jordan3.miz



    begin

    reserve r1,r2 for Real;

    reserve n,i,i1,i2,j for Nat;

    reserve D for non empty set;

    reserve f for FinSequence of D;

    theorem :: JORDAN3:1

    for f be FinSequence of ( TOP-REAL n) st 2 <= ( len f) holds (f . 1) in ( L~ f) & (f /. 1) in ( L~ f) & (f . ( len f)) in ( L~ f) & (f /. ( len f)) in ( L~ f)

    proof

      let f be FinSequence of ( TOP-REAL n);

      assume

       A1: 2 <= ( len f);

      then

       A2: (1 + 1) <= ( len f);

      then

       A3: ( LSeg (f,1)) in { ( LSeg (f,i)) : 1 <= i & (i + 1) <= ( len f) };

      (f /. 1) in ( LSeg ((f /. 1),(f /. (1 + 1)))) by RLTOPSP1: 68;

      then (f /. 1) in ( LSeg (f,1)) by A1, TOPREAL1:def 3;

      then (f /. 1) in ( union { ( LSeg (f,i)) : 1 <= i & (i + 1) <= ( len f) }) by A3, TARSKI:def 4;

      then

       A4: (f /. 1) in ( L~ f) by TOPREAL1:def 4;

      

       A5: ((( len f) -' 1) + 1) = ( len f) by A2, NAT_D: 46, XREAL_1: 235;

      

       A6: 1 <= (( len f) -' 1) by A2, NAT_D: 49;

      then

       A7: ( LSeg (f,(( len f) -' 1))) in { ( LSeg (f,i)) : 1 <= i & (i + 1) <= ( len f) } by A5;

      (f /. ( len f)) in ( LSeg ((f /. (( len f) -' 1)),(f /. ((( len f) -' 1) + 1)))) by A5, RLTOPSP1: 68;

      then (f /. ( len f)) in ( LSeg (f,(( len f) -' 1))) by A6, A5, TOPREAL1:def 3;

      then (f /. ( len f)) in ( union { ( LSeg (f,i)) : 1 <= i & (i + 1) <= ( len f) }) by A7, TARSKI:def 4;

      then

       A8: (f /. ( len f)) in ( L~ f) by TOPREAL1:def 4;

      1 <= ( len f) by A2, NAT_D: 46;

      hence thesis by A4, A8, FINSEQ_4: 15;

    end;

    theorem :: JORDAN3:2

    

     Th2: for p1,p2,q1,q2 be Point of ( TOP-REAL 2) st ((p1 `1 ) = (p2 `1 ) or (p1 `2 ) = (p2 `2 )) & q1 in ( LSeg (p1,p2)) & q2 in ( LSeg (p1,p2)) holds (q1 `1 ) = (q2 `1 ) or (q1 `2 ) = (q2 `2 )

    proof

      let p1,p2,q1,q2 be Point of ( TOP-REAL 2);

      assume that

       A1: (p1 `1 ) = (p2 `1 ) or (p1 `2 ) = (p2 `2 ) and

       A2: q1 in ( LSeg (p1,p2)) and

       A3: q2 in ( LSeg (p1,p2));

      consider r2 such that

       A4: q2 = (((1 - r2) * p1) + (r2 * p2)) and 0 <= r2 and r2 <= 1 by A3;

      consider r1 such that

       A5: q1 = (((1 - r1) * p1) + (r1 * p2)) and 0 <= r1 and r1 <= 1 by A2;

      (q1 `1 ) = ((((1 - r1) * p1) `1 ) + ((r1 * p2) `1 )) by A5, TOPREAL3: 2;

      then (q1 `1 ) = (((1 - r1) * (p1 `1 )) + ((r1 * p2) `1 )) by TOPREAL3: 4;

      then

       A6: (q1 `1 ) = (((1 - r1) * (p1 `1 )) + (r1 * (p2 `1 ))) by TOPREAL3: 4;

      (q2 `1 ) = ((((1 - r2) * p1) `1 ) + ((r2 * p2) `1 )) by A4, TOPREAL3: 2;

      then (q2 `1 ) = (((1 - r2) * (p1 `1 )) + ((r2 * p2) `1 )) by TOPREAL3: 4;

      then

       A7: (q2 `1 ) = (((1 - r2) * (p1 `1 )) + (r2 * (p2 `1 ))) by TOPREAL3: 4;

      (q1 `2 ) = ((((1 - r1) * p1) `2 ) + ((r1 * p2) `2 )) by A5, TOPREAL3: 2;

      then (q1 `2 ) = (((1 - r1) * (p1 `2 )) + ((r1 * p2) `2 )) by TOPREAL3: 4;

      then

       A8: (q1 `2 ) = (((1 - r1) * (p1 `2 )) + (r1 * (p2 `2 ))) by TOPREAL3: 4;

      (q2 `2 ) = ((((1 - r2) * p1) `2 ) + ((r2 * p2) `2 )) by A4, TOPREAL3: 2;

      then (q2 `2 ) = (((1 - r2) * (p1 `2 )) + ((r2 * p2) `2 )) by TOPREAL3: 4;

      then

       A9: (q2 `2 ) = (((1 - r2) * (p1 `2 )) + (r2 * (p2 `2 ))) by TOPREAL3: 4;

      per cases by A1;

        suppose (p1 `1 ) = (p2 `1 );

        hence thesis by A6, A7;

      end;

        suppose (p1 `2 ) = (p2 `2 );

        hence thesis by A8, A9;

      end;

    end;

    theorem :: JORDAN3:3

    

     Th3: for p1,p2,q1,q2 be Point of ( TOP-REAL 2) st ((p1 `1 ) = (p2 `1 ) or (p1 `2 ) = (p2 `2 )) & ( LSeg (q1,q2)) c= ( LSeg (p1,p2)) holds (q1 `1 ) = (q2 `1 ) or (q1 `2 ) = (q2 `2 )

    proof

      let p1,p2,q1,q2 be Point of ( TOP-REAL 2);

      

       A1: q2 in ( LSeg (q1,q2)) by RLTOPSP1: 68;

      q1 in ( LSeg (q1,q2)) by RLTOPSP1: 68;

      hence thesis by A1, Th2;

    end;

    theorem :: JORDAN3:4

    

     Th4: for f be FinSequence of ( TOP-REAL 2), n be Element of NAT st 2 <= n & f is being_S-Seq holds (f | n) is being_S-Seq

    proof

      let f be FinSequence of ( TOP-REAL 2), n be Element of NAT ;

      assume that

       A1: 2 <= n and

       A2: f is being_S-Seq;

      

       A3: ( len f) >= 2 by A2, TOPREAL1:def 8;

       A4:

      now

        per cases ;

          case n <= ( len f);

          hence ( len (f | n)) >= 2 by A1, FINSEQ_1: 59;

        end;

          case n > ( len f);

          hence ( len (f | n)) >= 2 by A3, FINSEQ_1: 58;

        end;

      end;

      reconsider f9 = f as s.n.c. special unfolded one-to-one FinSequence of ( TOP-REAL 2) by A2;

      (f9 | n) is one-to-one;

      hence thesis by A4, TOPREAL1:def 8;

    end;

    theorem :: JORDAN3:5

    

     Th5: for f be FinSequence of ( TOP-REAL 2), n be Element of NAT st n <= ( len f) & 2 <= (( len f) -' n) & f is being_S-Seq holds (f /^ n) is being_S-Seq

    proof

      let f be FinSequence of ( TOP-REAL 2), n be Element of NAT ;

      assume that

       A1: n <= ( len f) and

       A2: 2 <= (( len f) -' n) and

       A3: f is being_S-Seq;

      reconsider f9 = f as one-to-one special s.n.c. unfolded FinSequence of ( TOP-REAL 2) by A3;

      ( len (f /^ n)) = (( len f) - n) by A1, RFINSEQ:def 1;

      then ( len (f9 /^ n)) >= 2 by A1, A2, XREAL_1: 233;

      hence thesis by TOPREAL1:def 8;

    end;

    theorem :: JORDAN3:6

    for f be FinSequence of ( TOP-REAL 2), k1,k2 be Element of NAT st f is being_S-Seq & 1 <= k1 & k1 <= ( len f) & 1 <= k2 & k2 <= ( len f) & k1 <> k2 holds ( mid (f,k1,k2)) is being_S-Seq

    proof

      let f be FinSequence of ( TOP-REAL 2), k1,k2 be Element of NAT ;

      assume that

       A1: f is being_S-Seq and

       A2: 1 <= k1 and

       A3: k1 <= ( len f) and

       A4: 1 <= k2 and

       A5: k2 <= ( len f) and

       A6: k1 <> k2;

      per cases ;

        suppose

         A7: k1 <= k2;

        then k1 < k2 by A6, XXREAL_0: 1;

        then

         A8: (k1 + 1) <= k2 by NAT_1: 13;

        then ((k1 + 1) - k1) <= (k2 - k1) by XREAL_1: 9;

        then 1 <= (k2 -' k1) by NAT_D: 39;

        then

         A9: (1 + 1) <= ((k2 -' k1) + 1) by XREAL_1: 6;

        (k1 + 1) <= ( len f) by A5, A8, XXREAL_0: 2;

        then ((k1 + 1) - k1) <= (( len f) - k1) by XREAL_1: 9;

        then

         A10: (1 + 1) <= ((( len f) - k1) + 1) by XREAL_1: 6;

        (( len f) -' (k1 -' 1)) = (( len f) - (k1 -' 1)) by A3, NAT_D: 50, XREAL_1: 233

        .= (( len f) - (k1 - 1)) by A2, XREAL_1: 233

        .= ((( len f) - k1) + 1);

        then

         A11: (f /^ (k1 -' 1)) is being_S-Seq by A1, A3, A10, Th5, NAT_D: 50;

        ( mid (f,k1,k2)) = ((f /^ (k1 -' 1)) | ((k2 -' k1) + 1)) by A7, FINSEQ_6:def 3;

        hence thesis by A11, A9, Th4;

      end;

        suppose

         A12: k1 > k2;

        then

         A13: (k2 + 1) <= k1 by NAT_1: 13;

        then ((k2 + 1) - k2) <= (k1 - k2) by XREAL_1: 9;

        then 1 <= (k1 -' k2) by NAT_D: 39;

        then

         A14: (1 + 1) <= ((k1 -' k2) + 1) by XREAL_1: 6;

        (k2 + 1) <= ( len f) by A3, A13, XXREAL_0: 2;

        then ((k2 + 1) - k2) <= (( len f) - k2) by XREAL_1: 9;

        then

         A15: (1 + 1) <= ((( len f) - k2) + 1) by XREAL_1: 6;

        (( len f) -' (k2 -' 1)) = (( len f) - (k2 -' 1)) by A5, NAT_D: 50, XREAL_1: 233

        .= (( len f) - (k2 - 1)) by A4, XREAL_1: 233

        .= ((( len f) - k2) + 1);

        then (f /^ (k2 -' 1)) is being_S-Seq by A1, A5, A15, Th5, NAT_D: 50;

        then

         A16: ((f /^ (k2 -' 1)) | ((k1 -' k2) + 1)) is S-Sequence_in_R2 by A14, Th4;

        ( mid (f,k1,k2)) = ( Rev ((f /^ (k2 -' 1)) | ((k1 -' k2) + 1))) by A12, FINSEQ_6:def 3;

        hence thesis by A16;

      end;

    end;

    begin

    definition

      let f be FinSequence of ( TOP-REAL 2), p be Point of ( TOP-REAL 2);

      assume

       A1: p in ( L~ f);

      :: JORDAN3:def1

      func Index (p,f) -> Element of NAT means

      : Def1: ex S be non empty Subset of NAT st it = ( min S) & S = { i : p in ( LSeg (f,i)) };

      existence

      proof

        set S = { i : p in ( LSeg (f,i)) };

        

         A2: S c= NAT

        proof

          let x be object;

          assume x in S;

          then ex i st x = i & p in ( LSeg (f,i));

          hence thesis by ORDINAL1:def 12;

        end;

        consider i2 be Nat such that 1 <= i2 and (i2 + 1) <= ( len f) and

         A3: p in ( LSeg (f,i2)) by A1, SPPOL_2: 13;

        i2 in S by A3;

        then

        reconsider S as non empty Subset of NAT by A2;

        take ( min S), S;

        thus thesis;

      end;

      uniqueness ;

    end

    theorem :: JORDAN3:7

    

     Th7: for f be FinSequence of ( TOP-REAL 2), p be Point of ( TOP-REAL 2), i be Element of NAT st p in ( LSeg (f,i)) holds ( Index (p,f)) <= i

    proof

      let f be FinSequence of ( TOP-REAL 2);

      let p be Point of ( TOP-REAL 2), i0 be Element of NAT ;

      assume

       A1: p in ( LSeg (f,i0));

      ( LSeg (f,i0)) c= ( L~ f) by TOPREAL3: 19;

      then

      consider S be non empty Subset of NAT such that

       A2: ( Index (p,f)) = ( min S) and

       A3: S = { i : p in ( LSeg (f,i)) } by A1, Def1;

      i0 in S by A1, A3;

      hence thesis by A2, XXREAL_2:def 7;

    end;

    theorem :: JORDAN3:8

    

     Th8: for f be FinSequence of ( TOP-REAL 2), p be Point of ( TOP-REAL 2) st p in ( L~ f) holds 1 <= ( Index (p,f)) & ( Index (p,f)) < ( len f)

    proof

      let f be FinSequence of ( TOP-REAL 2), p be Point of ( TOP-REAL 2);

      assume p in ( L~ f);

      then

      consider S be non empty Subset of NAT such that

       A1: ( Index (p,f)) = ( min S) and

       A2: S = { i : p in ( LSeg (f,i)) } by Def1;

      ( Index (p,f)) in S by A1, XXREAL_2:def 7;

      then

       A3: ex i st i = ( Index (p,f)) & p in ( LSeg (f,i)) by A2;

      hence 1 <= ( Index (p,f)) by TOPREAL1:def 3;

      (( Index (p,f)) + 1) <= ( len f) by A3, TOPREAL1:def 3;

      hence thesis by NAT_1: 13;

    end;

    theorem :: JORDAN3:9

    

     Th9: for f be FinSequence of ( TOP-REAL 2), p be Point of ( TOP-REAL 2) st p in ( L~ f) holds p in ( LSeg (f,( Index (p,f))))

    proof

      let f be FinSequence of ( TOP-REAL 2);

      let p be Point of ( TOP-REAL 2);

      assume p in ( L~ f);

      then

      consider S be non empty Subset of NAT such that

       A1: ( Index (p,f)) = ( min S) and

       A2: S = { i : p in ( LSeg (f,i)) } by Def1;

      ( Index (p,f)) in S by A1, XXREAL_2:def 7;

      then ex i st i = ( Index (p,f)) & p in ( LSeg (f,i)) by A2;

      hence thesis;

    end;

    theorem :: JORDAN3:10

    

     Th10: for f be FinSequence of ( TOP-REAL 2), p be Point of ( TOP-REAL 2) st p in ( LSeg (f,1)) holds ( Index (p,f)) = 1

    proof

      let f be FinSequence of ( TOP-REAL 2), p be Point of ( TOP-REAL 2);

      assume

       A1: p in ( LSeg (f,1));

      then

       A2: ( Index (p,f)) <= 1 by Th7;

      ( LSeg (f,1)) c= ( L~ f) by TOPREAL3: 19;

      then ( Index (p,f)) >= 1 by A1, Th8;

      hence thesis by A2, XXREAL_0: 1;

    end;

    theorem :: JORDAN3:11

    

     Th11: for f be FinSequence of ( TOP-REAL 2), p be Point of ( TOP-REAL 2) st ( len f) >= 2 holds ( Index ((f /. 1),f)) = 1

    proof

      let f be FinSequence of ( TOP-REAL 2), p be Point of ( TOP-REAL 2);

      assume ( len f) >= 2;

      then ( len f) >= (1 + 1);

      then (f /. 1) in ( LSeg (f,1)) by TOPREAL1: 21;

      hence thesis by Th10;

    end;

    theorem :: JORDAN3:12

    

     Th12: for f be FinSequence of ( TOP-REAL 2), p be Point of ( TOP-REAL 2), i1 be Nat st f is being_S-Seq & 1 < i1 & i1 <= ( len f) & p = (f . i1) holds (( Index (p,f)) + 1) = i1

    proof

      let f be FinSequence of ( TOP-REAL 2), p be Point of ( TOP-REAL 2), i1 be Nat;

      assume

       A1: f is being_S-Seq;

      assume that

       A2: 1 < i1 and

       A3: i1 <= ( len f);

      

       A4: i1 in ( dom f) by A2, A3, FINSEQ_3: 25;

      assume p = (f . i1);

      then

       A5: p = (f /. i1) by A4, PARTFUN1:def 6;

      assume

       A6: (( Index (p,f)) + 1) <> i1;

      consider j be Nat such that

       A7: i1 = (j + 1) by A2, NAT_1: 6;

      reconsider j as Element of NAT by ORDINAL1:def 12;

      

       A8: (1 + 0 ) <= j by A2, A7, NAT_1: 13;

      then

       A9: p in ( LSeg (f,j)) by A3, A7, A5, TOPREAL1: 21;

      then ( Index (p,f)) <= j by Th7;

      then ( Index (p,f)) < j by A7, A6, XXREAL_0: 1;

      then

       A10: (( Index (p,f)) + 1) <= j by NAT_1: 13;

      

       A11: ( LSeg (f,j)) c= ( L~ f) by TOPREAL3: 19;

      then

       A12: p in ( LSeg (f,( Index (p,f)))) by A9, Th9;

      per cases by A10, XXREAL_0: 1;

        suppose

         A13: (( Index (p,f)) + 1) = j;

        then

         A14: (( Index (p,f)) + (1 + 1)) <= ( len f) by A3, A7;

        1 <= ( Index (p,f)) by A9, A11, Th8;

        then (( LSeg (f,( Index (p,f)))) /\ ( LSeg (f,j))) = {(f /. j)} by A1, A13, A14, TOPREAL1:def 6;

        then p in {(f /. j)} by A9, A12, XBOOLE_0:def 4;

        then

         A15: p = (f /. j) by TARSKI:def 1;

        j < ( len f) by A3, A7, NAT_1: 13;

        then

         A16: j in ( dom f) by A8, FINSEQ_3: 25;

        j < i1 by A7, NAT_1: 13;

        hence contradiction by A1, A4, A5, A15, A16, PARTFUN2: 10;

      end;

        suppose

         A17: (( Index (p,f)) + 1) < j;

        p in (( LSeg (f,( Index (p,f)))) /\ ( LSeg (f,j))) by A9, A12, XBOOLE_0:def 4;

        then ( LSeg (f,( Index (p,f)))) meets ( LSeg (f,j)) by XBOOLE_0: 4;

        hence contradiction by A1, A17, TOPREAL1:def 7;

      end;

    end;

    theorem :: JORDAN3:13

    

     Th13: for f be FinSequence of ( TOP-REAL 2) holds for p be Point of ( TOP-REAL 2) holds for i1 be Element of NAT st f is s.n.c. & p in ( LSeg (f,i1)) holds i1 = ( Index (p,f)) or i1 = (( Index (p,f)) + 1)

    proof

      let f be FinSequence of ( TOP-REAL 2);

      let p be Point of ( TOP-REAL 2);

      let i1 be Element of NAT ;

      assume that

       A1: f is s.n.c. and

       A2: p in ( LSeg (f,i1));

      p in ( L~ f) by A2, SPPOL_2: 17;

      then p in ( LSeg (f,( Index (p,f)))) by Th9;

      then p in (( LSeg (f,( Index (p,f)))) /\ ( LSeg (f,i1))) by A2, XBOOLE_0:def 4;

      then

       A3: ( LSeg (f,( Index (p,f)))) meets ( LSeg (f,i1)) by XBOOLE_0: 4;

      assume

       A4: not thesis;

      ( Index (p,f)) <= i1 by A2, Th7;

      then ( Index (p,f)) < i1 by A4, XXREAL_0: 1;

      then (( Index (p,f)) + 1) <= i1 by NAT_1: 13;

      then (( Index (p,f)) + 1) < i1 by A4, XXREAL_0: 1;

      hence contradiction by A1, A3, TOPREAL1:def 7;

    end;

    theorem :: JORDAN3:14

    

     Th14: for f be FinSequence of ( TOP-REAL 2) holds for p be Point of ( TOP-REAL 2) holds for i1 be Element of NAT st f is unfolded s.n.c. & (i1 + 1) <= ( len f) & p in ( LSeg (f,i1)) & p <> (f . i1) holds i1 = ( Index (p,f))

    proof

      let f be FinSequence of ( TOP-REAL 2);

      let p be Point of ( TOP-REAL 2);

      let i1 be Element of NAT ;

      assume that

       A1: f is unfolded s.n.c. and

       A2: (i1 + 1) <= ( len f) and

       A3: p in ( LSeg (f,i1));

      

       A4: i1 < ( len f) by A2, NAT_1: 13;

      

       A5: 1 <= (( Index (p,f)) + 1) by NAT_1: 11;

      ( Index (p,f)) <= i1 by A3, Th7;

      then ( Index (p,f)) < ( len f) by A4, XXREAL_0: 2;

      then (( Index (p,f)) + 1) <= ( len f) by NAT_1: 13;

      then

       A6: (( Index (p,f)) + 1) in ( dom f) by A5, FINSEQ_3: 25;

      assume

       A7: p <> (f . i1);

      

       A8: p in ( L~ f) by A3, SPPOL_2: 17;

      then p in ( LSeg (f,( Index (p,f)))) by Th9;

      then

       A9: p in (( LSeg (f,( Index (p,f)))) /\ ( LSeg (f,i1))) by A3, XBOOLE_0:def 4;

      

       A10: 1 <= ( Index (p,f)) by A8, Th8;

      now

        assume

         A11: i1 = (( Index (p,f)) + 1);

        then (( Index (p,f)) + (1 + 1)) <= ( len f) by A2;

        then p in {(f /. (( Index (p,f)) + 1))} by A1, A9, A10, A11, TOPREAL1:def 6;

        then p = (f /. (( Index (p,f)) + 1)) by TARSKI:def 1;

        hence contradiction by A7, A6, A11, PARTFUN1:def 6;

      end;

      hence thesis by A1, A3, Th13;

    end;

    definition

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

      :: JORDAN3:def2

      pred g is_S-Seq_joining p1,p2 means g is being_S-Seq & (g . 1) = p1 & (g . ( len g)) = p2;

    end

    theorem :: JORDAN3:15

    

     Th15: for g be FinSequence of ( TOP-REAL 2), p1,p2 be Point of ( TOP-REAL 2) st g is_S-Seq_joining (p1,p2) holds ( Rev g) is_S-Seq_joining (p2,p1)

    proof

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

      assume that

       A1: g is being_S-Seq and

       A2: (g . 1) = p1 and

       A3: (g . ( len g)) = p2;

      thus ( Rev g) is being_S-Seq by A1;

      thus (( Rev g) . 1) = p2 by A3, FINSEQ_5: 62;

      ( dom g) = ( dom ( Rev g)) by FINSEQ_5: 57;

      

      hence (( Rev g) . ( len ( Rev g))) = (( Rev g) . ( len g)) by FINSEQ_3: 29

      .= p1 by A2, FINSEQ_5: 62;

    end;

    theorem :: JORDAN3:16

    

     Th16: for f,g be FinSequence of ( TOP-REAL 2), p be Point of ( TOP-REAL 2), j be Nat st p in ( L~ f) & g = ( <*p*> ^ ( mid (f,(( Index (p,f)) + 1),( len f)))) & 1 <= j & (j + 1) <= ( len g) holds ( LSeg (g,j)) c= ( LSeg (f,((( Index (p,f)) + j) -' 1)))

    proof

      let f,g be FinSequence of ( TOP-REAL 2), p be Point of ( TOP-REAL 2), j be Nat;

      assume that

       A1: p in ( L~ f) and

       A2: g = ( <*p*> ^ ( mid (f,(( Index (p,f)) + 1),( len f)))) and

       A3: 1 <= j and

       A4: (j + 1) <= ( len g);

      

       A5: j <= ( len g) by A4, NAT_1: 13;

      ( len g) = (( len <*p*>) + ( len ( mid (f,(( Index (p,f)) + 1),( len f))))) by A2, FINSEQ_1: 22;

      then

       A6: ( len g) = (1 + ( len ( mid (f,(( Index (p,f)) + 1),( len f))))) by FINSEQ_1: 39;

      then

       A7: ((j + 1) - 1) <= ((1 + ( len ( mid (f,(( Index (p,f)) + 1),( len f))))) - 1) by A4, XREAL_1: 9;

      (j -' 1) <= j by NAT_D: 35;

      then

       A8: (j -' 1) <= ( len ( mid (f,(( Index (p,f)) + 1),( len f)))) by A7, XXREAL_0: 2;

      1 <= (( Index (p,f)) + j) by A3, NAT_1: 12;

      then

       A9: (1 - 1) <= ((( Index (p,f)) + j) - 1) by XREAL_1: 9;

      

       A10: (j -' 1) = (j - 1) by A3, XREAL_1: 233;

      

       A11: j = (1 + (j - 1))

      .= (( len <*p*>) + (j -' 1)) by A10, FINSEQ_1: 39;

      1 <= ( Index (p,f)) by A1, Th8;

      then (1 + 1) <= (( Index (p,f)) + j) by A3, XREAL_1: 7;

      then 1 <= ((( Index (p,f)) + j) - 1) by XREAL_1: 19;

      then

       A12: 1 <= ((( Index (p,f)) + j) -' 1) by NAT_D: 39;

      consider i such that 1 <= i and

       A13: (i + 1) <= ( len f) and p in ( LSeg (f,i)) by A1, SPPOL_2: 13;

      1 <= (i + 1) by NAT_1: 12;

      then

       A14: 1 <= ( len f) by A13, XXREAL_0: 2;

      

       A15: ( Index (p,f)) < ( len f) by A1, Th8;

      then

       A16: (( Index (p,f)) + 1) <= ( len f) by NAT_1: 13;

      (( Index (p,f)) + 1) <= ( len f) by A15, NAT_1: 13;

      then ((( Index (p,f)) + 1) - ( Index (p,f))) <= (( len f) - ( Index (p,f))) by XREAL_1: 9;

      then

       A17: (1 - 1) <= ((( len f) - ( Index (p,f))) - 1) by XREAL_1: 9;

      

      then

       A18: (( len f) -' (( Index (p,f)) + 1)) = (( len f) - (( Index (p,f)) + 1)) by XREAL_0:def 2

      .= ((( len f) - ( Index (p,f))) - 1);

      

       A19: ( 0 + 1) <= (( Index (p,f)) + 1) by NAT_1: 13;

      then

       A20: 1 <= ( len f) by A15, NAT_1: 13;

      (( Index (p,f)) + 1) <= ( len f) by A15, NAT_1: 13;

      then

       A21: ( len ( mid (f,(( Index (p,f)) + 1),( len f)))) = ((( len f) -' (( Index (p,f)) + 1)) + 1) by A14, A19, FINSEQ_6: 118;

      

       A22: ( len g) = (( len <*p*>) + ( len ( mid (f,(( Index (p,f)) + 1),( len f))))) by A2, FINSEQ_1: 22

      .= (1 + ( len ( mid (f,(( Index (p,f)) + 1),( len f))))) by FINSEQ_1: 39;

      

      then ( len g) = (1 + ((( len f) - (( Index (p,f)) + 1)) + 1)) by A17, A21, XREAL_0:def 2

      .= (1 + (( len f) - ( Index (p,f))));

      then j <= (( len f) - ( Index (p,f))) by A4, XREAL_1: 6;

      then

       A23: (j + ( Index (p,f))) <= ((( len f) - ( Index (p,f))) + ( Index (p,f))) by XREAL_1: 6;

      then

       A24: (((( Index (p,f)) + j) -' 1) + 1) <= ( len f) by A3, NAT_1: 12, XREAL_1: 235;

      

       A25: 1 <= (j + 1) by A3, NAT_1: 13;

      then

       A26: (g /. (j + 1)) = (g . (j + 1)) by A4, FINSEQ_4: 15;

      

       A27: (j + 1) = (( len <*p*>) + ((j + 1) - 1)) by FINSEQ_1: 39

      .= (( len <*p*>) + ((j + 1) -' 1)) by A25, XREAL_1: 233;

      

       A28: ((j + 1) -' 1) = ((j + 1) - 1) by A25, XREAL_1: 233;

      then ((j + 1) -' 1) in ( dom ( mid (f,(( Index (p,f)) + 1),( len f)))) by A3, A7, FINSEQ_3: 25;

      

      then (g . (j + 1)) = (( mid (f,(( Index (p,f)) + 1),( len f))) . ((j + 1) -' 1)) by A2, A27, FINSEQ_1:def 7

      .= (f . ((((j + 1) -' 1) + (( Index (p,f)) + 1)) -' 1)) by A3, A19, A16, A20, A28, A7, FINSEQ_6: 118

      .= (f . (((((j + 1) -' 1) + 1) + ( Index (p,f))) -' 1))

      .= (f . (((j + 1) + ( Index (p,f))) -' 1)) by A25, XREAL_1: 235

      .= (f . (((( Index (p,f)) + j) + 1) -' 1))

      .= (f . (( Index (p,f)) + j)) by NAT_D: 34

      .= (f . (((( Index (p,f)) + j) -' 1) + 1)) by A3, NAT_1: 12, XREAL_1: 235;

      then

       A29: (f /. (((( Index (p,f)) + j) -' 1) + 1)) = (g /. (j + 1)) by A24, A26, FINSEQ_4: 15, NAT_1: 11;

      ((j + 1) - 1) <= ((1 + ( len ( mid (f,(( Index (p,f)) + 1),( len f))))) - 1) by A4, A6, XREAL_1: 9;

      then (j + ( Index (p,f))) <= ((( len f) - ( Index (p,f))) + ( Index (p,f))) by A21, A18, XREAL_1: 6;

      then ((( Index (p,f)) + (j - 1)) + 1) <= ( len f);

      then (((( Index (p,f)) + j) -' 1) + 1) <= ( len f) by A9, XREAL_0:def 2;

      then

       A30: ( LSeg (f,((( Index (p,f)) + j) -' 1))) = ( LSeg ((f /. ((( Index (p,f)) + j) -' 1)),(f /. (((( Index (p,f)) + j) -' 1) + 1)))) by A12, TOPREAL1:def 3;

      

       A31: 1 <= ( len g) by A22, NAT_1: 11;

      now

        per cases by A3, XXREAL_0: 1;

          case

           A32: 1 < j;

          then

           A33: (j -' 1) = (j - 1) by XREAL_1: 233;

          then

           A34: 1 <= (j -' 1) by A32, SPPOL_1: 1;

          (j - 1) <= ((1 + ( len ( mid (f,(( Index (p,f)) + 1),( len f))))) - 1) by A6, A5, XREAL_1: 9;

          then (j -' 1) in ( dom ( mid (f,(( Index (p,f)) + 1),( len f)))) by A33, A34, FINSEQ_3: 25;

          

          then

           A35: (g . j) = (( mid (f,(( Index (p,f)) + 1),( len f))) . (j -' 1)) by A2, A11, FINSEQ_1:def 7

          .= (f . (((j -' 1) + (( Index (p,f)) + 1)) -' 1)) by A19, A16, A20, A8, A34, FINSEQ_6: 118

          .= (f . ((((j -' 1) + 1) + ( Index (p,f))) -' 1))

          .= (f . ((( Index (p,f)) + j) -' 1)) by A3, XREAL_1: 235;

          (g /. j) = (g . j) by A3, A5, FINSEQ_4: 15;

          

          then ( LSeg (f,((( Index (p,f)) + j) -' 1))) = ( LSeg ((g /. j),(g /. (j + 1)))) by A23, A29, A12, A30, A35, FINSEQ_4: 15, NAT_D: 50

          .= ( LSeg (g,j)) by A3, A4, TOPREAL1:def 3;

          hence thesis;

        end;

          case

           A36: 1 = j;

          then j <= ( len <*p*>) by FINSEQ_1: 39;

          then j in ( dom <*p*>) by A36, FINSEQ_3: 25;

          

          then

           A37: (g . j) = ( <*p*> . j) by A2, FINSEQ_1:def 7

          .= p by A36, FINSEQ_1: 40;

          

           A38: (f /. (((( Index (p,f)) + j) -' 1) + 1)) in ( LSeg ((f /. ((( Index (p,f)) + j) -' 1)),(f /. (((( Index (p,f)) + j) -' 1) + 1)))) by RLTOPSP1: 68;

          

           A39: (g /. j) = (g . j) by A31, A36, FINSEQ_4: 15;

          

           A40: ((( Index (p,f)) + j) -' 1) = ( Index (p,f)) by A36, NAT_D: 34;

          p in ( LSeg (f,( Index (p,f)))) by A1, Th9;

          then ( LSeg (p,(g /. (j + 1)))) c= ( LSeg (f,((( Index (p,f)) + j) -' 1))) by A29, A30, A38, A40, TOPREAL1: 6;

          hence thesis by A3, A4, A37, A39, TOPREAL1:def 3;

        end;

      end;

      hence thesis;

    end;

    theorem :: JORDAN3:17

    for f,g be FinSequence of ( TOP-REAL 2), p be Point of ( TOP-REAL 2) st f is being_S-Seq & p in ( L~ f) & p <> (f . (( Index (p,f)) + 1)) & g = ( <*p*> ^ ( mid (f,(( Index (p,f)) + 1),( len f)))) holds g is_S-Seq_joining (p,(f /. ( len f)))

    proof

      let f,g be FinSequence of ( TOP-REAL 2), p be Point of ( TOP-REAL 2);

      assume that

       A1: f is being_S-Seq and

       A2: p in ( L~ f) and

       A3: p <> (f . (( Index (p,f)) + 1)) and

       A4: g = ( <*p*> ^ ( mid (f,(( Index (p,f)) + 1),( len f))));

      ( len g) = (( len <*p*>) + ( len ( mid (f,(( Index (p,f)) + 1),( len f))))) by A4, FINSEQ_1: 22;

      then

       A5: ( len g) = (1 + ( len ( mid (f,(( Index (p,f)) + 1),( len f))))) by FINSEQ_1: 39;

      consider i such that 1 <= i and

       A6: (i + 1) <= ( len f) and p in ( LSeg (f,i)) by A2, SPPOL_2: 13;

      1 <= (1 + i) by NAT_1: 12;

      then

       A7: 1 <= ( len f) by A6, XXREAL_0: 2;

      

       A8: for j1,j2 be Nat st (j1 + 1) < j2 holds ( LSeg (g,j1)) misses ( LSeg (g,j2))

      proof

        let j1,j2 be Nat;

        assume

         A9: (j1 + 1) < j2;

        

         A10: j1 = 0 or j1 >= ( 0 + 1) by NAT_1: 13;

        now

          per cases by A10, XXREAL_0: 1;

            case j1 = 0 ;

            then ( LSeg (g,j1)) = {} by TOPREAL1:def 3;

            then (( LSeg (g,j1)) /\ ( LSeg (g,j2))) = {} ;

            hence thesis by XBOOLE_0:def 7;

          end;

            case that

             A11: j1 = 1 or j1 > 1 and

             A12: (j2 + 1) <= ( len g);

            1 < (j1 + 1) by A11, NAT_1: 13;

            then 1 <= j2 by A9, XXREAL_0: 2;

            then

             A13: ( LSeg (g,j2)) c= ( LSeg (f,((( Index (p,f)) + j2) -' 1))) by A2, A4, A12, Th16;

            1 <= (( Index (p,f)) + j1) by A2, Th8, NAT_1: 12;

            then (1 - 1) <= ((( Index (p,f)) + j1) - 1) by XREAL_1: 9;

            then

             A14: ((( Index (p,f)) + j1) - 1) = ((( Index (p,f)) + j1) -' 1) by XREAL_0:def 2;

            (( Index (p,f)) + (j1 + 1)) < (( Index (p,f)) + j2) by A9, XREAL_1: 6;

            then (((( Index (p,f)) + j1) + 1) - 1) < ((( Index (p,f)) + j2) - 1) by XREAL_1: 9;

            then (((( Index (p,f)) + j1) -' 1) + 1) < ((( Index (p,f)) + j2) -' 1) by A14, XREAL_0:def 2;

            then ( LSeg (f,((( Index (p,f)) + j1) -' 1))) misses ( LSeg (f,((( Index (p,f)) + j2) -' 1))) by A1, TOPREAL1:def 7;

            then

             A15: (( LSeg (f,((( Index (p,f)) + j1) -' 1))) /\ ( LSeg (f,((( Index (p,f)) + j2) -' 1)))) = {} by XBOOLE_0:def 7;

            j2 < ( len g) by A12, NAT_1: 13;

            then (j1 + 1) <= ( len g) by A9, XXREAL_0: 2;

            then ( LSeg (g,j1)) c= ( LSeg (f,((( Index (p,f)) + j1) -' 1))) by A2, A4, A11, Th16;

            then (( LSeg (g,j1)) /\ ( LSeg (g,j2))) = {} by A13, A15, XBOOLE_1: 3, XBOOLE_1: 27;

            hence thesis by XBOOLE_0:def 7;

          end;

            case (j2 + 1) > ( len g);

            then ( LSeg (g,j2)) = {} by TOPREAL1:def 3;

            then (( LSeg (g,j1)) /\ ( LSeg (g,j2))) = {} ;

            hence thesis by XBOOLE_0:def 7;

          end;

        end;

        hence thesis;

      end;

      

       A16: ( Index (p,f)) < ( len f) by A2, Th8;

      then

       A17: (( Index (p,f)) + 1) <= ( len f) by NAT_1: 13;

      (( Index (p,f)) + 1) <= ( len f) by A16, NAT_1: 13;

      then

       A18: ((( Index (p,f)) + 1) - ( Index (p,f))) <= (( len f) - ( Index (p,f))) by XREAL_1: 9;

      then

       A19: (1 - 1) <= ((( len f) - ( Index (p,f))) - 1) by XREAL_1: 9;

      

      then

       A20: (( len f) -' (( Index (p,f)) + 1)) = (( len f) - (( Index (p,f)) + 1)) by XREAL_0:def 2

      .= ((( len f) - ( Index (p,f))) - 1);

      

       A21: ( 0 + 1) <= (( Index (p,f)) + 1) by NAT_1: 11;

      then

       A22: ( len ( mid (f,(( Index (p,f)) + 1),( len f)))) = ((( len f) -' (( Index (p,f)) + 1)) + 1) by A7, A17, FINSEQ_6: 118;

      

       A23: for j be Nat st 1 <= j & (j + 2) <= ( len g) holds (( LSeg (g,j)) /\ ( LSeg (g,(j + 1)))) = {(g /. (j + 1))}

      proof

        let j be Nat;

        assume that

         A24: 1 <= j and

         A25: (j + 2) <= ( len g);

        

         A26: (j + 2) = ((j + 1) + 1);

        then

         A27: (j + 1) <= ( len g) by A25, NAT_1: 13;

        then

         A28: ( LSeg (g,j)) c= ( LSeg (f,((( Index (p,f)) + j) -' 1))) by A2, A4, A24, Th16;

        1 <= (j + 1) by A24, NAT_1: 13;

        then ( LSeg (g,(j + 1))) c= ( LSeg (f,((( Index (p,f)) + (j + 1)) -' 1))) by A2, A4, A25, A26, Th16;

        then

         A29: (( LSeg (g,j)) /\ ( LSeg (g,(j + 1)))) c= (( LSeg (f,((( Index (p,f)) + j) -' 1))) /\ ( LSeg (f,((( Index (p,f)) + (j + 1)) -' 1)))) by A28, XBOOLE_1: 27;

        

         A30: 1 <= ( Index (p,f)) by A2, Th8;

        1 <= ( Index (p,f)) by A2, Th8;

        then (1 + 1) <= (( Index (p,f)) + j) by A24, XREAL_1: 7;

        then 1 <= ((( Index (p,f)) + j) - 1) by XREAL_1: 19;

        then

         A31: 1 <= ((( Index (p,f)) + j) -' 1) by NAT_D: 39;

        1 <= (( Index (p,f)) + j) by A2, Th8, NAT_1: 12;

        then

         A32: (1 - 1) <= ((( Index (p,f)) + j) - 1) by XREAL_1: 9;

        (((j + 1) + 1) - 1) <= ((1 + ( len ( mid (f,(( Index (p,f)) + 1),( len f))))) - 1) by A5, A25, XREAL_1: 9;

        then

         A33: ((j + 1) + ( Index (p,f))) <= ((( len f) - ( Index (p,f))) + ( Index (p,f))) by A22, A20, XREAL_1: 6;

        then ((( Index (p,f)) + j) + 1) <= ( len f);

        then ((( Index (p,f)) + (j - 1)) + 1) <= ( len f) by NAT_D: 46;

        then

         A34: (((( Index (p,f)) + j) -' 1) + 1) <= ( len f) by A32, XREAL_0:def 2;

        (((( Index (p,f)) + j) - 1) + (1 + 1)) <= ( len f) by A33;

        then (((( Index (p,f)) + j) -' 1) + 2) <= ( len f) by A32, XREAL_0:def 2;

        then

         A35: {(f /. (((( Index (p,f)) + j) -' 1) + 1))} = (( LSeg (f,((( Index (p,f)) + j) -' 1))) /\ ( LSeg (f,(((( Index (p,f)) + j) -' 1) + 1)))) by A1, A31, TOPREAL1:def 6;

        

         A36: 1 < (j + 1) by A24, NAT_1: 13;

        then

         A37: (g /. (j + 1)) = (g . (j + 1)) by A27, FINSEQ_4: 15;

        

         A38: (g /. (j + 1)) in ( LSeg ((g /. (j + 1)),(g /. ((j + 1) + 1)))) by RLTOPSP1: 68;

        (g /. (j + 1)) in ( LSeg ((g /. j),(g /. (j + 1)))) by RLTOPSP1: 68;

        then

         A39: (g /. (j + 1)) in (( LSeg ((g /. j),(g /. (j + 1)))) /\ ( LSeg ((g /. (j + 1)),(g /. ((j + 1) + 1))))) by A38, XBOOLE_0:def 4;

        

         A40: ( LSeg (g,j)) = ( LSeg ((g /. j),(g /. (j + 1)))) by A24, A27, TOPREAL1:def 3;

        ( LSeg ((g /. (j + 1)),(g /. ((j + 1) + 1)))) = ( LSeg (g,(j + 1))) by A25, A36, TOPREAL1:def 3;

        then

         A41: {(g /. (j + 1))} c= (( LSeg (g,j)) /\ ( LSeg (g,(j + 1)))) by A40, A39, ZFMISC_1: 31;

        

         A42: (j + 1) = (((j + 1) - 1) + 1)

        .= (((j + 1) -' 1) + 1) by A36, XREAL_1: 233;

        then

         A43: (j + 1) = (( len <*p*>) + ((j + 1) -' 1)) by FINSEQ_1: 39;

        

         A44: ((j + 1) -' 1) <= ( len ( mid (f,(( Index (p,f)) + 1),( len f)))) by A5, A27, A42, XREAL_1: 6;

        then ((j + 1) -' 1) in ( dom ( mid (f,(( Index (p,f)) + 1),( len f)))) by A24, A42, FINSEQ_3: 25;

        

        then (g . (j + 1)) = (( mid (f,(( Index (p,f)) + 1),( len f))) . ((j + 1) -' 1)) by A4, A43, FINSEQ_1:def 7

        .= (f . ((((j + 1) -' 1) + (( Index (p,f)) + 1)) -' 1)) by A7, A17, A21, A24, A42, A44, FINSEQ_6: 118

        .= (f . (((((j + 1) -' 1) + 1) + ( Index (p,f))) -' 1))

        .= (f . (((j + 1) + ( Index (p,f))) -' 1)) by A36, XREAL_1: 235

        .= (f . (((( Index (p,f)) + j) + 1) -' 1))

        .= (f . (( Index (p,f)) + j)) by NAT_D: 34

        .= (f . (((( Index (p,f)) + j) -' 1) + 1)) by A30, NAT_1: 12, XREAL_1: 235;

        then

         A45: (f /. (((( Index (p,f)) + j) -' 1) + 1)) = (g /. (j + 1)) by A37, A34, FINSEQ_4: 15, NAT_1: 11;

        ((( Index (p,f)) + (j + 1)) -' 1) = (((( Index (p,f)) + j) + 1) - 1) by NAT_1: 11, XREAL_1: 233

        .= (((( Index (p,f)) + j) - 1) + 1)

        .= (((( Index (p,f)) + j) -' 1) + 1) by A30, NAT_1: 12, XREAL_1: 233;

        hence thesis by A29, A35, A45, A41, XBOOLE_0:def 10;

      end;

      

       A46: ( len g) = (( len <*p*>) + ( len ( mid (f,(( Index (p,f)) + 1),( len f))))) by A4, FINSEQ_1: 22

      .= (1 + ( len ( mid (f,(( Index (p,f)) + 1),( len f))))) by FINSEQ_1: 39;

      

      then

       A47: ( len g) = (1 + ((( len f) - (( Index (p,f)) + 1)) + 1)) by A19, A22, XREAL_0:def 2

      .= (1 + (( len f) - ( Index (p,f))));

      then

       A48: (( len g) -' 1) = (( len g) - 1) by A18, XREAL_0:def 2;

      then

       A49: (( len g) -' 1) in ( dom ( mid (f,(( Index (p,f)) + 1),( len f)))) by A18, A22, A47, A20, FINSEQ_3: 25;

      

       A50: (( len f) - ( Index (p,f))) >= 0 by A2, Th8, XREAL_1: 50;

      then

       A51: (( len f) - ( Index (p,f))) = (( len f) -' ( Index (p,f))) by XREAL_0:def 2;

      then

       A52: (( mid (f,(( Index (p,f)) + 1),( len f))) . (( len f) -' ( Index (p,f)))) = (f . (((( len f) -' ( Index (p,f))) + (( Index (p,f)) + 1)) -' 1)) by A7, A18, A17, A21, A22, A20, FINSEQ_6: 118;

      

       A53: (( len g) -' 1) = (( len f) -' ( Index (p,f))) by A47, A48, XREAL_0:def 2;

      for x1,x2 be object st x1 in ( dom g) & x2 in ( dom g) & (g . x1) = (g . x2) holds x1 = x2

      proof

        let x1,x2 be object;

        assume that

         A54: x1 in ( dom g) and

         A55: x2 in ( dom g) and

         A56: (g . x1) = (g . x2);

        reconsider n1 = x1, n2 = x2 as Element of NAT by A54, A55;

        

         A57: n1 <= ( len g) by A54, FINSEQ_3: 25;

        

         A58: 1 <= n2 by A55, FINSEQ_3: 25;

        

         A59: n2 <= ( len g) by A55, FINSEQ_3: 25;

        

         A60: 1 <= n1 by A54, FINSEQ_3: 25;

        now

          per cases by A60, A58, XXREAL_0: 1;

            case n1 = 1 & n2 = 1;

            hence thesis;

          end;

            case that

             A61: n1 = 1 and

             A62: n2 > 1;

            

             A63: (n2 - 1) <= ((1 + ( len ( mid (f,(( Index (p,f)) + 1),( len f))))) - 1) by A5, A59, XREAL_1: 9;

            n1 <= ( len <*p*>) by A61, FINSEQ_1: 39;

            then n1 in ( dom <*p*>) by A61, FINSEQ_3: 25;

            then

             A64: (g . n1) = ( <*p*> . n1) by A4, FINSEQ_1:def 7;

            (n2 - 1) > 0 by A62, XREAL_1: 50;

            then

             A65: (n2 -' 1) = (n2 - 1) by XREAL_0:def 2;

            

            then

             A66: (( len <*p*>) + (n2 -' 1)) = (1 + (n2 - 1)) by FINSEQ_1: 39

            .= n2;

            

             A67: 1 <= (n2 -' 1) by A62, A65, SPPOL_1: 1;

            then (n2 -' 1) in ( dom ( mid (f,(( Index (p,f)) + 1),( len f)))) by A65, A63, FINSEQ_3: 25;

            

            then (g . n2) = (( mid (f,(( Index (p,f)) + 1),( len f))) . (n2 -' 1)) by A4, A66, FINSEQ_1:def 7

            .= (f . (((n2 -' 1) + (( Index (p,f)) + 1)) -' 1)) by A7, A17, A21, A65, A63, A67, FINSEQ_6: 118

            .= (f . ((n2 + ( Index (p,f))) -' 1)) by A65;

            then

             A68: (f . ((n2 + ( Index (p,f))) -' 1)) = p by A56, A61, A64, FINSEQ_1: 40;

            (n2 -' 1) <= (( len f) - ( Index (p,f))) by A47, A48, A59, NAT_D: 42;

            then (n2 - 1) <= (( len f) - ( Index (p,f))) by A62, XREAL_1: 233;

            then

             A69: ((n2 - 1) + ( Index (p,f))) <= ((( len f) - ( Index (p,f))) + ( Index (p,f))) by XREAL_1: 6;

            (1 + 1) < (n2 + ( Index (p,f))) by A2, A62, Th8, XREAL_1: 8;

            then

             A70: 1 < ((n2 + ( Index (p,f))) - 1) by XREAL_1: 20;

            then ((n2 + ( Index (p,f))) -' 1) = ((n2 + ( Index (p,f))) - 1) by XREAL_0:def 2;

            hence contradiction by A1, A3, A70, A68, A69, Th12;

          end;

            case that

             A71: n1 > 1 and

             A72: n2 = 1;

            

             A73: (n1 - 1) <= ((1 + ( len ( mid (f,(( Index (p,f)) + 1),( len f))))) - 1) by A5, A57, XREAL_1: 9;

            n2 <= ( len <*p*>) by A72, FINSEQ_1: 39;

            then n2 in ( dom <*p*>) by A72, FINSEQ_3: 25;

            then

             A74: (g . n2) = ( <*p*> . n2) by A4, FINSEQ_1:def 7;

            (n1 - 1) > 0 by A71, XREAL_1: 50;

            then

             A75: (n1 -' 1) = (n1 - 1) by XREAL_0:def 2;

            

            then

             A76: (( len <*p*>) + (n1 -' 1)) = (1 + (n1 - 1)) by FINSEQ_1: 39

            .= n1;

            

             A77: 1 <= (n1 -' 1) by A71, A75, SPPOL_1: 1;

            then (n1 -' 1) in ( dom ( mid (f,(( Index (p,f)) + 1),( len f)))) by A75, A73, FINSEQ_3: 25;

            

            then (g . n1) = (( mid (f,(( Index (p,f)) + 1),( len f))) . (n1 -' 1)) by A4, A76, FINSEQ_1:def 7

            .= (f . (((n1 -' 1) + (( Index (p,f)) + 1)) -' 1)) by A7, A17, A21, A75, A73, A77, FINSEQ_6: 118

            .= (f . ((n1 + ( Index (p,f))) -' 1)) by A75;

            then

             A78: (f . ((n1 + ( Index (p,f))) -' 1)) = p by A56, A72, A74, FINSEQ_1: 40;

            (n1 -' 1) <= (( len f) - ( Index (p,f))) by A47, A48, A57, NAT_D: 42;

            then (n1 - 1) <= (( len f) - ( Index (p,f))) by A71, XREAL_1: 233;

            then

             A79: ((n1 - 1) + ( Index (p,f))) <= ((( len f) - ( Index (p,f))) + ( Index (p,f))) by XREAL_1: 6;

            (1 + 1) < (n1 + ( Index (p,f))) by A2, A71, Th8, XREAL_1: 8;

            then

             A80: 1 < ((n1 + ( Index (p,f))) - 1) by XREAL_1: 20;

            then ((n1 + ( Index (p,f))) -' 1) = ((n1 + ( Index (p,f))) - 1) by XREAL_0:def 2;

            hence contradiction by A1, A3, A80, A78, A79, Th12;

          end;

            case that

             A81: n1 > 1 and

             A82: n2 > 1;

            

             A83: (n2 - 1) > 0 by A82, XREAL_1: 50;

            then

             A84: (n2 -' 1) = (n2 - 1) by XREAL_0:def 2;

            

            then

             A85: (( len <*p*>) + (n2 -' 1)) = (1 + (n2 - 1)) by FINSEQ_1: 39

            .= n2;

            

             A86: (n1 - 1) > 0 by A81, XREAL_1: 50;

            then

             A87: (n1 -' 1) = (n1 - 1) by XREAL_0:def 2;

            then

             A88: ( 0 + 1) <= (n1 -' 1) by A86, NAT_1: 13;

            then

             A89: 1 <= ((n1 - 1) + ( Index (p,f))) by A87, NAT_1: 12;

            then

             A90: ((n1 + ( Index (p,f))) -' 1) = ((n1 + ( Index (p,f))) - 1) by XREAL_0:def 2;

            

             A91: (( len <*p*>) + (n1 -' 1)) = (1 + (n1 - 1)) by A87, FINSEQ_1: 39

            .= n1;

            

             A92: (n1 - 1) <= ((1 + ( len ( mid (f,(( Index (p,f)) + 1),( len f))))) - 1) by A5, A57, XREAL_1: 9;

            then (n1 -' 1) in ( dom ( mid (f,(( Index (p,f)) + 1),( len f)))) by A87, A88, FINSEQ_3: 25;

            

            then

             A93: (g . n1) = (( mid (f,(( Index (p,f)) + 1),( len f))) . (n1 -' 1)) by A4, A91, FINSEQ_1:def 7

            .= (f . (((n1 -' 1) + (( Index (p,f)) + 1)) -' 1)) by A7, A17, A21, A87, A88, A92, FINSEQ_6: 118

            .= (f . ((n1 + ( Index (p,f))) -' 1)) by A87;

            (n1 -' 1) <= (( len f) -' ( Index (p,f))) by A53, A57, NAT_D: 42;

            then ((n1 -' 1) + ( Index (p,f))) <= ((( len f) - ( Index (p,f))) + ( Index (p,f))) by A51, XREAL_1: 6;

            then

             A94: ((n1 + ( Index (p,f))) -' 1) in ( dom f) by A87, A89, A90, FINSEQ_3: 25;

            

             A95: ( 0 + 1) <= (n2 -' 1) by A83, A84, NAT_1: 13;

            then

             A96: 1 <= ((n2 -' 1) + ( Index (p,f))) by NAT_1: 12;

            then

             A97: ((n2 + ( Index (p,f))) -' 1) = ((n2 + ( Index (p,f))) - 1) by A84, XREAL_0:def 2;

            

             A98: (n2 - 1) <= ((1 + ( len ( mid (f,(( Index (p,f)) + 1),( len f))))) - 1) by A5, A59, XREAL_1: 9;

            then (n2 -' 1) in ( dom ( mid (f,(( Index (p,f)) + 1),( len f)))) by A84, A95, FINSEQ_3: 25;

            

            then

             A99: (g . n2) = (( mid (f,(( Index (p,f)) + 1),( len f))) . (n2 -' 1)) by A4, A85, FINSEQ_1:def 7

            .= (f . (((n2 -' 1) + (( Index (p,f)) + 1)) -' 1)) by A7, A17, A21, A84, A95, A98, FINSEQ_6: 118

            .= (f . ((n2 + ( Index (p,f))) -' 1)) by A84;

            (n2 -' 1) <= (( len f) -' ( Index (p,f))) by A53, A59, NAT_D: 42;

            then ((n2 -' 1) + ( Index (p,f))) <= ((( len f) - ( Index (p,f))) + ( Index (p,f))) by A51, XREAL_1: 6;

            then ((n2 + ( Index (p,f))) -' 1) in ( dom f) by A84, A96, A97, FINSEQ_3: 25;

            then ((n1 + ( Index (p,f))) -' 1) = ((n2 + ( Index (p,f))) -' 1) by A1, A56, A99, A93, A94, FUNCT_1:def 4;

            hence thesis by A97, A90;

          end;

        end;

        hence thesis;

      end;

      then

       A100: g is one-to-one by FUNCT_1:def 4;

      

       A101: ((( len g) - 1) + 1) >= (1 + 1) by A18, A47, XREAL_1: 6;

      

       A102: ((( len f) -' (( Index (p,f)) + 1)) + 1) = (( len f) - ( Index (p,f))) by A20;

      for j be Nat st 1 <= j & (j + 1) <= ( len g) holds ((g /. j) `1 ) = ((g /. (j + 1)) `1 ) or ((g /. j) `2 ) = ((g /. (j + 1)) `2 )

      proof

        1 <= ( Index (p,f)) by A2, Th8;

        then

         A103: 1 < (( Index (p,f)) + 1) by NAT_1: 13;

        let j be Nat;

        assume that

         A104: 1 <= j and

         A105: (j + 1) <= ( len g);

        

         A106: ( LSeg (g,j)) = ( LSeg ((g /. j),(g /. (j + 1)))) by A104, A105, TOPREAL1:def 3;

        ((j + 1) - 1) <= ((1 + ( len ( mid (f,(( Index (p,f)) + 1),( len f))))) - 1) by A5, A105, XREAL_1: 9;

        then ((j + 1) - 1) <= (( len f) - ( Index (p,f))) by A7, A17, A21, A102, FINSEQ_6: 118;

        then

         A107: (j + ( Index (p,f))) <= ((( len f) - ( Index (p,f))) + ( Index (p,f))) by XREAL_1: 6;

        (( Index (p,f)) + 1) <= (( Index (p,f)) + j) by A104, XREAL_1: 6;

        then 1 < (( Index (p,f)) + j) by A103, XXREAL_0: 2;

        then

         A108: 1 <= ((( Index (p,f)) + j) - 1) by SPPOL_1: 1;

        then

         A109: ((( Index (p,f)) + j) - 1) = ((( Index (p,f)) + j) -' 1) by XREAL_0:def 2;

        then

         A110: ( LSeg (f,((( Index (p,f)) + j) -' 1))) = ( LSeg ((f /. ((( Index (p,f)) + j) -' 1)),(f /. (((( Index (p,f)) + j) -' 1) + 1)))) by A108, A107, TOPREAL1:def 3;

        

         A111: ((f /. ((( Index (p,f)) + j) -' 1)) `1 ) = ((f /. (((( Index (p,f)) + j) -' 1) + 1)) `1 ) or ((f /. ((( Index (p,f)) + j) -' 1)) `2 ) = ((f /. (((( Index (p,f)) + j) -' 1) + 1)) `2 ) by A1, A108, A109, A107, TOPREAL1:def 5;

        ( LSeg (g,j)) c= ( LSeg (f,((( Index (p,f)) + j) -' 1))) by A2, A4, A104, A105, Th16;

        hence thesis by A106, A110, A111, Th3;

      end;

      then g is unfolded s.n.c. special by A23, A8, TOPREAL1:def 5, TOPREAL1:def 6, TOPREAL1:def 7;

      then

       A112: g is being_S-Seq by A101, A100, TOPREAL1:def 8;

      

       A113: ((( len f) -' ( Index (p,f))) + (( Index (p,f)) + 1)) = ((( len f) - ( Index (p,f))) + (( Index (p,f)) + 1)) by A50, XREAL_0:def 2

      .= (( len f) + 1);

      (1 + (( len g) -' 1)) = (1 + (( len g) - 1)) by A46, XREAL_0:def 2

      .= ( len g);

      

      then (g . ( len g)) = (g . (( len <*p*>) + (( len g) -' 1))) by FINSEQ_1: 39

      .= (( mid (f,(( Index (p,f)) + 1),( len f))) . (( len g) -' 1)) by A4, A49, FINSEQ_1:def 7;

      then (g . ( len g)) = (f . ( len f)) by A47, A48, A52, A113, NAT_D: 34;

      then

       A114: (g . ( len g)) = (f /. ( len f)) by A7, FINSEQ_4: 15;

      (g . 1) = p by A4, FINSEQ_1: 41;

      hence thesis by A112, A114;

    end;

    theorem :: JORDAN3:18

    

     Th18: for f,g be FinSequence of ( TOP-REAL 2), p be Point of ( TOP-REAL 2), j be Nat st p in ( L~ f) & 1 <= j & (j + 1) <= ( len g) & g = (( mid (f,1,( Index (p,f)))) ^ <*p*>) holds ( LSeg (g,j)) c= ( LSeg (f,j))

    proof

      let f,g be FinSequence of ( TOP-REAL 2), p be Point of ( TOP-REAL 2), j be Nat;

      assume that

       A1: p in ( L~ f) and

       A2: 1 <= j and

       A3: (j + 1) <= ( len g) and

       A4: g = (( mid (f,1,( Index (p,f)))) ^ <*p*>);

      

       A5: ( Index (p,f)) < ( len f) by A1, Th8;

      

       A6: 1 <= (j + 1) by NAT_1: 11;

      

       A7: 1 <= ( Index (p,f)) by A1, Th8;

      1 <= ( Index (p,f)) by A1, Th8;

      then

       A8: 1 <= ( len f) by A5, XXREAL_0: 2;

      j <= (j + 1) by NAT_1: 11;

      then

       A9: j <= ( len g) by A3, XXREAL_0: 2;

      now

        ( len g) = (( len ( mid (f,1,( Index (p,f))))) + ( len <*p*>)) by A4, FINSEQ_1: 22

        .= (( len ( mid (f,1,( Index (p,f))))) + 1) by FINSEQ_1: 39;

        then ( len g) = (((( Index (p,f)) -' 1) + 1) + 1) by A5, A8, A7, FINSEQ_6: 118;

        then

         A10: ( len g) = (( Index (p,f)) + 1) by A1, Th8, XREAL_1: 235;

        then

         A11: j <= ( Index (p,f)) by A3, XREAL_1: 6;

        (( Index (p,f)) + 1) <= (( len f) + 1) by A5, XREAL_1: 6;

        then (j + 1) <= (( len f) + 1) by A3, A10, XXREAL_0: 2;

        then

         A12: ((j + 1) - 1) <= ((( len f) + 1) - 1) by XREAL_1: 9;

        

         A13: ( len ( mid (f,1,( Index (p,f))))) = ((( Index (p,f)) -' 1) + 1) by A5, A8, A7, FINSEQ_6: 118

        .= ( Index (p,f)) by A1, Th8, XREAL_1: 235;

        then

         A14: j in ( dom ( mid (f,1,( Index (p,f))))) by A2, A11, FINSEQ_3: 25;

        

         A15: (g /. j) = (g . j) by A2, A9, FINSEQ_4: 15

        .= (( mid (f,1,( Index (p,f)))) . j) by A4, A14, FINSEQ_1:def 7

        .= (f . ((j + 1) -' 1)) by A2, A5, A8, A7, A11, A13, FINSEQ_6: 118

        .= (f . j) by NAT_D: 34

        .= (f /. j) by A2, A12, FINSEQ_4: 15;

        now

          per cases ;

            case

             A16: (j + 1) <= ( Index (p,f));

            

             A17: ( len ( mid (f,1,( Index (p,f))))) = ((( Index (p,f)) -' 1) + 1) by A5, A8, A7, FINSEQ_6: 118

            .= ( Index (p,f)) by A1, Th8, XREAL_1: 235;

            then

             A18: (j + 1) in ( dom ( mid (f,1,( Index (p,f))))) by A6, A16, FINSEQ_3: 25;

            

             A19: ( LSeg (g,j)) = ( LSeg ((g /. j),(g /. (j + 1)))) by A2, A3, TOPREAL1:def 3;

            

             A20: (j + 1) <= ( len f) by A5, A16, XXREAL_0: 2;

            (g /. (j + 1)) = (g . (j + 1)) by A3, FINSEQ_4: 15, NAT_1: 11

            .= (( mid (f,1,( Index (p,f)))) . (j + 1)) by A4, A18, FINSEQ_1:def 7

            .= (f . (((j + 1) + 1) -' 1)) by A5, A8, A7, A6, A16, A17, FINSEQ_6: 118

            .= (f . (j + 1)) by NAT_D: 34

            .= (f /. (j + 1)) by A20, FINSEQ_4: 15, NAT_1: 11;

            hence thesis by A2, A15, A20, A19, TOPREAL1:def 3;

          end;

            case (j + 1) > ( Index (p,f));

            then j >= ( Index (p,f)) by NAT_1: 13;

            then

             A21: j = ( Index (p,f)) by A11, XXREAL_0: 1;

            then

             A22: p in ( LSeg (f,j)) by A1, Th9;

            (j + 1) <= ( len f) by A1, A21, Th8, NAT_1: 13;

            then

             A23: ( LSeg (f,j)) = ( LSeg ((f /. j),(f /. (j + 1)))) by A2, TOPREAL1:def 3;

            1 <= ( len <*p*>) by FINSEQ_1: 40;

            then

             A24: 1 in ( dom <*p*>) by FINSEQ_3: 25;

            

             A25: ( len ( mid (f,1,( Index (p,f))))) = ((( Index (p,f)) -' 1) + 1) by A5, A8, A7, FINSEQ_6: 118

            .= ( Index (p,f)) by A1, Th8, XREAL_1: 235;

            

             A26: (f /. j) in ( LSeg ((f /. j),(f /. (j + 1)))) by RLTOPSP1: 68;

            (g /. (j + 1)) = (g . (j + 1)) by A3, FINSEQ_4: 15, NAT_1: 11

            .= ( <*p*> . 1) by A4, A21, A24, A25, FINSEQ_1:def 7

            .= p by FINSEQ_1:def 8;

            then ( LSeg ((g /. j),(g /. (j + 1)))) c= ( LSeg ((f /. j),(f /. (j + 1)))) by A15, A26, A22, A23, TOPREAL1: 6;

            hence thesis by A2, A3, A23, TOPREAL1:def 3;

          end;

        end;

        hence thesis;

      end;

      hence thesis;

    end;

    theorem :: JORDAN3:19

    

     Th19: for f,g be FinSequence of ( TOP-REAL 2), p be Point of ( TOP-REAL 2) st f is being_S-Seq & p in ( L~ f) & p <> (f . 1) & g = (( mid (f,1,( Index (p,f)))) ^ <*p*>) holds g is_S-Seq_joining ((f /. 1),p)

    proof

      let f,g be FinSequence of ( TOP-REAL 2), p be Point of ( TOP-REAL 2);

      assume that

       A1: f is being_S-Seq and

       A2: p in ( L~ f) and

       A3: p <> (f . 1) and

       A4: g = (( mid (f,1,( Index (p,f)))) ^ <*p*>);

      

       A5: ( Index (p,f)) <= ( len f) by A2, Th8;

      

       A6: for j1,j2 be Nat st (j1 + 1) < j2 holds ( LSeg (g,j1)) misses ( LSeg (g,j2))

      proof

        let j1,j2 be Nat;

        assume

         A7: (j1 + 1) < j2;

        

         A8: j1 = 0 or j1 >= ( 0 + 1) by NAT_1: 13;

        now

          per cases by A8, XXREAL_0: 1;

            case j1 = 0 ;

            then ( LSeg (g,j1)) = {} by TOPREAL1:def 3;

            then (( LSeg (g,j1)) /\ ( LSeg (g,j2))) = {} ;

            hence thesis by XBOOLE_0:def 7;

          end;

            case that

             A9: j1 = 1 or j1 > 1 and

             A10: (j2 + 1) <= ( len g);

            j2 < ( len g) by A10, NAT_1: 13;

            then (j1 + 1) < ( len g) by A7, XXREAL_0: 2;

            then

             A11: ( LSeg (g,j1)) c= ( LSeg (f,j1)) by A2, A4, A9, Th18;

            (1 + 1) <= (j1 + 1) by A9, XREAL_1: 6;

            then 2 <= j2 by A7, XXREAL_0: 2;

            then 1 <= j2 by XXREAL_0: 2;

            then

             A12: ( LSeg (g,j2)) c= ( LSeg (f,j2)) by A2, A4, A10, Th18;

            ( LSeg (f,j1)) misses ( LSeg (f,j2)) by A1, A7, TOPREAL1:def 7;

            then (( LSeg (f,j1)) /\ ( LSeg (f,j2))) = {} by XBOOLE_0:def 7;

            then (( LSeg (g,j1)) /\ ( LSeg (g,j2))) = {} by A11, A12, XBOOLE_1: 3, XBOOLE_1: 27;

            hence thesis by XBOOLE_0:def 7;

          end;

            case (j2 + 1) > ( len g);

            then ( LSeg (g,j2)) = {} by TOPREAL1:def 3;

            then (( LSeg (g,j1)) /\ ( LSeg (g,j2))) = {} ;

            hence thesis by XBOOLE_0:def 7;

          end;

        end;

        hence thesis;

      end;

      

       A13: for n1,n2 be Element of NAT st 1 <= n1 & n1 <= ( len f) & 1 <= n2 & n2 <= ( len f) & (f . n1) = (f . n2) holds n1 = n2

      proof

        let n1,n2 be Element of NAT ;

        assume that

         A14: 1 <= n1 and

         A15: n1 <= ( len f) and

         A16: 1 <= n2 and

         A17: n2 <= ( len f) and

         A18: (f . n1) = (f . n2);

        

         A19: n2 in ( dom f) by A16, A17, FINSEQ_3: 25;

        n1 in ( dom f) by A14, A15, FINSEQ_3: 25;

        hence thesis by A1, A18, A19, FUNCT_1:def 4;

      end;

      

       A20: ( len g) = (( len ( mid (f,1,( Index (p,f))))) + ( len <*p*>)) by A4, FINSEQ_1: 22

      .= (( len ( mid (f,1,( Index (p,f))))) + 1) by FINSEQ_1: 39;

      consider i such that 1 <= i and

       A21: (i + 1) <= ( len f) and p in ( LSeg (f,i)) by A2, SPPOL_2: 13;

      

       A22: 1 <= ( Index (p,f)) by A2, Th8;

      1 <= (1 + i) by NAT_1: 12;

      then

       A23: 1 <= ( len f) by A21, XXREAL_0: 2;

      then

       A24: ( len ( mid (f,1,( Index (p,f))))) = ((( Index (p,f)) -' 1) + 1) by A22, A5, FINSEQ_6: 118;

      then

       A25: ( len ( mid (f,1,( Index (p,f))))) = ( Index (p,f)) by A2, Th8, XREAL_1: 235;

      then (g . 1) = (( mid (f,1,( Index (p,f)))) . 1) by A4, A22, FINSEQ_6: 109;

      then (g . 1) = (f . 1) by A22, A5, A23, FINSEQ_6: 118;

      then

       A26: (g . 1) = (f /. 1) by A23, FINSEQ_4: 15;

      

       A27: for j be Nat st 1 <= j & (j + 2) <= ( len g) holds (( LSeg (g,j)) /\ ( LSeg (g,(j + 1)))) = {(g /. (j + 1))}

      proof

        let j be Nat;

        assume that

         A28: 1 <= j and

         A29: (j + 2) <= ( len g);

        

         A30: (j + 1) <= ( len g) by A29, NAT_D: 47;

        then ( LSeg (g,j)) = ( LSeg ((g /. j),(g /. (j + 1)))) by A28, TOPREAL1:def 3;

        then

         A31: (g /. (j + 1)) in ( LSeg (g,j)) by RLTOPSP1: 68;

        

         A32: 1 <= (j + 1) by A28, NAT_D: 48;

        then ( LSeg (g,(j + 1))) = ( LSeg ((g /. (j + 1)),(g /. ((j + 1) + 1)))) by A29, TOPREAL1:def 3;

        then (g /. (j + 1)) in ( LSeg (g,(j + 1))) by RLTOPSP1: 68;

        then (g /. (j + 1)) in (( LSeg (g,j)) /\ ( LSeg (g,(j + 1)))) by A31, XBOOLE_0:def 4;

        then

         A33: {(g /. (j + 1))} c= (( LSeg (g,j)) /\ ( LSeg (g,(j + 1)))) by ZFMISC_1: 31;

        (j + 1) <= ( len g) by A29, NAT_D: 47;

        then

         A34: ( LSeg (g,j)) c= ( LSeg (f,j)) by A2, A4, A28, Th18;

        

         A35: ( Index (p,f)) <= ( len f) by A2, Th8;

        

         A36: ((j + 1) + 1) <= ( len g) by A29;

        then ( LSeg (g,(j + 1))) c= ( LSeg (f,(j + 1))) by A2, A4, A32, Th18;

        then

         A37: (( LSeg (g,j)) /\ ( LSeg (g,(j + 1)))) c= (( LSeg (f,j)) /\ ( LSeg (f,(j + 1)))) by A34, XBOOLE_1: 27;

        

         A38: (g /. (j + 1)) = (g . (j + 1)) by A32, A30, FINSEQ_4: 15;

        now

          

           A39: ( len g) = (( len ( mid (f,1,( Index (p,f))))) + 1) by A4, FINSEQ_2: 16;

          ( Index (p,f)) <= ( len f) by A2, Th8;

          then

           A40: ( len g) <= (( len f) + 1) by A25, A39, XREAL_1: 6;

          now

            per cases by A40, XXREAL_0: 1;

              case ( len g) = (( len f) + 1);

              hence contradiction by A2, A25, A39, Th8;

            end;

              case ( len g) < (( len f) + 1);

              then ( len g) <= ( len f) by NAT_1: 13;

              then (j + 2) <= ( len f) by A29, XXREAL_0: 2;

              then

               A41: (( LSeg (g,j)) /\ ( LSeg (g,(j + 1)))) c= {(f /. (j + 1))} by A1, A28, A37, TOPREAL1:def 6;

              

               A42: (j + 1) <= ( Index (p,f)) by A25, A36, A39, XREAL_1: 6;

              then (j + 1) <= ( len f) by A35, XXREAL_0: 2;

              then

               A43: (f . (j + 1)) = (f /. (j + 1)) by A32, FINSEQ_4: 15;

              (g . (j + 1)) = (( mid (f,1,( Index (p,f)))) . (j + 1)) by A4, A25, A32, A42, FINSEQ_1: 64

              .= (f . (j + 1)) by A5, A32, A42, FINSEQ_6: 123;

              hence thesis by A38, A33, A41, A43, XBOOLE_0:def 10;

            end;

          end;

          hence thesis;

        end;

        hence thesis;

      end;

      for j be Nat st 1 <= j & (j + 1) <= ( len g) holds ((g /. j) `1 ) = ((g /. (j + 1)) `1 ) or ((g /. j) `2 ) = ((g /. (j + 1)) `2 )

      proof

        

         A44: ( Index (p,f)) < ( len f) by A2, Th8;

        let j be Nat;

        assume that

         A45: 1 <= j and

         A46: (j + 1) <= ( len g);

        

         A47: ( LSeg (g,j)) = ( LSeg ((g /. j),(g /. (j + 1)))) by A45, A46, TOPREAL1:def 3;

        (j + 1) <= (( Index (p,f)) + 1) by A4, A25, A46, FINSEQ_2: 16;

        then j <= ( Index (p,f)) by XREAL_1: 6;

        then j < ( len f) by A44, XXREAL_0: 2;

        then

         A48: (j + 1) <= ( len f) by NAT_1: 13;

        then

         A49: ( LSeg (f,j)) = ( LSeg ((f /. j),(f /. (j + 1)))) by A45, TOPREAL1:def 3;

        

         A50: ((f /. j) `1 ) = ((f /. (j + 1)) `1 ) or ((f /. j) `2 ) = ((f /. (j + 1)) `2 ) by A1, A45, A48, TOPREAL1:def 5;

        ( LSeg (g,j)) c= ( LSeg (f,j)) by A2, A4, A45, A46, Th18;

        hence thesis by A47, A49, A50, Th3;

      end;

      then

       A51: g is unfolded s.n.c. special by A27, A6, TOPREAL1:def 5, TOPREAL1:def 6, TOPREAL1:def 7;

      1 <= ( len <*p*>) by FINSEQ_1: 39;

      then

       A52: 1 in ( dom <*p*>) by FINSEQ_3: 25;

      for x1,x2 be object st x1 in ( dom g) & x2 in ( dom g) & (g . x1) = (g . x2) holds x1 = x2

      proof

        let x1,x2 be object;

        assume that

         A53: x1 in ( dom g) and

         A54: x2 in ( dom g) and

         A55: (g . x1) = (g . x2);

        reconsider n1 = x1, n2 = x2 as Element of NAT by A53, A54;

        

         A56: 1 <= n1 by A53, FINSEQ_3: 25;

        

         A57: n2 <= ( len g) by A54, FINSEQ_3: 25;

        

         A58: 1 <= n2 by A54, FINSEQ_3: 25;

        

         A59: n1 <= ( len g) by A53, FINSEQ_3: 25;

        now

          

           A60: (g . ( len g)) = ( <*p*> . 1) by A4, A52, A20, FINSEQ_1:def 7

          .= p by FINSEQ_1:def 8;

          now

            per cases ;

              case

               A61: n1 = ( len g);

              now

                assume

                 A62: n2 <> ( len g);

                then n2 < ( len g) by A57, XXREAL_0: 1;

                then

                 A63: n2 <= ( len ( mid (f,1,( Index (p,f))))) by A20, NAT_1: 13;

                then

                 A64: n2 <= ( len f) by A5, A25, XXREAL_0: 2;

                (g . n2) = (( mid (f,1,( Index (p,f)))) . n2) by A4, A58, A63, FINSEQ_1: 64;

                then (g . n2) = (f . ((n2 + 1) -' 1)) by A22, A5, A23, A58, A63, FINSEQ_6: 118;

                then

                 A65: p = (f . n2) by A55, A60, A61, NAT_D: 34;

                then 1 < n2 by A3, A58, XXREAL_0: 1;

                then (( Index (p,f)) + 1) = n2 by A1, A65, A64, Th12;

                hence contradiction by A2, A24, A20, A62, Th8, XREAL_1: 235;

              end;

              hence thesis by A61;

            end;

              case

               A66: n2 = ( len g);

              now

                assume

                 A67: n1 <> ( len g);

                then n1 < ( len g) by A59, XXREAL_0: 1;

                then

                 A68: n1 <= ( len ( mid (f,1,( Index (p,f))))) by A20, NAT_1: 13;

                then

                 A69: n1 <= ( len f) by A5, A25, XXREAL_0: 2;

                (g . n1) = (( mid (f,1,( Index (p,f)))) . n1) by A4, A56, A68, FINSEQ_1: 64;

                then (g . n1) = (f . ((n1 + 1) -' 1)) by A22, A5, A23, A56, A68, FINSEQ_6: 118;

                then

                 A70: p = (f . n1) by A55, A60, A66, NAT_D: 34;

                then 1 < n1 by A3, A56, XXREAL_0: 1;

                then (( Index (p,f)) + 1) = n1 by A1, A70, A69, Th12;

                hence contradiction by A2, A24, A20, A67, Th8, XREAL_1: 235;

              end;

              hence thesis by A66;

            end;

              case that

               A71: n1 <> ( len g) and

               A72: n2 <> ( len g);

              n1 < ( len g) by A59, A71, XXREAL_0: 1;

              then

               A73: n1 <= ( len ( mid (f,1,( Index (p,f))))) by A20, NAT_1: 13;

              then

               A74: n1 <= ( len f) by A5, A25, XXREAL_0: 2;

              n2 < ( len g) by A57, A72, XXREAL_0: 1;

              then

               A75: n2 <= ( len ( mid (f,1,( Index (p,f))))) by A20, NAT_1: 13;

              

              then

               A76: (g . n2) = (( mid (f,1,( Index (p,f)))) . n2) by A4, A58, FINSEQ_1: 64

              .= (f . n2) by A5, A25, A58, A75, FINSEQ_6: 123;

              

               A77: n2 <= ( len f) by A5, A25, A75, XXREAL_0: 2;

              (g . n1) = (( mid (f,1,( Index (p,f)))) . n1) by A4, A56, A73, FINSEQ_1: 64

              .= (f . n1) by A5, A25, A56, A73, FINSEQ_6: 123;

              hence thesis by A13, A55, A56, A58, A74, A77, A76;

            end;

          end;

          hence thesis;

        end;

        hence thesis;

      end;

      then

       A78: g is one-to-one by FUNCT_1:def 4;

      (1 + 1) <= ( len g) by A22, A25, A20, XREAL_1: 6;

      then

       A79: g is being_S-Seq by A78, A51, TOPREAL1:def 8;

      (g . ( len g)) = p by A4, A20, FINSEQ_1: 42;

      hence thesis by A26, A79;

    end;

    begin

    definition

      let f be FinSequence of ( TOP-REAL 2), p be Point of ( TOP-REAL 2);

      :: JORDAN3:def3

      func L_Cut (f,p) -> FinSequence of ( TOP-REAL 2) equals

      : Def3: ( <*p*> ^ ( mid (f,(( Index (p,f)) + 1),( len f)))) if p <> (f . (( Index (p,f)) + 1))

      otherwise ( mid (f,(( Index (p,f)) + 1),( len f)));

      correctness ;

      :: JORDAN3:def4

      func R_Cut (f,p) -> FinSequence of ( TOP-REAL 2) equals

      : Def4: (( mid (f,1,( Index (p,f)))) ^ <*p*>) if p <> (f . 1)

      otherwise <*p*>;

      correctness ;

    end

    theorem :: JORDAN3:20

    

     Th20: for f be FinSequence of ( TOP-REAL 2), p be Point of ( TOP-REAL 2) st f is being_S-Seq & p in ( L~ f) & p = (f . (( Index (p,f)) + 1)) & p <> (f . ( len f)) holds ((( Index (p,( Rev f))) + ( Index (p,f))) + 1) = ( len f)

    proof

      let f be FinSequence of ( TOP-REAL 2), p be Point of ( TOP-REAL 2) such that

       A1: f is being_S-Seq and

       A2: p in ( L~ f) and

       A3: p = (f . (( Index (p,f)) + 1)) and

       A4: p <> (f . ( len f));

      

       A5: ( len f) <= (( len f) + ( Index (p,f))) by NAT_1: 11;

      ( len f) = ( len ( Rev f)) by FINSEQ_5:def 3;

      then

       A6: (( len f) - ( Index (p,f))) <= ( len ( Rev f)) by A5, XREAL_1: 20;

      ( Index (p,f)) <= ( len f) by A2, Th8;

      then

       A7: (( len f) - ( Index (p,f))) = (( len f) -' ( Index (p,f))) by XREAL_1: 233;

      ( Index (p,f)) < ( len f) by A2, Th8;

      then

       A8: (( Index (p,f)) + 1) <= ( len f) by NAT_1: 13;

      then (( Index (p,f)) + 1) < ( len f) by A3, A4, XXREAL_0: 1;

      then

       A9: 1 < (( len f) - ( Index (p,f))) by XREAL_1: 20;

      1 <= (( Index (p,f)) + 1) by NAT_1: 11;

      then (( Index (p,f)) + 1) in ( dom f) by A8, FINSEQ_3: 25;

      then

       A10: (( Index (p,f)) + 1) in ( dom ( Rev f)) by FINSEQ_5: 57;

      p = (( Rev ( Rev f)) . (( Index (p,f)) + 1)) by A3

      .= (( Rev f) . ((( len ( Rev f)) - (( Index (p,f)) + 1)) + 1)) by A10, FINSEQ_5: 58

      .= (( Rev f) . (( len ( Rev f)) - ( Index (p,f))))

      .= (( Rev f) . (( len f) - ( Index (p,f)))) by FINSEQ_5:def 3;

      then (( Index (p,( Rev f))) + 1) = (( len f) -' ( Index (p,f))) by A1, A6, A9, A7, Th12;

      hence thesis by A7;

    end;

    theorem :: JORDAN3:21

    

     Th21: for f be FinSequence of ( TOP-REAL 2) holds for p be Point of ( TOP-REAL 2) st f is unfolded s.n.c. & p in ( L~ f) & p <> (f . (( Index (p,f)) + 1)) holds (( Index (p,( Rev f))) + ( Index (p,f))) = ( len f)

    proof

      let f be FinSequence of ( TOP-REAL 2);

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

       A1: f is unfolded s.n.c. and

       A2: p in ( L~ f) and

       A3: p <> (f . (( Index (p,f)) + 1));

      

       A4: ( Index (p,f)) < ( len f) by A2, Th8;

      then

       A5: ((( len f) -' ( Index (p,f))) + ( Index (p,f))) = ( len f) by XREAL_1: 235;

      ( 0 + 1) <= ( Index (p,f)) by A2, Th8;

      then (( len f) + 0 ) < (( len f) + ( Index (p,f))) by XREAL_1: 6;

      then (( len f) - ( Index (p,f))) < ( len f) by XREAL_1: 19;

      then

       A6: (( len f) -' ( Index (p,f))) < ( len f) by A4, XREAL_1: 233;

      

       A7: ( Index (p,f)) < ( len f) by A2, Th8;

      then (( Index (p,f)) + 1) <= ( len f) by NAT_1: 13;

      then 1 <= (( len f) - ( Index (p,f))) by XREAL_1: 19;

      then 1 <= (( len f) -' ( Index (p,f))) by NAT_D: 39;

      then (( len f) -' ( Index (p,f))) in ( dom f) by A6, FINSEQ_3: 25;

      

      then

       A8: (( Rev f) . (( len f) -' ( Index (p,f)))) = (f . ((( len f) - (( len f) -' ( Index (p,f)))) + 1)) by FINSEQ_5: 58

      .= (f . ((( len f) - (( len f) - ( Index (p,f)))) + 1)) by A7, XREAL_1: 233

      .= (f . (( 0 + ( Index (p,f))) + 1));

      p in ( LSeg (f,( Index (p,f)))) by A2, Th9;

      then

       A9: p in ( LSeg (( Rev f),(( len f) -' ( Index (p,f))))) by A5, SPPOL_2: 2;

      ( len f) = ( len ( Rev f)) by FINSEQ_5:def 3;

      then

       A10: ((( len f) -' ( Index (p,f))) + 1) <= ( len ( Rev f)) by A6, NAT_1: 13;

      ( Rev f) is s.n.c. by A1, SPPOL_2: 35;

      then (( len f) -' ( Index (p,f))) = ( Index (p,( Rev f))) by A1, A3, A9, A10, A8, Th14, SPPOL_2: 28;

      hence thesis by A7, XREAL_1: 235;

    end;

    theorem :: JORDAN3:22

    

     Th22: for f be FinSequence of ( TOP-REAL 2), p be Point of ( TOP-REAL 2) st f is being_S-Seq & p in ( L~ f) holds ( L_Cut (( Rev f),p)) = ( Rev ( R_Cut (f,p)))

    proof

      let f be FinSequence of ( TOP-REAL 2), p be Point of ( TOP-REAL 2) such that

       A1: f is being_S-Seq and

       A2: p in ( L~ f);

      

       A3: ( len f) = ( len ( Rev f)) by FINSEQ_5:def 3;

      

       A4: p in ( L~ ( Rev f)) by A2, SPPOL_2: 22;

      

       A5: 1 <= ( Index (p,f)) by A2, Th8;

      

       A6: ( Rev f) is being_S-Seq by A1;

      

       A7: ( Rev ( Rev f)) = f;

      

       A8: ( Index (p,f)) < ( len f) by A2, Th8;

      ( L~ f) = ( L~ ( Rev f)) by SPPOL_2: 22;

      then ( Index (p,( Rev f))) < ( len ( Rev f)) by A2, Th8;

      then

       A9: (( Index (p,( Rev f))) + 1) <= ( len f) by A3, NAT_1: 13;

      1 <= (( Index (p,( Rev f))) + 1) by NAT_1: 11;

      then

       A10: (( Index (p,( Rev f))) + 1) in ( dom f) by A9, FINSEQ_3: 25;

      

       A11: (1 + 1) <= ( len f) by A1, TOPREAL1:def 8;

      then

       A12: 1 < ( len f) by NAT_1: 13;

      then

       A13: 1 in ( dom f) by FINSEQ_3: 25;

      

       A14: ( len f) in ( dom f) by A12, FINSEQ_3: 25;

      

       A15: 2 in ( dom f) by A11, FINSEQ_3: 25;

      

       A16: ( dom ( Rev f)) = ( dom f) by FINSEQ_5: 57;

      per cases ;

        suppose

         A17: p = (f . ( len f));

        then

         A18: p <> (f . 1) by A1, A12, A13, A14, FUNCT_1:def 4;

        

         A19: p = (( Rev f) . 1) by A17, FINSEQ_5: 62;

        then

         A20: p <> (( Rev f) . (1 + 1)) by A1, A16, A13, A15, FUNCT_1:def 4;

        p = (( Rev f) /. 1) by A16, A13, A19, PARTFUN1:def 6;

        then

         A21: ( Index (p,( Rev f))) = 1 by A3, A11, Th11;

        then (( Index (p,( Rev f))) + ( Index (p,f))) = ( len f) by A6, A4, A7, A3, A20, Th21;

        then

         A22: ( Index (p,( Rev f))) = (( len f) - ( Index (p,f)));

        

        thus ( L_Cut (( Rev f),p)) = ( <*p*> ^ ( mid (( Rev f),(( Index (p,( Rev f))) + 1),( len f)))) by A3, A21, A20, Def3

        .= ( <*p*> ^ ( mid (( Rev f),((( len f) -' ( Index (p,f))) + 1),( len f)))) by A8, A22, XREAL_1: 233

        .= ( <*p*> ^ ( mid (( Rev f),((( len f) -' ( Index (p,f))) + 1),((( len f) -' 1) + 1)))) by A12, XREAL_1: 235

        .= ( <*p*> ^ ( Rev ( mid (f,1,( Index (p,f)))))) by A12, A5, A8, FINSEQ_6: 113

        .= ( Rev (( mid (f,1,( Index (p,f)))) ^ <*p*>)) by FINSEQ_5: 63

        .= ( Rev ( R_Cut (f,p))) by A18, Def4;

      end;

        suppose

         A23: p = (f . 1);

        

         A24: ((( len ( Rev f)) -' 1) + 1) = ( len ( Rev f)) by A3, A12, XREAL_1: 235;

        then

         A25: ((( Rev f) /^ (( len ( Rev f)) -' 1)) . 1) = (( Rev f) . ( len ( Rev f))) by FINSEQ_6: 114;

        

         A26: ( len (( Rev f) /^ (( len ( Rev f)) -' 1))) = (( len ( Rev f)) -' (( len ( Rev f)) -' 1)) by RFINSEQ: 29;

        1 <= (( len ( Rev f)) - (( len ( Rev f)) -' 1)) by A24;

        then

         A27: 1 <= ( len (( Rev f) /^ (( len ( Rev f)) -' 1))) by A26, NAT_D: 39;

        ((( len ( Rev f)) -' ( len ( Rev f))) + 1) = ((( len ( Rev f)) - ( len ( Rev f))) + 1) by XREAL_1: 233

        .= 1;

        

        then

         A28: ( mid (( Rev f),( len ( Rev f)),( len ( Rev f)))) = ((( Rev f) /^ (( len ( Rev f)) -' 1)) | 1) by FINSEQ_6:def 3

        .= <*((( Rev f) /^ (( len ( Rev f)) -' 1)) . 1)*> by A27, CARD_1: 27, FINSEQ_5: 20

        .= <*(( Rev f) . ( len ( Rev f)))*> by A25;

        

         A29: p = (( Rev f) . ( len f)) by A23, FINSEQ_5: 62;

        then (( Index (p,( Rev f))) + 1) = ( len f) by A1, A3, A12, Th12;

        

        hence ( L_Cut (( Rev f),p)) = <*p*> by A3, A29, A28, Def3

        .= ( Rev <*p*>) by FINSEQ_5: 60

        .= ( Rev ( R_Cut (f,p))) by A23, Def4;

      end;

        suppose that

         A30: p <> (f . 1) and

         A31: p <> (f . ( len f)) and

         A32: p = (f . (( Index (p,f)) + 1));

        

         A33: ( len f) = ((( Index (p,( Rev f))) + ( Index (p,f))) + 1) by A1, A2, A31, A32, Th20

        .= (( Index (p,f)) + (( Index (p,( Rev f))) + 1));

        ( len f) = ((( Index (p,( Rev f))) + ( Index (p,f))) + 1) by A1, A2, A31, A32, Th20

        .= (( Index (p,( Rev f))) + (( Index (p,f)) + 1));

        

        then

         A34: p = (f . ((( len f) - (( Index (p,( Rev f))) + 1)) + 1)) by A32

        .= (( Rev f) . (( Index (p,( Rev f))) + 1)) by A10, FINSEQ_5: 58;

        

         A35: (( len f) -' ( Index (p,f))) = (( len f) - ( Index (p,f))) by A8, XREAL_1: 233

        .= (( Index (p,( Rev f))) + 1) by A33;

        p <> (( Rev f) . ( len f)) by A30, FINSEQ_5: 62;

        then

         A36: (( Index (p,( Rev f))) + 1) < ( len f) by A9, A34, XXREAL_0: 1;

        

        thus ( L_Cut (( Rev f),p)) = ( mid (( Rev f),(( Index (p,( Rev f))) + 1),( len f))) by A3, A34, Def3

        .= ( <*p*> ^ ( mid (( Rev f),((( len f) -' ( Index (p,f))) + 1),( len f)))) by A16, A10, A34, A35, A36, FINSEQ_6: 126

        .= ( <*p*> ^ ( mid (( Rev f),((( len f) -' ( Index (p,f))) + 1),((( len f) -' 1) + 1)))) by A12, XREAL_1: 235

        .= ( <*p*> ^ ( Rev ( mid (f,1,( Index (p,f)))))) by A12, A5, A8, FINSEQ_6: 113

        .= ( Rev (( mid (f,1,( Index (p,f)))) ^ <*p*>)) by FINSEQ_5: 63

        .= ( Rev ( R_Cut (f,p))) by A30, Def4;

      end;

        suppose that

         A37: p <> (f . 1) and

         A38: p <> (f . (( Index (p,f)) + 1));

        

         A39: p <> (( Rev f) . ( len f)) by A37, FINSEQ_5: 62;

         A40:

        now

          assume

           A41: p = (( Rev f) . (( Index (p,( Rev f))) + 1));

          

          then

           A42: ( len ( Rev f)) = ((( Index (p,( Rev ( Rev f)))) + ( Index (p,( Rev f)))) + 1) by A1, A4, A3, A39, Th20

          .= ((( Index (p,f)) + 1) + ( Index (p,( Rev f))));

          p = (f . ((( len f) - (( Index (p,( Rev f))) + 1)) + 1)) by A10, A41, FINSEQ_5: 58

          .= (f . (( Index (p,f)) + 1)) by A3, A42;

          hence contradiction by A38;

        end;

        

         A43: ( Index (p,f)) < ( len f) by A2, Th8;

        ( len f) = (( Index (p,( Rev f))) + ( Index (p,f))) by A1, A2, A38, Th21;

        

        then ( Index (p,( Rev f))) = (( len f) - ( Index (p,f)))

        .= (( len f) -' ( Index (p,f))) by A43, XREAL_1: 233;

        

        hence ( L_Cut (( Rev f),p)) = ( <*p*> ^ ( mid (( Rev f),((( len f) -' ( Index (p,f))) + 1),( len f)))) by A3, A40, Def3

        .= ( <*p*> ^ ( mid (( Rev f),((( len f) -' ( Index (p,f))) + 1),((( len f) -' 1) + 1)))) by A12, XREAL_1: 235

        .= ( <*p*> ^ ( Rev ( mid (f,1,( Index (p,f)))))) by A12, A5, A8, FINSEQ_6: 113

        .= ( Rev (( mid (f,1,( Index (p,f)))) ^ <*p*>)) by FINSEQ_5: 63

        .= ( Rev ( R_Cut (f,p))) by A37, Def4;

      end;

    end;

    theorem :: JORDAN3:23

    

     Th23: for f be FinSequence of ( TOP-REAL 2), p be Point of ( TOP-REAL 2) st p in ( L~ f) holds (( L_Cut (f,p)) . 1) = p & for i st 1 < i & i <= ( len ( L_Cut (f,p))) holds (p = (f . (( Index (p,f)) + 1)) implies (( L_Cut (f,p)) . i) = (f . (( Index (p,f)) + i))) & (p <> (f . (( Index (p,f)) + 1)) implies (( L_Cut (f,p)) . i) = (f . ((( Index (p,f)) + i) - 1)))

    proof

      let f be FinSequence of ( TOP-REAL 2), p be Point of ( TOP-REAL 2);

      assume

       A1: p in ( L~ f);

      then ( Index (p,f)) < ( len f) by Th8;

      then

       A2: (( Index (p,f)) + 1) <= ( len f) by NAT_1: 13;

      

       A3: f is non empty by A1, CARD_1: 27, TOPREAL1: 22;

      now

        per cases ;

          suppose

           A4: p = (f . (( Index (p,f)) + 1));

          1 in ( dom f) by A3, FINSEQ_5: 6;

          then

           A5: 1 <= ( len f) by FINSEQ_3: 25;

          ( Index (p,f)) < ( len f) by A1, Th8;

          then

           A6: (( Index (p,f)) + 1) <= ( len f) by NAT_1: 13;

          

           A7: 1 <= (( Index (p,f)) + 1) by NAT_1: 11;

          ( L_Cut (f,p)) = ( mid (f,(( Index (p,f)) + 1),( len f))) by A4, Def3;

          hence (( L_Cut (f,p)) . 1) = p by A4, A7, A6, A5, FINSEQ_6: 118;

        end;

          suppose p <> (f . (( Index (p,f)) + 1));

          then ( L_Cut (f,p)) = ( <*p*> ^ ( mid (f,(( Index (p,f)) + 1),( len f)))) by Def3;

          hence (( L_Cut (f,p)) . 1) = p by FINSEQ_1: 41;

        end;

      end;

      hence (( L_Cut (f,p)) . 1) = p;

      let i;

      assume that

       A8: 1 < i and

       A9: i <= ( len ( L_Cut (f,p)));

      

       A10: ( len <*p*>) <= i by A8, FINSEQ_1: 40;

      

       A11: 1 <= (( Index (p,f)) + 1) by NAT_1: 11;

      then

       A12: 1 <= ( len f) by A2, XXREAL_0: 2;

      then ( len ( mid (f,(( Index (p,f)) + 1),( len f)))) = ((( len f) -' (( Index (p,f)) + 1)) + 1) by A11, A2, FINSEQ_6: 118;

      

      then

       A13: (( len <*p*>) + ( len ( mid (f,(( Index (p,f)) + 1),( len f))))) = (1 + ((( len f) -' (( Index (p,f)) + 1)) + 1)) by FINSEQ_1: 40

      .= (1 + ((( len f) - (( Index (p,f)) + 1)) + 1)) by A2, XREAL_1: 233

      .= ((( len f) - ( Index (p,f))) + 1);

      

       A14: ((i -' 1) + 1) = ((i - 1) + 1) by A8, XREAL_1: 233

      .= i;

      

       A15: 1 <= (i - 1) by A8, SPPOL_1: 1;

      then

       A16: 1 <= (i -' 1) by NAT_D: 39;

      hereby

        assume p = (f . (( Index (p,f)) + 1));

        then ( L_Cut (f,p)) = ( mid (f,(( Index (p,f)) + 1),( len f))) by Def3;

        

        hence (( L_Cut (f,p)) . i) = (f . ((i + (( Index (p,f)) + 1)) -' 1)) by A8, A9, A11, A2, A12, FINSEQ_6: 118

        .= (f . (((i + ( Index (p,f))) + 1) -' 1))

        .= (f . (( Index (p,f)) + i)) by NAT_D: 34;

      end;

      

       A17: i <= (i + ( Index (p,f))) by NAT_1: 11;

      assume p <> (f . (( Index (p,f)) + 1));

      then

       A18: ( L_Cut (f,p)) = ( <*p*> ^ ( mid (f,(( Index (p,f)) + 1),( len f)))) by Def3;

      then i <= (( len <*p*>) + ( len ( mid (f,(( Index (p,f)) + 1),( len f))))) by A9, FINSEQ_1: 22;

      then (i - 1) <= (((( len f) - ( Index (p,f))) + 1) - 1) by A13, XREAL_1: 9;

      then

       A19: (i -' 1) <= ((( len f) - (( Index (p,f)) + 1)) + 1) by A15, NAT_D: 39;

      ( len <*p*>) < i by A8, FINSEQ_1: 39;

      

      then (( L_Cut (f,p)) . i) = (( mid (f,(( Index (p,f)) + 1),( len f))) . (i - ( len <*p*>))) by A9, A18, FINSEQ_6: 108

      .= (( mid (f,(( Index (p,f)) + 1),( len f))) . (i -' ( len <*p*>))) by A10, XREAL_1: 233

      .= (( mid (f,(( Index (p,f)) + 1),( len f))) . (i -' 1)) by FINSEQ_1: 39

      .= (f . (((i -' 1) + (( Index (p,f)) + 1)) -' 1)) by A11, A2, A16, A19, FINSEQ_6: 122

      .= (f . ((( Index (p,f)) + i) -' 1)) by A14;

      hence thesis by A8, A17, XREAL_1: 233, XXREAL_0: 2;

    end;

    theorem :: JORDAN3:24

    

     Th24: for f be FinSequence of ( TOP-REAL 2) holds for p be Point of ( TOP-REAL 2) st p in ( L~ f) holds (( R_Cut (f,p)) . ( len ( R_Cut (f,p)))) = p & for i be Element of NAT st 1 <= i & i <= ( Index (p,f)) holds (( R_Cut (f,p)) . i) = (f . i)

    proof

      let f be FinSequence of ( TOP-REAL 2);

      let p be Point of ( TOP-REAL 2);

      assume

       A1: p in ( L~ f);

      then

       A2: ( Index (p,f)) < ( len f) by Th8;

      now

        per cases ;

          suppose

           A3: p <> (f . 1);

          

           A4: ( len (( mid (f,1,( Index (p,f)))) ^ <*p*>)) = (( len ( mid (f,1,( Index (p,f))))) + ( len <*p*>)) by FINSEQ_1: 22

          .= (( len ( mid (f,1,( Index (p,f))))) + 1) by FINSEQ_1: 39;

          ( R_Cut (f,p)) = (( mid (f,1,( Index (p,f)))) ^ <*p*>) by A3, Def4;

          hence (( R_Cut (f,p)) . ( len ( R_Cut (f,p)))) = p by A4, FINSEQ_1: 42;

        end;

          suppose p = (f . 1);

          then

           A5: ( R_Cut (f,p)) = <*p*> by Def4;

          then ( len ( R_Cut (f,p))) = 1 by FINSEQ_1: 40;

          hence (( R_Cut (f,p)) . ( len ( R_Cut (f,p)))) = p by A5, FINSEQ_1: 40;

        end;

      end;

      hence (( R_Cut (f,p)) . ( len ( R_Cut (f,p)))) = p;

      

       A6: 1 <= ( Index (p,f)) by A1, Th8;

      then ( len f) > 1 by A2, XXREAL_0: 2;

      

      then

       A7: ( len ( mid (f,1,( Index (p,f))))) = ((( Index (p,f)) -' 1) + 1) by A6, A2, FINSEQ_6: 118

      .= ( Index (p,f)) by A1, Th8, XREAL_1: 235;

      thus for i be Element of NAT st 1 <= i & i <= ( Index (p,f)) holds (( R_Cut (f,p)) . i) = (f . i)

      proof

        let i be Element of NAT ;

        assume that

         A8: 1 <= i and

         A9: i <= ( Index (p,f));

        now

          per cases ;

            case p <> (f . 1);

            

            then (( R_Cut (f,p)) . i) = ((( mid (f,1,( Index (p,f)))) ^ <*p*>) . i) by Def4

            .= (( mid (f,1,( Index (p,f)))) . i) by A7, A8, A9, FINSEQ_1: 64

            .= (f . i) by A2, A8, A9, FINSEQ_6: 123;

            hence thesis;

          end;

            case

             A10: p = (f . 1);

            

             A11: ( len f) > 1 by A6, A2, XXREAL_0: 2;

            then 1 in ( dom f) by FINSEQ_3: 25;

            then

             A12: p = (f /. 1) by A10, PARTFUN1:def 6;

            ( len f) >= (1 + 1) by A11, NAT_1: 13;

            then ( Index (p,f)) = 1 by A12, Th11;

            then

             A13: i = 1 by A8, A9, XXREAL_0: 1;

            ( R_Cut (f,p)) = <*p*> by A10, Def4;

            hence thesis by A10, A13, FINSEQ_1: 40;

          end;

        end;

        hence thesis;

      end;

    end;

    theorem :: JORDAN3:25

    for f be FinSequence of ( TOP-REAL 2) holds for p be Point of ( TOP-REAL 2) st p in ( L~ f) holds (p <> (f . 1) implies ( len ( R_Cut (f,p))) = (( Index (p,f)) + 1)) & (p = (f . 1) implies ( len ( R_Cut (f,p))) = ( Index (p,f)))

    proof

      let f be FinSequence of ( TOP-REAL 2);

      let p be Point of ( TOP-REAL 2);

      assume

       A1: p in ( L~ f);

      then

      consider i be Nat such that

       A2: 1 <= i and

       A3: (i + 1) <= ( len f) and p in ( LSeg (f,i)) by SPPOL_2: 13;

      

       A4: 1 <= ( Index (p,f)) by A1, Th8;

      

       A5: ( Index (p,f)) <= ( len f) by A1, Th8;

      i <= ( len f) by A3, NAT_D: 46;

      then

       A6: 1 <= ( len f) by A2, XXREAL_0: 2;

      now

        per cases ;

          case p <> (f . 1);

          then ( R_Cut (f,p)) = (( mid (f,1,( Index (p,f)))) ^ <*p*>) by Def4;

          

          hence ( len ( R_Cut (f,p))) = (( len ( mid (f,1,( Index (p,f))))) + ( len <*p*>)) by FINSEQ_1: 22

          .= (( len ( mid (f,1,( Index (p,f))))) + 1) by FINSEQ_1: 39

          .= (((( Index (p,f)) -' 1) + 1) + 1) by A6, A4, A5, FINSEQ_6: 118

          .= (( Index (p,f)) + 1) by A1, Th8, XREAL_1: 235;

        end;

          case

           A7: p = (f . 1);

          ( len f) > i by A3, NAT_1: 13;

          then ( len f) > 1 by A2, XXREAL_0: 2;

          then

           A8: ( len f) >= (1 + 1) by NAT_1: 13;

          1 in ( dom f) by A3, CARD_1: 27, FINSEQ_5: 6;

          then

           A9: p = (f /. 1) by A7, PARTFUN1:def 6;

          ( R_Cut (f,p)) = <*p*> by A7, Def4;

          

          hence ( len ( R_Cut (f,p))) = 1 by FINSEQ_1: 39

          .= ( Index (p,f)) by A8, A9, Th11;

        end;

      end;

      hence thesis;

    end;

    theorem :: JORDAN3:26

    

     Th26: for f be FinSequence of ( TOP-REAL 2) holds for p be Point of ( TOP-REAL 2) st p in ( L~ f) holds (p = (f . (( Index (p,f)) + 1)) implies ( len ( L_Cut (f,p))) = (( len f) - ( Index (p,f)))) & (p <> (f . (( Index (p,f)) + 1)) implies ( len ( L_Cut (f,p))) = ((( len f) - ( Index (p,f))) + 1))

    proof

      let f be FinSequence of ( TOP-REAL 2);

      let p be Point of ( TOP-REAL 2);

      assume

       A1: p in ( L~ f);

      then

      consider i be Nat such that

       A2: 1 <= i and

       A3: (i + 1) <= ( len f) and p in ( LSeg (f,i)) by SPPOL_2: 13;

      i <= ( len f) by A3, NAT_D: 46;

      then

       A4: 1 <= ( len f) by A2, XXREAL_0: 2;

      1 <= ( Index (p,f)) by A1, Th8;

      then

       A5: 1 < (( Index (p,f)) + 1) by NAT_1: 13;

      ( Index (p,f)) < ( len f) by A1, Th8;

      then

       A6: ((( Index (p,f)) + 1) + 0 ) <= ( len f) by NAT_1: 13;

      then

       A7: (( len f) - (( Index (p,f)) + 1)) >= 0 by XREAL_1: 19;

      now

        per cases ;

          case p <> (f . (( Index (p,f)) + 1));

          then ( L_Cut (f,p)) = ( <*p*> ^ ( mid (f,(( Index (p,f)) + 1),( len f)))) by Def3;

          

          hence ( len ( L_Cut (f,p))) = (1 + ( len ( mid (f,(( Index (p,f)) + 1),( len f))))) by FINSEQ_5: 8

          .= (((( len f) -' (( Index (p,f)) + 1)) + 1) + 1) by A4, A5, A6, FINSEQ_6: 118

          .= (((( len f) - (( Index (p,f)) + 1)) + 1) + 1) by A7, XREAL_0:def 2

          .= ((( len f) - ( Index (p,f))) + 1);

        end;

          case p = (f . (( Index (p,f)) + 1));

          then ( L_Cut (f,p)) = ( mid (f,(( Index (p,f)) + 1),( len f))) by Def3;

          

          hence ( len ( L_Cut (f,p))) = ((( len f) -' (( Index (p,f)) + 1)) + 1) by A4, A5, A6, FINSEQ_6: 118

          .= ((( len f) - (( Index (p,f)) + 1)) + 1) by A7, XREAL_0:def 2

          .= (( len f) - ( Index (p,f)));

        end;

      end;

      hence thesis;

    end;

    definition

      let p1,p2,q1,q2 be Point of ( TOP-REAL 2);

      :: JORDAN3:def5

      pred LE q1,q2,p1,p2 means q1 in ( LSeg (p1,p2)) & q2 in ( LSeg (p1,p2)) & for r1,r2 be Real st 0 <= r1 & r1 <= 1 & q1 = (((1 - r1) * p1) + (r1 * p2)) & 0 <= r2 & r2 <= 1 & q2 = (((1 - r2) * p1) + (r2 * p2)) holds r1 <= r2;

    end

    definition

      let p1,p2,q1,q2 be Point of ( TOP-REAL 2);

      :: JORDAN3:def6

      pred LT q1,q2,p1,p2 means LE (q1,q2,p1,p2) & q1 <> q2;

    end

    theorem :: JORDAN3:27

    for p1,p2,q1,q2 be Point of ( TOP-REAL 2) st LE (q1,q2,p1,p2) & LE (q2,q1,p1,p2) holds q1 = q2

    proof

      let p1,p2,q1,q2 be Point of ( TOP-REAL 2);

      assume that

       A1: LE (q1,q2,p1,p2) and

       A2: LE (q2,q1,p1,p2);

      q1 in ( LSeg (p1,p2)) by A1;

      then

      consider r1 such that

       A3: q1 = (((1 - r1) * p1) + (r1 * p2)) and

       A4: 0 <= r1 and

       A5: r1 <= 1;

      q2 in ( LSeg (p1,p2)) by A1;

      then

      consider r2 such that

       A6: q2 = (((1 - r2) * p1) + (r2 * p2)) and

       A7: 0 <= r2 and

       A8: r2 <= 1;

      

       A9: r2 <= r1 by A2, A3, A4, A5, A6, A8;

      r1 <= r2 by A1, A3, A5, A6, A7, A8;

      then r1 = r2 by A9, XXREAL_0: 1;

      hence thesis by A3, A6;

    end;

    theorem :: JORDAN3:28

    

     Th28: for p1,p2,q1,q2 be Point of ( TOP-REAL 2) st q1 in ( LSeg (p1,p2)) & q2 in ( LSeg (p1,p2)) & p1 <> p2 holds ( LE (q1,q2,p1,p2) or LT (q2,q1,p1,p2)) & not ( LE (q1,q2,p1,p2) & LT (q2,q1,p1,p2))

    proof

      let p1,p2,q1,q2 be Point of ( TOP-REAL 2);

      assume that

       A1: q1 in ( LSeg (p1,p2)) and

       A2: q2 in ( LSeg (p1,p2)) and

       A3: p1 <> p2;

      consider r1 such that

       A4: q1 = (((1 - r1) * p1) + (r1 * p2)) and

       A5: 0 <= r1 and

       A6: r1 <= 1 by A1;

      consider r2 such that

       A7: q2 = (((1 - r2) * p1) + (r2 * p2)) and

       A8: 0 <= r2 and

       A9: r2 <= 1 by A2;

       A10:

      now

        per cases ;

          case

           A11: r1 <= r2;

          for s1,s2 be Real st 0 <= s1 & s1 <= 1 & q1 = (((1 - s1) * p1) + (s1 * p2)) & 0 <= s2 & s2 <= 1 & q2 = (((1 - s2) * p1) + (s2 * p2)) holds s1 <= s2

          proof

            let s1,s2 be Real;

            assume that 0 <= s1 and s1 <= 1 and

             A12: q1 = (((1 - s1) * p1) + (s1 * p2)) and 0 <= s2 and s2 <= 1 and

             A13: q2 = (((1 - s2) * p1) + (s2 * p2));

            (((1 - s2) * p1) + ((s2 * p2) - (s2 * p2))) = ((((1 - r2) * p1) + (r2 * p2)) - (s2 * p2)) by A7, A13, RLVECT_1:def 3;

            then (((1 - s2) * p1) + ((s2 * p2) - (s2 * p2))) = (((1 - r2) * p1) + ((r2 * p2) - (s2 * p2))) by RLVECT_1:def 3;

            then (((1 - s2) * p1) + ( 0. ( TOP-REAL 2))) = (((1 - r2) * p1) + ((r2 * p2) - (s2 * p2))) by RLVECT_1: 5;

            then ((1 - s2) * p1) = (((1 - r2) * p1) + ((r2 * p2) - (s2 * p2))) by RLVECT_1: 4;

            then ((1 - s2) * p1) = (((1 - r2) * p1) + ((r2 - s2) * p2)) by RLVECT_1: 35;

            then (((1 - s2) * p1) - ((1 - r2) * p1)) = (((r2 - s2) * p2) + (((1 - r2) * p1) - ((1 - r2) * p1))) by RLVECT_1:def 3;

            then (((1 - s2) * p1) - ((1 - r2) * p1)) = (((r2 - s2) * p2) + ( 0. ( TOP-REAL 2))) by RLVECT_1: 5;

            then (((1 - s2) * p1) - ((1 - r2) * p1)) = ((r2 - s2) * p2) by RLVECT_1: 4;

            then (((1 - s2) - (1 - r2)) * p1) = ((r2 - s2) * p2) by RLVECT_1: 35;

            then

             A14: (r2 - s2) = 0 or p1 = p2 by RLVECT_1: 36;

            (((1 - s1) * p1) + ((s1 * p2) - (s1 * p2))) = ((((1 - r1) * p1) + (r1 * p2)) - (s1 * p2)) by A4, A12, RLVECT_1:def 3;

            then (((1 - s1) * p1) + ((s1 * p2) - (s1 * p2))) = (((1 - r1) * p1) + ((r1 * p2) - (s1 * p2))) by RLVECT_1:def 3;

            then (((1 - s1) * p1) + ( 0. ( TOP-REAL 2))) = (((1 - r1) * p1) + ((r1 * p2) - (s1 * p2))) by RLVECT_1: 5;

            then ((1 - s1) * p1) = (((1 - r1) * p1) + ((r1 * p2) - (s1 * p2))) by RLVECT_1: 4;

            then ((1 - s1) * p1) = (((1 - r1) * p1) + ((r1 - s1) * p2)) by RLVECT_1: 35;

            then (((1 - s1) * p1) - ((1 - r1) * p1)) = (((r1 - s1) * p2) + (((1 - r1) * p1) - ((1 - r1) * p1))) by RLVECT_1:def 3;

            then (((1 - s1) * p1) - ((1 - r1) * p1)) = (((r1 - s1) * p2) + ( 0. ( TOP-REAL 2))) by RLVECT_1: 5;

            then (((1 - s1) * p1) - ((1 - r1) * p1)) = ((r1 - s1) * p2) by RLVECT_1: 4;

            then (((1 - s1) - (1 - r1)) * p1) = ((r1 - s1) * p2) by RLVECT_1: 35;

            then (r1 - s1) = 0 or p1 = p2 by RLVECT_1: 36;

            hence thesis by A3, A11, A14;

          end;

          hence LE (q1,q2,p1,p2) or LT (q2,q1,p1,p2) by A1, A2;

        end;

          case

           A15: r1 > r2;

          for s2,s1 be Real st 0 <= s2 & s2 <= 1 & q2 = (((1 - s2) * p1) + (s2 * p2)) & 0 <= s1 & s1 <= 1 & q1 = (((1 - s1) * p1) + (s1 * p2)) holds s1 >= s2

          proof

            let s2,s1 be Real;

            assume that 0 <= s2 and s2 <= 1 and

             A16: q2 = (((1 - s2) * p1) + (s2 * p2)) and 0 <= s1 and s1 <= 1 and

             A17: q1 = (((1 - s1) * p1) + (s1 * p2));

            (((1 - s1) * p1) + ((s1 * p2) - (s1 * p2))) = ((((1 - r1) * p1) + (r1 * p2)) - (s1 * p2)) by A4, A17, RLVECT_1:def 3;

            then (((1 - s1) * p1) + ((s1 * p2) - (s1 * p2))) = (((1 - r1) * p1) + ((r1 * p2) - (s1 * p2))) by RLVECT_1:def 3;

            then (((1 - s1) * p1) + ( 0. ( TOP-REAL 2))) = (((1 - r1) * p1) + ((r1 * p2) - (s1 * p2))) by RLVECT_1: 5;

            then ((1 - s1) * p1) = (((1 - r1) * p1) + ((r1 * p2) - (s1 * p2))) by RLVECT_1: 4;

            then ((1 - s1) * p1) = (((1 - r1) * p1) + ((r1 - s1) * p2)) by RLVECT_1: 35;

            then (((1 - s1) * p1) - ((1 - r1) * p1)) = (((r1 - s1) * p2) + (((1 - r1) * p1) - ((1 - r1) * p1))) by RLVECT_1:def 3;

            then (((1 - s1) * p1) - ((1 - r1) * p1)) = (((r1 - s1) * p2) + ( 0. ( TOP-REAL 2))) by RLVECT_1: 5;

            then (((1 - s1) * p1) - ((1 - r1) * p1)) = ((r1 - s1) * p2) by RLVECT_1: 4;

            then (((1 - s1) - (1 - r1)) * p1) = ((r1 - s1) * p2) by RLVECT_1: 35;

            then

             A18: (r1 - s1) = 0 or p1 = p2 by RLVECT_1: 36;

            (((1 - s2) * p1) + ((s2 * p2) - (s2 * p2))) = ((((1 - r2) * p1) + (r2 * p2)) - (s2 * p2)) by A7, A16, RLVECT_1:def 3;

            then (((1 - s2) * p1) + ((s2 * p2) - (s2 * p2))) = (((1 - r2) * p1) + ((r2 * p2) - (s2 * p2))) by RLVECT_1:def 3;

            then (((1 - s2) * p1) + ( 0. ( TOP-REAL 2))) = (((1 - r2) * p1) + ((r2 * p2) - (s2 * p2))) by RLVECT_1: 5;

            

            then ((1 - s2) * p1) = (((1 - r2) * p1) + ((r2 * p2) - (s2 * p2))) by RLVECT_1: 4

            .= (((r2 - s2) * p2) + ((1 - r2) * p1)) by RLVECT_1: 35;

            then (((1 - s2) * p1) - ((1 - r2) * p1)) = (((r2 - s2) * p2) + (((1 - r2) * p1) - ((1 - r2) * p1))) by RLVECT_1:def 3;

            then (((1 - s2) * p1) - ((1 - r2) * p1)) = (((r2 - s2) * p2) + ( 0. ( TOP-REAL 2))) by RLVECT_1: 5;

            then (((1 - s2) * p1) - ((1 - r2) * p1)) = ((r2 - s2) * p2) by RLVECT_1: 4;

            then (((1 - s2) - (1 - r2)) * p1) = ((r2 - s2) * p2) by RLVECT_1: 35;

            then (r2 - s2) = 0 or p1 = p2 by RLVECT_1: 36;

            hence thesis by A3, A15, A18;

          end;

          then

           A19: LE (q2,q1,p1,p2) by A1, A2;

          thus LE (q1,q2,p1,p2) or LT (q2,q1,p1,p2) by A19;

        end;

      end;

      now

        assume that

         A20: LE (q1,q2,p1,p2) and

         A21: LT (q2,q1,p1,p2);

         LE (q2,q1,p1,p2) by A21;

        then

         A22: r2 <= r1 by A4, A5, A6, A7, A9;

        r1 <= r2 by A4, A6, A7, A8, A9, A20;

        then r1 = r2 by A22, XXREAL_0: 1;

        hence contradiction by A4, A7, A21;

      end;

      hence thesis by A10;

    end;

    theorem :: JORDAN3:29

    

     Th29: for f be FinSequence of ( TOP-REAL 2) holds for p,q be Point of ( TOP-REAL 2) st p in ( L~ f) & q in ( L~ f) & ( Index (p,f)) < ( Index (q,f)) holds q in ( L~ ( L_Cut (f,p)))

    proof

      let f be FinSequence of ( TOP-REAL 2);

      let p,q be Point of ( TOP-REAL 2);

      assume that

       A1: p in ( L~ f) and

       A2: q in ( L~ f) and

       A3: ( Index (p,f)) < ( Index (q,f));

      

       A4: ( Index (q,f)) < ( len f) by A2, Th8;

      then

       A5: (( Index (q,f)) - ( Index (p,f))) <= (( len f) - ( Index (p,f))) by XREAL_1: 9;

      then

       A6: ((( Index (q,f)) - ( Index (p,f))) + 1) <= ((( len f) - ( Index (p,f))) + 1) by XREAL_1: 6;

      (( Index (q,f)) - ( Index (p,f))) <= (((( len f) - ( Index (p,f))) - 1) + 1) by A4, XREAL_1: 9;

      then

       A7: (( Index (q,f)) -' ( Index (p,f))) <= ((( len f) - (( Index (p,f)) + 1)) + 1) by A3, XREAL_1: 233;

      set i1 = ((( Index (q,f)) -' ( Index (p,f))) + 1);

      

       A8: 1 <= (( Index (p,f)) + 1) by NAT_1: 11;

      

       A9: (( Index (p,f)) + 1) <= ( Index (q,f)) by A3, NAT_1: 13;

      then

       A10: ((( Index (p,f)) + 1) - ( Index (p,f))) <= (( Index (q,f)) - ( Index (p,f))) by XREAL_1: 9;

      then

       A11: 1 <= (( Index (q,f)) -' ( Index (p,f))) by XREAL_0:def 2;

      then

       A12: 1 <= ((( Index (q,f)) -' ( Index (p,f))) + 1) by NAT_D: 48;

      (1 + 1) <= ((( Index (q,f)) -' ( Index (p,f))) + 1) by A11, XREAL_1: 6;

      then

       A13: 1 < ((( Index (q,f)) -' ( Index (p,f))) + 1) by XXREAL_0: 2;

      then

       A14: ( len <*p*>) < ((( Index (q,f)) -' ( Index (p,f))) + 1) by FINSEQ_1: 40;

      then

       A15: ( len <*p*>) < (((( Index (q,f)) -' ( Index (p,f))) + 1) + 1) by NAT_1: 13;

      

       A16: (( Index (p,f)) + 1) <= ( len f) by A4, A9, XXREAL_0: 2;

      

       A17: 1 <= ( Index (q,f)) by A2, Th8;

      then 1 < ( len f) by A4, XXREAL_0: 2;

      then ( len ( mid (f,(( Index (p,f)) + 1),( len f)))) = ((( len f) -' (( Index (p,f)) + 1)) + 1) by A8, A16, FINSEQ_6: 118;

      

      then

       A18: (( len <*p*>) + ( len ( mid (f,(( Index (p,f)) + 1),( len f))))) = (1 + ((( len f) -' (( Index (p,f)) + 1)) + 1)) by FINSEQ_1: 40

      .= (1 + ((( len f) - (( Index (p,f)) + 1)) + 1)) by A4, A9, XREAL_1: 233, XXREAL_0: 2

      .= ((( len f) - ( Index (p,f))) + 1);

      then

       A19: ((( Index (q,f)) -' ( Index (p,f))) + 1) <= (( len <*p*>) + ( len ( mid (f,(( Index (p,f)) + 1),( len f))))) by A3, A6, XREAL_1: 233;

      per cases ;

        suppose

         A20: p = (f . (( Index (p,f)) + 1));

        then

         A21: ( len ( L_Cut (f,p))) = (( len f) - ( Index (p,f))) by A1, Th26;

        then ( len ( L_Cut (f,p))) >= (( Index (q,f)) -' ( Index (p,f))) by A3, A5, XREAL_1: 233;

        

        then (( L_Cut (f,p)) /. (( Index (q,f)) -' ( Index (p,f)))) = (( L_Cut (f,p)) . (( Index (q,f)) -' ( Index (p,f)))) by A11, FINSEQ_4: 15

        .= (( mid (f,(( Index (p,f)) + 1),( len f))) . (( Index (q,f)) -' ( Index (p,f)))) by A20, Def3

        .= (f . (((( Index (p,f)) + 1) + (( Index (q,f)) -' ( Index (p,f)))) - 1)) by A11, A8, A16, A7, FINSEQ_6: 122

        .= (f . (((( Index (p,f)) + 1) + (( Index (q,f)) - ( Index (p,f)))) - 1)) by A3, XREAL_1: 233

        .= (f . ( Index (q,f)));

        then

         A22: (( L_Cut (f,p)) /. (( Index (q,f)) -' ( Index (p,f)))) = (f /. ( Index (q,f))) by A2, A4, Th8, FINSEQ_4: 15;

        1 <= ( Index (q,f)) by A2, Th8;

        then

         A23: 1 <= (( Index (q,f)) + 1) by NAT_D: 48;

        

         A24: q in ( LSeg (f,( Index (q,f)))) by A2, Th9;

        

         A25: ( Index (q,f)) < ( len f) by A2, Th8;

        then

         A26: (( Index (q,f)) + 1) <= ( len f) by NAT_1: 13;

        then

         A27: ((( Index (q,f)) + 1) - ( Index (p,f))) <= (( len f) - ( Index (p,f))) by XREAL_1: 9;

        then ((( Index (q,f)) - ( Index (p,f))) + 1) <= (( len f) - ( Index (p,f)));

        then

         A28: ((( Index (q,f)) -' ( Index (p,f))) + 1) <= ((( len f) - (( Index (p,f)) + 1)) + 1) by A3, XREAL_1: 233;

        ((( Index (q,f)) + 1) - ( Index (p,f))) <= (( len f) - ( Index (p,f))) by A26, XREAL_1: 9;

        then ((( Index (q,f)) - ( Index (p,f))) + 1) <= (( len f) - ( Index (p,f)));

        then

         A29: i1 <= ( len ( L_Cut (f,p))) by A10, A21, XREAL_0:def 2;

        

         A30: (( Index (q,f)) + 1) <= ( len f) by A25, NAT_1: 13;

        ((( Index (q,f)) - ( Index (p,f))) + 1) <= (( len f) - ( Index (p,f))) by A27;

        then ( len ( L_Cut (f,p))) >= ((( Index (q,f)) -' ( Index (p,f))) + 1) by A3, A21, XREAL_1: 233;

        

        then (( L_Cut (f,p)) /. ((( Index (q,f)) -' ( Index (p,f))) + 1)) = (( L_Cut (f,p)) . ((( Index (q,f)) -' ( Index (p,f))) + 1)) by A13, FINSEQ_4: 15

        .= (( mid (f,(( Index (p,f)) + 1),( len f))) . ((( Index (q,f)) -' ( Index (p,f))) + 1)) by A20, Def3

        .= (f . (((( Index (p,f)) + 1) + ((( Index (q,f)) -' ( Index (p,f))) + 1)) - 1)) by A12, A8, A16, A28, FINSEQ_6: 122

        .= (f . (((( Index (p,f)) + 1) + ((( Index (q,f)) - ( Index (p,f))) + 1)) - 1)) by A3, XREAL_1: 233

        .= (f /. (( Index (q,f)) + 1)) by A23, A30, FINSEQ_4: 15;

        then q in ( LSeg ((( L_Cut (f,p)) /. (( Index (q,f)) -' ( Index (p,f)))),(( L_Cut (f,p)) /. ((( Index (q,f)) -' ( Index (p,f))) + 1)))) by A17, A22, A26, A24, TOPREAL1:def 3;

        hence thesis by A11, A29, SPPOL_2: 15;

      end;

        suppose that

         A31: p <> (f . (( Index (p,f)) + 1));

        

         A32: ( len ( L_Cut (f,p))) = ((( len f) - ( Index (p,f))) + 1) by A1, A31, Th26;

        then ( len ( L_Cut (f,p))) >= ((( Index (q,f)) -' ( Index (p,f))) + 1) by A3, A6, XREAL_1: 233;

        

        then (( L_Cut (f,p)) /. ((( Index (q,f)) -' ( Index (p,f))) + 1)) = (( L_Cut (f,p)) . ((( Index (q,f)) -' ( Index (p,f))) + 1)) by FINSEQ_4: 15, NAT_1: 11

        .= (( <*p*> ^ ( mid (f,(( Index (p,f)) + 1),( len f)))) . ((( Index (q,f)) -' ( Index (p,f))) + 1)) by A31, Def3

        .= (( mid (f,(( Index (p,f)) + 1),( len f))) . (((( Index (q,f)) -' ( Index (p,f))) + 1) - ( len <*p*>))) by A14, A19, FINSEQ_6: 108

        .= (( mid (f,(( Index (p,f)) + 1),( len f))) . (((( Index (q,f)) -' ( Index (p,f))) + 1) - 1)) by FINSEQ_1: 40

        .= (f . (((( Index (p,f)) + 1) + (( Index (q,f)) -' ( Index (p,f)))) - 1)) by A11, A8, A16, A7, FINSEQ_6: 122

        .= (f . (((( Index (p,f)) + 1) + (( Index (q,f)) - ( Index (p,f)))) - 1)) by A3, XREAL_1: 233

        .= (f . ( Index (q,f)));

        then

         A33: (( L_Cut (f,p)) /. ((( Index (q,f)) -' ( Index (p,f))) + 1)) = (f /. ( Index (q,f))) by A2, A4, Th8, FINSEQ_4: 15;

        

         A34: ( Index (q,f)) < ( len f) by A2, Th8;

        then

         A35: (( Index (q,f)) + 1) <= ( len f) by NAT_1: 13;

        then

         A36: ((( Index (q,f)) + 1) - ( Index (p,f))) <= (( len f) - ( Index (p,f))) by XREAL_1: 9;

        then ((( Index (q,f)) - ( Index (p,f))) + 1) <= (( len f) - ( Index (p,f)));

        then

         A37: ((( Index (q,f)) -' ( Index (p,f))) + 1) <= ((( len f) - (( Index (p,f)) + 1)) + 1) by A3, XREAL_1: 233;

        ((( Index (q,f)) + 1) - ( Index (p,f))) <= (( len f) - ( Index (p,f))) by A35, XREAL_1: 9;

        then ((( Index (q,f)) - ( Index (p,f))) + 1) <= (( len f) - ( Index (p,f)));

        then i1 <= (( len f) - ( Index (p,f))) by A10, XREAL_0:def 2;

        then

         A38: (i1 + 1) <= ( len ( L_Cut (f,p))) by A32, XREAL_1: 6;

        1 <= ( Index (q,f)) by A2, Th8;

        then

         A39: 1 <= (( Index (q,f)) + 1) by NAT_D: 48;

        

         A40: q in ( LSeg (f,( Index (q,f)))) by A2, Th9;

        

         A41: (( Index (q,f)) + 1) <= ( len f) by A34, NAT_1: 13;

        

         A42: (((( Index (q,f)) - ( Index (p,f))) + 1) + 1) <= ((( len f) - ( Index (p,f))) + 1) by A36, XREAL_1: 6;

        then

         A43: (((( Index (q,f)) -' ( Index (p,f))) + 1) + 1) <= (( len <*p*>) + ( len ( mid (f,(( Index (p,f)) + 1),( len f))))) by A3, A18, XREAL_1: 233;

        ( len ( L_Cut (f,p))) >= (((( Index (q,f)) -' ( Index (p,f))) + 1) + 1) by A3, A32, A42, XREAL_1: 233;

        

        then (( L_Cut (f,p)) /. (((( Index (q,f)) -' ( Index (p,f))) + 1) + 1)) = (( L_Cut (f,p)) . (((( Index (q,f)) -' ( Index (p,f))) + 1) + 1)) by FINSEQ_4: 15, NAT_1: 11

        .= (( <*p*> ^ ( mid (f,(( Index (p,f)) + 1),( len f)))) . (((( Index (q,f)) -' ( Index (p,f))) + 1) + 1)) by A31, Def3

        .= (( mid (f,(( Index (p,f)) + 1),( len f))) . ((((( Index (q,f)) -' ( Index (p,f))) + 1) + 1) - ( len <*p*>))) by A15, A43, FINSEQ_6: 108

        .= (( mid (f,(( Index (p,f)) + 1),( len f))) . ((((( Index (q,f)) -' ( Index (p,f))) + 1) + 1) - 1)) by FINSEQ_1: 40

        .= (f . (((( Index (p,f)) + 1) + ((( Index (q,f)) -' ( Index (p,f))) + 1)) - 1)) by A12, A8, A16, A37, FINSEQ_6: 122

        .= (f . (((( Index (p,f)) + 1) + ((( Index (q,f)) - ( Index (p,f))) + 1)) - 1)) by A3, XREAL_1: 233

        .= (f /. (( Index (q,f)) + 1)) by A39, A41, FINSEQ_4: 15;

        then q in ( LSeg ((( L_Cut (f,p)) /. ((( Index (q,f)) -' ( Index (p,f))) + 1)),(( L_Cut (f,p)) /. (((( Index (q,f)) -' ( Index (p,f))) + 1) + 1)))) by A17, A33, A35, A40, TOPREAL1:def 3;

        hence thesis by A38, NAT_1: 11, SPPOL_2: 15;

      end;

    end;

    theorem :: JORDAN3:30

    

     Th30: for p,q,p1,p2 be Point of ( TOP-REAL 2) st LE (p,q,p1,p2) holds q in ( LSeg (p,p2)) & p in ( LSeg (p1,q))

    proof

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

      assume

       A1: LE (p,q,p1,p2);

      then p in ( LSeg (p1,p2));

      then

      consider s1 be Real such that

       A2: p = (((1 - s1) * p1) + (s1 * p2)) and

       A3: 0 <= s1 and

       A4: s1 <= 1;

      q in ( LSeg (p1,p2)) by A1;

      then

      consider s2 be Real such that

       A5: q = (((1 - s2) * p1) + (s2 * p2)) and

       A6: 0 <= s2 and

       A7: s2 <= 1;

      

       A8: s1 <= s2 by A1, A2, A4, A5, A6, A7;

      

       A9: (1 - s1) >= 0 by A4, XREAL_1: 48;

       A10:

      now

        per cases ;

          case

           A11: (1 - s1) <> 0 ;

          set s = ((s2 - s1) / (1 - s1));

          

           A12: ((1 - s1) * ((1 - s2) / (1 - s1))) = (1 - s2) by A11, XCMPLX_1: 87;

          

           A13: ((1 - s1) * ((s2 - s1) / (1 - s1))) = (s2 - s1) by A11, XCMPLX_1: 87;

          (1 - ((s2 - s1) / (1 - s1))) = (((1 * (1 - s1)) - (s2 - s1)) / (1 - s1)) by A11, XCMPLX_1: 127

          .= ((((1 - s1) + s1) - s2) / (1 - s1));

          

          then ((1 - s1) * (((1 - s) * p) + (s * p2))) = (((1 - s1) * (((1 - s2) / (1 - s1)) * (((1 - s1) * p1) + (s1 * p2)))) + ((1 - s1) * (((s2 - s1) / (1 - s1)) * p2))) by A2, RLVECT_1:def 5

          .= ((((1 - s1) * ((1 - s2) / (1 - s1))) * (((1 - s1) * p1) + (s1 * p2))) + ((1 - s1) * (((s2 - s1) / (1 - s1)) * p2))) by RLVECT_1:def 7

          .= (((1 - s2) * (((1 - s1) * p1) + (s1 * p2))) + (((1 - s1) * ((s2 - s1) / (1 - s1))) * p2)) by A12, RLVECT_1:def 7

          .= ((((1 - s2) * ((1 - s1) * p1)) + ((1 - s2) * (s1 * p2))) + ((s2 - s1) * p2)) by A13, RLVECT_1:def 5

          .= (((((1 - s2) * (1 - s1)) * p1) + ((1 - s2) * (s1 * p2))) + ((s2 - s1) * p2)) by RLVECT_1:def 7

          .= (((((1 - s2) * (1 - s1)) * p1) + (((1 - s2) * s1) * p2)) + ((s2 - s1) * p2)) by RLVECT_1:def 7

          .= ((((1 - s2) * (1 - s1)) * p1) + ((((1 - s2) * s1) * p2) + ((s2 - s1) * p2))) by RLVECT_1:def 3

          .= ((((1 - s2) * (1 - s1)) * p1) + ((((1 * s1) - (s2 * s1)) + (s2 - s1)) * p2)) by RLVECT_1:def 6

          .= (((1 - s1) * ((1 - s2) * p1)) + (((1 - s1) * s2) * p2)) by RLVECT_1:def 7

          .= (((1 - s1) * ((1 - s2) * p1)) + ((1 - s1) * (s2 * p2))) by RLVECT_1:def 7

          .= ((1 - s1) * q) by A5, RLVECT_1:def 5;

          then

           A14: q = (((1 - s) * p) + (s * p2)) by A11, RLVECT_1: 36;

          (1 - s1) >= (s2 - s1) by A7, XREAL_1: 9;

          then ((1 - s1) / (1 - s1)) >= ((s2 - s1) / (1 - s1)) by A9, XREAL_1: 72;

          then

           A15: 1 >= s by A11, XCMPLX_1: 60;

          (s2 - s1) >= 0 by A8, XREAL_1: 48;

          hence q in ( LSeg (p,p2)) by A9, A15, A14;

        end;

          case (1 - s1) = 0 ;

          then s2 = 1 by A7, A8, XXREAL_0: 1;

          

          then q = (( 0. ( TOP-REAL 2)) + (1 * p2)) by A5, RLVECT_1: 10

          .= (( 0. ( TOP-REAL 2)) + p2) by RLVECT_1:def 8

          .= p2 by RLVECT_1: 4;

          hence q in ( LSeg (p,p2)) by RLTOPSP1: 68;

        end;

      end;

      now

        per cases ;

          case

           A16: s2 <> 0 ;

          set s = (s1 / s2);

          (s2 / s2) >= (s1 / s2) by A6, A8, XREAL_1: 72;

          then

           A17: 1 >= s by A16, XCMPLX_1: 60;

          

           A18: ((s2 - s1) + (s1 * (1 - s2))) = (s2 * (1 - s1));

          

           A19: (s2 * (s1 / s2)) = s1 by A16, XCMPLX_1: 87;

          

           A20: (s2 * ((s2 - s1) / s2)) = (s2 - s1) by A16, XCMPLX_1: 87;

          (s2 * (((1 - s) * p1) + (s * q))) = (s2 * (((((1 * s2) - s1) / s2) * p1) + ((s1 / s2) * (((1 - s2) * p1) + (s2 * p2))))) by A5, A16, XCMPLX_1: 127

          .= ((s2 * (((s2 - s1) / s2) * p1)) + (s2 * ((s1 / s2) * (((1 - s2) * p1) + (s2 * p2))))) by RLVECT_1:def 5

          .= (((s2 * ((s2 - s1) / s2)) * p1) + (s2 * ((s1 / s2) * (((1 - s2) * p1) + (s2 * p2))))) by RLVECT_1:def 7

          .= (((s2 - s1) * p1) + ((s2 * (s1 / s2)) * (((1 - s2) * p1) + (s2 * p2)))) by A20, RLVECT_1:def 7

          .= (((s2 - s1) * p1) + ((s1 * ((1 - s2) * p1)) + (s1 * (s2 * p2)))) by A19, RLVECT_1:def 5

          .= (((s2 - s1) * p1) + (((s1 * (1 - s2)) * p1) + (s1 * (s2 * p2)))) by RLVECT_1:def 7

          .= (((s2 - s1) * p1) + (((s1 * (1 - s2)) * p1) + ((s1 * s2) * p2))) by RLVECT_1:def 7

          .= ((((s2 - s1) * p1) + ((s1 * (1 - s2)) * p1)) + ((s1 * s2) * p2)) by RLVECT_1:def 3

          .= ((((s2 - s1) + (s1 * (1 - s2))) * p1) + ((s1 * s2) * p2)) by RLVECT_1:def 6

          .= ((s2 * ((1 - s1) * p1)) + ((s2 * s1) * p2)) by A18, RLVECT_1:def 7

          .= ((s2 * ((1 - s1) * p1)) + (s2 * (s1 * p2))) by RLVECT_1:def 7

          .= (s2 * p) by A2, RLVECT_1:def 5;

          then p = (((1 - s) * p1) + (s * q)) by A16, RLVECT_1: 36;

          hence p in ( LSeg (p1,q)) by A3, A6, A17;

        end;

          case s2 = 0 ;

          then s1 = 0 by A1, A2, A3, A4, A5;

          

          then p = ((1 * p1) + ( 0. ( TOP-REAL 2))) by A2, RLVECT_1: 10

          .= (p1 + ( 0. ( TOP-REAL 2))) by RLVECT_1:def 8

          .= p1 by RLVECT_1: 4;

          hence p in ( LSeg (p1,q)) by RLTOPSP1: 68;

        end;

      end;

      hence thesis by A10;

    end;

    theorem :: JORDAN3:31

    

     Th31: for f be FinSequence of ( TOP-REAL 2) holds for p,q be Point of ( TOP-REAL 2) st p in ( L~ f) & q in ( L~ f) & p <> q & ( Index (p,f)) = ( Index (q,f)) & LE (p,q,(f /. ( Index (p,f))),(f /. (( Index (p,f)) + 1))) holds q in ( L~ ( L_Cut (f,p)))

    proof

      let f be FinSequence of ( TOP-REAL 2);

      let p,q be Point of ( TOP-REAL 2);

      assume that

       A1: p in ( L~ f) and

       A2: q in ( L~ f) and

       A3: p <> q and

       A4: ( Index (p,f)) = ( Index (q,f)) and

       A5: LE (p,q,(f /. ( Index (p,f))),(f /. (( Index (p,f)) + 1)));

      

       A6: (( Index (q,f)) -' ( Index (p,f))) = (( Index (q,f)) - ( Index (p,f))) by A4, XREAL_1: 233

      .= 0 by A4;

      ( Index (q,f)) < ( len f) by A2, Th8;

      then

       A7: (( Index (q,f)) + 1) <= ( len f) by NAT_1: 13;

       A8:

      now

        q in ( LSeg ((f /. ( Index (p,f))),(f /. (( Index (p,f)) + 1)))) by A5;

        then

        consider r be Real such that

         A9: q = (((1 - r) * (f /. ( Index (p,f)))) + (r * (f /. (( Index (p,f)) + 1)))) and

         A10: 0 <= r and

         A11: r <= 1;

        

         A12: p = (1 * p) by RLVECT_1:def 8

        .= (( 0. ( TOP-REAL 2)) + (1 * p)) by RLVECT_1: 4

        .= (((1 - 1) * (f /. ( Index (p,f)))) + (1 * p)) by RLVECT_1: 10;

        assume

         A13: p = (f . (( Index (p,f)) + 1));

        then p = (f /. (( Index (p,f)) + 1)) by A4, A7, FINSEQ_4: 15, NAT_1: 11;

        then 1 <= r by A5, A9, A10, A12;

        then r = 1 by A11, XXREAL_0: 1;

        hence contradiction by A3, A4, A7, A13, A9, A12, FINSEQ_4: 15, NAT_1: 11;

      end;

      then

       A14: ( len ( L_Cut (f,p))) = ((( len f) - ( Index (p,f))) + 1) by A1, Th26;

      1 <= ( Index (q,f)) by A2, Th8;

      then

       A15: 1 <= (( Index (q,f)) + 1) by NAT_D: 48;

      1 < (( 0 + 1) + 1);

      then

       A16: ( len <*p*>) < (((( Index (q,f)) -' ( Index (p,f))) + 1) + 1) by A6, FINSEQ_1: 40;

      

       A17: ( Index (q,f)) < ( len f) by A2, Th8;

      then

       A18: (( Index (q,f)) + 1) <= ( len f) by NAT_1: 13;

      then

       A19: ((( Index (q,f)) + 1) - ( Index (p,f))) <= (( len f) - ( Index (p,f))) by XREAL_1: 9;

      then

       A20: (((( Index (q,f)) - ( Index (p,f))) + 1) + 1) <= ((( len f) - ( Index (p,f))) + 1) by XREAL_1: 6;

      ((( Index (q,f)) - ( Index (p,f))) + 1) <= (( len f) - ( Index (p,f))) by A19;

      then

       A21: ((( Index (q,f)) -' ( Index (p,f))) + 1) <= ((( len f) - (( Index (p,f)) + 1)) + 1) by A4, XREAL_1: 233;

      

       A22: 1 <= (( Index (p,f)) + 1) by NAT_1: 11;

      

       A23: ( Index (q,f)) < ( len f) by A2, Th8;

      then (( Index (q,f)) - ( Index (p,f))) <= (( len f) - ( Index (p,f))) by XREAL_1: 9;

      then ((( Index (q,f)) - ( Index (p,f))) + 1) <= ((( len f) - ( Index (p,f))) + 1) by XREAL_1: 6;

      

      then

       A24: (( L_Cut (f,p)) /. ((( Index (q,f)) -' ( Index (p,f))) + 1)) = (( L_Cut (f,p)) . ((( Index (q,f)) -' ( Index (p,f))) + 1)) by A4, A6, A14, FINSEQ_4: 15

      .= (( <*p*> ^ ( mid (f,(( Index (p,f)) + 1),( len f)))) . ((( Index (q,f)) -' ( Index (p,f))) + 1)) by A8, Def3

      .= p by A6, FINSEQ_1: 41;

      set i1 = ((( Index (q,f)) -' ( Index (p,f))) + 1);

      

       A25: (( Index (q,f)) + 1) <= ( len f) by A17, NAT_1: 13;

      ((( Index (q,f)) + 1) - ( Index (p,f))) <= (( len f) - ( Index (p,f))) by A18, XREAL_1: 9;

      then ((( Index (q,f)) - ( Index (p,f))) + 1) <= (( len f) - ( Index (p,f)));

      then i1 <= (( len f) - ( Index (p,f))) by A4, XREAL_0:def 2;

      then

       A26: (i1 + 1) <= ( len ( L_Cut (f,p))) by A14, XREAL_1: 6;

      1 <= ( Index (q,f)) by A2, Th8;

      then 1 < ( len f) by A23, XXREAL_0: 2;

      then ( len ( mid (f,(( Index (p,f)) + 1),( len f)))) = ((( len f) -' (( Index (p,f)) + 1)) + 1) by A4, A7, A22, FINSEQ_6: 118;

      

      then (( len <*p*>) + ( len ( mid (f,(( Index (p,f)) + 1),( len f))))) = (1 + ((( len f) -' (( Index (p,f)) + 1)) + 1)) by FINSEQ_1: 40

      .= (1 + ((( len f) - (( Index (p,f)) + 1)) + 1)) by A4, A7, XREAL_1: 233

      .= ((( len f) - ( Index (p,f))) + 1);

      then

       A27: (((( Index (q,f)) -' ( Index (p,f))) + 1) + 1) <= (( len <*p*>) + ( len ( mid (f,(( Index (p,f)) + 1),( len f))))) by A4, A20, XREAL_1: 233;

      (( L_Cut (f,p)) /. (((( Index (q,f)) -' ( Index (p,f))) + 1) + 1)) = (( L_Cut (f,p)) . (((( Index (q,f)) -' ( Index (p,f))) + 1) + 1)) by A4, A6, A14, A20, FINSEQ_4: 15

      .= (( <*p*> ^ ( mid (f,(( Index (p,f)) + 1),( len f)))) . (((( Index (q,f)) -' ( Index (p,f))) + 1) + 1)) by A8, Def3

      .= (( mid (f,(( Index (p,f)) + 1),( len f))) . ((((( Index (q,f)) -' ( Index (p,f))) + 1) + 1) - ( len <*p*>))) by A16, A27, FINSEQ_6: 108

      .= (( mid (f,(( Index (p,f)) + 1),( len f))) . ((((( Index (q,f)) -' ( Index (p,f))) + 1) + 1) - 1)) by FINSEQ_1: 40

      .= (f . (((( Index (p,f)) + 1) + ((( Index (q,f)) - ( Index (p,f))) + 1)) - 1)) by A4, A7, A6, A22, A21, FINSEQ_6: 122

      .= (f /. (( Index (q,f)) + 1)) by A15, A25, FINSEQ_4: 15;

      then q in ( LSeg ((( L_Cut (f,p)) /. ((( Index (q,f)) -' ( Index (p,f))) + 1)),(( L_Cut (f,p)) /. (((( Index (q,f)) -' ( Index (p,f))) + 1) + 1)))) by A4, A5, A24, Th30;

      hence thesis by A6, A26, SPPOL_2: 15;

    end;

    begin

    definition

      let f be FinSequence of ( TOP-REAL 2), p,q be Point of ( TOP-REAL 2);

      :: JORDAN3:def7

      func B_Cut (f,p,q) -> FinSequence of ( TOP-REAL 2) equals

      : Def7: ( R_Cut (( L_Cut (f,p)),q)) if p in ( L~ f) & q in ( L~ f) & ( Index (p,f)) < ( Index (q,f)) or ( Index (p,f)) = ( Index (q,f)) & LE (p,q,(f /. ( Index (p,f))),(f /. (( Index (p,f)) + 1)))

      otherwise ( Rev ( R_Cut (( L_Cut (f,q)),p)));

      correctness ;

    end

    theorem :: JORDAN3:32

    

     Th32: for f be FinSequence of ( TOP-REAL 2), p be Point of ( TOP-REAL 2) st f is being_S-Seq & p in ( L~ f) & p <> (f . 1) holds ( R_Cut (f,p)) is_S-Seq_joining ((f /. 1),p)

    proof

      let f be FinSequence of ( TOP-REAL 2), p be Point of ( TOP-REAL 2);

      assume that

       A1: f is being_S-Seq and

       A2: p in ( L~ f) and

       A3: p <> (f . 1);

      ( R_Cut (f,p)) = (( mid (f,1,( Index (p,f)))) ^ <*p*>) by A3, Def4;

      hence thesis by A1, A2, A3, Th19;

    end;

    theorem :: JORDAN3:33

    

     Th33: for f be FinSequence of ( TOP-REAL 2), p be Point of ( TOP-REAL 2) st f is being_S-Seq & p in ( L~ f) & p <> (f . ( len f)) holds ( L_Cut (f,p)) is_S-Seq_joining (p,(f /. ( len f)))

    proof

      let f be FinSequence of ( TOP-REAL 2), p be Point of ( TOP-REAL 2);

      assume that

       A1: f is being_S-Seq and

       A2: p in ( L~ f) and

       A3: p <> (f . ( len f));

      

       A4: f <> {} by A2, CARD_1: 27, TOPREAL1: 22;

      

       A5: ( Rev f) is being_S-Seq by A1;

      

       A6: p in ( L~ ( Rev f)) by A2, SPPOL_2: 22;

      

       A7: p <> (( Rev f) . 1) by A3, FINSEQ_5: 62;

      ( L_Cut (f,p)) = ( L_Cut (( Rev ( Rev f)),p))

      .= ( Rev ( R_Cut (( Rev f),p))) by A1, A6, Th22;

      then ( L_Cut (f,p)) is_S-Seq_joining (p,(( Rev f) /. 1)) by A5, A6, A7, Th15, Th32;

      hence thesis by A4, FINSEQ_5: 65;

    end;

    theorem :: JORDAN3:34

    

     Th34: for f be FinSequence of ( TOP-REAL 2), p be Point of ( TOP-REAL 2) st f is being_S-Seq & p in ( L~ f) & p <> (f . ( len f)) holds ( L_Cut (f,p)) is being_S-Seq

    proof

      let f be FinSequence of ( TOP-REAL 2), p be Point of ( TOP-REAL 2);

      assume that

       A1: f is being_S-Seq and

       A2: p in ( L~ f) and

       A3: p <> (f . ( len f));

      ( L_Cut (f,p)) is_S-Seq_joining (p,(f /. ( len f))) by A1, A2, A3, Th33;

      hence thesis;

    end;

    theorem :: JORDAN3:35

    

     Th35: for f be FinSequence of ( TOP-REAL 2), p be Point of ( TOP-REAL 2) st f is being_S-Seq & p in ( L~ f) & p <> (f . 1) holds ( R_Cut (f,p)) is being_S-Seq

    proof

      let f be FinSequence of ( TOP-REAL 2), p be Point of ( TOP-REAL 2);

      assume that

       A1: f is being_S-Seq and

       A2: p in ( L~ f) and

       A3: p <> (f . 1);

      ( R_Cut (f,p)) is_S-Seq_joining ((f /. 1),p) by A1, A2, A3, Th32;

      hence thesis;

    end;

    

     Lm1: for f be FinSequence of ( TOP-REAL 2), p,q be Point of ( TOP-REAL 2) st f is being_S-Seq & p in ( L~ f) & q in ( L~ f) & p <> q & (( Index (p,f)) < ( Index (q,f)) or ( Index (p,f)) = ( Index (q,f)) & LE (p,q,(f /. ( Index (p,f))),(f /. (( Index (p,f)) + 1)))) holds ( B_Cut (f,p,q)) is_S-Seq_joining (p,q)

    proof

      let f be FinSequence of ( TOP-REAL 2), p,q be Point of ( TOP-REAL 2);

      assume that

       A1: f is being_S-Seq and

       A2: p in ( L~ f) and

       A3: q in ( L~ f) and

       A4: p <> q;

      assume

       A5: ( Index (p,f)) < ( Index (q,f)) or ( Index (p,f)) = ( Index (q,f)) & LE (p,q,(f /. ( Index (p,f))),(f /. (( Index (p,f)) + 1)));

      then

       A6: ( B_Cut (f,p,q)) = ( R_Cut (( L_Cut (f,p)),q)) by A2, A3, Def7;

      ( Index (p,f)) < ( len f) by A2, Th8;

      then

       A7: (( Index (p,f)) + 1) <= ( len f) by NAT_1: 13;

      

       A8: ( Index (q,f)) < ( len f) by A3, Th8;

      1 <= ( Index (q,f)) by A3, Th8;

      then

       A9: 1 < ( len f) by A8, XXREAL_0: 2;

       A10:

      now

        per cases by A5;

          case

           A11: ( Index (p,f)) < ( Index (q,f));

          assume

           A12: p = (f . ( len f));

          (( Index (p,f)) + 1) <= ( Index (q,f)) by A11, NAT_1: 13;

          then ( len f) <= ( Index (q,f)) by A1, A9, A12, Th12;

          hence contradiction by A3, Th8;

        end;

          case

           A13: ( Index (p,f)) = ( Index (q,f)) & LE (p,q,(f /. ( Index (p,f))),(f /. (( Index (p,f)) + 1)));

           A14:

          now

            q in ( LSeg ((f /. ( Index (p,f))),(f /. (( Index (p,f)) + 1)))) by A13;

            then

            consider r be Real such that

             A15: q = (((1 - r) * (f /. ( Index (p,f)))) + (r * (f /. (( Index (p,f)) + 1)))) and

             A16: 0 <= r and

             A17: r <= 1;

            

             A18: p = (1 * p) by RLVECT_1:def 8

            .= (( 0. ( TOP-REAL 2)) + (1 * p)) by RLVECT_1: 4

            .= (((1 - 1) * (f /. ( Index (p,f)))) + (1 * p)) by RLVECT_1: 10;

            assume

             A19: p = (f . (( Index (p,f)) + 1));

            then p = (f /. (( Index (p,f)) + 1)) by A7, FINSEQ_4: 15, NAT_1: 11;

            then 1 <= r by A13, A15, A16, A18;

            then r = 1 by A17, XXREAL_0: 1;

            hence contradiction by A4, A7, A19, A15, A18, FINSEQ_4: 15, NAT_1: 11;

          end;

          assume p = (f . ( len f));

          hence contradiction by A1, A9, A14, Th12;

        end;

      end;

      then ( L_Cut (f,p)) is_S-Seq_joining (p,(f /. ( len f))) by A1, A2, Th33;

      then

       A20: (( L_Cut (f,p)) . 1) = p;

      now

        per cases by A5;

          case ( Index (p,f)) < ( Index (q,f));

          then q in ( L~ ( L_Cut (f,p))) by A2, A3, Th29;

          hence ex i1 be Nat st 1 <= i1 & (i1 + 1) <= ( len ( L_Cut (f,p))) & q in ( LSeg (( L_Cut (f,p)),i1)) by SPPOL_2: 13;

        end;

          case ( Index (p,f)) = ( Index (q,f)) & LE (p,q,(f /. ( Index (p,f))),(f /. (( Index (p,f)) + 1)));

          then q in ( L~ ( L_Cut (f,p))) by A2, A3, A4, Th31;

          hence ex i1 be Nat st 1 <= i1 & (i1 + 1) <= ( len ( L_Cut (f,p))) & q in ( LSeg (( L_Cut (f,p)),i1)) by SPPOL_2: 13;

        end;

      end;

      then

       A21: q in ( L~ ( L_Cut (f,p))) by SPPOL_2: 17;

      then

       A22: ( Index (q,( L_Cut (f,p)))) < ( len ( L_Cut (f,p))) by Th8;

      1 <= ( Index (q,( L_Cut (f,p)))) by A21, Th8;

      then 1 <= ( len ( L_Cut (f,p))) by A22, XXREAL_0: 2;

      then p = (( L_Cut (f,p)) /. 1) by A20, FINSEQ_4: 15;

      hence thesis by A1, A2, A4, A6, A10, A21, A20, Th32, Th34;

    end;

    theorem :: JORDAN3:36

    

     Th36: for f be FinSequence of ( TOP-REAL 2), p,q be Point of ( TOP-REAL 2) st f is being_S-Seq & p in ( L~ f) & q in ( L~ f) & p <> q holds ( B_Cut (f,p,q)) is_S-Seq_joining (p,q)

    proof

      let f be FinSequence of ( TOP-REAL 2), p,q be Point of ( TOP-REAL 2);

      assume that

       A1: f is being_S-Seq and

       A2: p in ( L~ f) and

       A3: q in ( L~ f) and

       A4: p <> q;

      per cases ;

        suppose ( Index (p,f)) < ( Index (q,f)) or ( Index (p,f)) = ( Index (q,f)) & LE (p,q,(f /. ( Index (p,f))),(f /. (( Index (p,f)) + 1)));

        hence thesis by A1, A2, A3, A4, Lm1;

      end;

        suppose

         A5: not (( Index (p,f)) < ( Index (q,f)) or ( Index (p,f)) = ( Index (q,f)) & LE (p,q,(f /. ( Index (p,f))),(f /. (( Index (p,f)) + 1))));

         A6:

        now

          

           A7: ( Index (p,f)) < ( len f) by A2, Th8;

          then

           A8: (( Index (p,f)) + 1) <= ( len f) by NAT_1: 13;

          1 <= (( Index (p,f)) + 1) by NAT_1: 11;

          then

           A9: (( Index (p,f)) + 1) in ( dom f) by A8, FINSEQ_3: 25;

          

           A10: (( Index (p,f)) + 0 ) <> (( Index (p,f)) + 1);

          

           A11: 1 <= ( Index (p,f)) by A2, Th8;

          then

           A12: ( LSeg (f,( Index (p,f)))) = ( LSeg ((f /. ( Index (p,f))),(f /. (( Index (p,f)) + 1)))) by A8, TOPREAL1:def 3;

          then

           A13: p in ( LSeg ((f /. ( Index (p,f))),(f /. (( Index (p,f)) + 1)))) by A2, Th9;

          ( Index (p,f)) in ( dom f) by A11, A7, FINSEQ_3: 25;

          then

           A14: (f /. ( Index (p,f))) <> (f /. (( Index (p,f)) + 1)) by A1, A9, A10, PARTFUN2: 10;

          assume that

           A15: ( Index (p,f)) = ( Index (q,f)) and

           A16: not LE (p,q,(f /. ( Index (p,f))),(f /. (( Index (p,f)) + 1)));

          q in ( LSeg ((f /. ( Index (p,f))),(f /. (( Index (p,f)) + 1)))) by A3, A15, A12, Th9;

          then LT (q,p,(f /. ( Index (p,f))),(f /. (( Index (p,f)) + 1))) by A16, A13, A14, Th28;

          hence LE (q,p,(f /. ( Index (q,f))),(f /. (( Index (q,f)) + 1))) by A15;

        end;

        

         A17: ( Index (q,f)) < ( Index (p,f)) or ( Index (p,f)) = ( Index (q,f)) & not LE (p,q,(f /. ( Index (p,f))),(f /. (( Index (p,f)) + 1))) by A5, XXREAL_0: 1;

        ( B_Cut (f,p,q)) = ( Rev ( R_Cut (( L_Cut (f,q)),p))) by A5, Def7;

        then

         A18: ( Rev ( B_Cut (f,q,p))) = ( B_Cut (f,p,q)) by A2, A3, A17, A6, Def7;

        ( B_Cut (f,q,p)) is_S-Seq_joining (q,p) by A1, A2, A3, A4, A17, A6, Lm1;

        hence thesis by A18, Th15;

      end;

    end;

    theorem :: JORDAN3:37

    for f be FinSequence of ( TOP-REAL 2), p,q be Point of ( TOP-REAL 2) st f is being_S-Seq & p in ( L~ f) & q in ( L~ f) & p <> q holds ( B_Cut (f,p,q)) is being_S-Seq

    proof

      let f be FinSequence of ( TOP-REAL 2), p,q be Point of ( TOP-REAL 2);

      assume that

       A1: f is being_S-Seq and

       A2: p in ( L~ f) and

       A3: q in ( L~ f) and

       A4: p <> q;

      ( B_Cut (f,p,q)) is_S-Seq_joining (p,q) by A1, A2, A3, A4, Th36;

      hence thesis;

    end;

    theorem :: JORDAN3:38

    

     Th38: for f,g be FinSequence of ( TOP-REAL 2) st (f . ( len f)) = (g . 1) & f is being_S-Seq & g is being_S-Seq & (( L~ f) /\ ( L~ g)) = {(g . 1)} holds (f ^ ( mid (g,2,( len g)))) is being_S-Seq

    proof

      let f,g be FinSequence of ( TOP-REAL 2);

      assume that

       A1: (f . ( len f)) = (g . 1) and

       A2: f is being_S-Seq and

       A3: g is being_S-Seq and

       A4: (( L~ f) /\ ( L~ g)) = {(g . 1)};

      

       A5: ( len f) >= 2 by A2, TOPREAL1:def 8;

      

       A6: ( len (f ^ ( mid (g,2,( len g))))) = (( len f) + ( len ( mid (g,2,( len g))))) by FINSEQ_1: 22;

      then ( len f) <= ( len (f ^ ( mid (g,2,( len g))))) by NAT_1: 11;

      then

       A7: ( len (f ^ ( mid (g,2,( len g))))) >= 2 by A5, XXREAL_0: 2;

      

       A8: ( len g) >= 2 by A3, TOPREAL1:def 8;

      then

       A9: 1 <= ( len g) by XXREAL_0: 2;

      

      then

       A10: ( len ( mid (g,2,( len g)))) = ((( len g) -' 2) + 1) by A8, FINSEQ_6: 118

      .= ((( len g) - 2) + 1) by A8, XREAL_1: 233

      .= (( len g) - 1);

      for x1,x2 be object st x1 in ( dom (f ^ ( mid (g,2,( len g))))) & x2 in ( dom (f ^ ( mid (g,2,( len g))))) & ((f ^ ( mid (g,2,( len g)))) . x1) = ((f ^ ( mid (g,2,( len g)))) . x2) holds x1 = x2

      proof

        

         A11: ( rng g) c= ( L~ g) by A8, SPPOL_2: 18;

        

         A12: ( rng (f ^ ( mid (g,2,( len g))))) c= the carrier of ( TOP-REAL 2) by FINSEQ_1:def 4;

        let x1,x2 be object;

        assume that

         A13: x1 in ( dom (f ^ ( mid (g,2,( len g))))) and

         A14: x2 in ( dom (f ^ ( mid (g,2,( len g))))) and

         A15: ((f ^ ( mid (g,2,( len g)))) . x1) = ((f ^ ( mid (g,2,( len g)))) . x2);

        reconsider n1 = x1, n2 = x2 as Element of NAT by A13, A14;

        

         A16: x2 in ( Seg ( len (f ^ ( mid (g,2,( len g)))))) by A14, FINSEQ_1:def 3;

        then

         A17: 1 <= n2 by FINSEQ_1: 1;

        ((f ^ ( mid (g,2,( len g)))) . x1) in ( rng (f ^ ( mid (g,2,( len g))))) by A13, FUNCT_1:def 3;

        then

        reconsider q = ((f ^ ( mid (g,2,( len g)))) . x1) as Point of ( TOP-REAL 2) by A12;

        

         A18: ( rng ( mid (g,2,( len g)))) c= ( rng g) by FINSEQ_6: 119;

        

         A19: ( rng f) c= ( L~ f) by A5, SPPOL_2: 18;

         A20:

        now

           A21:

          now

            (g | 1) = (g | ( Seg 1)) by FINSEQ_1:def 15;

            then

             A22: ((g | 1) . 1) = (g . 1) by FINSEQ_1: 3, FUNCT_1: 49;

            ( len (g | 1)) = 1 by A8, FINSEQ_1: 59, XXREAL_0: 2;

            then 1 in ( dom (g | 1)) by FINSEQ_3: 25;

            then

             A23: (g . 1) in ( rng (g | 1)) by A22, FUNCT_1:def 3;

            

             A24: (2 -' 1) = (2 - 1) by XREAL_1: 233;

            assume (g . 1) in ( rng ( mid (g,2,( len g))));

            then

             A25: (g . 1) in ( rng (g /^ 1)) by A8, A24, FINSEQ_6: 117;

            ( rng (g | 1)) misses ( rng (g /^ 1)) by A3, FINSEQ_5: 34;

            hence contradiction by A25, A23, XBOOLE_0: 3;

          end;

          assume that

           A26: q in ( rng f) and

           A27: q in ( rng ( mid (g,2,( len g))));

          q in ( rng g) by A18, A27;

          then q in (( L~ f) /\ ( L~ g)) by A19, A11, A26, XBOOLE_0:def 4;

          hence contradiction by A4, A27, A21, TARSKI:def 1;

        end;

        n2 <= ( len (f ^ ( mid (g,2,( len g))))) by A16, FINSEQ_1: 1;

        then

         A28: (n2 - ( len f)) <= ((( len f) + ( len ( mid (g,2,( len g))))) - ( len f)) by A6, XREAL_1: 9;

        

         A29: x1 in ( Seg ( len (f ^ ( mid (g,2,( len g)))))) by A13, FINSEQ_1:def 3;

        then n1 <= ( len (f ^ ( mid (g,2,( len g))))) by FINSEQ_1: 1;

        then

         A30: (n1 - ( len f)) <= ((( len f) + ( len ( mid (g,2,( len g))))) - ( len f)) by A6, XREAL_1: 9;

        

         A31: 1 <= n1 by A29, FINSEQ_1: 1;

        now

          per cases ;

            case n1 <= ( len f);

            then

             A32: n1 in ( dom f) by A31, FINSEQ_3: 25;

            then

             A33: ((f ^ ( mid (g,2,( len g)))) . x1) = (f . n1) by FINSEQ_1:def 7;

            now

              per cases ;

                case n2 <= ( len f);

                then

                 A34: n2 in ( dom f) by A17, FINSEQ_3: 25;

                then ((f ^ ( mid (g,2,( len g)))) . x2) = (f . n2) by FINSEQ_1:def 7;

                hence thesis by A2, A15, A32, A33, A34, FUNCT_1:def 4;

              end;

                case

                 A35: n2 > ( len f);

                then (( len f) + 1) <= n2 by NAT_1: 13;

                then

                 A36: ((( len f) + 1) - ( len f)) <= (n2 - ( len f)) by XREAL_1: 9;

                then

                 A37: 1 <= (n2 -' ( len f)) by NAT_D: 39;

                

                 A38: (( len f) + (n2 -' ( len f))) = (( len f) + (n2 - ( len f))) by A35, XREAL_1: 233

                .= n2;

                (n2 -' ( len f)) <= ( len ( mid (g,2,( len g)))) by A28, A36, NAT_D: 39;

                then

                 A39: (n2 -' ( len f)) in ( dom ( mid (g,2,( len g)))) by A37, FINSEQ_3: 25;

                then ((f ^ ( mid (g,2,( len g)))) . (( len f) + (n2 -' ( len f)))) = (( mid (g,2,( len g))) . (n2 -' ( len f))) by FINSEQ_1:def 7;

                hence contradiction by A15, A20, A32, A33, A39, A38, FUNCT_1:def 3;

              end;

            end;

            hence thesis;

          end;

            case

             A40: n1 > ( len f);

            then (( len f) + 1) <= n1 by NAT_1: 13;

            then

             A41: ((( len f) + 1) - ( len f)) <= (n1 - ( len f)) by XREAL_1: 9;

            then

             A42: 1 <= (n1 -' ( len f)) by NAT_D: 39;

            then

             A43: 1 <= ((n1 -' ( len f)) + 1) by NAT_D: 48;

            (n1 -' ( len f)) <= ((n1 -' ( len f)) + 2) by NAT_1: 11;

            

            then

             A44: (((n1 -' ( len f)) + 2) -' 1) = (((n1 -' ( len f)) + 2) - 1) by A42, XREAL_1: 233, XXREAL_0: 2

            .= ((n1 -' ( len f)) + ((1 + 1) - 1));

            

             A45: (( len f) + (n1 -' ( len f))) = (( len f) + (n1 - ( len f))) by A40, XREAL_1: 233

            .= n1;

            

             A46: (n1 -' ( len f)) <= ( len ( mid (g,2,( len g)))) by A30, A41, NAT_D: 39;

            then

             A47: (n1 -' ( len f)) in ( dom ( mid (g,2,( len g)))) by A42, FINSEQ_3: 25;

            then

             A48: ((f ^ ( mid (g,2,( len g)))) . (( len f) + (n1 -' ( len f)))) = (( mid (g,2,( len g))) . (n1 -' ( len f))) by FINSEQ_1:def 7;

            ((n1 -' ( len f)) + 1) <= ((( len g) - 1) + 1) by A10, A46, XREAL_1: 6;

            then

             A49: ((n1 -' ( len f)) + 1) in ( dom g) by A43, FINSEQ_3: 25;

            (( len f) + (n1 -' ( len f))) = (( len f) + (n1 - ( len f))) by A40, XREAL_1: 233

            .= n1;

            then

             A50: ((f ^ ( mid (g,2,( len g)))) . n1) = (g . ((n1 -' ( len f)) + 1)) by A8, A9, A30, A41, A48, A44, FINSEQ_6: 118;

            now

              per cases ;

                case n2 <= ( len f);

                then

                 A51: n2 in ( dom f) by A17, FINSEQ_3: 25;

                then ((f ^ ( mid (g,2,( len g)))) . x2) = (f . n2) by FINSEQ_1:def 7;

                hence contradiction by A15, A20, A47, A48, A45, A51, FUNCT_1:def 3;

              end;

                case

                 A52: n2 > ( len f);

                then (( len f) + 1) <= n2 by NAT_1: 13;

                then

                 A53: ((( len f) + 1) - ( len f)) <= (n2 - ( len f)) by XREAL_1: 9;

                then

                 A54: 1 <= (n2 -' ( len f)) by NAT_D: 39;

                then

                 A55: 1 <= ((n2 -' ( len f)) + 1) by NAT_D: 48;

                

                 A56: (n2 -' ( len f)) <= ( len ( mid (g,2,( len g)))) by A28, A53, NAT_D: 39;

                then ((n2 -' ( len f)) + 1) <= ((( len g) - 1) + 1) by A10, XREAL_1: 6;

                then

                 A57: ((n2 -' ( len f)) + 1) in ( dom g) by A55, FINSEQ_3: 25;

                (n2 -' ( len f)) <= ((n2 -' ( len f)) + 2) by NAT_1: 11;

                

                then

                 A58: (((n2 -' ( len f)) + 2) -' 1) = (((n2 -' ( len f)) + 2) - 1) by A54, XREAL_1: 233, XXREAL_0: 2

                .= ((n2 -' ( len f)) + 1);

                1 <= (n2 -' ( len f)) by A53, NAT_D: 39;

                then (n2 -' ( len f)) in ( dom ( mid (g,2,( len g)))) by A56, FINSEQ_3: 25;

                then

                 A59: ((f ^ ( mid (g,2,( len g)))) . (( len f) + (n2 -' ( len f)))) = (( mid (g,2,( len g))) . (n2 -' ( len f))) by FINSEQ_1:def 7;

                (( len f) + (n2 -' ( len f))) = (( len f) + (n2 - ( len f))) by A52, XREAL_1: 233

                .= n2;

                then ((f ^ ( mid (g,2,( len g)))) . n2) = (g . ((n2 -' ( len f)) + 1)) by A8, A9, A28, A53, A59, A58, FINSEQ_6: 118;

                then ((n1 -' ( len f)) + 1) = ((n2 -' ( len f)) + 1) by A3, A15, A49, A50, A57, FUNCT_1:def 4;

                then (n1 - ( len f)) = (n2 -' ( len f)) by A40, XREAL_1: 233;

                then (n1 - ( len f)) = (n2 - ( len f)) by A52, XREAL_1: 233;

                hence thesis;

              end;

            end;

            hence thesis;

          end;

        end;

        hence thesis;

      end;

      then

       A60: (f ^ ( mid (g,2,( len g)))) is one-to-one by FUNCT_1:def 4;

      

       A61: 1 <= ( len f) by A5, XXREAL_0: 2;

      

       A62: for i,j be Nat st (i + 1) < j holds ( LSeg ((f ^ ( mid (g,2,( len g)))),i)) misses ( LSeg ((f ^ ( mid (g,2,( len g)))),j))

      proof

        let i,j be Nat;

        assume

         A63: (i + 1) < j;

        now

          per cases ;

            case

             A64: j < ( len f) & (j + 1) <= ( len (f ^ ( mid (g,2,( len g)))));

            then

             A65: (i + 1) < ( len f) by A63, XXREAL_0: 2;

            then

             A66: i < ( len f) by NAT_1: 13;

            

             A67: j <= ( len (f ^ ( mid (g,2,( len g))))) by A64, NAT_D: 46;

            then

             A68: (i + 1) < ( len (f ^ ( mid (g,2,( len g))))) by A63, XXREAL_0: 2;

            then

             A69: i <= ( len (f ^ ( mid (g,2,( len g))))) by NAT_D: 46;

            now

              per cases ;

                case

                 A70: 1 <= i;

                then

                 A71: (f /. i) = (f . i) by A66, FINSEQ_4: 15;

                ((f ^ ( mid (g,2,( len g)))) /. i) = ((f ^ ( mid (g,2,( len g)))) . i) by A69, A70, FINSEQ_4: 15;

                then

                 A72: ((f ^ ( mid (g,2,( len g)))) /. i) = (f /. i) by A66, A70, A71, FINSEQ_1: 64;

                

                 A73: ( LSeg (((f ^ ( mid (g,2,( len g)))) /. i),((f ^ ( mid (g,2,( len g)))) /. (i + 1)))) = ( LSeg ((f ^ ( mid (g,2,( len g)))),i)) by A68, A70, TOPREAL1:def 3;

                

                 A74: 1 <= (i + 1) by A70, NAT_D: 48;

                then

                 A75: (f /. (i + 1)) = (f . (i + 1)) by A65, FINSEQ_4: 15;

                ((f ^ ( mid (g,2,( len g)))) /. (i + 1)) = ((f ^ ( mid (g,2,( len g)))) . (i + 1)) by A68, A74, FINSEQ_4: 15;

                then ( LSeg (((f ^ ( mid (g,2,( len g)))) /. i),((f ^ ( mid (g,2,( len g)))) /. (i + 1)))) = ( LSeg ((f /. i),(f /. (i + 1)))) by A65, A74, A72, A75, FINSEQ_1: 64;

                then

                 A76: ( LSeg ((f ^ ( mid (g,2,( len g)))),i)) = ( LSeg (f,i)) by A65, A70, A73, TOPREAL1:def 3;

                

                 A77: 1 < j by A63, A74, XXREAL_0: 2;

                then

                 A78: (f /. j) = (f . j) by A64, FINSEQ_4: 15;

                ((f ^ ( mid (g,2,( len g)))) /. j) = ((f ^ ( mid (g,2,( len g)))) . j) by A67, A77, FINSEQ_4: 15;

                then

                 A79: ((f ^ ( mid (g,2,( len g)))) /. j) = (f /. j) by A64, A77, A78, FINSEQ_1: 64;

                

                 A80: 1 <= (j + 1) by A77, NAT_D: 48;

                then

                 A81: ((f ^ ( mid (g,2,( len g)))) /. (j + 1)) = ((f ^ ( mid (g,2,( len g)))) . (j + 1)) by A64, FINSEQ_4: 15;

                

                 A82: (j + 1) <= ( len f) by A64, NAT_1: 13;

                then

                 A83: ( LSeg (f,j)) = ( LSeg ((f /. j),(f /. (j + 1)))) by A77, TOPREAL1:def 3;

                (f /. (j + 1)) = (f . (j + 1)) by A80, A82, FINSEQ_4: 15;

                then

                 A84: ( LSeg (((f ^ ( mid (g,2,( len g)))) /. j),((f ^ ( mid (g,2,( len g)))) /. (j + 1)))) = ( LSeg ((f /. j),(f /. (j + 1)))) by A80, A82, A79, A81, FINSEQ_1: 64;

                ( LSeg (((f ^ ( mid (g,2,( len g)))) /. j),((f ^ ( mid (g,2,( len g)))) /. (j + 1)))) = ( LSeg ((f ^ ( mid (g,2,( len g)))),j)) by A64, A77, TOPREAL1:def 3;

                then ( LSeg ((f ^ ( mid (g,2,( len g)))),i)) misses ( LSeg ((f ^ ( mid (g,2,( len g)))),j)) by A2, A63, A76, A84, A83, TOPREAL1:def 7;

                hence (( LSeg ((f ^ ( mid (g,2,( len g)))),i)) /\ ( LSeg ((f ^ ( mid (g,2,( len g)))),j))) = {} by XBOOLE_0:def 7;

              end;

                case i < 1;

                then ( LSeg ((f ^ ( mid (g,2,( len g)))),i)) = {} by TOPREAL1:def 3;

                hence (( LSeg ((f ^ ( mid (g,2,( len g)))),i)) /\ ( LSeg ((f ^ ( mid (g,2,( len g)))),j))) = {} ;

              end;

            end;

            hence thesis by XBOOLE_0:def 7;

          end;

            case

             A85: (i + 1) <= ( len f) & ( len f) <= j & (j + 1) <= ( len (f ^ ( mid (g,2,( len g)))));

            now

              per cases by A63, A85, XXREAL_0: 1;

                case

                 A86: (i + 1) < ( len f) & ( len f) <= j;

                ( len f) <= (( len f) + ( len ( mid (g,2,( len g))))) by NAT_1: 11;

                then

                 A87: (i + 1) < ( len (f ^ ( mid (g,2,( len g))))) by A6, A86, XXREAL_0: 2;

                

                 A88: ( len f) <= (j + 1) by A86, NAT_D: 48;

                

                 A89: (1 + 1) <= j by A5, A86, XXREAL_0: 2;

                then

                 A90: 1 <= j by NAT_D: 46;

                now

                  per cases ;

                    case

                     A91: 1 <= i;

                    i <= ( len f) by A85, NAT_D: 46;

                    then

                     A92: (f /. i) = (f . i) by A91, FINSEQ_4: 15;

                    i <= ( len (f ^ ( mid (g,2,( len g))))) by A87, NAT_D: 46;

                    then

                     A93: ((f ^ ( mid (g,2,( len g)))) /. i) = ((f ^ ( mid (g,2,( len g)))) . i) by A91, FINSEQ_4: 15;

                    i <= ( len f) by A85, NAT_D: 46;

                    then

                     A94: ((f ^ ( mid (g,2,( len g)))) /. i) = (f /. i) by A91, A93, A92, FINSEQ_1: 64;

                    

                     A95: j <= ( len (f ^ ( mid (g,2,( len g))))) by A85, NAT_D: 46;

                     A96:

                    now

                      assume 1 > (j -' ( len f));

                      then ((j -' ( len f)) + 1) <= ( 0 + 1) by NAT_1: 13;

                      then

                       A97: (j -' ( len f)) = 0 by XREAL_1: 6;

                      then (j - ( len f)) = 0 by A85, XREAL_1: 233;

                      hence ((f ^ ( mid (g,2,( len g)))) . j) = (g . ((j -' ( len f)) + 1)) by A1, A61, A97, FINSEQ_1: 64;

                    end;

                    1 <= (j + 1) by A90, NAT_D: 48;

                    then

                     A98: ((f ^ ( mid (g,2,( len g)))) /. (j + 1)) = ((f ^ ( mid (g,2,( len g)))) . (j + 1)) by A85, FINSEQ_4: 15;

                    

                     A99: ( LSeg (((f ^ ( mid (g,2,( len g)))) /. i),((f ^ ( mid (g,2,( len g)))) /. (i + 1)))) = ( LSeg ((f ^ ( mid (g,2,( len g)))),i)) by A87, A91, TOPREAL1:def 3;

                    

                     A100: 1 <= (i + 1) by A91, NAT_D: 48;

                    then

                     A101: (f /. (i + 1)) = (f . (i + 1)) by A85, FINSEQ_4: 15;

                     A102:

                    now

                      assume 1 > ((j + 1) -' ( len f));

                      then (((j + 1) -' ( len f)) + 1) <= ( 0 + 1) by NAT_1: 13;

                      then

                       A103: ((j + 1) -' ( len f)) = 0 by XREAL_1: 6;

                      then ((j + 1) - ( len f)) = 0 by A88, XREAL_1: 233;

                      hence ((f ^ ( mid (g,2,( len g)))) . (j + 1)) = (g . (((j + 1) -' ( len f)) + 1)) by A1, A61, A103, FINSEQ_1: 64;

                    end;

                    ((j + 1) + 1) <= ((( len f) + (( len g) - 1)) + 1) by A6, A10, A85, XREAL_1: 6;

                    then (((j + 1) + 1) - ( len f)) <= ((( len f) + ( len g)) - ( len f)) by XREAL_1: 9;

                    then (((j - ( len f)) + 1) + 1) <= ( len g);

                    then

                     A104: (((j -' ( len f)) + 1) + 1) <= ( len g) by A86, XREAL_1: 233;

                    then ((j -' ( len f)) + 1) <= ( len g) by NAT_D: 46;

                    then

                     A105: (g /. ((j -' ( len f)) + 1)) = (g . ((j -' ( len f)) + 1)) by FINSEQ_4: 15, NAT_1: 11;

                    ((((j -' ( len f)) + 1) + 1) - 1) <= (( len g) - 1) by A104, XREAL_1: 9;

                    then

                     A106: ((j -' ( len f)) + 1) <= ((( len g) - 2) + 1);

                    then ((j -' ( len f)) + 1) <= ((( len g) -' 2) + 1) by A8, XREAL_1: 233;

                    then

                     A107: (j -' ( len f)) <= ((( len g) -' 2) + 1) by NAT_D: 46;

                     A108:

                    now

                      assume

                       A109: 1 <= (j -' ( len f));

                      then 1 <= (j - ( len f)) by NAT_D: 39;

                      then (1 + ( len f)) <= ((j - ( len f)) + ( len f)) by XREAL_1: 6;

                      then

                       A110: ( len f) < j by NAT_1: 13;

                      then ((f ^ ( mid (g,2,( len g)))) . j) = (( mid (g,2,( len g))) . (j - ( len f))) by A95, FINSEQ_6: 108;

                      then ((f ^ ( mid (g,2,( len g)))) . j) = (( mid (g,2,( len g))) . (j -' ( len f))) by A110, XREAL_1: 233;

                      then ((f ^ ( mid (g,2,( len g)))) . j) = (g . (((j -' ( len f)) + 2) - 1)) by A8, A107, A109, FINSEQ_6: 122;

                      hence ((f ^ ( mid (g,2,( len g)))) . j) = (g . ((j -' ( len f)) + 1));

                    end;

                    

                     A111: ((j -' ( len f)) + 1) = ((j - ( len f)) + 1) by A85, XREAL_1: 233

                    .= ((j + 1) - ( len f))

                    .= ((j + 1) -' ( len f)) by A88, XREAL_1: 233;

                     A112:

                    now

                      assume

                       A113: 1 <= ((j + 1) -' ( len f));

                      then 1 <= ((j + 1) - ( len f)) by NAT_D: 39;

                      then (1 + ( len f)) <= (((j + 1) - ( len f)) + ( len f)) by XREAL_1: 6;

                      then

                       A114: ( len f) < (j + 1) by NAT_1: 13;

                      then ((f ^ ( mid (g,2,( len g)))) . (j + 1)) = (( mid (g,2,( len g))) . ((j + 1) - ( len f))) by A85, FINSEQ_6: 108;

                      then ((f ^ ( mid (g,2,( len g)))) . (j + 1)) = (( mid (g,2,( len g))) . ((j + 1) -' ( len f))) by A114, XREAL_1: 233;

                      then ((f ^ ( mid (g,2,( len g)))) . (j + 1)) = (g . ((((j + 1) -' ( len f)) + 2) - 1)) by A8, A106, A111, A113, FINSEQ_6: 122;

                      hence ((f ^ ( mid (g,2,( len g)))) . (j + 1)) = (g . (((j + 1) -' ( len f)) + 1));

                    end;

                    1 <= (1 + (j -' ( len f))) by NAT_1: 11;

                    then

                     A115: ( LSeg (g,((j -' ( len f)) + 1))) = ( LSeg ((g /. ((j -' ( len f)) + 1)),(g /. (((j -' ( len f)) + 1) + 1)))) by A104, TOPREAL1:def 3;

                    1 <= j by A89, NAT_D: 46;

                    then

                     A116: ((f ^ ( mid (g,2,( len g)))) /. j) = (g /. ((j -' ( len f)) + 1)) by A95, A105, A108, A96, FINSEQ_4: 15;

                    (g /. (((j -' ( len f)) + 1) + 1)) = (g . (((j -' ( len f)) + 1) + 1)) by A104, FINSEQ_4: 15, NAT_1: 11;

                    then ( LSeg ((f ^ ( mid (g,2,( len g)))),j)) = ( LSeg (g,((j -' ( len f)) + 1))) by A85, A90, A111, A115, A116, A98, A112, A102, TOPREAL1:def 3;

                    then

                     A117: ( LSeg ((f ^ ( mid (g,2,( len g)))),j)) c= ( L~ g) by TOPREAL3: 19;

                    

                     A118: ((i + 1) + 1) <= ( len f) by A86, NAT_1: 13;

                    ((f ^ ( mid (g,2,( len g)))) /. (i + 1)) = ((f ^ ( mid (g,2,( len g)))) . (i + 1)) by A87, A100, FINSEQ_4: 15;

                    then ( LSeg (((f ^ ( mid (g,2,( len g)))) /. i),((f ^ ( mid (g,2,( len g)))) /. (i + 1)))) = ( LSeg ((f /. i),(f /. (i + 1)))) by A85, A100, A94, A101, FINSEQ_1: 64;

                    then

                     A119: ( LSeg ((f ^ ( mid (g,2,( len g)))),i)) = ( LSeg (f,i)) by A85, A91, A99, TOPREAL1:def 3;

                    then ( LSeg ((f ^ ( mid (g,2,( len g)))),i)) c= ( L~ f) by TOPREAL3: 19;

                    then

                     A120: (( LSeg ((f ^ ( mid (g,2,( len g)))),i)) /\ ( LSeg ((f ^ ( mid (g,2,( len g)))),j))) c= {(g . 1)} by A4, A117, XBOOLE_1: 27;

                    now

                      per cases ;

                        case

                         A121: (i + 1) < (( len f) -' 1);

                        

                         A122: 1 <= ( len f) by A5, XXREAL_0: 2;

                        

                         A123: ((( len f) -' 1) + 1) = ((( len f) - 1) + 1) by A5, XREAL_1: 233, XXREAL_0: 2

                        .= ( len f);

                        

                         A124: ((1 + 1) - 1) <= (( len f) - 1) by A5, XREAL_1: 9;

                        now

                          (f /. ( len f)) in ( LSeg (f,(( len f) -' 1))) by A124, A123, TOPREAL1: 21;

                          then

                           A125: (g . 1) in ( LSeg (f,(( len f) -' 1))) by A1, A122, FINSEQ_4: 15;

                          given x be object such that

                           A126: x in (( LSeg ((f ^ ( mid (g,2,( len g)))),i)) /\ ( LSeg ((f ^ ( mid (g,2,( len g)))),j)));

                          

                           A127: x in ( LSeg (f,i)) by A119, A126, XBOOLE_0:def 4;

                          x = (g . 1) by A120, A126, TARSKI:def 1;

                          then x in (( LSeg (f,i)) /\ ( LSeg (f,(( len f) -' 1)))) by A127, A125, XBOOLE_0:def 4;

                          then ( LSeg (f,i)) meets ( LSeg (f,(( len f) -' 1))) by XBOOLE_0: 4;

                          hence contradiction by A2, A121, TOPREAL1:def 7;

                        end;

                        hence thesis by XBOOLE_0: 4;

                      end;

                        case (i + 1) >= (( len f) -' 1);

                        then (i + 1) >= (( len f) - 1) by A5, XREAL_1: 233, XXREAL_0: 2;

                        then

                         A128: ((i + 1) + 1) >= ((( len f) - 1) + 1) by XREAL_1: 6;

                        then

                         A129: ((i + 1) + 1) = ( len f) by A118, XXREAL_0: 1;

                        then

                         A130: (i + 1) <= ( len f) by NAT_1: 11;

                        (i + 1) = (( len f) - 1) by A129;

                        then

                         A131: (i + 1) = (( len f) -' 1) by A5, XREAL_1: 233, XXREAL_0: 2;

                        

                         A132: ((( len f) -' 1) + 1) = ((( len f) - 1) + 1) by A5, XREAL_1: 233, XXREAL_0: 2

                        .= ( len f);

                        then (1 + 1) <= ((( len f) -' 1) + 1) by A2, TOPREAL1:def 8;

                        then

                         A133: 1 <= (( len f) -' 1) by XREAL_1: 6;

                        

                         A134: (i + (1 + 1)) = ( len f) by A118, A128, XXREAL_0: 1;

                        now

                          ((1 + 1) - 1) <= (( len f) - 1) by A5, XREAL_1: 9;

                          then

                           A135: 1 <= (( len f) -' 1) by NAT_D: 39;

                          (( len f) -' 1) <= ( len f) by NAT_D: 35;

                          then

                           A136: (( len f) -' 1) in ( dom f) by A135, FINSEQ_3: 25;

                          

                           A137: (( LSeg (f,i)) /\ ( LSeg (f,(( len f) -' 1)))) = {(f /. (i + 1))} by A2, A91, A131, A134, TOPREAL1:def 6;

                          (f /. ( len f)) in ( LSeg (f,(( len f) -' 1))) by A132, A133, TOPREAL1: 21;

                          then

                           A138: (g . 1) in ( LSeg (f,(( len f) -' 1))) by A1, A61, FINSEQ_4: 15;

                          given x be object such that

                           A139: x in (( LSeg ((f ^ ( mid (g,2,( len g)))),i)) /\ ( LSeg ((f ^ ( mid (g,2,( len g)))),j)));

                          

                           A140: x = (g . 1) by A120, A139, TARSKI:def 1;

                          x in ( LSeg (f,i)) by A119, A139, XBOOLE_0:def 4;

                          then x in (( LSeg (f,i)) /\ ( LSeg (f,(( len f) -' 1)))) by A140, A138, XBOOLE_0:def 4;

                          then (f . ( len f)) = (f /. (i + 1)) by A1, A140, A137, TARSKI:def 1;

                          then

                           A141: (f . ( len f)) = (f . (( len f) -' 1)) by A131, A130, FINSEQ_4: 15, NAT_1: 11;

                          ( len f) in ( dom f) by A61, FINSEQ_3: 25;

                          then ( len f) = (( len f) -' 1) by A2, A141, A136, FUNCT_1:def 4;

                          then ( len f) = (( len f) - 1) by A5, XREAL_1: 233, XXREAL_0: 2;

                          hence contradiction;

                        end;

                        hence (( LSeg ((f ^ ( mid (g,2,( len g)))),i)) /\ ( LSeg ((f ^ ( mid (g,2,( len g)))),j))) = {} by XBOOLE_0:def 1;

                      end;

                    end;

                    hence (( LSeg ((f ^ ( mid (g,2,( len g)))),i)) /\ ( LSeg ((f ^ ( mid (g,2,( len g)))),j))) = {} by XBOOLE_0:def 7;

                  end;

                    case i < 1;

                    then ( LSeg ((f ^ ( mid (g,2,( len g)))),i)) = {} by TOPREAL1:def 3;

                    hence (( LSeg ((f ^ ( mid (g,2,( len g)))),i)) /\ ( LSeg ((f ^ ( mid (g,2,( len g)))),j))) = {} ;

                  end;

                end;

                hence (( LSeg ((f ^ ( mid (g,2,( len g)))),i)) /\ ( LSeg ((f ^ ( mid (g,2,( len g)))),j))) = {} ;

              end;

                case

                 A142: (i + 1) <= ( len f) & ( len f) < j;

                ((1 + 1) - 1) <= (( len g) - 1) by A8, XREAL_1: 9;

                then (( len f) + 1) <= (( len f) + ( len ( mid (g,2,( len g))))) by A10, XREAL_1: 7;

                then ( len f) < (( len f) + ( len ( mid (g,2,( len g))))) by NAT_1: 13;

                then

                 A143: (i + 1) < ( len (f ^ ( mid (g,2,( len g))))) by A6, A142, XXREAL_0: 2;

                

                 A144: ( len f) <= (j + 1) by A142, NAT_D: 48;

                

                 A145: (1 + 1) <= j by A5, A142, XXREAL_0: 2;

                then

                 A146: 1 <= j by NAT_D: 46;

                now

                  per cases ;

                    case

                     A147: 1 <= i;

                    i <= ( len f) by A85, NAT_D: 46;

                    then

                     A148: (f /. i) = (f . i) by A147, FINSEQ_4: 15;

                    i <= ( len (f ^ ( mid (g,2,( len g))))) by A143, NAT_D: 46;

                    then

                     A149: ((f ^ ( mid (g,2,( len g)))) /. i) = ((f ^ ( mid (g,2,( len g)))) . i) by A147, FINSEQ_4: 15;

                    i <= ( len f) by A85, NAT_D: 46;

                    then

                     A150: ((f ^ ( mid (g,2,( len g)))) /. i) = (f /. i) by A147, A149, A148, FINSEQ_1: 64;

                    

                     A151: ( LSeg (((f ^ ( mid (g,2,( len g)))) /. i),((f ^ ( mid (g,2,( len g)))) /. (i + 1)))) = ( LSeg ((f ^ ( mid (g,2,( len g)))),i)) by A143, A147, TOPREAL1:def 3;

                    

                     A152: 1 <= (i + 1) by A147, NAT_D: 48;

                    then

                     A153: (f /. (i + 1)) = (f . (i + 1)) by A85, FINSEQ_4: 15;

                    ((f ^ ( mid (g,2,( len g)))) /. (i + 1)) = ((f ^ ( mid (g,2,( len g)))) . (i + 1)) by A143, A152, FINSEQ_4: 15;

                    then ( LSeg (((f ^ ( mid (g,2,( len g)))) /. i),((f ^ ( mid (g,2,( len g)))) /. (i + 1)))) = ( LSeg ((f /. i),(f /. (i + 1)))) by A85, A152, A150, A153, FINSEQ_1: 64;

                    then ( LSeg ((f ^ ( mid (g,2,( len g)))),i)) = ( LSeg (f,i)) by A85, A147, A151, TOPREAL1:def 3;

                    then

                     A154: ( LSeg ((f ^ ( mid (g,2,( len g)))),i)) c= ( L~ f) by TOPREAL3: 19;

                    

                     A155: j <= ( len (f ^ ( mid (g,2,( len g))))) by A85, NAT_D: 46;

                     A156:

                    now

                      assume 1 > (j -' ( len f));

                      then ((j -' ( len f)) + 1) <= ( 0 + 1) by NAT_1: 13;

                      then

                       A157: (j -' ( len f)) = 0 by XREAL_1: 6;

                      then (j - ( len f)) = 0 by A85, XREAL_1: 233;

                      hence ((f ^ ( mid (g,2,( len g)))) . j) = (g . ((j -' ( len f)) + 1)) by A1, A61, A157, FINSEQ_1: 64;

                    end;

                    1 <= (j + 1) by A146, NAT_D: 48;

                    then

                     A158: ((f ^ ( mid (g,2,( len g)))) /. (j + 1)) = ((f ^ ( mid (g,2,( len g)))) . (j + 1)) by A85, FINSEQ_4: 15;

                     A159:

                    now

                      assume 1 > ((j + 1) -' ( len f));

                      then (((j + 1) -' ( len f)) + 1) <= ( 0 + 1) by NAT_1: 13;

                      then

                       A160: ((j + 1) -' ( len f)) = 0 by XREAL_1: 6;

                      then ((j + 1) - ( len f)) = 0 by A144, XREAL_1: 233;

                      hence ((f ^ ( mid (g,2,( len g)))) . (j + 1)) = (g . (((j + 1) -' ( len f)) + 1)) by A1, A61, A160, FINSEQ_1: 64;

                    end;

                    ((j + 1) + 1) <= ((( len f) + (( len g) - 1)) + 1) by A6, A10, A85, XREAL_1: 6;

                    then (((j + 1) + 1) - ( len f)) <= ((( len f) + ( len g)) - ( len f)) by XREAL_1: 9;

                    then (((j - ( len f)) + 1) + 1) <= ( len g);

                    then

                     A161: (((j -' ( len f)) + 1) + 1) <= ( len g) by A142, XREAL_1: 233;

                    then ((j -' ( len f)) + 1) <= ( len g) by NAT_D: 46;

                    then

                     A162: (g /. ((j -' ( len f)) + 1)) = (g . ((j -' ( len f)) + 1)) by FINSEQ_4: 15, NAT_1: 11;

                    ((((j -' ( len f)) + 1) + 1) - 1) <= (( len g) - 1) by A161, XREAL_1: 9;

                    then

                     A163: ((j -' ( len f)) + 1) <= ((( len g) - 2) + 1);

                    then ((j -' ( len f)) + 1) <= ((( len g) -' 2) + 1) by A8, XREAL_1: 233;

                    then

                     A164: (j -' ( len f)) <= ((( len g) -' 2) + 1) by NAT_D: 46;

                     A165:

                    now

                      assume

                       A166: 1 <= (j -' ( len f));

                      then 1 <= (j - ( len f)) by NAT_D: 39;

                      then (1 + ( len f)) <= ((j - ( len f)) + ( len f)) by XREAL_1: 6;

                      then

                       A167: ( len f) < j by NAT_1: 13;

                      then ((f ^ ( mid (g,2,( len g)))) . j) = (( mid (g,2,( len g))) . (j - ( len f))) by A155, FINSEQ_6: 108;

                      then ((f ^ ( mid (g,2,( len g)))) . j) = (( mid (g,2,( len g))) . (j -' ( len f))) by A167, XREAL_1: 233;

                      then ((f ^ ( mid (g,2,( len g)))) . j) = (g . (((j -' ( len f)) + 2) - 1)) by A8, A164, A166, FINSEQ_6: 122;

                      hence ((f ^ ( mid (g,2,( len g)))) . j) = (g . ((j -' ( len f)) + 1));

                    end;

                    

                     A168: ((j -' ( len f)) + 1) = ((j - ( len f)) + 1) by A85, XREAL_1: 233

                    .= ((j + 1) - ( len f))

                    .= ((j + 1) -' ( len f)) by A144, XREAL_1: 233;

                     A169:

                    now

                      assume

                       A170: 1 <= ((j + 1) -' ( len f));

                      then 1 <= ((j + 1) - ( len f)) by NAT_D: 39;

                      then (1 + ( len f)) <= (((j + 1) - ( len f)) + ( len f)) by XREAL_1: 6;

                      then

                       A171: ( len f) < (j + 1) by NAT_1: 13;

                      then ((f ^ ( mid (g,2,( len g)))) . (j + 1)) = (( mid (g,2,( len g))) . ((j + 1) - ( len f))) by A85, FINSEQ_6: 108;

                      then ((f ^ ( mid (g,2,( len g)))) . (j + 1)) = (( mid (g,2,( len g))) . ((j + 1) -' ( len f))) by A171, XREAL_1: 233;

                      then ((f ^ ( mid (g,2,( len g)))) . (j + 1)) = (g . ((((j + 1) -' ( len f)) + 2) - 1)) by A8, A163, A168, A170, FINSEQ_6: 122;

                      hence ((f ^ ( mid (g,2,( len g)))) . (j + 1)) = (g . (((j + 1) -' ( len f)) + 1));

                    end;

                    1 <= (1 + (j -' ( len f))) by NAT_1: 11;

                    then

                     A172: ( LSeg (g,((j -' ( len f)) + 1))) = ( LSeg ((g /. ((j -' ( len f)) + 1)),(g /. (((j -' ( len f)) + 1) + 1)))) by A161, TOPREAL1:def 3;

                    1 <= j by A145, NAT_D: 46;

                    then

                     A173: ((f ^ ( mid (g,2,( len g)))) /. j) = (g /. ((j -' ( len f)) + 1)) by A155, A162, A165, A156, FINSEQ_4: 15;

                    (g /. (((j -' ( len f)) + 1) + 1)) = (g . (((j -' ( len f)) + 1) + 1)) by A161, FINSEQ_4: 15, NAT_1: 11;

                    then

                     A174: ( LSeg ((f ^ ( mid (g,2,( len g)))),j)) = ( LSeg (g,((j -' ( len f)) + 1))) by A85, A146, A168, A172, A173, A158, A169, A159, TOPREAL1:def 3;

                    then ( LSeg ((f ^ ( mid (g,2,( len g)))),j)) c= ( L~ g) by TOPREAL3: 19;

                    then

                     A175: (( LSeg ((f ^ ( mid (g,2,( len g)))),i)) /\ ( LSeg ((f ^ ( mid (g,2,( len g)))),j))) c= {(g . 1)} by A4, A154, XBOOLE_1: 27;

                    now

                      

                       A176: (1 + 1) in ( dom g) by A8, FINSEQ_3: 25;

                      

                       A177: ((j -' ( len f)) + 1) = ((j - ( len f)) + 1) by A142, XREAL_1: 233;

                      

                       A178: (1 + 1) <= ( len g) by A3, TOPREAL1:def 8;

                      given x be object such that

                       A179: x in (( LSeg ((f ^ ( mid (g,2,( len g)))),i)) /\ ( LSeg ((f ^ ( mid (g,2,( len g)))),j)));

                      

                       A180: x in ( LSeg (g,((j -' ( len f)) + 1))) by A174, A179, XBOOLE_0:def 4;

                      

                       A181: x = (g . 1) by A175, A179, TARSKI:def 1;

                      then (g /. 1) = x by A9, FINSEQ_4: 15;

                      then x in ( LSeg (g,1)) by A178, TOPREAL1: 21;

                      then

                       A182: x in (( LSeg (g,1)) /\ ( LSeg (g,((j -' ( len f)) + 1)))) by A180, XBOOLE_0:def 4;

                      then ( LSeg (g,1)) meets ( LSeg (g,((j -' ( len f)) + 1))) by XBOOLE_0: 4;

                      then (1 + 1) >= ((j -' ( len f)) + 1) by A3, TOPREAL1:def 7;

                      then 1 >= (j -' ( len f)) by XREAL_1: 6;

                      then 1 >= (j - ( len f)) by NAT_D: 39;

                      then

                       A183: (1 + ( len f)) >= ((j - ( len f)) + ( len f)) by XREAL_1: 6;

                      j >= (( len f) + 1) by A142, NAT_1: 13;

                      then

                       A184: j = (( len f) + 1) by A183, XXREAL_0: 1;

                      ( LSeg (g,((j -' ( len f)) + 1))) <> {} by A174, A179;

                      then (1 + 2) <= ( len g) by A184, A177, TOPREAL1:def 3;

                      then (( LSeg (g,1)) /\ ( LSeg (g,((j -' ( len f)) + 1)))) = {(g /. (1 + 1))} by A3, A184, A177, TOPREAL1:def 6;

                      

                      then

                       A185: x = (g /. (1 + 1)) by A182, TARSKI:def 1

                      .= (g . (1 + 1)) by A8, FINSEQ_4: 15;

                      1 in ( dom g) by A9, FINSEQ_3: 25;

                      hence contradiction by A3, A181, A185, A176, FUNCT_1:def 4;

                    end;

                    hence (( LSeg ((f ^ ( mid (g,2,( len g)))),i)) /\ ( LSeg ((f ^ ( mid (g,2,( len g)))),j))) = {} by XBOOLE_0:def 1;

                  end;

                    case i < 1;

                    then ( LSeg ((f ^ ( mid (g,2,( len g)))),i)) = {} by TOPREAL1:def 3;

                    hence (( LSeg ((f ^ ( mid (g,2,( len g)))),i)) /\ ( LSeg ((f ^ ( mid (g,2,( len g)))),j))) = {} ;

                  end;

                end;

                hence (( LSeg ((f ^ ( mid (g,2,( len g)))),i)) /\ ( LSeg ((f ^ ( mid (g,2,( len g)))),j))) = {} ;

              end;

            end;

            hence thesis by XBOOLE_0:def 7;

          end;

            case

             A186: ( len f) < (i + 1) & (j + 1) <= ( len (f ^ ( mid (g,2,( len g)))));

            then

             A187: ( len f) <= i by NAT_1: 13;

            then

             A188: (i -' ( len f)) = (i - ( len f)) by XREAL_1: 233;

            

             A189: (1 + 1) <= i by A5, A187, XXREAL_0: 2;

            then

             A190: 1 <= i by NAT_D: 46;

            then

             A191: 1 <= (i + 1) by NAT_D: 48;

            then

             A192: 1 <= j by A63, XXREAL_0: 2;

            

             A193: 1 <= ((j -' ( len f)) + 1) by NAT_1: 11;

            

             A194: ( len f) < j by A63, A186, XXREAL_0: 2;

            j <= (j + 1) by NAT_1: 11;

            then

             A195: ( len f) < (j + 1) by A194, XXREAL_0: 2;

            

             A196: 1 <= ((i -' ( len f)) + 1) by NAT_1: 11;

            

             A197: (j -' ( len f)) = (j - ( len f)) by A63, A186, XREAL_1: 233, XXREAL_0: 2;

            ((i + 1) - ( len f)) < (j - ( len f)) by A63, XREAL_1: 9;

            then

             A198: (((i -' ( len f)) + 1) + 1) < ((j -' ( len f)) + 1) by A188, A197, XREAL_1: 6;

            now

              per cases ;

                case

                 A199: (j + 1) <= ( len (f ^ ( mid (g,2,( len g)))));

                

                 A200: 1 <= j by A63, A191, XXREAL_0: 2;

                then 1 <= (j + 1) by NAT_D: 48;

                then

                 A201: ((f ^ ( mid (g,2,( len g)))) /. (j + 1)) = ((f ^ ( mid (g,2,( len g)))) . (j + 1)) by A199, FINSEQ_4: 15;

                (( len f) + 1) <= j by A194, NAT_1: 13;

                then

                 A202: ((( len f) + 1) - ( len f)) <= (j - ( len f)) by XREAL_1: 9;

                

                 A203: 1 <= i by A189, NAT_D: 46;

                then

                 A204: 1 <= (i + 1) by NAT_D: 48;

                

                 A205: j <= ( len (f ^ ( mid (g,2,( len g))))) by A199, NAT_D: 46;

                then

                 A206: (i + 1) < ( len (f ^ ( mid (g,2,( len g))))) by A63, XXREAL_0: 2;

                then

                 A207: i <= ( len (f ^ ( mid (g,2,( len g))))) by NAT_D: 46;

                (i + 1) < (( len f) + (( len g) - 1)) by A10, A206, FINSEQ_1: 22;

                then

                 A208: ((i + 1) - ( len f)) < ((( len f) + (( len g) - 1)) - ( len f)) by XREAL_1: 9;

                then

                 A209: (((i - ( len f)) + 1) + 1) < ((( len g) - 1) + 1) by XREAL_1: 6;

                then (((i -' ( len f)) + 1) + 1) <= ( len g) by A187, XREAL_1: 233;

                then

                 A210: (g /. (((i -' ( len f)) + 1) + 1)) = (g . (((i -' ( len f)) + 1) + 1)) by FINSEQ_4: 15, NAT_1: 11;

                (i + 1) <= ( len (f ^ ( mid (g,2,( len g))))) by A63, A205, XXREAL_0: 2;

                then

                 A211: ((f ^ ( mid (g,2,( len g)))) /. (i + 1)) = ((f ^ ( mid (g,2,( len g)))) . (i + 1)) by A204, FINSEQ_4: 15;

                

                 A212: ( LSeg (((f ^ ( mid (g,2,( len g)))) /. j),((f ^ ( mid (g,2,( len g)))) /. (j + 1)))) = ( LSeg ((f ^ ( mid (g,2,( len g)))),j)) by A192, A199, TOPREAL1:def 3;

                

                 A213: ( LSeg (((f ^ ( mid (g,2,( len g)))) /. i),((f ^ ( mid (g,2,( len g)))) /. (i + 1)))) = ( LSeg ((f ^ ( mid (g,2,( len g)))),i)) by A190, A206, TOPREAL1:def 3;

                

                 A214: ((i -' ( len f)) + 1) <= ( len g) by A188, A209, NAT_D: 46;

                then

                 A215: (g /. ((i -' ( len f)) + 1)) = (g . ((i -' ( len f)) + 1)) by FINSEQ_4: 15, NAT_1: 11;

                 A216:

                now

                  per cases ;

                    case i <= ( len f);

                    then

                     A217: i = ( len f) by A187, XXREAL_0: 1;

                    

                    then ((f ^ ( mid (g,2,( len g)))) . i) = (g . ( 0 + 1)) by A1, A190, FINSEQ_1: 64

                    .= (g . ((i -' ( len f)) + 1)) by A217, XREAL_1: 232;

                    hence ((f ^ ( mid (g,2,( len g)))) /. i) = (g /. ((i -' ( len f)) + 1)) by A203, A207, A215, FINSEQ_4: 15;

                  end;

                    case

                     A218: i > ( len f);

                    then (( len f) + 1) <= i by NAT_1: 13;

                    then

                     A219: ((( len f) + 1) - ( len f)) <= (i - ( len f)) by XREAL_1: 9;

                    (((i -' ( len f)) + 1) - 1) <= (( len g) - 1) by A214, XREAL_1: 9;

                    then

                     A220: (i -' ( len f)) <= ((( len g) - 2) + 1);

                    ((f ^ ( mid (g,2,( len g)))) . i) = (( mid (g,2,( len g))) . (i -' ( len f))) by A188, A207, A218, FINSEQ_6: 108

                    .= (g . (((i -' ( len f)) + 2) - 1)) by A8, A188, A219, A220, FINSEQ_6: 122

                    .= (g . ((i -' ( len f)) + 1));

                    hence ((f ^ ( mid (g,2,( len g)))) /. i) = (g /. ((i -' ( len f)) + 1)) by A203, A207, A215, FINSEQ_4: 15;

                  end;

                end;

                (j + 1) <= (( len f) + (( len g) - 1)) by A10, A199, FINSEQ_1: 22;

                then

                 A221: ((j + 1) - ( len f)) <= ((( len f) + (( len g) - 1)) - ( len f)) by XREAL_1: 9;

                then

                 A222: ((j -' ( len f)) + 1) <= ((( len g) - 2) + 1) by A197;

                

                 A223: ((((j -' ( len f)) + 1) + 2) - 1) = (((j -' ( len f)) + 1) + 1);

                

                 A224: ((f ^ ( mid (g,2,( len g)))) . (j + 1)) = (( mid (g,2,( len g))) . ((j + 1) - ( len f))) by A195, A199, FINSEQ_6: 108

                .= (g . (((j -' ( len f)) + 1) + 1)) by A8, A197, A193, A222, A223, FINSEQ_6: 122;

                

                 A225: ((((i -' ( len f)) + 1) + 2) - 1) = (((i -' ( len f)) + 1) + 1);

                

                 A226: ((i -' ( len f)) + 1) <= ((( len g) - 2) + 1) by A188, A208;

                ((f ^ ( mid (g,2,( len g)))) . (i + 1)) = (( mid (g,2,( len g))) . ((i + 1) - ( len f))) by A186, A206, FINSEQ_6: 108

                .= (g . (((i -' ( len f)) + 1) + 1)) by A8, A188, A196, A226, A225, FINSEQ_6: 122;

                then

                 A227: ( LSeg ((f ^ ( mid (g,2,( len g)))),i)) = ( LSeg (g,((i -' ( len f)) + 1))) by A188, A196, A209, A216, A211, A210, A213, TOPREAL1:def 3;

                

                 A228: (((j - ( len f)) + 1) + 1) <= ((( len g) - 1) + 1) by A221, XREAL_1: 6;

                then

                 A229: ((j -' ( len f)) + 1) <= ( len g) by A197, NAT_D: 46;

                then

                 A230: (g /. ((j -' ( len f)) + 1)) = (g . ((j -' ( len f)) + 1)) by FINSEQ_4: 15, NAT_1: 11;

                

                 A231: j <= ( len (f ^ ( mid (g,2,( len g))))) by A199, NAT_D: 46;

                then

                 A232: ((f ^ ( mid (g,2,( len g)))) /. j) = ((f ^ ( mid (g,2,( len g)))) . j) by A200, FINSEQ_4: 15;

                (((j -' ( len f)) + 1) + 1) <= ( len g) by A63, A186, A228, XREAL_1: 233, XXREAL_0: 2;

                then

                 A233: (g /. (((j -' ( len f)) + 1) + 1)) = (g . (((j -' ( len f)) + 1) + 1)) by FINSEQ_4: 15, NAT_1: 11;

                (((j -' ( len f)) + 1) - 1) <= (( len g) - 1) by A229, XREAL_1: 9;

                then

                 A234: (j -' ( len f)) <= ((( len g) - 2) + 1);

                ((f ^ ( mid (g,2,( len g)))) . j) = (( mid (g,2,( len g))) . (j - ( len f))) by A194, A231, FINSEQ_6: 108

                .= (g . (((j -' ( len f)) + 2) - 1)) by A8, A197, A202, A234, FINSEQ_6: 122

                .= (g . ((j -' ( len f)) + 1));

                then ( LSeg ((f ^ ( mid (g,2,( len g)))),j)) = ( LSeg (g,((j -' ( len f)) + 1))) by A197, A193, A228, A232, A230, A201, A233, A224, A212, TOPREAL1:def 3;

                then ( LSeg ((f ^ ( mid (g,2,( len g)))),i)) misses ( LSeg ((f ^ ( mid (g,2,( len g)))),j)) by A3, A198, A227, TOPREAL1:def 7;

                hence (( LSeg ((f ^ ( mid (g,2,( len g)))),i)) /\ ( LSeg ((f ^ ( mid (g,2,( len g)))),j))) = {} by XBOOLE_0:def 7;

              end;

                case (j + 1) > ( len (f ^ ( mid (g,2,( len g)))));

                then ( LSeg ((f ^ ( mid (g,2,( len g)))),j)) = {} by TOPREAL1:def 3;

                hence (( LSeg ((f ^ ( mid (g,2,( len g)))),i)) /\ ( LSeg ((f ^ ( mid (g,2,( len g)))),j))) = {} ;

              end;

            end;

            hence (( LSeg ((f ^ ( mid (g,2,( len g)))),i)) /\ ( LSeg ((f ^ ( mid (g,2,( len g)))),j))) = {} ;

          end;

            case (j + 1) > ( len (f ^ ( mid (g,2,( len g)))));

            then ( LSeg ((f ^ ( mid (g,2,( len g)))),j)) = {} by TOPREAL1:def 3;

            hence (( LSeg ((f ^ ( mid (g,2,( len g)))),i)) /\ ( LSeg ((f ^ ( mid (g,2,( len g)))),j))) = {} ;

          end;

        end;

        hence thesis by XBOOLE_0:def 7;

      end;

      

       A235: for i be Nat st 1 <= i & (i + 2) <= ( len (f ^ ( mid (g,2,( len g))))) holds (( LSeg ((f ^ ( mid (g,2,( len g)))),i)) /\ ( LSeg ((f ^ ( mid (g,2,( len g)))),(i + 1)))) = {((f ^ ( mid (g,2,( len g)))) /. (i + 1))}

      proof

        let i be Nat;

        assume that

         A236: 1 <= i and

         A237: (i + 2) <= ( len (f ^ ( mid (g,2,( len g)))));

        

         A238: 1 <= (i + 1) by A236, NAT_D: 48;

        

         A239: i <= ( len (f ^ ( mid (g,2,( len g))))) by A237, NAT_D: 47;

        

         A240: 1 <= ((i + 1) + 1) by A236, NAT_D: 48;

        

         A241: (i + 1) <= ( len (f ^ ( mid (g,2,( len g))))) by A237, NAT_D: 47;

        ((i + 1) + 1) <= (( len f) + ( len ( mid (g,2,( len g))))) by A237, FINSEQ_1: 22;

        then ((i + 1) + 1) <= (( len f) + ((( len g) -' 2) + 1)) by A8, A9, FINSEQ_6: 118;

        then ((i + 1) + 1) <= (( len f) + ((( len g) - (1 + 1)) + 1)) by A8, XREAL_1: 233;

        then

         A242: (((i + 1) + 1) - ( len f)) <= ((( len f) + ((( len g) - (1 + 1)) + 1)) - ( len f)) by XREAL_1: 9;

        then

         A243: ((((i + 1) - ( len f)) + 1) + 1) <= ((( len g) - 1) + 1) by XREAL_1: 6;

        then

         A244: ((((i - ( len f)) + 1) + 1) + 1) <= ( len g);

        now

          per cases ;

            case

             A245: (i + 2) <= ( len f);

            

             A246: ((f ^ ( mid (g,2,( len g)))) /. (i + 1)) = ((f ^ ( mid (g,2,( len g)))) . (i + 1)) by A238, A241, FINSEQ_4: 15;

            ((i + 1) + 1) <= ( len f) by A245;

            then

             A247: (i + 1) <= ( len f) by NAT_D: 46;

            then (f /. (i + 1)) = (f . (i + 1)) by A238, FINSEQ_4: 15;

            then

             A248: ((f ^ ( mid (g,2,( len g)))) /. (i + 1)) = (f /. (i + 1)) by A238, A247, A246, FINSEQ_1: 64;

            

             A249: (f /. ((i + 1) + 1)) = (f . ((i + 1) + 1)) by A240, A245, FINSEQ_4: 15;

            

             A250: ( LSeg (((f ^ ( mid (g,2,( len g)))) /. (i + 1)),((f ^ ( mid (g,2,( len g)))) /. ((i + 1) + 1)))) = ( LSeg ((f ^ ( mid (g,2,( len g)))),(i + 1))) by A237, A238, TOPREAL1:def 3;

            

             A251: ((f ^ ( mid (g,2,( len g)))) /. i) = ((f ^ ( mid (g,2,( len g)))) . i) by A236, A239, FINSEQ_4: 15;

            

             A252: i <= ( len f) by A247, NAT_D: 46;

            then (f /. i) = (f . i) by A236, FINSEQ_4: 15;

            then

             A253: ((f ^ ( mid (g,2,( len g)))) /. i) = (f /. i) by A236, A252, A251, FINSEQ_1: 64;

            ((f ^ ( mid (g,2,( len g)))) /. ((i + 1) + 1)) = ((f ^ ( mid (g,2,( len g)))) . ((i + 1) + 1)) by A237, A240, FINSEQ_4: 15;

            then ( LSeg (((f ^ ( mid (g,2,( len g)))) /. (i + 1)),((f ^ ( mid (g,2,( len g)))) /. ((i + 1) + 1)))) = ( LSeg ((f /. (i + 1)),(f /. ((i + 1) + 1)))) by A240, A245, A248, A249, FINSEQ_1: 64;

            then

             A254: ( LSeg ((f ^ ( mid (g,2,( len g)))),(i + 1))) = ( LSeg (f,(i + 1))) by A238, A245, A250, TOPREAL1:def 3;

            ( LSeg (((f ^ ( mid (g,2,( len g)))) /. i),((f ^ ( mid (g,2,( len g)))) /. (i + 1)))) = ( LSeg ((f ^ ( mid (g,2,( len g)))),i)) by A236, A241, TOPREAL1:def 3;

            then ( LSeg ((f ^ ( mid (g,2,( len g)))),i)) = ( LSeg (f,i)) by A236, A247, A253, A248, TOPREAL1:def 3;

            hence thesis by A2, A236, A245, A248, A254, TOPREAL1:def 6;

          end;

            case (i + 2) > ( len f);

            then

             A255: (i + 2) >= (( len f) + 1) by NAT_1: 13;

            now

              per cases by A255, XXREAL_0: 1;

                case

                 A256: (i + 2) > (( len f) + 1);

                then ((i + 1) + 1) > (( len f) + 1);

                then

                 A257: (i + 1) >= (( len f) + 1) by NAT_1: 13;

                then

                 A258: i >= ( len f) by XREAL_1: 6;

                 A259:

                now

                  assume 1 > (i -' ( len f));

                  then ((i -' ( len f)) + 1) <= ( 0 + 1) by NAT_1: 13;

                  then

                   A260: (i -' ( len f)) = 0 by XREAL_1: 6;

                  then (i - ( len f)) = 0 by A258, XREAL_1: 233;

                  hence ((f ^ ( mid (g,2,( len g)))) . i) = (g . ((i -' ( len f)) + 1)) by A1, A61, A260, FINSEQ_1: 64;

                end;

                

                 A261: (i + 1) >= ( len f) by A257, NAT_D: 48;

                 A262:

                now

                  assume 1 > ((i + 1) -' ( len f));

                  then (((i + 1) -' ( len f)) + 1) <= ( 0 + 1) by NAT_1: 13;

                  then

                   A263: ((i + 1) -' ( len f)) = 0 by XREAL_1: 6;

                  then ((i + 1) - ( len f)) = 0 by A261, XREAL_1: 233;

                  hence ((f ^ ( mid (g,2,( len g)))) . (i + 1)) = (g . (((i + 1) -' ( len f)) + 1)) by A1, A61, A263, FINSEQ_1: 64;

                end;

                ((i + 1) + 1) >= ((( len f) + 1) + 1) by A256, NAT_1: 13;

                then (((i + 1) + 1) - (( len f) + 1)) >= (((( len f) + 1) + 1) - (( len f) + 1)) by XREAL_1: 9;

                then ((i - ( len f)) + 1) >= 1;

                then

                 A264: ((i -' ( len f)) + 1) >= 1 by A258, XREAL_1: 233;

                then

                 A265: (((i -' ( len f)) + 1) + 1) >= 1 by NAT_D: 48;

                then

                 A266: (((i - ( len f)) + 1) + 1) >= 1 by A258, XREAL_1: 233;

                then (((i + 1) - ( len f)) + 1) >= 1;

                then

                 A267: (((i + 1) -' ( len f)) + 1) >= 1 by A261, XREAL_1: 233;

                then

                 A268: ((((i + 1) -' ( len f)) + 1) + 1) >= 1 by NAT_D: 48;

                (((i + 1) - ( len f)) + 1) >= ( 0 + 1) by A266;

                then

                 A269: ((i + 1) - ( len f)) >= 0 by XREAL_1: 6;

                then ((((i + 1) -' ( len f)) + 1) + 1) <= ( len g) by A243, XREAL_0:def 2;

                then

                 A270: (g /. ((((i + 1) -' ( len f)) + 1) + 1)) = (g . ((((i + 1) -' ( len f)) + 1) + 1)) by A268, FINSEQ_4: 15;

                ((((i + 1) -' ( len f)) + 1) + 1) <= ( len g) by A243, A261, XREAL_1: 233;

                then

                 A271: ( LSeg (g,(((i + 1) -' ( len f)) + 1))) = ( LSeg ((g /. (((i + 1) -' ( len f)) + 1)),(g /. ((((i + 1) -' ( len f)) + 1) + 1)))) by A267, TOPREAL1:def 3;

                ((((i + 1) + 1) - ( len f)) + 1) = ((((i + 1) - ( len f)) + 1) + 1);

                then

                 A272: ((((i + 1) + 1) - ( len f)) + 1) = ((((i + 1) -' ( len f)) + 1) + 1) by A269, XREAL_0:def 2;

                

                 A273: ((((i -' ( len f)) + 1) + 1) + 1) <= ( len g) by A244, A258, XREAL_1: 233;

                then

                 A274: (((i -' ( len f)) + 1) + 1) <= ( len g) by NAT_D: 46;

                then

                 A275: ( LSeg (g,((i -' ( len f)) + 1))) = ( LSeg ((g /. ((i -' ( len f)) + 1)),(g /. (((i -' ( len f)) + 1) + 1)))) by A264, TOPREAL1:def 3;

                ((((i -' ( len f)) + 1) + 1) - 1) <= (( len g) - 1) by A274, XREAL_1: 9;

                then ((i -' ( len f)) + 1) <= ((( len g) - 2) + 1);

                then

                 A276: ((i -' ( len f)) + 1) <= ((( len g) -' 2) + 1) by A8, XREAL_1: 233;

                then

                 A277: (i -' ( len f)) <= ((( len g) -' 2) + 1) by NAT_D: 46;

                 A278:

                now

                  assume

                   A279: 1 <= (i -' ( len f));

                  then 1 <= (i - ( len f)) by NAT_D: 39;

                  then (1 + ( len f)) <= ((i - ( len f)) + ( len f)) by XREAL_1: 6;

                  then

                   A280: ( len f) < i by NAT_1: 13;

                  then ((f ^ ( mid (g,2,( len g)))) . i) = (( mid (g,2,( len g))) . (i - ( len f))) by A239, FINSEQ_6: 108;

                  then ((f ^ ( mid (g,2,( len g)))) . i) = (( mid (g,2,( len g))) . (i -' ( len f))) by A280, XREAL_1: 233;

                  then ((f ^ ( mid (g,2,( len g)))) . i) = (g . (((i -' ( len f)) + 2) - 1)) by A8, A277, A279, FINSEQ_6: 122;

                  hence ((f ^ ( mid (g,2,( len g)))) . i) = (g . ((i -' ( len f)) + 1));

                end;

                ((i -' ( len f)) + 1) <= ( len g) by A274, NAT_D: 46;

                then (g /. ((i -' ( len f)) + 1)) = (g . ((i -' ( len f)) + 1)) by A264, FINSEQ_4: 15;

                then

                 A281: ((f ^ ( mid (g,2,( len g)))) /. i) = (g /. ((i -' ( len f)) + 1)) by A236, A239, A278, A259, FINSEQ_4: 15;

                

                 A282: (((i -' ( len f)) + 1) + (1 + 1)) <= ( len g) by A273;

                

                 A283: (g /. (((i -' ( len f)) + 1) + 1)) = (g . (((i -' ( len f)) + 1) + 1)) by A274, A265, FINSEQ_4: 15;

                ((i - ( len f)) + 1) <= ((( len g) -' 2) + 1) by A258, A276, XREAL_1: 233;

                then ((i + 1) - ( len f)) <= ((( len g) -' 2) + 1);

                then

                 A284: ((i + 1) -' ( len f)) <= ((( len g) -' 2) + 1) by A261, XREAL_1: 233;

                 A285:

                now

                  assume

                   A286: 1 <= ((i + 1) -' ( len f));

                  then 1 <= ((i + 1) - ( len f)) by NAT_D: 39;

                  then (1 + ( len f)) <= (((i + 1) - ( len f)) + ( len f)) by XREAL_1: 6;

                  then

                   A287: ( len f) < (i + 1) by NAT_1: 13;

                  then ((f ^ ( mid (g,2,( len g)))) . (i + 1)) = (( mid (g,2,( len g))) . ((i + 1) - ( len f))) by A241, FINSEQ_6: 108;

                  then ((f ^ ( mid (g,2,( len g)))) . (i + 1)) = (( mid (g,2,( len g))) . ((i + 1) -' ( len f))) by A287, XREAL_1: 233;

                  then ((f ^ ( mid (g,2,( len g)))) . (i + 1)) = (g . ((((i + 1) -' ( len f)) + 2) - 1)) by A8, A284, A286, FINSEQ_6: 122;

                  hence ((f ^ ( mid (g,2,( len g)))) . (i + 1)) = (g . (((i + 1) -' ( len f)) + 1));

                end;

                 A288:

                now

                  assume 1 > (((i + 1) + 1) -' ( len f));

                  then

                   A289: ((((i + 1) + 1) -' ( len f)) + 1) <= ( 0 + 1) by NAT_1: 13;

                  then (((i + 1) + 1) -' ( len f)) <= 0 by XREAL_1: 6;

                  then

                   A290: (((i + 1) + 1) - ( len f)) = 0 by A266, XREAL_0:def 2;

                  (((i + 1) + 1) -' ( len f)) = 0 by A289, XREAL_1: 6;

                  hence ((f ^ ( mid (g,2,( len g)))) . ((i + 1) + 1)) = (g . ((((i + 1) + 1) -' ( len f)) + 1)) by A1, A61, A290, FINSEQ_1: 64;

                end;

                (((i + 1) - ( len f)) + 1) = (((i - ( len f)) + 1) + 1);

                then

                 A291: (((i + 1) - ( len f)) + 1) = (((i -' ( len f)) + 1) + 1) by A258, XREAL_1: 233;

                then

                 A292: (((i + 1) -' ( len f)) + 1) = (((i -' ( len f)) + 1) + 1) by A261, XREAL_1: 233;

                

                 A293: (((i + 1) + 1) -' ( len f)) <= ((( len g) - 2) + 1) by A242, A266, XREAL_0:def 2;

                 A294:

                now

                  assume

                   A295: 1 <= (((i + 1) + 1) -' ( len f));

                  then 1 <= (((i + 1) + 1) - ( len f)) by NAT_D: 39;

                  then (1 + ( len f)) <= ((((i + 1) + 1) - ( len f)) + ( len f)) by XREAL_1: 6;

                  then

                   A296: ( len f) < ((i + 1) + 1) by NAT_1: 13;

                  then ((f ^ ( mid (g,2,( len g)))) . ((i + 1) + 1)) = (( mid (g,2,( len g))) . (((i + 1) + 1) - ( len f))) by A237, FINSEQ_6: 108;

                  then ((f ^ ( mid (g,2,( len g)))) . ((i + 1) + 1)) = (( mid (g,2,( len g))) . (((i + 1) + 1) -' ( len f))) by A296, XREAL_1: 233;

                  then ((f ^ ( mid (g,2,( len g)))) . ((i + 1) + 1)) = (g . (((((i + 1) + 1) -' ( len f)) + 2) - 1)) by A8, A293, A295, FINSEQ_6: 122;

                  hence ((f ^ ( mid (g,2,( len g)))) . ((i + 1) + 1)) = (g . ((((i + 1) + 1) -' ( len f)) + 1));

                end;

                

                 A297: ((f ^ ( mid (g,2,( len g)))) /. ((i + 1) + 1)) = ((f ^ ( mid (g,2,( len g)))) . ((i + 1) + 1)) by A237, A240, FINSEQ_4: 15;

                

                 A298: ( LSeg ((f ^ ( mid (g,2,( len g)))),i)) = ( LSeg (((f ^ ( mid (g,2,( len g)))) /. i),((f ^ ( mid (g,2,( len g)))) /. (i + 1)))) by A236, A241, TOPREAL1:def 3;

                

                 A299: ((f ^ ( mid (g,2,( len g)))) /. (i + 1)) = ((f ^ ( mid (g,2,( len g)))) . (i + 1)) by A238, A241, FINSEQ_4: 15;

                ( LSeg ((f ^ ( mid (g,2,( len g)))),(i + 1))) = ( LSeg (((f ^ ( mid (g,2,( len g)))) /. (i + 1)),((f ^ ( mid (g,2,( len g)))) /. ((i + 1) + 1)))) by A237, A238, TOPREAL1:def 3;

                then ( LSeg ((f ^ ( mid (g,2,( len g)))),(i + 1))) = ( LSeg (g,(((i + 1) -' ( len f)) + 1))) by A291, A272, A299, A283, A285, A262, A271, A297, A270, A294, A288, XREAL_0:def 2;

                hence thesis by A3, A264, A292, A298, A275, A281, A299, A283, A285, A282, TOPREAL1:def 6;

              end;

                case

                 A300: (i + 2) = (( len f) + 1);

                then

                 A301: (f /. (i + 1)) = (f . (i + 1)) by A238, FINSEQ_4: 15;

                then

                 A302: (f /. (i + 1)) = (g /. 1) by A1, A9, A300, FINSEQ_4: 15;

                ((f ^ ( mid (g,2,( len g)))) /. (i + 1)) = ((f ^ ( mid (g,2,( len g)))) . (i + 1)) by A238, A241, FINSEQ_4: 15;

                then

                 A303: ((f ^ ( mid (g,2,( len g)))) /. (i + 1)) = (f /. (i + 1)) by A238, A300, A301, FINSEQ_1: 64;

                

                 A304: ( LSeg (f,i)) = ( LSeg ((f /. i),(f /. (i + 1)))) by A236, A300, TOPREAL1:def 3;

                

                 A305: ((f ^ ( mid (g,2,( len g)))) /. i) = ((f ^ ( mid (g,2,( len g)))) . i) by A236, A239, FINSEQ_4: 15;

                ((i + 1) + 1) = (( len f) + 1) by A300;

                then

                 A306: i <= ( len f) by NAT_D: 46;

                then (f /. i) = (f . i) by A236, FINSEQ_4: 15;

                then

                 A307: ((f ^ ( mid (g,2,( len g)))) /. i) = (f /. i) by A236, A306, A305, FINSEQ_1: 64;

                

                 A308: ( LSeg (((f ^ ( mid (g,2,( len g)))) /. i),((f ^ ( mid (g,2,( len g)))) /. (i + 1)))) = ( LSeg ((f ^ ( mid (g,2,( len g)))),i)) by A236, A241, TOPREAL1:def 3;

                i = (( len f) - 1) by A300;

                then

                 A309: i = (( len f) -' 1) by A5, XREAL_1: 233, XXREAL_0: 2;

                

                 A310: (g /. 1) in ( LSeg ((g /. 1),(g /. (1 + 1)))) by RLTOPSP1: 68;

                

                 A311: (g /. 1) = (g . 1) by A9, FINSEQ_4: 15;

                then (g /. 1) = (f /. ( len f)) by A1, A61, FINSEQ_4: 15;

                then

                 A312: (g /. 1) in ( LSeg ((f /. (( len f) -' 1)),(f /. ( len f)))) by RLTOPSP1: 68;

                (( len g) - 2) >= 0 by A8, XREAL_1: 48;

                then

                 A313: ( 0 + 1) <= ((( len g) - 2) + 1) by XREAL_1: 6;

                ( len f) < ((i + 1) + 1) by A300, NAT_1: 13;

                then ((f ^ ( mid (g,2,( len g)))) . ((i + 1) + 1)) = (( mid (g,2,( len g))) . (((i + 1) + 1) - ( len f))) by A237, FINSEQ_6: 108;

                

                then

                 A314: ((f ^ ( mid (g,2,( len g)))) . ((i + 1) + 1)) = (g . ((2 + 1) -' 1)) by A8, A300, A313, FINSEQ_6: 122

                .= (g . 2) by NAT_D: 34;

                

                 A315: ( LSeg (g,1)) c= ( L~ g) by TOPREAL3: 19;

                ( LSeg (f,i)) c= ( L~ f) by TOPREAL3: 19;

                then

                 A316: (( LSeg (f,i)) /\ ( LSeg (g,1))) c= {(g /. 1)} by A4, A311, A315, XBOOLE_1: 27;

                

                 A317: (((i + 1) -' ( len f)) + 1) = ( 0 + 1) by A300, XREAL_1: 232

                .= 1;

                then

                 A318: (g /. ((((i + 1) -' ( len f)) + 1) + 1)) = (g . ((((i + 1) -' ( len f)) + 1) + 1)) by A8, FINSEQ_4: 15;

                ( LSeg (g,1)) = ( LSeg ((g /. 1),(g /. (1 + 1)))) by A8, TOPREAL1:def 3;

                then (g /. 1) in (( LSeg (f,i)) /\ ( LSeg (g,1))) by A300, A309, A304, A312, A310, XBOOLE_0:def 4;

                then

                 A319: {(g /. 1)} c= (( LSeg (f,i)) /\ ( LSeg (g,1))) by ZFMISC_1: 31;

                

                 A320: ((f ^ ( mid (g,2,( len g)))) /. ((i + 1) + 1)) = ((f ^ ( mid (g,2,( len g)))) . ((i + 1) + 1)) by A237, A240, FINSEQ_4: 15;

                

                 A321: ( LSeg ((f ^ ( mid (g,2,( len g)))),(i + 1))) = ( LSeg (((f ^ ( mid (g,2,( len g)))) /. (i + 1)),((f ^ ( mid (g,2,( len g)))) /. ((i + 1) + 1)))) by A237, A238, TOPREAL1:def 3;

                ( LSeg (g,(((i + 1) -' ( len f)) + 1))) = ( LSeg ((g /. (((i + 1) -' ( len f)) + 1)),(g /. ((((i + 1) -' ( len f)) + 1) + 1)))) by A8, A317, TOPREAL1:def 3;

                hence thesis by A307, A302, A303, A308, A304, A321, A317, A320, A318, A314, A319, A316, XBOOLE_0:def 10;

              end;

            end;

            hence thesis;

          end;

        end;

        hence thesis;

      end;

      for i be Nat st 1 <= i & (i + 1) <= ( len (f ^ ( mid (g,2,( len g))))) holds (((f ^ ( mid (g,2,( len g)))) /. i) `1 ) = (((f ^ ( mid (g,2,( len g)))) /. (i + 1)) `1 ) or (((f ^ ( mid (g,2,( len g)))) /. i) `2 ) = (((f ^ ( mid (g,2,( len g)))) /. (i + 1)) `2 )

      proof

        let i be Nat;

        assume that

         A322: 1 <= i and

         A323: (i + 1) <= ( len (f ^ ( mid (g,2,( len g)))));

        now

          per cases ;

            case

             A324: i < ( len f);

            i <= ( len (f ^ ( mid (g,2,( len g))))) by A323, NAT_D: 46;

            then

             A325: ((f ^ ( mid (g,2,( len g)))) /. i) = ((f ^ ( mid (g,2,( len g)))) . i) by A322, FINSEQ_4: 15;

            (f /. i) = (f . i) by A322, A324, FINSEQ_4: 15;

            then

             A326: ((f ^ ( mid (g,2,( len g)))) /. i) = (f /. i) by A322, A324, A325, FINSEQ_1: 64;

            

             A327: 1 <= (i + 1) by A322, NAT_D: 48;

            then

             A328: ((f ^ ( mid (g,2,( len g)))) /. (i + 1)) = ((f ^ ( mid (g,2,( len g)))) . (i + 1)) by A323, FINSEQ_4: 15;

            

             A329: (i + 1) <= ( len f) by A324, NAT_1: 13;

            then

             A330: ((f ^ ( mid (g,2,( len g)))) . (i + 1)) = (f . (i + 1)) by A327, FINSEQ_1: 64;

            (f /. (i + 1)) = (f . (i + 1)) by A327, A329, FINSEQ_4: 15;

            hence thesis by A2, A322, A329, A326, A328, A330, TOPREAL1:def 5;

          end;

            case

             A331: i >= ( len f);

            1 <= (1 + (i -' ( len f))) by NAT_1: 11;

            then 1 <= (1 + (i - ( len f))) by A331, XREAL_1: 233;

            then 1 <= ((1 + i) - ( len f));

            then

             A332: 1 <= ((i + 1) -' ( len f)) by NAT_D: 39;

            

             A333: i <= ( len (f ^ ( mid (g,2,( len g))))) by A323, NAT_D: 46;

            

             A334: (i - ( len f)) >= 0 by A331, XREAL_1: 48;

            then

             A335: (i -' ( len f)) = (i - ( len f)) by XREAL_0:def 2;

             A336:

            now

              assume 1 > (i -' ( len f));

              then ((i -' ( len f)) + 1) <= ( 0 + 1) by NAT_1: 13;

              then (i -' ( len f)) = 0 by XREAL_1: 6;

              hence ((f ^ ( mid (g,2,( len g)))) . i) = (g . ((i -' ( len f)) + 1)) by A1, A61, A335, FINSEQ_1: 64;

            end;

            

             A337: (i + 1) >= ( len f) by A331, NAT_D: 48;

            

            then

             A338: (((i + 1) -' ( len f)) + 1) = (((i + 1) - ( len f)) + 1) by XREAL_1: 233

            .= (((i - ( len f)) + 1) + 1)

            .= (((i -' ( len f)) + 1) + 1) by A331, XREAL_1: 233;

            

             A339: ((i + 1) - ( len f)) <= ((( len f) + (( len g) - 1)) - ( len f)) by A6, A10, A323, XREAL_1: 9;

            then

             A340: (((i - ( len f)) + 1) + 1) <= ((( len g) - 1) + 1) by XREAL_1: 6;

            then

             A341: (((i -' ( len f)) + 1) + 1) <= ( len g) by A334, XREAL_0:def 2;

            (i -' ( len f)) <= ((i -' ( len f)) + 1) by NAT_1: 11;

            then

             A342: (i -' ( len f)) <= ((( len g) - 2) + 1) by A335, A339, XXREAL_0: 2;

             A343:

            now

              assume

               A344: 1 <= (i -' ( len f));

              then 1 <= (i - ( len f)) by NAT_D: 39;

              then (1 + ( len f)) <= ((i - ( len f)) + ( len f)) by XREAL_1: 6;

              then

               A345: ( len f) < i by NAT_1: 13;

              then ((f ^ ( mid (g,2,( len g)))) . i) = (( mid (g,2,( len g))) . (i - ( len f))) by A333, FINSEQ_6: 108;

              then ((f ^ ( mid (g,2,( len g)))) . i) = (( mid (g,2,( len g))) . (i -' ( len f))) by A345, XREAL_1: 233;

              then ((f ^ ( mid (g,2,( len g)))) . i) = (g . (((i -' ( len f)) + 2) - 1)) by A8, A342, A344, FINSEQ_6: 122;

              hence ((f ^ ( mid (g,2,( len g)))) . i) = (g . ((i -' ( len f)) + 1));

            end;

            1 <= (i + 1) by A322, NAT_D: 48;

            then

             A346: ((f ^ ( mid (g,2,( len g)))) /. (i + 1)) = ((f ^ ( mid (g,2,( len g)))) . (i + 1)) by A323, FINSEQ_4: 15;

            

             A347: 1 <= ((i -' ( len f)) + 1) by NAT_1: 11;

            ((i + 1) - ( len f)) <= ((( len g) - 2) + 1) by A339;

            then ((i + 1) - ( len f)) <= ((( len g) -' 2) + 1) by A8, XREAL_1: 233;

            then

             A348: ((i + 1) -' ( len f)) <= ((( len g) -' 2) + 1) by A337, XREAL_1: 233;

            ( len f) < (i + 1) by A331, NAT_1: 13;

            then ((f ^ ( mid (g,2,( len g)))) . (i + 1)) = (( mid (g,2,( len g))) . ((i + 1) - ( len f))) by A323, FINSEQ_6: 108;

            then ((f ^ ( mid (g,2,( len g)))) . (i + 1)) = (( mid (g,2,( len g))) . ((i + 1) -' ( len f))) by A337, XREAL_1: 233;

            then

             A349: ((f ^ ( mid (g,2,( len g)))) . (i + 1)) = (g . ((((i + 1) -' ( len f)) + 2) - 1)) by A8, A348, A332, FINSEQ_6: 122;

            (((i -' ( len f)) + 1) + 1) <= ( len g) by A334, A340, XREAL_0:def 2;

            then

             A350: (g /. (((i -' ( len f)) + 1) + 1)) = (g . (((i -' ( len f)) + 1) + 1)) by FINSEQ_4: 15, NAT_1: 11;

            ((i -' ( len f)) + 1) <= ( len g) by A335, A340, NAT_D: 46;

            then (g /. ((i -' ( len f)) + 1)) = (g . ((i -' ( len f)) + 1)) by FINSEQ_4: 15, NAT_1: 11;

            then ((f ^ ( mid (g,2,( len g)))) /. i) = (g /. ((i -' ( len f)) + 1)) by A322, A333, A343, A336, FINSEQ_4: 15;

            hence thesis by A3, A347, A341, A338, A346, A350, A349, TOPREAL1:def 5;

          end;

        end;

        hence thesis;

      end;

      then (f ^ ( mid (g,2,( len g)))) is unfolded s.n.c. special by A235, A62, TOPREAL1:def 5, TOPREAL1:def 6, TOPREAL1:def 7;

      hence thesis by A60, A7, TOPREAL1:def 8;

    end;

    theorem :: JORDAN3:39

    

     Th39: for f,g be FinSequence of ( TOP-REAL 2) st (f . ( len f)) = (g . 1) & f is being_S-Seq & g is being_S-Seq & (( L~ f) /\ ( L~ g)) = {(g . 1)} holds (f ^ ( mid (g,2,( len g)))) is_S-Seq_joining ((f /. 1),(g /. ( len g)))

    proof

      let f,g be FinSequence of ( TOP-REAL 2);

      assume that

       A1: (f . ( len f)) = (g . 1) and

       A2: f is being_S-Seq and

       A3: g is being_S-Seq and

       A4: (( L~ f) /\ ( L~ g)) = {(g . 1)};

      

       A5: (f ^ ( mid (g,2,( len g)))) is being_S-Seq by A1, A2, A3, A4, Th38;

      

       A6: ( len g) >= 2 by A3, TOPREAL1:def 8;

      then

       A7: ((1 + 1) - 1) <= (( len g) - 1) by XREAL_1: 9;

      ( len f) >= 2 by A2, TOPREAL1:def 8;

      then

       A8: 1 <= ( len f) by XXREAL_0: 2;

      

      then

       A9: ((f ^ ( mid (g,2,( len g)))) . 1) = (f . 1) by FINSEQ_1: 64

      .= (f /. 1) by A8, FINSEQ_4: 15;

      

       A10: ( len (f ^ ( mid (g,2,( len g))))) = (( len f) + ( len ( mid (g,2,( len g))))) by FINSEQ_1: 22;

      

       A11: 1 <= ( len g) by A6, XXREAL_0: 2;

      then

       A12: ( len ( mid (g,2,( len g)))) = ((( len g) -' 2) + 1) by A6, FINSEQ_6: 118;

      

      then

       A13: ( len ( mid (g,2,( len g)))) = ((( len g) - 2) + 1) by A6, XREAL_1: 233

      .= (( len g) - 1);

      then

       A14: ((( len ( mid (g,2,( len g)))) + 2) - 1) = ( len g);

      (( len g) - 1) >= ((1 + 1) - 1) by A6, XREAL_1: 9;

      then (( len f) + 1) <= ( len (f ^ ( mid (g,2,( len g))))) by A10, A13, XREAL_1: 6;

      then ( len f) < ( len (f ^ ( mid (g,2,( len g))))) by NAT_1: 13;

      

      then ((f ^ ( mid (g,2,( len g)))) . ( len (f ^ ( mid (g,2,( len g)))))) = (( mid (g,2,( len g))) . (( len (f ^ ( mid (g,2,( len g))))) - ( len f))) by FINSEQ_6: 108

      .= (g . ( len g)) by A6, A10, A12, A7, A14, FINSEQ_6: 122

      .= (g /. ( len g)) by A11, FINSEQ_4: 15;

      hence thesis by A5, A9;

    end;

    theorem :: JORDAN3:40

    for f be FinSequence of ( TOP-REAL 2), n be Element of NAT holds ( L~ (f /^ n)) c= ( L~ f)

    proof

      let f be FinSequence of ( TOP-REAL 2), n be Element of NAT ;

      let x be object;

      assume x in ( L~ (f /^ n));

      then x in ( union { ( LSeg ((f /^ n),i)) : 1 <= i & (i + 1) <= ( len (f /^ n)) }) by TOPREAL1:def 4;

      then

      consider Y be set such that

       A1: x in Y & Y in { ( LSeg ((f /^ n),i)) : 1 <= i & (i + 1) <= ( len (f /^ n)) } by TARSKI:def 4;

      consider i such that

       A2: Y = ( LSeg ((f /^ n),i)) and

       A3: 1 <= i and

       A4: (i + 1) <= ( len (f /^ n)) by A1;

      now

        per cases ;

          case n <= ( len f);

          then ( LSeg ((f /^ n),i)) = ( LSeg (f,(n + i))) by A3, SPPOL_2: 4;

          then Y c= ( L~ f) by A2, TOPREAL3: 19;

          hence thesis by A1;

        end;

          case n > ( len f);

          then (f /^ n) = ( <*> the carrier of ( TOP-REAL 2)) by RFINSEQ:def 1;

          hence contradiction by A4;

        end;

      end;

      hence thesis;

    end;

    theorem :: JORDAN3:41

    

     Th41: for f be FinSequence of ( TOP-REAL 2), p be Point of ( TOP-REAL 2) st p in ( L~ f) holds ( L~ ( R_Cut (f,p))) c= ( L~ f)

    proof

      let f be FinSequence of ( TOP-REAL 2), p be Point of ( TOP-REAL 2) such that

       A1: p in ( L~ f);

      

       A2: 1 <= ( Index (p,f)) by A1, Th8;

      ( len f) <> 0 by A1, TOPREAL1: 22;

      then

       A3: ( len f) >= ( 0 + 1) by NAT_1: 13;

      

       A4: ( Index (p,f)) <= ( len f) by A1, Th8;

      per cases ;

        suppose p = (f . 1);

        then ( R_Cut (f,p)) = <*p*> by Def4;

        then ( len ( R_Cut (f,p))) = 1 by FINSEQ_1: 39;

        then ( L~ ( R_Cut (f,p))) = {} by TOPREAL1: 22;

        hence thesis;

      end;

        suppose

         A5: p <> (f . 1);

        

         A6: (f /. ( Index (p,f))) in ( LSeg ((f /. ( Index (p,f))),(f /. (( Index (p,f)) + 1)))) by RLTOPSP1: 68;

        

         A7: ( len ( mid (f,1,( Index (p,f))))) = ((( Index (p,f)) -' 1) + 1) by A3, A2, A4, FINSEQ_6: 118

        .= ( Index (p,f)) by A1, Th8, XREAL_1: 235;

        then ( mid (f,1,( Index (p,f)))) <> ( <*> the carrier of ( TOP-REAL 2)) by A2;

        then

         A8: ( L~ (( mid (f,1,( Index (p,f)))) ^ <*p*>)) = (( L~ ( mid (f,1,( Index (p,f))))) \/ ( LSeg ((( mid (f,1,( Index (p,f)))) /. ( Index (p,f))),p))) by A7, SPPOL_2: 19;

        ( mid (f,1,( Index (p,f)))) = ((f /^ (1 -' 1)) | ((( Index (p,f)) -' 1) + 1)) by A2, FINSEQ_6:def 3

        .= ((f /^ 0 ) | ((( Index (p,f)) -' 1) + 1)) by XREAL_1: 232

        .= (f | ((( Index (p,f)) -' 1) + 1))

        .= (f | ( Index (p,f))) by A1, Th8, XREAL_1: 235;

        then

         A9: ( L~ ( mid (f,1,( Index (p,f))))) c= ( L~ f) by TOPREAL3: 20;

        ( Index (p,f)) < ( len f) by A1, Th8;

        then

         A10: (( Index (p,f)) + 1) <= ( len f) by NAT_1: 13;

        then

         A11: ( LSeg ((f /. ( Index (p,f))),(f /. (( Index (p,f)) + 1)))) c= ( L~ f) by A1, Th8, SPPOL_2: 16;

        p in ( LSeg (f,( Index (p,f)))) by A1, Th9;

        then

         A12: p in ( LSeg ((f /. ( Index (p,f))),(f /. (( Index (p,f)) + 1)))) by A2, A10, TOPREAL1:def 3;

        (( mid (f,1,( Index (p,f)))) /. ( Index (p,f))) = (( mid (f,1,( Index (p,f)))) . ( Index (p,f))) by A2, A7, FINSEQ_4: 15

        .= (f . ( Index (p,f))) by A2, A4, FINSEQ_6: 123

        .= (f /. ( Index (p,f))) by A1, A4, Th8, FINSEQ_4: 15;

        then ( LSeg ((( mid (f,1,( Index (p,f)))) /. ( Index (p,f))),p)) c= ( LSeg ((f /. ( Index (p,f))),(f /. (( Index (p,f)) + 1)))) by A12, A6, TOPREAL1: 6;

        then

         A13: ( LSeg ((( mid (f,1,( Index (p,f)))) /. ( Index (p,f))),p)) c= ( L~ f) by A11;

        ( R_Cut (f,p)) = (( mid (f,1,( Index (p,f)))) ^ <*p*>) by A5, Def4;

        hence thesis by A8, A13, A9, XBOOLE_1: 8;

      end;

    end;

    

     Lm2: (( <*> D) | i) = ( <*> D)

    proof

      ( len ( <*> D)) = 0 ;

      hence thesis by FINSEQ_1: 58;

    end;

    

     Lm3: for f1 be FinSequence of D, k be Nat st 1 <= k & k <= ( len f1) holds ( mid (f1,k,k)) = <*(f1 /. k)*> & ( len ( mid (f1,k,k))) = 1

    proof

      let f1 be FinSequence of D, k be Nat;

      assume that

       A1: 1 <= k and

       A2: k <= ( len f1);

      

       A3: (f1 /. k) = (f1 . k) by A1, A2, FINSEQ_4: 15;

      ((k -' 1) + 1) <= ( len f1) by A1, A2, XREAL_1: 235;

      then

       A4: (((k -' 1) + 1) - (k -' 1)) <= (( len f1) - (k -' 1)) by XREAL_1: 9;

      ( len (f1 /^ (k -' 1))) = (( len f1) -' (k -' 1)) by RFINSEQ: 29;

      then

       A5: 1 <= ( len (f1 /^ (k -' 1))) by A4, NAT_D: 39;

      ((k -' 1) + 1) = k by A1, XREAL_1: 235;

      then

       A6: ((f1 /^ (k -' 1)) . 1) = (f1 . k) by A2, FINSEQ_6: 114;

      ((k -' k) + 1) = ((k - k) + 1) by XREAL_1: 233

      .= 1;

      

      then ( mid (f1,k,k)) = ((f1 /^ (k -' 1)) | 1) by FINSEQ_6:def 3

      .= <*((f1 /^ (k -' 1)) . 1)*> by A5, CARD_1: 27, FINSEQ_5: 20;

      hence thesis by A6, A3, FINSEQ_1: 39;

    end;

    

     Lm4: for f1 be FinSequence of D holds ( mid (f1, 0 , 0 )) = (f1 | 1)

    proof

      let f1 be FinSequence of D;

      ( 0 - 1) < 0 ;

      then

       A1: ( 0 -' 1) = 0 by XREAL_0:def 2;

      (( 0 -' 0 ) + 1) = (( 0 - 0 ) + 1) by XREAL_1: 233

      .= 1;

      then ( mid (f1, 0 , 0 )) = ((f1 /^ ( 0 -' 1)) | 1) by FINSEQ_6:def 3;

      hence thesis by A1;

    end;

    

     Lm5: for f1 be FinSequence of D, k be Nat st ( len f1) < k holds ( mid (f1,k,k)) = ( <*> D)

    proof

      let f1 be FinSequence of D, k be Nat;

      assume

       A1: ( len f1) < k;

      then (( len f1) + 1) <= k by NAT_1: 13;

      then

       A2: ((( len f1) + 1) - 1) <= (k - 1) by XREAL_1: 9;

      ( 0 + 1) <= k by A1, NAT_1: 13;

      then ( len f1) <= (k -' 1) by A2, XREAL_1: 233;

      then

       A3: (f1 /^ (k -' 1)) = ( <*> D) by FINSEQ_5: 32;

      ((k -' k) + 1) = ((k - k) + 1) by XREAL_1: 233

      .= 1;

      then ( mid (f1,k,k)) = ((f1 /^ (k -' 1)) | 1) by FINSEQ_6:def 3;

      hence thesis by A3, Lm2;

    end;

    

     Lm6: for f1 be FinSequence of D, i1, i2 holds ( mid (f1,i1,i2)) = ( Rev ( mid (f1,i2,i1)))

    proof

      let f1 be FinSequence of D;

      let k1,k2 be Nat;

      now

        per cases ;

          case

           A1: k1 <= k2;

          then

           A2: ( mid (f1,k1,k2)) = ((f1 /^ (k1 -' 1)) | ((k2 -' k1) + 1)) by FINSEQ_6:def 3;

          now

            per cases by A1, XXREAL_0: 1;

              case k1 < k2;

              then ( mid (f1,k2,k1)) = ( Rev ( mid (f1,k1,k2))) by A2, FINSEQ_6:def 3;

              hence thesis;

            end;

              case

               A3: k1 = k2;

              

               A4: k1 = 0 or ( 0 + 1) <= k1 by NAT_1: 13;

              now

                per cases by A4;

                  case k1 = 0 ;

                  then

                   A5: ( mid (f1,k1,k2)) = (f1 | 1) by A3, Lm4;

                  now

                    per cases ;

                      case ( len f1) = 0 ;

                      then f1 = ( <*> D);

                      then (f1 | 1) = ( <*> D) by Lm2;

                      hence thesis by A3, A5;

                    end;

                      case ( len f1) <> 0 ;

                      then f1 <> ( <*> D);

                      then (f1 | 1) = <*(f1 . 1)*> by FINSEQ_5: 20;

                      hence thesis by A3, A5, FINSEQ_5: 60;

                    end;

                  end;

                  hence thesis;

                end;

                  case 1 <= k1 & k1 <= ( len f1);

                  then ( mid (f1,k1,k1)) = <*(f1 /. k1)*> by Lm3;

                  hence thesis by A3, FINSEQ_5: 60;

                end;

                  case ( len f1) < k1;

                  then ( mid (f1,k1,k1)) = ( <*> D) by Lm5;

                  hence thesis by A3;

                end;

              end;

              hence thesis;

            end;

          end;

          hence thesis;

        end;

          case

           A6: k1 > k2;

          then ( mid (f1,k1,k2)) = ( Rev ((f1 /^ (k2 -' 1)) | ((k1 -' k2) + 1))) by FINSEQ_6:def 3;

          hence thesis by A6, FINSEQ_6:def 3;

        end;

      end;

      hence thesis;

    end;

    

     Lm7: for h be FinSequence of ( TOP-REAL 2), i1, i2 st 1 <= i1 & i1 <= ( len h) & 1 <= i2 & i2 <= ( len h) holds ( L~ ( mid (h,i1,i2))) c= ( L~ h)

    proof

      let h be FinSequence of ( TOP-REAL 2), i1, i2;

      assume that

       A1: 1 <= i1 and

       A2: i1 <= ( len h) and

       A3: 1 <= i2 and

       A4: i2 <= ( len h);

      thus ( L~ ( mid (h,i1,i2))) c= ( L~ h)

      proof

        let x be object;

        assume

         A5: x in ( L~ ( mid (h,i1,i2)));

        now

          per cases ;

            case

             A6: i1 <= i2;

            x in ( union { ( LSeg (( mid (h,i1,i2)),i)) : 1 <= i & (i + 1) <= ( len ( mid (h,i1,i2))) }) by A5, TOPREAL1:def 4;

            then

            consider Y be set such that

             A7: x in Y & Y in { ( LSeg (( mid (h,i1,i2)),i)) : 1 <= i & (i + 1) <= ( len ( mid (h,i1,i2))) } by TARSKI:def 4;

            consider i such that

             A8: Y = ( LSeg (( mid (h,i1,i2)),i)) and

             A9: 1 <= i and

             A10: (i + 1) <= ( len ( mid (h,i1,i2))) by A7;

            

             A11: ( LSeg (( mid (h,i1,i2)),i)) = ( LSeg ((( mid (h,i1,i2)) /. i),(( mid (h,i1,i2)) /. (i + 1)))) by A9, A10, TOPREAL1:def 3;

            ( len ( mid (h,i1,i2))) = ((i2 -' i1) + 1) by A1, A2, A3, A4, A6, FINSEQ_6: 118;

            then ((i + 1) - 1) <= (((i2 -' i1) + 1) - 1) by A10, XREAL_1: 9;

            then i <= (i2 - i1) by A6, XREAL_1: 233;

            then

             A12: (i + i1) <= ((i2 - i1) + i1) by XREAL_1: 6;

            then

             A13: (i + i1) <= ( len h) by A4, XXREAL_0: 2;

            (1 + 1) <= (i + i1) by A1, A9, XREAL_1: 7;

            then

             A14: ((1 + 1) - 1) <= ((i + i1) - 1) by XREAL_1: 9;

            then 1 <= ((i + i1) -' 1) by A1, NAT_1: 12, XREAL_1: 233;

            then

             A15: (h /. ((i + i1) -' 1)) = (h . ((i + i1) -' 1)) by A13, FINSEQ_4: 15, NAT_D: 44;

            1 <= (i + 1) by NAT_1: 11;

            then

             A16: (( mid (h,i1,i2)) . (i + 1)) = (h . (((i + 1) + i1) -' 1)) by A1, A2, A3, A4, A6, A10, FINSEQ_6: 118;

            

             A17: (((i + 1) + i1) -' 1) = (((i + 1) + i1) - 1) by A1, NAT_1: 12, XREAL_1: 233

            .= (i + i1);

            

            then

             A18: (((i + 1) + i1) -' 1) = (((i + i1) - 1) + 1)

            .= (((i + i1) -' 1) + 1) by A1, NAT_1: 12, XREAL_1: 233;

            i <= (i + 1) by NAT_1: 11;

            then

             A19: i <= ( len ( mid (h,i1,i2))) by A10, XXREAL_0: 2;

            then

             A20: (( mid (h,i1,i2)) /. i) = (( mid (h,i1,i2)) . i) by A9, FINSEQ_4: 15;

            

             A21: (( mid (h,i1,i2)) /. (i + 1)) = (( mid (h,i1,i2)) . (i + 1)) by A10, FINSEQ_4: 15, NAT_1: 11;

            

             A22: (i + i1) <= ( len h) by A4, A12, XXREAL_0: 2;

            (( mid (h,i1,i2)) . i) = (h . ((i + i1) -' 1)) by A1, A2, A3, A4, A6, A9, A19, FINSEQ_6: 118;

            

            then ( LSeg (( mid (h,i1,i2)),i)) = ( LSeg ((h /. ((i + i1) -' 1)),(h /. (((i + 1) + i1) -' 1)))) by A1, A11, A16, A20, A21, A15, A17, A22, FINSEQ_4: 15, NAT_1: 12

            .= ( LSeg (h,((i + i1) -' 1))) by A14, A13, A17, A18, TOPREAL1:def 3;

            then ( LSeg (( mid (h,i1,i2)),i)) in { ( LSeg (h,j)) : 1 <= j & (j + 1) <= ( len h) } by A14, A17, A18, A22;

            then x in ( union { ( LSeg (h,j)) : 1 <= j & (j + 1) <= ( len h) }) by A7, A8, TARSKI:def 4;

            hence thesis by TOPREAL1:def 4;

          end;

            case

             A23: i1 > i2;

            ( mid (h,i1,i2)) = ( Rev ( mid (h,i2,i1))) by Lm6;

            then x in ( L~ ( mid (h,i2,i1))) by A5, SPPOL_2: 22;

            then x in ( union { ( LSeg (( mid (h,i2,i1)),i)) : 1 <= i & (i + 1) <= ( len ( mid (h,i2,i1))) }) by TOPREAL1:def 4;

            then

            consider Y be set such that

             A24: x in Y & Y in { ( LSeg (( mid (h,i2,i1)),i)) : 1 <= i & (i + 1) <= ( len ( mid (h,i2,i1))) } by TARSKI:def 4;

            consider i such that

             A25: Y = ( LSeg (( mid (h,i2,i1)),i)) and

             A26: 1 <= i and

             A27: (i + 1) <= ( len ( mid (h,i2,i1))) by A24;

            

             A28: ( LSeg (( mid (h,i2,i1)),i)) = ( LSeg ((( mid (h,i2,i1)) /. i),(( mid (h,i2,i1)) /. (i + 1)))) by A26, A27, TOPREAL1:def 3;

            ( len ( mid (h,i2,i1))) = ((i1 -' i2) + 1) by A1, A2, A3, A4, A23, FINSEQ_6: 118;

            then ((i + 1) - 1) <= (((i1 -' i2) + 1) - 1) by A27, XREAL_1: 9;

            then i <= (i1 - i2) by A23, XREAL_1: 233;

            then

             A29: (i + i2) <= ((i1 - i2) + i2) by XREAL_1: 6;

            then

             A30: (i + i2) <= ( len h) by A2, XXREAL_0: 2;

            (1 + 1) <= (i + i2) by A3, A26, XREAL_1: 7;

            then

             A31: ((1 + 1) - 1) <= ((i + i2) - 1) by XREAL_1: 9;

            then 1 <= ((i + i2) -' 1) by A3, NAT_1: 12, XREAL_1: 233;

            then

             A32: (h /. ((i + i2) -' 1)) = (h . ((i + i2) -' 1)) by A30, FINSEQ_4: 15, NAT_D: 44;

            1 <= (i + 1) by NAT_1: 11;

            then

             A33: (( mid (h,i2,i1)) . (i + 1)) = (h . (((i + 1) + i2) -' 1)) by A1, A2, A3, A4, A23, A27, FINSEQ_6: 118;

            

             A34: (((i + 1) + i2) -' 1) = (((i + 1) + i2) - 1) by A3, NAT_1: 12, XREAL_1: 233

            .= (i + i2);

            

            then

             A35: (((i + 1) + i2) -' 1) = (((i + i2) - 1) + 1)

            .= (((i + i2) -' 1) + 1) by A3, NAT_1: 12, XREAL_1: 233;

            i <= (i + 1) by NAT_1: 11;

            then

             A36: i <= ( len ( mid (h,i2,i1))) by A27, XXREAL_0: 2;

            then

             A37: (( mid (h,i2,i1)) /. i) = (( mid (h,i2,i1)) . i) by A26, FINSEQ_4: 15;

            

             A38: (( mid (h,i2,i1)) /. (i + 1)) = (( mid (h,i2,i1)) . (i + 1)) by A27, FINSEQ_4: 15, NAT_1: 11;

            

             A39: (i + i2) <= ( len h) by A2, A29, XXREAL_0: 2;

            (( mid (h,i2,i1)) . i) = (h . ((i + i2) -' 1)) by A1, A2, A3, A4, A23, A26, A36, FINSEQ_6: 118;

            

            then ( LSeg (( mid (h,i2,i1)),i)) = ( LSeg ((h /. ((i + i2) -' 1)),(h /. (((i + 1) + i2) -' 1)))) by A3, A28, A33, A37, A38, A32, A34, A39, FINSEQ_4: 15, NAT_1: 12

            .= ( LSeg (h,((i + i2) -' 1))) by A31, A30, A34, A35, TOPREAL1:def 3;

            then ( LSeg (( mid (h,i2,i1)),i)) in { ( LSeg (h,j)) : 1 <= j & (j + 1) <= ( len h) } by A31, A34, A35, A39;

            then x in ( union { ( LSeg (h,j)) : 1 <= j & (j + 1) <= ( len h) }) by A24, A25, TARSKI:def 4;

            hence thesis by TOPREAL1:def 4;

          end;

        end;

        hence thesis;

      end;

    end;

    

     Lm8: i in ( dom f) & j in ( dom f) implies ( len ( mid (f,i,j))) >= 1

    proof

      

       A1: i <= j or j < i;

      assume

       A2: i in ( dom f);

      then

       A3: i <= ( len f) by FINSEQ_3: 25;

      assume

       A4: j in ( dom f);

      then

       A5: 1 <= j by FINSEQ_3: 25;

      

       A6: j <= ( len f) by A4, FINSEQ_3: 25;

      1 <= i by A2, FINSEQ_3: 25;

      then ( len ( mid (f,i,j))) = ((i -' j) + 1) or ( len ( mid (f,i,j))) = ((j -' i) + 1) by A3, A5, A6, A1, FINSEQ_6: 118;

      hence thesis by NAT_1: 11;

    end;

    

     Lm9: i in ( dom f) & j in ( dom f) implies ( mid (f,i,j)) is non empty

    proof

      assume that

       A1: i in ( dom f) and

       A2: j in ( dom f);

      ( len ( mid (f,i,j))) >= 1 by A1, A2, Lm8;

      hence thesis;

    end;

    

     Lm10: i in ( dom f) & j in ( dom f) implies (( mid (f,i,j)) /. 1) = (f /. i)

    proof

      assume

       A1: i in ( dom f);

      then

       A2: 1 <= i by FINSEQ_3: 25;

      

       A3: i <= ( len f) by A1, FINSEQ_3: 25;

      assume

       A4: j in ( dom f);

      then

       A5: 1 <= j by FINSEQ_3: 25;

      

       A6: j <= ( len f) by A4, FINSEQ_3: 25;

      ( mid (f,i,j)) is non empty by A1, A4, Lm9;

      then 1 in ( dom ( mid (f,i,j))) by FINSEQ_5: 6;

      

      hence (( mid (f,i,j)) /. 1) = (( mid (f,i,j)) . 1) by PARTFUN1:def 6

      .= (f . i) by A2, A3, A5, A6, FINSEQ_6: 118

      .= (f /. i) by A1, PARTFUN1:def 6;

    end;

    theorem :: JORDAN3:42

    

     Th42: for f be FinSequence of ( TOP-REAL 2) holds for p be Point of ( TOP-REAL 2) st p in ( L~ f) holds ( L~ ( L_Cut (f,p))) c= ( L~ f)

    proof

      let f be FinSequence of ( TOP-REAL 2);

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

       A1: p in ( L~ f);

      ( Index (p,f)) < ( len f) by A1, Th8;

      then

       A2: (( Index (p,f)) + 1) <= ( len f) by NAT_1: 13;

      

       A3: 1 <= ( Index (p,f)) by A1, Th8;

      then

       A4: 1 < (( Index (p,f)) + 1) by NAT_1: 13;

      then

       A5: (( Index (p,f)) + 1) in ( dom f) by A2, FINSEQ_3: 25;

      ( len f) <> 0 by A1, TOPREAL1: 22;

      then

       A6: ( len f) >= ( 0 + 1) by NAT_1: 13;

      then

       A7: ( len f) in ( dom f) by FINSEQ_3: 25;

      per cases ;

        suppose p = (f . (( Index (p,f)) + 1));

        then ( L_Cut (f,p)) = ( mid (f,(( Index (p,f)) + 1),( len f))) by Def3;

        hence thesis by A6, A4, A2, Lm7;

      end;

        suppose p <> (f . (( Index (p,f)) + 1));

        then

         A8: ( L_Cut (f,p)) = ( <*p*> ^ ( mid (f,(( Index (p,f)) + 1),( len f)))) by Def3;

        

         A9: (f /. (( Index (p,f)) + 1)) in ( LSeg ((f /. ( Index (p,f))),(f /. (( Index (p,f)) + 1)))) by RLTOPSP1: 68;

        p in ( LSeg (f,( Index (p,f)))) by A1, Th9;

        then

         A10: p in ( LSeg ((f /. ( Index (p,f))),(f /. (( Index (p,f)) + 1)))) by A3, A2, TOPREAL1:def 3;

        

         A11: ( LSeg ((f /. ( Index (p,f))),(f /. (( Index (p,f)) + 1)))) c= ( L~ f) by A1, A2, Th8, SPPOL_2: 16;

        (( mid (f,(( Index (p,f)) + 1),( len f))) /. 1) = (f /. (( Index (p,f)) + 1)) by A7, A5, Lm10;

        then ( LSeg (p,(( mid (f,(( Index (p,f)) + 1),( len f))) /. 1))) c= ( LSeg ((f /. ( Index (p,f))),(f /. (( Index (p,f)) + 1)))) by A10, A9, TOPREAL1: 6;

        then

         A12: ( LSeg (p,(( mid (f,(( Index (p,f)) + 1),( len f))) /. 1))) c= ( L~ f) by A11;

        ( mid (f,(( Index (p,f)) + 1),( len f))) <> {} by A7, A5, Lm8, CARD_1: 27;

        then

         A13: ( L~ ( <*p*> ^ ( mid (f,(( Index (p,f)) + 1),( len f))))) = (( LSeg (p,(( mid (f,(( Index (p,f)) + 1),( len f))) /. 1))) \/ ( L~ ( mid (f,(( Index (p,f)) + 1),( len f))))) by SPPOL_2: 20;

        ( L~ ( mid (f,(( Index (p,f)) + 1),( len f)))) c= ( L~ f) by A6, A4, A2, Lm7;

        hence thesis by A8, A13, A12, XBOOLE_1: 8;

      end;

    end;

    theorem :: JORDAN3:43

    

     Th43: for f,g be FinSequence of ( TOP-REAL 2), p be Point of ( TOP-REAL 2) st (f . ( len f)) = (g . 1) & p in ( L~ f) & f is being_S-Seq & g is being_S-Seq & (( L~ f) /\ ( L~ g)) = {(g . 1)} & p <> (f . ( len f)) holds (( L_Cut (f,p)) ^ ( mid (g,2,( len g)))) is_S-Seq_joining (p,(g /. ( len g)))

    proof

      let f,g be FinSequence of ( TOP-REAL 2), p be Point of ( TOP-REAL 2);

      assume that

       A1: (f . ( len f)) = (g . 1) and

       A2: p in ( L~ f) and

       A3: f is being_S-Seq and

       A4: g is being_S-Seq and

       A5: (( L~ f) /\ ( L~ g)) = {(g . 1)} and

       A6: p <> (f . ( len f));

      ( L_Cut (f,p)) is_S-Seq_joining (p,(f /. ( len f))) by A2, A3, A6, Th33;

      then

       A7: (( L_Cut (f,p)) . ( len ( L_Cut (f,p)))) = (f /. ( len f));

      

       A8: ( len g) >= 2 by A4, TOPREAL1:def 8;

      then

       A9: 1 <= ( len g) by XXREAL_0: 2;

      (g /. 1) in ( LSeg ((g /. 1),(g /. (1 + 1)))) by RLTOPSP1: 68;

      then (g /. 1) in ( LSeg (g,1)) by A8, TOPREAL1:def 3;

      then (g . 1) in ( LSeg (g,1)) by A9, FINSEQ_4: 15;

      then

       A10: (g . 1) in ( L~ g) by SPPOL_2: 17;

      ( L~ ( L_Cut (f,p))) c= ( L~ f) by A2, Th42;

      then

       A11: (( L~ ( L_Cut (f,p))) /\ ( L~ g)) c= (( L~ f) /\ ( L~ g)) by XBOOLE_1: 27;

      ( len f) >= 2 by A3, TOPREAL1:def 8;

      then

       A12: 1 <= ( len f) by XXREAL_0: 2;

      

       A13: ( L_Cut (f,p)) is being_S-Seq by A2, A3, A6, Th34;

      then

       A14: (1 + 1) <= ( len ( L_Cut (f,p))) by TOPREAL1:def 8;

      then

       A15: ((1 + 1) - 1) <= (( len ( L_Cut (f,p))) - 1) by XREAL_1: 9;

      

       A16: 1 <= ( len ( L_Cut (f,p))) by A14, XXREAL_0: 2;

      then (( L_Cut (f,p)) . 1) = (( L_Cut (f,p)) /. 1) by FINSEQ_4: 15;

      then

       A17: (( L_Cut (f,p)) /. 1) = p by A2, Th23;

      

       A18: ((( len ( L_Cut (f,p))) -' 1) + 1) = ( len ( L_Cut (f,p))) by A14, XREAL_1: 235, XXREAL_0: 2;

      then (( L_Cut (f,p)) /. ( len ( L_Cut (f,p)))) in ( LSeg ((( L_Cut (f,p)) /. (( len ( L_Cut (f,p))) -' 1)),(( L_Cut (f,p)) /. ((( len ( L_Cut (f,p))) -' 1) + 1)))) by RLTOPSP1: 68;

      then (( L_Cut (f,p)) . ( len ( L_Cut (f,p)))) in ( LSeg ((( L_Cut (f,p)) /. (( len ( L_Cut (f,p))) -' 1)),(( L_Cut (f,p)) /. ((( len ( L_Cut (f,p))) -' 1) + 1)))) by A16, FINSEQ_4: 15;

      then (( L_Cut (f,p)) . ( len ( L_Cut (f,p)))) in ( LSeg (( L_Cut (f,p)),(( len ( L_Cut (f,p))) -' 1))) by A15, A18, TOPREAL1:def 3;

      then (f /. ( len f)) in ( L~ ( L_Cut (f,p))) by A7, SPPOL_2: 17;

      then (f . ( len f)) in ( L~ ( L_Cut (f,p))) by A12, FINSEQ_4: 15;

      then (g . 1) in (( L~ ( L_Cut (f,p))) /\ ( L~ g)) by A1, A10, XBOOLE_0:def 4;

      then {(g . 1)} c= (( L~ ( L_Cut (f,p))) /\ ( L~ g)) by ZFMISC_1: 31;

      then (( L~ ( L_Cut (f,p))) /\ ( L~ g)) = {(g . 1)} by A5, A11, XBOOLE_0:def 10;

      hence thesis by A1, A4, A12, A13, A7, A17, Th39, FINSEQ_4: 15;

    end;

    theorem :: JORDAN3:44

    for f,g be FinSequence of ( TOP-REAL 2), p be Point of ( TOP-REAL 2) st (f . ( len f)) = (g . 1) & p in ( L~ f) & f is being_S-Seq & g is being_S-Seq & (( L~ f) /\ ( L~ g)) = {(g . 1)} & p <> (f . ( len f)) holds (( L_Cut (f,p)) ^ ( mid (g,2,( len g)))) is being_S-Seq

    proof

      let f,g be FinSequence of ( TOP-REAL 2), p be Point of ( TOP-REAL 2);

      assume that

       A1: (f . ( len f)) = (g . 1) and

       A2: p in ( L~ f) and

       A3: f is being_S-Seq and

       A4: g is being_S-Seq and

       A5: (( L~ f) /\ ( L~ g)) = {(g . 1)} and

       A6: p <> (f . ( len f));

      (( L_Cut (f,p)) ^ ( mid (g,2,( len g)))) is_S-Seq_joining (p,(g /. ( len g))) by A1, A2, A3, A4, A5, A6, Th43;

      hence thesis;

    end;

    theorem :: JORDAN3:45

    

     Th45: for f,g be FinSequence of ( TOP-REAL 2) st (f . ( len f)) = (g . 1) & f is being_S-Seq & g is being_S-Seq & (( L~ f) /\ ( L~ g)) = {(g . 1)} holds (( mid (f,1,(( len f) -' 1))) ^ g) is being_S-Seq

    proof

      let f,g be FinSequence of ( TOP-REAL 2);

      assume that

       A1: (f . ( len f)) = (g . 1) and

       A2: f is being_S-Seq and

       A3: g is being_S-Seq and

       A4: (( L~ f) /\ ( L~ g)) = {(g . 1)};

      

       A5: ( Rev f) is being_S-Seq by A2;

      ( L~ ( Rev f)) = ( L~ f) by SPPOL_2: 22;

      then

       A6: (( L~ ( Rev g)) /\ ( L~ ( Rev f))) = {(g . 1)} by A4, SPPOL_2: 22;

      

       A7: (( Rev f) . 1) = (f . ( len f)) by FINSEQ_5: 62;

      

       A8: ( Rev g) is being_S-Seq by A3;

      (( Rev g) . ( len ( Rev g))) = (( Rev ( Rev g)) . 1) by FINSEQ_5: 62

      .= (( Rev f) . 1) by A1, A7;

      then

       A9: (( Rev g) ^ ( mid (( Rev f),2,( len ( Rev f))))) is being_S-Seq by A1, A5, A8, A6, A7, Th38;

      

       A10: (( len f) -' 1) <= ( len f) by NAT_D: 50;

      

       A11: ( len ( Rev f)) = ( len f) by FINSEQ_5:def 3;

      

       A12: ( len f) >= 2 by A2, TOPREAL1:def 8;

      then

       A13: (( len f) - 1) >= ((1 + 1) - 1) by XREAL_1: 9;

      

       A14: ((( len f) -' 1) + 1) = ((( len f) - 1) + 1) by A12, XREAL_1: 233, XXREAL_0: 2

      .= ( len f);

      

       A15: ((( len f) -' (( len f) -' 1)) + 1) = ((( len f) - (( len f) -' 1)) + 1) by NAT_D: 50, XREAL_1: 233

      .= ((( len f) - (( len f) - 1)) + 1) by A12, XREAL_1: 233, XXREAL_0: 2

      .= 2;

      1 <= ( len f) by A12, XXREAL_0: 2;

      then (( Rev g) ^ ( Rev ( mid (f,1,(( len f) -' 1))))) is being_S-Seq by A13, A10, A15, A11, A14, A9, FINSEQ_6: 113;

      then ( Rev (( mid (f,1,(( len f) -' 1))) ^ g)) is being_S-Seq by FINSEQ_5: 64;

      then ( Rev ( Rev (( mid (f,1,(( len f) -' 1))) ^ g))) is being_S-Seq;

      hence thesis;

    end;

    theorem :: JORDAN3:46

    

     Th46: for f,g be FinSequence of ( TOP-REAL 2) st (f . ( len f)) = (g . 1) & f is being_S-Seq & g is being_S-Seq & (( L~ f) /\ ( L~ g)) = {(g . 1)} holds (( mid (f,1,(( len f) -' 1))) ^ g) is_S-Seq_joining ((f /. 1),(g /. ( len g)))

    proof

      let f,g be FinSequence of ( TOP-REAL 2);

      assume that

       A1: (f . ( len f)) = (g . 1) and

       A2: f is being_S-Seq and

       A3: g is being_S-Seq and

       A4: (( L~ f) /\ ( L~ g)) = {(g . 1)};

      

       A5: (( len f) -' 1) <= ( len f) by NAT_D: 50;

      

       A6: ( len f) >= 2 by A2, TOPREAL1:def 8;

      then ((1 + 1) - 1) <= (( len f) - 1) by XREAL_1: 9;

      then

       A7: 1 <= (( len f) -' 1) by NAT_D: 39;

      

       A8: 1 <= ( len f) by A6, XXREAL_0: 2;

      

      then ( len ( mid (f,1,(( len f) -' 1)))) = (((( len f) -' 1) -' 1) + 1) by A5, A7, FINSEQ_6: 118

      .= (((( len f) -' 1) - 1) + 1) by A7, XREAL_1: 233

      .= (( len f) -' 1);

      

      then

       A9: ((( mid (f,1,(( len f) -' 1))) ^ g) . 1) = (( mid (f,1,(( len f) -' 1))) . 1) by A7, FINSEQ_1: 64

      .= (f . 1) by A5, A7, FINSEQ_6: 123

      .= (f /. 1) by A8, FINSEQ_4: 15;

      

       A10: ( len (( mid (f,1,(( len f) -' 1))) ^ g)) = (( len ( mid (f,1,(( len f) -' 1)))) + ( len g)) by FINSEQ_1: 22;

      

       A11: ( len g) >= 2 by A3, TOPREAL1:def 8;

      then

       A12: 1 <= ( len g) by XXREAL_0: 2;

      ( 0 + ( len ( mid (f,1,(( len f) -' 1))))) < (( len g) + ( len ( mid (f,1,(( len f) -' 1))))) by A11, XREAL_1: 6;

      

      then

       A13: ((( mid (f,1,(( len f) -' 1))) ^ g) . ( len (( mid (f,1,(( len f) -' 1))) ^ g))) = (g . (( len (( mid (f,1,(( len f) -' 1))) ^ g)) - ( len ( mid (f,1,(( len f) -' 1)))))) by A10, FINSEQ_6: 108

      .= (g /. ( len g)) by A12, A10, FINSEQ_4: 15;

      (( mid (f,1,(( len f) -' 1))) ^ g) is being_S-Seq by A1, A2, A3, A4, Th45;

      hence thesis by A9, A13;

    end;

    theorem :: JORDAN3:47

    

     Th47: for f,g be FinSequence of ( TOP-REAL 2), p be Point of ( TOP-REAL 2) st (f . ( len f)) = (g . 1) & p in ( L~ g) & f is being_S-Seq & g is being_S-Seq & (( L~ f) /\ ( L~ g)) = {(g . 1)} & p <> (g . 1) holds (( mid (f,1,(( len f) -' 1))) ^ ( R_Cut (g,p))) is_S-Seq_joining ((f /. 1),p)

    proof

      let f,g be FinSequence of ( TOP-REAL 2), p be Point of ( TOP-REAL 2);

      assume that

       A1: (f . ( len f)) = (g . 1) and

       A2: p in ( L~ g) and

       A3: f is being_S-Seq and

       A4: g is being_S-Seq and

       A5: (( L~ f) /\ ( L~ g)) = {(g . 1)} and

       A6: p <> (g . 1);

      ( len g) >= 2 by A4, TOPREAL1:def 8;

      then

       A7: 1 <= ( len g) by XXREAL_0: 2;

      ( R_Cut (g,p)) is_S-Seq_joining ((g /. 1),p) by A2, A4, A6, Th32;

      then

       A8: (( R_Cut (g,p)) . 1) = (g /. 1);

      then

       A9: (( R_Cut (g,p)) . 1) = (f . ( len f)) by A1, A7, FINSEQ_4: 15;

      

       A10: ( len f) >= 2 by A3, TOPREAL1:def 8;

      then

       A11: 1 <= ( len f) by XXREAL_0: 2;

      

       A12: ((1 + 1) - 1) <= (( len f) - 1) by A10, XREAL_1: 9;

      

       A13: ((( len f) -' 1) + 1) = ( len f) by A10, XREAL_1: 235, XXREAL_0: 2;

      then (f /. ( len f)) in ( LSeg ((f /. (( len f) -' 1)),(f /. ((( len f) -' 1) + 1)))) by RLTOPSP1: 68;

      then (f /. ( len f)) in ( LSeg (f,(( len f) -' 1))) by A12, A13, TOPREAL1:def 3;

      then (f . ( len f)) in ( LSeg (f,(( len f) -' 1))) by A11, FINSEQ_4: 15;

      then

       A14: (f . ( len f)) in ( L~ f) by SPPOL_2: 17;

      

       A15: ( R_Cut (g,p)) is being_S-Seq by A2, A4, A6, Th35;

      then

       A16: (1 + 1) <= ( len ( R_Cut (g,p))) by TOPREAL1:def 8;

      then

       A17: 1 <= ( len ( R_Cut (g,p))) by XXREAL_0: 2;

      then (( R_Cut (g,p)) . ( len ( R_Cut (g,p)))) = (( R_Cut (g,p)) /. ( len ( R_Cut (g,p)))) by FINSEQ_4: 15;

      then

       A18: (( R_Cut (g,p)) /. ( len ( R_Cut (g,p)))) = p by A2, Th24;

      (( R_Cut (g,p)) /. 1) in ( LSeg ((( R_Cut (g,p)) /. 1),(( R_Cut (g,p)) /. (1 + 1)))) by RLTOPSP1: 68;

      then (( R_Cut (g,p)) . 1) in ( LSeg ((( R_Cut (g,p)) /. 1),(( R_Cut (g,p)) /. (1 + 1)))) by A17, FINSEQ_4: 15;

      then (( R_Cut (g,p)) . 1) in ( LSeg (( R_Cut (g,p)),1)) by A16, TOPREAL1:def 3;

      then (g /. 1) in ( L~ ( R_Cut (g,p))) by A8, SPPOL_2: 17;

      then (g . 1) in ( L~ ( R_Cut (g,p))) by A7, FINSEQ_4: 15;

      then (f . ( len f)) in (( L~ f) /\ ( L~ ( R_Cut (g,p)))) by A1, A14, XBOOLE_0:def 4;

      then

       A19: {(f . ( len f))} c= (( L~ f) /\ ( L~ ( R_Cut (g,p)))) by ZFMISC_1: 31;

      ( L~ ( R_Cut (g,p))) c= ( L~ g) by A2, Th41;

      then (( L~ f) /\ ( L~ ( R_Cut (g,p)))) c= (( L~ f) /\ ( L~ g)) by XBOOLE_1: 27;

      then (( L~ f) /\ ( L~ ( R_Cut (g,p)))) = {(( R_Cut (g,p)) . 1)} by A1, A5, A9, A19, XBOOLE_0:def 10;

      hence thesis by A3, A15, A9, A18, Th46;

    end;

    theorem :: JORDAN3:48

    for f,g be FinSequence of ( TOP-REAL 2), p be Point of ( TOP-REAL 2) st (f . ( len f)) = (g . 1) & p in ( L~ g) & f is being_S-Seq & g is being_S-Seq & (( L~ f) /\ ( L~ g)) = {(g . 1)} & p <> (g . 1) holds (( mid (f,1,(( len f) -' 1))) ^ ( R_Cut (g,p))) is being_S-Seq

    proof

      let f,g be FinSequence of ( TOP-REAL 2), p be Point of ( TOP-REAL 2);

      assume that

       A1: (f . ( len f)) = (g . 1) and

       A2: p in ( L~ g) and

       A3: f is being_S-Seq and

       A4: g is being_S-Seq and

       A5: (( L~ f) /\ ( L~ g)) = {(g . 1)} and

       A6: p <> (g . 1);

      (( mid (f,1,(( len f) -' 1))) ^ ( R_Cut (g,p))) is_S-Seq_joining ((f /. 1),p) by A1, A2, A3, A4, A5, A6, Th47;

      hence thesis;

    end;