

       A53: (g . z2) in ( dom h) by FUNCT_2:def 1;

      (f . z1) in the carrier of ( TOP-REAL 2) by A49, FUNCT_2: 5;

      then

       A54: (f . z1) in ( dom h) by FUNCT_2:def 1;

      x = (h . (f . z1)) by A37, A49, A50, FUNCT_1: 13;

      then (f . z1) = (g . z2) by A47, A54, A53, A48, FUNCT_1:def 4;

      hence thesis by A51, A52, XBOOLE_0: 3;

    end;

    theorem :: JGRAPH_7:119

    for p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real, P,Q be Subset of ( TOP-REAL 2) st a < b & c < d & (p1 `2 ) = d & (p2 `2 ) = d & (p3 `2 ) = c & (p4 `2 ) = c & a <= (p1 `1 ) & (p1 `1 ) < (p2 `1 ) & (p2 `1 ) <= b & a < (p4 `1 ) & (p4 `1 ) < (p3 `1 ) & (p3 `1 ) <= b & P is_an_arc_of (p1,p3) & Q is_an_arc_of (p2,p4) & P c= ( closed_inside_of_rectangle (a,b,c,d)) & Q c= ( closed_inside_of_rectangle (a,b,c,d)) holds P meets Q

    proof

      let p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real, P,Q be Subset of ( TOP-REAL 2);

      assume that

       A1: a < b and

       A2: c < d and

       A3: (p1 `2 ) = d and

       A4: (p2 `2 ) = d and

       A5: (p3 `2 ) = c and

       A6: (p4 `2 ) = c and

       A7: a <= (p1 `1 ) and

       A8: (p1 `1 ) < (p2 `1 ) and

       A9: (p2 `1 ) <= b and

       A10: a < (p4 `1 ) and

       A11: (p4 `1 ) < (p3 `1 ) and

       A12: (p3 `1 ) <= b and

       A13: P is_an_arc_of (p1,p3) and

       A14: Q is_an_arc_of (p2,p4) and

       A15: P c= ( closed_inside_of_rectangle (a,b,c,d)) and

       A16: Q c= ( closed_inside_of_rectangle (a,b,c,d));

      

       A17: ex g be Function of I[01] , ( TOP-REAL 2) st g is continuous one-to-one & ( rng g) = Q & (g . 0 ) = p2 & (g . 1) = p4 by A14, Th2;

      ex f be Function of I[01] , ( TOP-REAL 2) st f is continuous one-to-one & ( rng f) = P & (f . 0 ) = p1 & (f . 1) = p3 by A13, Th2;

      hence thesis by A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, A12, A15, A16, A17, Th118;

    end;

    theorem :: JGRAPH_7:120

    

     Th120: for p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real, f,g be Function of I[01] , ( TOP-REAL 2) st a < b & c < d & (p1 `2 ) = d & (p2 `1 ) = b & (p3 `1 ) = b & (p4 `1 ) = b & a <= (p1 `1 ) & (p1 `1 ) <= b & d >= (p2 `2 ) & (p2 `2 ) > (p3 `2 ) & (p3 `2 ) > (p4 `2 ) & (p4 `2 ) >= c & (f . 0 ) = p1 & (f . 1) = p3 & (g . 0 ) = p2 & (g . 1) = p4 & f is continuous one-to-one & g is continuous one-to-one & ( rng f) c= ( closed_inside_of_rectangle (a,b,c,d)) & ( rng g) c= ( closed_inside_of_rectangle (a,b,c,d)) holds ( rng f) meets ( rng g)

    proof

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

      let p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real, f,g be Function of I[01] , ( TOP-REAL 2);

      assume that

       A1: a < b and

       A2: c < d and

       A3: (p1 `2 ) = d and

       A4: (p2 `1 ) = b and

       A5: (p3 `1 ) = b and

       A6: (p4 `1 ) = b and

       A7: a <= (p1 `1 ) and

       A8: (p1 `1 ) <= b and

       A9: d >= (p2 `2 ) and

       A10: (p2 `2 ) > (p3 `2 ) and

       A11: (p3 `2 ) > (p4 `2 ) and

       A12: (p4 `2 ) >= c and

       A13: (f . 0 ) = p1 and

       A14: (f . 1) = p3 and

       A15: (g . 0 ) = p2 and

       A16: (g . 1) = p4 and

       A17: f is continuous one-to-one and

       A18: g is continuous one-to-one and

       A19: ( rng f) c= ( closed_inside_of_rectangle (a,b,c,d)) and

       A20: ( rng g) c= ( closed_inside_of_rectangle (a,b,c,d));

      set A = (2 / (b - a)), B = ( - ((b + a) / (b - a))), C = (2 / (d - c)), D = ( - ((d + c) / (d - c)));

      set h = ( AffineMap (A,B,C,D));

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

      

       A21: (p2 `2 ) > (p4 `2 ) by A10, A11, XXREAL_0: 2;

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

      

       A22: ( - 1) <= ((f2 . O) `1 ) by A1, A2, A3, A7, A8, A13, Th63;

      

       A23: ((f2 . O) `2 ) = 1 by A2, A3, A13, Th55;

      

       A24: f2 is continuous one-to-one by A1, A2, A17, Th53;

      

       A25: ( rng f2) c= ( closed_inside_of_rectangle (( - 1),1,( - 1),1)) by A1, A2, A19, Th52;

      

       A26: ((f2 . I) `1 ) = 1 by A1, A5, A14, Th56;

      

       A27: ((f2 . O) `1 ) <= 1 by A1, A2, A3, A7, A8, A13, Th63;

      set x = the Element of (( rng f2) /\ ( rng g2));

      

       A28: ( dom f) = the carrier of I[01] by FUNCT_2:def 1;

      then

       A29: (h . p3) = (f2 . I) by A14, FUNCT_1: 13;

      (d - c) > 0 by A2, XREAL_1: 50;

      then

       A30: C > 0 by XREAL_1: 139;

      (b - a) > 0 by A1, XREAL_1: 50;

      then

       A31: A > 0 by XREAL_1: 139;

      then

       A32: ((h . p2) `2 ) > ((h . p3) `2 ) by A10, A30, Th51;

      ((g . O) `1 ) = b by A4, A15;

      then

       A33: ((g2 . I) `2 ) >= ( - 1) by A2, A9, A12, A15, A16, A21, Th65;

      h is being_homeomorphism by A31, A30, Th51;

      then

       A34: h is one-to-one by TOPS_2:def 5;

      

       A35: ( rng g2) c= ( closed_inside_of_rectangle (( - 1),1,( - 1),1)) by A1, A2, A20, Th52;

      

       A36: ((h . p3) `2 ) > ((h . p4) `2 ) by A11, A31, A30, Th51;

      

       A37: ((g2 . I) `1 ) = 1 by A1, A6, A16, Th56;

      

       A38: ((g2 . O) `1 ) = 1 by A1, A4, A15, Th56;

      ((g . I) `1 ) = b by A6, A16;

      then

       A39: 1 >= ((g2 . O) `2 ) by A2, A9, A12, A15, A16, A21, Th65;

      

       A40: g2 is continuous one-to-one by A1, A2, A18, Th53;

      

       A41: ( dom g) = the carrier of I[01] by FUNCT_2:def 1;

      then

       A42: (h . p4) = (g2 . I) by A16, FUNCT_1: 13;

      (h . p2) = (g2 . O) by A15, A41, FUNCT_1: 13;

      then ( rng f2) meets ( rng g2) by A32, A36, A29, A42, A24, A23, A26, A22, A27, A25, A40, A38, A37, A39, A33, A35, Th40, JGRAPH_6: 79;

      then

       A43: (( rng f2) /\ ( rng g2)) <> {} by XBOOLE_0:def 7;

      then x in ( rng g2) by XBOOLE_0:def 4;

      then

      consider z2 be object such that

       A44: z2 in ( dom g2) and

       A45: x = (g2 . z2) by FUNCT_1:def 3;

      

       A46: x = (h . (g . z2)) by A41, A44, A45, FUNCT_1: 13;

      

       A47: (g . z2) in ( rng g) by A41, A44, FUNCT_1:def 3;

      (g . z2) in the carrier of ( TOP-REAL 2) by A44, FUNCT_2: 5;

      then

       A48: (g . z2) in ( dom h) by FUNCT_2:def 1;

      x in ( rng f2) by A43, XBOOLE_0:def 4;

      then

      consider z1 be object such that

       A49: z1 in ( dom f2) and

       A50: x = (f2 . z1) by FUNCT_1:def 3;

      

       A51: (f . z1) in ( rng f) by A28, A49, FUNCT_1:def 3;

      (f . z1) in the carrier of ( TOP-REAL 2) by A49, FUNCT_2: 5;

      then

       A52: (f . z1) in ( dom h) by FUNCT_2:def 1;

      x = (h . (f . z1)) by A28, A49, A50, FUNCT_1: 13;

      then (f . z1) = (g . z2) by A46, A52, A48, A34, FUNCT_1:def 4;

      hence thesis by A51, A47, XBOOLE_0: 3;

    end;

    theorem :: JGRAPH_7:121

    for p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real, P,Q be Subset of ( TOP-REAL 2) st a < b & c < d & (p1 `2 ) = d & (p2 `1 ) = b & (p3 `1 ) = b & (p4 `1 ) = b & a <= (p1 `1 ) & (p1 `1 ) <= b & d >= (p2 `2 ) & (p2 `2 ) > (p3 `2 ) & (p3 `2 ) > (p4 `2 ) & (p4 `2 ) >= c & P is_an_arc_of (p1,p3) & Q is_an_arc_of (p2,p4) & P c= ( closed_inside_of_rectangle (a,b,c,d)) & Q c= ( closed_inside_of_rectangle (a,b,c,d)) holds P meets Q

    proof

      let p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real, P,Q be Subset of ( TOP-REAL 2);

      assume that

       A1: a < b and

       A2: c < d and

       A3: (p1 `2 ) = d and

       A4: (p2 `1 ) = b and

       A5: (p3 `1 ) = b and

       A6: (p4 `1 ) = b and

       A7: a <= (p1 `1 ) and

       A8: (p1 `1 ) <= b and

       A9: d >= (p2 `2 ) and

       A10: (p2 `2 ) > (p3 `2 ) and

       A11: (p3 `2 ) > (p4 `2 ) and

       A12: (p4 `2 ) >= c and

       A13: P is_an_arc_of (p1,p3) and

       A14: Q is_an_arc_of (p2,p4) and

       A15: P c= ( closed_inside_of_rectangle (a,b,c,d)) and

       A16: Q c= ( closed_inside_of_rectangle (a,b,c,d));

      

       A17: ex g be Function of I[01] , ( TOP-REAL 2) st g is continuous one-to-one & ( rng g) = Q & (g . 0 ) = p2 & (g . 1) = p4 by A14, Th2;

      ex f be Function of I[01] , ( TOP-REAL 2) st f is continuous one-to-one & ( rng f) = P & (f . 0 ) = p1 & (f . 1) = p3 by A13, Th2;

      hence thesis by A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, A12, A15, A16, A17, Th120;

    end;

    theorem :: JGRAPH_7:122

    

     Th122: for p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real, f,g be Function of I[01] , ( TOP-REAL 2) st a < b & c < d & (p1 `2 ) = d & (p2 `1 ) = b & (p3 `1 ) = b & (p4 `2 ) = c & a <= (p1 `1 ) & (p1 `1 ) <= b & d >= (p2 `2 ) & (p2 `2 ) > (p3 `2 ) & (p3 `2 ) >= c & a < (p4 `1 ) & (p4 `1 ) <= b & (f . 0 ) = p1 & (f . 1) = p3 & (g . 0 ) = p2 & (g . 1) = p4 & f is continuous one-to-one & g is continuous one-to-one & ( rng f) c= ( closed_inside_of_rectangle (a,b,c,d)) & ( rng g) c= ( closed_inside_of_rectangle (a,b,c,d)) holds ( rng f) meets ( rng g)

    proof

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

      let p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real, f,g be Function of I[01] , ( TOP-REAL 2);

      assume that

       A1: a < b and

       A2: c < d and

       A3: (p1 `2 ) = d and

       A4: (p2 `1 ) = b and

       A5: (p3 `1 ) = b and

       A6: (p4 `2 ) = c and

       A7: a <= (p1 `1 ) and

       A8: (p1 `1 ) <= b and

       A9: d >= (p2 `2 ) and

       A10: (p2 `2 ) > (p3 `2 ) and

       A11: (p3 `2 ) >= c and

       A12: a < (p4 `1 ) and

       A13: (p4 `1 ) <= b and

       A14: (f . 0 ) = p1 and

       A15: (f . 1) = p3 and

       A16: (g . 0 ) = p2 and

       A17: (g . 1) = p4 and

       A18: f is continuous one-to-one and

       A19: g is continuous one-to-one and

       A20: ( rng f) c= ( closed_inside_of_rectangle (a,b,c,d)) and

       A21: ( rng g) c= ( closed_inside_of_rectangle (a,b,c,d));

      set A = (2 / (b - a)), B = ( - ((b + a) / (b - a))), C = (2 / (d - c)), D = ( - ((d + c) / (d - c)));

      set h = ( AffineMap (A,B,C,D));

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

      

       A22: ( - 1) <= ((f2 . O) `1 ) by A1, A2, A3, A7, A8, A14, Th63;

      

       A23: ((f2 . O) `1 ) <= 1 by A1, A2, A3, A7, A8, A14, Th63;

      

       A24: f2 is continuous one-to-one by A1, A2, A18, Th53;

      

       A25: ((f2 . I) `1 ) = 1 by A1, A5, A15, Th56;

      

       A26: ((f2 . O) `2 ) = 1 by A2, A3, A14, Th55;

      

       A27: ( rng f2) c= ( closed_inside_of_rectangle (( - 1),1,( - 1),1)) by A1, A2, A20, Th52;

      (d - c) > 0 by A2, XREAL_1: 50;

      then

       A28: C > 0 by XREAL_1: 139;

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

      

       A29: ( - 1) < ((g2 . I) `1 ) by A1, A2, A6, A12, A13, A17, Th66;

      

       A30: ( rng g2) c= ( closed_inside_of_rectangle (( - 1),1,( - 1),1)) by A1, A2, A21, Th52;

      

       A31: ((g2 . I) `2 ) = ( - 1) by A2, A6, A17, Th57;

      c < (p2 `2 ) by A10, A11, XXREAL_0: 2;

      then

       A32: ((g2 . O) `2 ) <= 1 by A1, A2, A4, A9, A16, Th66;

      

       A33: ((g2 . O) `1 ) = 1 by A1, A4, A16, Th56;

      

       A34: g2 is continuous one-to-one by A1, A2, A19, Th53;

      

       A35: ((g2 . I) `1 ) <= 1 by A1, A2, A6, A12, A13, A17, Th66;

      set x = the Element of (( rng f2) /\ ( rng g2));

      

       A36: ( dom f) = the carrier of I[01] by FUNCT_2:def 1;

      then

       A37: (h . p3) = (f2 . I) by A15, FUNCT_1: 13;

      (b - a) > 0 by A1, XREAL_1: 50;

      then

       A38: A > 0 by XREAL_1: 139;

      then

       A39: ((h . p2) `2 ) > ((h . p3) `2 ) by A10, A28, Th51;

      d >= (p3 `2 ) by A9, A10, XXREAL_0: 2;

      then

       A40: ( - 1) <= ((f2 . I) `2 ) by A1, A2, A5, A11, A15, Th63;

      h is being_homeomorphism by A38, A28, Th51;

      then

       A41: h is one-to-one by TOPS_2:def 5;

      

       A42: ( dom g) = the carrier of I[01] by FUNCT_2:def 1;

      then (h . p2) = (g2 . O) by A16, FUNCT_1: 13;

      then ( rng f2) meets ( rng g2) by A39, A37, A24, A26, A25, A22, A23, A40, A27, A34, A33, A31, A32, A29, A35, A30, Th41, JGRAPH_6: 79;

      then

       A43: (( rng f2) /\ ( rng g2)) <> {} by XBOOLE_0:def 7;

      then x in ( rng g2) by XBOOLE_0:def 4;

      then

      consider z2 be object such that

       A44: z2 in ( dom g2) and

       A45: x = (g2 . z2) by FUNCT_1:def 3;

      

       A46: x = (h . (g . z2)) by A42, A44, A45, FUNCT_1: 13;

      

       A47: (g . z2) in ( rng g) by A42, A44, FUNCT_1:def 3;

      (g . z2) in the carrier of ( TOP-REAL 2) by A44, FUNCT_2: 5;

      then

       A48: (g . z2) in ( dom h) by FUNCT_2:def 1;

      x in ( rng f2) by A43, XBOOLE_0:def 4;

      then

      consider z1 be object such that

       A49: z1 in ( dom f2) and

       A50: x = (f2 . z1) by FUNCT_1:def 3;

      

       A51: (f . z1) in ( rng f) by A36, A49, FUNCT_1:def 3;

      (f . z1) in the carrier of ( TOP-REAL 2) by A49, FUNCT_2: 5;

      then

       A52: (f . z1) in ( dom h) by FUNCT_2:def 1;

      x = (h . (f . z1)) by A36, A49, A50, FUNCT_1: 13;

      then (f . z1) = (g . z2) by A46, A52, A48, A41, FUNCT_1:def 4;

      hence thesis by A51, A47, XBOOLE_0: 3;

    end;

    theorem :: JGRAPH_7:123

    for p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real, P,Q be Subset of ( TOP-REAL 2) st a < b & c < d & (p1 `2 ) = d & (p2 `1 ) = b & (p3 `1 ) = b & (p4 `2 ) = c & a <= (p1 `1 ) & (p1 `1 ) <= b & d >= (p2 `2 ) & (p2 `2 ) > (p3 `2 ) & (p3 `2 ) >= c & a < (p4 `1 ) & (p4 `1 ) <= b & P is_an_arc_of (p1,p3) & Q is_an_arc_of (p2,p4) & P c= ( closed_inside_of_rectangle (a,b,c,d)) & Q c= ( closed_inside_of_rectangle (a,b,c,d)) holds P meets Q

    proof

      let p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real, P,Q be Subset of ( TOP-REAL 2);

      assume that

       A1: a < b and

       A2: c < d and

       A3: (p1 `2 ) = d and

       A4: (p2 `1 ) = b and

       A5: (p3 `1 ) = b and

       A6: (p4 `2 ) = c and

       A7: a <= (p1 `1 ) and

       A8: (p1 `1 ) <= b and

       A9: d >= (p2 `2 ) and

       A10: (p2 `2 ) > (p3 `2 ) and

       A11: (p3 `2 ) >= c and

       A12: a < (p4 `1 ) and

       A13: (p4 `1 ) <= b and

       A14: P is_an_arc_of (p1,p3) and

       A15: Q is_an_arc_of (p2,p4) and

       A16: P c= ( closed_inside_of_rectangle (a,b,c,d)) and

       A17: Q c= ( closed_inside_of_rectangle (a,b,c,d));

      

       A18: ex g be Function of I[01] , ( TOP-REAL 2) st g is continuous one-to-one & ( rng g) = Q & (g . 0 ) = p2 & (g . 1) = p4 by A15, Th2;

      ex f be Function of I[01] , ( TOP-REAL 2) st f is continuous one-to-one & ( rng f) = P & (f . 0 ) = p1 & (f . 1) = p3 by A14, Th2;

      hence thesis by A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, A12, A13, A16, A17, A18, Th122;

    end;

    theorem :: JGRAPH_7:124

    

     Th124: for p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real, f,g be Function of I[01] , ( TOP-REAL 2) st a < b & c < d & (p1 `2 ) = d & (p2 `1 ) = b & (p3 `2 ) = c & (p4 `2 ) = c & a <= (p1 `1 ) & (p1 `1 ) <= b & c <= (p2 `2 ) & (p2 `2 ) <= d & a < (p4 `1 ) & (p4 `1 ) < (p3 `1 ) & (p3 `1 ) <= b & (f . 0 ) = p1 & (f . 1) = p3 & (g . 0 ) = p2 & (g . 1) = p4 & f is continuous one-to-one & g is continuous one-to-one & ( rng f) c= ( closed_inside_of_rectangle (a,b,c,d)) & ( rng g) c= ( closed_inside_of_rectangle (a,b,c,d)) holds ( rng f) meets ( rng g)

    proof

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

      let p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real, f,g be Function of I[01] , ( TOP-REAL 2);

      assume that

       A1: a < b and

       A2: c < d and

       A3: (p1 `2 ) = d and

       A4: (p2 `1 ) = b and

       A5: (p3 `2 ) = c and

       A6: (p4 `2 ) = c and

       A7: a <= (p1 `1 ) and

       A8: (p1 `1 ) <= b and

       A9: c <= (p2 `2 ) and

       A10: (p2 `2 ) <= d and

       A11: a < (p4 `1 ) and

       A12: (p4 `1 ) < (p3 `1 ) and

       A13: (p3 `1 ) <= b and

       A14: (f . 0 ) = p1 and

       A15: (f . 1) = p3 and

       A16: (g . 0 ) = p2 and

       A17: (g . 1) = p4 and

       A18: f is continuous one-to-one and

       A19: g is continuous one-to-one and

       A20: ( rng f) c= ( closed_inside_of_rectangle (a,b,c,d)) and

       A21: ( rng g) c= ( closed_inside_of_rectangle (a,b,c,d));

      set A = (2 / (b - a)), B = ( - ((b + a) / (b - a))), C = (2 / (d - c)), D = ( - ((d + c) / (d - c)));

      set h = ( AffineMap (A,B,C,D));

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

      

       A22: ( - 1) <= ((g2 . O) `2 ) by A1, A2, A4, A9, A10, A16, Th66;

      

       A23: ((g2 . O) `2 ) <= 1 by A1, A2, A4, A9, A10, A16, Th66;

      

       A24: g2 is continuous one-to-one by A1, A2, A19, Th53;

      

       A25: ((g2 . I) `2 ) = ( - 1) by A2, A6, A17, Th57;

      

       A26: ((g2 . O) `1 ) = 1 by A1, A4, A16, Th56;

      

       A27: ( rng g2) c= ( closed_inside_of_rectangle (( - 1),1,( - 1),1)) by A1, A2, A21, Th52;

      (d - c) > 0 by A2, XREAL_1: 50;

      then

       A28: C > 0 by XREAL_1: 139;

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

      

       A29: (p3 `1 ) > a by A11, A12, XXREAL_0: 2;

      then

       A30: ((f2 . I) `1 ) <= 1 by A1, A13, A15, Th64;

      

       A31: ((f2 . I) `2 ) = ( - 1) by A2, A5, A15, Th57;

      

       A32: ((f2 . O) `2 ) = 1 by A2, A3, A14, Th55;

      

       A33: f2 is continuous one-to-one by A1, A2, A18, Th53;

      set x = the Element of (( rng f2) /\ ( rng g2));

      

       A34: ( dom g) = the carrier of I[01] by FUNCT_2:def 1;

      then

       A35: (h . p4) = (g2 . I) by A17, FUNCT_1: 13;

      (b - a) > 0 by A1, XREAL_1: 50;

      then

       A36: A > 0 by XREAL_1: 139;

      then

       A37: ((h . p4) `1 ) < ((h . p3) `1 ) by A12, A28, Th50;

      (p4 `1 ) < b by A12, A13, XXREAL_0: 2;

      then

       A38: ( - 1) < ((g2 . I) `1 ) by A1, A2, A6, A11, A17, Th66;

      h is being_homeomorphism by A36, A28, Th50;

      then

       A39: h is one-to-one by TOPS_2:def 5;

      

       A40: ((f . I) `2 ) = c by A5, A15;

      then

       A41: ( - 1) <= ((f2 . O) `1 ) by A1, A7, A8, A13, A14, A15, A29, Th64;

      

       A42: ((f2 . O) `1 ) <= 1 by A1, A7, A8, A13, A14, A15, A29, A40, Th64;

      

       A43: ( rng f2) c= ( closed_inside_of_rectangle (( - 1),1,( - 1),1)) by A1, A2, A20, Th52;

      

       A44: ( dom f) = the carrier of I[01] by FUNCT_2:def 1;

      then (h . p3) = (f2 . I) by A15, FUNCT_1: 13;

      then ( rng f2) meets ( rng g2) by A37, A35, A33, A32, A31, A41, A42, A30, A43, A24, A26, A25, A22, A23, A38, A27, Th42, JGRAPH_6: 79;

      then

       A45: (( rng f2) /\ ( rng g2)) <> {} by XBOOLE_0:def 7;

      then x in ( rng g2) by XBOOLE_0:def 4;

      then

      consider z2 be object such that

       A46: z2 in ( dom g2) and

       A47: x = (g2 . z2) by FUNCT_1:def 3;

      

       A48: x = (h . (g . z2)) by A34, A46, A47, FUNCT_1: 13;

      

       A49: (g . z2) in ( rng g) by A34, A46, FUNCT_1:def 3;

      (g . z2) in the carrier of ( TOP-REAL 2) by A46, FUNCT_2: 5;

      then

       A50: (g . z2) in ( dom h) by FUNCT_2:def 1;

      x in ( rng f2) by A45, XBOOLE_0:def 4;

      then

      consider z1 be object such that

       A51: z1 in ( dom f2) and

       A52: x = (f2 . z1) by FUNCT_1:def 3;

      

       A53: (f . z1) in ( rng f) by A44, A51, FUNCT_1:def 3;

      (f . z1) in the carrier of ( TOP-REAL 2) by A51, FUNCT_2: 5;

      then

       A54: (f . z1) in ( dom h) by FUNCT_2:def 1;

      x = (h . (f . z1)) by A44, A51, A52, FUNCT_1: 13;

      then (f . z1) = (g . z2) by A48, A54, A50, A39, FUNCT_1:def 4;

      hence thesis by A53, A49, XBOOLE_0: 3;

    end;

    theorem :: JGRAPH_7:125

    for p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real, P,Q be Subset of ( TOP-REAL 2) st a < b & c < d & (p1 `2 ) = d & (p2 `1 ) = b & (p3 `2 ) = c & (p4 `2 ) = c & a <= (p1 `1 ) & (p1 `1 ) <= b & c <= (p2 `2 ) & (p2 `2 ) <= d & a < (p4 `1 ) & (p4 `1 ) < (p3 `1 ) & (p3 `1 ) <= b & P is_an_arc_of (p1,p3) & Q is_an_arc_of (p2,p4) & P c= ( closed_inside_of_rectangle (a,b,c,d)) & Q c= ( closed_inside_of_rectangle (a,b,c,d)) holds P meets Q

    proof

      let p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real, P,Q be Subset of ( TOP-REAL 2);

      assume that

       A1: a < b and

       A2: c < d and

       A3: (p1 `2 ) = d and

       A4: (p2 `1 ) = b and

       A5: (p3 `2 ) = c and

       A6: (p4 `2 ) = c and

       A7: a <= (p1 `1 ) and

       A8: (p1 `1 ) <= b and

       A9: c <= (p2 `2 ) and

       A10: (p2 `2 ) <= d and

       A11: a < (p4 `1 ) and

       A12: (p4 `1 ) < (p3 `1 ) and

       A13: (p3 `1 ) <= b and

       A14: P is_an_arc_of (p1,p3) and

       A15: Q is_an_arc_of (p2,p4) and

       A16: P c= ( closed_inside_of_rectangle (a,b,c,d)) and

       A17: Q c= ( closed_inside_of_rectangle (a,b,c,d));

      

       A18: ex g be Function of I[01] , ( TOP-REAL 2) st g is continuous one-to-one & ( rng g) = Q & (g . 0 ) = p2 & (g . 1) = p4 by A15, Th2;

      ex f be Function of I[01] , ( TOP-REAL 2) st f is continuous one-to-one & ( rng f) = P & (f . 0 ) = p1 & (f . 1) = p3 by A14, Th2;

      hence thesis by A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, A12, A13, A16, A17, A18, Th124;

    end;

    theorem :: JGRAPH_7:126

    

     Th126: for p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real, f,g be Function of I[01] , ( TOP-REAL 2) st a < b & c < d & (p1 `2 ) = d & (p2 `2 ) = c & (p3 `2 ) = c & (p4 `2 ) = c & a <= (p1 `1 ) & (p1 `1 ) <= b & a < (p4 `1 ) & (p4 `1 ) < (p3 `1 ) & (p3 `1 ) < (p2 `1 ) & (p2 `1 ) <= b & (f . 0 ) = p1 & (f . 1) = p3 & (g . 0 ) = p2 & (g . 1) = p4 & f is continuous one-to-one & g is continuous one-to-one & ( rng f) c= ( closed_inside_of_rectangle (a,b,c,d)) & ( rng g) c= ( closed_inside_of_rectangle (a,b,c,d)) holds ( rng f) meets ( rng g)

    proof

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

      let p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real, f,g be Function of I[01] , ( TOP-REAL 2);

      assume that

       A1: a < b and

       A2: c < d and

       A3: (p1 `2 ) = d and

       A4: (p2 `2 ) = c and

       A5: (p3 `2 ) = c and

       A6: (p4 `2 ) = c and

       A7: a <= (p1 `1 ) and

       A8: (p1 `1 ) <= b and

       A9: a < (p4 `1 ) and

       A10: (p4 `1 ) < (p3 `1 ) and

       A11: (p3 `1 ) < (p2 `1 ) and

       A12: (p2 `1 ) <= b and

       A13: (f . 0 ) = p1 and

       A14: (f . 1) = p3 and

       A15: (g . 0 ) = p2 and

       A16: (g . 1) = p4 and

       A17: f is continuous one-to-one and

       A18: g is continuous one-to-one and

       A19: ( rng f) c= ( closed_inside_of_rectangle (a,b,c,d)) and

       A20: ( rng g) c= ( closed_inside_of_rectangle (a,b,c,d));

      set A = (2 / (b - a)), B = ( - ((b + a) / (b - a))), C = (2 / (d - c)), D = ( - ((d + c) / (d - c)));

      set h = ( AffineMap (A,B,C,D));

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

      

       A21: (p2 `1 ) > (p4 `1 ) by A10, A11, XXREAL_0: 2;

      ((g . O) `2 ) = c by A4, A15;

      then

       A22: ((g2 . I) `1 ) > ( - 1) by A1, A9, A12, A15, A16, A21, Th67;

      (d - c) > 0 by A2, XREAL_1: 50;

      then

       A23: C > 0 by XREAL_1: 139;

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

      

       A24: b > (p3 `1 ) by A11, A12, XXREAL_0: 2;

      ((g . I) `2 ) = c by A6, A16;

      then

       A25: 1 >= ((g2 . O) `1 ) by A1, A9, A12, A15, A16, A21, Th67;

      

       A26: ((g2 . O) `2 ) = ( - 1) by A2, A4, A15, Th57;

      

       A27: g2 is continuous one-to-one by A1, A2, A18, Th53;

      

       A28: ((f2 . O) `2 ) = 1 by A2, A3, A13, Th55;

      

       A29: f2 is continuous one-to-one by A1, A2, A17, Th53;

      set x = the Element of (( rng f2) /\ ( rng g2));

      

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

      then

       A31: (h . p3) = (f2 . I) by A14, FUNCT_1: 13;

      

       A32: ( rng g2) c= ( closed_inside_of_rectangle (( - 1),1,( - 1),1)) by A1, A2, A20, Th52;

      

       A33: ((g2 . I) `2 ) = ( - 1) by A2, A6, A16, Th57;

      (b - a) > 0 by A1, XREAL_1: 50;

      then

       A34: A > 0 by XREAL_1: 139;

      then h is being_homeomorphism by A23, Th51;

      then

       A35: h is one-to-one by TOPS_2:def 5;

      

       A36: ((f . I) `2 ) = c by A5, A14;

      

       A37: ( dom g) = the carrier of I[01] by FUNCT_2:def 1;

      then (h . p2) = (g2 . O) by A15, FUNCT_1: 13;

      then

       A38: ((g2 . O) `1 ) > ((f2 . I) `1 ) by A11, A34, A23, A31, Th50;

      

       A39: (p3 `1 ) > a by A9, A10, XXREAL_0: 2;

      then

       A40: ((f2 . O) `1 ) <= 1 by A1, A7, A8, A13, A14, A24, A36, Th64;

      

       A41: ( rng f2) c= ( closed_inside_of_rectangle (( - 1),1,( - 1),1)) by A1, A2, A19, Th52;

      

       A42: ((f2 . I) `2 ) = ( - 1) by A2, A5, A14, Th57;

      (h . p4) = (g2 . I) by A16, A37, FUNCT_1: 13;

      then

       A43: ((g2 . I) `1 ) < ((f2 . I) `1 ) by A10, A34, A23, A31, Th50;

      ( - 1) <= ((f2 . O) `1 ) by A1, A7, A8, A13, A14, A39, A24, A36, Th64;

      then ( rng f2) meets ( rng g2) by A29, A28, A42, A40, A41, A27, A26, A33, A25, A22, A32, A38, A43, Th43, JGRAPH_6: 79;

      then

       A44: (( rng f2) /\ ( rng g2)) <> {} by XBOOLE_0:def 7;

      then x in ( rng g2) by XBOOLE_0:def 4;

      then

      consider z2 be object such that

       A45: z2 in ( dom g2) and

       A46: x = (g2 . z2) by FUNCT_1:def 3;

      

       A47: x = (h . (g . z2)) by A37, A45, A46, FUNCT_1: 13;

      

       A48: (g . z2) in ( rng g) by A37, A45, FUNCT_1:def 3;

      (g . z2) in the carrier of ( TOP-REAL 2) by A45, FUNCT_2: 5;

      then

       A49: (g . z2) in ( dom h) by FUNCT_2:def 1;

      x in ( rng f2) by A44, XBOOLE_0:def 4;

      then

      consider z1 be object such that

       A50: z1 in ( dom f2) and

       A51: x = (f2 . z1) by FUNCT_1:def 3;

      

       A52: (f . z1) in ( rng f) by A30, A50, FUNCT_1:def 3;

      (f . z1) in the carrier of ( TOP-REAL 2) by A50, FUNCT_2: 5;

      then

       A53: (f . z1) in ( dom h) by FUNCT_2:def 1;

      x = (h . (f . z1)) by A30, A50, A51, FUNCT_1: 13;

      then (f . z1) = (g . z2) by A47, A53, A49, A35, FUNCT_1:def 4;

      hence thesis by A52, A48, XBOOLE_0: 3;

    end;

    theorem :: JGRAPH_7:127

    for p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real, P,Q be Subset of ( TOP-REAL 2) st a < b & c < d & (p1 `2 ) = d & (p2 `2 ) = c & (p3 `2 ) = c & (p4 `2 ) = c & a <= (p1 `1 ) & (p1 `1 ) <= b & a < (p4 `1 ) & (p4 `1 ) < (p3 `1 ) & (p3 `1 ) < (p2 `1 ) & (p2 `1 ) <= b & P is_an_arc_of (p1,p3) & Q is_an_arc_of (p2,p4) & P c= ( closed_inside_of_rectangle (a,b,c,d)) & Q c= ( closed_inside_of_rectangle (a,b,c,d)) holds P meets Q

    proof

      let p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real, P,Q be Subset of ( TOP-REAL 2);

      assume that

       A1: a < b and

       A2: c < d and

       A3: (p1 `2 ) = d and

       A4: (p2 `2 ) = c and

       A5: (p3 `2 ) = c and

       A6: (p4 `2 ) = c and

       A7: a <= (p1 `1 ) and

       A8: (p1 `1 ) <= b and

       A9: a < (p4 `1 ) and

       A10: (p4 `1 ) < (p3 `1 ) and

       A11: (p3 `1 ) < (p2 `1 ) and

       A12: (p2 `1 ) <= b and

       A13: P is_an_arc_of (p1,p3) and

       A14: Q is_an_arc_of (p2,p4) and

       A15: P c= ( closed_inside_of_rectangle (a,b,c,d)) and

       A16: Q c= ( closed_inside_of_rectangle (a,b,c,d));

      

       A17: ex g be Function of I[01] , ( TOP-REAL 2) st g is continuous one-to-one & ( rng g) = Q & (g . 0 ) = p2 & (g . 1) = p4 by A14, Th2;

      ex f be Function of I[01] , ( TOP-REAL 2) st f is continuous one-to-one & ( rng f) = P & (f . 0 ) = p1 & (f . 1) = p3 by A13, Th2;

      hence thesis by A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, A12, A15, A16, A17, Th126;

    end;

    theorem :: JGRAPH_7:128

    

     Th128: for p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real, f,g be Function of I[01] , ( TOP-REAL 2) st a < b & c < d & (p1 `1 ) = b & (p2 `1 ) = b & (p3 `1 ) = b & (p4 `1 ) = b & d >= (p1 `2 ) & (p1 `2 ) > (p2 `2 ) & (p2 `2 ) > (p3 `2 ) & (p3 `2 ) > (p4 `2 ) & (p4 `2 ) >= c & (f . 0 ) = p1 & (f . 1) = p3 & (g . 0 ) = p2 & (g . 1) = p4 & f is continuous one-to-one & g is continuous one-to-one & ( rng f) c= ( closed_inside_of_rectangle (a,b,c,d)) & ( rng g) c= ( closed_inside_of_rectangle (a,b,c,d)) holds ( rng f) meets ( rng g)

    proof

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

      let p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real, f,g be Function of I[01] , ( TOP-REAL 2);

      assume that

       A1: a < b and

       A2: c < d and

       A3: (p1 `1 ) = b and

       A4: (p2 `1 ) = b and

       A5: (p3 `1 ) = b and

       A6: (p4 `1 ) = b and

       A7: d >= (p1 `2 ) and

       A8: (p1 `2 ) > (p2 `2 ) and

       A9: (p2 `2 ) > (p3 `2 ) and

       A10: (p3 `2 ) > (p4 `2 ) and

       A11: (p4 `2 ) >= c and

       A12: (f . 0 ) = p1 and

       A13: (f . 1) = p3 and

       A14: (g . 0 ) = p2 and

       A15: (g . 1) = p4 and

       A16: f is continuous one-to-one and

       A17: g is continuous one-to-one and

       A18: ( rng f) c= ( closed_inside_of_rectangle (a,b,c,d)) and

       A19: ( rng g) c= ( closed_inside_of_rectangle (a,b,c,d));

      set A = (2 / (b - a)), B = ( - ((b + a) / (b - a))), C = (2 / (d - c)), D = ( - ((d + c) / (d - c)));

      set h = ( AffineMap (A,B,C,D));

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

      

       A20: g2 is continuous one-to-one by A1, A2, A17, Th53;

      

       A21: ((g . O) `1 ) = b by A4, A14;

      

       A22: d > (p2 `2 ) by A7, A8, XXREAL_0: 2;

      (p2 `2 ) > (p4 `2 ) by A9, A10, XXREAL_0: 2;

      then

       A23: ((g2 . I) `2 ) >= ( - 1) by A2, A11, A14, A15, A22, A21, Th65;

      

       A24: ( rng g2) c= ( closed_inside_of_rectangle (( - 1),1,( - 1),1)) by A1, A2, A19, Th52;

      

       A25: ((g2 . I) `1 ) = 1 by A1, A6, A15, Th56;

      

       A26: ((g2 . O) `1 ) = 1 by A1, A4, A14, Th56;

      

       A27: ( dom g) = the carrier of I[01] by FUNCT_2:def 1;

      then

       A28: (h . p2) = (g2 . O) by A14, FUNCT_1: 13;

      

       A29: (h . p4) = (g2 . I) by A15, A27, FUNCT_1: 13;

      (d - c) > 0 by A2, XREAL_1: 50;

      then

       A30: C > 0 by XREAL_1: 139;

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

      

       A31: f2 is continuous one-to-one by A1, A2, A16, Th53;

      

       A32: ( rng f2) c= ( closed_inside_of_rectangle (( - 1),1,( - 1),1)) by A1, A2, A18, Th52;

      

       A33: ((f2 . I) `1 ) = 1 by A1, A5, A13, Th56;

      

       A34: ((f . I) `1 ) = b by A5, A13;

      

       A35: (p3 `2 ) > c by A10, A11, XXREAL_0: 2;

      (p1 `2 ) > (p3 `2 ) by A8, A9, XXREAL_0: 2;

      then

       A36: 1 >= ((f2 . O) `2 ) by A2, A7, A12, A13, A35, A34, Th65;

      

       A37: ((f2 . O) `1 ) = 1 by A1, A3, A12, Th56;

      set x = the Element of (( rng f2) /\ ( rng g2));

      

       A38: ( dom f) = the carrier of I[01] by FUNCT_2:def 1;

      then

       A39: (h . p3) = (f2 . I) by A13, FUNCT_1: 13;

      (b - a) > 0 by A1, XREAL_1: 50;

      then

       A40: A > 0 by XREAL_1: 139;

      then

       A41: ((h . p1) `2 ) > ((h . p2) `2 ) by A8, A30, Th51;

      

       A42: ((h . p3) `2 ) > ((h . p4) `2 ) by A10, A40, A30, Th51;

      

       A43: ((h . p2) `2 ) > ((h . p3) `2 ) by A9, A40, A30, Th51;

      (h . p1) = (f2 . O) by A12, A38, FUNCT_1: 13;

      then ((f2 . O),(g2 . O),(f2 . I),(g2 . I)) are_in_this_order_on ( rectangle (( - 1),1,( - 1),1)) by A41, A43, A42, A28, A39, A29, A37, A33, A36, A26, A25, A23, Th44;

      then ( rng f2) meets ( rng g2) by A31, A32, A20, A24, JGRAPH_6: 79;

      then

       A44: (( rng f2) /\ ( rng g2)) <> {} by XBOOLE_0:def 7;

      then x in ( rng g2) by XBOOLE_0:def 4;

      then

      consider z2 be object such that

       A45: z2 in ( dom g2) and

       A46: x = (g2 . z2) by FUNCT_1:def 3;

      

       A47: x = (h . (g . z2)) by A27, A45, A46, FUNCT_1: 13;

      h is being_homeomorphism by A40, A30, Th51;

      then

       A48: h is one-to-one by TOPS_2:def 5;

      x in ( rng f2) by A44, XBOOLE_0:def 4;

      then

      consider z1 be object such that

       A49: z1 in ( dom f2) and

       A50: x = (f2 . z1) by FUNCT_1:def 3;

      

       A51: (f . z1) in ( rng f) by A38, A49, FUNCT_1:def 3;

      

       A52: (g . z2) in ( rng g) by A27, A45, FUNCT_1:def 3;

      (g . z2) in the carrier of ( TOP-REAL 2) by A45, FUNCT_2: 5;

      then

       A53: (g . z2) in ( dom h) by FUNCT_2:def 1;

      (f . z1) in the carrier of ( TOP-REAL 2) by A49, FUNCT_2: 5;

      then

       A54: (f . z1) in ( dom h) by FUNCT_2:def 1;

      x = (h . (f . z1)) by A38, A49, A50, FUNCT_1: 13;

      then (f . z1) = (g . z2) by A47, A54, A53, A48, FUNCT_1:def 4;

      hence thesis by A51, A52, XBOOLE_0: 3;

    end;

    theorem :: JGRAPH_7:129

    for p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real, P,Q be Subset of ( TOP-REAL 2) st a < b & c < d & (p1 `1 ) = b & (p2 `1 ) = b & (p3 `1 ) = b & (p4 `1 ) = b & d >= (p1 `2 ) & (p1 `2 ) > (p2 `2 ) & (p2 `2 ) > (p3 `2 ) & (p3 `2 ) > (p4 `2 ) & (p4 `2 ) >= c & P is_an_arc_of (p1,p3) & Q is_an_arc_of (p2,p4) & P c= ( closed_inside_of_rectangle (a,b,c,d)) & Q c= ( closed_inside_of_rectangle (a,b,c,d)) holds P meets Q

    proof

      let p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real, P,Q be Subset of ( TOP-REAL 2);

      assume that

       A1: a < b and

       A2: c < d and

       A3: (p1 `1 ) = b and

       A4: (p2 `1 ) = b and

       A5: (p3 `1 ) = b and

       A6: (p4 `1 ) = b and

       A7: d >= (p1 `2 ) and

       A8: (p1 `2 ) > (p2 `2 ) and

       A9: (p2 `2 ) > (p3 `2 ) and

       A10: (p3 `2 ) > (p4 `2 ) and

       A11: (p4 `2 ) >= c and

       A12: P is_an_arc_of (p1,p3) and

       A13: Q is_an_arc_of (p2,p4) and

       A14: P c= ( closed_inside_of_rectangle (a,b,c,d)) and

       A15: Q c= ( closed_inside_of_rectangle (a,b,c,d));

      

       A16: ex g be Function of I[01] , ( TOP-REAL 2) st g is continuous one-to-one & ( rng g) = Q & (g . 0 ) = p2 & (g . 1) = p4 by A13, Th2;

      ex f be Function of I[01] , ( TOP-REAL 2) st f is continuous one-to-one & ( rng f) = P & (f . 0 ) = p1 & (f . 1) = p3 by A12, Th2;

      hence thesis by A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, A14, A15, A16, Th128;

    end;

    theorem :: JGRAPH_7:130

    

     Th130: for p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real, f,g be Function of I[01] , ( TOP-REAL 2) st a < b & c < d & (p1 `1 ) = b & (p2 `1 ) = b & (p3 `1 ) = b & (p4 `2 ) = c & d >= (p1 `2 ) & (p1 `2 ) > (p2 `2 ) & (p2 `2 ) > (p3 `2 ) & (p3 `2 ) >= c & a < (p4 `1 ) & (p4 `1 ) <= b & (f . 0 ) = p1 & (f . 1) = p3 & (g . 0 ) = p2 & (g . 1) = p4 & f is continuous one-to-one & g is continuous one-to-one & ( rng f) c= ( closed_inside_of_rectangle (a,b,c,d)) & ( rng g) c= ( closed_inside_of_rectangle (a,b,c,d)) holds ( rng f) meets ( rng g)

    proof

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

      let p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real, f,g be Function of I[01] , ( TOP-REAL 2);

      assume that

       A1: a < b and

       A2: c < d and

       A3: (p1 `1 ) = b and

       A4: (p2 `1 ) = b and

       A5: (p3 `1 ) = b and

       A6: (p4 `2 ) = c and

       A7: d >= (p1 `2 ) and

       A8: (p1 `2 ) > (p2 `2 ) and

       A9: (p2 `2 ) > (p3 `2 ) and

       A10: (p3 `2 ) >= c and

       A11: a < (p4 `1 ) and

       A12: (p4 `1 ) <= b and

       A13: (f . 0 ) = p1 and

       A14: (f . 1) = p3 and

       A15: (g . 0 ) = p2 and

       A16: (g . 1) = p4 and

       A17: f is continuous one-to-one and

       A18: g is continuous one-to-one and

       A19: ( rng f) c= ( closed_inside_of_rectangle (a,b,c,d)) and

       A20: ( rng g) c= ( closed_inside_of_rectangle (a,b,c,d));

      set A = (2 / (b - a)), B = ( - ((b + a) / (b - a))), C = (2 / (d - c)), D = ( - ((d + c) / (d - c)));

      set h = ( AffineMap (A,B,C,D));

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

      

       A21: (p1 `2 ) > (p3 `2 ) by A8, A9, XXREAL_0: 2;

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

      

       A22: ( - 1) < ((g2 . I) `1 ) by A1, A2, A6, A11, A12, A16, Th66;

      

       A23: ((g2 . I) `2 ) = ( - 1) by A2, A6, A16, Th57;

      

       A24: ((g2 . O) `1 ) = 1 by A1, A4, A15, Th56;

      ((f . I) `1 ) = b by A5, A14;

      then

       A25: 1 >= ((f2 . O) `2 ) by A2, A7, A10, A13, A14, A21, Th65;

      

       A26: g2 is continuous one-to-one by A1, A2, A18, Th53;

      set x = the Element of (( rng f2) /\ ( rng g2));

      

       A27: ( dom f) = the carrier of I[01] by FUNCT_2:def 1;

      then

       A28: (h . p3) = (f2 . I) by A14, FUNCT_1: 13;

      (d - c) > 0 by A2, XREAL_1: 50;

      then

       A29: C > 0 by XREAL_1: 139;

      (b - a) > 0 by A1, XREAL_1: 50;

      then

       A30: A > 0 by XREAL_1: 139;

      then

       A31: ((h . p1) `2 ) > ((h . p2) `2 ) by A8, A29, Th51;

      ((f . O) `1 ) = b by A3, A13;

      then

       A32: ((f2 . I) `2 ) >= ( - 1) by A2, A7, A10, A13, A14, A21, Th65;

      h is being_homeomorphism by A30, A29, Th51;

      then

       A33: h is one-to-one by TOPS_2:def 5;

      

       A34: ( rng f2) c= ( closed_inside_of_rectangle (( - 1),1,( - 1),1)) by A1, A2, A19, Th52;

      

       A35: ((f2 . I) `1 ) = 1 by A1, A5, A14, Th56;

      

       A36: ((f2 . O) `1 ) = 1 by A1, A3, A13, Th56;

      

       A37: f2 is continuous one-to-one by A1, A2, A17, Th53;

      

       A38: ( rng g2) c= ( closed_inside_of_rectangle (( - 1),1,( - 1),1)) by A1, A2, A20, Th52;

      

       A39: ((h . p2) `2 ) > ((h . p3) `2 ) by A9, A30, A29, Th51;

      

       A40: ((g2 . I) `1 ) <= 1 by A1, A2, A6, A11, A12, A16, Th66;

      

       A41: ( dom g) = the carrier of I[01] by FUNCT_2:def 1;

      then

       A42: (h . p2) = (g2 . O) by A15, FUNCT_1: 13;

      (h . p1) = (f2 . O) by A13, A27, FUNCT_1: 13;

      then ( rng f2) meets ( rng g2) by A31, A39, A42, A28, A37, A36, A35, A25, A32, A34, A26, A24, A23, A22, A40, A38, Th45, JGRAPH_6: 79;

      then

       A43: (( rng f2) /\ ( rng g2)) <> {} by XBOOLE_0:def 7;

      then x in ( rng g2) by XBOOLE_0:def 4;

      then

      consider z2 be object such that

       A44: z2 in ( dom g2) and

       A45: x = (g2 . z2) by FUNCT_1:def 3;

      

       A46: x = (h . (g . z2)) by A41, A44, A45, FUNCT_1: 13;

      

       A47: (g . z2) in ( rng g) by A41, A44, FUNCT_1:def 3;

      (g . z2) in the carrier of ( TOP-REAL 2) by A44, FUNCT_2: 5;

      then

       A48: (g . z2) in ( dom h) by FUNCT_2:def 1;

      x in ( rng f2) by A43, XBOOLE_0:def 4;

      then

      consider z1 be object such that

       A49: z1 in ( dom f2) and

       A50: x = (f2 . z1) by FUNCT_1:def 3;

      

       A51: (f . z1) in ( rng f) by A27, A49, FUNCT_1:def 3;

      (f . z1) in the carrier of ( TOP-REAL 2) by A49, FUNCT_2: 5;

      then

       A52: (f . z1) in ( dom h) by FUNCT_2:def 1;

      x = (h . (f . z1)) by A27, A49, A50, FUNCT_1: 13;

      then (f . z1) = (g . z2) by A46, A52, A48, A33, FUNCT_1:def 4;

      hence thesis by A51, A47, XBOOLE_0: 3;

    end;

    theorem :: JGRAPH_7:131

    for p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real, P,Q be Subset of ( TOP-REAL 2) st a < b & c < d & (p1 `1 ) = b & (p2 `1 ) = b & (p3 `1 ) = b & (p4 `2 ) = c & d >= (p1 `2 ) & (p1 `2 ) > (p2 `2 ) & (p2 `2 ) > (p3 `2 ) & (p3 `2 ) >= c & a < (p4 `1 ) & (p4 `1 ) <= b & P is_an_arc_of (p1,p3) & Q is_an_arc_of (p2,p4) & P c= ( closed_inside_of_rectangle (a,b,c,d)) & Q c= ( closed_inside_of_rectangle (a,b,c,d)) holds P meets Q

    proof

      let p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real, P,Q be Subset of ( TOP-REAL 2);

      assume that

       A1: a < b and

       A2: c < d and

       A3: (p1 `1 ) = b and

       A4: (p2 `1 ) = b and

       A5: (p3 `1 ) = b and

       A6: (p4 `2 ) = c and

       A7: d >= (p1 `2 ) and

       A8: (p1 `2 ) > (p2 `2 ) and

       A9: (p2 `2 ) > (p3 `2 ) and

       A10: (p3 `2 ) >= c and

       A11: a < (p4 `1 ) and

       A12: (p4 `1 ) <= b and

       A13: P is_an_arc_of (p1,p3) and

       A14: Q is_an_arc_of (p2,p4) and

       A15: P c= ( closed_inside_of_rectangle (a,b,c,d)) and

       A16: Q c= ( closed_inside_of_rectangle (a,b,c,d));

      

       A17: ex g be Function of I[01] , ( TOP-REAL 2) st g is continuous one-to-one & ( rng g) = Q & (g . 0 ) = p2 & (g . 1) = p4 by A14, Th2;

      ex f be Function of I[01] , ( TOP-REAL 2) st f is continuous one-to-one & ( rng f) = P & (f . 0 ) = p1 & (f . 1) = p3 by A13, Th2;

      hence thesis by A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, A12, A15, A16, A17, Th130;

    end;

    theorem :: JGRAPH_7:132

    

     Th132: for p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real, f,g be Function of I[01] , ( TOP-REAL 2) st a < b & c < d & (p1 `1 ) = b & (p2 `1 ) = b & (p3 `2 ) = c & (p4 `2 ) = c & d >= (p1 `2 ) & (p1 `2 ) > (p2 `2 ) & (p2 `2 ) >= c & b >= (p3 `1 ) & (p3 `1 ) > (p4 `1 ) & (p4 `1 ) > a & (f . 0 ) = p1 & (f . 1) = p3 & (g . 0 ) = p2 & (g . 1) = p4 & f is continuous one-to-one & g is continuous one-to-one & ( rng f) c= ( closed_inside_of_rectangle (a,b,c,d)) & ( rng g) c= ( closed_inside_of_rectangle (a,b,c,d)) holds ( rng f) meets ( rng g)

    proof

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

      let p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real, f,g be Function of I[01] , ( TOP-REAL 2);

      assume that

       A1: a < b and

       A2: c < d and

       A3: (p1 `1 ) = b and

       A4: (p2 `1 ) = b and

       A5: (p3 `2 ) = c and

       A6: (p4 `2 ) = c and

       A7: d >= (p1 `2 ) and

       A8: (p1 `2 ) > (p2 `2 ) and

       A9: (p2 `2 ) >= c and

       A10: b >= (p3 `1 ) and

       A11: (p3 `1 ) > (p4 `1 ) and

       A12: (p4 `1 ) > a and

       A13: (f . 0 ) = p1 and

       A14: (f . 1) = p3 and

       A15: (g . 0 ) = p2 and

       A16: (g . 1) = p4 and

       A17: f is continuous one-to-one and

       A18: g is continuous one-to-one and

       A19: ( rng f) c= ( closed_inside_of_rectangle (a,b,c,d)) and

       A20: ( rng g) c= ( closed_inside_of_rectangle (a,b,c,d));

      set A = (2 / (b - a)), B = ( - ((b + a) / (b - a))), C = (2 / (d - c)), D = ( - ((d + c) / (d - c)));

      set h = ( AffineMap (A,B,C,D));

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

      

       A21: g2 is continuous one-to-one by A1, A2, A18, Th53;

      d > (p2 `2 ) by A7, A8, XXREAL_0: 2;

      then

       A22: ( - 1) <= ((g2 . O) `2 ) by A1, A2, A4, A9, A15, Th66;

      

       A23: ( rng g2) c= ( closed_inside_of_rectangle (( - 1),1,( - 1),1)) by A1, A2, A20, Th52;

      

       A24: ((g2 . I) `2 ) = ( - 1) by A2, A6, A16, Th57;

      (d - c) > 0 by A2, XREAL_1: 50;

      then

       A25: C > 0 by XREAL_1: 139;

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

      

       A26: f2 is continuous one-to-one by A1, A2, A17, Th53;

      (p3 `1 ) > a by A11, A12, XXREAL_0: 2;

      then

       A27: ((f2 . I) `1 ) <= 1 by A1, A2, A5, A10, A14, Th66;

      

       A28: ( rng f2) c= ( closed_inside_of_rectangle (( - 1),1,( - 1),1)) by A1, A2, A19, Th52;

      

       A29: ((f2 . I) `2 ) = ( - 1) by A2, A5, A14, Th57;

      (p1 `2 ) >= c by A8, A9, XXREAL_0: 2;

      then

       A30: ((f2 . O) `2 ) <= 1 by A1, A2, A3, A7, A13, Th66;

      

       A31: ((f2 . O) `1 ) = 1 by A1, A3, A13, Th56;

      set x = the Element of (( rng f2) /\ ( rng g2));

      

       A32: ( dom f) = the carrier of I[01] by FUNCT_2:def 1;

      then

       A33: (h . p3) = (f2 . I) by A14, FUNCT_1: 13;

      (b - a) > 0 by A1, XREAL_1: 50;

      then

       A34: A > 0 by XREAL_1: 139;

      then

       A35: ((h . p1) `2 ) > ((h . p2) `2 ) by A8, A25, Th51;

      b > (p4 `1 ) by A10, A11, XXREAL_0: 2;

      then

       A36: ( - 1) < ((g2 . I) `1 ) by A1, A2, A6, A12, A16, Th66;

      h is being_homeomorphism by A34, A25, Th51;

      then

       A37: h is one-to-one by TOPS_2:def 5;

      

       A38: ((g2 . O) `1 ) = 1 by A1, A4, A15, Th56;

      

       A39: ( dom g) = the carrier of I[01] by FUNCT_2:def 1;

      then

       A40: (h . p2) = (g2 . O) by A15, FUNCT_1: 13;

      

       A41: ((h . p3) `1 ) > ((h . p4) `1 ) by A11, A34, A25, Th50;

      

       A42: (h . p4) = (g2 . I) by A16, A39, FUNCT_1: 13;

      (h . p1) = (f2 . O) by A13, A32, FUNCT_1: 13;

      then ((f2 . O),(g2 . O),(f2 . I),(g2 . I)) are_in_this_order_on ( rectangle (( - 1),1,( - 1),1)) by A35, A41, A40, A33, A42, A31, A29, A30, A27, A38, A24, A22, A36, Th46;

      then ( rng f2) meets ( rng g2) by A26, A28, A21, A23, JGRAPH_6: 79;

      then

       A43: (( rng f2) /\ ( rng g2)) <> {} by XBOOLE_0:def 7;

      then x in ( rng g2) by XBOOLE_0:def 4;

      then

      consider z2 be object such that

       A44: z2 in ( dom g2) and

       A45: x = (g2 . z2) by FUNCT_1:def 3;

      

       A46: x = (h . (g . z2)) by A39, A44, A45, FUNCT_1: 13;

      

       A47: (g . z2) in ( rng g) by A39, A44, FUNCT_1:def 3;

      (g . z2) in the carrier of ( TOP-REAL 2) by A44, FUNCT_2: 5;

      then

       A48: (g . z2) in ( dom h) by FUNCT_2:def 1;

      x in ( rng f2) by A43, XBOOLE_0:def 4;

      then

      consider z1 be object such that

       A49: z1 in ( dom f2) and

       A50: x = (f2 . z1) by FUNCT_1:def 3;

      

       A51: (f . z1) in ( rng f) by A32, A49, FUNCT_1:def 3;

      (f . z1) in the carrier of ( TOP-REAL 2) by A49, FUNCT_2: 5;

      then

       A52: (f . z1) in ( dom h) by FUNCT_2:def 1;

      x = (h . (f . z1)) by A32, A49, A50, FUNCT_1: 13;

      then (f . z1) = (g . z2) by A46, A52, A48, A37, FUNCT_1:def 4;

      hence thesis by A51, A47, XBOOLE_0: 3;

    end;

    theorem :: JGRAPH_7:133

    for p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real, P,Q be Subset of ( TOP-REAL 2) st a < b & c < d & (p1 `1 ) = b & (p2 `1 ) = b & (p3 `2 ) = c & (p4 `2 ) = c & d >= (p1 `2 ) & (p1 `2 ) > (p2 `2 ) & (p2 `2 ) >= c & b >= (p3 `1 ) & (p3 `1 ) > (p4 `1 ) & (p4 `1 ) > a & P is_an_arc_of (p1,p3) & Q is_an_arc_of (p2,p4) & P c= ( closed_inside_of_rectangle (a,b,c,d)) & Q c= ( closed_inside_of_rectangle (a,b,c,d)) holds P meets Q

    proof

      let p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real, P,Q be Subset of ( TOP-REAL 2);

      assume that

       A1: a < b and

       A2: c < d and

       A3: (p1 `1 ) = b and

       A4: (p2 `1 ) = b and

       A5: (p3 `2 ) = c and

       A6: (p4 `2 ) = c and

       A7: d >= (p1 `2 ) and

       A8: (p1 `2 ) > (p2 `2 ) and

       A9: (p2 `2 ) >= c and

       A10: b >= (p3 `1 ) and

       A11: (p3 `1 ) > (p4 `1 ) and

       A12: (p4 `1 ) > a and

       A13: P is_an_arc_of (p1,p3) and

       A14: Q is_an_arc_of (p2,p4) and

       A15: P c= ( closed_inside_of_rectangle (a,b,c,d)) and

       A16: Q c= ( closed_inside_of_rectangle (a,b,c,d));

      

       A17: ex g be Function of I[01] , ( TOP-REAL 2) st g is continuous one-to-one & ( rng g) = Q & (g . 0 ) = p2 & (g . 1) = p4 by A14, Th2;

      ex f be Function of I[01] , ( TOP-REAL 2) st f is continuous one-to-one & ( rng f) = P & (f . 0 ) = p1 & (f . 1) = p3 by A13, Th2;

      hence thesis by A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, A12, A15, A16, A17, Th132;

    end;

    theorem :: JGRAPH_7:134

    

     Th134: for p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real, f,g be Function of I[01] , ( TOP-REAL 2) st a < b & c < d & (p1 `1 ) = b & (p2 `2 ) = c & (p3 `2 ) = c & (p4 `2 ) = c & c <= (p1 `2 ) & (p1 `2 ) <= d & b >= (p2 `1 ) & (p2 `1 ) > (p3 `1 ) & (p3 `1 ) > (p4 `1 ) & (p4 `1 ) > a & (f . 0 ) = p1 & (f . 1) = p3 & (g . 0 ) = p2 & (g . 1) = p4 & f is continuous one-to-one & g is continuous one-to-one & ( rng f) c= ( closed_inside_of_rectangle (a,b,c,d)) & ( rng g) c= ( closed_inside_of_rectangle (a,b,c,d)) holds ( rng f) meets ( rng g)

    proof

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

      let p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real, f,g be Function of I[01] , ( TOP-REAL 2);

      assume that

       A1: a < b and

       A2: c < d and

       A3: (p1 `1 ) = b and

       A4: (p2 `2 ) = c and

       A5: (p3 `2 ) = c and

       A6: (p4 `2 ) = c and

       A7: c <= (p1 `2 ) and

       A8: (p1 `2 ) <= d and

       A9: b >= (p2 `1 ) and

       A10: (p2 `1 ) > (p3 `1 ) and

       A11: (p3 `1 ) > (p4 `1 ) and

       A12: (p4 `1 ) > a and

       A13: (f . 0 ) = p1 and

       A14: (f . 1) = p3 and

       A15: (g . 0 ) = p2 and

       A16: (g . 1) = p4 and

       A17: f is continuous one-to-one and

       A18: g is continuous one-to-one and

       A19: ( rng f) c= ( closed_inside_of_rectangle (a,b,c,d)) and

       A20: ( rng g) c= ( closed_inside_of_rectangle (a,b,c,d));

      set A = (2 / (b - a)), B = ( - ((b + a) / (b - a))), C = (2 / (d - c)), D = ( - ((d + c) / (d - c)));

      set h = ( AffineMap (A,B,C,D));

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

      

       A21: (p2 `1 ) > (p4 `1 ) by A10, A11, XXREAL_0: 2;

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

      

       A22: ( - 1) <= ((f2 . O) `2 ) by A1, A2, A3, A7, A8, A13, Th66;

      

       A23: ((f2 . O) `1 ) = 1 by A1, A3, A13, Th56;

      

       A24: f2 is continuous one-to-one by A1, A2, A17, Th53;

      

       A25: ( rng f2) c= ( closed_inside_of_rectangle (( - 1),1,( - 1),1)) by A1, A2, A19, Th52;

      

       A26: ((f2 . I) `2 ) = ( - 1) by A2, A5, A14, Th57;

      

       A27: ((f2 . O) `2 ) <= 1 by A1, A2, A3, A7, A8, A13, Th66;

      set x = the Element of (( rng f2) /\ ( rng g2));

      

       A28: ( dom f) = the carrier of I[01] by FUNCT_2:def 1;

      then

       A29: (h . p3) = (f2 . I) by A14, FUNCT_1: 13;

      (d - c) > 0 by A2, XREAL_1: 50;

      then

       A30: C > 0 by XREAL_1: 139;

      (b - a) > 0 by A1, XREAL_1: 50;

      then

       A31: A > 0 by XREAL_1: 139;

      then

       A32: ((h . p2) `1 ) > ((h . p3) `1 ) by A10, A30, Th50;

      ((g . O) `2 ) = c by A4, A15;

      then

       A33: ((g2 . I) `1 ) > ( - 1) by A1, A9, A12, A15, A16, A21, Th67;

      h is being_homeomorphism by A31, A30, Th50;

      then

       A34: h is one-to-one by TOPS_2:def 5;

      

       A35: ( rng g2) c= ( closed_inside_of_rectangle (( - 1),1,( - 1),1)) by A1, A2, A20, Th52;

      

       A36: ((h . p3) `1 ) > ((h . p4) `1 ) by A11, A31, A30, Th50;

      

       A37: ((g2 . I) `2 ) = ( - 1) by A2, A6, A16, Th57;

      

       A38: ((g2 . O) `2 ) = ( - 1) by A2, A4, A15, Th57;

      ((g . I) `2 ) = c by A6, A16;

      then

       A39: 1 >= ((g2 . O) `1 ) by A1, A9, A12, A15, A16, A21, Th67;

      

       A40: g2 is continuous one-to-one by A1, A2, A18, Th53;

      

       A41: ( dom g) = the carrier of I[01] by FUNCT_2:def 1;

      then

       A42: (h . p4) = (g2 . I) by A16, FUNCT_1: 13;

      (h . p2) = (g2 . O) by A15, A41, FUNCT_1: 13;

      then ( rng f2) meets ( rng g2) by A32, A36, A29, A42, A24, A23, A26, A22, A27, A25, A40, A38, A37, A39, A33, A35, Th47, JGRAPH_6: 79;

      then

       A43: (( rng f2) /\ ( rng g2)) <> {} by XBOOLE_0:def 7;

      then x in ( rng g2) by XBOOLE_0:def 4;

      then

      consider z2 be object such that

       A44: z2 in ( dom g2) and

       A45: x = (g2 . z2) by FUNCT_1:def 3;

      

       A46: x = (h . (g . z2)) by A41, A44, A45, FUNCT_1: 13;

      

       A47: (g . z2) in ( rng g) by A41, A44, FUNCT_1:def 3;

      (g . z2) in the carrier of ( TOP-REAL 2) by A44, FUNCT_2: 5;

      then

       A48: (g . z2) in ( dom h) by FUNCT_2:def 1;

      x in ( rng f2) by A43, XBOOLE_0:def 4;

      then

      consider z1 be object such that

       A49: z1 in ( dom f2) and

       A50: x = (f2 . z1) by FUNCT_1:def 3;

      

       A51: (f . z1) in ( rng f) by A28, A49, FUNCT_1:def 3;

      (f . z1) in the carrier of ( TOP-REAL 2) by A49, FUNCT_2: 5;

      then

       A52: (f . z1) in ( dom h) by FUNCT_2:def 1;

      x = (h . (f . z1)) by A28, A49, A50, FUNCT_1: 13;

      then (f . z1) = (g . z2) by A46, A52, A48, A34, FUNCT_1:def 4;

      hence thesis by A51, A47, XBOOLE_0: 3;

    end;

    theorem :: JGRAPH_7:135

    for p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real, P,Q be Subset of ( TOP-REAL 2) st a < b & c < d & (p1 `1 ) = b & (p2 `2 ) = c & (p3 `2 ) = c & (p4 `2 ) = c & c <= (p1 `2 ) & (p1 `2 ) <= d & b >= (p2 `1 ) & (p2 `1 ) > (p3 `1 ) & (p3 `1 ) > (p4 `1 ) & (p4 `1 ) > a & P is_an_arc_of (p1,p3) & Q is_an_arc_of (p2,p4) & P c= ( closed_inside_of_rectangle (a,b,c,d)) & Q c= ( closed_inside_of_rectangle (a,b,c,d)) holds P meets Q

    proof

      let p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real, P,Q be Subset of ( TOP-REAL 2);

      assume that

       A1: a < b and

       A2: c < d and

       A3: (p1 `1 ) = b and

       A4: (p2 `2 ) = c and

       A5: (p3 `2 ) = c and

       A6: (p4 `2 ) = c and

       A7: c <= (p1 `2 ) and

       A8: (p1 `2 ) <= d and

       A9: b >= (p2 `1 ) and

       A10: (p2 `1 ) > (p3 `1 ) and

       A11: (p3 `1 ) > (p4 `1 ) and

       A12: (p4 `1 ) > a and

       A13: P is_an_arc_of (p1,p3) and

       A14: Q is_an_arc_of (p2,p4) and

       A15: P c= ( closed_inside_of_rectangle (a,b,c,d)) and

       A16: Q c= ( closed_inside_of_rectangle (a,b,c,d));

      

       A17: ex g be Function of I[01] , ( TOP-REAL 2) st g is continuous one-to-one & ( rng g) = Q & (g . 0 ) = p2 & (g . 1) = p4 by A14, Th2;

      ex f be Function of I[01] , ( TOP-REAL 2) st f is continuous one-to-one & ( rng f) = P & (f . 0 ) = p1 & (f . 1) = p3 by A13, Th2;

      hence thesis by A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, A12, A15, A16, A17, Th134;

    end;

    theorem :: JGRAPH_7:136

    

     Th136: for p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real, f,g be Function of I[01] , ( TOP-REAL 2) st a < b & c < d & (p1 `2 ) = c & (p2 `2 ) = c & (p3 `2 ) = c & (p4 `2 ) = c & b >= (p1 `1 ) & (p1 `1 ) > (p2 `1 ) & (p2 `1 ) > (p3 `1 ) & (p3 `1 ) > (p4 `1 ) & (p4 `1 ) > a & (f . 0 ) = p1 & (f . 1) = p3 & (g . 0 ) = p2 & (g . 1) = p4 & f is continuous one-to-one & g is continuous one-to-one & ( rng f) c= ( closed_inside_of_rectangle (a,b,c,d)) & ( rng g) c= ( closed_inside_of_rectangle (a,b,c,d)) holds ( rng f) meets ( rng g)

    proof

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

      let p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real, f,g be Function of I[01] , ( TOP-REAL 2);

      assume that

       A1: a < b and

       A2: c < d and

       A3: (p1 `2 ) = c and

       A4: (p2 `2 ) = c and

       A5: (p3 `2 ) = c and

       A6: (p4 `2 ) = c and

       A7: b >= (p1 `1 ) and

       A8: (p1 `1 ) > (p2 `1 ) and

       A9: (p2 `1 ) > (p3 `1 ) and

       A10: (p3 `1 ) > (p4 `1 ) and

       A11: (p4 `1 ) > a and

       A12: (f . 0 ) = p1 and

       A13: (f . 1) = p3 and

       A14: (g . 0 ) = p2 and

       A15: (g . 1) = p4 and

       A16: f is continuous one-to-one and

       A17: g is continuous one-to-one and

       A18: ( rng f) c= ( closed_inside_of_rectangle (a,b,c,d)) and

       A19: ( rng g) c= ( closed_inside_of_rectangle (a,b,c,d));

      set A = (2 / (b - a)), B = ( - ((b + a) / (b - a))), C = (2 / (d - c)), D = ( - ((d + c) / (d - c)));

      set h = ( AffineMap (A,B,C,D));

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

      

       A20: g2 is continuous one-to-one by A1, A2, A17, Th53;

      

       A21: ((g . O) `2 ) = c by A4, A14;

      

       A22: b > (p2 `1 ) by A7, A8, XXREAL_0: 2;

      (p2 `1 ) > (p4 `1 ) by A9, A10, XXREAL_0: 2;

      then

       A23: ((g2 . I) `1 ) > ( - 1) by A1, A11, A14, A15, A22, A21, Th67;

      

       A24: ( rng g2) c= ( closed_inside_of_rectangle (( - 1),1,( - 1),1)) by A1, A2, A19, Th52;

      

       A25: ((g2 . I) `2 ) = ( - 1) by A2, A6, A15, Th57;

      

       A26: ((g2 . O) `2 ) = ( - 1) by A2, A4, A14, Th57;

      

       A27: ( dom g) = the carrier of I[01] by FUNCT_2:def 1;

      then

       A28: (h . p2) = (g2 . O) by A14, FUNCT_1: 13;

      

       A29: (h . p4) = (g2 . I) by A15, A27, FUNCT_1: 13;

      (d - c) > 0 by A2, XREAL_1: 50;

      then

       A30: C > 0 by XREAL_1: 139;

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

      

       A31: f2 is continuous one-to-one by A1, A2, A16, Th53;

      

       A32: ( rng f2) c= ( closed_inside_of_rectangle (( - 1),1,( - 1),1)) by A1, A2, A18, Th52;

      

       A33: ((f2 . I) `2 ) = ( - 1) by A2, A5, A13, Th57;

      

       A34: ((f . I) `2 ) = c by A5, A13;

      

       A35: (p3 `1 ) > a by A10, A11, XXREAL_0: 2;

      (p1 `1 ) > (p3 `1 ) by A8, A9, XXREAL_0: 2;

      then

       A36: 1 >= ((f2 . O) `1 ) by A1, A7, A12, A13, A35, A34, Th67;

      

       A37: ((f2 . O) `2 ) = ( - 1) by A2, A3, A12, Th57;

      set x = the Element of (( rng f2) /\ ( rng g2));

      

       A38: ( dom f) = the carrier of I[01] by FUNCT_2:def 1;

      then

       A39: (h . p3) = (f2 . I) by A13, FUNCT_1: 13;

      (b - a) > 0 by A1, XREAL_1: 50;

      then

       A40: A > 0 by XREAL_1: 139;

      then

       A41: ((h . p1) `1 ) > ((h . p2) `1 ) by A8, A30, Th50;

      

       A42: ((h . p3) `1 ) > ((h . p4) `1 ) by A10, A40, A30, Th50;

      

       A43: ((h . p2) `1 ) > ((h . p3) `1 ) by A9, A40, A30, Th50;

      (h . p1) = (f2 . O) by A12, A38, FUNCT_1: 13;

      then ((f2 . O),(g2 . O),(f2 . I),(g2 . I)) are_in_this_order_on ( rectangle (( - 1),1,( - 1),1)) by A41, A43, A42, A28, A39, A29, A37, A33, A36, A26, A25, A23, Th48;

      then ( rng f2) meets ( rng g2) by A31, A32, A20, A24, JGRAPH_6: 79;

      then

       A44: (( rng f2) /\ ( rng g2)) <> {} by XBOOLE_0:def 7;

      then x in ( rng g2) by XBOOLE_0:def 4;

      then

      consider z2 be object such that

       A45: z2 in ( dom g2) and

       A46: x = (g2 . z2) by FUNCT_1:def 3;

      

       A47: x = (h . (g . z2)) by A27, A45, A46, FUNCT_1: 13;

      h is being_homeomorphism by A40, A30, Th51;

      then

       A48: h is one-to-one by TOPS_2:def 5;

      x in ( rng f2) by A44, XBOOLE_0:def 4;

      then

      consider z1 be object such that

       A49: z1 in ( dom f2) and

       A50: x = (f2 . z1) by FUNCT_1:def 3;

      

       A51: (f . z1) in ( rng f) by A38, A49, FUNCT_1:def 3;

      

       A52: (g . z2) in ( rng g) by A27, A45, FUNCT_1:def 3;

      (g . z2) in the carrier of ( TOP-REAL 2) by A45, FUNCT_2: 5;

      then

       A53: (g . z2) in ( dom h) by FUNCT_2:def 1;

      (f . z1) in the carrier of ( TOP-REAL 2) by A49, FUNCT_2: 5;

      then

       A54: (f . z1) in ( dom h) by FUNCT_2:def 1;

      x = (h . (f . z1)) by A38, A49, A50, FUNCT_1: 13;

      then (f . z1) = (g . z2) by A47, A54, A53, A48, FUNCT_1:def 4;

      hence thesis by A51, A52, XBOOLE_0: 3;

    end;

    theorem :: JGRAPH_7:137

    for p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real, P,Q be Subset of ( TOP-REAL 2) st a < b & c < d & (p1 `2 ) = c & (p2 `2 ) = c & (p3 `2 ) = c & (p4 `2 ) = c & b >= (p1 `1 ) & (p1 `1 ) > (p2 `1 ) & (p2 `1 ) > (p3 `1 ) & (p3 `1 ) > (p4 `1 ) & (p4 `1 ) > a & P is_an_arc_of (p1,p3) & Q is_an_arc_of (p2,p4) & P c= ( closed_inside_of_rectangle (a,b,c,d)) & Q c= ( closed_inside_of_rectangle (a,b,c,d)) holds P meets Q

    proof

      let p1,p2,p3,p4 be Point of ( TOP-REAL 2), a,b,c,d be Real, P,Q be Subset of ( TOP-REAL 2);

      assume that

       A1: a < b and

       A2: c < d and

       A3: (p1 `2 ) = c and

       A4: (p2 `2 ) = c and

       A5: (p3 `2 ) = c and

       A6: (p4 `2 ) = c and

       A7: b >= (p1 `1 ) and

       A8: (p1 `1 ) > (p2 `1 ) and

       A9: (p2 `1 ) > (p3 `1 ) and

       A10: (p3 `1 ) > (p4 `1 ) and

       A11: (p4 `1 ) > a and

       A12: P is_an_arc_of (p1,p3) and

       A13: Q is_an_arc_of (p2,p4) and

       A14: P c= ( closed_inside_of_rectangle (a,b,c,d)) and

       A15: Q c= ( closed_inside_of_rectangle (a,b,c,d));

      

       A16: ex g be Function of I[01] , ( TOP-REAL 2) st g is continuous one-to-one & ( rng g) = Q & (g . 0 ) = p2 & (g . 1) = p4 by A13, Th2;

      ex f be Function of I[01] , ( TOP-REAL 2) st f is continuous one-to-one & ( rng f) = P & (f . 0 ) = p1 & (f . 1) = p3 by A12, Th2;

      hence thesis by A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, A14, A15, A16, Th136;

    end;