jordan15.miz



    begin

    reserve n for Nat;

    theorem :: JORDAN15:1

    for A,B be Subset of ( TOP-REAL 2) st A meets B holds ( proj1 .: A) meets ( proj1 .: B)

    proof

      let A,B be Subset of ( TOP-REAL 2);

      assume A meets B;

      then

      consider e be object such that

       A1: e in A and

       A2: e in B by XBOOLE_0: 3;

      reconsider e as Point of ( TOP-REAL 2) by A1;

      

       A3: (e `1 ) = ( proj1 . e) by PSCOMP_1:def 5;

      then

       A4: (e `1 ) in ( proj1 .: B) by A2, FUNCT_2: 35;

      (e `1 ) in ( proj1 .: A) by A1, A3, FUNCT_2: 35;

      hence thesis by A4, XBOOLE_0: 3;

    end;

    theorem :: JORDAN15:2

    for A,B be Subset of ( TOP-REAL 2) holds for s be Real st A misses B & A c= ( Horizontal_Line s) & B c= ( Horizontal_Line s) holds ( proj1 .: A) misses ( proj1 .: B)

    proof

      let A,B be Subset of ( TOP-REAL 2);

      let s be Real such that

       A1: A misses B and

       A2: A c= ( Horizontal_Line s) and

       A3: B c= ( Horizontal_Line s);

      assume ( proj1 .: A) meets ( proj1 .: B);

      then

      consider e be object such that

       A4: e in ( proj1 .: A) and

       A5: e in ( proj1 .: B) by XBOOLE_0: 3;

      reconsider e as Real by A4;

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

       A6: d in B and

       A7: e = ( proj1 . d) by A5, FUNCT_2: 65;

      

       A8: (d `2 ) = s by A3, A6, JORDAN6: 32;

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

       A9: c in A and

       A10: e = ( proj1 . c) by A4, FUNCT_2: 65;

      (c `2 ) = s by A2, A9, JORDAN6: 32;

      

      then c = |[(c `1 ), (d `2 )]| by A8, EUCLID: 53

      .= |[e, (d `2 )]| by A10, PSCOMP_1:def 5

      .= |[(d `1 ), (d `2 )]| by A7, PSCOMP_1:def 5

      .= d by EUCLID: 53;

      hence contradiction by A1, A9, A6, XBOOLE_0: 3;

    end;

    theorem :: JORDAN15:3

    

     Th3: for S be closed Subset of ( TOP-REAL 2) st S is bounded holds ( proj1 .: S) is closed

    proof

      let S be closed Subset of ( TOP-REAL 2);

      assume S is bounded;

      

      then ( Cl ( proj1 .: S)) = ( proj1 .: ( Cl S)) by TOPREAL6: 83

      .= ( proj1 .: S) by PRE_TOPC: 22;

      hence thesis;

    end;

    theorem :: JORDAN15:4

    

     Th4: for S be compact Subset of ( TOP-REAL 2) holds ( proj1 .: S) is compact

    proof

      let S be compact Subset of ( TOP-REAL 2);

      ( proj1 .: S) is closed by Th3;

      hence thesis by JORDAN1C: 3, RCOMP_1: 11;

    end;

    theorem :: JORDAN15:5

    

     Th5: for G be Go-board holds for i,j,k,j1,k1 be Nat st 1 <= i & i <= ( len G) & 1 <= j & j <= j1 & j1 <= k1 & k1 <= k & k <= ( width G) holds ( LSeg ((G * (i,j1)),(G * (i,k1)))) c= ( LSeg ((G * (i,j)),(G * (i,k))))

    proof

      let G be Go-board;

      let i,j,k,j1,k1 be Nat;

      assume that

       A1: 1 <= i and

       A2: i <= ( len G) and

       A3: 1 <= j and

       A4: j <= j1 and

       A5: j1 <= k1 and

       A6: k1 <= k and

       A7: k <= ( width G);

      

       A8: j1 <= k by A5, A6, XXREAL_0: 2;

      j <= k1 by A4, A5, XXREAL_0: 2;

      then

       A9: 1 <= k1 by A3, XXREAL_0: 2;

      then

       A10: ((G * (i,k1)) `2 ) <= ((G * (i,k)) `2 ) by A1, A2, A6, A7, SPRECT_3: 12;

      

       A11: 1 <= j1 by A3, A4, XXREAL_0: 2;

      1 <= j1 by A3, A4, XXREAL_0: 2;

      then

       A12: 1 <= k by A8, XXREAL_0: 2;

      

       A13: k1 <= ( width G) by A6, A7, XXREAL_0: 2;

      j <= k1 by A4, A5, XXREAL_0: 2;

      then

       A14: j <= ( width G) by A13, XXREAL_0: 2;

      

      then ((G * (i,j)) `1 ) = ((G * (i,1)) `1 ) by A1, A2, A3, GOBOARD5: 2

      .= ((G * (i,k)) `1 ) by A1, A2, A7, A12, GOBOARD5: 2;

      then

       A15: ( LSeg ((G * (i,j)),(G * (i,k)))) is vertical by SPPOL_1: 16;

      j1 <= k by A5, A6, XXREAL_0: 2;

      then

       A16: j1 <= ( width G) by A7, XXREAL_0: 2;

      then

       A17: ((G * (i,j)) `2 ) <= ((G * (i,j1)) `2 ) by A1, A2, A3, A4, SPRECT_3: 12;

      

       A18: k1 <= ( width G) by A6, A7, XXREAL_0: 2;

      then

       A19: ((G * (i,j1)) `2 ) <= ((G * (i,k1)) `2 ) by A1, A2, A5, A11, SPRECT_3: 12;

      ((G * (i,j1)) `1 ) = ((G * (i,1)) `1 ) by A1, A2, A11, A16, GOBOARD5: 2

      .= ((G * (i,k1)) `1 ) by A1, A2, A9, A18, GOBOARD5: 2;

      then

       A20: ( LSeg ((G * (i,j1)),(G * (i,k1)))) is vertical by SPPOL_1: 16;

      ((G * (i,j)) `1 ) = ((G * (i,1)) `1 ) by A1, A2, A3, A14, GOBOARD5: 2

      .= ((G * (i,j1)) `1 ) by A1, A2, A11, A16, GOBOARD5: 2;

      hence thesis by A15, A20, A17, A19, A10, GOBOARD7: 63;

    end;

    theorem :: JORDAN15:6

    

     Th6: for G be Go-board holds for i,j,k,j1,k1 be Nat st 1 <= i & i <= ( width G) & 1 <= j & j <= j1 & j1 <= k1 & k1 <= k & k <= ( len G) holds ( LSeg ((G * (j1,i)),(G * (k1,i)))) c= ( LSeg ((G * (j,i)),(G * (k,i))))

    proof

      let G be Go-board;

      let i,j,k,j1,k1 be Nat;

      assume that

       A1: 1 <= i and

       A2: i <= ( width G) and

       A3: 1 <= j and

       A4: j <= j1 and

       A5: j1 <= k1 and

       A6: k1 <= k and

       A7: k <= ( len G);

      

       A8: j1 <= k by A5, A6, XXREAL_0: 2;

      j <= k1 by A4, A5, XXREAL_0: 2;

      then

       A9: 1 <= k1 by A3, XXREAL_0: 2;

      then

       A10: ((G * (k1,i)) `1 ) <= ((G * (k,i)) `1 ) by A1, A2, A6, A7, SPRECT_3: 13;

      

       A11: 1 <= j1 by A3, A4, XXREAL_0: 2;

      1 <= j1 by A3, A4, XXREAL_0: 2;

      then

       A12: 1 <= k by A8, XXREAL_0: 2;

      

       A13: k1 <= ( len G) by A6, A7, XXREAL_0: 2;

      j <= k1 by A4, A5, XXREAL_0: 2;

      then

       A14: j <= ( len G) by A13, XXREAL_0: 2;

      

      then ((G * (j,i)) `2 ) = ((G * (1,i)) `2 ) by A1, A2, A3, GOBOARD5: 1

      .= ((G * (k,i)) `2 ) by A1, A2, A7, A12, GOBOARD5: 1;

      then

       A15: ( LSeg ((G * (j,i)),(G * (k,i)))) is horizontal by SPPOL_1: 15;

      j1 <= k by A5, A6, XXREAL_0: 2;

      then

       A16: j1 <= ( len G) by A7, XXREAL_0: 2;

      then

       A17: ((G * (j,i)) `1 ) <= ((G * (j1,i)) `1 ) by A1, A2, A3, A4, SPRECT_3: 13;

      

       A18: k1 <= ( len G) by A6, A7, XXREAL_0: 2;

      then

       A19: ((G * (j1,i)) `1 ) <= ((G * (k1,i)) `1 ) by A1, A2, A5, A11, SPRECT_3: 13;

      ((G * (j1,i)) `2 ) = ((G * (1,i)) `2 ) by A1, A2, A11, A16, GOBOARD5: 1

      .= ((G * (k1,i)) `2 ) by A1, A2, A9, A18, GOBOARD5: 1;

      then

       A20: ( LSeg ((G * (j1,i)),(G * (k1,i)))) is horizontal by SPPOL_1: 15;

      ((G * (j,i)) `2 ) = ((G * (1,i)) `2 ) by A1, A2, A3, A14, GOBOARD5: 1

      .= ((G * (j1,i)) `2 ) by A1, A2, A11, A16, GOBOARD5: 1;

      hence thesis by A15, A20, A17, A19, A10, GOBOARD7: 64;

    end;

    theorem :: JORDAN15:7

    for G be Go-board holds for j,k,j1,k1 be Nat st 1 <= j & j <= j1 & j1 <= k1 & k1 <= k & k <= ( width G) holds ( LSeg ((G * (( Center G),j1)),(G * (( Center G),k1)))) c= ( LSeg ((G * (( Center G),j)),(G * (( Center G),k))))

    proof

      let G be Go-board;

      let j,k,j1,k1 be Nat;

      assume that

       A1: 1 <= j and

       A2: j <= j1 and

       A3: j1 <= k1 and

       A4: k1 <= k and

       A5: k <= ( width G);

      

       A6: ( Center G) <= ( len G) by JORDAN1B: 13;

      1 <= ( Center G) by JORDAN1B: 11;

      hence thesis by A1, A2, A3, A4, A5, A6, Th5;

    end;

    theorem :: JORDAN15:8

    for G be Go-board st ( len G) = ( width G) holds for j,k,j1,k1 be Nat st 1 <= j & j <= j1 & j1 <= k1 & k1 <= k & k <= ( len G) holds ( LSeg ((G * (j1,( Center G))),(G * (k1,( Center G))))) c= ( LSeg ((G * (j,( Center G))),(G * (k,( Center G)))))

    proof

      let G be Go-board;

      assume ( len G) = ( width G);

      then

       A1: ( Center G) <= ( width G) by JORDAN1B: 13;

      let j,k,j1,k1 be Nat;

      assume that

       A2: 1 <= j and

       A3: j <= j1 and

       A4: j1 <= k1 and

       A5: k1 <= k and

       A6: k <= ( len G);

      1 <= ( Center G) by JORDAN1B: 11;

      hence thesis by A2, A3, A4, A5, A6, A1, Th6;

    end;

    theorem :: JORDAN15:9

    

     Th9: for C be compact connected non vertical non horizontal Subset of ( TOP-REAL 2) holds for i,j,k be Nat st 1 <= i & i <= ( len ( Gauge (C,n))) & 1 <= j & j <= k & k <= ( width ( Gauge (C,n))) & (( Gauge (C,n)) * (i,j)) in ( L~ ( Lower_Seq (C,n))) holds ex j1 be Nat st j <= j1 & j1 <= k & (( LSeg ((( Gauge (C,n)) * (i,j1)),(( Gauge (C,n)) * (i,k)))) /\ ( L~ ( Lower_Seq (C,n)))) = {(( Gauge (C,n)) * (i,j1))}

    proof

      let C be compact connected non vertical non horizontal Subset of ( TOP-REAL 2);

      let i,j,k be Nat;

      assume that

       A1: 1 <= i and

       A2: i <= ( len ( Gauge (C,n))) and

       A3: 1 <= j and

       A4: j <= k and

       A5: k <= ( width ( Gauge (C,n))) and

       A6: (( Gauge (C,n)) * (i,j)) in ( L~ ( Lower_Seq (C,n)));

      set G = ( Gauge (C,n));

      

       A7: k >= 1 by A3, A4, XXREAL_0: 2;

      then

       A8: [i, k] in ( Indices G) by A1, A2, A5, MATRIX_0: 30;

      set X = (( LSeg ((G * (i,j)),(G * (i,k)))) /\ ( L~ ( Lower_Seq (C,n))));

      

       A9: (G * (i,j)) in ( LSeg ((G * (i,j)),(G * (i,k)))) by RLTOPSP1: 68;

      then

      reconsider X1 = X as non empty compact Subset of ( TOP-REAL 2) by A6, XBOOLE_0:def 4;

      

       A10: ( LSeg ((G * (i,j)),(G * (i,k)))) meets ( L~ ( Lower_Seq (C,n))) by A6, A9, XBOOLE_0: 3;

      set s = ((G * (i,1)) `1 );

      set e = (G * (i,k));

      set f = (G * (i,j));

      set w2 = ( upper_bound ( proj2 .: (( LSeg (f,e)) /\ ( L~ ( Lower_Seq (C,n))))));

      

       A11: j <= ( width G) by A4, A5, XXREAL_0: 2;

      then [i, j] in ( Indices G) by A1, A2, A3, MATRIX_0: 30;

      then

      consider j1 be Nat such that

       A12: j <= j1 and

       A13: j1 <= k and

       A14: ((G * (i,j1)) `2 ) = w2 by A4, A10, A8, JORDAN1F: 2, JORDAN1G: 5;

      set q = |[s, w2]|;

      

       A15: j1 <= ( width G) by A5, A13, XXREAL_0: 2;

      

       A16: ((G * (i,k)) `1 ) = s by A1, A2, A5, A7, GOBOARD5: 2;

      then (f `1 ) = (e `1 ) by A1, A2, A3, A11, GOBOARD5: 2;

      then

       A17: ( LSeg (f,e)) is vertical by SPPOL_1: 16;

      take j1;

      thus j <= j1 & j1 <= k by A12, A13;

      consider pp be object such that

       A18: pp in ( N-most X1) by XBOOLE_0:def 1;

      reconsider pp as Point of ( TOP-REAL 2) by A18;

      

       A19: pp in X by A18, XBOOLE_0:def 4;

      then

       A20: pp in ( L~ ( Lower_Seq (C,n))) by XBOOLE_0:def 4;

      

       A21: 1 <= j1 by A3, A12, XXREAL_0: 2;

      then

       A22: ((G * (i,j1)) `1 ) = s by A1, A2, A15, GOBOARD5: 2;

      then

       A23: q = (G * (i,j1)) by A14, EUCLID: 53;

      then

       A24: (q `2 ) <= (e `2 ) by A1, A2, A5, A13, A21, SPRECT_3: 12;

      

       A25: (q `2 ) = ( N-bound X) by A14, A23, SPRECT_1: 45

      .= (( N-min X) `2 ) by EUCLID: 52

      .= (pp `2 ) by A18, PSCOMP_1: 39;

      pp in ( LSeg ((G * (i,j)),(G * (i,k)))) by A19, XBOOLE_0:def 4;

      then (pp `1 ) = (q `1 ) by A16, A22, A23, A17, SPPOL_1: 41;

      then

       A26: q in ( L~ ( Lower_Seq (C,n))) by A20, A25, TOPREAL3: 6;

      for x be object holds x in (( LSeg (e,q)) /\ ( L~ ( Lower_Seq (C,n)))) iff x = q

      proof

        let x be object;

        thus x in (( LSeg (e,q)) /\ ( L~ ( Lower_Seq (C,n)))) implies x = q

        proof

          reconsider EE = (( LSeg (f,e)) /\ ( L~ ( Lower_Seq (C,n)))) as compact Subset of ( TOP-REAL 2);

          reconsider E0 = ( proj2 .: EE) as compact Subset of REAL by JCT_MISC: 15;

          

           A27: e in ( LSeg (f,e)) by RLTOPSP1: 68;

          

           A28: (f `2 ) <= (q `2 ) by A1, A2, A3, A12, A15, A23, SPRECT_3: 12;

          (f `1 ) = (q `1 ) by A1, A2, A3, A11, A22, A23, GOBOARD5: 2;

          then q in ( LSeg (e,f)) by A16, A22, A23, A24, A28, GOBOARD7: 7;

          then

           A29: ( LSeg (e,q)) c= ( LSeg (f,e)) by A27, TOPREAL1: 6;

          assume

           A30: x in (( LSeg (e,q)) /\ ( L~ ( Lower_Seq (C,n))));

          then

          reconsider pp = x as Point of ( TOP-REAL 2);

          

           A31: pp in ( LSeg (e,q)) by A30, XBOOLE_0:def 4;

          then

           A32: (pp `2 ) >= (q `2 ) by A24, TOPREAL1: 4;

          pp in ( L~ ( Lower_Seq (C,n))) by A30, XBOOLE_0:def 4;

          then pp in EE by A31, A29, XBOOLE_0:def 4;

          then ( proj2 . pp) in E0 by FUNCT_2: 35;

          then

           A33: (pp `2 ) in E0 by PSCOMP_1:def 6;

          E0 is real-bounded by RCOMP_1: 10;

          then E0 is bounded_above by XXREAL_2:def 11;

          then (q `2 ) >= (pp `2 ) by A14, A23, A33, SEQ_4:def 1;

          then

           A34: (pp `2 ) = (q `2 ) by A32, XXREAL_0: 1;

          (pp `1 ) = (q `1 ) by A16, A22, A23, A31, GOBOARD7: 5;

          hence thesis by A34, TOPREAL3: 6;

        end;

        assume

         A35: x = q;

        then x in ( LSeg (e,q)) by RLTOPSP1: 68;

        hence thesis by A26, A35, XBOOLE_0:def 4;

      end;

      hence thesis by A23, TARSKI:def 1;

    end;

    theorem :: JORDAN15:10

    for C be compact connected non vertical non horizontal Subset of ( TOP-REAL 2) holds for i,j,k be Nat st 1 <= i & i <= ( len ( Gauge (C,n))) & 1 <= j & j <= k & k <= ( width ( Gauge (C,n))) & (( Gauge (C,n)) * (i,k)) in ( L~ ( Upper_Seq (C,n))) holds ex k1 be Nat st j <= k1 & k1 <= k & (( LSeg ((( Gauge (C,n)) * (i,j)),(( Gauge (C,n)) * (i,k1)))) /\ ( L~ ( Upper_Seq (C,n)))) = {(( Gauge (C,n)) * (i,k1))}

    proof

      let C be compact connected non vertical non horizontal Subset of ( TOP-REAL 2);

      let i,j,k be Nat;

      assume that

       A1: 1 <= i and

       A2: i <= ( len ( Gauge (C,n))) and

       A3: 1 <= j and

       A4: j <= k and

       A5: k <= ( width ( Gauge (C,n))) and

       A6: (( Gauge (C,n)) * (i,k)) in ( L~ ( Upper_Seq (C,n)));

      set G = ( Gauge (C,n));

      

       A7: k >= 1 by A3, A4, XXREAL_0: 2;

      then

       A8: [i, k] in ( Indices G) by A1, A2, A5, MATRIX_0: 30;

      set X = (( LSeg ((G * (i,j)),(G * (i,k)))) /\ ( L~ ( Upper_Seq (C,n))));

      

       A9: (G * (i,k)) in ( LSeg ((G * (i,j)),(G * (i,k)))) by RLTOPSP1: 68;

      then

      reconsider X1 = X as non empty compact Subset of ( TOP-REAL 2) by A6, XBOOLE_0:def 4;

      

       A10: ( LSeg ((G * (i,j)),(G * (i,k)))) meets ( L~ ( Upper_Seq (C,n))) by A6, A9, XBOOLE_0: 3;

      set s = ((G * (i,1)) `1 );

      set e = (G * (i,k));

      set f = (G * (i,j));

      set w1 = ( lower_bound ( proj2 .: (( LSeg (f,e)) /\ ( L~ ( Upper_Seq (C,n))))));

      

       A11: j <= ( width G) by A4, A5, XXREAL_0: 2;

      then [i, j] in ( Indices G) by A1, A2, A3, MATRIX_0: 30;

      then

      consider k1 be Nat such that

       A12: j <= k1 and

       A13: k1 <= k and

       A14: ((G * (i,k1)) `2 ) = w1 by A4, A10, A8, JORDAN1F: 1, JORDAN1G: 4;

      set p = |[s, w1]|;

      

       A15: k1 <= ( width G) by A5, A13, XXREAL_0: 2;

      (f `1 ) = s by A1, A2, A3, A11, GOBOARD5: 2

      .= (e `1 ) by A1, A2, A5, A7, GOBOARD5: 2;

      then

       A16: ( LSeg (f,e)) is vertical by SPPOL_1: 16;

      take k1;

      thus j <= k1 & k1 <= k by A12, A13;

      consider pp be object such that

       A17: pp in ( S-most X1) by XBOOLE_0:def 1;

      

       A18: 1 <= k1 by A3, A12, XXREAL_0: 2;

      then

       A19: ((G * (i,k1)) `1 ) = s by A1, A2, A15, GOBOARD5: 2;

      then

       A20: p = (G * (i,k1)) by A14, EUCLID: 53;

      then

       A21: (f `2 ) <= (p `2 ) by A1, A2, A3, A12, A15, SPRECT_3: 12;

      

       A22: (f `1 ) = (p `1 ) by A1, A2, A3, A11, A19, A20, GOBOARD5: 2;

      reconsider pp as Point of ( TOP-REAL 2) by A17;

      

       A23: pp in X by A17, XBOOLE_0:def 4;

      then

       A24: pp in ( L~ ( Upper_Seq (C,n))) by XBOOLE_0:def 4;

      

       A25: (p `2 ) = ( S-bound X) by A14, A20, SPRECT_1: 44

      .= (( S-min X) `2 ) by EUCLID: 52

      .= (pp `2 ) by A17, PSCOMP_1: 55;

      pp in ( LSeg ((G * (i,j)),(G * (i,k)))) by A23, XBOOLE_0:def 4;

      then (pp `1 ) = (p `1 ) by A22, A16, SPPOL_1: 41;

      then

       A26: p in ( L~ ( Upper_Seq (C,n))) by A24, A25, TOPREAL3: 6;

      for x be object holds x in (( LSeg (p,f)) /\ ( L~ ( Upper_Seq (C,n)))) iff x = p

      proof

        let x be object;

        thus x in (( LSeg (p,f)) /\ ( L~ ( Upper_Seq (C,n)))) implies x = p

        proof

          reconsider EE = (( LSeg (f,e)) /\ ( L~ ( Upper_Seq (C,n)))) as compact Subset of ( TOP-REAL 2);

          reconsider E0 = ( proj2 .: EE) as compact Subset of REAL by JCT_MISC: 15;

          

           A27: f in ( LSeg (f,e)) by RLTOPSP1: 68;

          

           A28: (e `1 ) = (p `1 ) by A1, A2, A5, A7, A19, A20, GOBOARD5: 2;

          

           A29: (p `2 ) <= (e `2 ) by A1, A2, A5, A13, A18, A20, SPRECT_3: 12;

          

           A30: (f `2 ) <= (p `2 ) by A1, A2, A3, A12, A15, A20, SPRECT_3: 12;

          (f `1 ) = (p `1 ) by A1, A2, A3, A11, A19, A20, GOBOARD5: 2;

          then p in ( LSeg (f,e)) by A28, A30, A29, GOBOARD7: 7;

          then

           A31: ( LSeg (p,f)) c= ( LSeg (f,e)) by A27, TOPREAL1: 6;

          assume

           A32: x in (( LSeg (p,f)) /\ ( L~ ( Upper_Seq (C,n))));

          then

          reconsider pp = x as Point of ( TOP-REAL 2);

          

           A33: pp in ( LSeg (p,f)) by A32, XBOOLE_0:def 4;

          then

           A34: (pp `2 ) <= (p `2 ) by A21, TOPREAL1: 4;

          pp in ( L~ ( Upper_Seq (C,n))) by A32, XBOOLE_0:def 4;

          then pp in EE by A33, A31, XBOOLE_0:def 4;

          then ( proj2 . pp) in E0 by FUNCT_2: 35;

          then

           A35: (pp `2 ) in E0 by PSCOMP_1:def 6;

          E0 is real-bounded by RCOMP_1: 10;

          then E0 is bounded_below by XXREAL_2:def 11;

          then (p `2 ) <= (pp `2 ) by A14, A20, A35, SEQ_4:def 2;

          then

           A36: (pp `2 ) = (p `2 ) by A34, XXREAL_0: 1;

          (pp `1 ) = (p `1 ) by A22, A33, GOBOARD7: 5;

          hence thesis by A36, TOPREAL3: 6;

        end;

        assume

         A37: x = p;

        then x in ( LSeg (p,f)) by RLTOPSP1: 68;

        hence thesis by A26, A37, XBOOLE_0:def 4;

      end;

      hence thesis by A20, TARSKI:def 1;

    end;

    theorem :: JORDAN15:11

    

     Th11: for C be compact connected non vertical non horizontal Subset of ( TOP-REAL 2) holds for i,j,k be Nat st 1 <= i & i <= ( len ( Gauge (C,n))) & 1 <= j & j <= k & k <= ( width ( Gauge (C,n))) & (( Gauge (C,n)) * (i,j)) in ( L~ ( Lower_Seq (C,n))) & (( Gauge (C,n)) * (i,k)) in ( L~ ( Upper_Seq (C,n))) holds ex j1,k1 be Nat st j <= j1 & j1 <= k1 & k1 <= k & (( LSeg ((( Gauge (C,n)) * (i,j1)),(( Gauge (C,n)) * (i,k1)))) /\ ( L~ ( Lower_Seq (C,n)))) = {(( Gauge (C,n)) * (i,j1))} & (( LSeg ((( Gauge (C,n)) * (i,j1)),(( Gauge (C,n)) * (i,k1)))) /\ ( L~ ( Upper_Seq (C,n)))) = {(( Gauge (C,n)) * (i,k1))}

    proof

      let C be compact connected non vertical non horizontal Subset of ( TOP-REAL 2);

      let i,j,k be Nat;

      assume that

       A1: 1 <= i and

       A2: i <= ( len ( Gauge (C,n))) and

       A3: 1 <= j and

       A4: j <= k and

       A5: k <= ( width ( Gauge (C,n))) and

       A6: (( Gauge (C,n)) * (i,j)) in ( L~ ( Lower_Seq (C,n))) and

       A7: (( Gauge (C,n)) * (i,k)) in ( L~ ( Upper_Seq (C,n)));

      set G = ( Gauge (C,n));

      

       A8: j <= ( width G) by A4, A5, XXREAL_0: 2;

      then

       A9: [i, j] in ( Indices G) by A1, A2, A3, MATRIX_0: 30;

      set s = ((G * (i,1)) `1 );

      set e = (G * (i,k));

      set f = (G * (i,j));

      set w1 = ( lower_bound ( proj2 .: (( LSeg (f,e)) /\ ( L~ ( Upper_Seq (C,n))))));

      

       A10: (G * (i,k)) in ( LSeg ((G * (i,j)),(G * (i,k)))) by RLTOPSP1: 68;

      then

       A11: ( LSeg ((G * (i,j)),(G * (i,k)))) meets ( L~ ( Upper_Seq (C,n))) by A7, XBOOLE_0: 3;

      

       A12: k >= 1 by A3, A4, XXREAL_0: 2;

      then [i, k] in ( Indices G) by A1, A2, A5, MATRIX_0: 30;

      then

      consider k1 be Nat such that

       A13: j <= k1 and

       A14: k1 <= k and

       A15: ((G * (i,k1)) `2 ) = w1 by A4, A11, A9, JORDAN1F: 1, JORDAN1G: 4;

      

       A16: k1 <= ( width G) by A5, A14, XXREAL_0: 2;

      

       A17: (G * (i,j)) in ( LSeg ((G * (i,j)),(G * (i,k1)))) by RLTOPSP1: 68;

      then

       A18: ( LSeg ((G * (i,j)),(G * (i,k1)))) meets ( L~ ( Lower_Seq (C,n))) by A6, XBOOLE_0: 3;

      set X = (( LSeg ((G * (i,j)),(G * (i,k1)))) /\ ( L~ ( Lower_Seq (C,n))));

      reconsider X1 = X as non empty compact Subset of ( TOP-REAL 2) by A6, A17, XBOOLE_0:def 4;

      consider pp be object such that

       A19: pp in ( N-most X1) by XBOOLE_0:def 1;

      reconsider pp as Point of ( TOP-REAL 2) by A19;

      

       A20: pp in X by A19, XBOOLE_0:def 4;

      then

       A21: pp in ( L~ ( Lower_Seq (C,n))) by XBOOLE_0:def 4;

      set p = |[s, w1]|;

      set w2 = ( upper_bound ( proj2 .: (( LSeg (f,p)) /\ ( L~ ( Lower_Seq (C,n))))));

      set q = |[s, w2]|;

      

       A22: pp in ( LSeg ((G * (i,j)),(G * (i,k1)))) by A20, XBOOLE_0:def 4;

      

       A23: 1 <= k1 by A3, A13, XXREAL_0: 2;

      then

       A24: ((G * (i,k1)) `1 ) = s by A1, A2, A16, GOBOARD5: 2;

      then

       A25: p = (G * (i,k1)) by A15, EUCLID: 53;

       [i, k1] in ( Indices G) by A1, A2, A23, A16, MATRIX_0: 30;

      then

      consider j1 be Nat such that

       A26: j <= j1 and

       A27: j1 <= k1 and

       A28: ((G * (i,j1)) `2 ) = w2 by A9, A13, A25, A18, JORDAN1F: 2, JORDAN1G: 5;

      take j1, k1;

      thus j <= j1 & j1 <= k1 & k1 <= k by A14, A26, A27;

      

       A29: j1 <= ( width G) by A16, A27, XXREAL_0: 2;

      

       A30: 1 <= j1 by A3, A26, XXREAL_0: 2;

      then

       A31: ((G * (i,j1)) `1 ) = s by A1, A2, A29, GOBOARD5: 2;

      then

       A32: q = (G * (i,j1)) by A28, EUCLID: 53;

      then

       A33: (q `2 ) <= (p `2 ) by A1, A2, A16, A25, A27, A30, SPRECT_3: 12;

      

       A34: (q `2 ) = ( N-bound X) by A25, A28, A32, SPRECT_1: 45

      .= (( N-min X) `2 ) by EUCLID: 52

      .= (pp `2 ) by A19, PSCOMP_1: 39;

      

       A35: (f `1 ) = (p `1 ) by A1, A2, A3, A8, A24, A25, GOBOARD5: 2;

      then ( LSeg (f,p)) is vertical by SPPOL_1: 16;

      then (pp `1 ) = (q `1 ) by A24, A25, A31, A32, A22, SPPOL_1: 41;

      then

       A36: q in ( L~ ( Lower_Seq (C,n))) by A21, A34, TOPREAL3: 6;

      for x be object holds x in (( LSeg (p,q)) /\ ( L~ ( Lower_Seq (C,n)))) iff x = q

      proof

        let x be object;

        thus x in (( LSeg (p,q)) /\ ( L~ ( Lower_Seq (C,n)))) implies x = q

        proof

          reconsider EE = (( LSeg (f,p)) /\ ( L~ ( Lower_Seq (C,n)))) as compact Subset of ( TOP-REAL 2);

          reconsider E0 = ( proj2 .: EE) as compact Subset of REAL by JCT_MISC: 15;

          

           A37: p in ( LSeg (f,p)) by RLTOPSP1: 68;

          

           A38: (f `2 ) <= (q `2 ) by A1, A2, A3, A26, A29, A32, SPRECT_3: 12;

          (f `1 ) = (q `1 ) by A1, A2, A3, A8, A31, A32, GOBOARD5: 2;

          then q in ( LSeg (p,f)) by A24, A25, A31, A32, A33, A38, GOBOARD7: 7;

          then

           A39: ( LSeg (p,q)) c= ( LSeg (f,p)) by A37, TOPREAL1: 6;

          assume

           A40: x in (( LSeg (p,q)) /\ ( L~ ( Lower_Seq (C,n))));

          then

          reconsider pp = x as Point of ( TOP-REAL 2);

          

           A41: pp in ( LSeg (p,q)) by A40, XBOOLE_0:def 4;

          then

           A42: (pp `2 ) >= (q `2 ) by A33, TOPREAL1: 4;

          pp in ( L~ ( Lower_Seq (C,n))) by A40, XBOOLE_0:def 4;

          then pp in EE by A41, A39, XBOOLE_0:def 4;

          then ( proj2 . pp) in E0 by FUNCT_2: 35;

          then

           A43: (pp `2 ) in E0 by PSCOMP_1:def 6;

          E0 is real-bounded by RCOMP_1: 10;

          then E0 is bounded_above by XXREAL_2:def 11;

          then (q `2 ) >= (pp `2 ) by A28, A32, A43, SEQ_4:def 1;

          then

           A44: (pp `2 ) = (q `2 ) by A42, XXREAL_0: 1;

          (pp `1 ) = (q `1 ) by A24, A25, A31, A32, A41, GOBOARD7: 5;

          hence thesis by A44, TOPREAL3: 6;

        end;

        assume

         A45: x = q;

        then x in ( LSeg (p,q)) by RLTOPSP1: 68;

        hence thesis by A36, A45, XBOOLE_0:def 4;

      end;

      hence (( LSeg ((( Gauge (C,n)) * (i,j1)),(( Gauge (C,n)) * (i,k1)))) /\ ( L~ ( Lower_Seq (C,n)))) = {(( Gauge (C,n)) * (i,j1))} by A25, A32, TARSKI:def 1;

      set X = (( LSeg ((G * (i,j)),(G * (i,k)))) /\ ( L~ ( Upper_Seq (C,n))));

      reconsider X1 = X as non empty compact Subset of ( TOP-REAL 2) by A7, A10, XBOOLE_0:def 4;

      consider pp be object such that

       A46: pp in ( S-most X1) by XBOOLE_0:def 1;

      reconsider pp as Point of ( TOP-REAL 2) by A46;

      

       A47: pp in X by A46, XBOOLE_0:def 4;

      then

       A48: pp in ( L~ ( Upper_Seq (C,n))) by XBOOLE_0:def 4;

      (f `1 ) = s by A1, A2, A3, A8, GOBOARD5: 2

      .= (e `1 ) by A1, A2, A5, A12, GOBOARD5: 2;

      then

       A49: ( LSeg (f,e)) is vertical by SPPOL_1: 16;

      pp in ( LSeg ((G * (i,j)),(G * (i,k)))) by A47, XBOOLE_0:def 4;

      then

       A50: (pp `1 ) = (p `1 ) by A35, A49, SPPOL_1: 41;

      (p `2 ) = ( S-bound X) by A15, A25, SPRECT_1: 44

      .= (( S-min X) `2 ) by EUCLID: 52

      .= (pp `2 ) by A46, PSCOMP_1: 55;

      then

       A51: p in ( L~ ( Upper_Seq (C,n))) by A48, A50, TOPREAL3: 6;

      for x be object holds x in (( LSeg (p,q)) /\ ( L~ ( Upper_Seq (C,n)))) iff x = p

      proof

        let x be object;

        thus x in (( LSeg (p,q)) /\ ( L~ ( Upper_Seq (C,n)))) implies x = p

        proof

          

           A52: (p `2 ) <= (e `2 ) by A1, A2, A5, A14, A23, A25, SPRECT_3: 12;

          

           A53: (f `2 ) <= (p `2 ) by A1, A2, A3, A13, A16, A25, SPRECT_3: 12;

          

           A54: (e `1 ) = (p `1 ) by A1, A2, A5, A12, A24, A25, GOBOARD5: 2;

          (f `1 ) = (p `1 ) by A1, A2, A3, A8, A24, A25, GOBOARD5: 2;

          then

           A55: p in ( LSeg (f,e)) by A54, A53, A52, GOBOARD7: 7;

          

           A56: (e `1 ) = (q `1 ) by A1, A2, A5, A12, A31, A32, GOBOARD5: 2;

          j1 <= k by A14, A27, XXREAL_0: 2;

          then

           A57: (q `2 ) <= (e `2 ) by A1, A2, A5, A30, A32, SPRECT_3: 12;

          

           A58: (f `2 ) <= (q `2 ) by A1, A2, A3, A26, A29, A32, SPRECT_3: 12;

          (f `1 ) = (q `1 ) by A1, A2, A3, A8, A31, A32, GOBOARD5: 2;

          then q in ( LSeg (f,e)) by A56, A58, A57, GOBOARD7: 7;

          then

           A59: ( LSeg (p,q)) c= ( LSeg (f,e)) by A55, TOPREAL1: 6;

          reconsider EE = (( LSeg (f,e)) /\ ( L~ ( Upper_Seq (C,n)))) as compact Subset of ( TOP-REAL 2);

          reconsider E0 = ( proj2 .: EE) as compact Subset of REAL by JCT_MISC: 15;

          assume

           A60: x in (( LSeg (p,q)) /\ ( L~ ( Upper_Seq (C,n))));

          then

          reconsider pp = x as Point of ( TOP-REAL 2);

          

           A61: pp in ( LSeg (p,q)) by A60, XBOOLE_0:def 4;

          then

           A62: (pp `2 ) <= (p `2 ) by A33, TOPREAL1: 4;

          pp in ( L~ ( Upper_Seq (C,n))) by A60, XBOOLE_0:def 4;

          then pp in EE by A61, A59, XBOOLE_0:def 4;

          then ( proj2 . pp) in E0 by FUNCT_2: 35;

          then

           A63: (pp `2 ) in E0 by PSCOMP_1:def 6;

          E0 is real-bounded by RCOMP_1: 10;

          then E0 is bounded_below by XXREAL_2:def 11;

          then (p `2 ) <= (pp `2 ) by A15, A25, A63, SEQ_4:def 2;

          then

           A64: (pp `2 ) = (p `2 ) by A62, XXREAL_0: 1;

          (pp `1 ) = (p `1 ) by A24, A25, A31, A32, A61, GOBOARD7: 5;

          hence thesis by A64, TOPREAL3: 6;

        end;

        assume

         A65: x = p;

        then x in ( LSeg (p,q)) by RLTOPSP1: 68;

        hence thesis by A51, A65, XBOOLE_0:def 4;

      end;

      hence thesis by A25, A32, TARSKI:def 1;

    end;

    theorem :: JORDAN15:12

    for C be compact connected non vertical non horizontal Subset of ( TOP-REAL 2) holds for i,j,k be Nat st 1 <= j & j <= k & k <= ( len ( Gauge (C,n))) & 1 <= i & i <= ( width ( Gauge (C,n))) & (( Gauge (C,n)) * (j,i)) in ( L~ ( Lower_Seq (C,n))) holds ex j1 be Nat st j <= j1 & j1 <= k & (( LSeg ((( Gauge (C,n)) * (j1,i)),(( Gauge (C,n)) * (k,i)))) /\ ( L~ ( Lower_Seq (C,n)))) = {(( Gauge (C,n)) * (j1,i))}

    proof

      let C be compact connected non vertical non horizontal Subset of ( TOP-REAL 2);

      let i,j,k be Nat;

      assume that

       A1: 1 <= j and

       A2: j <= k and

       A3: k <= ( len ( Gauge (C,n))) and

       A4: 1 <= i and

       A5: i <= ( width ( Gauge (C,n))) and

       A6: (( Gauge (C,n)) * (j,i)) in ( L~ ( Lower_Seq (C,n)));

      set G = ( Gauge (C,n));

      

       A7: k >= 1 by A1, A2, XXREAL_0: 2;

      then

       A8: [k, i] in ( Indices G) by A3, A4, A5, MATRIX_0: 30;

      set X = (( LSeg ((G * (j,i)),(G * (k,i)))) /\ ( L~ ( Lower_Seq (C,n))));

      

       A9: (G * (j,i)) in ( LSeg ((G * (j,i)),(G * (k,i)))) by RLTOPSP1: 68;

      then

      reconsider X1 = X as non empty compact Subset of ( TOP-REAL 2) by A6, XBOOLE_0:def 4;

      

       A10: ( LSeg ((G * (j,i)),(G * (k,i)))) meets ( L~ ( Lower_Seq (C,n))) by A6, A9, XBOOLE_0: 3;

      set s = ((G * (1,i)) `2 );

      set e = (G * (k,i));

      set f = (G * (j,i));

      set w2 = ( upper_bound ( proj1 .: (( LSeg (f,e)) /\ ( L~ ( Lower_Seq (C,n))))));

      

       A11: ( len G) = ( width G) by JORDAN8:def 1;

      then

       A12: j <= ( width G) by A2, A3, XXREAL_0: 2;

      then [j, i] in ( Indices G) by A1, A4, A5, A11, MATRIX_0: 30;

      then

      consider j1 be Nat such that

       A13: j <= j1 and

       A14: j1 <= k and

       A15: ((G * (j1,i)) `1 ) = w2 by A2, A10, A8, JORDAN1F: 4, JORDAN1G: 5;

      set q = |[w2, s]|;

      

       A16: 1 <= j1 by A1, A13, XXREAL_0: 2;

      take j1;

      thus j <= j1 & j1 <= k by A13, A14;

      consider pp be object such that

       A17: pp in ( E-most X1) by XBOOLE_0:def 1;

      reconsider pp as Point of ( TOP-REAL 2) by A17;

      

       A18: pp in X by A17, XBOOLE_0:def 4;

      then

       A19: pp in ( L~ ( Lower_Seq (C,n))) by XBOOLE_0:def 4;

      

       A20: j1 <= ( width G) by A3, A11, A14, XXREAL_0: 2;

      then

       A21: ((G * (j1,i)) `2 ) = s by A4, A5, A11, A16, GOBOARD5: 1;

      then

       A22: q = (G * (j1,i)) by A15, EUCLID: 53;

      then

       A23: (q `1 ) <= (e `1 ) by A3, A4, A5, A14, A16, SPRECT_3: 13;

      

       A24: ((G * (k,i)) `2 ) = s by A3, A4, A5, A7, GOBOARD5: 1;

      then (f `2 ) = (e `2 ) by A1, A4, A5, A11, A12, GOBOARD5: 1;

      then

       A25: ( LSeg (f,e)) is horizontal by SPPOL_1: 15;

      

       A26: (q `1 ) = ( E-bound X) by A15, A22, SPRECT_1: 46

      .= (( E-min X) `1 ) by EUCLID: 52

      .= (pp `1 ) by A17, PSCOMP_1: 47;

      pp in ( LSeg ((G * (j,i)),(G * (k,i)))) by A18, XBOOLE_0:def 4;

      then (pp `2 ) = (q `2 ) by A24, A21, A22, A25, SPPOL_1: 40;

      then

       A27: q in ( L~ ( Lower_Seq (C,n))) by A19, A26, TOPREAL3: 6;

      for x be object holds x in (( LSeg (e,q)) /\ ( L~ ( Lower_Seq (C,n)))) iff x = q

      proof

        let x be object;

        thus x in (( LSeg (e,q)) /\ ( L~ ( Lower_Seq (C,n)))) implies x = q

        proof

          

           A28: (f `1 ) <= (q `1 ) by A1, A4, A5, A11, A13, A20, A22, SPRECT_3: 13;

          (f `2 ) = (q `2 ) by A1, A4, A5, A11, A12, A21, A22, GOBOARD5: 1;

          then

           A29: q in ( LSeg (e,f)) by A24, A21, A22, A23, A28, GOBOARD7: 8;

          e in ( LSeg (f,e)) by RLTOPSP1: 68;

          then

           A30: ( LSeg (e,q)) c= ( LSeg (f,e)) by A29, TOPREAL1: 6;

          reconsider EE = (( LSeg (f,e)) /\ ( L~ ( Lower_Seq (C,n)))) as compact Subset of ( TOP-REAL 2);

          reconsider E0 = ( proj1 .: EE) as compact Subset of REAL by Th4;

          assume

           A31: x in (( LSeg (e,q)) /\ ( L~ ( Lower_Seq (C,n))));

          then

          reconsider pp = x as Point of ( TOP-REAL 2);

          

           A32: pp in ( LSeg (e,q)) by A31, XBOOLE_0:def 4;

          then

           A33: (pp `1 ) >= (q `1 ) by A23, TOPREAL1: 3;

          pp in ( L~ ( Lower_Seq (C,n))) by A31, XBOOLE_0:def 4;

          then pp in EE by A32, A30, XBOOLE_0:def 4;

          then ( proj1 . pp) in E0 by FUNCT_2: 35;

          then

           A34: (pp `1 ) in E0 by PSCOMP_1:def 5;

          E0 is real-bounded by RCOMP_1: 10;

          then E0 is bounded_above by XXREAL_2:def 11;

          then (q `1 ) >= (pp `1 ) by A15, A22, A34, SEQ_4:def 1;

          then

           A35: (pp `1 ) = (q `1 ) by A33, XXREAL_0: 1;

          (pp `2 ) = (q `2 ) by A24, A21, A22, A32, GOBOARD7: 6;

          hence thesis by A35, TOPREAL3: 6;

        end;

        assume

         A36: x = q;

        then x in ( LSeg (e,q)) by RLTOPSP1: 68;

        hence thesis by A27, A36, XBOOLE_0:def 4;

      end;

      hence thesis by A22, TARSKI:def 1;

    end;

    theorem :: JORDAN15:13

    

     Th13: for C be compact connected non vertical non horizontal Subset of ( TOP-REAL 2) holds for i,j,k be Nat st 1 <= j & j <= k & k <= ( len ( Gauge (C,n))) & 1 <= i & i <= ( width ( Gauge (C,n))) & (( Gauge (C,n)) * (k,i)) in ( L~ ( Upper_Seq (C,n))) holds ex k1 be Nat st j <= k1 & k1 <= k & (( LSeg ((( Gauge (C,n)) * (j,i)),(( Gauge (C,n)) * (k1,i)))) /\ ( L~ ( Upper_Seq (C,n)))) = {(( Gauge (C,n)) * (k1,i))}

    proof

      let C be compact connected non vertical non horizontal Subset of ( TOP-REAL 2);

      let i,j,k be Nat;

      assume that

       A1: 1 <= j and

       A2: j <= k and

       A3: k <= ( len ( Gauge (C,n))) and

       A4: 1 <= i and

       A5: i <= ( width ( Gauge (C,n))) and

       A6: (( Gauge (C,n)) * (k,i)) in ( L~ ( Upper_Seq (C,n)));

      set G = ( Gauge (C,n));

      

       A7: k >= 1 by A1, A2, XXREAL_0: 2;

      then

       A8: [k, i] in ( Indices G) by A3, A4, A5, MATRIX_0: 30;

      set X = (( LSeg ((G * (j,i)),(G * (k,i)))) /\ ( L~ ( Upper_Seq (C,n))));

      

       A9: (G * (k,i)) in ( LSeg ((G * (j,i)),(G * (k,i)))) by RLTOPSP1: 68;

      then

      reconsider X1 = X as non empty compact Subset of ( TOP-REAL 2) by A6, XBOOLE_0:def 4;

      

       A10: ( LSeg ((G * (j,i)),(G * (k,i)))) meets ( L~ ( Upper_Seq (C,n))) by A6, A9, XBOOLE_0: 3;

      set s = ((G * (1,i)) `2 );

      set e = (G * (k,i));

      set f = (G * (j,i));

      set w1 = ( lower_bound ( proj1 .: (( LSeg (f,e)) /\ ( L~ ( Upper_Seq (C,n))))));

      

       A11: ( len G) = ( width G) by JORDAN8:def 1;

      then

       A12: j <= ( width G) by A2, A3, XXREAL_0: 2;

      then [j, i] in ( Indices G) by A1, A4, A5, A11, MATRIX_0: 30;

      then

      consider k1 be Nat such that

       A13: j <= k1 and

       A14: k1 <= k and

       A15: ((G * (k1,i)) `1 ) = w1 by A2, A10, A8, JORDAN1F: 3, JORDAN1G: 4;

      set p = |[w1, s]|;

      

       A16: k1 <= ( width G) by A3, A11, A14, XXREAL_0: 2;

      (f `2 ) = s by A1, A4, A5, A11, A12, GOBOARD5: 1

      .= (e `2 ) by A3, A4, A5, A7, GOBOARD5: 1;

      then

       A17: ( LSeg (f,e)) is horizontal by SPPOL_1: 15;

      take k1;

      thus j <= k1 & k1 <= k by A13, A14;

      consider pp be object such that

       A18: pp in ( W-most X1) by XBOOLE_0:def 1;

      

       A19: 1 <= k1 by A1, A13, XXREAL_0: 2;

      then

       A20: ((G * (k1,i)) `2 ) = s by A4, A5, A11, A16, GOBOARD5: 1;

      then

       A21: p = (G * (k1,i)) by A15, EUCLID: 53;

      then

       A22: (f `1 ) <= (p `1 ) by A1, A4, A5, A11, A13, A16, SPRECT_3: 13;

      

       A23: (f `2 ) = (p `2 ) by A1, A4, A5, A11, A12, A20, A21, GOBOARD5: 1;

      reconsider pp as Point of ( TOP-REAL 2) by A18;

      

       A24: pp in X by A18, XBOOLE_0:def 4;

      then

       A25: pp in ( L~ ( Upper_Seq (C,n))) by XBOOLE_0:def 4;

      

       A26: (p `1 ) = ( W-bound X) by A15, A21, SPRECT_1: 43

      .= (( W-min X) `1 ) by EUCLID: 52

      .= (pp `1 ) by A18, PSCOMP_1: 31;

      pp in ( LSeg ((G * (j,i)),(G * (k,i)))) by A24, XBOOLE_0:def 4;

      then (pp `2 ) = (p `2 ) by A23, A17, SPPOL_1: 40;

      then

       A27: p in ( L~ ( Upper_Seq (C,n))) by A25, A26, TOPREAL3: 6;

      for x be object holds x in (( LSeg (p,f)) /\ ( L~ ( Upper_Seq (C,n)))) iff x = p

      proof

        let x be object;

        thus x in (( LSeg (p,f)) /\ ( L~ ( Upper_Seq (C,n)))) implies x = p

        proof

          reconsider EE = (( LSeg (f,e)) /\ ( L~ ( Upper_Seq (C,n)))) as compact Subset of ( TOP-REAL 2);

          assume

           A28: x in (( LSeg (p,f)) /\ ( L~ ( Upper_Seq (C,n))));

          then

          reconsider pp = x as Point of ( TOP-REAL 2);

          

           A29: pp in ( LSeg (p,f)) by A28, XBOOLE_0:def 4;

          then

           A30: (pp `1 ) <= (p `1 ) by A22, TOPREAL1: 3;

          

           A31: (p `1 ) <= (e `1 ) by A3, A4, A5, A14, A19, A21, SPRECT_3: 13;

          

           A32: (f `1 ) <= (p `1 ) by A1, A4, A5, A11, A13, A16, A21, SPRECT_3: 13;

          

           A33: (e `2 ) = (p `2 ) by A3, A4, A5, A7, A20, A21, GOBOARD5: 1;

          reconsider E0 = ( proj1 .: EE) as compact Subset of REAL by Th4;

          

           A34: f in ( LSeg (f,e)) by RLTOPSP1: 68;

          (f `2 ) = (p `2 ) by A1, A4, A5, A11, A12, A20, A21, GOBOARD5: 1;

          then p in ( LSeg (f,e)) by A33, A32, A31, GOBOARD7: 8;

          then

           A35: ( LSeg (p,f)) c= ( LSeg (f,e)) by A34, TOPREAL1: 6;

          pp in ( L~ ( Upper_Seq (C,n))) by A28, XBOOLE_0:def 4;

          then pp in EE by A29, A35, XBOOLE_0:def 4;

          then ( proj1 . pp) in E0 by FUNCT_2: 35;

          then

           A36: (pp `1 ) in E0 by PSCOMP_1:def 5;

          E0 is real-bounded by RCOMP_1: 10;

          then E0 is bounded_below by XXREAL_2:def 11;

          then (p `1 ) <= (pp `1 ) by A15, A21, A36, SEQ_4:def 2;

          then

           A37: (pp `1 ) = (p `1 ) by A30, XXREAL_0: 1;

          (pp `2 ) = (p `2 ) by A23, A29, GOBOARD7: 6;

          hence thesis by A37, TOPREAL3: 6;

        end;

        assume

         A38: x = p;

        then x in ( LSeg (p,f)) by RLTOPSP1: 68;

        hence thesis by A27, A38, XBOOLE_0:def 4;

      end;

      hence thesis by A21, TARSKI:def 1;

    end;

    theorem :: JORDAN15:14

    

     Th14: for C be compact connected non vertical non horizontal Subset of ( TOP-REAL 2) holds for i,j,k be Nat st 1 <= j & j <= k & k <= ( len ( Gauge (C,n))) & 1 <= i & i <= ( width ( Gauge (C,n))) & (( Gauge (C,n)) * (j,i)) in ( L~ ( Lower_Seq (C,n))) & (( Gauge (C,n)) * (k,i)) in ( L~ ( Upper_Seq (C,n))) holds ex j1,k1 be Nat st j <= j1 & j1 <= k1 & k1 <= k & (( LSeg ((( Gauge (C,n)) * (j1,i)),(( Gauge (C,n)) * (k1,i)))) /\ ( L~ ( Lower_Seq (C,n)))) = {(( Gauge (C,n)) * (j1,i))} & (( LSeg ((( Gauge (C,n)) * (j1,i)),(( Gauge (C,n)) * (k1,i)))) /\ ( L~ ( Upper_Seq (C,n)))) = {(( Gauge (C,n)) * (k1,i))}

    proof

      let C be compact connected non vertical non horizontal Subset of ( TOP-REAL 2);

      let i,j,k be Nat;

      assume that

       A1: 1 <= j and

       A2: j <= k and

       A3: k <= ( len ( Gauge (C,n))) and

       A4: 1 <= i and

       A5: i <= ( width ( Gauge (C,n))) and

       A6: (( Gauge (C,n)) * (j,i)) in ( L~ ( Lower_Seq (C,n))) and

       A7: (( Gauge (C,n)) * (k,i)) in ( L~ ( Upper_Seq (C,n)));

      set G = ( Gauge (C,n));

      

       A8: ( len G) = ( width G) by JORDAN8:def 1;

      then

       A9: j <= ( width G) by A2, A3, XXREAL_0: 2;

      then

       A10: [j, i] in ( Indices G) by A1, A4, A5, A8, MATRIX_0: 30;

      set s = ((G * (1,i)) `2 );

      set e = (G * (k,i));

      set f = (G * (j,i));

      set w1 = ( lower_bound ( proj1 .: (( LSeg (f,e)) /\ ( L~ ( Upper_Seq (C,n))))));

      

       A11: (G * (k,i)) in ( LSeg ((G * (j,i)),(G * (k,i)))) by RLTOPSP1: 68;

      then

       A12: ( LSeg ((G * (j,i)),(G * (k,i)))) meets ( L~ ( Upper_Seq (C,n))) by A7, XBOOLE_0: 3;

      

       A13: k >= 1 by A1, A2, XXREAL_0: 2;

      then [k, i] in ( Indices G) by A3, A4, A5, MATRIX_0: 30;

      then

      consider k1 be Nat such that

       A14: j <= k1 and

       A15: k1 <= k and

       A16: ((G * (k1,i)) `1 ) = w1 by A2, A12, A10, JORDAN1F: 3, JORDAN1G: 4;

      

       A17: k1 <= ( width G) by A3, A8, A15, XXREAL_0: 2;

      set p = |[w1, s]|;

      set w2 = ( upper_bound ( proj1 .: (( LSeg (f,p)) /\ ( L~ ( Lower_Seq (C,n))))));

      set q = |[w2, s]|;

      

       A18: (G * (j,i)) in ( LSeg ((G * (j,i)),(G * (k1,i)))) by RLTOPSP1: 68;

      then

       A19: ( LSeg ((G * (j,i)),(G * (k1,i)))) meets ( L~ ( Lower_Seq (C,n))) by A6, XBOOLE_0: 3;

      

       A20: 1 <= k1 by A1, A14, XXREAL_0: 2;

      then

       A21: ((G * (k1,i)) `2 ) = s by A4, A5, A8, A17, GOBOARD5: 1;

      then

       A22: p = (G * (k1,i)) by A16, EUCLID: 53;

      (f `2 ) = s by A1, A4, A5, A8, A9, GOBOARD5: 1

      .= (e `2 ) by A3, A4, A5, A13, GOBOARD5: 1;

      then

       A23: ( LSeg (f,e)) is horizontal by SPPOL_1: 15;

      set X = (( LSeg ((G * (j,i)),(G * (k1,i)))) /\ ( L~ ( Lower_Seq (C,n))));

      reconsider X1 = X as non empty compact Subset of ( TOP-REAL 2) by A6, A18, XBOOLE_0:def 4;

      consider pp be object such that

       A24: pp in ( E-most X1) by XBOOLE_0:def 1;

       [k1, i] in ( Indices G) by A4, A5, A8, A20, A17, MATRIX_0: 30;

      then

      consider j1 be Nat such that

       A25: j <= j1 and

       A26: j1 <= k1 and

       A27: ((G * (j1,i)) `1 ) = w2 by A10, A14, A22, A19, JORDAN1F: 4, JORDAN1G: 5;

      

       A28: j1 <= ( width G) by A17, A26, XXREAL_0: 2;

      reconsider pp as Point of ( TOP-REAL 2) by A24;

      

       A29: pp in X by A24, XBOOLE_0:def 4;

      then

       A30: pp in ( L~ ( Lower_Seq (C,n))) by XBOOLE_0:def 4;

      take j1, k1;

      thus j <= j1 & j1 <= k1 & k1 <= k by A15, A25, A26;

      

       A31: pp in ( LSeg ((G * (j,i)),(G * (k1,i)))) by A29, XBOOLE_0:def 4;

      

       A32: 1 <= j1 by A1, A25, XXREAL_0: 2;

      then

       A33: ((G * (j1,i)) `2 ) = s by A4, A5, A8, A28, GOBOARD5: 1;

      then

       A34: q = (G * (j1,i)) by A27, EUCLID: 53;

      then

       A35: (q `1 ) <= (p `1 ) by A4, A5, A8, A17, A22, A26, A32, SPRECT_3: 13;

      

       A36: (q `1 ) = ( E-bound X) by A22, A27, A34, SPRECT_1: 46

      .= (( E-min X) `1 ) by EUCLID: 52

      .= (pp `1 ) by A24, PSCOMP_1: 47;

      

       A37: (f `2 ) = (p `2 ) by A1, A4, A5, A8, A9, A21, A22, GOBOARD5: 1;

      then ( LSeg (f,p)) is horizontal by SPPOL_1: 15;

      then (pp `2 ) = (q `2 ) by A21, A22, A33, A34, A31, SPPOL_1: 40;

      then

       A38: q in ( L~ ( Lower_Seq (C,n))) by A30, A36, TOPREAL3: 6;

      for x be object holds x in (( LSeg (p,q)) /\ ( L~ ( Lower_Seq (C,n)))) iff x = q

      proof

        let x be object;

        thus x in (( LSeg (p,q)) /\ ( L~ ( Lower_Seq (C,n)))) implies x = q

        proof

          reconsider EE = (( LSeg (f,p)) /\ ( L~ ( Lower_Seq (C,n)))) as compact Subset of ( TOP-REAL 2);

          assume

           A39: x in (( LSeg (p,q)) /\ ( L~ ( Lower_Seq (C,n))));

          then

          reconsider pp = x as Point of ( TOP-REAL 2);

          

           A40: pp in ( LSeg (p,q)) by A39, XBOOLE_0:def 4;

          then

           A41: (pp `1 ) >= (q `1 ) by A35, TOPREAL1: 3;

          

           A42: (f `1 ) <= (q `1 ) by A1, A4, A5, A8, A25, A28, A34, SPRECT_3: 13;

          reconsider E0 = ( proj1 .: EE) as compact Subset of REAL by Th4;

          

           A43: p in ( LSeg (f,p)) by RLTOPSP1: 68;

          (f `2 ) = (q `2 ) by A1, A4, A5, A8, A9, A33, A34, GOBOARD5: 1;

          then q in ( LSeg (p,f)) by A21, A22, A33, A34, A35, A42, GOBOARD7: 8;

          then

           A44: ( LSeg (p,q)) c= ( LSeg (f,p)) by A43, TOPREAL1: 6;

          pp in ( L~ ( Lower_Seq (C,n))) by A39, XBOOLE_0:def 4;

          then pp in EE by A40, A44, XBOOLE_0:def 4;

          then ( proj1 . pp) in E0 by FUNCT_2: 35;

          then

           A45: (pp `1 ) in E0 by PSCOMP_1:def 5;

          E0 is real-bounded by RCOMP_1: 10;

          then E0 is bounded_above by XXREAL_2:def 11;

          then (q `1 ) >= (pp `1 ) by A27, A34, A45, SEQ_4:def 1;

          then

           A46: (pp `1 ) = (q `1 ) by A41, XXREAL_0: 1;

          (pp `2 ) = (q `2 ) by A21, A22, A33, A34, A40, GOBOARD7: 6;

          hence thesis by A46, TOPREAL3: 6;

        end;

        assume

         A47: x = q;

        then x in ( LSeg (p,q)) by RLTOPSP1: 68;

        hence thesis by A38, A47, XBOOLE_0:def 4;

      end;

      hence (( LSeg ((( Gauge (C,n)) * (j1,i)),(( Gauge (C,n)) * (k1,i)))) /\ ( L~ ( Lower_Seq (C,n)))) = {(( Gauge (C,n)) * (j1,i))} by A22, A34, TARSKI:def 1;

      set X = (( LSeg ((G * (j,i)),(G * (k,i)))) /\ ( L~ ( Upper_Seq (C,n))));

      reconsider X1 = X as non empty compact Subset of ( TOP-REAL 2) by A7, A11, XBOOLE_0:def 4;

      consider pp be object such that

       A48: pp in ( W-most X1) by XBOOLE_0:def 1;

      reconsider pp as Point of ( TOP-REAL 2) by A48;

      

       A49: pp in X by A48, XBOOLE_0:def 4;

      then

       A50: pp in ( L~ ( Upper_Seq (C,n))) by XBOOLE_0:def 4;

      pp in ( LSeg ((G * (j,i)),(G * (k,i)))) by A49, XBOOLE_0:def 4;

      then

       A51: (pp `2 ) = (p `2 ) by A37, A23, SPPOL_1: 40;

      (p `1 ) = ( W-bound X) by A16, A22, SPRECT_1: 43

      .= (( W-min X) `1 ) by EUCLID: 52

      .= (pp `1 ) by A48, PSCOMP_1: 31;

      then

       A52: p in ( L~ ( Upper_Seq (C,n))) by A50, A51, TOPREAL3: 6;

      for x be object holds x in (( LSeg (p,q)) /\ ( L~ ( Upper_Seq (C,n)))) iff x = p

      proof

        let x be object;

        thus x in (( LSeg (p,q)) /\ ( L~ ( Upper_Seq (C,n)))) implies x = p

        proof

          j1 <= k by A15, A26, XXREAL_0: 2;

          then

           A53: (q `1 ) <= (e `1 ) by A3, A4, A5, A32, A34, SPRECT_3: 13;

          

           A54: (e `2 ) = (p `2 ) by A3, A4, A5, A13, A21, A22, GOBOARD5: 1;

          

           A55: (f `1 ) <= (p `1 ) by A1, A4, A5, A8, A14, A17, A22, SPRECT_3: 13;

          

           A56: (f `1 ) <= (q `1 ) by A1, A4, A5, A8, A25, A28, A34, SPRECT_3: 13;

          

           A57: (p `1 ) <= (e `1 ) by A3, A4, A5, A15, A20, A22, SPRECT_3: 13;

          (f `2 ) = (p `2 ) by A1, A4, A5, A8, A9, A21, A22, GOBOARD5: 1;

          then

           A58: p in ( LSeg (f,e)) by A54, A55, A57, GOBOARD7: 8;

          

           A59: (e `2 ) = (q `2 ) by A3, A4, A5, A13, A33, A34, GOBOARD5: 1;

          (f `2 ) = (q `2 ) by A1, A4, A5, A8, A9, A33, A34, GOBOARD5: 1;

          then q in ( LSeg (f,e)) by A59, A56, A53, GOBOARD7: 8;

          then

           A60: ( LSeg (p,q)) c= ( LSeg (f,e)) by A58, TOPREAL1: 6;

          reconsider EE = (( LSeg (f,e)) /\ ( L~ ( Upper_Seq (C,n)))) as compact Subset of ( TOP-REAL 2);

          reconsider E0 = ( proj1 .: EE) as compact Subset of REAL by Th4;

          assume

           A61: x in (( LSeg (p,q)) /\ ( L~ ( Upper_Seq (C,n))));

          then

          reconsider pp = x as Point of ( TOP-REAL 2);

          

           A62: pp in ( LSeg (p,q)) by A61, XBOOLE_0:def 4;

          then

           A63: (pp `1 ) <= (p `1 ) by A35, TOPREAL1: 3;

          pp in ( L~ ( Upper_Seq (C,n))) by A61, XBOOLE_0:def 4;

          then pp in EE by A62, A60, XBOOLE_0:def 4;

          then ( proj1 . pp) in E0 by FUNCT_2: 35;

          then

           A64: (pp `1 ) in E0 by PSCOMP_1:def 5;

          E0 is real-bounded by RCOMP_1: 10;

          then E0 is bounded_below by XXREAL_2:def 11;

          then (p `1 ) <= (pp `1 ) by A16, A22, A64, SEQ_4:def 2;

          then

           A65: (pp `1 ) = (p `1 ) by A63, XXREAL_0: 1;

          (pp `2 ) = (p `2 ) by A21, A22, A33, A34, A62, GOBOARD7: 6;

          hence thesis by A65, TOPREAL3: 6;

        end;

        assume

         A66: x = p;

        then x in ( LSeg (p,q)) by RLTOPSP1: 68;

        hence thesis by A52, A66, XBOOLE_0:def 4;

      end;

      hence thesis by A22, A34, TARSKI:def 1;

    end;

    theorem :: JORDAN15:15

    for C be compact connected non vertical non horizontal Subset of ( TOP-REAL 2) holds for i,j,k be Nat st 1 <= i & i <= ( len ( Gauge (C,n))) & 1 <= j & j <= k & k <= ( width ( Gauge (C,n))) & (( Gauge (C,n)) * (i,j)) in ( L~ ( Upper_Seq (C,n))) holds ex j1 be Nat st j <= j1 & j1 <= k & (( LSeg ((( Gauge (C,n)) * (i,j1)),(( Gauge (C,n)) * (i,k)))) /\ ( L~ ( Upper_Seq (C,n)))) = {(( Gauge (C,n)) * (i,j1))}

    proof

      let C be compact connected non vertical non horizontal Subset of ( TOP-REAL 2);

      let i,j,k be Nat;

      assume that

       A1: 1 <= i and

       A2: i <= ( len ( Gauge (C,n))) and

       A3: 1 <= j and

       A4: j <= k and

       A5: k <= ( width ( Gauge (C,n))) and

       A6: (( Gauge (C,n)) * (i,j)) in ( L~ ( Upper_Seq (C,n)));

      set G = ( Gauge (C,n));

      

       A7: k >= 1 by A3, A4, XXREAL_0: 2;

      then

       A8: [i, k] in ( Indices G) by A1, A2, A5, MATRIX_0: 30;

      set X = (( LSeg ((G * (i,j)),(G * (i,k)))) /\ ( L~ ( Upper_Seq (C,n))));

      

       A9: (G * (i,j)) in ( LSeg ((G * (i,j)),(G * (i,k)))) by RLTOPSP1: 68;

      then

      reconsider X1 = X as non empty compact Subset of ( TOP-REAL 2) by A6, XBOOLE_0:def 4;

      

       A10: ( LSeg ((G * (i,j)),(G * (i,k)))) meets ( L~ ( Upper_Seq (C,n))) by A6, A9, XBOOLE_0: 3;

      set s = ((G * (i,1)) `1 );

      set e = (G * (i,k));

      set f = (G * (i,j));

      set w2 = ( upper_bound ( proj2 .: (( LSeg (f,e)) /\ ( L~ ( Upper_Seq (C,n))))));

      

       A11: j <= ( width G) by A4, A5, XXREAL_0: 2;

      then [i, j] in ( Indices G) by A1, A2, A3, MATRIX_0: 30;

      then

      consider j1 be Nat such that

       A12: j <= j1 and

       A13: j1 <= k and

       A14: ((G * (i,j1)) `2 ) = w2 by A4, A10, A8, JORDAN1F: 2, JORDAN1G: 4;

      set q = |[s, w2]|;

      

       A15: j1 <= ( width G) by A5, A13, XXREAL_0: 2;

      

       A16: ((G * (i,k)) `1 ) = s by A1, A2, A5, A7, GOBOARD5: 2;

      then (f `1 ) = (e `1 ) by A1, A2, A3, A11, GOBOARD5: 2;

      then

       A17: ( LSeg (f,e)) is vertical by SPPOL_1: 16;

      take j1;

      thus j <= j1 & j1 <= k by A12, A13;

      consider pp be object such that

       A18: pp in ( N-most X1) by XBOOLE_0:def 1;

      reconsider pp as Point of ( TOP-REAL 2) by A18;

      

       A19: pp in X by A18, XBOOLE_0:def 4;

      then

       A20: pp in ( L~ ( Upper_Seq (C,n))) by XBOOLE_0:def 4;

      

       A21: 1 <= j1 by A3, A12, XXREAL_0: 2;

      then

       A22: ((G * (i,j1)) `1 ) = s by A1, A2, A15, GOBOARD5: 2;

      then

       A23: q = (G * (i,j1)) by A14, EUCLID: 53;

      then

       A24: (q `2 ) <= (e `2 ) by A1, A2, A5, A13, A21, SPRECT_3: 12;

      

       A25: (q `2 ) = ( N-bound X) by A14, A23, SPRECT_1: 45

      .= (( N-min X) `2 ) by EUCLID: 52

      .= (pp `2 ) by A18, PSCOMP_1: 39;

      pp in ( LSeg ((G * (i,j)),(G * (i,k)))) by A19, XBOOLE_0:def 4;

      then (pp `1 ) = (q `1 ) by A16, A22, A23, A17, SPPOL_1: 41;

      then

       A26: q in ( L~ ( Upper_Seq (C,n))) by A20, A25, TOPREAL3: 6;

      for x be object holds x in (( LSeg (e,q)) /\ ( L~ ( Upper_Seq (C,n)))) iff x = q

      proof

        let x be object;

        thus x in (( LSeg (e,q)) /\ ( L~ ( Upper_Seq (C,n)))) implies x = q

        proof

          reconsider EE = (( LSeg (f,e)) /\ ( L~ ( Upper_Seq (C,n)))) as compact Subset of ( TOP-REAL 2);

          reconsider E0 = ( proj2 .: EE) as compact Subset of REAL by JCT_MISC: 15;

          

           A27: e in ( LSeg (f,e)) by RLTOPSP1: 68;

          

           A28: (f `2 ) <= (q `2 ) by A1, A2, A3, A12, A15, A23, SPRECT_3: 12;

          (f `1 ) = (q `1 ) by A1, A2, A3, A11, A22, A23, GOBOARD5: 2;

          then q in ( LSeg (e,f)) by A16, A22, A23, A24, A28, GOBOARD7: 7;

          then

           A29: ( LSeg (e,q)) c= ( LSeg (f,e)) by A27, TOPREAL1: 6;

          assume

           A30: x in (( LSeg (e,q)) /\ ( L~ ( Upper_Seq (C,n))));

          then

          reconsider pp = x as Point of ( TOP-REAL 2);

          

           A31: pp in ( LSeg (e,q)) by A30, XBOOLE_0:def 4;

          then

           A32: (pp `2 ) >= (q `2 ) by A24, TOPREAL1: 4;

          pp in ( L~ ( Upper_Seq (C,n))) by A30, XBOOLE_0:def 4;

          then pp in EE by A31, A29, XBOOLE_0:def 4;

          then ( proj2 . pp) in E0 by FUNCT_2: 35;

          then

           A33: (pp `2 ) in E0 by PSCOMP_1:def 6;

          E0 is real-bounded by RCOMP_1: 10;

          then E0 is bounded_above by XXREAL_2:def 11;

          then (q `2 ) >= (pp `2 ) by A14, A23, A33, SEQ_4:def 1;

          then

           A34: (pp `2 ) = (q `2 ) by A32, XXREAL_0: 1;

          (pp `1 ) = (q `1 ) by A16, A22, A23, A31, GOBOARD7: 5;

          hence thesis by A34, TOPREAL3: 6;

        end;

        assume

         A35: x = q;

        then x in ( LSeg (e,q)) by RLTOPSP1: 68;

        hence thesis by A26, A35, XBOOLE_0:def 4;

      end;

      hence thesis by A23, TARSKI:def 1;

    end;

    theorem :: JORDAN15:16

    for C be compact connected non vertical non horizontal Subset of ( TOP-REAL 2) holds for i,j,k be Nat st 1 <= i & i <= ( len ( Gauge (C,n))) & 1 <= j & j <= k & k <= ( width ( Gauge (C,n))) & (( Gauge (C,n)) * (i,k)) in ( L~ ( Lower_Seq (C,n))) holds ex k1 be Nat st j <= k1 & k1 <= k & (( LSeg ((( Gauge (C,n)) * (i,j)),(( Gauge (C,n)) * (i,k1)))) /\ ( L~ ( Lower_Seq (C,n)))) = {(( Gauge (C,n)) * (i,k1))}

    proof

      let C be compact connected non vertical non horizontal Subset of ( TOP-REAL 2);

      let i,j,k be Nat;

      assume that

       A1: 1 <= i and

       A2: i <= ( len ( Gauge (C,n))) and

       A3: 1 <= j and

       A4: j <= k and

       A5: k <= ( width ( Gauge (C,n))) and

       A6: (( Gauge (C,n)) * (i,k)) in ( L~ ( Lower_Seq (C,n)));

      set G = ( Gauge (C,n));

      

       A7: k >= 1 by A3, A4, XXREAL_0: 2;

      then

       A8: [i, k] in ( Indices G) by A1, A2, A5, MATRIX_0: 30;

      set X = (( LSeg ((G * (i,j)),(G * (i,k)))) /\ ( L~ ( Lower_Seq (C,n))));

      

       A9: (G * (i,k)) in ( LSeg ((G * (i,j)),(G * (i,k)))) by RLTOPSP1: 68;

      then

      reconsider X1 = X as non empty compact Subset of ( TOP-REAL 2) by A6, XBOOLE_0:def 4;

      

       A10: ( LSeg ((G * (i,j)),(G * (i,k)))) meets ( L~ ( Lower_Seq (C,n))) by A6, A9, XBOOLE_0: 3;

      set s = ((G * (i,1)) `1 );

      set e = (G * (i,k));

      set f = (G * (i,j));

      set w1 = ( lower_bound ( proj2 .: (( LSeg (f,e)) /\ ( L~ ( Lower_Seq (C,n))))));

      

       A11: j <= ( width G) by A4, A5, XXREAL_0: 2;

      then [i, j] in ( Indices G) by A1, A2, A3, MATRIX_0: 30;

      then

      consider k1 be Nat such that

       A12: j <= k1 and

       A13: k1 <= k and

       A14: ((G * (i,k1)) `2 ) = w1 by A4, A10, A8, JORDAN1F: 1, JORDAN1G: 5;

      set p = |[s, w1]|;

      

       A15: k1 <= ( width G) by A5, A13, XXREAL_0: 2;

      (f `1 ) = s by A1, A2, A3, A11, GOBOARD5: 2

      .= (e `1 ) by A1, A2, A5, A7, GOBOARD5: 2;

      then

       A16: ( LSeg (f,e)) is vertical by SPPOL_1: 16;

      take k1;

      thus j <= k1 & k1 <= k by A12, A13;

      consider pp be object such that

       A17: pp in ( S-most X1) by XBOOLE_0:def 1;

      

       A18: 1 <= k1 by A3, A12, XXREAL_0: 2;

      then

       A19: ((G * (i,k1)) `1 ) = s by A1, A2, A15, GOBOARD5: 2;

      then

       A20: p = (G * (i,k1)) by A14, EUCLID: 53;

      then

       A21: (f `2 ) <= (p `2 ) by A1, A2, A3, A12, A15, SPRECT_3: 12;

      

       A22: (f `1 ) = (p `1 ) by A1, A2, A3, A11, A19, A20, GOBOARD5: 2;

      reconsider pp as Point of ( TOP-REAL 2) by A17;

      

       A23: pp in X by A17, XBOOLE_0:def 4;

      then

       A24: pp in ( L~ ( Lower_Seq (C,n))) by XBOOLE_0:def 4;

      

       A25: (p `2 ) = ( S-bound X) by A14, A20, SPRECT_1: 44

      .= (( S-min X) `2 ) by EUCLID: 52

      .= (pp `2 ) by A17, PSCOMP_1: 55;

      pp in ( LSeg ((G * (i,j)),(G * (i,k)))) by A23, XBOOLE_0:def 4;

      then (pp `1 ) = (p `1 ) by A22, A16, SPPOL_1: 41;

      then

       A26: p in ( L~ ( Lower_Seq (C,n))) by A24, A25, TOPREAL3: 6;

      for x be object holds x in (( LSeg (p,f)) /\ ( L~ ( Lower_Seq (C,n)))) iff x = p

      proof

        let x be object;

        thus x in (( LSeg (p,f)) /\ ( L~ ( Lower_Seq (C,n)))) implies x = p

        proof

          reconsider EE = (( LSeg (f,e)) /\ ( L~ ( Lower_Seq (C,n)))) as compact Subset of ( TOP-REAL 2);

          reconsider E0 = ( proj2 .: EE) as compact Subset of REAL by JCT_MISC: 15;

          

           A27: f in ( LSeg (f,e)) by RLTOPSP1: 68;

          

           A28: (e `1 ) = (p `1 ) by A1, A2, A5, A7, A19, A20, GOBOARD5: 2;

          

           A29: (p `2 ) <= (e `2 ) by A1, A2, A5, A13, A18, A20, SPRECT_3: 12;

          

           A30: (f `2 ) <= (p `2 ) by A1, A2, A3, A12, A15, A20, SPRECT_3: 12;

          (f `1 ) = (p `1 ) by A1, A2, A3, A11, A19, A20, GOBOARD5: 2;

          then p in ( LSeg (f,e)) by A28, A30, A29, GOBOARD7: 7;

          then

           A31: ( LSeg (p,f)) c= ( LSeg (f,e)) by A27, TOPREAL1: 6;

          assume

           A32: x in (( LSeg (p,f)) /\ ( L~ ( Lower_Seq (C,n))));

          then

          reconsider pp = x as Point of ( TOP-REAL 2);

          

           A33: pp in ( LSeg (p,f)) by A32, XBOOLE_0:def 4;

          then

           A34: (pp `2 ) <= (p `2 ) by A21, TOPREAL1: 4;

          pp in ( L~ ( Lower_Seq (C,n))) by A32, XBOOLE_0:def 4;

          then pp in EE by A33, A31, XBOOLE_0:def 4;

          then ( proj2 . pp) in E0 by FUNCT_2: 35;

          then

           A35: (pp `2 ) in E0 by PSCOMP_1:def 6;

          E0 is real-bounded by RCOMP_1: 10;

          then E0 is bounded_below by XXREAL_2:def 11;

          then (p `2 ) <= (pp `2 ) by A14, A20, A35, SEQ_4:def 2;

          then

           A36: (pp `2 ) = (p `2 ) by A34, XXREAL_0: 1;

          (pp `1 ) = (p `1 ) by A22, A33, GOBOARD7: 5;

          hence thesis by A36, TOPREAL3: 6;

        end;

        assume

         A37: x = p;

        then x in ( LSeg (p,f)) by RLTOPSP1: 68;

        hence thesis by A26, A37, XBOOLE_0:def 4;

      end;

      hence thesis by A20, TARSKI:def 1;

    end;

    theorem :: JORDAN15:17

    for C be compact connected non vertical non horizontal Subset of ( TOP-REAL 2) holds for i,j,k be Nat st 1 <= i & i <= ( len ( Gauge (C,n))) & 1 <= j & j <= k & k <= ( width ( Gauge (C,n))) & (( Gauge (C,n)) * (i,j)) in ( L~ ( Upper_Seq (C,n))) & (( Gauge (C,n)) * (i,k)) in ( L~ ( Lower_Seq (C,n))) holds ex j1,k1 be Nat st j <= j1 & j1 <= k1 & k1 <= k & (( LSeg ((( Gauge (C,n)) * (i,j1)),(( Gauge (C,n)) * (i,k1)))) /\ ( L~ ( Upper_Seq (C,n)))) = {(( Gauge (C,n)) * (i,j1))} & (( LSeg ((( Gauge (C,n)) * (i,j1)),(( Gauge (C,n)) * (i,k1)))) /\ ( L~ ( Lower_Seq (C,n)))) = {(( Gauge (C,n)) * (i,k1))}

    proof

      let C be compact connected non vertical non horizontal Subset of ( TOP-REAL 2);

      let i,j,k be Nat;

      assume that

       A1: 1 <= i and

       A2: i <= ( len ( Gauge (C,n))) and

       A3: 1 <= j and

       A4: j <= k and

       A5: k <= ( width ( Gauge (C,n))) and

       A6: (( Gauge (C,n)) * (i,j)) in ( L~ ( Upper_Seq (C,n))) and

       A7: (( Gauge (C,n)) * (i,k)) in ( L~ ( Lower_Seq (C,n)));

      set G = ( Gauge (C,n));

      

       A8: j <= ( width G) by A4, A5, XXREAL_0: 2;

      then

       A9: [i, j] in ( Indices G) by A1, A2, A3, MATRIX_0: 30;

      set s = ((G * (i,1)) `1 );

      set e = (G * (i,k));

      set f = (G * (i,j));

      set w1 = ( lower_bound ( proj2 .: (( LSeg (f,e)) /\ ( L~ ( Lower_Seq (C,n))))));

      

       A10: (G * (i,k)) in ( LSeg ((G * (i,j)),(G * (i,k)))) by RLTOPSP1: 68;

      then

       A11: ( LSeg ((G * (i,j)),(G * (i,k)))) meets ( L~ ( Lower_Seq (C,n))) by A7, XBOOLE_0: 3;

      

       A12: k >= 1 by A3, A4, XXREAL_0: 2;

      then [i, k] in ( Indices G) by A1, A2, A5, MATRIX_0: 30;

      then

      consider k1 be Nat such that

       A13: j <= k1 and

       A14: k1 <= k and

       A15: ((G * (i,k1)) `2 ) = w1 by A4, A11, A9, JORDAN1F: 1, JORDAN1G: 5;

      

       A16: k1 <= ( width G) by A5, A14, XXREAL_0: 2;

      

       A17: (G * (i,j)) in ( LSeg ((G * (i,j)),(G * (i,k1)))) by RLTOPSP1: 68;

      then

       A18: ( LSeg ((G * (i,j)),(G * (i,k1)))) meets ( L~ ( Upper_Seq (C,n))) by A6, XBOOLE_0: 3;

      set X = (( LSeg ((G * (i,j)),(G * (i,k1)))) /\ ( L~ ( Upper_Seq (C,n))));

      reconsider X1 = X as non empty compact Subset of ( TOP-REAL 2) by A6, A17, XBOOLE_0:def 4;

      consider pp be object such that

       A19: pp in ( N-most X1) by XBOOLE_0:def 1;

      reconsider pp as Point of ( TOP-REAL 2) by A19;

      

       A20: pp in X by A19, XBOOLE_0:def 4;

      then

       A21: pp in ( L~ ( Upper_Seq (C,n))) by XBOOLE_0:def 4;

      set p = |[s, w1]|;

      set w2 = ( upper_bound ( proj2 .: (( LSeg (f,p)) /\ ( L~ ( Upper_Seq (C,n))))));

      set q = |[s, w2]|;

      

       A22: pp in ( LSeg ((G * (i,j)),(G * (i,k1)))) by A20, XBOOLE_0:def 4;

      

       A23: 1 <= k1 by A3, A13, XXREAL_0: 2;

      then

       A24: ((G * (i,k1)) `1 ) = s by A1, A2, A16, GOBOARD5: 2;

      then

       A25: p = (G * (i,k1)) by A15, EUCLID: 53;

       [i, k1] in ( Indices G) by A1, A2, A23, A16, MATRIX_0: 30;

      then

      consider j1 be Nat such that

       A26: j <= j1 and

       A27: j1 <= k1 and

       A28: ((G * (i,j1)) `2 ) = w2 by A9, A13, A25, A18, JORDAN1F: 2, JORDAN1G: 4;

      take j1, k1;

      thus j <= j1 & j1 <= k1 & k1 <= k by A14, A26, A27;

      

       A29: j1 <= ( width G) by A16, A27, XXREAL_0: 2;

      

       A30: 1 <= j1 by A3, A26, XXREAL_0: 2;

      then

       A31: ((G * (i,j1)) `1 ) = s by A1, A2, A29, GOBOARD5: 2;

      then

       A32: q = (G * (i,j1)) by A28, EUCLID: 53;

      then

       A33: (q `2 ) <= (p `2 ) by A1, A2, A16, A25, A27, A30, SPRECT_3: 12;

      

       A34: (q `2 ) = ( N-bound X) by A25, A28, A32, SPRECT_1: 45

      .= (( N-min X) `2 ) by EUCLID: 52

      .= (pp `2 ) by A19, PSCOMP_1: 39;

      

       A35: (f `1 ) = (p `1 ) by A1, A2, A3, A8, A24, A25, GOBOARD5: 2;

      then ( LSeg (f,p)) is vertical by SPPOL_1: 16;

      then (pp `1 ) = (q `1 ) by A24, A25, A31, A32, A22, SPPOL_1: 41;

      then

       A36: q in ( L~ ( Upper_Seq (C,n))) by A21, A34, TOPREAL3: 6;

      for x be object holds x in (( LSeg (p,q)) /\ ( L~ ( Upper_Seq (C,n)))) iff x = q

      proof

        let x be object;

        thus x in (( LSeg (p,q)) /\ ( L~ ( Upper_Seq (C,n)))) implies x = q

        proof

          reconsider EE = (( LSeg (f,p)) /\ ( L~ ( Upper_Seq (C,n)))) as compact Subset of ( TOP-REAL 2);

          reconsider E0 = ( proj2 .: EE) as compact Subset of REAL by JCT_MISC: 15;

          

           A37: p in ( LSeg (f,p)) by RLTOPSP1: 68;

          

           A38: (f `2 ) <= (q `2 ) by A1, A2, A3, A26, A29, A32, SPRECT_3: 12;

          (f `1 ) = (q `1 ) by A1, A2, A3, A8, A31, A32, GOBOARD5: 2;

          then q in ( LSeg (p,f)) by A24, A25, A31, A32, A33, A38, GOBOARD7: 7;

          then

           A39: ( LSeg (p,q)) c= ( LSeg (f,p)) by A37, TOPREAL1: 6;

          assume

           A40: x in (( LSeg (p,q)) /\ ( L~ ( Upper_Seq (C,n))));

          then

          reconsider pp = x as Point of ( TOP-REAL 2);

          

           A41: pp in ( LSeg (p,q)) by A40, XBOOLE_0:def 4;

          then

           A42: (pp `2 ) >= (q `2 ) by A33, TOPREAL1: 4;

          pp in ( L~ ( Upper_Seq (C,n))) by A40, XBOOLE_0:def 4;

          then pp in EE by A41, A39, XBOOLE_0:def 4;

          then ( proj2 . pp) in E0 by FUNCT_2: 35;

          then

           A43: (pp `2 ) in E0 by PSCOMP_1:def 6;

          E0 is real-bounded by RCOMP_1: 10;

          then E0 is bounded_above by XXREAL_2:def 11;

          then (q `2 ) >= (pp `2 ) by A28, A32, A43, SEQ_4:def 1;

          then

           A44: (pp `2 ) = (q `2 ) by A42, XXREAL_0: 1;

          (pp `1 ) = (q `1 ) by A24, A25, A31, A32, A41, GOBOARD7: 5;

          hence thesis by A44, TOPREAL3: 6;

        end;

        assume

         A45: x = q;

        then x in ( LSeg (p,q)) by RLTOPSP1: 68;

        hence thesis by A36, A45, XBOOLE_0:def 4;

      end;

      hence (( LSeg ((( Gauge (C,n)) * (i,j1)),(( Gauge (C,n)) * (i,k1)))) /\ ( L~ ( Upper_Seq (C,n)))) = {(( Gauge (C,n)) * (i,j1))} by A25, A32, TARSKI:def 1;

      set X = (( LSeg ((G * (i,j)),(G * (i,k)))) /\ ( L~ ( Lower_Seq (C,n))));

      reconsider X1 = X as non empty compact Subset of ( TOP-REAL 2) by A7, A10, XBOOLE_0:def 4;

      consider pp be object such that

       A46: pp in ( S-most X1) by XBOOLE_0:def 1;

      reconsider pp as Point of ( TOP-REAL 2) by A46;

      

       A47: pp in X by A46, XBOOLE_0:def 4;

      then

       A48: pp in ( L~ ( Lower_Seq (C,n))) by XBOOLE_0:def 4;

      (f `1 ) = s by A1, A2, A3, A8, GOBOARD5: 2

      .= (e `1 ) by A1, A2, A5, A12, GOBOARD5: 2;

      then

       A49: ( LSeg (f,e)) is vertical by SPPOL_1: 16;

      pp in ( LSeg ((G * (i,j)),(G * (i,k)))) by A47, XBOOLE_0:def 4;

      then

       A50: (pp `1 ) = (p `1 ) by A35, A49, SPPOL_1: 41;

      (p `2 ) = ( S-bound X) by A15, A25, SPRECT_1: 44

      .= (( S-min X) `2 ) by EUCLID: 52

      .= (pp `2 ) by A46, PSCOMP_1: 55;

      then

       A51: p in ( L~ ( Lower_Seq (C,n))) by A48, A50, TOPREAL3: 6;

      for x be object holds x in (( LSeg (p,q)) /\ ( L~ ( Lower_Seq (C,n)))) iff x = p

      proof

        let x be object;

        thus x in (( LSeg (p,q)) /\ ( L~ ( Lower_Seq (C,n)))) implies x = p

        proof

          

           A52: (p `2 ) <= (e `2 ) by A1, A2, A5, A14, A23, A25, SPRECT_3: 12;

          

           A53: (f `2 ) <= (p `2 ) by A1, A2, A3, A13, A16, A25, SPRECT_3: 12;

          

           A54: (e `1 ) = (p `1 ) by A1, A2, A5, A12, A24, A25, GOBOARD5: 2;

          (f `1 ) = (p `1 ) by A1, A2, A3, A8, A24, A25, GOBOARD5: 2;

          then

           A55: p in ( LSeg (f,e)) by A54, A53, A52, GOBOARD7: 7;

          

           A56: (e `1 ) = (q `1 ) by A1, A2, A5, A12, A31, A32, GOBOARD5: 2;

          j1 <= k by A14, A27, XXREAL_0: 2;

          then

           A57: (q `2 ) <= (e `2 ) by A1, A2, A5, A30, A32, SPRECT_3: 12;

          

           A58: (f `2 ) <= (q `2 ) by A1, A2, A3, A26, A29, A32, SPRECT_3: 12;

          (f `1 ) = (q `1 ) by A1, A2, A3, A8, A31, A32, GOBOARD5: 2;

          then q in ( LSeg (f,e)) by A56, A58, A57, GOBOARD7: 7;

          then

           A59: ( LSeg (p,q)) c= ( LSeg (f,e)) by A55, TOPREAL1: 6;

          reconsider EE = (( LSeg (f,e)) /\ ( L~ ( Lower_Seq (C,n)))) as compact Subset of ( TOP-REAL 2);

          reconsider E0 = ( proj2 .: EE) as compact Subset of REAL by JCT_MISC: 15;

          assume

           A60: x in (( LSeg (p,q)) /\ ( L~ ( Lower_Seq (C,n))));

          then

          reconsider pp = x as Point of ( TOP-REAL 2);

          

           A61: pp in ( LSeg (p,q)) by A60, XBOOLE_0:def 4;

          then

           A62: (pp `2 ) <= (p `2 ) by A33, TOPREAL1: 4;

          pp in ( L~ ( Lower_Seq (C,n))) by A60, XBOOLE_0:def 4;

          then pp in EE by A61, A59, XBOOLE_0:def 4;

          then ( proj2 . pp) in E0 by FUNCT_2: 35;

          then

           A63: (pp `2 ) in E0 by PSCOMP_1:def 6;

          E0 is real-bounded by RCOMP_1: 10;

          then E0 is bounded_below by XXREAL_2:def 11;

          then (p `2 ) <= (pp `2 ) by A15, A25, A63, SEQ_4:def 2;

          then

           A64: (pp `2 ) = (p `2 ) by A62, XXREAL_0: 1;

          (pp `1 ) = (p `1 ) by A24, A25, A31, A32, A61, GOBOARD7: 5;

          hence thesis by A64, TOPREAL3: 6;

        end;

        assume

         A65: x = p;

        then x in ( LSeg (p,q)) by RLTOPSP1: 68;

        hence thesis by A51, A65, XBOOLE_0:def 4;

      end;

      hence thesis by A25, A32, TARSKI:def 1;

    end;

    theorem :: JORDAN15:18

    

     Th18: for C be compact connected non vertical non horizontal Subset of ( TOP-REAL 2) holds for i,j,k be Nat st 1 <= j & j <= k & k <= ( len ( Gauge (C,n))) & 1 <= i & i <= ( width ( Gauge (C,n))) & (( Gauge (C,n)) * (j,i)) in ( L~ ( Upper_Seq (C,n))) holds ex j1 be Nat st j <= j1 & j1 <= k & (( LSeg ((( Gauge (C,n)) * (j1,i)),(( Gauge (C,n)) * (k,i)))) /\ ( L~ ( Upper_Seq (C,n)))) = {(( Gauge (C,n)) * (j1,i))}

    proof

      let C be compact connected non vertical non horizontal Subset of ( TOP-REAL 2);

      let i,j,k be Nat;

      assume that

       A1: 1 <= j and

       A2: j <= k and

       A3: k <= ( len ( Gauge (C,n))) and

       A4: 1 <= i and

       A5: i <= ( width ( Gauge (C,n))) and

       A6: (( Gauge (C,n)) * (j,i)) in ( L~ ( Upper_Seq (C,n)));

      set G = ( Gauge (C,n));

      

       A7: k >= 1 by A1, A2, XXREAL_0: 2;

      then

       A8: [k, i] in ( Indices G) by A3, A4, A5, MATRIX_0: 30;

      set X = (( LSeg ((G * (j,i)),(G * (k,i)))) /\ ( L~ ( Upper_Seq (C,n))));

      

       A9: (G * (j,i)) in ( LSeg ((G * (j,i)),(G * (k,i)))) by RLTOPSP1: 68;

      then

      reconsider X1 = X as non empty compact Subset of ( TOP-REAL 2) by A6, XBOOLE_0:def 4;

      

       A10: ( LSeg ((G * (j,i)),(G * (k,i)))) meets ( L~ ( Upper_Seq (C,n))) by A6, A9, XBOOLE_0: 3;

      set s = ((G * (1,i)) `2 );

      set e = (G * (k,i));

      set f = (G * (j,i));

      set w2 = ( upper_bound ( proj1 .: (( LSeg (f,e)) /\ ( L~ ( Upper_Seq (C,n))))));

      

       A11: ( len G) = ( width G) by JORDAN8:def 1;

      then

       A12: j <= ( width G) by A2, A3, XXREAL_0: 2;

      then [j, i] in ( Indices G) by A1, A4, A5, A11, MATRIX_0: 30;

      then

      consider j1 be Nat such that

       A13: j <= j1 and

       A14: j1 <= k and

       A15: ((G * (j1,i)) `1 ) = w2 by A2, A10, A8, JORDAN1F: 4, JORDAN1G: 4;

      set q = |[w2, s]|;

      

       A16: 1 <= j1 by A1, A13, XXREAL_0: 2;

      take j1;

      thus j <= j1 & j1 <= k by A13, A14;

      consider pp be object such that

       A17: pp in ( E-most X1) by XBOOLE_0:def 1;

      reconsider pp as Point of ( TOP-REAL 2) by A17;

      

       A18: pp in X by A17, XBOOLE_0:def 4;

      then

       A19: pp in ( L~ ( Upper_Seq (C,n))) by XBOOLE_0:def 4;

      

       A20: j1 <= ( width G) by A3, A11, A14, XXREAL_0: 2;

      then

       A21: ((G * (j1,i)) `2 ) = s by A4, A5, A11, A16, GOBOARD5: 1;

      then

       A22: q = (G * (j1,i)) by A15, EUCLID: 53;

      then

       A23: (q `1 ) <= (e `1 ) by A3, A4, A5, A14, A16, SPRECT_3: 13;

      

       A24: ((G * (k,i)) `2 ) = s by A3, A4, A5, A7, GOBOARD5: 1;

      then (f `2 ) = (e `2 ) by A1, A4, A5, A11, A12, GOBOARD5: 1;

      then

       A25: ( LSeg (f,e)) is horizontal by SPPOL_1: 15;

      

       A26: (q `1 ) = ( E-bound X) by A15, A22, SPRECT_1: 46

      .= (( E-min X) `1 ) by EUCLID: 52

      .= (pp `1 ) by A17, PSCOMP_1: 47;

      pp in ( LSeg ((G * (j,i)),(G * (k,i)))) by A18, XBOOLE_0:def 4;

      then (pp `2 ) = (q `2 ) by A24, A21, A22, A25, SPPOL_1: 40;

      then

       A27: q in ( L~ ( Upper_Seq (C,n))) by A19, A26, TOPREAL3: 6;

      for x be object holds x in (( LSeg (e,q)) /\ ( L~ ( Upper_Seq (C,n)))) iff x = q

      proof

        let x be object;

        thus x in (( LSeg (e,q)) /\ ( L~ ( Upper_Seq (C,n)))) implies x = q

        proof

          

           A28: (f `1 ) <= (q `1 ) by A1, A4, A5, A11, A13, A20, A22, SPRECT_3: 13;

          (f `2 ) = (q `2 ) by A1, A4, A5, A11, A12, A21, A22, GOBOARD5: 1;

          then

           A29: q in ( LSeg (e,f)) by A24, A21, A22, A23, A28, GOBOARD7: 8;

          e in ( LSeg (f,e)) by RLTOPSP1: 68;

          then

           A30: ( LSeg (e,q)) c= ( LSeg (f,e)) by A29, TOPREAL1: 6;

          reconsider EE = (( LSeg (f,e)) /\ ( L~ ( Upper_Seq (C,n)))) as compact Subset of ( TOP-REAL 2);

          reconsider E0 = ( proj1 .: EE) as compact Subset of REAL by Th4;

          assume

           A31: x in (( LSeg (e,q)) /\ ( L~ ( Upper_Seq (C,n))));

          then

          reconsider pp = x as Point of ( TOP-REAL 2);

          

           A32: pp in ( LSeg (e,q)) by A31, XBOOLE_0:def 4;

          then

           A33: (pp `1 ) >= (q `1 ) by A23, TOPREAL1: 3;

          pp in ( L~ ( Upper_Seq (C,n))) by A31, XBOOLE_0:def 4;

          then pp in EE by A32, A30, XBOOLE_0:def 4;

          then ( proj1 . pp) in E0 by FUNCT_2: 35;

          then

           A34: (pp `1 ) in E0 by PSCOMP_1:def 5;

          E0 is real-bounded by RCOMP_1: 10;

          then E0 is bounded_above by XXREAL_2:def 11;

          then (q `1 ) >= (pp `1 ) by A15, A22, A34, SEQ_4:def 1;

          then

           A35: (pp `1 ) = (q `1 ) by A33, XXREAL_0: 1;

          (pp `2 ) = (q `2 ) by A24, A21, A22, A32, GOBOARD7: 6;

          hence thesis by A35, TOPREAL3: 6;

        end;

        assume

         A36: x = q;

        then x in ( LSeg (e,q)) by RLTOPSP1: 68;

        hence thesis by A27, A36, XBOOLE_0:def 4;

      end;

      hence thesis by A22, TARSKI:def 1;

    end;

    theorem :: JORDAN15:19

    for C be compact connected non vertical non horizontal Subset of ( TOP-REAL 2) holds for i,j,k be Nat st 1 <= j & j <= k & k <= ( len ( Gauge (C,n))) & 1 <= i & i <= ( width ( Gauge (C,n))) & (( Gauge (C,n)) * (k,i)) in ( L~ ( Lower_Seq (C,n))) holds ex k1 be Nat st j <= k1 & k1 <= k & (( LSeg ((( Gauge (C,n)) * (j,i)),(( Gauge (C,n)) * (k1,i)))) /\ ( L~ ( Lower_Seq (C,n)))) = {(( Gauge (C,n)) * (k1,i))}

    proof

      let C be compact connected non vertical non horizontal Subset of ( TOP-REAL 2);

      let i,j,k be Nat;

      assume that

       A1: 1 <= j and

       A2: j <= k and

       A3: k <= ( len ( Gauge (C,n))) and

       A4: 1 <= i and

       A5: i <= ( width ( Gauge (C,n))) and

       A6: (( Gauge (C,n)) * (k,i)) in ( L~ ( Lower_Seq (C,n)));

      set G = ( Gauge (C,n));

      

       A7: k >= 1 by A1, A2, XXREAL_0: 2;

      then

       A8: [k, i] in ( Indices G) by A3, A4, A5, MATRIX_0: 30;

      set X = (( LSeg ((G * (j,i)),(G * (k,i)))) /\ ( L~ ( Lower_Seq (C,n))));

      

       A9: (G * (k,i)) in ( LSeg ((G * (j,i)),(G * (k,i)))) by RLTOPSP1: 68;

      then

      reconsider X1 = X as non empty compact Subset of ( TOP-REAL 2) by A6, XBOOLE_0:def 4;

      

       A10: ( LSeg ((G * (j,i)),(G * (k,i)))) meets ( L~ ( Lower_Seq (C,n))) by A6, A9, XBOOLE_0: 3;

      set s = ((G * (1,i)) `2 );

      set e = (G * (k,i));

      set f = (G * (j,i));

      set w1 = ( lower_bound ( proj1 .: (( LSeg (f,e)) /\ ( L~ ( Lower_Seq (C,n))))));

      

       A11: ( len G) = ( width G) by JORDAN8:def 1;

      then

       A12: j <= ( width G) by A2, A3, XXREAL_0: 2;

      then [j, i] in ( Indices G) by A1, A4, A5, A11, MATRIX_0: 30;

      then

      consider k1 be Nat such that

       A13: j <= k1 and

       A14: k1 <= k and

       A15: ((G * (k1,i)) `1 ) = w1 by A2, A10, A8, JORDAN1F: 3, JORDAN1G: 5;

      set p = |[w1, s]|;

      

       A16: k1 <= ( width G) by A3, A11, A14, XXREAL_0: 2;

      (f `2 ) = s by A1, A4, A5, A11, A12, GOBOARD5: 1

      .= (e `2 ) by A3, A4, A5, A7, GOBOARD5: 1;

      then

       A17: ( LSeg (f,e)) is horizontal by SPPOL_1: 15;

      take k1;

      thus j <= k1 & k1 <= k by A13, A14;

      consider pp be object such that

       A18: pp in ( W-most X1) by XBOOLE_0:def 1;

      

       A19: 1 <= k1 by A1, A13, XXREAL_0: 2;

      then

       A20: ((G * (k1,i)) `2 ) = s by A4, A5, A11, A16, GOBOARD5: 1;

      then

       A21: p = (G * (k1,i)) by A15, EUCLID: 53;

      then

       A22: (f `1 ) <= (p `1 ) by A1, A4, A5, A11, A13, A16, SPRECT_3: 13;

      

       A23: (f `2 ) = (p `2 ) by A1, A4, A5, A11, A12, A20, A21, GOBOARD5: 1;

      reconsider pp as Point of ( TOP-REAL 2) by A18;

      

       A24: pp in X by A18, XBOOLE_0:def 4;

      then

       A25: pp in ( L~ ( Lower_Seq (C,n))) by XBOOLE_0:def 4;

      

       A26: (p `1 ) = ( W-bound X) by A15, A21, SPRECT_1: 43

      .= (( W-min X) `1 ) by EUCLID: 52

      .= (pp `1 ) by A18, PSCOMP_1: 31;

      pp in ( LSeg ((G * (j,i)),(G * (k,i)))) by A24, XBOOLE_0:def 4;

      then (pp `2 ) = (p `2 ) by A23, A17, SPPOL_1: 40;

      then

       A27: p in ( L~ ( Lower_Seq (C,n))) by A25, A26, TOPREAL3: 6;

      for x be object holds x in (( LSeg (p,f)) /\ ( L~ ( Lower_Seq (C,n)))) iff x = p

      proof

        let x be object;

        thus x in (( LSeg (p,f)) /\ ( L~ ( Lower_Seq (C,n)))) implies x = p

        proof

          reconsider EE = (( LSeg (f,e)) /\ ( L~ ( Lower_Seq (C,n)))) as compact Subset of ( TOP-REAL 2);

          assume

           A28: x in (( LSeg (p,f)) /\ ( L~ ( Lower_Seq (C,n))));

          then

          reconsider pp = x as Point of ( TOP-REAL 2);

          

           A29: pp in ( LSeg (p,f)) by A28, XBOOLE_0:def 4;

          then

           A30: (pp `1 ) <= (p `1 ) by A22, TOPREAL1: 3;

          

           A31: (p `1 ) <= (e `1 ) by A3, A4, A5, A14, A19, A21, SPRECT_3: 13;

          

           A32: (f `1 ) <= (p `1 ) by A1, A4, A5, A11, A13, A16, A21, SPRECT_3: 13;

          

           A33: (e `2 ) = (p `2 ) by A3, A4, A5, A7, A20, A21, GOBOARD5: 1;

          reconsider E0 = ( proj1 .: EE) as compact Subset of REAL by Th4;

          

           A34: f in ( LSeg (f,e)) by RLTOPSP1: 68;

          (f `2 ) = (p `2 ) by A1, A4, A5, A11, A12, A20, A21, GOBOARD5: 1;

          then p in ( LSeg (f,e)) by A33, A32, A31, GOBOARD7: 8;

          then

           A35: ( LSeg (p,f)) c= ( LSeg (f,e)) by A34, TOPREAL1: 6;

          pp in ( L~ ( Lower_Seq (C,n))) by A28, XBOOLE_0:def 4;

          then pp in EE by A29, A35, XBOOLE_0:def 4;

          then ( proj1 . pp) in E0 by FUNCT_2: 35;

          then

           A36: (pp `1 ) in E0 by PSCOMP_1:def 5;

          E0 is real-bounded by RCOMP_1: 10;

          then E0 is bounded_below by XXREAL_2:def 11;

          then (p `1 ) <= (pp `1 ) by A15, A21, A36, SEQ_4:def 2;

          then

           A37: (pp `1 ) = (p `1 ) by A30, XXREAL_0: 1;

          (pp `2 ) = (p `2 ) by A23, A29, GOBOARD7: 6;

          hence thesis by A37, TOPREAL3: 6;

        end;

        assume

         A38: x = p;

        then x in ( LSeg (p,f)) by RLTOPSP1: 68;

        hence thesis by A27, A38, XBOOLE_0:def 4;

      end;

      hence thesis by A21, TARSKI:def 1;

    end;

    theorem :: JORDAN15:20

    

     Th20: for C be compact connected non vertical non horizontal Subset of ( TOP-REAL 2) holds for i,j,k be Nat st 1 <= j & j <= k & k <= ( len ( Gauge (C,n))) & 1 <= i & i <= ( width ( Gauge (C,n))) & (( Gauge (C,n)) * (j,i)) in ( L~ ( Upper_Seq (C,n))) & (( Gauge (C,n)) * (k,i)) in ( L~ ( Lower_Seq (C,n))) holds ex j1,k1 be Nat st j <= j1 & j1 <= k1 & k1 <= k & (( LSeg ((( Gauge (C,n)) * (j1,i)),(( Gauge (C,n)) * (k1,i)))) /\ ( L~ ( Upper_Seq (C,n)))) = {(( Gauge (C,n)) * (j1,i))} & (( LSeg ((( Gauge (C,n)) * (j1,i)),(( Gauge (C,n)) * (k1,i)))) /\ ( L~ ( Lower_Seq (C,n)))) = {(( Gauge (C,n)) * (k1,i))}

    proof

      let C be compact connected non vertical non horizontal Subset of ( TOP-REAL 2);

      let i,j,k be Nat;

      assume that

       A1: 1 <= j and

       A2: j <= k and

       A3: k <= ( len ( Gauge (C,n))) and

       A4: 1 <= i and

       A5: i <= ( width ( Gauge (C,n))) and

       A6: (( Gauge (C,n)) * (j,i)) in ( L~ ( Upper_Seq (C,n))) and

       A7: (( Gauge (C,n)) * (k,i)) in ( L~ ( Lower_Seq (C,n)));

      set G = ( Gauge (C,n));

      

       A8: ( len G) = ( width G) by JORDAN8:def 1;

      then

       A9: j <= ( width G) by A2, A3, XXREAL_0: 2;

      then

       A10: [j, i] in ( Indices G) by A1, A4, A5, A8, MATRIX_0: 30;

      set s = ((G * (1,i)) `2 );

      set e = (G * (k,i));

      set f = (G * (j,i));

      set w1 = ( lower_bound ( proj1 .: (( LSeg (f,e)) /\ ( L~ ( Lower_Seq (C,n))))));

      

       A11: (G * (k,i)) in ( LSeg ((G * (j,i)),(G * (k,i)))) by RLTOPSP1: 68;

      then

       A12: ( LSeg ((G * (j,i)),(G * (k,i)))) meets ( L~ ( Lower_Seq (C,n))) by A7, XBOOLE_0: 3;

      

       A13: k >= 1 by A1, A2, XXREAL_0: 2;

      then [k, i] in ( Indices G) by A3, A4, A5, MATRIX_0: 30;

      then

      consider k1 be Nat such that

       A14: j <= k1 and

       A15: k1 <= k and

       A16: ((G * (k1,i)) `1 ) = w1 by A2, A12, A10, JORDAN1F: 3, JORDAN1G: 5;

      

       A17: k1 <= ( width G) by A3, A8, A15, XXREAL_0: 2;

      set p = |[w1, s]|;

      set w2 = ( upper_bound ( proj1 .: (( LSeg (f,p)) /\ ( L~ ( Upper_Seq (C,n))))));

      set q = |[w2, s]|;

      

       A18: (G * (j,i)) in ( LSeg ((G * (j,i)),(G * (k1,i)))) by RLTOPSP1: 68;

      then

       A19: ( LSeg ((G * (j,i)),(G * (k1,i)))) meets ( L~ ( Upper_Seq (C,n))) by A6, XBOOLE_0: 3;

      

       A20: 1 <= k1 by A1, A14, XXREAL_0: 2;

      then

       A21: ((G * (k1,i)) `2 ) = s by A4, A5, A8, A17, GOBOARD5: 1;

      then

       A22: p = (G * (k1,i)) by A16, EUCLID: 53;

      (f `2 ) = s by A1, A4, A5, A8, A9, GOBOARD5: 1

      .= (e `2 ) by A3, A4, A5, A13, GOBOARD5: 1;

      then

       A23: ( LSeg (f,e)) is horizontal by SPPOL_1: 15;

      set X = (( LSeg ((G * (j,i)),(G * (k1,i)))) /\ ( L~ ( Upper_Seq (C,n))));

      reconsider X1 = X as non empty compact Subset of ( TOP-REAL 2) by A6, A18, XBOOLE_0:def 4;

      consider pp be object such that

       A24: pp in ( E-most X1) by XBOOLE_0:def 1;

       [k1, i] in ( Indices G) by A4, A5, A8, A20, A17, MATRIX_0: 30;

      then

      consider j1 be Nat such that

       A25: j <= j1 and

       A26: j1 <= k1 and

       A27: ((G * (j1,i)) `1 ) = w2 by A10, A14, A22, A19, JORDAN1F: 4, JORDAN1G: 4;

      

       A28: j1 <= ( width G) by A17, A26, XXREAL_0: 2;

      reconsider pp as Point of ( TOP-REAL 2) by A24;

      

       A29: pp in X by A24, XBOOLE_0:def 4;

      then

       A30: pp in ( L~ ( Upper_Seq (C,n))) by XBOOLE_0:def 4;

      take j1, k1;

      thus j <= j1 & j1 <= k1 & k1 <= k by A15, A25, A26;

      

       A31: pp in ( LSeg ((G * (j,i)),(G * (k1,i)))) by A29, XBOOLE_0:def 4;

      

       A32: 1 <= j1 by A1, A25, XXREAL_0: 2;

      then

       A33: ((G * (j1,i)) `2 ) = s by A4, A5, A8, A28, GOBOARD5: 1;

      then

       A34: q = (G * (j1,i)) by A27, EUCLID: 53;

      then

       A35: (q `1 ) <= (p `1 ) by A4, A5, A8, A17, A22, A26, A32, SPRECT_3: 13;

      

       A36: (q `1 ) = ( E-bound X) by A22, A27, A34, SPRECT_1: 46

      .= (( E-min X) `1 ) by EUCLID: 52

      .= (pp `1 ) by A24, PSCOMP_1: 47;

      

       A37: (f `2 ) = (p `2 ) by A1, A4, A5, A8, A9, A21, A22, GOBOARD5: 1;

      then ( LSeg (f,p)) is horizontal by SPPOL_1: 15;

      then (pp `2 ) = (q `2 ) by A21, A22, A33, A34, A31, SPPOL_1: 40;

      then

       A38: q in ( L~ ( Upper_Seq (C,n))) by A30, A36, TOPREAL3: 6;

      for x be object holds x in (( LSeg (p,q)) /\ ( L~ ( Upper_Seq (C,n)))) iff x = q

      proof

        let x be object;

        thus x in (( LSeg (p,q)) /\ ( L~ ( Upper_Seq (C,n)))) implies x = q

        proof

          reconsider EE = (( LSeg (f,p)) /\ ( L~ ( Upper_Seq (C,n)))) as compact Subset of ( TOP-REAL 2);

          assume

           A39: x in (( LSeg (p,q)) /\ ( L~ ( Upper_Seq (C,n))));

          then

          reconsider pp = x as Point of ( TOP-REAL 2);

          

           A40: pp in ( LSeg (p,q)) by A39, XBOOLE_0:def 4;

          then

           A41: (pp `1 ) >= (q `1 ) by A35, TOPREAL1: 3;

          

           A42: (f `1 ) <= (q `1 ) by A1, A4, A5, A8, A25, A28, A34, SPRECT_3: 13;

          reconsider E0 = ( proj1 .: EE) as compact Subset of REAL by Th4;

          

           A43: p in ( LSeg (f,p)) by RLTOPSP1: 68;

          (f `2 ) = (q `2 ) by A1, A4, A5, A8, A9, A33, A34, GOBOARD5: 1;

          then q in ( LSeg (p,f)) by A21, A22, A33, A34, A35, A42, GOBOARD7: 8;

          then

           A44: ( LSeg (p,q)) c= ( LSeg (f,p)) by A43, TOPREAL1: 6;

          pp in ( L~ ( Upper_Seq (C,n))) by A39, XBOOLE_0:def 4;

          then pp in EE by A40, A44, XBOOLE_0:def 4;

          then ( proj1 . pp) in E0 by FUNCT_2: 35;

          then

           A45: (pp `1 ) in E0 by PSCOMP_1:def 5;

          E0 is real-bounded by RCOMP_1: 10;

          then E0 is bounded_above by XXREAL_2:def 11;

          then (q `1 ) >= (pp `1 ) by A27, A34, A45, SEQ_4:def 1;

          then

           A46: (pp `1 ) = (q `1 ) by A41, XXREAL_0: 1;

          (pp `2 ) = (q `2 ) by A21, A22, A33, A34, A40, GOBOARD7: 6;

          hence thesis by A46, TOPREAL3: 6;

        end;

        assume

         A47: x = q;

        then x in ( LSeg (p,q)) by RLTOPSP1: 68;

        hence thesis by A38, A47, XBOOLE_0:def 4;

      end;

      hence (( LSeg ((( Gauge (C,n)) * (j1,i)),(( Gauge (C,n)) * (k1,i)))) /\ ( L~ ( Upper_Seq (C,n)))) = {(( Gauge (C,n)) * (j1,i))} by A22, A34, TARSKI:def 1;

      set X = (( LSeg ((G * (j,i)),(G * (k,i)))) /\ ( L~ ( Lower_Seq (C,n))));

      reconsider X1 = X as non empty compact Subset of ( TOP-REAL 2) by A7, A11, XBOOLE_0:def 4;

      consider pp be object such that

       A48: pp in ( W-most X1) by XBOOLE_0:def 1;

      reconsider pp as Point of ( TOP-REAL 2) by A48;

      

       A49: pp in X by A48, XBOOLE_0:def 4;

      then

       A50: pp in ( L~ ( Lower_Seq (C,n))) by XBOOLE_0:def 4;

      pp in ( LSeg ((G * (j,i)),(G * (k,i)))) by A49, XBOOLE_0:def 4;

      then

       A51: (pp `2 ) = (p `2 ) by A37, A23, SPPOL_1: 40;

      (p `1 ) = ( W-bound X) by A16, A22, SPRECT_1: 43

      .= (( W-min X) `1 ) by EUCLID: 52

      .= (pp `1 ) by A48, PSCOMP_1: 31;

      then

       A52: p in ( L~ ( Lower_Seq (C,n))) by A50, A51, TOPREAL3: 6;

      for x be object holds x in (( LSeg (p,q)) /\ ( L~ ( Lower_Seq (C,n)))) iff x = p

      proof

        let x be object;

        thus x in (( LSeg (p,q)) /\ ( L~ ( Lower_Seq (C,n)))) implies x = p

        proof

          j1 <= k by A15, A26, XXREAL_0: 2;

          then

           A53: (q `1 ) <= (e `1 ) by A3, A4, A5, A32, A34, SPRECT_3: 13;

          

           A54: (e `2 ) = (p `2 ) by A3, A4, A5, A13, A21, A22, GOBOARD5: 1;

          

           A55: (f `1 ) <= (p `1 ) by A1, A4, A5, A8, A14, A17, A22, SPRECT_3: 13;

          

           A56: (f `1 ) <= (q `1 ) by A1, A4, A5, A8, A25, A28, A34, SPRECT_3: 13;

          

           A57: (p `1 ) <= (e `1 ) by A3, A4, A5, A15, A20, A22, SPRECT_3: 13;

          (f `2 ) = (p `2 ) by A1, A4, A5, A8, A9, A21, A22, GOBOARD5: 1;

          then

           A58: p in ( LSeg (f,e)) by A54, A55, A57, GOBOARD7: 8;

          

           A59: (e `2 ) = (q `2 ) by A3, A4, A5, A13, A33, A34, GOBOARD5: 1;

          (f `2 ) = (q `2 ) by A1, A4, A5, A8, A9, A33, A34, GOBOARD5: 1;

          then q in ( LSeg (f,e)) by A59, A56, A53, GOBOARD7: 8;

          then

           A60: ( LSeg (p,q)) c= ( LSeg (f,e)) by A58, TOPREAL1: 6;

          reconsider EE = (( LSeg (f,e)) /\ ( L~ ( Lower_Seq (C,n)))) as compact Subset of ( TOP-REAL 2);

          reconsider E0 = ( proj1 .: EE) as compact Subset of REAL by Th4;

          assume

           A61: x in (( LSeg (p,q)) /\ ( L~ ( Lower_Seq (C,n))));

          then

          reconsider pp = x as Point of ( TOP-REAL 2);

          

           A62: pp in ( LSeg (p,q)) by A61, XBOOLE_0:def 4;

          then

           A63: (pp `1 ) <= (p `1 ) by A35, TOPREAL1: 3;

          pp in ( L~ ( Lower_Seq (C,n))) by A61, XBOOLE_0:def 4;

          then pp in EE by A62, A60, XBOOLE_0:def 4;

          then ( proj1 . pp) in E0 by FUNCT_2: 35;

          then

           A64: (pp `1 ) in E0 by PSCOMP_1:def 5;

          E0 is real-bounded by RCOMP_1: 10;

          then E0 is bounded_below by XXREAL_2:def 11;

          then (p `1 ) <= (pp `1 ) by A16, A22, A64, SEQ_4:def 2;

          then

           A65: (pp `1 ) = (p `1 ) by A63, XXREAL_0: 1;

          (pp `2 ) = (p `2 ) by A21, A22, A33, A34, A62, GOBOARD7: 6;

          hence thesis by A65, TOPREAL3: 6;

        end;

        assume

         A66: x = p;

        then x in ( LSeg (p,q)) by RLTOPSP1: 68;

        hence thesis by A52, A66, XBOOLE_0:def 4;

      end;

      hence thesis by A22, A34, TARSKI:def 1;

    end;

    theorem :: JORDAN15:21

    

     Th21: for C be Simple_closed_curve holds for i,j,k be Nat st 1 < i & i < ( len ( Gauge (C,n))) & 1 <= j & j <= k & k <= ( width ( Gauge (C,n))) & (( Gauge (C,n)) * (i,k)) in ( L~ ( Upper_Seq (C,n))) & (( Gauge (C,n)) * (i,j)) in ( L~ ( Lower_Seq (C,n))) holds ( LSeg ((( Gauge (C,n)) * (i,j)),(( Gauge (C,n)) * (i,k)))) meets ( Lower_Arc C)

    proof

      let C be Simple_closed_curve;

      let i,j,k be Nat;

      assume that

       A1: 1 < i and

       A2: i < ( len ( Gauge (C,n))) and

       A3: 1 <= j and

       A4: j <= k and

       A5: k <= ( width ( Gauge (C,n))) and

       A6: (( Gauge (C,n)) * (i,k)) in ( L~ ( Upper_Seq (C,n))) and

       A7: (( Gauge (C,n)) * (i,j)) in ( L~ ( Lower_Seq (C,n)));

      consider j1,k1 be Nat such that

       A8: j <= j1 and

       A9: j1 <= k1 and

       A10: k1 <= k and

       A11: (( LSeg ((( Gauge (C,n)) * (i,j1)),(( Gauge (C,n)) * (i,k1)))) /\ ( L~ ( Lower_Seq (C,n)))) = {(( Gauge (C,n)) * (i,j1))} and

       A12: (( LSeg ((( Gauge (C,n)) * (i,j1)),(( Gauge (C,n)) * (i,k1)))) /\ ( L~ ( Upper_Seq (C,n)))) = {(( Gauge (C,n)) * (i,k1))} by A1, A2, A3, A4, A5, A6, A7, Th11;

      

       A13: k1 <= ( width ( Gauge (C,n))) by A5, A10, XXREAL_0: 2;

      1 <= j1 by A3, A8, XXREAL_0: 2;

      then ( LSeg ((( Gauge (C,n)) * (i,j1)),(( Gauge (C,n)) * (i,k1)))) meets ( Lower_Arc C) by A1, A2, A9, A11, A12, A13, JORDAN1J: 58;

      hence thesis by A1, A2, A3, A5, A8, A9, A10, Th5, XBOOLE_1: 63;

    end;

    theorem :: JORDAN15:22

    

     Th22: for C be Simple_closed_curve holds for i,j,k be Nat st 1 < i & i < ( len ( Gauge (C,n))) & 1 <= j & j <= k & k <= ( width ( Gauge (C,n))) & (( Gauge (C,n)) * (i,k)) in ( L~ ( Upper_Seq (C,n))) & (( Gauge (C,n)) * (i,j)) in ( L~ ( Lower_Seq (C,n))) holds ( LSeg ((( Gauge (C,n)) * (i,j)),(( Gauge (C,n)) * (i,k)))) meets ( Upper_Arc C)

    proof

      let C be Simple_closed_curve;

      let i,j,k be Nat;

      assume that

       A1: 1 < i and

       A2: i < ( len ( Gauge (C,n))) and

       A3: 1 <= j and

       A4: j <= k and

       A5: k <= ( width ( Gauge (C,n))) and

       A6: (( Gauge (C,n)) * (i,k)) in ( L~ ( Upper_Seq (C,n))) and

       A7: (( Gauge (C,n)) * (i,j)) in ( L~ ( Lower_Seq (C,n)));

      consider j1,k1 be Nat such that

       A8: j <= j1 and

       A9: j1 <= k1 and

       A10: k1 <= k and

       A11: (( LSeg ((( Gauge (C,n)) * (i,j1)),(( Gauge (C,n)) * (i,k1)))) /\ ( L~ ( Lower_Seq (C,n)))) = {(( Gauge (C,n)) * (i,j1))} and

       A12: (( LSeg ((( Gauge (C,n)) * (i,j1)),(( Gauge (C,n)) * (i,k1)))) /\ ( L~ ( Upper_Seq (C,n)))) = {(( Gauge (C,n)) * (i,k1))} by A1, A2, A3, A4, A5, A6, A7, Th11;

      

       A13: k1 <= ( width ( Gauge (C,n))) by A5, A10, XXREAL_0: 2;

      1 <= j1 by A3, A8, XXREAL_0: 2;

      then ( LSeg ((( Gauge (C,n)) * (i,j1)),(( Gauge (C,n)) * (i,k1)))) meets ( Upper_Arc C) by A1, A2, A9, A11, A12, A13, JORDAN1J: 59;

      hence thesis by A1, A2, A3, A5, A8, A9, A10, Th5, XBOOLE_1: 63;

    end;

    theorem :: JORDAN15:23

    

     Th23: for C be Simple_closed_curve holds for i,j,k be Nat st 1 < i & i < ( len ( Gauge (C,n))) & 1 <= j & j <= k & k <= ( width ( Gauge (C,n))) & n > 0 & (( Gauge (C,n)) * (i,k)) in ( Upper_Arc ( L~ ( Cage (C,n)))) & (( Gauge (C,n)) * (i,j)) in ( Lower_Arc ( L~ ( Cage (C,n)))) holds ( LSeg ((( Gauge (C,n)) * (i,j)),(( Gauge (C,n)) * (i,k)))) meets ( Lower_Arc C)

    proof

      let C be Simple_closed_curve;

      let i,j,k be Nat;

      assume that

       A1: 1 < i and

       A2: i < ( len ( Gauge (C,n))) and

       A3: 1 <= j and

       A4: j <= k and

       A5: k <= ( width ( Gauge (C,n))) and

       A6: n > 0 and

       A7: (( Gauge (C,n)) * (i,k)) in ( Upper_Arc ( L~ ( Cage (C,n)))) and

       A8: (( Gauge (C,n)) * (i,j)) in ( Lower_Arc ( L~ ( Cage (C,n))));

      

       A9: ( L~ ( Lower_Seq (C,n))) = ( Lower_Arc ( L~ ( Cage (C,n)))) by A6, JORDAN1G: 56;

      ( L~ ( Upper_Seq (C,n))) = ( Upper_Arc ( L~ ( Cage (C,n)))) by A6, JORDAN1G: 55;

      hence thesis by A1, A2, A3, A4, A5, A7, A8, A9, Th21;

    end;

    theorem :: JORDAN15:24

    

     Th24: for C be Simple_closed_curve holds for i,j,k be Nat st 1 < i & i < ( len ( Gauge (C,n))) & 1 <= j & j <= k & k <= ( width ( Gauge (C,n))) & n > 0 & (( Gauge (C,n)) * (i,k)) in ( Upper_Arc ( L~ ( Cage (C,n)))) & (( Gauge (C,n)) * (i,j)) in ( Lower_Arc ( L~ ( Cage (C,n)))) holds ( LSeg ((( Gauge (C,n)) * (i,j)),(( Gauge (C,n)) * (i,k)))) meets ( Upper_Arc C)

    proof

      let C be Simple_closed_curve;

      let i,j,k be Nat;

      assume that

       A1: 1 < i and

       A2: i < ( len ( Gauge (C,n))) and

       A3: 1 <= j and

       A4: j <= k and

       A5: k <= ( width ( Gauge (C,n))) and

       A6: n > 0 and

       A7: (( Gauge (C,n)) * (i,k)) in ( Upper_Arc ( L~ ( Cage (C,n)))) and

       A8: (( Gauge (C,n)) * (i,j)) in ( Lower_Arc ( L~ ( Cage (C,n))));

      

       A9: ( L~ ( Lower_Seq (C,n))) = ( Lower_Arc ( L~ ( Cage (C,n)))) by A6, JORDAN1G: 56;

      ( L~ ( Upper_Seq (C,n))) = ( Upper_Arc ( L~ ( Cage (C,n)))) by A6, JORDAN1G: 55;

      hence thesis by A1, A2, A3, A4, A5, A7, A8, A9, Th22;

    end;

    theorem :: JORDAN15:25

    for C be Simple_closed_curve holds for j,k be Nat holds 1 <= j & j <= k & k <= ( width ( Gauge (C,(n + 1)))) & (( Gauge (C,(n + 1))) * (( Center ( Gauge (C,(n + 1)))),k)) in ( Upper_Arc ( L~ ( Cage (C,(n + 1))))) & (( Gauge (C,(n + 1))) * (( Center ( Gauge (C,(n + 1)))),j)) in ( Lower_Arc ( L~ ( Cage (C,(n + 1))))) implies ( LSeg ((( Gauge (C,(n + 1))) * (( Center ( Gauge (C,(n + 1)))),j)),(( Gauge (C,(n + 1))) * (( Center ( Gauge (C,(n + 1)))),k)))) meets ( Lower_Arc C)

    proof

      let C be Simple_closed_curve;

      let j,k be Nat;

      assume that

       A1: 1 <= j and

       A2: j <= k and

       A3: k <= ( width ( Gauge (C,(n + 1)))) and

       A4: (( Gauge (C,(n + 1))) * (( Center ( Gauge (C,(n + 1)))),k)) in ( Upper_Arc ( L~ ( Cage (C,(n + 1))))) and

       A5: (( Gauge (C,(n + 1))) * (( Center ( Gauge (C,(n + 1)))),j)) in ( Lower_Arc ( L~ ( Cage (C,(n + 1)))));

      

       A6: ( len ( Gauge (C,(n + 1)))) >= 4 by JORDAN8: 10;

      then ( len ( Gauge (C,(n + 1)))) >= 2 by XXREAL_0: 2;

      then

       A7: 1 < ( Center ( Gauge (C,(n + 1)))) by JORDAN1B: 14;

      ( len ( Gauge (C,(n + 1)))) >= 3 by A6, XXREAL_0: 2;

      hence thesis by A1, A2, A3, A4, A5, A7, Th23, JORDAN1B: 15;

    end;

    theorem :: JORDAN15:26

    for C be Simple_closed_curve holds for j,k be Nat holds 1 <= j & j <= k & k <= ( width ( Gauge (C,(n + 1)))) & (( Gauge (C,(n + 1))) * (( Center ( Gauge (C,(n + 1)))),k)) in ( Upper_Arc ( L~ ( Cage (C,(n + 1))))) & (( Gauge (C,(n + 1))) * (( Center ( Gauge (C,(n + 1)))),j)) in ( Lower_Arc ( L~ ( Cage (C,(n + 1))))) implies ( LSeg ((( Gauge (C,(n + 1))) * (( Center ( Gauge (C,(n + 1)))),j)),(( Gauge (C,(n + 1))) * (( Center ( Gauge (C,(n + 1)))),k)))) meets ( Upper_Arc C)

    proof

      let C be Simple_closed_curve;

      let j,k be Nat;

      assume that

       A1: 1 <= j and

       A2: j <= k and

       A3: k <= ( width ( Gauge (C,(n + 1)))) and

       A4: (( Gauge (C,(n + 1))) * (( Center ( Gauge (C,(n + 1)))),k)) in ( Upper_Arc ( L~ ( Cage (C,(n + 1))))) and

       A5: (( Gauge (C,(n + 1))) * (( Center ( Gauge (C,(n + 1)))),j)) in ( Lower_Arc ( L~ ( Cage (C,(n + 1)))));

      

       A6: ( len ( Gauge (C,(n + 1)))) >= 4 by JORDAN8: 10;

      then ( len ( Gauge (C,(n + 1)))) >= 2 by XXREAL_0: 2;

      then

       A7: 1 < ( Center ( Gauge (C,(n + 1)))) by JORDAN1B: 14;

      ( len ( Gauge (C,(n + 1)))) >= 3 by A6, XXREAL_0: 2;

      hence thesis by A1, A2, A3, A4, A5, A7, Th24, JORDAN1B: 15;

    end;

    theorem :: JORDAN15:27

    

     Th27: for C be compact connected non vertical non horizontal Subset of ( TOP-REAL 2) holds for i,j,k be Nat st 1 < j & k < ( len ( Gauge (C,n))) & 1 <= i & i <= ( width ( Gauge (C,n))) & (( Gauge (C,n)) * (k,i)) in ( L~ ( Upper_Seq (C,n))) & (( Gauge (C,n)) * (j,i)) in ( L~ ( Lower_Seq (C,n))) holds j <> k

    proof

      let C be compact connected non vertical non horizontal Subset of ( TOP-REAL 2);

      let i,j,k be Nat;

      assume that

       A1: 1 < j and

       A2: k < ( len ( Gauge (C,n))) and

       A3: 1 <= i and

       A4: i <= ( width ( Gauge (C,n))) and

       A5: (( Gauge (C,n)) * (k,i)) in ( L~ ( Upper_Seq (C,n))) and

       A6: (( Gauge (C,n)) * (j,i)) in ( L~ ( Lower_Seq (C,n))) and

       A7: j = k;

      

       A8: [j, i] in ( Indices ( Gauge (C,n))) by A1, A2, A3, A4, A7, MATRIX_0: 30;

      (( Gauge (C,n)) * (k,i)) in (( L~ ( Upper_Seq (C,n))) /\ ( L~ ( Lower_Seq (C,n)))) by A5, A6, A7, XBOOLE_0:def 4;

      then

       A9: (( Gauge (C,n)) * (k,i)) in {( W-min ( L~ ( Cage (C,n)))), ( E-max ( L~ ( Cage (C,n))))} by JORDAN1E: 16;

      

       A10: ( len ( Gauge (C,n))) = ( width ( Gauge (C,n))) by JORDAN8:def 1;

      ( len ( Gauge (C,n))) >= 4 by JORDAN8: 10;

      then

       A11: ( len ( Gauge (C,n))) >= 1 by XXREAL_0: 2;

      then

       A12: [( len ( Gauge (C,n))), i] in ( Indices ( Gauge (C,n))) by A3, A4, MATRIX_0: 30;

      

       A13: [1, i] in ( Indices ( Gauge (C,n))) by A3, A4, A11, MATRIX_0: 30;

      per cases by A9, TARSKI:def 2;

        suppose

         A14: (( Gauge (C,n)) * (k,i)) = ( W-min ( L~ ( Cage (C,n))));

        ((( Gauge (C,n)) * (1,i)) `1 ) = ( W-bound ( L~ ( Cage (C,n)))) by A3, A4, A10, JORDAN1A: 73;

        then (( W-min ( L~ ( Cage (C,n)))) `1 ) <> ( W-bound ( L~ ( Cage (C,n)))) by A1, A7, A8, A13, A14, JORDAN1G: 7;

        hence contradiction by EUCLID: 52;

      end;

        suppose

         A15: (( Gauge (C,n)) * (k,i)) = ( E-max ( L~ ( Cage (C,n))));

        ((( Gauge (C,n)) * (( len ( Gauge (C,n))),i)) `1 ) = ( E-bound ( L~ ( Cage (C,n)))) by A3, A4, A10, JORDAN1A: 71;

        then (( E-max ( L~ ( Cage (C,n)))) `1 ) <> ( E-bound ( L~ ( Cage (C,n)))) by A2, A7, A8, A12, A15, JORDAN1G: 7;

        hence contradiction by EUCLID: 52;

      end;

    end;

    theorem :: JORDAN15:28

    

     Th28: for C be Simple_closed_curve holds for i,j,k be Nat st 1 < j & j <= k & k < ( len ( Gauge (C,n))) & 1 <= i & i <= ( width ( Gauge (C,n))) & (( LSeg ((( Gauge (C,n)) * (j,i)),(( Gauge (C,n)) * (k,i)))) /\ ( L~ ( Upper_Seq (C,n)))) = {(( Gauge (C,n)) * (k,i))} & (( LSeg ((( Gauge (C,n)) * (j,i)),(( Gauge (C,n)) * (k,i)))) /\ ( L~ ( Lower_Seq (C,n)))) = {(( Gauge (C,n)) * (j,i))} holds ( LSeg ((( Gauge (C,n)) * (j,i)),(( Gauge (C,n)) * (k,i)))) meets ( Lower_Arc C)

    proof

      let C be Simple_closed_curve;

      let i,j,k be Nat;

      set Ga = ( Gauge (C,n));

      set US = ( Upper_Seq (C,n));

      set LS = ( Lower_Seq (C,n));

      set LA = ( Lower_Arc C);

      set Wmin = ( W-min ( L~ ( Cage (C,n))));

      set Emax = ( E-max ( L~ ( Cage (C,n))));

      set Wbo = ( W-bound ( L~ ( Cage (C,n))));

      set Ebo = ( E-bound ( L~ ( Cage (C,n))));

      set Gij = (Ga * (j,i));

      set Gik = (Ga * (k,i));

      assume that

       A1: 1 < j and

       A2: j <= k and

       A3: k < ( len Ga) and

       A4: 1 <= i and

       A5: i <= ( width Ga) and

       A6: (( LSeg (Gij,Gik)) /\ ( L~ US)) = {Gik} and

       A7: (( LSeg (Gij,Gik)) /\ ( L~ LS)) = {Gij} and

       A8: ( LSeg (Gij,Gik)) misses LA;

      Gij in {Gij} by TARSKI:def 1;

      then

       A9: Gij in ( L~ LS) by A7, XBOOLE_0:def 4;

      Gik in {Gik} by TARSKI:def 1;

      then

       A10: Gik in ( L~ US) by A6, XBOOLE_0:def 4;

      

       A11: ( len Ga) = ( width Ga) by JORDAN8:def 1;

      

       A12: j <> k by A1, A3, A4, A5, A9, A10, Th27;

      

       A13: j <= ( width Ga) by A2, A3, A11, XXREAL_0: 2;

      

       A14: 1 <= k by A1, A2, XXREAL_0: 2;

      

       A15: k <= ( width Ga) by A3, JORDAN8:def 1;

      

       A16: [j, i] in ( Indices Ga) by A1, A4, A5, A11, A13, MATRIX_0: 30;

      

       A17: [k, i] in ( Indices Ga) by A3, A4, A5, A14, MATRIX_0: 30;

      set go = ( R_Cut (US,Gik));

      set co = ( L_Cut (LS,Gij));

      

       A18: ( len US) >= 3 by JORDAN1E: 15;

      then ( len US) >= 1 by XXREAL_0: 2;

      then 1 in ( dom US) by FINSEQ_3: 25;

      

      then

       A19: (US . 1) = (US /. 1) by PARTFUN1:def 6

      .= Wmin by JORDAN1F: 5;

      

       A20: (Wmin `1 ) = Wbo by EUCLID: 52

      .= ((Ga * (1,k)) `1 ) by A3, A14, JORDAN1A: 73;

      ( len Ga) >= 4 by JORDAN8: 10;

      then

       A21: ( len Ga) >= 1 by XXREAL_0: 2;

      then

       A22: [1, k] in ( Indices Ga) by A14, A15, MATRIX_0: 30;

      then

       A23: Gik <> (US . 1) by A1, A2, A17, A19, A20, JORDAN1G: 7;

      then

      reconsider go as being_S-Seq FinSequence of ( TOP-REAL 2) by A10, JORDAN3: 35;

      

       A24: [1, j] in ( Indices Ga) by A1, A13, A21, MATRIX_0: 30;

      

       A25: ( len LS) >= (1 + 2) by JORDAN1E: 15;

      then

       A26: ( len LS) >= 1 by XXREAL_0: 2;

      then

       A27: 1 in ( dom LS) by FINSEQ_3: 25;

      ( len LS) in ( dom LS) by A26, FINSEQ_3: 25;

      

      then

       A28: (LS . ( len LS)) = (LS /. ( len LS)) by PARTFUN1:def 6

      .= Wmin by JORDAN1F: 8;

      

       A29: (Wmin `1 ) = Wbo by EUCLID: 52

      .= ((Ga * (1,k)) `1 ) by A3, A14, JORDAN1A: 73;

      

       A30: [j, i] in ( Indices Ga) by A1, A4, A5, A11, A13, MATRIX_0: 30;

      then

       A31: Gij <> (LS . ( len LS)) by A1, A22, A28, A29, JORDAN1G: 7;

      then

      reconsider co as being_S-Seq FinSequence of ( TOP-REAL 2) by A9, JORDAN3: 34;

      

       A32: [( len Ga), k] in ( Indices Ga) by A14, A15, A21, MATRIX_0: 30;

      

       A33: (LS . 1) = (LS /. 1) by A27, PARTFUN1:def 6

      .= Emax by JORDAN1F: 6;

      (Emax `1 ) = Ebo by EUCLID: 52

      .= ((Ga * (( len Ga),k)) `1 ) by A3, A14, JORDAN1A: 71;

      then

       A34: Gij <> (LS . 1) by A2, A3, A30, A32, A33, JORDAN1G: 7;

      

       A35: ( len go) >= (1 + 1) by TOPREAL1:def 8;

      

       A36: Gik in ( rng US) by A4, A5, A10, A11, A14, A15, JORDAN1G: 4, JORDAN1J: 40;

      then

       A37: go is_sequence_on Ga by JORDAN1G: 4, JORDAN1J: 38;

      

       A38: ( len co) >= (1 + 1) by TOPREAL1:def 8;

      

       A39: Gij in ( rng LS) by A1, A4, A5, A9, A11, A13, JORDAN1G: 5, JORDAN1J: 40;

      then

       A40: co is_sequence_on Ga by JORDAN1G: 5, JORDAN1J: 39;

      reconsider go as non constant s.c.c. being_S-Seq FinSequence of ( TOP-REAL 2) by A35, A37, JGRAPH_1: 12, JORDAN8: 5;

      reconsider co as non constant s.c.c. being_S-Seq FinSequence of ( TOP-REAL 2) by A38, A40, JGRAPH_1: 12, JORDAN8: 5;

      

       A41: ( len go) > 1 by A35, NAT_1: 13;

      then

       A42: ( len go) in ( dom go) by FINSEQ_3: 25;

      

      then

       A43: (go /. ( len go)) = (go . ( len go)) by PARTFUN1:def 6

      .= Gik by A10, JORDAN3: 24;

      ( len co) >= 1 by A38, XXREAL_0: 2;

      then 1 in ( dom co) by FINSEQ_3: 25;

      

      then

       A44: (co /. 1) = (co . 1) by PARTFUN1:def 6

      .= Gij by A9, JORDAN3: 23;

      reconsider m = (( len go) - 1) as Nat by A42, FINSEQ_3: 26;

      

       A45: (m + 1) = ( len go);

      then

       A46: (( len go) -' 1) = m by NAT_D: 34;

      

       A47: ( LSeg (go,m)) c= ( L~ go) by TOPREAL3: 19;

      

       A48: ( L~ go) c= ( L~ US) by A10, JORDAN3: 41;

      then ( LSeg (go,m)) c= ( L~ US) by A47;

      then

       A49: (( LSeg (go,m)) /\ ( LSeg (Gik,Gij))) c= {Gik} by A6, XBOOLE_1: 26;

      m >= 1 by A35, XREAL_1: 19;

      then

       A50: ( LSeg (go,m)) = ( LSeg ((go /. m),Gik)) by A43, A45, TOPREAL1:def 3;

       {Gik} c= (( LSeg (go,m)) /\ ( LSeg (Gik,Gij)))

      proof

        let x be object;

        

         A51: Gik in ( LSeg (Gik,Gij)) by RLTOPSP1: 68;

        assume x in {Gik};

        then

         A52: x = Gik by TARSKI:def 1;

        Gik in ( LSeg (go,m)) by A50, RLTOPSP1: 68;

        hence thesis by A52, A51, XBOOLE_0:def 4;

      end;

      then

       A53: (( LSeg (go,m)) /\ ( LSeg (Gik,Gij))) = {Gik} by A49;

      

       A54: ( LSeg (co,1)) c= ( L~ co) by TOPREAL3: 19;

      

       A55: ( L~ co) c= ( L~ LS) by A9, JORDAN3: 42;

      then ( LSeg (co,1)) c= ( L~ LS) by A54;

      then

       A56: (( LSeg (co,1)) /\ ( LSeg (Gik,Gij))) c= {Gij} by A7, XBOOLE_1: 26;

      

       A57: ( LSeg (co,1)) = ( LSeg (Gij,(co /. (1 + 1)))) by A38, A44, TOPREAL1:def 3;

       {Gij} c= (( LSeg (co,1)) /\ ( LSeg (Gik,Gij)))

      proof

        let x be object;

        

         A58: Gij in ( LSeg (Gik,Gij)) by RLTOPSP1: 68;

        assume x in {Gij};

        then

         A59: x = Gij by TARSKI:def 1;

        Gij in ( LSeg (co,1)) by A57, RLTOPSP1: 68;

        hence thesis by A59, A58, XBOOLE_0:def 4;

      end;

      then

       A60: (( LSeg (Gik,Gij)) /\ ( LSeg (co,1))) = {Gij} by A56;

      

       A61: (go /. 1) = (US /. 1) by A10, SPRECT_3: 22

      .= Wmin by JORDAN1F: 5;

      

      then

       A62: (go /. 1) = (LS /. ( len LS)) by JORDAN1F: 8

      .= (co /. ( len co)) by A9, JORDAN1J: 35;

      

       A63: ( rng go) c= ( L~ go) by A35, SPPOL_2: 18;

      

       A64: ( rng co) c= ( L~ co) by A38, SPPOL_2: 18;

      

       A65: {(go /. 1)} c= (( L~ go) /\ ( L~ co))

      proof

        let x be object;

        assume x in {(go /. 1)};

        then

         A66: x = (go /. 1) by TARSKI:def 1;

        then

         A67: x in ( rng go) by FINSEQ_6: 42;

        x in ( rng co) by A62, A66, FINSEQ_6: 168;

        hence thesis by A63, A64, A67, XBOOLE_0:def 4;

      end;

      

       A68: (LS . 1) = (LS /. 1) by A27, PARTFUN1:def 6

      .= Emax by JORDAN1F: 6;

      

       A69: [( len Ga), j] in ( Indices Ga) by A1, A13, A21, MATRIX_0: 30;

      (( L~ go) /\ ( L~ co)) c= {(go /. 1)}

      proof

        let x be object;

        assume

         A70: x in (( L~ go) /\ ( L~ co));

        then

         A71: x in ( L~ co) by XBOOLE_0:def 4;

         A72:

        now

          assume x = Emax;

          then

           A73: Emax = Gij by A9, A68, A71, JORDAN1E: 7;

          ((Ga * (( len Ga),j)) `1 ) = Ebo by A1, A11, A13, JORDAN1A: 71;

          then (Emax `1 ) <> Ebo by A2, A3, A16, A69, A73, JORDAN1G: 7;

          hence contradiction by EUCLID: 52;

        end;

        x in ( L~ go) by A70, XBOOLE_0:def 4;

        then x in (( L~ US) /\ ( L~ LS)) by A48, A55, A71, XBOOLE_0:def 4;

        then x in {Wmin, Emax} by JORDAN1E: 16;

        then x = Wmin or x = Emax by TARSKI:def 2;

        hence thesis by A61, A72, TARSKI:def 1;

      end;

      then

       A74: (( L~ go) /\ ( L~ co)) = {(go /. 1)} by A65;

      set W2 = (go /. 2);

      

       A75: 2 in ( dom go) by A35, FINSEQ_3: 25;

       A76:

      now

        assume (Gij `1 ) = Wbo;

        then ((Ga * (1,j)) `1 ) = ((Ga * (j,i)) `1 ) by A1, A11, A13, JORDAN1A: 73;

        hence contradiction by A1, A16, A24, JORDAN1G: 7;

      end;

      go = ( mid (US,1,(Gik .. US))) by A36, JORDAN1G: 49

      .= (US | (Gik .. US)) by A36, FINSEQ_4: 21, FINSEQ_6: 116;

      then

       A77: W2 = (US /. 2) by A75, FINSEQ_4: 70;

      

       A78: Wmin in ( rng go) by A61, FINSEQ_6: 42;

      set pion = <*Gik, Gij*>;

       A79:

      now

        let n be Nat;

        assume n in ( dom pion);

        then n in {1, 2} by FINSEQ_1: 2, FINSEQ_1: 89;

        then n = 1 or n = 2 by TARSKI:def 2;

        hence ex j,i be Nat st [j, i] in ( Indices Ga) & (pion /. n) = (Ga * (j,i)) by A16, A17, FINSEQ_4: 17;

      end;

      

       A80: Gik <> Gij by A12, A16, A17, GOBOARD1: 5;

      (Gik `2 ) = ((Ga * (1,i)) `2 ) by A3, A4, A5, A14, GOBOARD5: 1

      .= (Gij `2 ) by A1, A4, A5, A11, A13, GOBOARD5: 1;

      then ( LSeg (Gik,Gij)) is horizontal by SPPOL_1: 15;

      then pion is being_S-Seq by A80, JORDAN1B: 8;

      then

      consider pion1 be FinSequence of ( TOP-REAL 2) such that

       A81: pion1 is_sequence_on Ga and

       A82: pion1 is being_S-Seq and

       A83: ( L~ pion) = ( L~ pion1) and

       A84: (pion /. 1) = (pion1 /. 1) and

       A85: (pion /. ( len pion)) = (pion1 /. ( len pion1)) and

       A86: ( len pion) <= ( len pion1) by A79, GOBOARD3: 2;

      reconsider pion1 as being_S-Seq FinSequence of ( TOP-REAL 2) by A82;

      set godo = ((go ^' pion1) ^' co);

      

       A87: (1 + 1) <= ( len ( Cage (C,n))) by GOBOARD7: 34, XXREAL_0: 2;

      

       A88: (1 + 1) <= ( len ( Rotate (( Cage (C,n)),Wmin))) by GOBOARD7: 34, XXREAL_0: 2;

      ( len (go ^' pion1)) >= ( len go) by TOPREAL8: 7;

      then

       A89: ( len (go ^' pion1)) >= (1 + 1) by A35, XXREAL_0: 2;

      then

       A90: ( len (go ^' pion1)) > (1 + 0 ) by NAT_1: 13;

      

       A91: ( len godo) >= ( len (go ^' pion1)) by TOPREAL8: 7;

      then

       A92: (1 + 1) <= ( len godo) by A89, XXREAL_0: 2;

      

       A93: US is_sequence_on Ga by JORDAN1G: 4;

      

       A94: (go /. ( len go)) = (pion1 /. 1) by A43, A84, FINSEQ_4: 17;

      then

       A95: (go ^' pion1) is_sequence_on Ga by A37, A81, TOPREAL8: 12;

      

       A96: ((go ^' pion1) /. ( len (go ^' pion1))) = (pion /. ( len pion)) by A85, FINSEQ_6: 156

      .= (pion /. 2) by FINSEQ_1: 44

      .= (co /. 1) by A44, FINSEQ_4: 17;

      then

       A97: godo is_sequence_on Ga by A40, A95, TOPREAL8: 12;

      ( LSeg (pion1,1)) c= ( L~ <*Gik, Gij*>) by A83, TOPREAL3: 19;

      then ( LSeg (pion1,1)) c= ( LSeg (Gik,Gij)) by SPPOL_2: 21;

      then

       A98: (( LSeg (go,(( len go) -' 1))) /\ ( LSeg (pion1,1))) c= {Gik} by A46, A53, XBOOLE_1: 27;

      

       A99: ( len pion1) >= (1 + 1) by A86, FINSEQ_1: 44;

       {Gik} c= (( LSeg (go,m)) /\ ( LSeg (pion1,1)))

      proof

        let x be object;

        assume x in {Gik};

        then

         A100: x = Gik by TARSKI:def 1;

        

         A101: Gik in ( LSeg (go,m)) by A50, RLTOPSP1: 68;

        Gik in ( LSeg (pion1,1)) by A43, A94, A99, TOPREAL1: 21;

        hence thesis by A100, A101, XBOOLE_0:def 4;

      end;

      then (( LSeg (go,(( len go) -' 1))) /\ ( LSeg (pion1,1))) = {(go /. ( len go))} by A43, A46, A98;

      then

       A102: (go ^' pion1) is unfolded by A94, TOPREAL8: 34;

      ( len pion1) >= (2 + 0 ) by A86, FINSEQ_1: 44;

      then

       A103: (( len pion1) - 2) >= 0 by XREAL_1: 19;

      ((( len (go ^' pion1)) + 1) - 1) = ((( len go) + ( len pion1)) - 1) by FINSEQ_6: 139;

      

      then (( len (go ^' pion1)) - 1) = (( len go) + (( len pion1) - 2))

      .= (( len go) + (( len pion1) -' 2)) by A103, XREAL_0:def 2;

      then

       A104: (( len (go ^' pion1)) -' 1) = (( len go) + (( len pion1) -' 2)) by XREAL_0:def 2;

      

       A105: (( len pion1) - 1) >= 1 by A99, XREAL_1: 19;

      then

       A106: (( len pion1) -' 1) = (( len pion1) - 1) by XREAL_0:def 2;

      

       A107: ((( len pion1) -' 2) + 1) = ((( len pion1) - 2) + 1) by A103, XREAL_0:def 2

      .= (( len pion1) -' 1) by A105, XREAL_0:def 2;

      ((( len pion1) - 1) + 1) <= ( len pion1);

      then

       A108: (( len pion1) -' 1) < ( len pion1) by A106, NAT_1: 13;

      ( LSeg (pion1,(( len pion1) -' 1))) c= ( L~ <*Gik, Gij*>) by A83, TOPREAL3: 19;

      then ( LSeg (pion1,(( len pion1) -' 1))) c= ( LSeg (Gik,Gij)) by SPPOL_2: 21;

      then

       A109: (( LSeg (pion1,(( len pion1) -' 1))) /\ ( LSeg (co,1))) c= {Gij} by A60, XBOOLE_1: 27;

       {Gij} c= (( LSeg (pion1,(( len pion1) -' 1))) /\ ( LSeg (co,1)))

      proof

        let x be object;

        assume x in {Gij};

        then

         A110: x = Gij by TARSKI:def 1;

        (pion1 /. ((( len pion1) -' 1) + 1)) = (pion /. 2) by A85, A106, FINSEQ_1: 44

        .= Gij by FINSEQ_4: 17;

        then

         A111: Gij in ( LSeg (pion1,(( len pion1) -' 1))) by A105, A106, TOPREAL1: 21;

        Gij in ( LSeg (co,1)) by A57, RLTOPSP1: 68;

        hence thesis by A110, A111, XBOOLE_0:def 4;

      end;

      then (( LSeg (pion1,(( len pion1) -' 1))) /\ ( LSeg (co,1))) = {Gij} by A109;

      then

       A112: (( LSeg ((go ^' pion1),(( len go) + (( len pion1) -' 2)))) /\ ( LSeg (co,1))) = {((go ^' pion1) /. ( len (go ^' pion1)))} by A44, A94, A96, A107, A108, TOPREAL8: 31;

      

       A113: (go ^' pion1) is non trivial by A89, NAT_D: 60;

      

       A114: ( rng pion1) c= ( L~ pion1) by A99, SPPOL_2: 18;

      

       A115: {(pion1 /. 1)} c= (( L~ go) /\ ( L~ pion1))

      proof

        let x be object;

        assume x in {(pion1 /. 1)};

        then

         A116: x = (pion1 /. 1) by TARSKI:def 1;

        then

         A117: x in ( rng pion1) by FINSEQ_6: 42;

        x in ( rng go) by A94, A116, FINSEQ_6: 168;

        hence thesis by A63, A114, A117, XBOOLE_0:def 4;

      end;

      (( L~ go) /\ ( L~ pion1)) c= {(pion1 /. 1)}

      proof

        let x be object;

        assume

         A118: x in (( L~ go) /\ ( L~ pion1));

        then

         A119: x in ( L~ pion1) by XBOOLE_0:def 4;

        x in ( L~ go) by A118, XBOOLE_0:def 4;

        then x in (( L~ pion1) /\ ( L~ US)) by A48, A119, XBOOLE_0:def 4;

        hence thesis by A6, A43, A83, A94, SPPOL_2: 21;

      end;

      then

       A120: (( L~ go) /\ ( L~ pion1)) = {(pion1 /. 1)} by A115;

      then

       A121: (go ^' pion1) is s.n.c. by A94, JORDAN1J: 54;

      (( rng go) /\ ( rng pion1)) c= {(pion1 /. 1)} by A63, A114, A120, XBOOLE_1: 27;

      then

       A122: (go ^' pion1) is one-to-one by JORDAN1J: 55;

      

       A123: (pion /. ( len pion)) = (pion /. 2) by FINSEQ_1: 44

      .= (co /. 1) by A44, FINSEQ_4: 17;

      

       A124: {(pion1 /. ( len pion1))} c= (( L~ co) /\ ( L~ pion1))

      proof

        let x be object;

        assume x in {(pion1 /. ( len pion1))};

        then

         A125: x = (pion1 /. ( len pion1)) by TARSKI:def 1;

        then

         A126: x in ( rng pion1) by FINSEQ_6: 168;

        x in ( rng co) by A85, A123, A125, FINSEQ_6: 42;

        hence thesis by A64, A114, A126, XBOOLE_0:def 4;

      end;

      (( L~ co) /\ ( L~ pion1)) c= {(pion1 /. ( len pion1))}

      proof

        let x be object;

        assume

         A127: x in (( L~ co) /\ ( L~ pion1));

        then

         A128: x in ( L~ pion1) by XBOOLE_0:def 4;

        x in ( L~ co) by A127, XBOOLE_0:def 4;

        then x in (( L~ pion1) /\ ( L~ LS)) by A55, A128, XBOOLE_0:def 4;

        hence thesis by A7, A44, A83, A85, A123, SPPOL_2: 21;

      end;

      then

       A129: (( L~ co) /\ ( L~ pion1)) = {(pion1 /. ( len pion1))} by A124;

      

       A130: (( L~ (go ^' pion1)) /\ ( L~ co)) = ((( L~ go) \/ ( L~ pion1)) /\ ( L~ co)) by A94, TOPREAL8: 35

      .= ( {(go /. 1)} \/ {(co /. 1)}) by A74, A85, A123, A129, XBOOLE_1: 23

      .= ( {((go ^' pion1) /. 1)} \/ {(co /. 1)}) by FINSEQ_6: 155

      .= {((go ^' pion1) /. 1), (co /. 1)} by ENUMSET1: 1;

      (co /. ( len co)) = ((go ^' pion1) /. 1) by A62, FINSEQ_6: 155;

      then

      reconsider godo as non constant standard special_circular_sequence by A92, A96, A97, A102, A104, A112, A113, A121, A122, A130, JORDAN8: 4, JORDAN8: 5, TOPREAL8: 11, TOPREAL8: 33, TOPREAL8: 34;

      

       A131: LA is_an_arc_of (( E-max C),( W-min C)) by JORDAN6:def 9;

      then

       A132: LA is connected by JORDAN6: 10;

      

       A133: ( W-min C) in LA by A131, TOPREAL1: 1;

      

       A134: ( E-max C) in LA by A131, TOPREAL1: 1;

      set ff = ( Rotate (( Cage (C,n)),Wmin));

      Wmin in ( rng ( Cage (C,n))) by SPRECT_2: 43;

      then

       A135: (ff /. 1) = Wmin by FINSEQ_6: 92;

      

       A136: ( L~ ff) = ( L~ ( Cage (C,n))) by REVROT_1: 33;

      then (( W-max ( L~ ff)) .. ff) > 1 by A135, SPRECT_5: 22;

      then (( N-min ( L~ ff)) .. ff) > 1 by A135, A136, SPRECT_5: 23, XXREAL_0: 2;

      then (( N-max ( L~ ff)) .. ff) > 1 by A135, A136, SPRECT_5: 24, XXREAL_0: 2;

      then

       A137: (Emax .. ff) > 1 by A135, A136, SPRECT_5: 25, XXREAL_0: 2;

       A138:

      now

        assume

         A139: (Gik .. US) <= 1;

        (Gik .. US) >= 1 by A36, FINSEQ_4: 21;

        then (Gik .. US) = 1 by A139, XXREAL_0: 1;

        then Gik = (US /. 1) by A36, FINSEQ_5: 38;

        hence contradiction by A19, A23, JORDAN1F: 5;

      end;

      

       A140: ( Cage (C,n)) is_sequence_on Ga by JORDAN9:def 1;

      then

       A141: ff is_sequence_on Ga by REVROT_1: 34;

      

       A142: (( right_cell (godo,1,Ga)) \ ( L~ godo)) c= ( RightComp godo) by A92, A97, JORDAN9: 27;

      

       A143: ( L~ godo) = (( L~ (go ^' pion1)) \/ ( L~ co)) by A96, TOPREAL8: 35

      .= ((( L~ go) \/ ( L~ pion1)) \/ ( L~ co)) by A94, TOPREAL8: 35;

      

       A144: ( L~ ( Cage (C,n))) = (( L~ US) \/ ( L~ LS)) by JORDAN1E: 13;

      then

       A145: ( L~ US) c= ( L~ ( Cage (C,n))) by XBOOLE_1: 7;

      

       A146: ( L~ LS) c= ( L~ ( Cage (C,n))) by A144, XBOOLE_1: 7;

      

       A147: ( L~ go) c= ( L~ ( Cage (C,n))) by A48, A145;

      

       A148: ( L~ co) c= ( L~ ( Cage (C,n))) by A55, A146;

      

       A149: ( W-min C) in C by SPRECT_1: 13;

      

       A150: ( L~ pion) = ( LSeg (Gik,Gij)) by SPPOL_2: 21;

       A151:

      now

        assume ( W-min C) in ( L~ godo);

        then

         A152: ( W-min C) in (( L~ go) \/ ( L~ pion1)) or ( W-min C) in ( L~ co) by A143, XBOOLE_0:def 3;

        per cases by A152, XBOOLE_0:def 3;

          suppose ( W-min C) in ( L~ go);

          then C meets ( L~ ( Cage (C,n))) by A147, A149, XBOOLE_0: 3;

          hence contradiction by JORDAN10: 5;

        end;

          suppose ( W-min C) in ( L~ pion1);

          hence contradiction by A8, A83, A133, A150, XBOOLE_0: 3;

        end;

          suppose ( W-min C) in ( L~ co);

          then C meets ( L~ ( Cage (C,n))) by A148, A149, XBOOLE_0: 3;

          hence contradiction by JORDAN10: 5;

        end;

      end;

      ( right_cell (( Rotate (( Cage (C,n)),Wmin)),1)) = ( right_cell (ff,1,( GoB ff))) by A88, JORDAN1H: 23

      .= ( right_cell (ff,1,( GoB ( Cage (C,n))))) by REVROT_1: 28

      .= ( right_cell (ff,1,Ga)) by JORDAN1H: 44

      .= ( right_cell ((ff -: Emax),1,Ga)) by A137, A141, JORDAN1J: 53

      .= ( right_cell (US,1,Ga)) by JORDAN1E:def 1

      .= ( right_cell (( R_Cut (US,Gik)),1,Ga)) by A36, A93, A138, JORDAN1J: 52

      .= ( right_cell ((go ^' pion1),1,Ga)) by A41, A95, JORDAN1J: 51

      .= ( right_cell (godo,1,Ga)) by A90, A97, JORDAN1J: 51;

      then ( W-min C) in ( right_cell (godo,1,Ga)) by JORDAN1I: 6;

      then

       A153: ( W-min C) in (( right_cell (godo,1,Ga)) \ ( L~ godo)) by A151, XBOOLE_0:def 5;

      

       A154: (godo /. 1) = ((go ^' pion1) /. 1) by FINSEQ_6: 155

      .= Wmin by A61, FINSEQ_6: 155;

      

       A155: ( len US) >= 2 by A18, XXREAL_0: 2;

      

       A156: (godo /. 2) = ((go ^' pion1) /. 2) by A89, FINSEQ_6: 159

      .= (US /. 2) by A35, A77, FINSEQ_6: 159

      .= ((US ^' LS) /. 2) by A155, FINSEQ_6: 159

      .= (( Rotate (( Cage (C,n)),Wmin)) /. 2) by JORDAN1E: 11;

      

       A157: (( L~ go) \/ ( L~ co)) is compact by COMPTS_1: 10;

      Wmin in (( L~ go) \/ ( L~ co)) by A63, A78, XBOOLE_0:def 3;

      then

       A158: ( W-min (( L~ go) \/ ( L~ co))) = Wmin by A147, A148, A157, JORDAN1J: 21, XBOOLE_1: 8;

      

       A159: (( W-min (( L~ go) \/ ( L~ co))) `1 ) = ( W-bound (( L~ go) \/ ( L~ co))) by EUCLID: 52;

      

       A160: (Wmin `1 ) = Wbo by EUCLID: 52;

      

       A161: (Gij `1 ) <= (Gik `1 ) by A1, A2, A3, A4, A5, SPRECT_3: 13;

      then ( W-bound ( LSeg (Gik,Gij))) = (Gij `1 ) by SPRECT_1: 54;

      then

       A162: ( W-bound ( L~ pion1)) = (Gij `1 ) by A83, SPPOL_2: 21;

      (Gij `1 ) >= Wbo by A9, A146, PSCOMP_1: 24;

      then (Gij `1 ) > Wbo by A76, XXREAL_0: 1;

      then ( W-min ((( L~ go) \/ ( L~ co)) \/ ( L~ pion1))) = ( W-min (( L~ go) \/ ( L~ co))) by A157, A158, A159, A160, A162, JORDAN1J: 33;

      then

       A163: ( W-min ( L~ godo)) = Wmin by A143, A158, XBOOLE_1: 4;

      

       A164: ( rng godo) c= ( L~ godo) by A89, A91, SPPOL_2: 18, XXREAL_0: 2;

      2 in ( dom godo) by A92, FINSEQ_3: 25;

      then

       A165: (godo /. 2) in ( rng godo) by PARTFUN2: 2;

      (godo /. 2) in ( W-most ( L~ ( Cage (C,n)))) by A156, JORDAN1I: 25;

      

      then ((godo /. 2) `1 ) = (( W-min ( L~ godo)) `1 ) by A163, PSCOMP_1: 31

      .= ( W-bound ( L~ godo)) by EUCLID: 52;

      then (godo /. 2) in ( W-most ( L~ godo)) by A164, A165, SPRECT_2: 12;

      then (( Rotate (godo,( W-min ( L~ godo)))) /. 2) in ( W-most ( L~ godo)) by A154, A163, FINSEQ_6: 89;

      then

      reconsider godo as clockwise_oriented non constant standard special_circular_sequence by JORDAN1I: 25;

      ( len US) in ( dom US) by FINSEQ_5: 6;

      

      then

       A166: (US . ( len US)) = (US /. ( len US)) by PARTFUN1:def 6

      .= Emax by JORDAN1F: 7;

      

       A167: ( east_halfline ( E-max C)) misses ( L~ go)

      proof

        assume ( east_halfline ( E-max C)) meets ( L~ go);

        then

        consider p be object such that

         A168: p in ( east_halfline ( E-max C)) and

         A169: p in ( L~ go) by XBOOLE_0: 3;

        reconsider p as Point of ( TOP-REAL 2) by A168;

        p in ( L~ US) by A48, A169;

        then p in (( east_halfline ( E-max C)) /\ ( L~ ( Cage (C,n)))) by A145, A168, XBOOLE_0:def 4;

        then

         A170: (p `1 ) = Ebo by JORDAN1A: 83, PSCOMP_1: 50;

        then

         A171: p = Emax by A48, A169, JORDAN1J: 46;

        then Emax = Gik by A10, A166, A169, JORDAN1J: 43;

        then (Gik `1 ) = ((Ga * (( len Ga),k)) `1 ) by A3, A14, A170, A171, JORDAN1A: 71;

        hence contradiction by A3, A17, A32, JORDAN1G: 7;

      end;

      now

        assume ( east_halfline ( E-max C)) meets ( L~ godo);

        then

         A172: ( east_halfline ( E-max C)) meets (( L~ go) \/ ( L~ pion1)) or ( east_halfline ( E-max C)) meets ( L~ co) by A143, XBOOLE_1: 70;

        per cases by A172, XBOOLE_1: 70;

          suppose ( east_halfline ( E-max C)) meets ( L~ go);

          hence contradiction by A167;

        end;

          suppose ( east_halfline ( E-max C)) meets ( L~ pion1);

          then

          consider p be object such that

           A173: p in ( east_halfline ( E-max C)) and

           A174: p in ( L~ pion1) by XBOOLE_0: 3;

          reconsider p as Point of ( TOP-REAL 2) by A173;

          

           A175: (p `2 ) = (( E-max C) `2 ) by A173, TOPREAL1:def 11;

          (k + 1) <= ( len Ga) by A3, NAT_1: 13;

          then ((k + 1) - 1) <= (( len Ga) - 1) by XREAL_1: 9;

          then

           A176: k <= (( len Ga) -' 1) by XREAL_0:def 2;

          (( len Ga) -' 1) <= ( len Ga) by NAT_D: 35;

          then

           A177: (Gik `1 ) <= ((Ga * ((( len Ga) -' 1),1)) `1 ) by A4, A5, A11, A14, A21, A176, JORDAN1A: 18;

          (p `1 ) <= (Gik `1 ) by A83, A150, A161, A174, TOPREAL1: 3;

          then (p `1 ) <= ((Ga * ((( len Ga) -' 1),1)) `1 ) by A177, XXREAL_0: 2;

          then (p `1 ) <= ( E-bound C) by A21, JORDAN8: 12;

          then

           A178: (p `1 ) <= (( E-max C) `1 ) by EUCLID: 52;

          (p `1 ) >= (( E-max C) `1 ) by A173, TOPREAL1:def 11;

          then (p `1 ) = (( E-max C) `1 ) by A178, XXREAL_0: 1;

          then p = ( E-max C) by A175, TOPREAL3: 6;

          hence contradiction by A8, A83, A134, A150, A174, XBOOLE_0: 3;

        end;

          suppose ( east_halfline ( E-max C)) meets ( L~ co);

          then

          consider p be object such that

           A179: p in ( east_halfline ( E-max C)) and

           A180: p in ( L~ co) by XBOOLE_0: 3;

          reconsider p as Point of ( TOP-REAL 2) by A179;

          

           A181: p in ( LSeg (co,( Index (p,co)))) by A180, JORDAN3: 9;

          consider t be Nat such that

           A182: t in ( dom LS) and

           A183: (LS . t) = Gij by A39, FINSEQ_2: 10;

          1 <= t by A182, FINSEQ_3: 25;

          then

           A184: 1 < t by A34, A183, XXREAL_0: 1;

          t <= ( len LS) by A182, FINSEQ_3: 25;

          then (( Index (Gij,LS)) + 1) = t by A183, A184, JORDAN3: 12;

          then

           A185: ( len ( L_Cut (LS,Gij))) = (( len LS) - ( Index (Gij,LS))) by A9, A183, JORDAN3: 26;

          ( Index (p,co)) < ( len co) by A180, JORDAN3: 8;

          then ( Index (p,co)) < (( len LS) -' ( Index (Gij,LS))) by A185, XREAL_0:def 2;

          then (( Index (p,co)) + 1) <= (( len LS) -' ( Index (Gij,LS))) by NAT_1: 13;

          then

           A186: ( Index (p,co)) <= ((( len LS) -' ( Index (Gij,LS))) - 1) by XREAL_1: 19;

          

           A187: co = ( mid (LS,(Gij .. LS),( len LS))) by A39, JORDAN1J: 37;

          p in ( L~ LS) by A55, A180;

          then p in (( east_halfline ( E-max C)) /\ ( L~ ( Cage (C,n)))) by A146, A179, XBOOLE_0:def 4;

          then

           A188: (p `1 ) = Ebo by JORDAN1A: 83, PSCOMP_1: 50;

          

           A189: (( Index (Gij,LS)) + 1) = (Gij .. LS) by A34, A39, JORDAN1J: 56;

          ( 0 + ( Index (Gij,LS))) < ( len LS) by A9, JORDAN3: 8;

          then (( len LS) - ( Index (Gij,LS))) > 0 by XREAL_1: 20;

          then ( Index (p,co)) <= ((( len LS) - ( Index (Gij,LS))) - 1) by A186, XREAL_0:def 2;

          then ( Index (p,co)) <= (( len LS) - (Gij .. LS)) by A189;

          then ( Index (p,co)) <= (( len LS) -' (Gij .. LS)) by XREAL_0:def 2;

          then

           A190: ( Index (p,co)) < ((( len LS) -' (Gij .. LS)) + 1) by NAT_1: 13;

          

           A191: 1 <= ( Index (p,co)) by A180, JORDAN3: 8;

          

           A192: (Gij .. LS) <= ( len LS) by A39, FINSEQ_4: 21;

          (Gij .. LS) <> ( len LS) by A31, A39, FINSEQ_4: 19;

          then

           A193: (Gij .. LS) < ( len LS) by A192, XXREAL_0: 1;

          

           A194: (1 + 1) <= ( len LS) by A25, XXREAL_0: 2;

          then

           A195: 2 in ( dom LS) by FINSEQ_3: 25;

          set tt = ((( Index (p,co)) + (Gij .. LS)) -' 1);

          set RC = ( Rotate (( Cage (C,n)),Emax));

          

           A196: ( E-max C) in ( right_cell (RC,1)) by JORDAN1I: 7;

          

           A197: ( GoB RC) = ( GoB ( Cage (C,n))) by REVROT_1: 28

          .= Ga by JORDAN1H: 44;

          

           A198: ( L~ RC) = ( L~ ( Cage (C,n))) by REVROT_1: 33;

          consider jj2 be Nat such that

           A199: 1 <= jj2 and

           A200: jj2 <= ( width Ga) and

           A201: Emax = (Ga * (( len Ga),jj2)) by JORDAN1D: 25;

          

           A202: ( len Ga) >= 4 by JORDAN8: 10;

          then ( len Ga) >= 1 by XXREAL_0: 2;

          then

           A203: [( len Ga), jj2] in ( Indices Ga) by A199, A200, MATRIX_0: 30;

          

           A204: ( len RC) = ( len ( Cage (C,n))) by FINSEQ_6: 179;

          LS = (RC -: Wmin) by JORDAN1G: 18;

          then

           A205: ( LSeg (LS,1)) = ( LSeg (RC,1)) by A194, SPPOL_2: 9;

          

           A206: Emax in ( rng ( Cage (C,n))) by SPRECT_2: 46;

          RC is_sequence_on Ga by A140, REVROT_1: 34;

          then

          consider ii,jj be Nat such that

           A207: [ii, (jj + 1)] in ( Indices Ga) and

           A208: [ii, jj] in ( Indices Ga) and

           A209: (RC /. 1) = (Ga * (ii,(jj + 1))) and

           A210: (RC /. (1 + 1)) = (Ga * (ii,jj)) by A87, A198, A204, A206, FINSEQ_6: 92, JORDAN1I: 23;

          

           A211: ((jj + 1) + 1) <> jj;

          

           A212: 1 <= jj by A208, MATRIX_0: 32;

          (RC /. 1) = ( E-max ( L~ RC)) by A198, A206, FINSEQ_6: 92;

          then

           A213: ii = ( len Ga) by A198, A207, A209, A201, A203, GOBOARD1: 5;

          then (ii - 1) >= (4 - 1) by A202, XREAL_1: 9;

          then

           A214: (ii - 1) >= 1 by XXREAL_0: 2;

          then

           A215: 1 <= (ii -' 1) by XREAL_0:def 2;

          

           A216: jj <= ( width Ga) by A208, MATRIX_0: 32;

          then

           A217: ((Ga * (( len Ga),jj)) `1 ) = Ebo by A11, A212, JORDAN1A: 71;

          

           A218: (jj + 1) <= ( width Ga) by A207, MATRIX_0: 32;

          (ii + 1) <> ii;

          then

           A219: ( right_cell (RC,1)) = ( cell (Ga,(ii -' 1),jj)) by A87, A204, A197, A207, A208, A209, A210, A211, GOBOARD5:def 6;

          

           A220: ii <= ( len Ga) by A208, MATRIX_0: 32;

          

           A221: 1 <= ii by A208, MATRIX_0: 32;

          

           A222: ii <= ( len Ga) by A207, MATRIX_0: 32;

          

           A223: 1 <= (jj + 1) by A207, MATRIX_0: 32;

          then

           A224: Ebo = ((Ga * (( len Ga),(jj + 1))) `1 ) by A11, A218, JORDAN1A: 71;

          

           A225: 1 <= ii by A207, MATRIX_0: 32;

          then

           A226: ((ii -' 1) + 1) = ii by XREAL_1: 235;

          then

           A227: (ii -' 1) < ( len Ga) by A222, NAT_1: 13;

          

          then

           A228: ((Ga * ((ii -' 1),(jj + 1))) `2 ) = ((Ga * (1,(jj + 1))) `2 ) by A223, A218, A215, GOBOARD5: 1

          .= ((Ga * (ii,(jj + 1))) `2 ) by A225, A222, A223, A218, GOBOARD5: 1;

          

           A229: (( E-max C) `2 ) = (p `2 ) by A179, TOPREAL1:def 11;

          then

           A230: (p `2 ) <= ((Ga * ((ii -' 1),(jj + 1))) `2 ) by A196, A222, A218, A212, A219, A226, A214, JORDAN9: 17;

          

           A231: ((Ga * ((ii -' 1),jj)) `2 ) = ((Ga * (1,jj)) `2 ) by A212, A216, A215, A227, GOBOARD5: 1

          .= ((Ga * (ii,jj)) `2 ) by A221, A220, A212, A216, GOBOARD5: 1;

          ((Ga * ((ii -' 1),jj)) `2 ) <= (p `2 ) by A229, A196, A222, A218, A212, A219, A226, A214, JORDAN9: 17;

          then p in ( LSeg ((RC /. 1),(RC /. (1 + 1)))) by A188, A209, A210, A213, A230, A231, A228, A217, A224, GOBOARD7: 7;

          then

           A232: p in ( LSeg (LS,1)) by A87, A205, A204, TOPREAL1:def 3;

          1 <= (Gij .. LS) by A39, FINSEQ_4: 21;

          then

           A233: ( LSeg (( mid (LS,(Gij .. LS),( len LS))),( Index (p,co)))) = ( LSeg (LS,((( Index (p,co)) + (Gij .. LS)) -' 1))) by A193, A191, A190, JORDAN4: 19;

          1 <= ( Index (Gij,LS)) by A9, JORDAN3: 8;

          then

           A234: (1 + 1) <= (Gij .. LS) by A189, XREAL_1: 7;

          then (( Index (p,co)) + (Gij .. LS)) >= ((1 + 1) + 1) by A191, XREAL_1: 7;

          then ((( Index (p,co)) + (Gij .. LS)) - 1) >= (((1 + 1) + 1) - 1) by XREAL_1: 9;

          then

           A235: tt >= (1 + 1) by XREAL_0:def 2;

          now

            per cases by A235, XXREAL_0: 1;

              suppose tt > (1 + 1);

              then ( LSeg (LS,1)) misses ( LSeg (LS,tt)) by TOPREAL1:def 7;

              hence contradiction by A232, A181, A187, A233, XBOOLE_0: 3;

            end;

              suppose

               A236: tt = (1 + 1);

              then (1 + 1) = ((( Index (p,co)) + (Gij .. LS)) - 1) by XREAL_0:def 2;

              then ((1 + 1) + 1) = (( Index (p,co)) + (Gij .. LS));

              then

               A237: (Gij .. LS) = 2 by A191, A234, JORDAN1E: 6;

              (( LSeg (LS,1)) /\ ( LSeg (LS,tt))) = {(LS /. 2)} by A25, A236, TOPREAL1:def 6;

              then p in {(LS /. 2)} by A232, A181, A187, A233, XBOOLE_0:def 4;

              then

               A238: p = (LS /. 2) by TARSKI:def 1;

              then

               A239: p in ( rng LS) by A195, PARTFUN2: 2;

              (p .. LS) = 2 by A195, A238, FINSEQ_5: 41;

              then p = Gij by A39, A237, A239, FINSEQ_5: 9;

              then (Gij `1 ) = Ebo by A238, JORDAN1G: 32;

              then (Gij `1 ) = ((Ga * (( len Ga),j)) `1 ) by A1, A11, A13, JORDAN1A: 71;

              hence contradiction by A2, A3, A16, A69, JORDAN1G: 7;

            end;

          end;

          hence contradiction;

        end;

      end;

      then ( east_halfline ( E-max C)) c= (( L~ godo) ` ) by SUBSET_1: 23;

      then

      consider W be Subset of ( TOP-REAL 2) such that

       A240: W is_a_component_of (( L~ godo) ` ) and

       A241: ( east_halfline ( E-max C)) c= W by GOBOARD9: 3;

       not W is bounded by A241, JORDAN2C: 121, RLTOPSP1: 42;

      then W is_outside_component_of ( L~ godo) by A240, JORDAN2C:def 3;

      then W c= ( UBD ( L~ godo)) by JORDAN2C: 23;

      then

       A242: ( east_halfline ( E-max C)) c= ( UBD ( L~ godo)) by A241;

      ( E-max C) in ( east_halfline ( E-max C)) by TOPREAL1: 38;

      then ( E-max C) in ( UBD ( L~ godo)) by A242;

      then ( E-max C) in ( LeftComp godo) by GOBRD14: 36;

      then LA meets ( L~ godo) by A132, A133, A134, A142, A153, JORDAN1J: 36;

      then

       A243: LA meets (( L~ go) \/ ( L~ pion1)) or LA meets ( L~ co) by A143, XBOOLE_1: 70;

      

       A244: LA c= C by JORDAN6: 61;

      per cases by A243, XBOOLE_1: 70;

        suppose LA meets ( L~ go);

        then LA meets ( L~ ( Cage (C,n))) by A48, A145, XBOOLE_1: 1, XBOOLE_1: 63;

        hence contradiction by A244, JORDAN10: 5, XBOOLE_1: 63;

      end;

        suppose LA meets ( L~ pion1);

        hence contradiction by A8, A83, A150;

      end;

        suppose LA meets ( L~ co);

        then LA meets ( L~ ( Cage (C,n))) by A55, A146, XBOOLE_1: 1, XBOOLE_1: 63;

        hence contradiction by A244, JORDAN10: 5, XBOOLE_1: 63;

      end;

    end;

    theorem :: JORDAN15:29

    

     Th29: for C be Simple_closed_curve holds for i,j,k be Nat st 1 < j & j <= k & k < ( len ( Gauge (C,n))) & 1 <= i & i <= ( width ( Gauge (C,n))) & (( LSeg ((( Gauge (C,n)) * (j,i)),(( Gauge (C,n)) * (k,i)))) /\ ( L~ ( Upper_Seq (C,n)))) = {(( Gauge (C,n)) * (k,i))} & (( LSeg ((( Gauge (C,n)) * (j,i)),(( Gauge (C,n)) * (k,i)))) /\ ( L~ ( Lower_Seq (C,n)))) = {(( Gauge (C,n)) * (j,i))} holds ( LSeg ((( Gauge (C,n)) * (j,i)),(( Gauge (C,n)) * (k,i)))) meets ( Upper_Arc C)

    proof

      let C be Simple_closed_curve;

      let i,j,k be Nat;

      set Ga = ( Gauge (C,n));

      set US = ( Upper_Seq (C,n));

      set LS = ( Lower_Seq (C,n));

      set UA = ( Upper_Arc C);

      set Wmin = ( W-min ( L~ ( Cage (C,n))));

      set Emax = ( E-max ( L~ ( Cage (C,n))));

      set Wbo = ( W-bound ( L~ ( Cage (C,n))));

      set Ebo = ( E-bound ( L~ ( Cage (C,n))));

      set Gij = (Ga * (j,i));

      set Gik = (Ga * (k,i));

      assume that

       A1: 1 < j and

       A2: j <= k and

       A3: k < ( len Ga) and

       A4: 1 <= i and

       A5: i <= ( width Ga) and

       A6: (( LSeg (Gij,Gik)) /\ ( L~ US)) = {Gik} and

       A7: (( LSeg (Gij,Gik)) /\ ( L~ LS)) = {Gij} and

       A8: ( LSeg (Gij,Gik)) misses UA;

      Gij in {Gij} by TARSKI:def 1;

      then

       A9: Gij in ( L~ LS) by A7, XBOOLE_0:def 4;

      Gik in {Gik} by TARSKI:def 1;

      then

       A10: Gik in ( L~ US) by A6, XBOOLE_0:def 4;

      

       A11: ( len Ga) = ( width Ga) by JORDAN8:def 1;

      

       A12: j <> k by A1, A3, A4, A5, A9, A10, Th27;

      

       A13: j <= ( width Ga) by A2, A3, A11, XXREAL_0: 2;

      

       A14: 1 <= k by A1, A2, XXREAL_0: 2;

      

       A15: k <= ( width Ga) by A3, JORDAN8:def 1;

      

       A16: [j, i] in ( Indices Ga) by A1, A4, A5, A11, A13, MATRIX_0: 30;

      

       A17: [k, i] in ( Indices Ga) by A3, A4, A5, A14, MATRIX_0: 30;

      set go = ( R_Cut (US,Gik));

      set co = ( L_Cut (LS,Gij));

      

       A18: ( len US) >= 3 by JORDAN1E: 15;

      then ( len US) >= 1 by XXREAL_0: 2;

      then 1 in ( dom US) by FINSEQ_3: 25;

      

      then

       A19: (US . 1) = (US /. 1) by PARTFUN1:def 6

      .= Wmin by JORDAN1F: 5;

      

       A20: (Wmin `1 ) = Wbo by EUCLID: 52

      .= ((Ga * (1,k)) `1 ) by A3, A14, JORDAN1A: 73;

      ( len Ga) >= 4 by JORDAN8: 10;

      then

       A21: ( len Ga) >= 1 by XXREAL_0: 2;

      then

       A22: [1, k] in ( Indices Ga) by A14, A15, MATRIX_0: 30;

      then

       A23: Gik <> (US . 1) by A1, A2, A17, A19, A20, JORDAN1G: 7;

      then

      reconsider go as being_S-Seq FinSequence of ( TOP-REAL 2) by A10, JORDAN3: 35;

      

       A24: [1, j] in ( Indices Ga) by A1, A13, A21, MATRIX_0: 30;

      

       A25: ( len LS) >= (1 + 2) by JORDAN1E: 15;

      then

       A26: ( len LS) >= 1 by XXREAL_0: 2;

      then

       A27: 1 in ( dom LS) by FINSEQ_3: 25;

      ( len LS) in ( dom LS) by A26, FINSEQ_3: 25;

      

      then

       A28: (LS . ( len LS)) = (LS /. ( len LS)) by PARTFUN1:def 6

      .= Wmin by JORDAN1F: 8;

      

       A29: (Wmin `1 ) = Wbo by EUCLID: 52

      .= ((Ga * (1,k)) `1 ) by A3, A14, JORDAN1A: 73;

      

       A30: [j, i] in ( Indices Ga) by A1, A4, A5, A11, A13, MATRIX_0: 30;

      then

       A31: Gij <> (LS . ( len LS)) by A1, A22, A28, A29, JORDAN1G: 7;

      then

      reconsider co as being_S-Seq FinSequence of ( TOP-REAL 2) by A9, JORDAN3: 34;

      

       A32: [( len Ga), k] in ( Indices Ga) by A14, A15, A21, MATRIX_0: 30;

      

       A33: (LS . 1) = (LS /. 1) by A27, PARTFUN1:def 6

      .= Emax by JORDAN1F: 6;

      (Emax `1 ) = Ebo by EUCLID: 52

      .= ((Ga * (( len Ga),k)) `1 ) by A3, A14, JORDAN1A: 71;

      then

       A34: Gij <> (LS . 1) by A2, A3, A30, A32, A33, JORDAN1G: 7;

      

       A35: ( len go) >= (1 + 1) by TOPREAL1:def 8;

      

       A36: Gik in ( rng US) by A4, A5, A10, A11, A14, A15, JORDAN1G: 4, JORDAN1J: 40;

      then

       A37: go is_sequence_on Ga by JORDAN1G: 4, JORDAN1J: 38;

      

       A38: ( len co) >= (1 + 1) by TOPREAL1:def 8;

      

       A39: Gij in ( rng LS) by A1, A4, A5, A9, A11, A13, JORDAN1G: 5, JORDAN1J: 40;

      then

       A40: co is_sequence_on Ga by JORDAN1G: 5, JORDAN1J: 39;

      reconsider go as non constant s.c.c. being_S-Seq FinSequence of ( TOP-REAL 2) by A35, A37, JGRAPH_1: 12, JORDAN8: 5;

      reconsider co as non constant s.c.c. being_S-Seq FinSequence of ( TOP-REAL 2) by A38, A40, JGRAPH_1: 12, JORDAN8: 5;

      

       A41: ( len go) > 1 by A35, NAT_1: 13;

      then

       A42: ( len go) in ( dom go) by FINSEQ_3: 25;

      

      then

       A43: (go /. ( len go)) = (go . ( len go)) by PARTFUN1:def 6

      .= Gik by A10, JORDAN3: 24;

      ( len co) >= 1 by A38, XXREAL_0: 2;

      then 1 in ( dom co) by FINSEQ_3: 25;

      

      then

       A44: (co /. 1) = (co . 1) by PARTFUN1:def 6

      .= Gij by A9, JORDAN3: 23;

      reconsider m = (( len go) - 1) as Nat by A42, FINSEQ_3: 26;

      

       A45: (m + 1) = ( len go);

      then

       A46: (( len go) -' 1) = m by NAT_D: 34;

      

       A47: ( LSeg (go,m)) c= ( L~ go) by TOPREAL3: 19;

      

       A48: ( L~ go) c= ( L~ US) by A10, JORDAN3: 41;

      then ( LSeg (go,m)) c= ( L~ US) by A47;

      then

       A49: (( LSeg (go,m)) /\ ( LSeg (Gik,Gij))) c= {Gik} by A6, XBOOLE_1: 26;

      m >= 1 by A35, XREAL_1: 19;

      then

       A50: ( LSeg (go,m)) = ( LSeg ((go /. m),Gik)) by A43, A45, TOPREAL1:def 3;

       {Gik} c= (( LSeg (go,m)) /\ ( LSeg (Gik,Gij)))

      proof

        let x be object;

        

         A51: Gik in ( LSeg (Gik,Gij)) by RLTOPSP1: 68;

        assume x in {Gik};

        then

         A52: x = Gik by TARSKI:def 1;

        Gik in ( LSeg (go,m)) by A50, RLTOPSP1: 68;

        hence thesis by A52, A51, XBOOLE_0:def 4;

      end;

      then

       A53: (( LSeg (go,m)) /\ ( LSeg (Gik,Gij))) = {Gik} by A49;

      

       A54: ( LSeg (co,1)) c= ( L~ co) by TOPREAL3: 19;

      

       A55: ( L~ co) c= ( L~ LS) by A9, JORDAN3: 42;

      then ( LSeg (co,1)) c= ( L~ LS) by A54;

      then

       A56: (( LSeg (co,1)) /\ ( LSeg (Gik,Gij))) c= {Gij} by A7, XBOOLE_1: 26;

      

       A57: ( LSeg (co,1)) = ( LSeg (Gij,(co /. (1 + 1)))) by A38, A44, TOPREAL1:def 3;

       {Gij} c= (( LSeg (co,1)) /\ ( LSeg (Gik,Gij)))

      proof

        let x be object;

        

         A58: Gij in ( LSeg (Gik,Gij)) by RLTOPSP1: 68;

        assume x in {Gij};

        then

         A59: x = Gij by TARSKI:def 1;

        Gij in ( LSeg (co,1)) by A57, RLTOPSP1: 68;

        hence thesis by A59, A58, XBOOLE_0:def 4;

      end;

      then

       A60: (( LSeg (Gik,Gij)) /\ ( LSeg (co,1))) = {Gij} by A56;

      

       A61: (go /. 1) = (US /. 1) by A10, SPRECT_3: 22

      .= Wmin by JORDAN1F: 5;

      

      then

       A62: (go /. 1) = (LS /. ( len LS)) by JORDAN1F: 8

      .= (co /. ( len co)) by A9, JORDAN1J: 35;

      

       A63: ( rng go) c= ( L~ go) by A35, SPPOL_2: 18;

      

       A64: ( rng co) c= ( L~ co) by A38, SPPOL_2: 18;

      

       A65: {(go /. 1)} c= (( L~ go) /\ ( L~ co))

      proof

        let x be object;

        assume x in {(go /. 1)};

        then

         A66: x = (go /. 1) by TARSKI:def 1;

        then

         A67: x in ( rng go) by FINSEQ_6: 42;

        x in ( rng co) by A62, A66, FINSEQ_6: 168;

        hence thesis by A63, A64, A67, XBOOLE_0:def 4;

      end;

      

       A68: (LS . 1) = (LS /. 1) by A27, PARTFUN1:def 6

      .= Emax by JORDAN1F: 6;

      

       A69: [( len Ga), j] in ( Indices Ga) by A1, A13, A21, MATRIX_0: 30;

      (( L~ go) /\ ( L~ co)) c= {(go /. 1)}

      proof

        let x be object;

        assume

         A70: x in (( L~ go) /\ ( L~ co));

        then

         A71: x in ( L~ co) by XBOOLE_0:def 4;

         A72:

        now

          assume x = Emax;

          then

           A73: Emax = Gij by A9, A68, A71, JORDAN1E: 7;

          ((Ga * (( len Ga),j)) `1 ) = Ebo by A1, A11, A13, JORDAN1A: 71;

          then (Emax `1 ) <> Ebo by A2, A3, A16, A69, A73, JORDAN1G: 7;

          hence contradiction by EUCLID: 52;

        end;

        x in ( L~ go) by A70, XBOOLE_0:def 4;

        then x in (( L~ US) /\ ( L~ LS)) by A48, A55, A71, XBOOLE_0:def 4;

        then x in {Wmin, Emax} by JORDAN1E: 16;

        then x = Wmin or x = Emax by TARSKI:def 2;

        hence thesis by A61, A72, TARSKI:def 1;

      end;

      then

       A74: (( L~ go) /\ ( L~ co)) = {(go /. 1)} by A65;

      set W2 = (go /. 2);

      

       A75: 2 in ( dom go) by A35, FINSEQ_3: 25;

       A76:

      now

        assume (Gij `1 ) = Wbo;

        then ((Ga * (1,j)) `1 ) = ((Ga * (j,i)) `1 ) by A1, A11, A13, JORDAN1A: 73;

        hence contradiction by A1, A16, A24, JORDAN1G: 7;

      end;

      go = ( mid (US,1,(Gik .. US))) by A36, JORDAN1G: 49

      .= (US | (Gik .. US)) by A36, FINSEQ_4: 21, FINSEQ_6: 116;

      then

       A77: W2 = (US /. 2) by A75, FINSEQ_4: 70;

      

       A78: Wmin in ( rng go) by A61, FINSEQ_6: 42;

      set pion = <*Gik, Gij*>;

       A79:

      now

        let n be Nat;

        assume n in ( dom pion);

        then n in {1, 2} by FINSEQ_1: 2, FINSEQ_1: 89;

        then n = 1 or n = 2 by TARSKI:def 2;

        hence ex j,i be Nat st [j, i] in ( Indices Ga) & (pion /. n) = (Ga * (j,i)) by A16, A17, FINSEQ_4: 17;

      end;

      

       A80: Gik <> Gij by A12, A16, A17, GOBOARD1: 5;

      (Gik `2 ) = ((Ga * (1,i)) `2 ) by A3, A4, A5, A14, GOBOARD5: 1

      .= (Gij `2 ) by A1, A4, A5, A11, A13, GOBOARD5: 1;

      then ( LSeg (Gik,Gij)) is horizontal by SPPOL_1: 15;

      then pion is being_S-Seq by A80, JORDAN1B: 8;

      then

      consider pion1 be FinSequence of ( TOP-REAL 2) such that

       A81: pion1 is_sequence_on Ga and

       A82: pion1 is being_S-Seq and

       A83: ( L~ pion) = ( L~ pion1) and

       A84: (pion /. 1) = (pion1 /. 1) and

       A85: (pion /. ( len pion)) = (pion1 /. ( len pion1)) and

       A86: ( len pion) <= ( len pion1) by A79, GOBOARD3: 2;

      reconsider pion1 as being_S-Seq FinSequence of ( TOP-REAL 2) by A82;

      set godo = ((go ^' pion1) ^' co);

      

       A87: (1 + 1) <= ( len ( Cage (C,n))) by GOBOARD7: 34, XXREAL_0: 2;

      

       A88: (1 + 1) <= ( len ( Rotate (( Cage (C,n)),Wmin))) by GOBOARD7: 34, XXREAL_0: 2;

      ( len (go ^' pion1)) >= ( len go) by TOPREAL8: 7;

      then

       A89: ( len (go ^' pion1)) >= (1 + 1) by A35, XXREAL_0: 2;

      then

       A90: ( len (go ^' pion1)) > (1 + 0 ) by NAT_1: 13;

      

       A91: ( len godo) >= ( len (go ^' pion1)) by TOPREAL8: 7;

      then

       A92: (1 + 1) <= ( len godo) by A89, XXREAL_0: 2;

      

       A93: US is_sequence_on Ga by JORDAN1G: 4;

      

       A94: (go /. ( len go)) = (pion1 /. 1) by A43, A84, FINSEQ_4: 17;

      then

       A95: (go ^' pion1) is_sequence_on Ga by A37, A81, TOPREAL8: 12;

      

       A96: ((go ^' pion1) /. ( len (go ^' pion1))) = (pion /. ( len pion)) by A85, FINSEQ_6: 156

      .= (pion /. 2) by FINSEQ_1: 44

      .= (co /. 1) by A44, FINSEQ_4: 17;

      then

       A97: godo is_sequence_on Ga by A40, A95, TOPREAL8: 12;

      ( LSeg (pion1,1)) c= ( L~ <*Gik, Gij*>) by A83, TOPREAL3: 19;

      then ( LSeg (pion1,1)) c= ( LSeg (Gik,Gij)) by SPPOL_2: 21;

      then

       A98: (( LSeg (go,(( len go) -' 1))) /\ ( LSeg (pion1,1))) c= {Gik} by A46, A53, XBOOLE_1: 27;

      

       A99: ( len pion1) >= (1 + 1) by A86, FINSEQ_1: 44;

       {Gik} c= (( LSeg (go,m)) /\ ( LSeg (pion1,1)))

      proof

        let x be object;

        assume x in {Gik};

        then

         A100: x = Gik by TARSKI:def 1;

        

         A101: Gik in ( LSeg (go,m)) by A50, RLTOPSP1: 68;

        Gik in ( LSeg (pion1,1)) by A43, A94, A99, TOPREAL1: 21;

        hence thesis by A100, A101, XBOOLE_0:def 4;

      end;

      then (( LSeg (go,(( len go) -' 1))) /\ ( LSeg (pion1,1))) = {(go /. ( len go))} by A43, A46, A98;

      then

       A102: (go ^' pion1) is unfolded by A94, TOPREAL8: 34;

      ( len pion1) >= (2 + 0 ) by A86, FINSEQ_1: 44;

      then

       A103: (( len pion1) - 2) >= 0 by XREAL_1: 19;

      ((( len (go ^' pion1)) + 1) - 1) = ((( len go) + ( len pion1)) - 1) by FINSEQ_6: 139;

      

      then (( len (go ^' pion1)) - 1) = (( len go) + (( len pion1) - 2))

      .= (( len go) + (( len pion1) -' 2)) by A103, XREAL_0:def 2;

      then

       A104: (( len (go ^' pion1)) -' 1) = (( len go) + (( len pion1) -' 2)) by XREAL_0:def 2;

      

       A105: (( len pion1) - 1) >= 1 by A99, XREAL_1: 19;

      then

       A106: (( len pion1) -' 1) = (( len pion1) - 1) by XREAL_0:def 2;

      

       A107: ((( len pion1) -' 2) + 1) = ((( len pion1) - 2) + 1) by A103, XREAL_0:def 2

      .= (( len pion1) -' 1) by A105, XREAL_0:def 2;

      ((( len pion1) - 1) + 1) <= ( len pion1);

      then

       A108: (( len pion1) -' 1) < ( len pion1) by A106, NAT_1: 13;

      ( LSeg (pion1,(( len pion1) -' 1))) c= ( L~ <*Gik, Gij*>) by A83, TOPREAL3: 19;

      then ( LSeg (pion1,(( len pion1) -' 1))) c= ( LSeg (Gik,Gij)) by SPPOL_2: 21;

      then

       A109: (( LSeg (pion1,(( len pion1) -' 1))) /\ ( LSeg (co,1))) c= {Gij} by A60, XBOOLE_1: 27;

       {Gij} c= (( LSeg (pion1,(( len pion1) -' 1))) /\ ( LSeg (co,1)))

      proof

        let x be object;

        assume x in {Gij};

        then

         A110: x = Gij by TARSKI:def 1;

        (pion1 /. ((( len pion1) -' 1) + 1)) = (pion /. 2) by A85, A106, FINSEQ_1: 44

        .= Gij by FINSEQ_4: 17;

        then

         A111: Gij in ( LSeg (pion1,(( len pion1) -' 1))) by A105, A106, TOPREAL1: 21;

        Gij in ( LSeg (co,1)) by A57, RLTOPSP1: 68;

        hence thesis by A110, A111, XBOOLE_0:def 4;

      end;

      then (( LSeg (pion1,(( len pion1) -' 1))) /\ ( LSeg (co,1))) = {Gij} by A109;

      then

       A112: (( LSeg ((go ^' pion1),(( len go) + (( len pion1) -' 2)))) /\ ( LSeg (co,1))) = {((go ^' pion1) /. ( len (go ^' pion1)))} by A44, A94, A96, A107, A108, TOPREAL8: 31;

      

       A113: (go ^' pion1) is non trivial by A89, NAT_D: 60;

      

       A114: ( rng pion1) c= ( L~ pion1) by A99, SPPOL_2: 18;

      

       A115: {(pion1 /. 1)} c= (( L~ go) /\ ( L~ pion1))

      proof

        let x be object;

        assume x in {(pion1 /. 1)};

        then

         A116: x = (pion1 /. 1) by TARSKI:def 1;

        then

         A117: x in ( rng pion1) by FINSEQ_6: 42;

        x in ( rng go) by A94, A116, FINSEQ_6: 168;

        hence thesis by A63, A114, A117, XBOOLE_0:def 4;

      end;

      (( L~ go) /\ ( L~ pion1)) c= {(pion1 /. 1)}

      proof

        let x be object;

        assume

         A118: x in (( L~ go) /\ ( L~ pion1));

        then

         A119: x in ( L~ pion1) by XBOOLE_0:def 4;

        x in ( L~ go) by A118, XBOOLE_0:def 4;

        then x in (( L~ pion1) /\ ( L~ US)) by A48, A119, XBOOLE_0:def 4;

        hence thesis by A6, A43, A83, A94, SPPOL_2: 21;

      end;

      then

       A120: (( L~ go) /\ ( L~ pion1)) = {(pion1 /. 1)} by A115;

      then

       A121: (go ^' pion1) is s.n.c. by A94, JORDAN1J: 54;

      (( rng go) /\ ( rng pion1)) c= {(pion1 /. 1)} by A63, A114, A120, XBOOLE_1: 27;

      then

       A122: (go ^' pion1) is one-to-one by JORDAN1J: 55;

      

       A123: (pion /. ( len pion)) = (pion /. 2) by FINSEQ_1: 44

      .= (co /. 1) by A44, FINSEQ_4: 17;

      

       A124: {(pion1 /. ( len pion1))} c= (( L~ co) /\ ( L~ pion1))

      proof

        let x be object;

        assume x in {(pion1 /. ( len pion1))};

        then

         A125: x = (pion1 /. ( len pion1)) by TARSKI:def 1;

        then

         A126: x in ( rng pion1) by FINSEQ_6: 168;

        x in ( rng co) by A85, A123, A125, FINSEQ_6: 42;

        hence thesis by A64, A114, A126, XBOOLE_0:def 4;

      end;

      (( L~ co) /\ ( L~ pion1)) c= {(pion1 /. ( len pion1))}

      proof

        let x be object;

        assume

         A127: x in (( L~ co) /\ ( L~ pion1));

        then

         A128: x in ( L~ pion1) by XBOOLE_0:def 4;

        x in ( L~ co) by A127, XBOOLE_0:def 4;

        then x in (( L~ pion1) /\ ( L~ LS)) by A55, A128, XBOOLE_0:def 4;

        hence thesis by A7, A44, A83, A85, A123, SPPOL_2: 21;

      end;

      then

       A129: (( L~ co) /\ ( L~ pion1)) = {(pion1 /. ( len pion1))} by A124;

      

       A130: (( L~ (go ^' pion1)) /\ ( L~ co)) = ((( L~ go) \/ ( L~ pion1)) /\ ( L~ co)) by A94, TOPREAL8: 35

      .= ( {(go /. 1)} \/ {(co /. 1)}) by A74, A85, A123, A129, XBOOLE_1: 23

      .= ( {((go ^' pion1) /. 1)} \/ {(co /. 1)}) by FINSEQ_6: 155

      .= {((go ^' pion1) /. 1), (co /. 1)} by ENUMSET1: 1;

      (co /. ( len co)) = ((go ^' pion1) /. 1) by A62, FINSEQ_6: 155;

      then

      reconsider godo as non constant standard special_circular_sequence by A92, A96, A97, A102, A104, A112, A113, A121, A122, A130, JORDAN8: 4, JORDAN8: 5, TOPREAL8: 11, TOPREAL8: 33, TOPREAL8: 34;

      

       A131: UA is_an_arc_of (( W-min C),( E-max C)) by JORDAN6:def 8;

      then

       A132: UA is connected by JORDAN6: 10;

      

       A133: ( W-min C) in UA by A131, TOPREAL1: 1;

      

       A134: ( E-max C) in UA by A131, TOPREAL1: 1;

      set ff = ( Rotate (( Cage (C,n)),Wmin));

      Wmin in ( rng ( Cage (C,n))) by SPRECT_2: 43;

      then

       A135: (ff /. 1) = Wmin by FINSEQ_6: 92;

      

       A136: ( L~ ff) = ( L~ ( Cage (C,n))) by REVROT_1: 33;

      then (( W-max ( L~ ff)) .. ff) > 1 by A135, SPRECT_5: 22;

      then (( N-min ( L~ ff)) .. ff) > 1 by A135, A136, SPRECT_5: 23, XXREAL_0: 2;

      then (( N-max ( L~ ff)) .. ff) > 1 by A135, A136, SPRECT_5: 24, XXREAL_0: 2;

      then

       A137: (Emax .. ff) > 1 by A135, A136, SPRECT_5: 25, XXREAL_0: 2;

       A138:

      now

        assume

         A139: (Gik .. US) <= 1;

        (Gik .. US) >= 1 by A36, FINSEQ_4: 21;

        then (Gik .. US) = 1 by A139, XXREAL_0: 1;

        then Gik = (US /. 1) by A36, FINSEQ_5: 38;

        hence contradiction by A19, A23, JORDAN1F: 5;

      end;

      

       A140: ( Cage (C,n)) is_sequence_on Ga by JORDAN9:def 1;

      then

       A141: ff is_sequence_on Ga by REVROT_1: 34;

      

       A142: (( right_cell (godo,1,Ga)) \ ( L~ godo)) c= ( RightComp godo) by A92, A97, JORDAN9: 27;

      

       A143: ( L~ godo) = (( L~ (go ^' pion1)) \/ ( L~ co)) by A96, TOPREAL8: 35

      .= ((( L~ go) \/ ( L~ pion1)) \/ ( L~ co)) by A94, TOPREAL8: 35;

      

       A144: ( L~ ( Cage (C,n))) = (( L~ US) \/ ( L~ LS)) by JORDAN1E: 13;

      then

       A145: ( L~ US) c= ( L~ ( Cage (C,n))) by XBOOLE_1: 7;

      

       A146: ( L~ LS) c= ( L~ ( Cage (C,n))) by A144, XBOOLE_1: 7;

      

       A147: ( L~ go) c= ( L~ ( Cage (C,n))) by A48, A145;

      

       A148: ( L~ co) c= ( L~ ( Cage (C,n))) by A55, A146;

      

       A149: ( W-min C) in C by SPRECT_1: 13;

      

       A150: ( L~ pion) = ( LSeg (Gik,Gij)) by SPPOL_2: 21;

       A151:

      now

        assume ( W-min C) in ( L~ godo);

        then

         A152: ( W-min C) in (( L~ go) \/ ( L~ pion1)) or ( W-min C) in ( L~ co) by A143, XBOOLE_0:def 3;

        per cases by A152, XBOOLE_0:def 3;

          suppose ( W-min C) in ( L~ go);

          then C meets ( L~ ( Cage (C,n))) by A147, A149, XBOOLE_0: 3;

          hence contradiction by JORDAN10: 5;

        end;

          suppose ( W-min C) in ( L~ pion1);

          hence contradiction by A8, A83, A133, A150, XBOOLE_0: 3;

        end;

          suppose ( W-min C) in ( L~ co);

          then C meets ( L~ ( Cage (C,n))) by A148, A149, XBOOLE_0: 3;

          hence contradiction by JORDAN10: 5;

        end;

      end;

      ( right_cell (( Rotate (( Cage (C,n)),Wmin)),1)) = ( right_cell (ff,1,( GoB ff))) by A88, JORDAN1H: 23

      .= ( right_cell (ff,1,( GoB ( Cage (C,n))))) by REVROT_1: 28

      .= ( right_cell (ff,1,Ga)) by JORDAN1H: 44

      .= ( right_cell ((ff -: Emax),1,Ga)) by A137, A141, JORDAN1J: 53

      .= ( right_cell (US,1,Ga)) by JORDAN1E:def 1

      .= ( right_cell (( R_Cut (US,Gik)),1,Ga)) by A36, A93, A138, JORDAN1J: 52

      .= ( right_cell ((go ^' pion1),1,Ga)) by A41, A95, JORDAN1J: 51

      .= ( right_cell (godo,1,Ga)) by A90, A97, JORDAN1J: 51;

      then ( W-min C) in ( right_cell (godo,1,Ga)) by JORDAN1I: 6;

      then

       A153: ( W-min C) in (( right_cell (godo,1,Ga)) \ ( L~ godo)) by A151, XBOOLE_0:def 5;

      

       A154: (godo /. 1) = ((go ^' pion1) /. 1) by FINSEQ_6: 155

      .= Wmin by A61, FINSEQ_6: 155;

      

       A155: ( len US) >= 2 by A18, XXREAL_0: 2;

      

       A156: (godo /. 2) = ((go ^' pion1) /. 2) by A89, FINSEQ_6: 159

      .= (US /. 2) by A35, A77, FINSEQ_6: 159

      .= ((US ^' LS) /. 2) by A155, FINSEQ_6: 159

      .= (( Rotate (( Cage (C,n)),Wmin)) /. 2) by JORDAN1E: 11;

      

       A157: (( L~ go) \/ ( L~ co)) is compact by COMPTS_1: 10;

      Wmin in (( L~ go) \/ ( L~ co)) by A63, A78, XBOOLE_0:def 3;

      then

       A158: ( W-min (( L~ go) \/ ( L~ co))) = Wmin by A147, A148, A157, JORDAN1J: 21, XBOOLE_1: 8;

      

       A159: (( W-min (( L~ go) \/ ( L~ co))) `1 ) = ( W-bound (( L~ go) \/ ( L~ co))) by EUCLID: 52;

      

       A160: (Wmin `1 ) = Wbo by EUCLID: 52;

      

       A161: (Gij `1 ) <= (Gik `1 ) by A1, A2, A3, A4, A5, SPRECT_3: 13;

      then ( W-bound ( LSeg (Gik,Gij))) = (Gij `1 ) by SPRECT_1: 54;

      then

       A162: ( W-bound ( L~ pion1)) = (Gij `1 ) by A83, SPPOL_2: 21;

      (Gij `1 ) >= Wbo by A9, A146, PSCOMP_1: 24;

      then (Gij `1 ) > Wbo by A76, XXREAL_0: 1;

      then ( W-min ((( L~ go) \/ ( L~ co)) \/ ( L~ pion1))) = ( W-min (( L~ go) \/ ( L~ co))) by A157, A158, A159, A160, A162, JORDAN1J: 33;

      then

       A163: ( W-min ( L~ godo)) = Wmin by A143, A158, XBOOLE_1: 4;

      

       A164: ( rng godo) c= ( L~ godo) by A89, A91, SPPOL_2: 18, XXREAL_0: 2;

      2 in ( dom godo) by A92, FINSEQ_3: 25;

      then

       A165: (godo /. 2) in ( rng godo) by PARTFUN2: 2;

      (godo /. 2) in ( W-most ( L~ ( Cage (C,n)))) by A156, JORDAN1I: 25;

      

      then ((godo /. 2) `1 ) = (( W-min ( L~ godo)) `1 ) by A163, PSCOMP_1: 31

      .= ( W-bound ( L~ godo)) by EUCLID: 52;

      then (godo /. 2) in ( W-most ( L~ godo)) by A164, A165, SPRECT_2: 12;

      then (( Rotate (godo,( W-min ( L~ godo)))) /. 2) in ( W-most ( L~ godo)) by A154, A163, FINSEQ_6: 89;

      then

      reconsider godo as clockwise_oriented non constant standard special_circular_sequence by JORDAN1I: 25;

      ( len US) in ( dom US) by FINSEQ_5: 6;

      

      then

       A166: (US . ( len US)) = (US /. ( len US)) by PARTFUN1:def 6

      .= Emax by JORDAN1F: 7;

      

       A167: ( east_halfline ( E-max C)) misses ( L~ go)

      proof

        assume ( east_halfline ( E-max C)) meets ( L~ go);

        then

        consider p be object such that

         A168: p in ( east_halfline ( E-max C)) and

         A169: p in ( L~ go) by XBOOLE_0: 3;

        reconsider p as Point of ( TOP-REAL 2) by A168;

        p in ( L~ US) by A48, A169;

        then p in (( east_halfline ( E-max C)) /\ ( L~ ( Cage (C,n)))) by A145, A168, XBOOLE_0:def 4;

        then

         A170: (p `1 ) = Ebo by JORDAN1A: 83, PSCOMP_1: 50;

        then

         A171: p = Emax by A48, A169, JORDAN1J: 46;

        then Emax = Gik by A10, A166, A169, JORDAN1J: 43;

        then (Gik `1 ) = ((Ga * (( len Ga),k)) `1 ) by A3, A14, A170, A171, JORDAN1A: 71;

        hence contradiction by A3, A17, A32, JORDAN1G: 7;

      end;

      now

        assume ( east_halfline ( E-max C)) meets ( L~ godo);

        then

         A172: ( east_halfline ( E-max C)) meets (( L~ go) \/ ( L~ pion1)) or ( east_halfline ( E-max C)) meets ( L~ co) by A143, XBOOLE_1: 70;

        per cases by A172, XBOOLE_1: 70;

          suppose ( east_halfline ( E-max C)) meets ( L~ go);

          hence contradiction by A167;

        end;

          suppose ( east_halfline ( E-max C)) meets ( L~ pion1);

          then

          consider p be object such that

           A173: p in ( east_halfline ( E-max C)) and

           A174: p in ( L~ pion1) by XBOOLE_0: 3;

          reconsider p as Point of ( TOP-REAL 2) by A173;

          

           A175: (p `2 ) = (( E-max C) `2 ) by A173, TOPREAL1:def 11;

          (k + 1) <= ( len Ga) by A3, NAT_1: 13;

          then ((k + 1) - 1) <= (( len Ga) - 1) by XREAL_1: 9;

          then

           A176: k <= (( len Ga) -' 1) by XREAL_0:def 2;

          (( len Ga) -' 1) <= ( len Ga) by NAT_D: 35;

          then

           A177: (Gik `1 ) <= ((Ga * ((( len Ga) -' 1),1)) `1 ) by A4, A5, A11, A14, A21, A176, JORDAN1A: 18;

          (p `1 ) <= (Gik `1 ) by A83, A150, A161, A174, TOPREAL1: 3;

          then (p `1 ) <= ((Ga * ((( len Ga) -' 1),1)) `1 ) by A177, XXREAL_0: 2;

          then (p `1 ) <= ( E-bound C) by A21, JORDAN8: 12;

          then

           A178: (p `1 ) <= (( E-max C) `1 ) by EUCLID: 52;

          (p `1 ) >= (( E-max C) `1 ) by A173, TOPREAL1:def 11;

          then (p `1 ) = (( E-max C) `1 ) by A178, XXREAL_0: 1;

          then p = ( E-max C) by A175, TOPREAL3: 6;

          hence contradiction by A8, A83, A134, A150, A174, XBOOLE_0: 3;

        end;

          suppose ( east_halfline ( E-max C)) meets ( L~ co);

          then

          consider p be object such that

           A179: p in ( east_halfline ( E-max C)) and

           A180: p in ( L~ co) by XBOOLE_0: 3;

          reconsider p as Point of ( TOP-REAL 2) by A179;

          

           A181: p in ( LSeg (co,( Index (p,co)))) by A180, JORDAN3: 9;

          consider t be Nat such that

           A182: t in ( dom LS) and

           A183: (LS . t) = Gij by A39, FINSEQ_2: 10;

          1 <= t by A182, FINSEQ_3: 25;

          then

           A184: 1 < t by A34, A183, XXREAL_0: 1;

          t <= ( len LS) by A182, FINSEQ_3: 25;

          then (( Index (Gij,LS)) + 1) = t by A183, A184, JORDAN3: 12;

          then

           A185: ( len ( L_Cut (LS,Gij))) = (( len LS) - ( Index (Gij,LS))) by A9, A183, JORDAN3: 26;

          ( Index (p,co)) < ( len co) by A180, JORDAN3: 8;

          then ( Index (p,co)) < (( len LS) -' ( Index (Gij,LS))) by A185, XREAL_0:def 2;

          then (( Index (p,co)) + 1) <= (( len LS) -' ( Index (Gij,LS))) by NAT_1: 13;

          then

           A186: ( Index (p,co)) <= ((( len LS) -' ( Index (Gij,LS))) - 1) by XREAL_1: 19;

          

           A187: co = ( mid (LS,(Gij .. LS),( len LS))) by A39, JORDAN1J: 37;

          p in ( L~ LS) by A55, A180;

          then p in (( east_halfline ( E-max C)) /\ ( L~ ( Cage (C,n)))) by A146, A179, XBOOLE_0:def 4;

          then

           A188: (p `1 ) = Ebo by JORDAN1A: 83, PSCOMP_1: 50;

          

           A189: (( Index (Gij,LS)) + 1) = (Gij .. LS) by A34, A39, JORDAN1J: 56;

          ( 0 + ( Index (Gij,LS))) < ( len LS) by A9, JORDAN3: 8;

          then (( len LS) - ( Index (Gij,LS))) > 0 by XREAL_1: 20;

          then ( Index (p,co)) <= ((( len LS) - ( Index (Gij,LS))) - 1) by A186, XREAL_0:def 2;

          then ( Index (p,co)) <= (( len LS) - (Gij .. LS)) by A189;

          then ( Index (p,co)) <= (( len LS) -' (Gij .. LS)) by XREAL_0:def 2;

          then

           A190: ( Index (p,co)) < ((( len LS) -' (Gij .. LS)) + 1) by NAT_1: 13;

          

           A191: 1 <= ( Index (p,co)) by A180, JORDAN3: 8;

          

           A192: (Gij .. LS) <= ( len LS) by A39, FINSEQ_4: 21;

          (Gij .. LS) <> ( len LS) by A31, A39, FINSEQ_4: 19;

          then

           A193: (Gij .. LS) < ( len LS) by A192, XXREAL_0: 1;

          

           A194: (1 + 1) <= ( len LS) by A25, XXREAL_0: 2;

          then

           A195: 2 in ( dom LS) by FINSEQ_3: 25;

          set tt = ((( Index (p,co)) + (Gij .. LS)) -' 1);

          set RC = ( Rotate (( Cage (C,n)),Emax));

          

           A196: ( E-max C) in ( right_cell (RC,1)) by JORDAN1I: 7;

          

           A197: ( GoB RC) = ( GoB ( Cage (C,n))) by REVROT_1: 28

          .= Ga by JORDAN1H: 44;

          

           A198: ( L~ RC) = ( L~ ( Cage (C,n))) by REVROT_1: 33;

          consider jj2 be Nat such that

           A199: 1 <= jj2 and

           A200: jj2 <= ( width Ga) and

           A201: Emax = (Ga * (( len Ga),jj2)) by JORDAN1D: 25;

          

           A202: ( len Ga) >= 4 by JORDAN8: 10;

          then ( len Ga) >= 1 by XXREAL_0: 2;

          then

           A203: [( len Ga), jj2] in ( Indices Ga) by A199, A200, MATRIX_0: 30;

          

           A204: ( len RC) = ( len ( Cage (C,n))) by FINSEQ_6: 179;

          LS = (RC -: Wmin) by JORDAN1G: 18;

          then

           A205: ( LSeg (LS,1)) = ( LSeg (RC,1)) by A194, SPPOL_2: 9;

          

           A206: Emax in ( rng ( Cage (C,n))) by SPRECT_2: 46;

          RC is_sequence_on Ga by A140, REVROT_1: 34;

          then

          consider ii,jj be Nat such that

           A207: [ii, (jj + 1)] in ( Indices Ga) and

           A208: [ii, jj] in ( Indices Ga) and

           A209: (RC /. 1) = (Ga * (ii,(jj + 1))) and

           A210: (RC /. (1 + 1)) = (Ga * (ii,jj)) by A87, A198, A204, A206, FINSEQ_6: 92, JORDAN1I: 23;

          

           A211: ((jj + 1) + 1) <> jj;

          

           A212: 1 <= jj by A208, MATRIX_0: 32;

          (RC /. 1) = ( E-max ( L~ RC)) by A198, A206, FINSEQ_6: 92;

          then

           A213: ii = ( len Ga) by A198, A207, A209, A201, A203, GOBOARD1: 5;

          then (ii - 1) >= (4 - 1) by A202, XREAL_1: 9;

          then

           A214: (ii - 1) >= 1 by XXREAL_0: 2;

          then

           A215: 1 <= (ii -' 1) by XREAL_0:def 2;

          

           A216: jj <= ( width Ga) by A208, MATRIX_0: 32;

          then

           A217: ((Ga * (( len Ga),jj)) `1 ) = Ebo by A11, A212, JORDAN1A: 71;

          

           A218: (jj + 1) <= ( width Ga) by A207, MATRIX_0: 32;

          (ii + 1) <> ii;

          then

           A219: ( right_cell (RC,1)) = ( cell (Ga,(ii -' 1),jj)) by A87, A204, A197, A207, A208, A209, A210, A211, GOBOARD5:def 6;

          

           A220: ii <= ( len Ga) by A208, MATRIX_0: 32;

          

           A221: 1 <= ii by A208, MATRIX_0: 32;

          

           A222: ii <= ( len Ga) by A207, MATRIX_0: 32;

          

           A223: 1 <= (jj + 1) by A207, MATRIX_0: 32;

          then

           A224: Ebo = ((Ga * (( len Ga),(jj + 1))) `1 ) by A11, A218, JORDAN1A: 71;

          

           A225: 1 <= ii by A207, MATRIX_0: 32;

          then

           A226: ((ii -' 1) + 1) = ii by XREAL_1: 235;

          then

           A227: (ii -' 1) < ( len Ga) by A222, NAT_1: 13;

          

          then

           A228: ((Ga * ((ii -' 1),(jj + 1))) `2 ) = ((Ga * (1,(jj + 1))) `2 ) by A223, A218, A215, GOBOARD5: 1

          .= ((Ga * (ii,(jj + 1))) `2 ) by A225, A222, A223, A218, GOBOARD5: 1;

          

           A229: (( E-max C) `2 ) = (p `2 ) by A179, TOPREAL1:def 11;

          then

           A230: (p `2 ) <= ((Ga * ((ii -' 1),(jj + 1))) `2 ) by A196, A222, A218, A212, A219, A226, A214, JORDAN9: 17;

          

           A231: ((Ga * ((ii -' 1),jj)) `2 ) = ((Ga * (1,jj)) `2 ) by A212, A216, A215, A227, GOBOARD5: 1

          .= ((Ga * (ii,jj)) `2 ) by A221, A220, A212, A216, GOBOARD5: 1;

          ((Ga * ((ii -' 1),jj)) `2 ) <= (p `2 ) by A229, A196, A222, A218, A212, A219, A226, A214, JORDAN9: 17;

          then p in ( LSeg ((RC /. 1),(RC /. (1 + 1)))) by A188, A209, A210, A213, A230, A231, A228, A217, A224, GOBOARD7: 7;

          then

           A232: p in ( LSeg (LS,1)) by A87, A205, A204, TOPREAL1:def 3;

          1 <= (Gij .. LS) by A39, FINSEQ_4: 21;

          then

           A233: ( LSeg (( mid (LS,(Gij .. LS),( len LS))),( Index (p,co)))) = ( LSeg (LS,((( Index (p,co)) + (Gij .. LS)) -' 1))) by A193, A191, A190, JORDAN4: 19;

          1 <= ( Index (Gij,LS)) by A9, JORDAN3: 8;

          then

           A234: (1 + 1) <= (Gij .. LS) by A189, XREAL_1: 7;

          then (( Index (p,co)) + (Gij .. LS)) >= ((1 + 1) + 1) by A191, XREAL_1: 7;

          then ((( Index (p,co)) + (Gij .. LS)) - 1) >= (((1 + 1) + 1) - 1) by XREAL_1: 9;

          then

           A235: tt >= (1 + 1) by XREAL_0:def 2;

          now

            per cases by A235, XXREAL_0: 1;

              suppose tt > (1 + 1);

              then ( LSeg (LS,1)) misses ( LSeg (LS,tt)) by TOPREAL1:def 7;

              hence contradiction by A232, A181, A187, A233, XBOOLE_0: 3;

            end;

              suppose

               A236: tt = (1 + 1);

              then (1 + 1) = ((( Index (p,co)) + (Gij .. LS)) - 1) by XREAL_0:def 2;

              then ((1 + 1) + 1) = (( Index (p,co)) + (Gij .. LS));

              then

               A237: (Gij .. LS) = 2 by A191, A234, JORDAN1E: 6;

              (( LSeg (LS,1)) /\ ( LSeg (LS,tt))) = {(LS /. 2)} by A25, A236, TOPREAL1:def 6;

              then p in {(LS /. 2)} by A232, A181, A187, A233, XBOOLE_0:def 4;

              then

               A238: p = (LS /. 2) by TARSKI:def 1;

              then

               A239: p in ( rng LS) by A195, PARTFUN2: 2;

              (p .. LS) = 2 by A195, A238, FINSEQ_5: 41;

              then p = Gij by A39, A237, A239, FINSEQ_5: 9;

              then (Gij `1 ) = Ebo by A238, JORDAN1G: 32;

              then (Gij `1 ) = ((Ga * (( len Ga),j)) `1 ) by A1, A11, A13, JORDAN1A: 71;

              hence contradiction by A2, A3, A16, A69, JORDAN1G: 7;

            end;

          end;

          hence contradiction;

        end;

      end;

      then ( east_halfline ( E-max C)) c= (( L~ godo) ` ) by SUBSET_1: 23;

      then

      consider W be Subset of ( TOP-REAL 2) such that

       A240: W is_a_component_of (( L~ godo) ` ) and

       A241: ( east_halfline ( E-max C)) c= W by GOBOARD9: 3;

       not W is bounded by A241, JORDAN2C: 121, RLTOPSP1: 42;

      then W is_outside_component_of ( L~ godo) by A240, JORDAN2C:def 3;

      then W c= ( UBD ( L~ godo)) by JORDAN2C: 23;

      then

       A242: ( east_halfline ( E-max C)) c= ( UBD ( L~ godo)) by A241;

      ( E-max C) in ( east_halfline ( E-max C)) by TOPREAL1: 38;

      then ( E-max C) in ( UBD ( L~ godo)) by A242;

      then ( E-max C) in ( LeftComp godo) by GOBRD14: 36;

      then UA meets ( L~ godo) by A132, A133, A134, A142, A153, JORDAN1J: 36;

      then

       A243: UA meets (( L~ go) \/ ( L~ pion1)) or UA meets ( L~ co) by A143, XBOOLE_1: 70;

      

       A244: UA c= C by JORDAN6: 61;

      per cases by A243, XBOOLE_1: 70;

        suppose UA meets ( L~ go);

        then UA meets ( L~ ( Cage (C,n))) by A48, A145, XBOOLE_1: 1, XBOOLE_1: 63;

        hence contradiction by A244, JORDAN10: 5, XBOOLE_1: 63;

      end;

        suppose UA meets ( L~ pion1);

        hence contradiction by A8, A83, A150;

      end;

        suppose UA meets ( L~ co);

        then UA meets ( L~ ( Cage (C,n))) by A55, A146, XBOOLE_1: 1, XBOOLE_1: 63;

        hence contradiction by A244, JORDAN10: 5, XBOOLE_1: 63;

      end;

    end;

    theorem :: JORDAN15:30

    

     Th30: for C be Simple_closed_curve holds for i,j,k be Nat st 1 < j & j <= k & k < ( len ( Gauge (C,n))) & 1 <= i & i <= ( width ( Gauge (C,n))) & (( Gauge (C,n)) * (k,i)) in ( L~ ( Upper_Seq (C,n))) & (( Gauge (C,n)) * (j,i)) in ( L~ ( Lower_Seq (C,n))) holds ( LSeg ((( Gauge (C,n)) * (j,i)),(( Gauge (C,n)) * (k,i)))) meets ( Lower_Arc C)

    proof

      let C be Simple_closed_curve;

      let i,j,k be Nat;

      assume that

       A1: 1 < j and

       A2: j <= k and

       A3: k < ( len ( Gauge (C,n))) and

       A4: 1 <= i and

       A5: i <= ( width ( Gauge (C,n))) and

       A6: (( Gauge (C,n)) * (k,i)) in ( L~ ( Upper_Seq (C,n))) and

       A7: (( Gauge (C,n)) * (j,i)) in ( L~ ( Lower_Seq (C,n)));

      consider j1,k1 be Nat such that

       A8: j <= j1 and

       A9: j1 <= k1 and

       A10: k1 <= k and

       A11: (( LSeg ((( Gauge (C,n)) * (j1,i)),(( Gauge (C,n)) * (k1,i)))) /\ ( L~ ( Lower_Seq (C,n)))) = {(( Gauge (C,n)) * (j1,i))} and

       A12: (( LSeg ((( Gauge (C,n)) * (j1,i)),(( Gauge (C,n)) * (k1,i)))) /\ ( L~ ( Upper_Seq (C,n)))) = {(( Gauge (C,n)) * (k1,i))} by A1, A2, A3, A4, A5, A6, A7, Th14;

      

       A13: k1 < ( len ( Gauge (C,n))) by A3, A10, XXREAL_0: 2;

      1 < j1 by A1, A8, XXREAL_0: 2;

      then ( LSeg ((( Gauge (C,n)) * (j1,i)),(( Gauge (C,n)) * (k1,i)))) meets ( Lower_Arc C) by A4, A5, A9, A11, A12, A13, Th28;

      hence thesis by A1, A3, A4, A5, A8, A9, A10, Th6, XBOOLE_1: 63;

    end;

    theorem :: JORDAN15:31

    

     Th31: for C be Simple_closed_curve holds for i,j,k be Nat st 1 < j & j <= k & k < ( len ( Gauge (C,n))) & 1 <= i & i <= ( width ( Gauge (C,n))) & (( Gauge (C,n)) * (k,i)) in ( L~ ( Upper_Seq (C,n))) & (( Gauge (C,n)) * (j,i)) in ( L~ ( Lower_Seq (C,n))) holds ( LSeg ((( Gauge (C,n)) * (j,i)),(( Gauge (C,n)) * (k,i)))) meets ( Upper_Arc C)

    proof

      let C be Simple_closed_curve;

      let i,j,k be Nat;

      assume that

       A1: 1 < j and

       A2: j <= k and

       A3: k < ( len ( Gauge (C,n))) and

       A4: 1 <= i and

       A5: i <= ( width ( Gauge (C,n))) and

       A6: (( Gauge (C,n)) * (k,i)) in ( L~ ( Upper_Seq (C,n))) and

       A7: (( Gauge (C,n)) * (j,i)) in ( L~ ( Lower_Seq (C,n)));

      consider j1,k1 be Nat such that

       A8: j <= j1 and

       A9: j1 <= k1 and

       A10: k1 <= k and

       A11: (( LSeg ((( Gauge (C,n)) * (j1,i)),(( Gauge (C,n)) * (k1,i)))) /\ ( L~ ( Lower_Seq (C,n)))) = {(( Gauge (C,n)) * (j1,i))} and

       A12: (( LSeg ((( Gauge (C,n)) * (j1,i)),(( Gauge (C,n)) * (k1,i)))) /\ ( L~ ( Upper_Seq (C,n)))) = {(( Gauge (C,n)) * (k1,i))} by A1, A2, A3, A4, A5, A6, A7, Th14;

      

       A13: k1 < ( len ( Gauge (C,n))) by A3, A10, XXREAL_0: 2;

      1 < j1 by A1, A8, XXREAL_0: 2;

      then ( LSeg ((( Gauge (C,n)) * (j1,i)),(( Gauge (C,n)) * (k1,i)))) meets ( Upper_Arc C) by A4, A5, A9, A11, A12, A13, Th29;

      hence thesis by A1, A3, A4, A5, A8, A9, A10, Th6, XBOOLE_1: 63;

    end;

    theorem :: JORDAN15:32

    

     Th32: for C be Simple_closed_curve holds for i,j,k be Nat st 1 < j & j <= k & k < ( len ( Gauge (C,n))) & 1 <= i & i <= ( width ( Gauge (C,n))) & n > 0 & (( Gauge (C,n)) * (k,i)) in ( Upper_Arc ( L~ ( Cage (C,n)))) & (( Gauge (C,n)) * (j,i)) in ( Lower_Arc ( L~ ( Cage (C,n)))) holds ( LSeg ((( Gauge (C,n)) * (j,i)),(( Gauge (C,n)) * (k,i)))) meets ( Lower_Arc C)

    proof

      let C be Simple_closed_curve;

      let i,j,k be Nat;

      assume that

       A1: 1 < j and

       A2: j <= k and

       A3: k < ( len ( Gauge (C,n))) and

       A4: 1 <= i and

       A5: i <= ( width ( Gauge (C,n))) and

       A6: n > 0 and

       A7: (( Gauge (C,n)) * (k,i)) in ( Upper_Arc ( L~ ( Cage (C,n)))) and

       A8: (( Gauge (C,n)) * (j,i)) in ( Lower_Arc ( L~ ( Cage (C,n))));

      

       A9: ( L~ ( Lower_Seq (C,n))) = ( Lower_Arc ( L~ ( Cage (C,n)))) by A6, JORDAN1G: 56;

      ( L~ ( Upper_Seq (C,n))) = ( Upper_Arc ( L~ ( Cage (C,n)))) by A6, JORDAN1G: 55;

      hence thesis by A1, A2, A3, A4, A5, A7, A8, A9, Th30;

    end;

    theorem :: JORDAN15:33

    

     Th33: for C be Simple_closed_curve holds for i,j,k be Nat st 1 < j & j <= k & k < ( len ( Gauge (C,n))) & 1 <= i & i <= ( width ( Gauge (C,n))) & n > 0 & (( Gauge (C,n)) * (k,i)) in ( Upper_Arc ( L~ ( Cage (C,n)))) & (( Gauge (C,n)) * (j,i)) in ( Lower_Arc ( L~ ( Cage (C,n)))) holds ( LSeg ((( Gauge (C,n)) * (j,i)),(( Gauge (C,n)) * (k,i)))) meets ( Upper_Arc C)

    proof

      let C be Simple_closed_curve;

      let i,j,k be Nat;

      assume that

       A1: 1 < j and

       A2: j <= k and

       A3: k < ( len ( Gauge (C,n))) and

       A4: 1 <= i and

       A5: i <= ( width ( Gauge (C,n))) and

       A6: n > 0 and

       A7: (( Gauge (C,n)) * (k,i)) in ( Upper_Arc ( L~ ( Cage (C,n)))) and

       A8: (( Gauge (C,n)) * (j,i)) in ( Lower_Arc ( L~ ( Cage (C,n))));

      

       A9: ( L~ ( Lower_Seq (C,n))) = ( Lower_Arc ( L~ ( Cage (C,n)))) by A6, JORDAN1G: 56;

      ( L~ ( Upper_Seq (C,n))) = ( Upper_Arc ( L~ ( Cage (C,n)))) by A6, JORDAN1G: 55;

      hence thesis by A1, A2, A3, A4, A5, A7, A8, A9, Th31;

    end;

    theorem :: JORDAN15:34

    for C be Simple_closed_curve holds for j,k be Nat holds 1 < j & j <= k & k < ( len ( Gauge (C,(n + 1)))) & (( Gauge (C,(n + 1))) * (k,( Center ( Gauge (C,(n + 1)))))) in ( Upper_Arc ( L~ ( Cage (C,(n + 1))))) & (( Gauge (C,(n + 1))) * (j,( Center ( Gauge (C,(n + 1)))))) in ( Lower_Arc ( L~ ( Cage (C,(n + 1))))) implies ( LSeg ((( Gauge (C,(n + 1))) * (j,( Center ( Gauge (C,(n + 1)))))),(( Gauge (C,(n + 1))) * (k,( Center ( Gauge (C,(n + 1)))))))) meets ( Lower_Arc C)

    proof

      let C be Simple_closed_curve;

      let j,k be Nat;

      assume that

       A1: 1 < j and

       A2: j <= k and

       A3: k < ( len ( Gauge (C,(n + 1)))) and

       A4: (( Gauge (C,(n + 1))) * (k,( Center ( Gauge (C,(n + 1)))))) in ( Upper_Arc ( L~ ( Cage (C,(n + 1))))) and

       A5: (( Gauge (C,(n + 1))) * (j,( Center ( Gauge (C,(n + 1)))))) in ( Lower_Arc ( L~ ( Cage (C,(n + 1)))));

      

       A6: ( len ( Gauge (C,(n + 1)))) >= 4 by JORDAN8: 10;

      then ( len ( Gauge (C,(n + 1)))) >= 3 by XXREAL_0: 2;

      then ( Center ( Gauge (C,(n + 1)))) < ( len ( Gauge (C,(n + 1)))) by JORDAN1B: 15;

      then

       A7: ( Center ( Gauge (C,(n + 1)))) < ( width ( Gauge (C,(n + 1)))) by JORDAN8:def 1;

      ( len ( Gauge (C,(n + 1)))) >= 2 by A6, XXREAL_0: 2;

      then 1 < ( Center ( Gauge (C,(n + 1)))) by JORDAN1B: 14;

      hence thesis by A1, A2, A3, A4, A5, A7, Th32;

    end;

    theorem :: JORDAN15:35

    for C be Simple_closed_curve holds for j,k be Nat holds 1 < j & j <= k & k < ( len ( Gauge (C,(n + 1)))) & (( Gauge (C,(n + 1))) * (k,( Center ( Gauge (C,(n + 1)))))) in ( Upper_Arc ( L~ ( Cage (C,(n + 1))))) & (( Gauge (C,(n + 1))) * (j,( Center ( Gauge (C,(n + 1)))))) in ( Lower_Arc ( L~ ( Cage (C,(n + 1))))) implies ( LSeg ((( Gauge (C,(n + 1))) * (j,( Center ( Gauge (C,(n + 1)))))),(( Gauge (C,(n + 1))) * (k,( Center ( Gauge (C,(n + 1)))))))) meets ( Upper_Arc C)

    proof

      let C be Simple_closed_curve;

      let j,k be Nat;

      assume that

       A1: 1 < j and

       A2: j <= k and

       A3: k < ( len ( Gauge (C,(n + 1)))) and

       A4: (( Gauge (C,(n + 1))) * (k,( Center ( Gauge (C,(n + 1)))))) in ( Upper_Arc ( L~ ( Cage (C,(n + 1))))) and

       A5: (( Gauge (C,(n + 1))) * (j,( Center ( Gauge (C,(n + 1)))))) in ( Lower_Arc ( L~ ( Cage (C,(n + 1)))));

      

       A6: ( len ( Gauge (C,(n + 1)))) >= 4 by JORDAN8: 10;

      then ( len ( Gauge (C,(n + 1)))) >= 3 by XXREAL_0: 2;

      then ( Center ( Gauge (C,(n + 1)))) < ( len ( Gauge (C,(n + 1)))) by JORDAN1B: 15;

      then

       A7: ( Center ( Gauge (C,(n + 1)))) < ( width ( Gauge (C,(n + 1)))) by JORDAN8:def 1;

      ( len ( Gauge (C,(n + 1)))) >= 2 by A6, XXREAL_0: 2;

      then 1 < ( Center ( Gauge (C,(n + 1)))) by JORDAN1B: 14;

      hence thesis by A1, A2, A3, A4, A5, A7, Th33;

    end;

    theorem :: JORDAN15:36

    

     Th36: for C be Simple_closed_curve holds for i,j,k be Nat st 1 < j & j <= k & k < ( len ( Gauge (C,n))) & 1 <= i & i <= ( width ( Gauge (C,n))) & (( LSeg ((( Gauge (C,n)) * (j,i)),(( Gauge (C,n)) * (k,i)))) /\ ( L~ ( Upper_Seq (C,n)))) = {(( Gauge (C,n)) * (j,i))} & (( LSeg ((( Gauge (C,n)) * (j,i)),(( Gauge (C,n)) * (k,i)))) /\ ( L~ ( Lower_Seq (C,n)))) = {(( Gauge (C,n)) * (k,i))} holds ( LSeg ((( Gauge (C,n)) * (j,i)),(( Gauge (C,n)) * (k,i)))) meets ( Lower_Arc C)

    proof

      let C be Simple_closed_curve;

      let i,j,k be Nat;

      set Ga = ( Gauge (C,n));

      set US = ( Upper_Seq (C,n));

      set LS = ( Lower_Seq (C,n));

      set LA = ( Lower_Arc C);

      set Wmin = ( W-min ( L~ ( Cage (C,n))));

      set Emax = ( E-max ( L~ ( Cage (C,n))));

      set Wbo = ( W-bound ( L~ ( Cage (C,n))));

      set Ebo = ( E-bound ( L~ ( Cage (C,n))));

      set Gij = (Ga * (j,i));

      set Gik = (Ga * (k,i));

      assume that

       A1: 1 < j and

       A2: j <= k and

       A3: k < ( len Ga) and

       A4: 1 <= i and

       A5: i <= ( width Ga) and

       A6: (( LSeg (Gij,Gik)) /\ ( L~ US)) = {Gij} and

       A7: (( LSeg (Gij,Gik)) /\ ( L~ LS)) = {Gik} and

       A8: ( LSeg (Gij,Gik)) misses LA;

      Gik in {Gik} by TARSKI:def 1;

      then

       A9: Gik in ( L~ LS) by A7, XBOOLE_0:def 4;

      Gij in {Gij} by TARSKI:def 1;

      then

       A10: Gij in ( L~ US) by A6, XBOOLE_0:def 4;

      

       A11: ( len Ga) = ( width Ga) by JORDAN8:def 1;

      

       A12: j <> k by A1, A3, A4, A5, A9, A10, Th27;

      

       A13: j < ( width Ga) by A2, A3, A11, XXREAL_0: 2;

      

       A14: 1 < k by A1, A2, XXREAL_0: 2;

      

       A15: k < ( width Ga) by A3, JORDAN8:def 1;

      

       A16: [j, i] in ( Indices Ga) by A1, A4, A5, A11, A13, MATRIX_0: 30;

      

       A17: [k, i] in ( Indices Ga) by A3, A4, A5, A14, MATRIX_0: 30;

      set go = ( R_Cut (US,Gij));

      set co = ( L_Cut (LS,Gik));

      

       A18: ( len US) >= 3 by JORDAN1E: 15;

      then ( len US) >= 1 by XXREAL_0: 2;

      then 1 in ( dom US) by FINSEQ_3: 25;

      

      then

       A19: (US . 1) = (US /. 1) by PARTFUN1:def 6

      .= Wmin by JORDAN1F: 5;

      

       A20: (Wmin `1 ) = Wbo by EUCLID: 52

      .= ((Ga * (1,i)) `1 ) by A4, A5, A11, JORDAN1A: 73;

      ( len Ga) >= 4 by JORDAN8: 10;

      then

       A21: ( len Ga) >= 1 by XXREAL_0: 2;

      then

       A22: [1, k] in ( Indices Ga) by A14, A15, MATRIX_0: 30;

      

       A23: [1, i] in ( Indices Ga) by A4, A5, A21, MATRIX_0: 30;

      then

       A24: Gij <> (US . 1) by A1, A16, A19, A20, JORDAN1G: 7;

      then

      reconsider go as being_S-Seq FinSequence of ( TOP-REAL 2) by A10, JORDAN3: 35;

      

       A25: ( len LS) >= (1 + 2) by JORDAN1E: 15;

      then

       A26: ( len LS) >= 1 by XXREAL_0: 2;

      then

       A27: 1 in ( dom LS) by FINSEQ_3: 25;

      ( len LS) in ( dom LS) by A26, FINSEQ_3: 25;

      

      then

       A28: (LS . ( len LS)) = (LS /. ( len LS)) by PARTFUN1:def 6

      .= Wmin by JORDAN1F: 8;

      (Wmin `1 ) = Wbo by EUCLID: 52

      .= ((Ga * (1,i)) `1 ) by A4, A5, A11, JORDAN1A: 73;

      then

       A29: Gik <> (LS . ( len LS)) by A1, A2, A17, A23, A28, JORDAN1G: 7;

      then

      reconsider co as being_S-Seq FinSequence of ( TOP-REAL 2) by A9, JORDAN3: 34;

      

       A30: [( len Ga), k] in ( Indices Ga) by A14, A15, A21, MATRIX_0: 30;

      

       A31: (LS . 1) = (LS /. 1) by A27, PARTFUN1:def 6

      .= Emax by JORDAN1F: 6;

      (Emax `1 ) = Ebo by EUCLID: 52

      .= ((Ga * (( len Ga),k)) `1 ) by A3, A14, JORDAN1A: 71;

      then

       A32: Gik <> (LS . 1) by A3, A17, A30, A31, JORDAN1G: 7;

      

       A33: ( len go) >= (1 + 1) by TOPREAL1:def 8;

      

       A34: Gij in ( rng US) by A1, A4, A5, A10, A11, A13, JORDAN1G: 4, JORDAN1J: 40;

      then

       A35: go is_sequence_on Ga by JORDAN1G: 4, JORDAN1J: 38;

      

       A36: ( len co) >= (1 + 1) by TOPREAL1:def 8;

      

       A37: Gik in ( rng LS) by A4, A5, A9, A11, A14, A15, JORDAN1G: 5, JORDAN1J: 40;

      then

       A38: co is_sequence_on Ga by JORDAN1G: 5, JORDAN1J: 39;

      reconsider go as non constant s.c.c. being_S-Seq FinSequence of ( TOP-REAL 2) by A33, A35, JGRAPH_1: 12, JORDAN8: 5;

      reconsider co as non constant s.c.c. being_S-Seq FinSequence of ( TOP-REAL 2) by A36, A38, JGRAPH_1: 12, JORDAN8: 5;

      

       A39: ( len go) > 1 by A33, NAT_1: 13;

      then

       A40: ( len go) in ( dom go) by FINSEQ_3: 25;

      

      then

       A41: (go /. ( len go)) = (go . ( len go)) by PARTFUN1:def 6

      .= Gij by A10, JORDAN3: 24;

      ( len co) >= 1 by A36, XXREAL_0: 2;

      then 1 in ( dom co) by FINSEQ_3: 25;

      

      then

       A42: (co /. 1) = (co . 1) by PARTFUN1:def 6

      .= Gik by A9, JORDAN3: 23;

      reconsider m = (( len go) - 1) as Nat by A40, FINSEQ_3: 26;

      

       A43: (m + 1) = ( len go);

      then

       A44: (( len go) -' 1) = m by NAT_D: 34;

      

       A45: ( LSeg (go,m)) c= ( L~ go) by TOPREAL3: 19;

      

       A46: ( L~ go) c= ( L~ US) by A10, JORDAN3: 41;

      then ( LSeg (go,m)) c= ( L~ US) by A45;

      then

       A47: (( LSeg (go,m)) /\ ( LSeg (Gik,Gij))) c= {Gij} by A6, XBOOLE_1: 26;

      m >= 1 by A33, XREAL_1: 19;

      then

       A48: ( LSeg (go,m)) = ( LSeg ((go /. m),Gij)) by A41, A43, TOPREAL1:def 3;

       {Gij} c= (( LSeg (go,m)) /\ ( LSeg (Gik,Gij)))

      proof

        let x be object;

        

         A49: Gij in ( LSeg (Gik,Gij)) by RLTOPSP1: 68;

        assume x in {Gij};

        then

         A50: x = Gij by TARSKI:def 1;

        Gij in ( LSeg (go,m)) by A48, RLTOPSP1: 68;

        hence thesis by A50, A49, XBOOLE_0:def 4;

      end;

      then

       A51: (( LSeg (go,m)) /\ ( LSeg (Gik,Gij))) = {Gij} by A47;

      

       A52: ( LSeg (co,1)) c= ( L~ co) by TOPREAL3: 19;

      

       A53: ( L~ co) c= ( L~ LS) by A9, JORDAN3: 42;

      then ( LSeg (co,1)) c= ( L~ LS) by A52;

      then

       A54: (( LSeg (co,1)) /\ ( LSeg (Gik,Gij))) c= {Gik} by A7, XBOOLE_1: 26;

      

       A55: ( LSeg (co,1)) = ( LSeg (Gik,(co /. (1 + 1)))) by A36, A42, TOPREAL1:def 3;

       {Gik} c= (( LSeg (co,1)) /\ ( LSeg (Gik,Gij)))

      proof

        let x be object;

        

         A56: Gik in ( LSeg (Gik,Gij)) by RLTOPSP1: 68;

        assume x in {Gik};

        then

         A57: x = Gik by TARSKI:def 1;

        Gik in ( LSeg (co,1)) by A55, RLTOPSP1: 68;

        hence thesis by A57, A56, XBOOLE_0:def 4;

      end;

      then

       A58: (( LSeg (Gik,Gij)) /\ ( LSeg (co,1))) = {Gik} by A54;

      

       A59: (go /. 1) = (US /. 1) by A10, SPRECT_3: 22

      .= Wmin by JORDAN1F: 5;

      

      then

       A60: (go /. 1) = (LS /. ( len LS)) by JORDAN1F: 8

      .= (co /. ( len co)) by A9, JORDAN1J: 35;

      

       A61: ( rng go) c= ( L~ go) by A33, SPPOL_2: 18;

      

       A62: ( rng co) c= ( L~ co) by A36, SPPOL_2: 18;

      

       A63: {(go /. 1)} c= (( L~ go) /\ ( L~ co))

      proof

        let x be object;

        assume x in {(go /. 1)};

        then

         A64: x = (go /. 1) by TARSKI:def 1;

        then

         A65: x in ( rng go) by FINSEQ_6: 42;

        x in ( rng co) by A60, A64, FINSEQ_6: 168;

        hence thesis by A61, A62, A65, XBOOLE_0:def 4;

      end;

      

       A66: (LS . 1) = (LS /. 1) by A27, PARTFUN1:def 6

      .= Emax by JORDAN1F: 6;

      

       A67: [( len Ga), j] in ( Indices Ga) by A1, A13, A21, MATRIX_0: 30;

      (( L~ go) /\ ( L~ co)) c= {(go /. 1)}

      proof

        let x be object;

        assume

         A68: x in (( L~ go) /\ ( L~ co));

        then

         A69: x in ( L~ co) by XBOOLE_0:def 4;

         A70:

        now

          assume x = Emax;

          then

           A71: Emax = Gik by A9, A66, A69, JORDAN1E: 7;

          ((Ga * (( len Ga),j)) `1 ) = Ebo by A1, A11, A13, JORDAN1A: 71;

          then (Emax `1 ) <> Ebo by A3, A17, A67, A71, JORDAN1G: 7;

          hence contradiction by EUCLID: 52;

        end;

        x in ( L~ go) by A68, XBOOLE_0:def 4;

        then x in (( L~ US) /\ ( L~ LS)) by A46, A53, A69, XBOOLE_0:def 4;

        then x in {Wmin, Emax} by JORDAN1E: 16;

        then x = Wmin or x = Emax by TARSKI:def 2;

        hence thesis by A59, A70, TARSKI:def 1;

      end;

      then

       A72: (( L~ go) /\ ( L~ co)) = {(go /. 1)} by A63;

      set W2 = (go /. 2);

      

       A73: 2 in ( dom go) by A33, FINSEQ_3: 25;

       A74:

      now

        assume (Gij `1 ) = Wbo;

        then ((Ga * (1,k)) `1 ) = ((Ga * (j,i)) `1 ) by A3, A14, JORDAN1A: 73;

        hence contradiction by A1, A16, A22, JORDAN1G: 7;

      end;

      go = ( mid (US,1,(Gij .. US))) by A34, JORDAN1G: 49

      .= (US | (Gij .. US)) by A34, FINSEQ_4: 21, FINSEQ_6: 116;

      then

       A75: W2 = (US /. 2) by A73, FINSEQ_4: 70;

      

       A76: Wmin in ( rng go) by A59, FINSEQ_6: 42;

      set pion = <*Gij, Gik*>;

       A77:

      now

        let n be Nat;

        assume n in ( dom pion);

        then n in {1, 2} by FINSEQ_1: 2, FINSEQ_1: 89;

        then n = 1 or n = 2 by TARSKI:def 2;

        hence ex j,i be Nat st [j, i] in ( Indices Ga) & (pion /. n) = (Ga * (j,i)) by A16, A17, FINSEQ_4: 17;

      end;

      

       A78: Gik <> Gij by A12, A16, A17, GOBOARD1: 5;

      (Gik `2 ) = ((Ga * (1,i)) `2 ) by A3, A4, A5, A14, GOBOARD5: 1

      .= (Gij `2 ) by A1, A4, A5, A11, A13, GOBOARD5: 1;

      then ( LSeg (Gik,Gij)) is horizontal by SPPOL_1: 15;

      then pion is being_S-Seq by A78, JORDAN1B: 8;

      then

      consider pion1 be FinSequence of ( TOP-REAL 2) such that

       A79: pion1 is_sequence_on Ga and

       A80: pion1 is being_S-Seq and

       A81: ( L~ pion) = ( L~ pion1) and

       A82: (pion /. 1) = (pion1 /. 1) and

       A83: (pion /. ( len pion)) = (pion1 /. ( len pion1)) and

       A84: ( len pion) <= ( len pion1) by A77, GOBOARD3: 2;

      reconsider pion1 as being_S-Seq FinSequence of ( TOP-REAL 2) by A80;

      set godo = ((go ^' pion1) ^' co);

      

       A85: (1 + 1) <= ( len ( Cage (C,n))) by GOBOARD7: 34, XXREAL_0: 2;

      

       A86: (1 + 1) <= ( len ( Rotate (( Cage (C,n)),Wmin))) by GOBOARD7: 34, XXREAL_0: 2;

      ( len (go ^' pion1)) >= ( len go) by TOPREAL8: 7;

      then

       A87: ( len (go ^' pion1)) >= (1 + 1) by A33, XXREAL_0: 2;

      then

       A88: ( len (go ^' pion1)) > (1 + 0 ) by NAT_1: 13;

      

       A89: ( len godo) >= ( len (go ^' pion1)) by TOPREAL8: 7;

      then

       A90: (1 + 1) <= ( len godo) by A87, XXREAL_0: 2;

      

       A91: US is_sequence_on Ga by JORDAN1G: 4;

      

       A92: (go /. ( len go)) = (pion1 /. 1) by A41, A82, FINSEQ_4: 17;

      then

       A93: (go ^' pion1) is_sequence_on Ga by A35, A79, TOPREAL8: 12;

      

       A94: ((go ^' pion1) /. ( len (go ^' pion1))) = (pion /. ( len pion)) by A83, FINSEQ_6: 156

      .= (pion /. 2) by FINSEQ_1: 44

      .= (co /. 1) by A42, FINSEQ_4: 17;

      then

       A95: godo is_sequence_on Ga by A38, A93, TOPREAL8: 12;

      ( LSeg (pion1,1)) c= ( L~ <*Gij, Gik*>) by A81, TOPREAL3: 19;

      then ( LSeg (pion1,1)) c= ( LSeg (Gij,Gik)) by SPPOL_2: 21;

      then

       A96: (( LSeg (go,(( len go) -' 1))) /\ ( LSeg (pion1,1))) c= {Gij} by A44, A51, XBOOLE_1: 27;

      

       A97: ( len pion1) >= (1 + 1) by A84, FINSEQ_1: 44;

       {Gij} c= (( LSeg (go,m)) /\ ( LSeg (pion1,1)))

      proof

        let x be object;

        assume x in {Gij};

        then

         A98: x = Gij by TARSKI:def 1;

        

         A99: Gij in ( LSeg (go,m)) by A48, RLTOPSP1: 68;

        Gij in ( LSeg (pion1,1)) by A41, A92, A97, TOPREAL1: 21;

        hence thesis by A98, A99, XBOOLE_0:def 4;

      end;

      then (( LSeg (go,(( len go) -' 1))) /\ ( LSeg (pion1,1))) = {(go /. ( len go))} by A41, A44, A96;

      then

       A100: (go ^' pion1) is unfolded by A92, TOPREAL8: 34;

      ( len pion1) >= (2 + 0 ) by A84, FINSEQ_1: 44;

      then

       A101: (( len pion1) - 2) >= 0 by XREAL_1: 19;

      ((( len (go ^' pion1)) + 1) - 1) = ((( len go) + ( len pion1)) - 1) by FINSEQ_6: 139;

      

      then (( len (go ^' pion1)) - 1) = (( len go) + (( len pion1) - 2))

      .= (( len go) + (( len pion1) -' 2)) by A101, XREAL_0:def 2;

      then

       A102: (( len (go ^' pion1)) -' 1) = (( len go) + (( len pion1) -' 2)) by XREAL_0:def 2;

      

       A103: (( len pion1) - 1) >= 1 by A97, XREAL_1: 19;

      then

       A104: (( len pion1) -' 1) = (( len pion1) - 1) by XREAL_0:def 2;

      

       A105: ((( len pion1) -' 2) + 1) = ((( len pion1) - 2) + 1) by A101, XREAL_0:def 2

      .= (( len pion1) -' 1) by A103, XREAL_0:def 2;

      ((( len pion1) - 1) + 1) <= ( len pion1);

      then

       A106: (( len pion1) -' 1) < ( len pion1) by A104, NAT_1: 13;

      ( LSeg (pion1,(( len pion1) -' 1))) c= ( L~ <*Gij, Gik*>) by A81, TOPREAL3: 19;

      then ( LSeg (pion1,(( len pion1) -' 1))) c= ( LSeg (Gij,Gik)) by SPPOL_2: 21;

      then

       A107: (( LSeg (pion1,(( len pion1) -' 1))) /\ ( LSeg (co,1))) c= {Gik} by A58, XBOOLE_1: 27;

       {Gik} c= (( LSeg (pion1,(( len pion1) -' 1))) /\ ( LSeg (co,1)))

      proof

        let x be object;

        assume x in {Gik};

        then

         A108: x = Gik by TARSKI:def 1;

        (pion1 /. ((( len pion1) -' 1) + 1)) = (pion /. 2) by A83, A104, FINSEQ_1: 44

        .= Gik by FINSEQ_4: 17;

        then

         A109: Gik in ( LSeg (pion1,(( len pion1) -' 1))) by A103, A104, TOPREAL1: 21;

        Gik in ( LSeg (co,1)) by A55, RLTOPSP1: 68;

        hence thesis by A108, A109, XBOOLE_0:def 4;

      end;

      then (( LSeg (pion1,(( len pion1) -' 1))) /\ ( LSeg (co,1))) = {Gik} by A107;

      then

       A110: (( LSeg ((go ^' pion1),(( len go) + (( len pion1) -' 2)))) /\ ( LSeg (co,1))) = {((go ^' pion1) /. ( len (go ^' pion1)))} by A42, A92, A94, A105, A106, TOPREAL8: 31;

      

       A111: (go ^' pion1) is non trivial by A87, NAT_D: 60;

      

       A112: ( rng pion1) c= ( L~ pion1) by A97, SPPOL_2: 18;

      

       A113: {(pion1 /. 1)} c= (( L~ go) /\ ( L~ pion1))

      proof

        let x be object;

        assume x in {(pion1 /. 1)};

        then

         A114: x = (pion1 /. 1) by TARSKI:def 1;

        then

         A115: x in ( rng pion1) by FINSEQ_6: 42;

        x in ( rng go) by A92, A114, FINSEQ_6: 168;

        hence thesis by A61, A112, A115, XBOOLE_0:def 4;

      end;

      (( L~ go) /\ ( L~ pion1)) c= {(pion1 /. 1)}

      proof

        let x be object;

        assume

         A116: x in (( L~ go) /\ ( L~ pion1));

        then

         A117: x in ( L~ pion1) by XBOOLE_0:def 4;

        x in ( L~ go) by A116, XBOOLE_0:def 4;

        then x in (( L~ pion1) /\ ( L~ US)) by A46, A117, XBOOLE_0:def 4;

        hence thesis by A6, A41, A81, A92, SPPOL_2: 21;

      end;

      then

       A118: (( L~ go) /\ ( L~ pion1)) = {(pion1 /. 1)} by A113;

      then

       A119: (go ^' pion1) is s.n.c. by A92, JORDAN1J: 54;

      (( rng go) /\ ( rng pion1)) c= {(pion1 /. 1)} by A61, A112, A118, XBOOLE_1: 27;

      then

       A120: (go ^' pion1) is one-to-one by JORDAN1J: 55;

      

       A121: (pion /. ( len pion)) = (pion /. 2) by FINSEQ_1: 44

      .= (co /. 1) by A42, FINSEQ_4: 17;

      

       A122: {(pion1 /. ( len pion1))} c= (( L~ co) /\ ( L~ pion1))

      proof

        let x be object;

        assume x in {(pion1 /. ( len pion1))};

        then

         A123: x = (pion1 /. ( len pion1)) by TARSKI:def 1;

        then

         A124: x in ( rng pion1) by FINSEQ_6: 168;

        x in ( rng co) by A83, A121, A123, FINSEQ_6: 42;

        hence thesis by A62, A112, A124, XBOOLE_0:def 4;

      end;

      (( L~ co) /\ ( L~ pion1)) c= {(pion1 /. ( len pion1))}

      proof

        let x be object;

        assume

         A125: x in (( L~ co) /\ ( L~ pion1));

        then

         A126: x in ( L~ pion1) by XBOOLE_0:def 4;

        x in ( L~ co) by A125, XBOOLE_0:def 4;

        then x in (( L~ pion1) /\ ( L~ LS)) by A53, A126, XBOOLE_0:def 4;

        hence thesis by A7, A42, A81, A83, A121, SPPOL_2: 21;

      end;

      then

       A127: (( L~ co) /\ ( L~ pion1)) = {(pion1 /. ( len pion1))} by A122;

      

       A128: (( L~ (go ^' pion1)) /\ ( L~ co)) = ((( L~ go) \/ ( L~ pion1)) /\ ( L~ co)) by A92, TOPREAL8: 35

      .= ( {(go /. 1)} \/ {(co /. 1)}) by A72, A83, A121, A127, XBOOLE_1: 23

      .= ( {((go ^' pion1) /. 1)} \/ {(co /. 1)}) by FINSEQ_6: 155

      .= {((go ^' pion1) /. 1), (co /. 1)} by ENUMSET1: 1;

      (co /. ( len co)) = ((go ^' pion1) /. 1) by A60, FINSEQ_6: 155;

      then

      reconsider godo as non constant standard special_circular_sequence by A90, A94, A95, A100, A102, A110, A111, A119, A120, A128, JORDAN8: 4, JORDAN8: 5, TOPREAL8: 11, TOPREAL8: 33, TOPREAL8: 34;

      

       A129: LA is_an_arc_of (( E-max C),( W-min C)) by JORDAN6:def 9;

      then

       A130: LA is connected by JORDAN6: 10;

      

       A131: ( W-min C) in LA by A129, TOPREAL1: 1;

      

       A132: ( E-max C) in LA by A129, TOPREAL1: 1;

      set ff = ( Rotate (( Cage (C,n)),Wmin));

      Wmin in ( rng ( Cage (C,n))) by SPRECT_2: 43;

      then

       A133: (ff /. 1) = Wmin by FINSEQ_6: 92;

      

       A134: ( L~ ff) = ( L~ ( Cage (C,n))) by REVROT_1: 33;

      then (( W-max ( L~ ff)) .. ff) > 1 by A133, SPRECT_5: 22;

      then (( N-min ( L~ ff)) .. ff) > 1 by A133, A134, SPRECT_5: 23, XXREAL_0: 2;

      then (( N-max ( L~ ff)) .. ff) > 1 by A133, A134, SPRECT_5: 24, XXREAL_0: 2;

      then

       A135: (Emax .. ff) > 1 by A133, A134, SPRECT_5: 25, XXREAL_0: 2;

       A136:

      now

        assume

         A137: (Gij .. US) <= 1;

        (Gij .. US) >= 1 by A34, FINSEQ_4: 21;

        then (Gij .. US) = 1 by A137, XXREAL_0: 1;

        then Gij = (US /. 1) by A34, FINSEQ_5: 38;

        hence contradiction by A19, A24, JORDAN1F: 5;

      end;

      

       A138: ( Cage (C,n)) is_sequence_on Ga by JORDAN9:def 1;

      then

       A139: ff is_sequence_on Ga by REVROT_1: 34;

      

       A140: (( right_cell (godo,1,Ga)) \ ( L~ godo)) c= ( RightComp godo) by A90, A95, JORDAN9: 27;

      

       A141: ( L~ godo) = (( L~ (go ^' pion1)) \/ ( L~ co)) by A94, TOPREAL8: 35

      .= ((( L~ go) \/ ( L~ pion1)) \/ ( L~ co)) by A92, TOPREAL8: 35;

      

       A142: ( L~ ( Cage (C,n))) = (( L~ US) \/ ( L~ LS)) by JORDAN1E: 13;

      then

       A143: ( L~ US) c= ( L~ ( Cage (C,n))) by XBOOLE_1: 7;

      

       A144: ( L~ LS) c= ( L~ ( Cage (C,n))) by A142, XBOOLE_1: 7;

      

       A145: ( L~ go) c= ( L~ ( Cage (C,n))) by A46, A143;

      

       A146: ( L~ co) c= ( L~ ( Cage (C,n))) by A53, A144;

      

       A147: ( W-min C) in C by SPRECT_1: 13;

      

       A148: ( L~ pion) = ( LSeg (Gik,Gij)) by SPPOL_2: 21;

       A149:

      now

        assume ( W-min C) in ( L~ godo);

        then

         A150: ( W-min C) in (( L~ go) \/ ( L~ pion1)) or ( W-min C) in ( L~ co) by A141, XBOOLE_0:def 3;

        per cases by A150, XBOOLE_0:def 3;

          suppose ( W-min C) in ( L~ go);

          then C meets ( L~ ( Cage (C,n))) by A145, A147, XBOOLE_0: 3;

          hence contradiction by JORDAN10: 5;

        end;

          suppose ( W-min C) in ( L~ pion1);

          hence contradiction by A8, A81, A131, A148, XBOOLE_0: 3;

        end;

          suppose ( W-min C) in ( L~ co);

          then C meets ( L~ ( Cage (C,n))) by A146, A147, XBOOLE_0: 3;

          hence contradiction by JORDAN10: 5;

        end;

      end;

      ( right_cell (( Rotate (( Cage (C,n)),Wmin)),1)) = ( right_cell (ff,1,( GoB ff))) by A86, JORDAN1H: 23

      .= ( right_cell (ff,1,( GoB ( Cage (C,n))))) by REVROT_1: 28

      .= ( right_cell (ff,1,Ga)) by JORDAN1H: 44

      .= ( right_cell ((ff -: Emax),1,Ga)) by A135, A139, JORDAN1J: 53

      .= ( right_cell (US,1,Ga)) by JORDAN1E:def 1

      .= ( right_cell (( R_Cut (US,Gij)),1,Ga)) by A34, A91, A136, JORDAN1J: 52

      .= ( right_cell ((go ^' pion1),1,Ga)) by A39, A93, JORDAN1J: 51

      .= ( right_cell (godo,1,Ga)) by A88, A95, JORDAN1J: 51;

      then ( W-min C) in ( right_cell (godo,1,Ga)) by JORDAN1I: 6;

      then

       A151: ( W-min C) in (( right_cell (godo,1,Ga)) \ ( L~ godo)) by A149, XBOOLE_0:def 5;

      

       A152: (godo /. 1) = ((go ^' pion1) /. 1) by FINSEQ_6: 155

      .= Wmin by A59, FINSEQ_6: 155;

      

       A153: ( len US) >= 2 by A18, XXREAL_0: 2;

      

       A154: (godo /. 2) = ((go ^' pion1) /. 2) by A87, FINSEQ_6: 159

      .= (US /. 2) by A33, A75, FINSEQ_6: 159

      .= ((US ^' LS) /. 2) by A153, FINSEQ_6: 159

      .= (( Rotate (( Cage (C,n)),Wmin)) /. 2) by JORDAN1E: 11;

      

       A155: (( L~ go) \/ ( L~ co)) is compact by COMPTS_1: 10;

      Wmin in (( L~ go) \/ ( L~ co)) by A61, A76, XBOOLE_0:def 3;

      then

       A156: ( W-min (( L~ go) \/ ( L~ co))) = Wmin by A145, A146, A155, JORDAN1J: 21, XBOOLE_1: 8;

      

       A157: (( W-min (( L~ go) \/ ( L~ co))) `1 ) = ( W-bound (( L~ go) \/ ( L~ co))) by EUCLID: 52;

      

       A158: (Wmin `1 ) = Wbo by EUCLID: 52;

      

       A159: (Gij `1 ) <= (Gik `1 ) by A1, A2, A3, A4, A5, SPRECT_3: 13;

      then ( W-bound ( LSeg (Gik,Gij))) = (Gij `1 ) by SPRECT_1: 54;

      then

       A160: ( W-bound ( L~ pion1)) = (Gij `1 ) by A81, SPPOL_2: 21;

      (Gij `1 ) >= Wbo by A10, A143, PSCOMP_1: 24;

      then (Gij `1 ) > Wbo by A74, XXREAL_0: 1;

      then ( W-min ((( L~ go) \/ ( L~ co)) \/ ( L~ pion1))) = ( W-min (( L~ go) \/ ( L~ co))) by A155, A156, A157, A158, A160, JORDAN1J: 33;

      then

       A161: ( W-min ( L~ godo)) = Wmin by A141, A156, XBOOLE_1: 4;

      

       A162: ( rng godo) c= ( L~ godo) by A87, A89, SPPOL_2: 18, XXREAL_0: 2;

      2 in ( dom godo) by A90, FINSEQ_3: 25;

      then

       A163: (godo /. 2) in ( rng godo) by PARTFUN2: 2;

      (godo /. 2) in ( W-most ( L~ ( Cage (C,n)))) by A154, JORDAN1I: 25;

      

      then ((godo /. 2) `1 ) = (( W-min ( L~ godo)) `1 ) by A161, PSCOMP_1: 31

      .= ( W-bound ( L~ godo)) by EUCLID: 52;

      then (godo /. 2) in ( W-most ( L~ godo)) by A162, A163, SPRECT_2: 12;

      then (( Rotate (godo,( W-min ( L~ godo)))) /. 2) in ( W-most ( L~ godo)) by A152, A161, FINSEQ_6: 89;

      then

      reconsider godo as clockwise_oriented non constant standard special_circular_sequence by JORDAN1I: 25;

      ( len US) in ( dom US) by FINSEQ_5: 6;

      

      then

       A164: (US . ( len US)) = (US /. ( len US)) by PARTFUN1:def 6

      .= Emax by JORDAN1F: 7;

      

       A165: ( east_halfline ( E-max C)) misses ( L~ go)

      proof

        assume ( east_halfline ( E-max C)) meets ( L~ go);

        then

        consider p be object such that

         A166: p in ( east_halfline ( E-max C)) and

         A167: p in ( L~ go) by XBOOLE_0: 3;

        reconsider p as Point of ( TOP-REAL 2) by A166;

        p in ( L~ US) by A46, A167;

        then p in (( east_halfline ( E-max C)) /\ ( L~ ( Cage (C,n)))) by A143, A166, XBOOLE_0:def 4;

        then

         A168: (p `1 ) = Ebo by JORDAN1A: 83, PSCOMP_1: 50;

        then

         A169: p = Emax by A46, A167, JORDAN1J: 46;

        then Emax = Gij by A10, A164, A167, JORDAN1J: 43;

        then (Gij `1 ) = ((Ga * (( len Ga),k)) `1 ) by A3, A14, A168, A169, JORDAN1A: 71;

        hence contradiction by A2, A3, A16, A30, JORDAN1G: 7;

      end;

      now

        assume ( east_halfline ( E-max C)) meets ( L~ godo);

        then

         A170: ( east_halfline ( E-max C)) meets (( L~ go) \/ ( L~ pion1)) or ( east_halfline ( E-max C)) meets ( L~ co) by A141, XBOOLE_1: 70;

        per cases by A170, XBOOLE_1: 70;

          suppose ( east_halfline ( E-max C)) meets ( L~ go);

          hence contradiction by A165;

        end;

          suppose ( east_halfline ( E-max C)) meets ( L~ pion1);

          then

          consider p be object such that

           A171: p in ( east_halfline ( E-max C)) and

           A172: p in ( L~ pion1) by XBOOLE_0: 3;

          reconsider p as Point of ( TOP-REAL 2) by A171;

          

           A173: (p `2 ) = (( E-max C) `2 ) by A171, TOPREAL1:def 11;

          (k + 1) <= ( len Ga) by A3, NAT_1: 13;

          then ((k + 1) - 1) <= (( len Ga) - 1) by XREAL_1: 9;

          then

           A174: k <= (( len Ga) -' 1) by XREAL_0:def 2;

          (( len Ga) -' 1) <= ( len Ga) by NAT_D: 35;

          then

           A175: (Gik `1 ) <= ((Ga * ((( len Ga) -' 1),1)) `1 ) by A4, A5, A11, A14, A21, A174, JORDAN1A: 18;

          (p `1 ) <= (Gik `1 ) by A81, A148, A159, A172, TOPREAL1: 3;

          then (p `1 ) <= ((Ga * ((( len Ga) -' 1),1)) `1 ) by A175, XXREAL_0: 2;

          then (p `1 ) <= ( E-bound C) by A21, JORDAN8: 12;

          then

           A176: (p `1 ) <= (( E-max C) `1 ) by EUCLID: 52;

          (p `1 ) >= (( E-max C) `1 ) by A171, TOPREAL1:def 11;

          then (p `1 ) = (( E-max C) `1 ) by A176, XXREAL_0: 1;

          then p = ( E-max C) by A173, TOPREAL3: 6;

          hence contradiction by A8, A81, A132, A148, A172, XBOOLE_0: 3;

        end;

          suppose ( east_halfline ( E-max C)) meets ( L~ co);

          then

          consider p be object such that

           A177: p in ( east_halfline ( E-max C)) and

           A178: p in ( L~ co) by XBOOLE_0: 3;

          reconsider p as Point of ( TOP-REAL 2) by A177;

          

           A179: p in ( LSeg (co,( Index (p,co)))) by A178, JORDAN3: 9;

          consider t be Nat such that

           A180: t in ( dom LS) and

           A181: (LS . t) = Gik by A37, FINSEQ_2: 10;

          1 <= t by A180, FINSEQ_3: 25;

          then

           A182: 1 < t by A32, A181, XXREAL_0: 1;

          t <= ( len LS) by A180, FINSEQ_3: 25;

          then (( Index (Gik,LS)) + 1) = t by A181, A182, JORDAN3: 12;

          then

           A183: ( len ( L_Cut (LS,Gik))) = (( len LS) - ( Index (Gik,LS))) by A9, A181, JORDAN3: 26;

          ( Index (p,co)) < ( len co) by A178, JORDAN3: 8;

          then ( Index (p,co)) < (( len LS) -' ( Index (Gik,LS))) by A183, XREAL_0:def 2;

          then (( Index (p,co)) + 1) <= (( len LS) -' ( Index (Gik,LS))) by NAT_1: 13;

          then

           A184: ( Index (p,co)) <= ((( len LS) -' ( Index (Gik,LS))) - 1) by XREAL_1: 19;

          

           A185: co = ( mid (LS,(Gik .. LS),( len LS))) by A37, JORDAN1J: 37;

          p in ( L~ LS) by A53, A178;

          then p in (( east_halfline ( E-max C)) /\ ( L~ ( Cage (C,n)))) by A144, A177, XBOOLE_0:def 4;

          then

           A186: (p `1 ) = Ebo by JORDAN1A: 83, PSCOMP_1: 50;

          

           A187: (( Index (Gik,LS)) + 1) = (Gik .. LS) by A32, A37, JORDAN1J: 56;

          ( 0 + ( Index (Gik,LS))) < ( len LS) by A9, JORDAN3: 8;

          then (( len LS) - ( Index (Gik,LS))) > 0 by XREAL_1: 20;

          then ( Index (p,co)) <= ((( len LS) - ( Index (Gik,LS))) - 1) by A184, XREAL_0:def 2;

          then ( Index (p,co)) <= (( len LS) - (Gik .. LS)) by A187;

          then ( Index (p,co)) <= (( len LS) -' (Gik .. LS)) by XREAL_0:def 2;

          then

           A188: ( Index (p,co)) < ((( len LS) -' (Gik .. LS)) + 1) by NAT_1: 13;

          

           A189: 1 <= ( Index (p,co)) by A178, JORDAN3: 8;

          

           A190: (Gik .. LS) <= ( len LS) by A37, FINSEQ_4: 21;

          (Gik .. LS) <> ( len LS) by A29, A37, FINSEQ_4: 19;

          then

           A191: (Gik .. LS) < ( len LS) by A190, XXREAL_0: 1;

          

           A192: (1 + 1) <= ( len LS) by A25, XXREAL_0: 2;

          then

           A193: 2 in ( dom LS) by FINSEQ_3: 25;

          set tt = ((( Index (p,co)) + (Gik .. LS)) -' 1);

          set RC = ( Rotate (( Cage (C,n)),Emax));

          

           A194: ( E-max C) in ( right_cell (RC,1)) by JORDAN1I: 7;

          

           A195: ( GoB RC) = ( GoB ( Cage (C,n))) by REVROT_1: 28

          .= Ga by JORDAN1H: 44;

          

           A196: ( L~ RC) = ( L~ ( Cage (C,n))) by REVROT_1: 33;

          consider g2 be Nat such that

           A197: 1 <= g2 and

           A198: g2 <= ( width Ga) and

           A199: Emax = (Ga * (( len Ga),g2)) by JORDAN1D: 25;

          

           A200: ( len Ga) >= 4 by JORDAN8: 10;

          then ( len Ga) >= 1 by XXREAL_0: 2;

          then

           A201: [( len Ga), g2] in ( Indices Ga) by A197, A198, MATRIX_0: 30;

          

           A202: ( len RC) = ( len ( Cage (C,n))) by FINSEQ_6: 179;

          LS = (RC -: Wmin) by JORDAN1G: 18;

          then

           A203: ( LSeg (LS,1)) = ( LSeg (RC,1)) by A192, SPPOL_2: 9;

          

           A204: Emax in ( rng ( Cage (C,n))) by SPRECT_2: 46;

          RC is_sequence_on Ga by A138, REVROT_1: 34;

          then

          consider ii,g be Nat such that

           A205: [ii, (g + 1)] in ( Indices Ga) and

           A206: [ii, g] in ( Indices Ga) and

           A207: (RC /. 1) = (Ga * (ii,(g + 1))) and

           A208: (RC /. (1 + 1)) = (Ga * (ii,g)) by A85, A196, A202, A204, FINSEQ_6: 92, JORDAN1I: 23;

          

           A209: ((g + 1) + 1) <> g;

          

           A210: 1 <= g by A206, MATRIX_0: 32;

          (RC /. 1) = ( E-max ( L~ RC)) by A196, A204, FINSEQ_6: 92;

          then

           A211: ii = ( len Ga) by A196, A205, A207, A199, A201, GOBOARD1: 5;

          then (ii - 1) >= (4 - 1) by A200, XREAL_1: 9;

          then

           A212: (ii - 1) >= 1 by XXREAL_0: 2;

          then

           A213: 1 <= (ii -' 1) by XREAL_0:def 2;

          

           A214: g <= ( width Ga) by A206, MATRIX_0: 32;

          then

           A215: ((Ga * (( len Ga),g)) `1 ) = Ebo by A11, A210, JORDAN1A: 71;

          

           A216: (g + 1) <= ( width Ga) by A205, MATRIX_0: 32;

          (ii + 1) <> ii;

          then

           A217: ( right_cell (RC,1)) = ( cell (Ga,(ii -' 1),g)) by A85, A202, A195, A205, A206, A207, A208, A209, GOBOARD5:def 6;

          

           A218: ii <= ( len Ga) by A206, MATRIX_0: 32;

          

           A219: 1 <= ii by A206, MATRIX_0: 32;

          

           A220: ii <= ( len Ga) by A205, MATRIX_0: 32;

          

           A221: 1 <= (g + 1) by A205, MATRIX_0: 32;

          then

           A222: Ebo = ((Ga * (( len Ga),(g + 1))) `1 ) by A11, A216, JORDAN1A: 71;

          

           A223: 1 <= ii by A205, MATRIX_0: 32;

          then

           A224: ((ii -' 1) + 1) = ii by XREAL_1: 235;

          then

           A225: (ii -' 1) < ( len Ga) by A220, NAT_1: 13;

          

          then

           A226: ((Ga * ((ii -' 1),(g + 1))) `2 ) = ((Ga * (1,(g + 1))) `2 ) by A221, A216, A213, GOBOARD5: 1

          .= ((Ga * (ii,(g + 1))) `2 ) by A223, A220, A221, A216, GOBOARD5: 1;

          

           A227: (( E-max C) `2 ) = (p `2 ) by A177, TOPREAL1:def 11;

          then

           A228: (p `2 ) <= ((Ga * ((ii -' 1),(g + 1))) `2 ) by A194, A220, A216, A210, A217, A224, A212, JORDAN9: 17;

          

           A229: ((Ga * ((ii -' 1),g)) `2 ) = ((Ga * (1,g)) `2 ) by A210, A214, A213, A225, GOBOARD5: 1

          .= ((Ga * (ii,g)) `2 ) by A219, A218, A210, A214, GOBOARD5: 1;

          ((Ga * ((ii -' 1),g)) `2 ) <= (p `2 ) by A227, A194, A220, A216, A210, A217, A224, A212, JORDAN9: 17;

          then p in ( LSeg ((RC /. 1),(RC /. (1 + 1)))) by A186, A207, A208, A211, A228, A229, A226, A215, A222, GOBOARD7: 7;

          then

           A230: p in ( LSeg (LS,1)) by A85, A203, A202, TOPREAL1:def 3;

          1 <= (Gik .. LS) by A37, FINSEQ_4: 21;

          then

           A231: ( LSeg (( mid (LS,(Gik .. LS),( len LS))),( Index (p,co)))) = ( LSeg (LS,((( Index (p,co)) + (Gik .. LS)) -' 1))) by A191, A189, A188, JORDAN4: 19;

          1 <= ( Index (Gik,LS)) by A9, JORDAN3: 8;

          then

           A232: (1 + 1) <= (Gik .. LS) by A187, XREAL_1: 7;

          then (( Index (p,co)) + (Gik .. LS)) >= ((1 + 1) + 1) by A189, XREAL_1: 7;

          then ((( Index (p,co)) + (Gik .. LS)) - 1) >= (((1 + 1) + 1) - 1) by XREAL_1: 9;

          then

           A233: tt >= (1 + 1) by XREAL_0:def 2;

          now

            per cases by A233, XXREAL_0: 1;

              suppose tt > (1 + 1);

              then ( LSeg (LS,1)) misses ( LSeg (LS,tt)) by TOPREAL1:def 7;

              hence contradiction by A230, A179, A185, A231, XBOOLE_0: 3;

            end;

              suppose

               A234: tt = (1 + 1);

              then (1 + 1) = ((( Index (p,co)) + (Gik .. LS)) - 1) by XREAL_0:def 2;

              then ((1 + 1) + 1) = (( Index (p,co)) + (Gik .. LS));

              then

               A235: (Gik .. LS) = 2 by A189, A232, JORDAN1E: 6;

              (( LSeg (LS,1)) /\ ( LSeg (LS,tt))) = {(LS /. 2)} by A25, A234, TOPREAL1:def 6;

              then p in {(LS /. 2)} by A230, A179, A185, A231, XBOOLE_0:def 4;

              then

               A236: p = (LS /. 2) by TARSKI:def 1;

              then

               A237: p in ( rng LS) by A193, PARTFUN2: 2;

              (p .. LS) = 2 by A193, A236, FINSEQ_5: 41;

              then p = Gik by A37, A235, A237, FINSEQ_5: 9;

              then (Gik `1 ) = Ebo by A236, JORDAN1G: 32;

              then (Gik `1 ) = ((Ga * (( len Ga),j)) `1 ) by A1, A11, A13, JORDAN1A: 71;

              hence contradiction by A3, A17, A67, JORDAN1G: 7;

            end;

          end;

          hence contradiction;

        end;

      end;

      then ( east_halfline ( E-max C)) c= (( L~ godo) ` ) by SUBSET_1: 23;

      then

      consider W be Subset of ( TOP-REAL 2) such that

       A238: W is_a_component_of (( L~ godo) ` ) and

       A239: ( east_halfline ( E-max C)) c= W by GOBOARD9: 3;

       not W is bounded by A239, JORDAN2C: 121, RLTOPSP1: 42;

      then W is_outside_component_of ( L~ godo) by A238, JORDAN2C:def 3;

      then W c= ( UBD ( L~ godo)) by JORDAN2C: 23;

      then

       A240: ( east_halfline ( E-max C)) c= ( UBD ( L~ godo)) by A239;

      ( E-max C) in ( east_halfline ( E-max C)) by TOPREAL1: 38;

      then ( E-max C) in ( UBD ( L~ godo)) by A240;

      then ( E-max C) in ( LeftComp godo) by GOBRD14: 36;

      then LA meets ( L~ godo) by A130, A131, A132, A140, A151, JORDAN1J: 36;

      then

       A241: LA meets (( L~ go) \/ ( L~ pion1)) or LA meets ( L~ co) by A141, XBOOLE_1: 70;

      

       A242: LA c= C by JORDAN6: 61;

      per cases by A241, XBOOLE_1: 70;

        suppose LA meets ( L~ go);

        then LA meets ( L~ ( Cage (C,n))) by A46, A143, XBOOLE_1: 1, XBOOLE_1: 63;

        hence contradiction by A242, JORDAN10: 5, XBOOLE_1: 63;

      end;

        suppose LA meets ( L~ pion1);

        hence contradiction by A8, A81, A148;

      end;

        suppose LA meets ( L~ co);

        then LA meets ( L~ ( Cage (C,n))) by A53, A144, XBOOLE_1: 1, XBOOLE_1: 63;

        hence contradiction by A242, JORDAN10: 5, XBOOLE_1: 63;

      end;

    end;

    theorem :: JORDAN15:37

    

     Th37: for C be Simple_closed_curve holds for i,j,k be Nat st 1 < j & j <= k & k < ( len ( Gauge (C,n))) & 1 <= i & i <= ( width ( Gauge (C,n))) & (( LSeg ((( Gauge (C,n)) * (j,i)),(( Gauge (C,n)) * (k,i)))) /\ ( L~ ( Upper_Seq (C,n)))) = {(( Gauge (C,n)) * (j,i))} & (( LSeg ((( Gauge (C,n)) * (j,i)),(( Gauge (C,n)) * (k,i)))) /\ ( L~ ( Lower_Seq (C,n)))) = {(( Gauge (C,n)) * (k,i))} holds ( LSeg ((( Gauge (C,n)) * (j,i)),(( Gauge (C,n)) * (k,i)))) meets ( Upper_Arc C)

    proof

      let C be Simple_closed_curve;

      let i,j,k be Nat;

      set Ga = ( Gauge (C,n));

      set US = ( Upper_Seq (C,n));

      set LS = ( Lower_Seq (C,n));

      set UA = ( Upper_Arc C);

      set Wmin = ( W-min ( L~ ( Cage (C,n))));

      set Emax = ( E-max ( L~ ( Cage (C,n))));

      set Wbo = ( W-bound ( L~ ( Cage (C,n))));

      set Ebo = ( E-bound ( L~ ( Cage (C,n))));

      set Gij = (Ga * (j,i));

      set Gik = (Ga * (k,i));

      assume that

       A1: 1 < j and

       A2: j <= k and

       A3: k < ( len Ga) and

       A4: 1 <= i and

       A5: i <= ( width Ga) and

       A6: (( LSeg (Gij,Gik)) /\ ( L~ US)) = {Gij} and

       A7: (( LSeg (Gij,Gik)) /\ ( L~ LS)) = {Gik} and

       A8: ( LSeg (Gij,Gik)) misses UA;

      Gik in {Gik} by TARSKI:def 1;

      then

       A9: Gik in ( L~ LS) by A7, XBOOLE_0:def 4;

      Gij in {Gij} by TARSKI:def 1;

      then

       A10: Gij in ( L~ US) by A6, XBOOLE_0:def 4;

      

       A11: ( len Ga) = ( width Ga) by JORDAN8:def 1;

      

       A12: j <> k by A1, A3, A4, A5, A9, A10, Th27;

      

       A13: j < ( width Ga) by A2, A3, A11, XXREAL_0: 2;

      

       A14: 1 < k by A1, A2, XXREAL_0: 2;

      

       A15: k < ( width Ga) by A3, JORDAN8:def 1;

      

       A16: [j, i] in ( Indices Ga) by A1, A4, A5, A11, A13, MATRIX_0: 30;

      

       A17: [k, i] in ( Indices Ga) by A3, A4, A5, A14, MATRIX_0: 30;

      set go = ( R_Cut (US,Gij));

      set co = ( L_Cut (LS,Gik));

      

       A18: ( len US) >= 3 by JORDAN1E: 15;

      then ( len US) >= 1 by XXREAL_0: 2;

      then 1 in ( dom US) by FINSEQ_3: 25;

      

      then

       A19: (US . 1) = (US /. 1) by PARTFUN1:def 6

      .= Wmin by JORDAN1F: 5;

      

       A20: (Wmin `1 ) = Wbo by EUCLID: 52

      .= ((Ga * (1,i)) `1 ) by A4, A5, A11, JORDAN1A: 73;

      ( len Ga) >= 4 by JORDAN8: 10;

      then

       A21: ( len Ga) >= 1 by XXREAL_0: 2;

      then

       A22: [1, k] in ( Indices Ga) by A14, A15, MATRIX_0: 30;

      

       A23: [1, i] in ( Indices Ga) by A4, A5, A21, MATRIX_0: 30;

      then

       A24: Gij <> (US . 1) by A1, A16, A19, A20, JORDAN1G: 7;

      then

      reconsider go as being_S-Seq FinSequence of ( TOP-REAL 2) by A10, JORDAN3: 35;

      

       A25: ( len LS) >= (1 + 2) by JORDAN1E: 15;

      then

       A26: ( len LS) >= 1 by XXREAL_0: 2;

      then

       A27: 1 in ( dom LS) by FINSEQ_3: 25;

      ( len LS) in ( dom LS) by A26, FINSEQ_3: 25;

      

      then

       A28: (LS . ( len LS)) = (LS /. ( len LS)) by PARTFUN1:def 6

      .= Wmin by JORDAN1F: 8;

      (Wmin `1 ) = Wbo by EUCLID: 52

      .= ((Ga * (1,i)) `1 ) by A4, A5, A11, JORDAN1A: 73;

      then

       A29: Gik <> (LS . ( len LS)) by A1, A2, A17, A23, A28, JORDAN1G: 7;

      then

      reconsider co as being_S-Seq FinSequence of ( TOP-REAL 2) by A9, JORDAN3: 34;

      

       A30: [( len Ga), k] in ( Indices Ga) by A14, A15, A21, MATRIX_0: 30;

      

       A31: (LS . 1) = (LS /. 1) by A27, PARTFUN1:def 6

      .= Emax by JORDAN1F: 6;

      (Emax `1 ) = Ebo by EUCLID: 52

      .= ((Ga * (( len Ga),k)) `1 ) by A3, A14, JORDAN1A: 71;

      then

       A32: Gik <> (LS . 1) by A3, A17, A30, A31, JORDAN1G: 7;

      

       A33: ( len go) >= (1 + 1) by TOPREAL1:def 8;

      

       A34: Gij in ( rng US) by A1, A4, A5, A10, A11, A13, JORDAN1G: 4, JORDAN1J: 40;

      then

       A35: go is_sequence_on Ga by JORDAN1G: 4, JORDAN1J: 38;

      

       A36: ( len co) >= (1 + 1) by TOPREAL1:def 8;

      

       A37: Gik in ( rng LS) by A4, A5, A9, A11, A14, A15, JORDAN1G: 5, JORDAN1J: 40;

      then

       A38: co is_sequence_on Ga by JORDAN1G: 5, JORDAN1J: 39;

      reconsider go as non constant s.c.c. being_S-Seq FinSequence of ( TOP-REAL 2) by A33, A35, JGRAPH_1: 12, JORDAN8: 5;

      reconsider co as non constant s.c.c. being_S-Seq FinSequence of ( TOP-REAL 2) by A36, A38, JGRAPH_1: 12, JORDAN8: 5;

      

       A39: ( len go) > 1 by A33, NAT_1: 13;

      then

       A40: ( len go) in ( dom go) by FINSEQ_3: 25;

      

      then

       A41: (go /. ( len go)) = (go . ( len go)) by PARTFUN1:def 6

      .= Gij by A10, JORDAN3: 24;

      ( len co) >= 1 by A36, XXREAL_0: 2;

      then 1 in ( dom co) by FINSEQ_3: 25;

      

      then

       A42: (co /. 1) = (co . 1) by PARTFUN1:def 6

      .= Gik by A9, JORDAN3: 23;

      reconsider m = (( len go) - 1) as Nat by A40, FINSEQ_3: 26;

      

       A43: (m + 1) = ( len go);

      then

       A44: (( len go) -' 1) = m by NAT_D: 34;

      

       A45: ( LSeg (go,m)) c= ( L~ go) by TOPREAL3: 19;

      

       A46: ( L~ go) c= ( L~ US) by A10, JORDAN3: 41;

      then ( LSeg (go,m)) c= ( L~ US) by A45;

      then

       A47: (( LSeg (go,m)) /\ ( LSeg (Gik,Gij))) c= {Gij} by A6, XBOOLE_1: 26;

      m >= 1 by A33, XREAL_1: 19;

      then

       A48: ( LSeg (go,m)) = ( LSeg ((go /. m),Gij)) by A41, A43, TOPREAL1:def 3;

       {Gij} c= (( LSeg (go,m)) /\ ( LSeg (Gik,Gij)))

      proof

        let x be object;

        

         A49: Gij in ( LSeg (Gik,Gij)) by RLTOPSP1: 68;

        assume x in {Gij};

        then

         A50: x = Gij by TARSKI:def 1;

        Gij in ( LSeg (go,m)) by A48, RLTOPSP1: 68;

        hence thesis by A50, A49, XBOOLE_0:def 4;

      end;

      then

       A51: (( LSeg (go,m)) /\ ( LSeg (Gik,Gij))) = {Gij} by A47;

      

       A52: ( LSeg (co,1)) c= ( L~ co) by TOPREAL3: 19;

      

       A53: ( L~ co) c= ( L~ LS) by A9, JORDAN3: 42;

      then ( LSeg (co,1)) c= ( L~ LS) by A52;

      then

       A54: (( LSeg (co,1)) /\ ( LSeg (Gik,Gij))) c= {Gik} by A7, XBOOLE_1: 26;

      

       A55: ( LSeg (co,1)) = ( LSeg (Gik,(co /. (1 + 1)))) by A36, A42, TOPREAL1:def 3;

       {Gik} c= (( LSeg (co,1)) /\ ( LSeg (Gik,Gij)))

      proof

        let x be object;

        

         A56: Gik in ( LSeg (Gik,Gij)) by RLTOPSP1: 68;

        assume x in {Gik};

        then

         A57: x = Gik by TARSKI:def 1;

        Gik in ( LSeg (co,1)) by A55, RLTOPSP1: 68;

        hence thesis by A57, A56, XBOOLE_0:def 4;

      end;

      then

       A58: (( LSeg (Gik,Gij)) /\ ( LSeg (co,1))) = {Gik} by A54;

      

       A59: (go /. 1) = (US /. 1) by A10, SPRECT_3: 22

      .= Wmin by JORDAN1F: 5;

      

      then

       A60: (go /. 1) = (LS /. ( len LS)) by JORDAN1F: 8

      .= (co /. ( len co)) by A9, JORDAN1J: 35;

      

       A61: ( rng go) c= ( L~ go) by A33, SPPOL_2: 18;

      

       A62: ( rng co) c= ( L~ co) by A36, SPPOL_2: 18;

      

       A63: {(go /. 1)} c= (( L~ go) /\ ( L~ co))

      proof

        let x be object;

        assume x in {(go /. 1)};

        then

         A64: x = (go /. 1) by TARSKI:def 1;

        then

         A65: x in ( rng go) by FINSEQ_6: 42;

        x in ( rng co) by A60, A64, FINSEQ_6: 168;

        hence thesis by A61, A62, A65, XBOOLE_0:def 4;

      end;

      

       A66: (LS . 1) = (LS /. 1) by A27, PARTFUN1:def 6

      .= Emax by JORDAN1F: 6;

      

       A67: [( len Ga), j] in ( Indices Ga) by A1, A13, A21, MATRIX_0: 30;

      (( L~ go) /\ ( L~ co)) c= {(go /. 1)}

      proof

        let x be object;

        assume

         A68: x in (( L~ go) /\ ( L~ co));

        then

         A69: x in ( L~ co) by XBOOLE_0:def 4;

         A70:

        now

          assume x = Emax;

          then

           A71: Emax = Gik by A9, A66, A69, JORDAN1E: 7;

          ((Ga * (( len Ga),j)) `1 ) = Ebo by A1, A11, A13, JORDAN1A: 71;

          then (Emax `1 ) <> Ebo by A3, A17, A67, A71, JORDAN1G: 7;

          hence contradiction by EUCLID: 52;

        end;

        x in ( L~ go) by A68, XBOOLE_0:def 4;

        then x in (( L~ US) /\ ( L~ LS)) by A46, A53, A69, XBOOLE_0:def 4;

        then x in {Wmin, Emax} by JORDAN1E: 16;

        then x = Wmin or x = Emax by TARSKI:def 2;

        hence thesis by A59, A70, TARSKI:def 1;

      end;

      then

       A72: (( L~ go) /\ ( L~ co)) = {(go /. 1)} by A63;

      set W2 = (go /. 2);

      

       A73: 2 in ( dom go) by A33, FINSEQ_3: 25;

       A74:

      now

        assume (Gij `1 ) = Wbo;

        then ((Ga * (1,k)) `1 ) = ((Ga * (j,i)) `1 ) by A3, A14, JORDAN1A: 73;

        hence contradiction by A1, A16, A22, JORDAN1G: 7;

      end;

      go = ( mid (US,1,(Gij .. US))) by A34, JORDAN1G: 49

      .= (US | (Gij .. US)) by A34, FINSEQ_4: 21, FINSEQ_6: 116;

      then

       A75: W2 = (US /. 2) by A73, FINSEQ_4: 70;

      

       A76: Wmin in ( rng go) by A59, FINSEQ_6: 42;

      set pion = <*Gij, Gik*>;

       A77:

      now

        let n be Nat;

        assume n in ( dom pion);

        then n in {1, 2} by FINSEQ_1: 2, FINSEQ_1: 89;

        then n = 1 or n = 2 by TARSKI:def 2;

        hence ex j,i be Nat st [j, i] in ( Indices Ga) & (pion /. n) = (Ga * (j,i)) by A16, A17, FINSEQ_4: 17;

      end;

      

       A78: Gik <> Gij by A12, A16, A17, GOBOARD1: 5;

      (Gik `2 ) = ((Ga * (1,i)) `2 ) by A3, A4, A5, A14, GOBOARD5: 1

      .= (Gij `2 ) by A1, A4, A5, A11, A13, GOBOARD5: 1;

      then ( LSeg (Gik,Gij)) is horizontal by SPPOL_1: 15;

      then pion is being_S-Seq by A78, JORDAN1B: 8;

      then

      consider pion1 be FinSequence of ( TOP-REAL 2) such that

       A79: pion1 is_sequence_on Ga and

       A80: pion1 is being_S-Seq and

       A81: ( L~ pion) = ( L~ pion1) and

       A82: (pion /. 1) = (pion1 /. 1) and

       A83: (pion /. ( len pion)) = (pion1 /. ( len pion1)) and

       A84: ( len pion) <= ( len pion1) by A77, GOBOARD3: 2;

      reconsider pion1 as being_S-Seq FinSequence of ( TOP-REAL 2) by A80;

      set godo = ((go ^' pion1) ^' co);

      

       A85: (1 + 1) <= ( len ( Cage (C,n))) by GOBOARD7: 34, XXREAL_0: 2;

      

       A86: (1 + 1) <= ( len ( Rotate (( Cage (C,n)),Wmin))) by GOBOARD7: 34, XXREAL_0: 2;

      ( len (go ^' pion1)) >= ( len go) by TOPREAL8: 7;

      then

       A87: ( len (go ^' pion1)) >= (1 + 1) by A33, XXREAL_0: 2;

      then

       A88: ( len (go ^' pion1)) > (1 + 0 ) by NAT_1: 13;

      

       A89: ( len godo) >= ( len (go ^' pion1)) by TOPREAL8: 7;

      then

       A90: (1 + 1) <= ( len godo) by A87, XXREAL_0: 2;

      

       A91: US is_sequence_on Ga by JORDAN1G: 4;

      

       A92: (go /. ( len go)) = (pion1 /. 1) by A41, A82, FINSEQ_4: 17;

      then

       A93: (go ^' pion1) is_sequence_on Ga by A35, A79, TOPREAL8: 12;

      

       A94: ((go ^' pion1) /. ( len (go ^' pion1))) = (pion /. ( len pion)) by A83, FINSEQ_6: 156

      .= (pion /. 2) by FINSEQ_1: 44

      .= (co /. 1) by A42, FINSEQ_4: 17;

      then

       A95: godo is_sequence_on Ga by A38, A93, TOPREAL8: 12;

      ( LSeg (pion1,1)) c= ( L~ <*Gij, Gik*>) by A81, TOPREAL3: 19;

      then ( LSeg (pion1,1)) c= ( LSeg (Gij,Gik)) by SPPOL_2: 21;

      then

       A96: (( LSeg (go,(( len go) -' 1))) /\ ( LSeg (pion1,1))) c= {Gij} by A44, A51, XBOOLE_1: 27;

      

       A97: ( len pion1) >= (1 + 1) by A84, FINSEQ_1: 44;

       {Gij} c= (( LSeg (go,m)) /\ ( LSeg (pion1,1)))

      proof

        let x be object;

        assume x in {Gij};

        then

         A98: x = Gij by TARSKI:def 1;

        

         A99: Gij in ( LSeg (go,m)) by A48, RLTOPSP1: 68;

        Gij in ( LSeg (pion1,1)) by A41, A92, A97, TOPREAL1: 21;

        hence thesis by A98, A99, XBOOLE_0:def 4;

      end;

      then (( LSeg (go,(( len go) -' 1))) /\ ( LSeg (pion1,1))) = {(go /. ( len go))} by A41, A44, A96;

      then

       A100: (go ^' pion1) is unfolded by A92, TOPREAL8: 34;

      ( len pion1) >= (2 + 0 ) by A84, FINSEQ_1: 44;

      then

       A101: (( len pion1) - 2) >= 0 by XREAL_1: 19;

      ((( len (go ^' pion1)) + 1) - 1) = ((( len go) + ( len pion1)) - 1) by FINSEQ_6: 139;

      

      then (( len (go ^' pion1)) - 1) = (( len go) + (( len pion1) - 2))

      .= (( len go) + (( len pion1) -' 2)) by A101, XREAL_0:def 2;

      then

       A102: (( len (go ^' pion1)) -' 1) = (( len go) + (( len pion1) -' 2)) by XREAL_0:def 2;

      

       A103: (( len pion1) - 1) >= 1 by A97, XREAL_1: 19;

      then

       A104: (( len pion1) -' 1) = (( len pion1) - 1) by XREAL_0:def 2;

      

       A105: ((( len pion1) -' 2) + 1) = ((( len pion1) - 2) + 1) by A101, XREAL_0:def 2

      .= (( len pion1) -' 1) by A103, XREAL_0:def 2;

      ((( len pion1) - 1) + 1) <= ( len pion1);

      then

       A106: (( len pion1) -' 1) < ( len pion1) by A104, NAT_1: 13;

      ( LSeg (pion1,(( len pion1) -' 1))) c= ( L~ <*Gij, Gik*>) by A81, TOPREAL3: 19;

      then ( LSeg (pion1,(( len pion1) -' 1))) c= ( LSeg (Gij,Gik)) by SPPOL_2: 21;

      then

       A107: (( LSeg (pion1,(( len pion1) -' 1))) /\ ( LSeg (co,1))) c= {Gik} by A58, XBOOLE_1: 27;

       {Gik} c= (( LSeg (pion1,(( len pion1) -' 1))) /\ ( LSeg (co,1)))

      proof

        let x be object;

        assume x in {Gik};

        then

         A108: x = Gik by TARSKI:def 1;

        (pion1 /. ((( len pion1) -' 1) + 1)) = (pion /. 2) by A83, A104, FINSEQ_1: 44

        .= Gik by FINSEQ_4: 17;

        then

         A109: Gik in ( LSeg (pion1,(( len pion1) -' 1))) by A103, A104, TOPREAL1: 21;

        Gik in ( LSeg (co,1)) by A55, RLTOPSP1: 68;

        hence thesis by A108, A109, XBOOLE_0:def 4;

      end;

      then (( LSeg (pion1,(( len pion1) -' 1))) /\ ( LSeg (co,1))) = {Gik} by A107;

      then

       A110: (( LSeg ((go ^' pion1),(( len go) + (( len pion1) -' 2)))) /\ ( LSeg (co,1))) = {((go ^' pion1) /. ( len (go ^' pion1)))} by A42, A92, A94, A105, A106, TOPREAL8: 31;

      

       A111: (go ^' pion1) is non trivial by A87, NAT_D: 60;

      

       A112: ( rng pion1) c= ( L~ pion1) by A97, SPPOL_2: 18;

      

       A113: {(pion1 /. 1)} c= (( L~ go) /\ ( L~ pion1))

      proof

        let x be object;

        assume x in {(pion1 /. 1)};

        then

         A114: x = (pion1 /. 1) by TARSKI:def 1;

        then

         A115: x in ( rng pion1) by FINSEQ_6: 42;

        x in ( rng go) by A92, A114, FINSEQ_6: 168;

        hence thesis by A61, A112, A115, XBOOLE_0:def 4;

      end;

      (( L~ go) /\ ( L~ pion1)) c= {(pion1 /. 1)}

      proof

        let x be object;

        assume

         A116: x in (( L~ go) /\ ( L~ pion1));

        then

         A117: x in ( L~ pion1) by XBOOLE_0:def 4;

        x in ( L~ go) by A116, XBOOLE_0:def 4;

        then x in (( L~ pion1) /\ ( L~ US)) by A46, A117, XBOOLE_0:def 4;

        hence thesis by A6, A41, A81, A92, SPPOL_2: 21;

      end;

      then

       A118: (( L~ go) /\ ( L~ pion1)) = {(pion1 /. 1)} by A113;

      then

       A119: (go ^' pion1) is s.n.c. by A92, JORDAN1J: 54;

      (( rng go) /\ ( rng pion1)) c= {(pion1 /. 1)} by A61, A112, A118, XBOOLE_1: 27;

      then

       A120: (go ^' pion1) is one-to-one by JORDAN1J: 55;

      

       A121: (pion /. ( len pion)) = (pion /. 2) by FINSEQ_1: 44

      .= (co /. 1) by A42, FINSEQ_4: 17;

      

       A122: {(pion1 /. ( len pion1))} c= (( L~ co) /\ ( L~ pion1))

      proof

        let x be object;

        assume x in {(pion1 /. ( len pion1))};

        then

         A123: x = (pion1 /. ( len pion1)) by TARSKI:def 1;

        then

         A124: x in ( rng pion1) by FINSEQ_6: 168;

        x in ( rng co) by A83, A121, A123, FINSEQ_6: 42;

        hence thesis by A62, A112, A124, XBOOLE_0:def 4;

      end;

      (( L~ co) /\ ( L~ pion1)) c= {(pion1 /. ( len pion1))}

      proof

        let x be object;

        assume

         A125: x in (( L~ co) /\ ( L~ pion1));

        then

         A126: x in ( L~ pion1) by XBOOLE_0:def 4;

        x in ( L~ co) by A125, XBOOLE_0:def 4;

        then x in (( L~ pion1) /\ ( L~ LS)) by A53, A126, XBOOLE_0:def 4;

        hence thesis by A7, A42, A81, A83, A121, SPPOL_2: 21;

      end;

      then

       A127: (( L~ co) /\ ( L~ pion1)) = {(pion1 /. ( len pion1))} by A122;

      

       A128: (( L~ (go ^' pion1)) /\ ( L~ co)) = ((( L~ go) \/ ( L~ pion1)) /\ ( L~ co)) by A92, TOPREAL8: 35

      .= ( {(go /. 1)} \/ {(co /. 1)}) by A72, A83, A121, A127, XBOOLE_1: 23

      .= ( {((go ^' pion1) /. 1)} \/ {(co /. 1)}) by FINSEQ_6: 155

      .= {((go ^' pion1) /. 1), (co /. 1)} by ENUMSET1: 1;

      (co /. ( len co)) = ((go ^' pion1) /. 1) by A60, FINSEQ_6: 155;

      then

      reconsider godo as non constant standard special_circular_sequence by A90, A94, A95, A100, A102, A110, A111, A119, A120, A128, JORDAN8: 4, JORDAN8: 5, TOPREAL8: 11, TOPREAL8: 33, TOPREAL8: 34;

      

       A129: UA is_an_arc_of (( W-min C),( E-max C)) by JORDAN6:def 8;

      then

       A130: UA is connected by JORDAN6: 10;

      

       A131: ( W-min C) in UA by A129, TOPREAL1: 1;

      

       A132: ( E-max C) in UA by A129, TOPREAL1: 1;

      set ff = ( Rotate (( Cage (C,n)),Wmin));

      Wmin in ( rng ( Cage (C,n))) by SPRECT_2: 43;

      then

       A133: (ff /. 1) = Wmin by FINSEQ_6: 92;

      

       A134: ( L~ ff) = ( L~ ( Cage (C,n))) by REVROT_1: 33;

      then (( W-max ( L~ ff)) .. ff) > 1 by A133, SPRECT_5: 22;

      then (( N-min ( L~ ff)) .. ff) > 1 by A133, A134, SPRECT_5: 23, XXREAL_0: 2;

      then (( N-max ( L~ ff)) .. ff) > 1 by A133, A134, SPRECT_5: 24, XXREAL_0: 2;

      then

       A135: (Emax .. ff) > 1 by A133, A134, SPRECT_5: 25, XXREAL_0: 2;

       A136:

      now

        assume

         A137: (Gij .. US) <= 1;

        (Gij .. US) >= 1 by A34, FINSEQ_4: 21;

        then (Gij .. US) = 1 by A137, XXREAL_0: 1;

        then Gij = (US /. 1) by A34, FINSEQ_5: 38;

        hence contradiction by A19, A24, JORDAN1F: 5;

      end;

      

       A138: ( Cage (C,n)) is_sequence_on Ga by JORDAN9:def 1;

      then

       A139: ff is_sequence_on Ga by REVROT_1: 34;

      

       A140: (( right_cell (godo,1,Ga)) \ ( L~ godo)) c= ( RightComp godo) by A90, A95, JORDAN9: 27;

      

       A141: ( L~ godo) = (( L~ (go ^' pion1)) \/ ( L~ co)) by A94, TOPREAL8: 35

      .= ((( L~ go) \/ ( L~ pion1)) \/ ( L~ co)) by A92, TOPREAL8: 35;

      

       A142: ( L~ ( Cage (C,n))) = (( L~ US) \/ ( L~ LS)) by JORDAN1E: 13;

      then

       A143: ( L~ US) c= ( L~ ( Cage (C,n))) by XBOOLE_1: 7;

      

       A144: ( L~ LS) c= ( L~ ( Cage (C,n))) by A142, XBOOLE_1: 7;

      

       A145: ( L~ go) c= ( L~ ( Cage (C,n))) by A46, A143;

      

       A146: ( L~ co) c= ( L~ ( Cage (C,n))) by A53, A144;

      

       A147: ( W-min C) in C by SPRECT_1: 13;

      

       A148: ( L~ pion) = ( LSeg (Gik,Gij)) by SPPOL_2: 21;

       A149:

      now

        assume ( W-min C) in ( L~ godo);

        then

         A150: ( W-min C) in (( L~ go) \/ ( L~ pion1)) or ( W-min C) in ( L~ co) by A141, XBOOLE_0:def 3;

        per cases by A150, XBOOLE_0:def 3;

          suppose ( W-min C) in ( L~ go);

          then C meets ( L~ ( Cage (C,n))) by A145, A147, XBOOLE_0: 3;

          hence contradiction by JORDAN10: 5;

        end;

          suppose ( W-min C) in ( L~ pion1);

          hence contradiction by A8, A81, A131, A148, XBOOLE_0: 3;

        end;

          suppose ( W-min C) in ( L~ co);

          then C meets ( L~ ( Cage (C,n))) by A146, A147, XBOOLE_0: 3;

          hence contradiction by JORDAN10: 5;

        end;

      end;

      ( right_cell (( Rotate (( Cage (C,n)),Wmin)),1)) = ( right_cell (ff,1,( GoB ff))) by A86, JORDAN1H: 23

      .= ( right_cell (ff,1,( GoB ( Cage (C,n))))) by REVROT_1: 28

      .= ( right_cell (ff,1,Ga)) by JORDAN1H: 44

      .= ( right_cell ((ff -: Emax),1,Ga)) by A135, A139, JORDAN1J: 53

      .= ( right_cell (US,1,Ga)) by JORDAN1E:def 1

      .= ( right_cell (( R_Cut (US,Gij)),1,Ga)) by A34, A91, A136, JORDAN1J: 52

      .= ( right_cell ((go ^' pion1),1,Ga)) by A39, A93, JORDAN1J: 51

      .= ( right_cell (godo,1,Ga)) by A88, A95, JORDAN1J: 51;

      then ( W-min C) in ( right_cell (godo,1,Ga)) by JORDAN1I: 6;

      then

       A151: ( W-min C) in (( right_cell (godo,1,Ga)) \ ( L~ godo)) by A149, XBOOLE_0:def 5;

      

       A152: (godo /. 1) = ((go ^' pion1) /. 1) by FINSEQ_6: 155

      .= Wmin by A59, FINSEQ_6: 155;

      

       A153: ( len US) >= 2 by A18, XXREAL_0: 2;

      

       A154: (godo /. 2) = ((go ^' pion1) /. 2) by A87, FINSEQ_6: 159

      .= (US /. 2) by A33, A75, FINSEQ_6: 159

      .= ((US ^' LS) /. 2) by A153, FINSEQ_6: 159

      .= (( Rotate (( Cage (C,n)),Wmin)) /. 2) by JORDAN1E: 11;

      

       A155: (( L~ go) \/ ( L~ co)) is compact by COMPTS_1: 10;

      Wmin in (( L~ go) \/ ( L~ co)) by A61, A76, XBOOLE_0:def 3;

      then

       A156: ( W-min (( L~ go) \/ ( L~ co))) = Wmin by A145, A146, A155, JORDAN1J: 21, XBOOLE_1: 8;

      

       A157: (( W-min (( L~ go) \/ ( L~ co))) `1 ) = ( W-bound (( L~ go) \/ ( L~ co))) by EUCLID: 52;

      

       A158: (Wmin `1 ) = Wbo by EUCLID: 52;

      

       A159: (Gij `1 ) <= (Gik `1 ) by A1, A2, A3, A4, A5, SPRECT_3: 13;

      then ( W-bound ( LSeg (Gik,Gij))) = (Gij `1 ) by SPRECT_1: 54;

      then

       A160: ( W-bound ( L~ pion1)) = (Gij `1 ) by A81, SPPOL_2: 21;

      (Gij `1 ) >= Wbo by A10, A143, PSCOMP_1: 24;

      then (Gij `1 ) > Wbo by A74, XXREAL_0: 1;

      then ( W-min ((( L~ go) \/ ( L~ co)) \/ ( L~ pion1))) = ( W-min (( L~ go) \/ ( L~ co))) by A155, A156, A157, A158, A160, JORDAN1J: 33;

      then

       A161: ( W-min ( L~ godo)) = Wmin by A141, A156, XBOOLE_1: 4;

      

       A162: ( rng godo) c= ( L~ godo) by A87, A89, SPPOL_2: 18, XXREAL_0: 2;

      2 in ( dom godo) by A90, FINSEQ_3: 25;

      then

       A163: (godo /. 2) in ( rng godo) by PARTFUN2: 2;

      (godo /. 2) in ( W-most ( L~ ( Cage (C,n)))) by A154, JORDAN1I: 25;

      

      then ((godo /. 2) `1 ) = (( W-min ( L~ godo)) `1 ) by A161, PSCOMP_1: 31

      .= ( W-bound ( L~ godo)) by EUCLID: 52;

      then (godo /. 2) in ( W-most ( L~ godo)) by A162, A163, SPRECT_2: 12;

      then (( Rotate (godo,( W-min ( L~ godo)))) /. 2) in ( W-most ( L~ godo)) by A152, A161, FINSEQ_6: 89;

      then

      reconsider godo as clockwise_oriented non constant standard special_circular_sequence by JORDAN1I: 25;

      ( len US) in ( dom US) by FINSEQ_5: 6;

      

      then

       A164: (US . ( len US)) = (US /. ( len US)) by PARTFUN1:def 6

      .= Emax by JORDAN1F: 7;

      

       A165: ( east_halfline ( E-max C)) misses ( L~ go)

      proof

        assume ( east_halfline ( E-max C)) meets ( L~ go);

        then

        consider p be object such that

         A166: p in ( east_halfline ( E-max C)) and

         A167: p in ( L~ go) by XBOOLE_0: 3;

        reconsider p as Point of ( TOP-REAL 2) by A166;

        p in ( L~ US) by A46, A167;

        then p in (( east_halfline ( E-max C)) /\ ( L~ ( Cage (C,n)))) by A143, A166, XBOOLE_0:def 4;

        then

         A168: (p `1 ) = Ebo by JORDAN1A: 83, PSCOMP_1: 50;

        then

         A169: p = Emax by A46, A167, JORDAN1J: 46;

        then Emax = Gij by A10, A164, A167, JORDAN1J: 43;

        then (Gij `1 ) = ((Ga * (( len Ga),k)) `1 ) by A3, A14, A168, A169, JORDAN1A: 71;

        hence contradiction by A2, A3, A16, A30, JORDAN1G: 7;

      end;

      now

        assume ( east_halfline ( E-max C)) meets ( L~ godo);

        then

         A170: ( east_halfline ( E-max C)) meets (( L~ go) \/ ( L~ pion1)) or ( east_halfline ( E-max C)) meets ( L~ co) by A141, XBOOLE_1: 70;

        per cases by A170, XBOOLE_1: 70;

          suppose ( east_halfline ( E-max C)) meets ( L~ go);

          hence contradiction by A165;

        end;

          suppose ( east_halfline ( E-max C)) meets ( L~ pion1);

          then

          consider p be object such that

           A171: p in ( east_halfline ( E-max C)) and

           A172: p in ( L~ pion1) by XBOOLE_0: 3;

          reconsider p as Point of ( TOP-REAL 2) by A171;

          

           A173: (p `2 ) = (( E-max C) `2 ) by A171, TOPREAL1:def 11;

          (k + 1) <= ( len Ga) by A3, NAT_1: 13;

          then ((k + 1) - 1) <= (( len Ga) - 1) by XREAL_1: 9;

          then

           A174: k <= (( len Ga) -' 1) by XREAL_0:def 2;

          (( len Ga) -' 1) <= ( len Ga) by NAT_D: 35;

          then

           A175: (Gik `1 ) <= ((Ga * ((( len Ga) -' 1),1)) `1 ) by A4, A5, A11, A14, A21, A174, JORDAN1A: 18;

          (p `1 ) <= (Gik `1 ) by A81, A148, A159, A172, TOPREAL1: 3;

          then (p `1 ) <= ((Ga * ((( len Ga) -' 1),1)) `1 ) by A175, XXREAL_0: 2;

          then (p `1 ) <= ( E-bound C) by A21, JORDAN8: 12;

          then

           A176: (p `1 ) <= (( E-max C) `1 ) by EUCLID: 52;

          (p `1 ) >= (( E-max C) `1 ) by A171, TOPREAL1:def 11;

          then (p `1 ) = (( E-max C) `1 ) by A176, XXREAL_0: 1;

          then p = ( E-max C) by A173, TOPREAL3: 6;

          hence contradiction by A8, A81, A132, A148, A172, XBOOLE_0: 3;

        end;

          suppose ( east_halfline ( E-max C)) meets ( L~ co);

          then

          consider p be object such that

           A177: p in ( east_halfline ( E-max C)) and

           A178: p in ( L~ co) by XBOOLE_0: 3;

          reconsider p as Point of ( TOP-REAL 2) by A177;

          

           A179: p in ( LSeg (co,( Index (p,co)))) by A178, JORDAN3: 9;

          consider t be Nat such that

           A180: t in ( dom LS) and

           A181: (LS . t) = Gik by A37, FINSEQ_2: 10;

          1 <= t by A180, FINSEQ_3: 25;

          then

           A182: 1 < t by A32, A181, XXREAL_0: 1;

          t <= ( len LS) by A180, FINSEQ_3: 25;

          then (( Index (Gik,LS)) + 1) = t by A181, A182, JORDAN3: 12;

          then

           A183: ( len ( L_Cut (LS,Gik))) = (( len LS) - ( Index (Gik,LS))) by A9, A181, JORDAN3: 26;

          ( Index (p,co)) < ( len co) by A178, JORDAN3: 8;

          then ( Index (p,co)) < (( len LS) -' ( Index (Gik,LS))) by A183, XREAL_0:def 2;

          then (( Index (p,co)) + 1) <= (( len LS) -' ( Index (Gik,LS))) by NAT_1: 13;

          then

           A184: ( Index (p,co)) <= ((( len LS) -' ( Index (Gik,LS))) - 1) by XREAL_1: 19;

          

           A185: co = ( mid (LS,(Gik .. LS),( len LS))) by A37, JORDAN1J: 37;

          p in ( L~ LS) by A53, A178;

          then p in (( east_halfline ( E-max C)) /\ ( L~ ( Cage (C,n)))) by A144, A177, XBOOLE_0:def 4;

          then

           A186: (p `1 ) = Ebo by JORDAN1A: 83, PSCOMP_1: 50;

          

           A187: (( Index (Gik,LS)) + 1) = (Gik .. LS) by A32, A37, JORDAN1J: 56;

          ( 0 + ( Index (Gik,LS))) < ( len LS) by A9, JORDAN3: 8;

          then (( len LS) - ( Index (Gik,LS))) > 0 by XREAL_1: 20;

          then ( Index (p,co)) <= ((( len LS) - ( Index (Gik,LS))) - 1) by A184, XREAL_0:def 2;

          then ( Index (p,co)) <= (( len LS) - (Gik .. LS)) by A187;

          then ( Index (p,co)) <= (( len LS) -' (Gik .. LS)) by XREAL_0:def 2;

          then

           A188: ( Index (p,co)) < ((( len LS) -' (Gik .. LS)) + 1) by NAT_1: 13;

          

           A189: 1 <= ( Index (p,co)) by A178, JORDAN3: 8;

          

           A190: (Gik .. LS) <= ( len LS) by A37, FINSEQ_4: 21;

          (Gik .. LS) <> ( len LS) by A29, A37, FINSEQ_4: 19;

          then

           A191: (Gik .. LS) < ( len LS) by A190, XXREAL_0: 1;

          

           A192: (1 + 1) <= ( len LS) by A25, XXREAL_0: 2;

          then

           A193: 2 in ( dom LS) by FINSEQ_3: 25;

          set tt = ((( Index (p,co)) + (Gik .. LS)) -' 1);

          set RC = ( Rotate (( Cage (C,n)),Emax));

          

           A194: ( E-max C) in ( right_cell (RC,1)) by JORDAN1I: 7;

          

           A195: ( GoB RC) = ( GoB ( Cage (C,n))) by REVROT_1: 28

          .= Ga by JORDAN1H: 44;

          

           A196: ( L~ RC) = ( L~ ( Cage (C,n))) by REVROT_1: 33;

          consider g2 be Nat such that

           A197: 1 <= g2 and

           A198: g2 <= ( width Ga) and

           A199: Emax = (Ga * (( len Ga),g2)) by JORDAN1D: 25;

          

           A200: ( len Ga) >= 4 by JORDAN8: 10;

          then ( len Ga) >= 1 by XXREAL_0: 2;

          then

           A201: [( len Ga), g2] in ( Indices Ga) by A197, A198, MATRIX_0: 30;

          

           A202: ( len RC) = ( len ( Cage (C,n))) by FINSEQ_6: 179;

          LS = (RC -: Wmin) by JORDAN1G: 18;

          then

           A203: ( LSeg (LS,1)) = ( LSeg (RC,1)) by A192, SPPOL_2: 9;

          

           A204: Emax in ( rng ( Cage (C,n))) by SPRECT_2: 46;

          RC is_sequence_on Ga by A138, REVROT_1: 34;

          then

          consider ii,g be Nat such that

           A205: [ii, (g + 1)] in ( Indices Ga) and

           A206: [ii, g] in ( Indices Ga) and

           A207: (RC /. 1) = (Ga * (ii,(g + 1))) and

           A208: (RC /. (1 + 1)) = (Ga * (ii,g)) by A85, A196, A202, A204, FINSEQ_6: 92, JORDAN1I: 23;

          

           A209: ((g + 1) + 1) <> g;

          

           A210: 1 <= g by A206, MATRIX_0: 32;

          (RC /. 1) = ( E-max ( L~ RC)) by A196, A204, FINSEQ_6: 92;

          then

           A211: ii = ( len Ga) by A196, A205, A207, A199, A201, GOBOARD1: 5;

          then (ii - 1) >= (4 - 1) by A200, XREAL_1: 9;

          then

           A212: (ii - 1) >= 1 by XXREAL_0: 2;

          then

           A213: 1 <= (ii -' 1) by XREAL_0:def 2;

          

           A214: g <= ( width Ga) by A206, MATRIX_0: 32;

          then

           A215: ((Ga * (( len Ga),g)) `1 ) = Ebo by A11, A210, JORDAN1A: 71;

          

           A216: (g + 1) <= ( width Ga) by A205, MATRIX_0: 32;

          (ii + 1) <> ii;

          then

           A217: ( right_cell (RC,1)) = ( cell (Ga,(ii -' 1),g)) by A85, A202, A195, A205, A206, A207, A208, A209, GOBOARD5:def 6;

          

           A218: ii <= ( len Ga) by A206, MATRIX_0: 32;

          

           A219: 1 <= ii by A206, MATRIX_0: 32;

          

           A220: ii <= ( len Ga) by A205, MATRIX_0: 32;

          

           A221: 1 <= (g + 1) by A205, MATRIX_0: 32;

          then

           A222: Ebo = ((Ga * (( len Ga),(g + 1))) `1 ) by A11, A216, JORDAN1A: 71;

          

           A223: 1 <= ii by A205, MATRIX_0: 32;

          then

           A224: ((ii -' 1) + 1) = ii by XREAL_1: 235;

          then

           A225: (ii -' 1) < ( len Ga) by A220, NAT_1: 13;

          

          then

           A226: ((Ga * ((ii -' 1),(g + 1))) `2 ) = ((Ga * (1,(g + 1))) `2 ) by A221, A216, A213, GOBOARD5: 1

          .= ((Ga * (ii,(g + 1))) `2 ) by A223, A220, A221, A216, GOBOARD5: 1;

          

           A227: (( E-max C) `2 ) = (p `2 ) by A177, TOPREAL1:def 11;

          then

           A228: (p `2 ) <= ((Ga * ((ii -' 1),(g + 1))) `2 ) by A194, A220, A216, A210, A217, A224, A212, JORDAN9: 17;

          

           A229: ((Ga * ((ii -' 1),g)) `2 ) = ((Ga * (1,g)) `2 ) by A210, A214, A213, A225, GOBOARD5: 1

          .= ((Ga * (ii,g)) `2 ) by A219, A218, A210, A214, GOBOARD5: 1;

          ((Ga * ((ii -' 1),g)) `2 ) <= (p `2 ) by A227, A194, A220, A216, A210, A217, A224, A212, JORDAN9: 17;

          then p in ( LSeg ((RC /. 1),(RC /. (1 + 1)))) by A186, A207, A208, A211, A228, A229, A226, A215, A222, GOBOARD7: 7;

          then

           A230: p in ( LSeg (LS,1)) by A85, A203, A202, TOPREAL1:def 3;

          1 <= (Gik .. LS) by A37, FINSEQ_4: 21;

          then

           A231: ( LSeg (( mid (LS,(Gik .. LS),( len LS))),( Index (p,co)))) = ( LSeg (LS,((( Index (p,co)) + (Gik .. LS)) -' 1))) by A191, A189, A188, JORDAN4: 19;

          1 <= ( Index (Gik,LS)) by A9, JORDAN3: 8;

          then

           A232: (1 + 1) <= (Gik .. LS) by A187, XREAL_1: 7;

          then (( Index (p,co)) + (Gik .. LS)) >= ((1 + 1) + 1) by A189, XREAL_1: 7;

          then ((( Index (p,co)) + (Gik .. LS)) - 1) >= (((1 + 1) + 1) - 1) by XREAL_1: 9;

          then

           A233: tt >= (1 + 1) by XREAL_0:def 2;

          now

            per cases by A233, XXREAL_0: 1;

              suppose tt > (1 + 1);

              then ( LSeg (LS,1)) misses ( LSeg (LS,tt)) by TOPREAL1:def 7;

              hence contradiction by A230, A179, A185, A231, XBOOLE_0: 3;

            end;

              suppose

               A234: tt = (1 + 1);

              then (1 + 1) = ((( Index (p,co)) + (Gik .. LS)) - 1) by XREAL_0:def 2;

              then ((1 + 1) + 1) = (( Index (p,co)) + (Gik .. LS));

              then

               A235: (Gik .. LS) = 2 by A189, A232, JORDAN1E: 6;

              (( LSeg (LS,1)) /\ ( LSeg (LS,tt))) = {(LS /. 2)} by A25, A234, TOPREAL1:def 6;

              then p in {(LS /. 2)} by A230, A179, A185, A231, XBOOLE_0:def 4;

              then

               A236: p = (LS /. 2) by TARSKI:def 1;

              then

               A237: p in ( rng LS) by A193, PARTFUN2: 2;

              (p .. LS) = 2 by A193, A236, FINSEQ_5: 41;

              then p = Gik by A37, A235, A237, FINSEQ_5: 9;

              then (Gik `1 ) = Ebo by A236, JORDAN1G: 32;

              then (Gik `1 ) = ((Ga * (( len Ga),j)) `1 ) by A1, A11, A13, JORDAN1A: 71;

              hence contradiction by A3, A17, A67, JORDAN1G: 7;

            end;

          end;

          hence contradiction;

        end;

      end;

      then ( east_halfline ( E-max C)) c= (( L~ godo) ` ) by SUBSET_1: 23;

      then

      consider W be Subset of ( TOP-REAL 2) such that

       A238: W is_a_component_of (( L~ godo) ` ) and

       A239: ( east_halfline ( E-max C)) c= W by GOBOARD9: 3;

       not W is bounded by A239, JORDAN2C: 121, RLTOPSP1: 42;

      then W is_outside_component_of ( L~ godo) by A238, JORDAN2C:def 3;

      then W c= ( UBD ( L~ godo)) by JORDAN2C: 23;

      then

       A240: ( east_halfline ( E-max C)) c= ( UBD ( L~ godo)) by A239;

      ( E-max C) in ( east_halfline ( E-max C)) by TOPREAL1: 38;

      then ( E-max C) in ( UBD ( L~ godo)) by A240;

      then ( E-max C) in ( LeftComp godo) by GOBRD14: 36;

      then UA meets ( L~ godo) by A130, A131, A132, A140, A151, JORDAN1J: 36;

      then

       A241: UA meets (( L~ go) \/ ( L~ pion1)) or UA meets ( L~ co) by A141, XBOOLE_1: 70;

      

       A242: UA c= C by JORDAN6: 61;

      per cases by A241, XBOOLE_1: 70;

        suppose UA meets ( L~ go);

        then UA meets ( L~ ( Cage (C,n))) by A46, A143, XBOOLE_1: 1, XBOOLE_1: 63;

        hence contradiction by A242, JORDAN10: 5, XBOOLE_1: 63;

      end;

        suppose UA meets ( L~ pion1);

        hence contradiction by A8, A81, A148;

      end;

        suppose UA meets ( L~ co);

        then UA meets ( L~ ( Cage (C,n))) by A53, A144, XBOOLE_1: 1, XBOOLE_1: 63;

        hence contradiction by A242, JORDAN10: 5, XBOOLE_1: 63;

      end;

    end;

    theorem :: JORDAN15:38

    

     Th38: for C be Simple_closed_curve holds for i,j,k be Nat st 1 < j & j <= k & k < ( len ( Gauge (C,n))) & 1 <= i & i <= ( width ( Gauge (C,n))) & (( Gauge (C,n)) * (j,i)) in ( L~ ( Upper_Seq (C,n))) & (( Gauge (C,n)) * (k,i)) in ( L~ ( Lower_Seq (C,n))) holds ( LSeg ((( Gauge (C,n)) * (j,i)),(( Gauge (C,n)) * (k,i)))) meets ( Lower_Arc C)

    proof

      let C be Simple_closed_curve;

      let i,j,k be Nat;

      assume that

       A1: 1 < j and

       A2: j <= k and

       A3: k < ( len ( Gauge (C,n))) and

       A4: 1 <= i and

       A5: i <= ( width ( Gauge (C,n))) and

       A6: (( Gauge (C,n)) * (j,i)) in ( L~ ( Upper_Seq (C,n))) and

       A7: (( Gauge (C,n)) * (k,i)) in ( L~ ( Lower_Seq (C,n)));

      consider j1,k1 be Nat such that

       A8: j <= j1 and

       A9: j1 <= k1 and

       A10: k1 <= k and

       A11: (( LSeg ((( Gauge (C,n)) * (j1,i)),(( Gauge (C,n)) * (k1,i)))) /\ ( L~ ( Upper_Seq (C,n)))) = {(( Gauge (C,n)) * (j1,i))} and

       A12: (( LSeg ((( Gauge (C,n)) * (j1,i)),(( Gauge (C,n)) * (k1,i)))) /\ ( L~ ( Lower_Seq (C,n)))) = {(( Gauge (C,n)) * (k1,i))} by A1, A2, A3, A4, A5, A6, A7, Th20;

      

       A13: k1 < ( len ( Gauge (C,n))) by A3, A10, XXREAL_0: 2;

      1 < j1 by A1, A8, XXREAL_0: 2;

      then ( LSeg ((( Gauge (C,n)) * (j1,i)),(( Gauge (C,n)) * (k1,i)))) meets ( Lower_Arc C) by A4, A5, A9, A11, A12, A13, Th36;

      hence thesis by A1, A3, A4, A5, A8, A9, A10, Th6, XBOOLE_1: 63;

    end;

    theorem :: JORDAN15:39

    

     Th39: for C be Simple_closed_curve holds for i,j,k be Nat st 1 < j & j <= k & k < ( len ( Gauge (C,n))) & 1 <= i & i <= ( width ( Gauge (C,n))) & (( Gauge (C,n)) * (j,i)) in ( L~ ( Upper_Seq (C,n))) & (( Gauge (C,n)) * (k,i)) in ( L~ ( Lower_Seq (C,n))) holds ( LSeg ((( Gauge (C,n)) * (j,i)),(( Gauge (C,n)) * (k,i)))) meets ( Upper_Arc C)

    proof

      let C be Simple_closed_curve;

      let i,j,k be Nat;

      assume that

       A1: 1 < j and

       A2: j <= k and

       A3: k < ( len ( Gauge (C,n))) and

       A4: 1 <= i and

       A5: i <= ( width ( Gauge (C,n))) and

       A6: (( Gauge (C,n)) * (j,i)) in ( L~ ( Upper_Seq (C,n))) and

       A7: (( Gauge (C,n)) * (k,i)) in ( L~ ( Lower_Seq (C,n)));

      consider j1,k1 be Nat such that

       A8: j <= j1 and

       A9: j1 <= k1 and

       A10: k1 <= k and

       A11: (( LSeg ((( Gauge (C,n)) * (j1,i)),(( Gauge (C,n)) * (k1,i)))) /\ ( L~ ( Upper_Seq (C,n)))) = {(( Gauge (C,n)) * (j1,i))} and

       A12: (( LSeg ((( Gauge (C,n)) * (j1,i)),(( Gauge (C,n)) * (k1,i)))) /\ ( L~ ( Lower_Seq (C,n)))) = {(( Gauge (C,n)) * (k1,i))} by A1, A2, A3, A4, A5, A6, A7, Th20;

      

       A13: k1 < ( len ( Gauge (C,n))) by A3, A10, XXREAL_0: 2;

      1 < j1 by A1, A8, XXREAL_0: 2;

      then ( LSeg ((( Gauge (C,n)) * (j1,i)),(( Gauge (C,n)) * (k1,i)))) meets ( Upper_Arc C) by A4, A5, A9, A11, A12, A13, Th37;

      hence thesis by A1, A3, A4, A5, A8, A9, A10, Th6, XBOOLE_1: 63;

    end;

    theorem :: JORDAN15:40

    

     Th40: for C be Simple_closed_curve holds for i,j,k be Nat st 1 < j & j <= k & k < ( len ( Gauge (C,n))) & 1 <= i & i <= ( width ( Gauge (C,n))) & n > 0 & (( Gauge (C,n)) * (j,i)) in ( Upper_Arc ( L~ ( Cage (C,n)))) & (( Gauge (C,n)) * (k,i)) in ( Lower_Arc ( L~ ( Cage (C,n)))) holds ( LSeg ((( Gauge (C,n)) * (j,i)),(( Gauge (C,n)) * (k,i)))) meets ( Lower_Arc C)

    proof

      let C be Simple_closed_curve;

      let i,j,k be Nat;

      assume that

       A1: 1 < j and

       A2: j <= k and

       A3: k < ( len ( Gauge (C,n))) and

       A4: 1 <= i and

       A5: i <= ( width ( Gauge (C,n))) and

       A6: n > 0 and

       A7: (( Gauge (C,n)) * (j,i)) in ( Upper_Arc ( L~ ( Cage (C,n)))) and

       A8: (( Gauge (C,n)) * (k,i)) in ( Lower_Arc ( L~ ( Cage (C,n))));

      

       A9: ( L~ ( Lower_Seq (C,n))) = ( Lower_Arc ( L~ ( Cage (C,n)))) by A6, JORDAN1G: 56;

      ( L~ ( Upper_Seq (C,n))) = ( Upper_Arc ( L~ ( Cage (C,n)))) by A6, JORDAN1G: 55;

      hence thesis by A1, A2, A3, A4, A5, A7, A8, A9, Th38;

    end;

    theorem :: JORDAN15:41

    

     Th41: for C be Simple_closed_curve holds for i,j,k be Nat st 1 < j & j <= k & k < ( len ( Gauge (C,n))) & 1 <= i & i <= ( width ( Gauge (C,n))) & n > 0 & (( Gauge (C,n)) * (j,i)) in ( Upper_Arc ( L~ ( Cage (C,n)))) & (( Gauge (C,n)) * (k,i)) in ( Lower_Arc ( L~ ( Cage (C,n)))) holds ( LSeg ((( Gauge (C,n)) * (j,i)),(( Gauge (C,n)) * (k,i)))) meets ( Upper_Arc C)

    proof

      let C be Simple_closed_curve;

      let i,j,k be Nat;

      assume that

       A1: 1 < j and

       A2: j <= k and

       A3: k < ( len ( Gauge (C,n))) and

       A4: 1 <= i and

       A5: i <= ( width ( Gauge (C,n))) and

       A6: n > 0 and

       A7: (( Gauge (C,n)) * (j,i)) in ( Upper_Arc ( L~ ( Cage (C,n)))) and

       A8: (( Gauge (C,n)) * (k,i)) in ( Lower_Arc ( L~ ( Cage (C,n))));

      

       A9: ( L~ ( Lower_Seq (C,n))) = ( Lower_Arc ( L~ ( Cage (C,n)))) by A6, JORDAN1G: 56;

      ( L~ ( Upper_Seq (C,n))) = ( Upper_Arc ( L~ ( Cage (C,n)))) by A6, JORDAN1G: 55;

      hence thesis by A1, A2, A3, A4, A5, A7, A8, A9, Th39;

    end;

    theorem :: JORDAN15:42

    for C be Simple_closed_curve holds for j,k be Nat holds 1 < j & j <= k & k < ( len ( Gauge (C,(n + 1)))) & (( Gauge (C,(n + 1))) * (j,( Center ( Gauge (C,(n + 1)))))) in ( Upper_Arc ( L~ ( Cage (C,(n + 1))))) & (( Gauge (C,(n + 1))) * (k,( Center ( Gauge (C,(n + 1)))))) in ( Lower_Arc ( L~ ( Cage (C,(n + 1))))) implies ( LSeg ((( Gauge (C,(n + 1))) * (j,( Center ( Gauge (C,(n + 1)))))),(( Gauge (C,(n + 1))) * (k,( Center ( Gauge (C,(n + 1)))))))) meets ( Lower_Arc C)

    proof

      let C be Simple_closed_curve;

      let j,k be Nat;

      assume that

       A1: 1 < j and

       A2: j <= k and

       A3: k < ( len ( Gauge (C,(n + 1)))) and

       A4: (( Gauge (C,(n + 1))) * (j,( Center ( Gauge (C,(n + 1)))))) in ( Upper_Arc ( L~ ( Cage (C,(n + 1))))) and

       A5: (( Gauge (C,(n + 1))) * (k,( Center ( Gauge (C,(n + 1)))))) in ( Lower_Arc ( L~ ( Cage (C,(n + 1)))));

      

       A6: ( len ( Gauge (C,(n + 1)))) >= 4 by JORDAN8: 10;

      then ( len ( Gauge (C,(n + 1)))) >= 3 by XXREAL_0: 2;

      then ( Center ( Gauge (C,(n + 1)))) < ( len ( Gauge (C,(n + 1)))) by JORDAN1B: 15;

      then

       A7: ( Center ( Gauge (C,(n + 1)))) < ( width ( Gauge (C,(n + 1)))) by JORDAN8:def 1;

      ( len ( Gauge (C,(n + 1)))) >= 2 by A6, XXREAL_0: 2;

      then 1 < ( Center ( Gauge (C,(n + 1)))) by JORDAN1B: 14;

      hence thesis by A1, A2, A3, A4, A5, A7, Th40;

    end;

    theorem :: JORDAN15:43

    for C be Simple_closed_curve holds for j,k be Nat holds 1 < j & j <= k & k < ( len ( Gauge (C,(n + 1)))) & (( Gauge (C,(n + 1))) * (j,( Center ( Gauge (C,(n + 1)))))) in ( Upper_Arc ( L~ ( Cage (C,(n + 1))))) & (( Gauge (C,(n + 1))) * (k,( Center ( Gauge (C,(n + 1)))))) in ( Lower_Arc ( L~ ( Cage (C,(n + 1))))) implies ( LSeg ((( Gauge (C,(n + 1))) * (j,( Center ( Gauge (C,(n + 1)))))),(( Gauge (C,(n + 1))) * (k,( Center ( Gauge (C,(n + 1)))))))) meets ( Upper_Arc C)

    proof

      let C be Simple_closed_curve;

      let j,k be Nat;

      assume that

       A1: 1 < j and

       A2: j <= k and

       A3: k < ( len ( Gauge (C,(n + 1)))) and

       A4: (( Gauge (C,(n + 1))) * (j,( Center ( Gauge (C,(n + 1)))))) in ( Upper_Arc ( L~ ( Cage (C,(n + 1))))) and

       A5: (( Gauge (C,(n + 1))) * (k,( Center ( Gauge (C,(n + 1)))))) in ( Lower_Arc ( L~ ( Cage (C,(n + 1)))));

      

       A6: ( len ( Gauge (C,(n + 1)))) >= 4 by JORDAN8: 10;

      then ( len ( Gauge (C,(n + 1)))) >= 3 by XXREAL_0: 2;

      then ( Center ( Gauge (C,(n + 1)))) < ( len ( Gauge (C,(n + 1)))) by JORDAN1B: 15;

      then

       A7: ( Center ( Gauge (C,(n + 1)))) < ( width ( Gauge (C,(n + 1)))) by JORDAN8:def 1;

      ( len ( Gauge (C,(n + 1)))) >= 2 by A6, XXREAL_0: 2;

      then 1 < ( Center ( Gauge (C,(n + 1)))) by JORDAN1B: 14;

      hence thesis by A1, A2, A3, A4, A5, A7, Th41;

    end;

    theorem :: JORDAN15:44

    

     Th44: for C be Simple_closed_curve holds for i1,i2,j,k be Nat st 1 < i1 & i1 <= i2 & i2 < ( len ( Gauge (C,n))) & 1 <= j & j <= k & k <= ( width ( Gauge (C,n))) & ((( LSeg ((( Gauge (C,n)) * (i1,j)),(( Gauge (C,n)) * (i1,k)))) \/ ( LSeg ((( Gauge (C,n)) * (i1,k)),(( Gauge (C,n)) * (i2,k))))) /\ ( L~ ( Upper_Seq (C,n)))) = {(( Gauge (C,n)) * (i2,k))} & ((( LSeg ((( Gauge (C,n)) * (i1,j)),(( Gauge (C,n)) * (i1,k)))) \/ ( LSeg ((( Gauge (C,n)) * (i1,k)),(( Gauge (C,n)) * (i2,k))))) /\ ( L~ ( Lower_Seq (C,n)))) = {(( Gauge (C,n)) * (i1,j))} holds (( LSeg ((( Gauge (C,n)) * (i1,j)),(( Gauge (C,n)) * (i1,k)))) \/ ( LSeg ((( Gauge (C,n)) * (i1,k)),(( Gauge (C,n)) * (i2,k))))) meets ( Upper_Arc C)

    proof

      let C be Simple_closed_curve;

      let i1,i2,j,k be Nat;

      set G = ( Gauge (C,n));

      set pio = ( LSeg ((G * (i1,j)),(G * (i1,k))));

      set poz = ( LSeg ((G * (i1,k)),(G * (i2,k))));

      set US = ( Upper_Seq (C,n));

      set LS = ( Lower_Seq (C,n));

      assume that

       A1: 1 < i1 and

       A2: i1 <= i2 and

       A3: i2 < ( len G) and

       A4: 1 <= j and

       A5: j <= k and

       A6: k <= ( width G) and

       A7: ((pio \/ poz) /\ ( L~ US)) = {(G * (i2,k))} and

       A8: ((pio \/ poz) /\ ( L~ LS)) = {(G * (i1,j))} and

       A9: (pio \/ poz) misses ( Upper_Arc C);

      set Gij = (G * (i1,j));

      

       A10: j <= ( width G) by A5, A6, XXREAL_0: 2;

      

       A11: i1 < ( len G) by A2, A3, XXREAL_0: 2;

      then

       A12: [i1, j] in ( Indices G) by A1, A4, A10, MATRIX_0: 30;

      set Gi1k = (G * (i1,k));

      set Gik = (G * (i2,k));

      

       A13: ( L~ <*Gik, Gi1k, Gij*>) = (poz \/ pio) by TOPREAL3: 16;

      ( len G) >= 4 by JORDAN8: 10;

      then

       A14: ( len G) >= 1 by XXREAL_0: 2;

      then

       A15: [( len G), j] in ( Indices G) by A4, A10, MATRIX_0: 30;

      

       A16: 1 <= k by A4, A5, XXREAL_0: 2;

      then

       A17: [1, k] in ( Indices G) by A6, A14, MATRIX_0: 30;

      

       A18: 1 < i2 by A1, A2, XXREAL_0: 2;

      then

       A19: [i2, k] in ( Indices G) by A3, A6, A16, MATRIX_0: 30;

      

       A20: (Gi1k `2 ) = ((G * (1,k)) `2 ) by A1, A6, A11, A16, GOBOARD5: 1

      .= (Gik `2 ) by A3, A6, A18, A16, GOBOARD5: 1;

      (Gi1k `1 ) = ((G * (i1,1)) `1 ) by A1, A6, A11, A16, GOBOARD5: 2

      .= (Gij `1 ) by A1, A4, A11, A10, GOBOARD5: 2;

      then

       A21: Gi1k = |[(Gij `1 ), (Gik `2 )]| by A20, EUCLID: 53;

      

       A22: [( len G), k] in ( Indices G) by A6, A16, A14, MATRIX_0: 30;

      

       A23: [i1, j] in ( Indices G) by A1, A4, A11, A10, MATRIX_0: 30;

      set Wbo = ( W-bound ( L~ ( Cage (C,n))));

      set Wmin = ( W-min ( L~ ( Cage (C,n))));

      

       A24: ( len G) = ( width G) by JORDAN8:def 1;

      set Ebo = ( E-bound ( L~ ( Cage (C,n))));

      set Emax = ( E-max ( L~ ( Cage (C,n))));

      

       A25: ( len LS) >= (1 + 2) by JORDAN1E: 15;

      then

       A26: ( len LS) >= 1 by XXREAL_0: 2;

      then

       A27: 1 in ( dom LS) by FINSEQ_3: 25;

      

      then

       A28: (LS . 1) = (LS /. 1) by PARTFUN1:def 6

      .= Emax by JORDAN1F: 6;

      ( len LS) in ( dom LS) by A26, FINSEQ_3: 25;

      

      then

       A29: (LS . ( len LS)) = (LS /. ( len LS)) by PARTFUN1:def 6

      .= Wmin by JORDAN1F: 8;

      set co = ( L_Cut (LS,Gij));

      Gij in {Gij} by TARSKI:def 1;

      then

       A30: Gij in ( L~ LS) by A8, XBOOLE_0:def 4;

      (Wmin `1 ) = Wbo by EUCLID: 52

      .= ((G * (1,k)) `1 ) by A6, A16, A24, JORDAN1A: 73;

      then

       A31: Gij <> (LS . ( len LS)) by A1, A17, A29, A12, JORDAN1G: 7;

      then

      reconsider co as being_S-Seq FinSequence of ( TOP-REAL 2) by A30, JORDAN3: 34;

      

       A32: Gij in ( rng LS) by A1, A4, A11, A30, A10, JORDAN1G: 5, JORDAN1J: 40;

      then

       A33: co is_sequence_on G by JORDAN1G: 5, JORDAN1J: 39;

      (Emax `1 ) = Ebo by EUCLID: 52

      .= ((G * (( len G),k)) `1 ) by A6, A16, A24, JORDAN1A: 71;

      then

       A34: Gij <> (LS . 1) by A2, A3, A12, A22, A28, JORDAN1G: 7;

      

       A35: ( len co) >= (1 + 1) by TOPREAL1:def 8;

      then

      reconsider co as non constant s.c.c. being_S-Seq FinSequence of ( TOP-REAL 2) by A33, JGRAPH_1: 12, JORDAN8: 5;

      

       A36: ( L~ co) c= ( L~ LS) by A30, JORDAN3: 42;

      

       A37: [1, j] in ( Indices G) by A4, A10, A14, MATRIX_0: 30;

       A38:

      now

        assume (Gij `1 ) = Wbo;

        then ((G * (1,j)) `1 ) = ((G * (i1,j)) `1 ) by A4, A10, A24, JORDAN1A: 73;

        hence contradiction by A1, A23, A37, JORDAN1G: 7;

      end;

      set pion = <*Gik, Gi1k, Gij*>;

      

       A39: Gi1k in poz by RLTOPSP1: 68;

      set UA = ( Upper_Arc C);

      

       A40: Gi1k in pio by RLTOPSP1: 68;

      set go = ( R_Cut (US,Gik));

      

       A41: ( len US) >= 3 by JORDAN1E: 15;

      then ( len US) >= 1 by XXREAL_0: 2;

      then 1 in ( dom US) by FINSEQ_3: 25;

      

      then

       A42: (US . 1) = (US /. 1) by PARTFUN1:def 6

      .= Wmin by JORDAN1F: 5;

      

       A43: [i1, k] in ( Indices G) by A1, A6, A11, A16, MATRIX_0: 30;

       A44:

      now

        let n be Nat;

        assume n in ( dom pion);

        then n in {1, 2, 3} by FINSEQ_1: 89, FINSEQ_3: 1;

        then n = 1 or n = 2 or n = 3 by ENUMSET1:def 1;

        hence ex i,j be Nat st [i, j] in ( Indices G) & (pion /. n) = (G * (i,j)) by A23, A19, A43, FINSEQ_4: 18;

      end;

      Gik in {Gik} by TARSKI:def 1;

      then

       A45: Gik in ( L~ US) by A7, XBOOLE_0:def 4;

      (Wmin `1 ) = Wbo by EUCLID: 52

      .= ((G * (1,k)) `1 ) by A6, A16, A24, JORDAN1A: 73;

      then

       A46: Gik <> (US . 1) by A1, A2, A19, A42, A17, JORDAN1G: 7;

      then

      reconsider go as being_S-Seq FinSequence of ( TOP-REAL 2) by A45, JORDAN3: 35;

      

       A47: Gik in ( rng US) by A3, A6, A18, A45, A16, JORDAN1G: 4, JORDAN1J: 40;

      then

       A48: go is_sequence_on G by JORDAN1G: 4, JORDAN1J: 38;

      ( len co) >= 1 by A35, XXREAL_0: 2;

      then 1 in ( dom co) by FINSEQ_3: 25;

      

      then

       A49: (co /. 1) = (co . 1) by PARTFUN1:def 6

      .= Gij by A30, JORDAN3: 23;

      then

       A50: ( LSeg (co,1)) = ( LSeg (Gij,(co /. (1 + 1)))) by A35, TOPREAL1:def 3;

      

       A51: {Gij} c= (( LSeg (co,1)) /\ ( L~ <*Gik, Gi1k, Gij*>))

      proof

        let x be object;

        assume x in {Gij};

        then

         A52: x = Gij by TARSKI:def 1;

        Gij in ( LSeg (Gi1k,Gij)) by RLTOPSP1: 68;

        then Gij in (( LSeg (Gik,Gi1k)) \/ ( LSeg (Gi1k,Gij))) by XBOOLE_0:def 3;

        then

         A53: Gij in ( L~ <*Gik, Gi1k, Gij*>) by SPRECT_1: 8;

        Gij in ( LSeg (co,1)) by A50, RLTOPSP1: 68;

        hence thesis by A52, A53, XBOOLE_0:def 4;

      end;

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

      then ( LSeg (co,1)) c= ( L~ LS) by A36;

      then (( LSeg (co,1)) /\ ( L~ <*Gik, Gi1k, Gij*>)) c= {Gij} by A8, A13, XBOOLE_1: 26;

      then

       A54: (( L~ <*Gik, Gi1k, Gij*>) /\ ( LSeg (co,1))) = {Gij} by A51;

      

       A55: ( rng co) c= ( L~ co) by A35, SPPOL_2: 18;

      

       A56: ( len go) >= (1 + 1) by TOPREAL1:def 8;

      then

      reconsider go as non constant s.c.c. being_S-Seq FinSequence of ( TOP-REAL 2) by A48, JGRAPH_1: 12, JORDAN8: 5;

      

       A57: ( L~ go) c= ( L~ US) by A45, JORDAN3: 41;

      

       A58: ( len go) > 1 by A56, NAT_1: 13;

      then

       A59: ( len go) in ( dom go) by FINSEQ_3: 25;

      

      then

       A60: (go /. ( len go)) = (go . ( len go)) by PARTFUN1:def 6

      .= Gik by A45, JORDAN3: 24;

      reconsider m = (( len go) - 1) as Nat by A59, FINSEQ_3: 26;

      

       A61: (m + 1) = ( len go);

      then

       A62: (( len go) -' 1) = m by NAT_D: 34;

      m >= 1 by A56, XREAL_1: 19;

      then

       A63: ( LSeg (go,m)) = ( LSeg ((go /. m),Gik)) by A60, A61, TOPREAL1:def 3;

      

       A64: {Gik} c= (( LSeg (go,m)) /\ ( L~ <*Gik, Gi1k, Gij*>))

      proof

        let x be object;

        assume x in {Gik};

        then

         A65: x = Gik by TARSKI:def 1;

        Gik in ( LSeg (Gik,Gi1k)) by RLTOPSP1: 68;

        then Gik in (( LSeg (Gik,Gi1k)) \/ ( LSeg (Gi1k,Gij))) by XBOOLE_0:def 3;

        then

         A66: Gik in ( L~ <*Gik, Gi1k, Gij*>) by SPRECT_1: 8;

        Gik in ( LSeg (go,m)) by A63, RLTOPSP1: 68;

        hence thesis by A65, A66, XBOOLE_0:def 4;

      end;

      ( LSeg (go,m)) c= ( L~ go) by TOPREAL3: 19;

      then ( LSeg (go,m)) c= ( L~ US) by A57;

      then (( LSeg (go,m)) /\ ( L~ <*Gik, Gi1k, Gij*>)) c= {Gik} by A7, A13, XBOOLE_1: 26;

      then

       A67: (( LSeg (go,m)) /\ ( L~ <*Gik, Gi1k, Gij*>)) = {Gik} by A64;

      

       A68: (go /. 1) = (US /. 1) by A45, SPRECT_3: 22

      .= Wmin by JORDAN1F: 5;

      

       A69: (LS . 1) = (LS /. 1) by A27, PARTFUN1:def 6

      .= Emax by JORDAN1F: 6;

      

       A70: (( L~ go) /\ ( L~ co)) c= {(go /. 1)}

      proof

        let x be object;

        assume

         A71: x in (( L~ go) /\ ( L~ co));

        then

         A72: x in ( L~ co) by XBOOLE_0:def 4;

         A73:

        now

          assume x = Emax;

          then

           A74: Emax = Gij by A30, A69, A72, JORDAN1E: 7;

          ((G * (( len G),j)) `1 ) = Ebo by A4, A10, A24, JORDAN1A: 71;

          then (Emax `1 ) <> Ebo by A2, A3, A23, A15, A74, JORDAN1G: 7;

          hence contradiction by EUCLID: 52;

        end;

        x in ( L~ go) by A71, XBOOLE_0:def 4;

        then x in (( L~ US) /\ ( L~ LS)) by A57, A36, A72, XBOOLE_0:def 4;

        then x in {Wmin, Emax} by JORDAN1E: 16;

        then x = Wmin or x = Emax by TARSKI:def 2;

        hence thesis by A68, A73, TARSKI:def 1;

      end;

      set W2 = (go /. 2);

      

       A75: 2 in ( dom go) by A56, FINSEQ_3: 25;

      go = ( mid (US,1,(Gik .. US))) by A47, JORDAN1G: 49

      .= (US | (Gik .. US)) by A47, FINSEQ_4: 21, FINSEQ_6: 116;

      then

       A76: W2 = (US /. 2) by A75, FINSEQ_4: 70;

      

       A77: ( rng go) c= ( L~ go) by A56, SPPOL_2: 18;

      

       A78: (go /. 1) = (LS /. ( len LS)) by A68, JORDAN1F: 8

      .= (co /. ( len co)) by A30, JORDAN1J: 35;

       {(go /. 1)} c= (( L~ go) /\ ( L~ co))

      proof

        let x be object;

        assume x in {(go /. 1)};

        then

         A79: x = (go /. 1) by TARSKI:def 1;

        then

         A80: x in ( rng go) by FINSEQ_6: 42;

        x in ( rng co) by A78, A79, FINSEQ_6: 168;

        hence thesis by A77, A55, A80, XBOOLE_0:def 4;

      end;

      then

       A81: (( L~ go) /\ ( L~ co)) = {(go /. 1)} by A70;

      now

        per cases ;

          suppose (Gij `1 ) <> (Gik `1 ) & (Gij `2 ) <> (Gik `2 );

          then pion is being_S-Seq by A21, TOPREAL3: 35;

          then

          consider pion1 be FinSequence of ( TOP-REAL 2) such that

           A82: pion1 is_sequence_on G and

           A83: pion1 is being_S-Seq and

           A84: ( L~ pion) = ( L~ pion1) and

           A85: (pion /. 1) = (pion1 /. 1) and

           A86: (pion /. ( len pion)) = (pion1 /. ( len pion1)) and

           A87: ( len pion) <= ( len pion1) by A44, GOBOARD3: 2;

          reconsider pion1 as being_S-Seq FinSequence of ( TOP-REAL 2) by A83;

          

           A88: ((go ^' pion1) /. ( len (go ^' pion1))) = (pion /. ( len pion)) by A86, FINSEQ_6: 156

          .= (pion /. 3) by FINSEQ_1: 45

          .= (co /. 1) by A49, FINSEQ_4: 18;

          

           A89: (go /. ( len go)) = (pion1 /. 1) by A60, A85, FINSEQ_4: 18;

          

           A90: (( L~ go) /\ ( L~ pion1)) c= {(pion1 /. 1)}

          proof

            let x be object;

            assume

             A91: x in (( L~ go) /\ ( L~ pion1));

            then

             A92: x in ( L~ pion1) by XBOOLE_0:def 4;

            x in ( L~ go) by A91, XBOOLE_0:def 4;

            hence thesis by A7, A13, A60, A57, A84, A89, A92, XBOOLE_0:def 4;

          end;

          ( len pion1) >= (2 + 1) by A87, FINSEQ_1: 45;

          then

           A93: ( len pion1) > (1 + 1) by NAT_1: 13;

          then

           A94: ( rng pion1) c= ( L~ pion1) by SPPOL_2: 18;

           {(pion1 /. 1)} c= (( L~ go) /\ ( L~ pion1))

          proof

            let x be object;

            assume x in {(pion1 /. 1)};

            then

             A95: x = (pion1 /. 1) by TARSKI:def 1;

            then

             A96: x in ( rng pion1) by FINSEQ_6: 42;

            x in ( rng go) by A89, A95, FINSEQ_6: 168;

            hence thesis by A77, A94, A96, XBOOLE_0:def 4;

          end;

          then

           A97: (( L~ go) /\ ( L~ pion1)) = {(pion1 /. 1)} by A90;

          then

           A98: (go ^' pion1) is s.n.c. by A89, JORDAN1J: 54;

          

           A99: (pion /. ( len pion)) = (pion /. 3) by FINSEQ_1: 45

          .= (co /. 1) by A49, FINSEQ_4: 18;

          

           A100: {(pion1 /. ( len pion1))} c= (( L~ co) /\ ( L~ pion1))

          proof

            let x be object;

            assume x in {(pion1 /. ( len pion1))};

            then

             A101: x = (pion1 /. ( len pion1)) by TARSKI:def 1;

            then

             A102: x in ( rng pion1) by FINSEQ_6: 168;

            x in ( rng co) by A86, A99, A101, FINSEQ_6: 42;

            hence thesis by A55, A94, A102, XBOOLE_0:def 4;

          end;

          (( L~ co) /\ ( L~ pion1)) c= {(pion1 /. ( len pion1))}

          proof

            let x be object;

            assume

             A103: x in (( L~ co) /\ ( L~ pion1));

            then

             A104: x in ( L~ pion1) by XBOOLE_0:def 4;

            x in ( L~ co) by A103, XBOOLE_0:def 4;

            hence thesis by A8, A13, A49, A36, A84, A86, A99, A104, XBOOLE_0:def 4;

          end;

          then

           A105: (( L~ co) /\ ( L~ pion1)) = {(pion1 /. ( len pion1))} by A100;

          

           A106: (( L~ (go ^' pion1)) /\ ( L~ co)) = ((( L~ go) \/ ( L~ pion1)) /\ ( L~ co)) by A89, TOPREAL8: 35

          .= ( {(go /. 1)} \/ {(co /. 1)}) by A81, A86, A99, A105, XBOOLE_1: 23

          .= ( {((go ^' pion1) /. 1)} \/ {(co /. 1)}) by FINSEQ_6: 155

          .= {((go ^' pion1) /. 1), (co /. 1)} by ENUMSET1: 1;

          

           A107: UA is_an_arc_of (( W-min C),( E-max C)) by JORDAN6:def 8;

          then

           A108: UA is connected by JORDAN6: 10;

          set godo = ((go ^' pion1) ^' co);

          

           A109: (co /. ( len co)) = ((go ^' pion1) /. 1) by A78, FINSEQ_6: 155;

          

           A110: (go ^' pion1) is_sequence_on G by A48, A82, A89, TOPREAL8: 12;

          then

           A111: godo is_sequence_on G by A33, A88, TOPREAL8: 12;

          

           A112: (( len pion1) - 1) >= 1 by A93, XREAL_1: 19;

          then

           A113: (( len pion1) -' 1) = (( len pion1) - 1) by XREAL_0:def 2;

          

           A114: {Gij} c= (( LSeg (pion1,(( len pion1) -' 1))) /\ ( LSeg (co,1)))

          proof

            let x be object;

            assume x in {Gij};

            then

             A115: x = Gij by TARSKI:def 1;

            (pion1 /. ((( len pion1) -' 1) + 1)) = (pion /. 3) by A86, A113, FINSEQ_1: 45

            .= Gij by FINSEQ_4: 18;

            then

             A116: Gij in ( LSeg (pion1,(( len pion1) -' 1))) by A112, A113, TOPREAL1: 21;

            Gij in ( LSeg (co,1)) by A50, RLTOPSP1: 68;

            hence thesis by A115, A116, XBOOLE_0:def 4;

          end;

          ( LSeg (pion1,(( len pion1) -' 1))) c= ( L~ pion) by A84, TOPREAL3: 19;

          then (( LSeg (pion1,(( len pion1) -' 1))) /\ ( LSeg (co,1))) c= {Gij} by A54, XBOOLE_1: 27;

          then

           A117: (( LSeg (pion1,(( len pion1) -' 1))) /\ ( LSeg (co,1))) = {Gij} by A114;

          ((( len pion1) - 1) + 1) <= ( len pion1);

          then

           A118: (( len pion1) -' 1) < ( len pion1) by A113, NAT_1: 13;

          ( len pion1) >= (2 + 1) by A87, FINSEQ_1: 45;

          then

           A119: (( len pion1) - 2) >= 0 by XREAL_1: 19;

          

          then ((( len pion1) -' 2) + 1) = ((( len pion1) - 2) + 1) by XREAL_0:def 2

          .= (( len pion1) -' 1) by A112, XREAL_0:def 2;

          then

           A120: (( LSeg ((go ^' pion1),(( len go) + (( len pion1) -' 2)))) /\ ( LSeg (co,1))) = {((go ^' pion1) /. ( len (go ^' pion1)))} by A49, A89, A88, A118, A117, TOPREAL8: 31;

          (( rng go) /\ ( rng pion1)) c= {(pion1 /. 1)} by A77, A94, A97, XBOOLE_1: 27;

          then

           A121: (go ^' pion1) is one-to-one by JORDAN1J: 55;

          ((( len (go ^' pion1)) + 1) - 1) = ((( len go) + ( len pion1)) - 1) by FINSEQ_6: 139;

          

          then (( len (go ^' pion1)) - 1) = (( len go) + (( len pion1) - 2))

          .= (( len go) + (( len pion1) -' 2)) by A119, XREAL_0:def 2;

          then

           A122: (( len (go ^' pion1)) -' 1) = (( len go) + (( len pion1) -' 2)) by XREAL_0:def 2;

          

           A123: ( L~ ( Cage (C,n))) = (( L~ US) \/ ( L~ LS)) by JORDAN1E: 13;

          then

           A124: ( L~ US) c= ( L~ ( Cage (C,n))) by XBOOLE_1: 7;

          then

           A125: ( L~ go) c= ( L~ ( Cage (C,n))) by A57;

          

           A126: {Gik} c= (( LSeg (go,m)) /\ ( LSeg (pion1,1)))

          proof

            let x be object;

            assume x in {Gik};

            then

             A127: x = Gik by TARSKI:def 1;

            

             A128: Gik in ( LSeg (go,m)) by A63, RLTOPSP1: 68;

            Gik in ( LSeg (pion1,1)) by A60, A89, A93, TOPREAL1: 21;

            hence thesis by A127, A128, XBOOLE_0:def 4;

          end;

          ( LSeg (pion1,1)) c= ( L~ pion) by A84, TOPREAL3: 19;

          then (( LSeg (go,(( len go) -' 1))) /\ ( LSeg (pion1,1))) c= {Gik} by A62, A67, XBOOLE_1: 27;

          then (( LSeg (go,(( len go) -' 1))) /\ ( LSeg (pion1,1))) = {(go /. ( len go))} by A60, A62, A126;

          then

           A129: (go ^' pion1) is unfolded by A89, TOPREAL8: 34;

          ( len (go ^' pion1)) >= ( len go) by TOPREAL8: 7;

          then

           A130: ( len (go ^' pion1)) >= (1 + 1) by A56, XXREAL_0: 2;

          then

           A131: ( len (go ^' pion1)) > (1 + 0 ) by NAT_1: 13;

           A132:

          now

            assume

             A133: (Gik .. US) <= 1;

            (Gik .. US) >= 1 by A47, FINSEQ_4: 21;

            then (Gik .. US) = 1 by A133, XXREAL_0: 1;

            then Gik = (US /. 1) by A47, FINSEQ_5: 38;

            hence contradiction by A42, A46, JORDAN1F: 5;

          end;

          

           A134: US is_sequence_on G by JORDAN1G: 4;

          

           A135: (Wmin `1 ) = Wbo by EUCLID: 52;

          set ff = ( Rotate (( Cage (C,n)),Wmin));

          

           A136: (1 + 1) <= ( len ( Cage (C,n))) by GOBOARD7: 34, XXREAL_0: 2;

          

           A137: ( len godo) >= ( len (go ^' pion1)) by TOPREAL8: 7;

          then

           A138: (1 + 1) <= ( len godo) by A130, XXREAL_0: 2;

          (go ^' pion1) is non trivial by A130, NAT_D: 60;

          then

          reconsider godo as non constant standard special_circular_sequence by A138, A88, A111, A129, A122, A120, A98, A121, A106, A109, JORDAN8: 4, JORDAN8: 5, TOPREAL8: 11, TOPREAL8: 33, TOPREAL8: 34;

          

           A139: ( L~ godo) = (( L~ (go ^' pion1)) \/ ( L~ co)) by A88, TOPREAL8: 35

          .= ((( L~ go) \/ ( L~ pion1)) \/ ( L~ co)) by A89, TOPREAL8: 35;

          

           A140: (( right_cell (godo,1,G)) \ ( L~ godo)) c= ( RightComp godo) by A138, A111, JORDAN9: 27;

          2 in ( dom godo) by A138, FINSEQ_3: 25;

          then

           A141: (godo /. 2) in ( rng godo) by PARTFUN2: 2;

          

           A142: ( W-min C) in UA by A107, TOPREAL1: 1;

          Wmin in ( rng ( Cage (C,n))) by SPRECT_2: 43;

          then

           A143: (ff /. 1) = Wmin by FINSEQ_6: 92;

          

           A144: ( L~ ff) = ( L~ ( Cage (C,n))) by REVROT_1: 33;

          then (( W-max ( L~ ff)) .. ff) > 1 by A143, SPRECT_5: 22;

          then (( N-min ( L~ ff)) .. ff) > 1 by A143, A144, SPRECT_5: 23, XXREAL_0: 2;

          then (( N-max ( L~ ff)) .. ff) > 1 by A143, A144, SPRECT_5: 24, XXREAL_0: 2;

          then

           A145: (Emax .. ff) > 1 by A143, A144, SPRECT_5: 25, XXREAL_0: 2;

          

           A146: ( Cage (C,n)) is_sequence_on G by JORDAN9:def 1;

          then

           A147: ff is_sequence_on G by REVROT_1: 34;

          

           A148: (Gi1k `1 ) = ((G * (i1,1)) `1 ) by A1, A6, A11, A16, GOBOARD5: 2

          .= (Gij `1 ) by A1, A4, A11, A10, GOBOARD5: 2;

          then

           A149: ( W-bound pio) = (Gij `1 ) by SPRECT_1: 54;

          

           A150: ( L~ LS) c= ( L~ ( Cage (C,n))) by A123, XBOOLE_1: 7;

          then

           A151: ( L~ co) c= ( L~ ( Cage (C,n))) by A36;

          

           A152: ( W-min C) in C by SPRECT_1: 13;

           A153:

          now

            assume ( W-min C) in ( L~ godo);

            then

             A154: ( W-min C) in (( L~ go) \/ ( L~ pion1)) or ( W-min C) in ( L~ co) by A139, XBOOLE_0:def 3;

            per cases by A154, XBOOLE_0:def 3;

              suppose ( W-min C) in ( L~ go);

              then C meets ( L~ ( Cage (C,n))) by A125, A152, XBOOLE_0: 3;

              hence contradiction by JORDAN10: 5;

            end;

              suppose ( W-min C) in ( L~ pion1);

              hence contradiction by A9, A13, A84, A142, XBOOLE_0: 3;

            end;

              suppose ( W-min C) in ( L~ co);

              then C meets ( L~ ( Cage (C,n))) by A151, A152, XBOOLE_0: 3;

              hence contradiction by JORDAN10: 5;

            end;

          end;

          

           A155: ( len US) >= 2 by A41, XXREAL_0: 2;

          

           A156: (( L~ go) \/ ( L~ co)) is compact by COMPTS_1: 10;

          (1 + 1) <= ( len ( Rotate (( Cage (C,n)),Wmin))) by GOBOARD7: 34, XXREAL_0: 2;

          

          then ( right_cell (( Rotate (( Cage (C,n)),Wmin)),1)) = ( right_cell (ff,1,( GoB ff))) by JORDAN1H: 23

          .= ( right_cell (ff,1,( GoB ( Cage (C,n))))) by REVROT_1: 28

          .= ( right_cell (ff,1,G)) by JORDAN1H: 44

          .= ( right_cell ((ff -: Emax),1,G)) by A145, A147, JORDAN1J: 53

          .= ( right_cell (US,1,G)) by JORDAN1E:def 1

          .= ( right_cell (( R_Cut (US,Gik)),1,G)) by A47, A134, A132, JORDAN1J: 52

          .= ( right_cell ((go ^' pion1),1,G)) by A58, A110, JORDAN1J: 51

          .= ( right_cell (godo,1,G)) by A131, A111, JORDAN1J: 51;

          then ( W-min C) in ( right_cell (godo,1,G)) by JORDAN1I: 6;

          then

           A157: ( W-min C) in (( right_cell (godo,1,G)) \ ( L~ godo)) by A153, XBOOLE_0:def 5;

          

           A158: ( rng godo) c= ( L~ godo) by A130, A137, SPPOL_2: 18, XXREAL_0: 2;

          

           A159: (godo /. 1) = ((go ^' pion1) /. 1) by FINSEQ_6: 155

          .= Wmin by A68, FINSEQ_6: 155;

          

           A160: (Gi1k `1 ) <= (Gik `1 ) by A1, A2, A3, A6, A16, JORDAN1A: 18;

          then

           A161: ( W-bound poz) = (Gi1k `1 ) by SPRECT_1: 54;

          ( W-bound (poz \/ pio)) = ( min (( W-bound poz),( W-bound pio))) by SPRECT_1: 47

          .= (Gij `1 ) by A148, A161, A149;

          then

           A162: ( W-bound ( L~ pion1)) = (Gij `1 ) by A84, TOPREAL3: 16;

          

           A163: UA c= C by JORDAN6: 61;

          (Gij `1 ) >= Wbo by A30, A150, PSCOMP_1: 24;

          then

           A164: (Gij `1 ) > Wbo by A38, XXREAL_0: 1;

          

           A165: ( E-max C) in UA by A107, TOPREAL1: 1;

          Wmin in ( rng go) by A68, FINSEQ_6: 42;

          then Wmin in (( L~ go) \/ ( L~ co)) by A77, XBOOLE_0:def 3;

          then

           A166: ( W-min (( L~ go) \/ ( L~ co))) = Wmin by A125, A151, A156, JORDAN1J: 21, XBOOLE_1: 8;

          (( W-min (( L~ go) \/ ( L~ co))) `1 ) = ( W-bound (( L~ go) \/ ( L~ co))) by EUCLID: 52;

          then ( W-min ((( L~ go) \/ ( L~ co)) \/ ( L~ pion1))) = ( W-min (( L~ go) \/ ( L~ co))) by A162, A156, A166, A135, A164, JORDAN1J: 33;

          then

           A167: ( W-min ( L~ godo)) = Wmin by A139, A166, XBOOLE_1: 4;

          (godo /. 2) = ((go ^' pion1) /. 2) by A130, FINSEQ_6: 159

          .= (US /. 2) by A56, A76, FINSEQ_6: 159

          .= ((US ^' LS) /. 2) by A155, FINSEQ_6: 159

          .= (( Rotate (( Cage (C,n)),Wmin)) /. 2) by JORDAN1E: 11;

          then (godo /. 2) in ( W-most ( L~ ( Cage (C,n)))) by JORDAN1I: 25;

          

          then ((godo /. 2) `1 ) = (( W-min ( L~ godo)) `1 ) by A167, PSCOMP_1: 31

          .= ( W-bound ( L~ godo)) by EUCLID: 52;

          then (godo /. 2) in ( W-most ( L~ godo)) by A158, A141, SPRECT_2: 12;

          then (( Rotate (godo,( W-min ( L~ godo)))) /. 2) in ( W-most ( L~ godo)) by A159, A167, FINSEQ_6: 89;

          then

          reconsider godo as clockwise_oriented non constant standard special_circular_sequence by JORDAN1I: 25;

          ( len US) in ( dom US) by FINSEQ_5: 6;

          

          then

           A168: (US . ( len US)) = (US /. ( len US)) by PARTFUN1:def 6

          .= Emax by JORDAN1F: 7;

          

           A169: ( east_halfline ( E-max C)) misses ( L~ go)

          proof

            assume ( east_halfline ( E-max C)) meets ( L~ go);

            then

            consider p be object such that

             A170: p in ( east_halfline ( E-max C)) and

             A171: p in ( L~ go) by XBOOLE_0: 3;

            reconsider p as Point of ( TOP-REAL 2) by A170;

            p in ( L~ US) by A57, A171;

            then p in (( east_halfline ( E-max C)) /\ ( L~ ( Cage (C,n)))) by A124, A170, XBOOLE_0:def 4;

            then

             A172: (p `1 ) = Ebo by JORDAN1A: 83, PSCOMP_1: 50;

            then

             A173: p = Emax by A57, A171, JORDAN1J: 46;

            then Emax = Gik by A45, A168, A171, JORDAN1J: 43;

            then (Gik `1 ) = ((G * (( len G),k)) `1 ) by A6, A16, A24, A172, A173, JORDAN1A: 71;

            hence contradiction by A3, A19, A22, JORDAN1G: 7;

          end;

          now

            assume ( east_halfline ( E-max C)) meets ( L~ godo);

            then

             A174: ( east_halfline ( E-max C)) meets (( L~ go) \/ ( L~ pion1)) or ( east_halfline ( E-max C)) meets ( L~ co) by A139, XBOOLE_1: 70;

            per cases by A174, XBOOLE_1: 70;

              suppose ( east_halfline ( E-max C)) meets ( L~ go);

              hence contradiction by A169;

            end;

              suppose ( east_halfline ( E-max C)) meets ( L~ pion1);

              then

              consider p be object such that

               A175: p in ( east_halfline ( E-max C)) and

               A176: p in ( L~ pion1) by XBOOLE_0: 3;

              reconsider p as Point of ( TOP-REAL 2) by A175;

              

               A177: (p `2 ) = (( E-max C) `2 ) by A175, TOPREAL1:def 11;

               A178:

              now

                per cases by A13, A84, A176, XBOOLE_0:def 3;

                  suppose p in poz;

                  hence (p `1 ) <= (Gik `1 ) by A160, TOPREAL1: 3;

                end;

                  suppose p in pio;

                  hence (p `1 ) <= (Gik `1 ) by A148, A160, GOBOARD7: 5;

                end;

              end;

              (i2 + 1) <= ( len G) by A3, NAT_1: 13;

              then ((i2 + 1) - 1) <= (( len G) - 1) by XREAL_1: 9;

              then

               A179: i2 <= (( len G) -' 1) by XREAL_0:def 2;

              (( len G) -' 1) <= ( len G) by NAT_D: 35;

              then (Gik `1 ) <= ((G * ((( len G) -' 1),1)) `1 ) by A6, A18, A16, A24, A14, A179, JORDAN1A: 18;

              then (p `1 ) <= ((G * ((( len G) -' 1),1)) `1 ) by A178, XXREAL_0: 2;

              then (p `1 ) <= ( E-bound C) by A14, JORDAN8: 12;

              then

               A180: (p `1 ) <= (( E-max C) `1 ) by EUCLID: 52;

              (p `1 ) >= (( E-max C) `1 ) by A175, TOPREAL1:def 11;

              then (p `1 ) = (( E-max C) `1 ) by A180, XXREAL_0: 1;

              then p = ( E-max C) by A177, TOPREAL3: 6;

              hence contradiction by A9, A13, A84, A165, A176, XBOOLE_0: 3;

            end;

              suppose ( east_halfline ( E-max C)) meets ( L~ co);

              then

              consider p be object such that

               A181: p in ( east_halfline ( E-max C)) and

               A182: p in ( L~ co) by XBOOLE_0: 3;

              reconsider p as Point of ( TOP-REAL 2) by A181;

              

               A183: p in ( LSeg (co,( Index (p,co)))) by A182, JORDAN3: 9;

              consider t be Nat such that

               A184: t in ( dom LS) and

               A185: (LS . t) = Gij by A32, FINSEQ_2: 10;

              1 <= t by A184, FINSEQ_3: 25;

              then

               A186: 1 < t by A34, A185, XXREAL_0: 1;

              t <= ( len LS) by A184, FINSEQ_3: 25;

              then (( Index (Gij,LS)) + 1) = t by A185, A186, JORDAN3: 12;

              then

               A187: ( len ( L_Cut (LS,Gij))) = (( len LS) - ( Index (Gij,LS))) by A30, A185, JORDAN3: 26;

              ( Index (p,co)) < ( len co) by A182, JORDAN3: 8;

              then ( Index (p,co)) < (( len LS) -' ( Index (Gij,LS))) by A187, XREAL_0:def 2;

              then (( Index (p,co)) + 1) <= (( len LS) -' ( Index (Gij,LS))) by NAT_1: 13;

              then

               A188: ( Index (p,co)) <= ((( len LS) -' ( Index (Gij,LS))) - 1) by XREAL_1: 19;

              

               A189: co = ( mid (LS,(Gij .. LS),( len LS))) by A32, JORDAN1J: 37;

              p in ( L~ LS) by A36, A182;

              then p in (( east_halfline ( E-max C)) /\ ( L~ ( Cage (C,n)))) by A150, A181, XBOOLE_0:def 4;

              then

               A190: (p `1 ) = Ebo by JORDAN1A: 83, PSCOMP_1: 50;

              

               A191: (( Index (Gij,LS)) + 1) = (Gij .. LS) by A34, A32, JORDAN1J: 56;

              ( 0 + ( Index (Gij,LS))) < ( len LS) by A30, JORDAN3: 8;

              then (( len LS) - ( Index (Gij,LS))) > 0 by XREAL_1: 20;

              then ( Index (p,co)) <= ((( len LS) - ( Index (Gij,LS))) - 1) by A188, XREAL_0:def 2;

              then ( Index (p,co)) <= (( len LS) - (Gij .. LS)) by A191;

              then ( Index (p,co)) <= (( len LS) -' (Gij .. LS)) by XREAL_0:def 2;

              then

               A192: ( Index (p,co)) < ((( len LS) -' (Gij .. LS)) + 1) by NAT_1: 13;

              

               A193: 1 <= ( Index (p,co)) by A182, JORDAN3: 8;

              

               A194: (Gij .. LS) <= ( len LS) by A32, FINSEQ_4: 21;

              (Gij .. LS) <> ( len LS) by A31, A32, FINSEQ_4: 19;

              then

               A195: (Gij .. LS) < ( len LS) by A194, XXREAL_0: 1;

              

               A196: (1 + 1) <= ( len LS) by A25, XXREAL_0: 2;

              then

               A197: 2 in ( dom LS) by FINSEQ_3: 25;

              set tt = ((( Index (p,co)) + (Gij .. LS)) -' 1);

              set RC = ( Rotate (( Cage (C,n)),Emax));

              

               A198: ( E-max C) in ( right_cell (RC,1)) by JORDAN1I: 7;

              

               A199: ( GoB RC) = ( GoB ( Cage (C,n))) by REVROT_1: 28

              .= G by JORDAN1H: 44;

              

               A200: ( L~ RC) = ( L~ ( Cage (C,n))) by REVROT_1: 33;

              consider jj2 be Nat such that

               A201: 1 <= jj2 and

               A202: jj2 <= ( width G) and

               A203: Emax = (G * (( len G),jj2)) by JORDAN1D: 25;

              

               A204: ( len G) >= 4 by JORDAN8: 10;

              then ( len G) >= 1 by XXREAL_0: 2;

              then

               A205: [( len G), jj2] in ( Indices G) by A201, A202, MATRIX_0: 30;

              

               A206: ( len RC) = ( len ( Cage (C,n))) by FINSEQ_6: 179;

              LS = (RC -: Wmin) by JORDAN1G: 18;

              then

               A207: ( LSeg (LS,1)) = ( LSeg (RC,1)) by A196, SPPOL_2: 9;

              

               A208: Emax in ( rng ( Cage (C,n))) by SPRECT_2: 46;

              RC is_sequence_on G by A146, REVROT_1: 34;

              then

              consider ii,jj be Nat such that

               A209: [ii, (jj + 1)] in ( Indices G) and

               A210: [ii, jj] in ( Indices G) and

               A211: (RC /. 1) = (G * (ii,(jj + 1))) and

               A212: (RC /. (1 + 1)) = (G * (ii,jj)) by A136, A200, A206, A208, FINSEQ_6: 92, JORDAN1I: 23;

              

               A213: ((jj + 1) + 1) <> jj;

              

               A214: 1 <= jj by A210, MATRIX_0: 32;

              (RC /. 1) = ( E-max ( L~ RC)) by A200, A208, FINSEQ_6: 92;

              then

               A215: ii = ( len G) by A200, A209, A211, A203, A205, GOBOARD1: 5;

              then (ii - 1) >= (4 - 1) by A204, XREAL_1: 9;

              then

               A216: (ii - 1) >= 1 by XXREAL_0: 2;

              then

               A217: 1 <= (ii -' 1) by XREAL_0:def 2;

              

               A218: jj <= ( width G) by A210, MATRIX_0: 32;

              then

               A219: ((G * (( len G),jj)) `1 ) = Ebo by A24, A214, JORDAN1A: 71;

              

               A220: (jj + 1) <= ( width G) by A209, MATRIX_0: 32;

              (ii + 1) <> ii;

              then

               A221: ( right_cell (RC,1)) = ( cell (G,(ii -' 1),jj)) by A136, A206, A199, A209, A210, A211, A212, A213, GOBOARD5:def 6;

              

               A222: ii <= ( len G) by A210, MATRIX_0: 32;

              

               A223: 1 <= ii by A210, MATRIX_0: 32;

              

               A224: ii <= ( len G) by A209, MATRIX_0: 32;

              

               A225: 1 <= (jj + 1) by A209, MATRIX_0: 32;

              then

               A226: Ebo = ((G * (( len G),(jj + 1))) `1 ) by A24, A220, JORDAN1A: 71;

              

               A227: 1 <= ii by A209, MATRIX_0: 32;

              then

               A228: ((ii -' 1) + 1) = ii by XREAL_1: 235;

              then

               A229: (ii -' 1) < ( len G) by A224, NAT_1: 13;

              

              then

               A230: ((G * ((ii -' 1),(jj + 1))) `2 ) = ((G * (1,(jj + 1))) `2 ) by A225, A220, A217, GOBOARD5: 1

              .= ((G * (ii,(jj + 1))) `2 ) by A227, A224, A225, A220, GOBOARD5: 1;

              

               A231: (( E-max C) `2 ) = (p `2 ) by A181, TOPREAL1:def 11;

              then

               A232: (p `2 ) <= ((G * ((ii -' 1),(jj + 1))) `2 ) by A198, A224, A220, A214, A221, A228, A216, JORDAN9: 17;

              

               A233: ((G * ((ii -' 1),jj)) `2 ) = ((G * (1,jj)) `2 ) by A214, A218, A217, A229, GOBOARD5: 1

              .= ((G * (ii,jj)) `2 ) by A223, A222, A214, A218, GOBOARD5: 1;

              ((G * ((ii -' 1),jj)) `2 ) <= (p `2 ) by A231, A198, A224, A220, A214, A221, A228, A216, JORDAN9: 17;

              then p in ( LSeg ((RC /. 1),(RC /. (1 + 1)))) by A190, A211, A212, A215, A232, A233, A230, A219, A226, GOBOARD7: 7;

              then

               A234: p in ( LSeg (LS,1)) by A136, A207, A206, TOPREAL1:def 3;

              1 <= (Gij .. LS) by A32, FINSEQ_4: 21;

              then

               A235: ( LSeg (( mid (LS,(Gij .. LS),( len LS))),( Index (p,co)))) = ( LSeg (LS,((( Index (p,co)) + (Gij .. LS)) -' 1))) by A195, A193, A192, JORDAN4: 19;

              1 <= ( Index (Gij,LS)) by A30, JORDAN3: 8;

              then

               A236: (1 + 1) <= (Gij .. LS) by A191, XREAL_1: 7;

              then (( Index (p,co)) + (Gij .. LS)) >= ((1 + 1) + 1) by A193, XREAL_1: 7;

              then ((( Index (p,co)) + (Gij .. LS)) - 1) >= (((1 + 1) + 1) - 1) by XREAL_1: 9;

              then

               A237: tt >= (1 + 1) by XREAL_0:def 2;

              now

                per cases by A237, XXREAL_0: 1;

                  suppose tt > (1 + 1);

                  then ( LSeg (LS,1)) misses ( LSeg (LS,tt)) by TOPREAL1:def 7;

                  hence contradiction by A234, A183, A189, A235, XBOOLE_0: 3;

                end;

                  suppose

                   A238: tt = (1 + 1);

                  then (1 + 1) = ((( Index (p,co)) + (Gij .. LS)) - 1) by XREAL_0:def 2;

                  then ((1 + 1) + 1) = (( Index (p,co)) + (Gij .. LS));

                  then

                   A239: (Gij .. LS) = 2 by A193, A236, JORDAN1E: 6;

                  (( LSeg (LS,1)) /\ ( LSeg (LS,tt))) = {(LS /. 2)} by A25, A238, TOPREAL1:def 6;

                  then p in {(LS /. 2)} by A234, A183, A189, A235, XBOOLE_0:def 4;

                  then

                   A240: p = (LS /. 2) by TARSKI:def 1;

                  then

                   A241: p in ( rng LS) by A197, PARTFUN2: 2;

                  (p .. LS) = 2 by A197, A240, FINSEQ_5: 41;

                  then p = Gij by A32, A239, A241, FINSEQ_5: 9;

                  then (Gij `1 ) = Ebo by A240, JORDAN1G: 32;

                  then (Gij `1 ) = ((G * (( len G),j)) `1 ) by A4, A10, A24, JORDAN1A: 71;

                  hence contradiction by A2, A3, A23, A15, JORDAN1G: 7;

                end;

              end;

              hence contradiction;

            end;

          end;

          then ( east_halfline ( E-max C)) c= (( L~ godo) ` ) by SUBSET_1: 23;

          then

          consider W be Subset of ( TOP-REAL 2) such that

           A242: W is_a_component_of (( L~ godo) ` ) and

           A243: ( east_halfline ( E-max C)) c= W by GOBOARD9: 3;

           not W is bounded by A243, JORDAN2C: 121, RLTOPSP1: 42;

          then W is_outside_component_of ( L~ godo) by A242, JORDAN2C:def 3;

          then W c= ( UBD ( L~ godo)) by JORDAN2C: 23;

          then

           A244: ( east_halfline ( E-max C)) c= ( UBD ( L~ godo)) by A243;

          ( E-max C) in ( east_halfline ( E-max C)) by TOPREAL1: 38;

          then ( E-max C) in ( UBD ( L~ godo)) by A244;

          then ( E-max C) in ( LeftComp godo) by GOBRD14: 36;

          then UA meets ( L~ godo) by A108, A142, A165, A140, A157, JORDAN1J: 36;

          then

           A245: UA meets (( L~ go) \/ ( L~ pion1)) or UA meets ( L~ co) by A139, XBOOLE_1: 70;

          now

            per cases by A245, XBOOLE_1: 70;

              suppose UA meets ( L~ go);

              then UA meets ( L~ ( Cage (C,n))) by A57, A124, XBOOLE_1: 1, XBOOLE_1: 63;

              hence contradiction by A163, JORDAN10: 5, XBOOLE_1: 63;

            end;

              suppose UA meets ( L~ pion1);

              hence contradiction by A9, A13, A84;

            end;

              suppose UA meets ( L~ co);

              then UA meets ( L~ ( Cage (C,n))) by A36, A150, XBOOLE_1: 1, XBOOLE_1: 63;

              hence contradiction by A163, JORDAN10: 5, XBOOLE_1: 63;

            end;

          end;

          hence contradiction;

        end;

          suppose (Gij `1 ) = (Gik `1 );

          then

           A246: i1 = i2 by A23, A19, JORDAN1G: 7;

          then poz = {Gi1k} by RLTOPSP1: 70;

          then poz c= pio by A40, ZFMISC_1: 31;

          then (pio \/ poz) = pio by XBOOLE_1: 12;

          hence contradiction by A1, A3, A4, A5, A6, A7, A8, A9, A246, JORDAN1J: 59;

        end;

          suppose (Gij `2 ) = (Gik `2 );

          then

           A247: j = k by A23, A19, JORDAN1G: 6;

          then pio = {Gi1k} by RLTOPSP1: 70;

          then pio c= poz by A39, ZFMISC_1: 31;

          then (pio \/ poz) = poz by XBOOLE_1: 12;

          hence contradiction by A1, A2, A3, A4, A6, A7, A8, A9, A247, Th29;

        end;

      end;

      hence contradiction;

    end;

    theorem :: JORDAN15:45

    

     Th45: for C be Simple_closed_curve holds for i1,i2,j,k be Nat st 1 < i1 & i1 <= i2 & i2 < ( len ( Gauge (C,n))) & 1 <= j & j <= k & k <= ( width ( Gauge (C,n))) & ((( LSeg ((( Gauge (C,n)) * (i1,j)),(( Gauge (C,n)) * (i1,k)))) \/ ( LSeg ((( Gauge (C,n)) * (i1,k)),(( Gauge (C,n)) * (i2,k))))) /\ ( L~ ( Upper_Seq (C,n)))) = {(( Gauge (C,n)) * (i2,k))} & ((( LSeg ((( Gauge (C,n)) * (i1,j)),(( Gauge (C,n)) * (i1,k)))) \/ ( LSeg ((( Gauge (C,n)) * (i1,k)),(( Gauge (C,n)) * (i2,k))))) /\ ( L~ ( Lower_Seq (C,n)))) = {(( Gauge (C,n)) * (i1,j))} holds (( LSeg ((( Gauge (C,n)) * (i1,j)),(( Gauge (C,n)) * (i1,k)))) \/ ( LSeg ((( Gauge (C,n)) * (i1,k)),(( Gauge (C,n)) * (i2,k))))) meets ( Lower_Arc C)

    proof

      let C be Simple_closed_curve;

      let i1,i2,j,k be Nat;

      set G = ( Gauge (C,n));

      set pio = ( LSeg ((G * (i1,j)),(G * (i1,k))));

      set poz = ( LSeg ((G * (i1,k)),(G * (i2,k))));

      set US = ( Upper_Seq (C,n));

      set LS = ( Lower_Seq (C,n));

      assume that

       A1: 1 < i1 and

       A2: i1 <= i2 and

       A3: i2 < ( len G) and

       A4: 1 <= j and

       A5: j <= k and

       A6: k <= ( width G) and

       A7: ((pio \/ poz) /\ ( L~ US)) = {(G * (i2,k))} and

       A8: ((pio \/ poz) /\ ( L~ LS)) = {(G * (i1,j))} and

       A9: (pio \/ poz) misses ( Lower_Arc C);

      set Gij = (G * (i1,j));

      

       A10: j <= ( width G) by A5, A6, XXREAL_0: 2;

      

       A11: i1 < ( len G) by A2, A3, XXREAL_0: 2;

      then

       A12: [i1, j] in ( Indices G) by A1, A4, A10, MATRIX_0: 30;

      set Gi1k = (G * (i1,k));

      set Gik = (G * (i2,k));

      

       A13: ( L~ <*Gik, Gi1k, Gij*>) = (poz \/ pio) by TOPREAL3: 16;

      ( len G) >= 4 by JORDAN8: 10;

      then

       A14: ( len G) >= 1 by XXREAL_0: 2;

      then

       A15: [( len G), j] in ( Indices G) by A4, A10, MATRIX_0: 30;

      

       A16: 1 <= k by A4, A5, XXREAL_0: 2;

      then

       A17: [1, k] in ( Indices G) by A6, A14, MATRIX_0: 30;

      

       A18: 1 < i2 by A1, A2, XXREAL_0: 2;

      then

       A19: [i2, k] in ( Indices G) by A3, A6, A16, MATRIX_0: 30;

      

       A20: (Gi1k `2 ) = ((G * (1,k)) `2 ) by A1, A6, A11, A16, GOBOARD5: 1

      .= (Gik `2 ) by A3, A6, A18, A16, GOBOARD5: 1;

      (Gi1k `1 ) = ((G * (i1,1)) `1 ) by A1, A6, A11, A16, GOBOARD5: 2

      .= (Gij `1 ) by A1, A4, A11, A10, GOBOARD5: 2;

      then

       A21: Gi1k = |[(Gij `1 ), (Gik `2 )]| by A20, EUCLID: 53;

      

       A22: [( len G), k] in ( Indices G) by A6, A16, A14, MATRIX_0: 30;

      

       A23: [i1, j] in ( Indices G) by A1, A4, A11, A10, MATRIX_0: 30;

      set Wbo = ( W-bound ( L~ ( Cage (C,n))));

      set Wmin = ( W-min ( L~ ( Cage (C,n))));

      

       A24: ( len G) = ( width G) by JORDAN8:def 1;

      set Ebo = ( E-bound ( L~ ( Cage (C,n))));

      set Emax = ( E-max ( L~ ( Cage (C,n))));

      

       A25: ( len LS) >= (1 + 2) by JORDAN1E: 15;

      then

       A26: ( len LS) >= 1 by XXREAL_0: 2;

      then

       A27: 1 in ( dom LS) by FINSEQ_3: 25;

      

      then

       A28: (LS . 1) = (LS /. 1) by PARTFUN1:def 6

      .= Emax by JORDAN1F: 6;

      ( len LS) in ( dom LS) by A26, FINSEQ_3: 25;

      

      then

       A29: (LS . ( len LS)) = (LS /. ( len LS)) by PARTFUN1:def 6

      .= Wmin by JORDAN1F: 8;

      set co = ( L_Cut (LS,Gij));

      Gij in {Gij} by TARSKI:def 1;

      then

       A30: Gij in ( L~ LS) by A8, XBOOLE_0:def 4;

      (Wmin `1 ) = Wbo by EUCLID: 52

      .= ((G * (1,k)) `1 ) by A6, A16, A24, JORDAN1A: 73;

      then

       A31: Gij <> (LS . ( len LS)) by A1, A17, A29, A12, JORDAN1G: 7;

      then

      reconsider co as being_S-Seq FinSequence of ( TOP-REAL 2) by A30, JORDAN3: 34;

      

       A32: Gij in ( rng LS) by A1, A4, A11, A30, A10, JORDAN1G: 5, JORDAN1J: 40;

      then

       A33: co is_sequence_on G by JORDAN1G: 5, JORDAN1J: 39;

      (Emax `1 ) = Ebo by EUCLID: 52

      .= ((G * (( len G),k)) `1 ) by A6, A16, A24, JORDAN1A: 71;

      then

       A34: Gij <> (LS . 1) by A2, A3, A12, A22, A28, JORDAN1G: 7;

      

       A35: ( len co) >= (1 + 1) by TOPREAL1:def 8;

      then

      reconsider co as non constant s.c.c. being_S-Seq FinSequence of ( TOP-REAL 2) by A33, JGRAPH_1: 12, JORDAN8: 5;

      

       A36: ( L~ co) c= ( L~ LS) by A30, JORDAN3: 42;

      

       A37: [1, j] in ( Indices G) by A4, A10, A14, MATRIX_0: 30;

       A38:

      now

        assume (Gij `1 ) = Wbo;

        then ((G * (1,j)) `1 ) = ((G * (i1,j)) `1 ) by A4, A10, A24, JORDAN1A: 73;

        hence contradiction by A1, A23, A37, JORDAN1G: 7;

      end;

      set pion = <*Gik, Gi1k, Gij*>;

      

       A39: Gi1k in poz by RLTOPSP1: 68;

      set LA = ( Lower_Arc C);

      

       A40: Gi1k in pio by RLTOPSP1: 68;

      set go = ( R_Cut (US,Gik));

      

       A41: ( len US) >= 3 by JORDAN1E: 15;

      then ( len US) >= 1 by XXREAL_0: 2;

      then 1 in ( dom US) by FINSEQ_3: 25;

      

      then

       A42: (US . 1) = (US /. 1) by PARTFUN1:def 6

      .= Wmin by JORDAN1F: 5;

      

       A43: [i1, k] in ( Indices G) by A1, A6, A11, A16, MATRIX_0: 30;

       A44:

      now

        let n be Nat;

        assume n in ( dom pion);

        then n in {1, 2, 3} by FINSEQ_1: 89, FINSEQ_3: 1;

        then n = 1 or n = 2 or n = 3 by ENUMSET1:def 1;

        hence ex i,j be Nat st [i, j] in ( Indices G) & (pion /. n) = (G * (i,j)) by A23, A19, A43, FINSEQ_4: 18;

      end;

      Gik in {Gik} by TARSKI:def 1;

      then

       A45: Gik in ( L~ US) by A7, XBOOLE_0:def 4;

      (Wmin `1 ) = Wbo by EUCLID: 52

      .= ((G * (1,k)) `1 ) by A6, A16, A24, JORDAN1A: 73;

      then

       A46: Gik <> (US . 1) by A1, A2, A19, A42, A17, JORDAN1G: 7;

      then

      reconsider go as being_S-Seq FinSequence of ( TOP-REAL 2) by A45, JORDAN3: 35;

      

       A47: Gik in ( rng US) by A3, A6, A18, A45, A16, JORDAN1G: 4, JORDAN1J: 40;

      then

       A48: go is_sequence_on G by JORDAN1G: 4, JORDAN1J: 38;

      ( len co) >= 1 by A35, XXREAL_0: 2;

      then 1 in ( dom co) by FINSEQ_3: 25;

      

      then

       A49: (co /. 1) = (co . 1) by PARTFUN1:def 6

      .= Gij by A30, JORDAN3: 23;

      then

       A50: ( LSeg (co,1)) = ( LSeg (Gij,(co /. (1 + 1)))) by A35, TOPREAL1:def 3;

      

       A51: {Gij} c= (( LSeg (co,1)) /\ ( L~ <*Gik, Gi1k, Gij*>))

      proof

        let x be object;

        assume x in {Gij};

        then

         A52: x = Gij by TARSKI:def 1;

        Gij in ( LSeg (Gi1k,Gij)) by RLTOPSP1: 68;

        then Gij in (( LSeg (Gik,Gi1k)) \/ ( LSeg (Gi1k,Gij))) by XBOOLE_0:def 3;

        then

         A53: Gij in ( L~ <*Gik, Gi1k, Gij*>) by SPRECT_1: 8;

        Gij in ( LSeg (co,1)) by A50, RLTOPSP1: 68;

        hence thesis by A52, A53, XBOOLE_0:def 4;

      end;

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

      then ( LSeg (co,1)) c= ( L~ LS) by A36;

      then (( LSeg (co,1)) /\ ( L~ <*Gik, Gi1k, Gij*>)) c= {Gij} by A8, A13, XBOOLE_1: 26;

      then

       A54: (( L~ <*Gik, Gi1k, Gij*>) /\ ( LSeg (co,1))) = {Gij} by A51;

      

       A55: ( rng co) c= ( L~ co) by A35, SPPOL_2: 18;

      

       A56: ( len go) >= (1 + 1) by TOPREAL1:def 8;

      then

      reconsider go as non constant s.c.c. being_S-Seq FinSequence of ( TOP-REAL 2) by A48, JGRAPH_1: 12, JORDAN8: 5;

      

       A57: ( L~ go) c= ( L~ US) by A45, JORDAN3: 41;

      

       A58: ( len go) > 1 by A56, NAT_1: 13;

      then

       A59: ( len go) in ( dom go) by FINSEQ_3: 25;

      

      then

       A60: (go /. ( len go)) = (go . ( len go)) by PARTFUN1:def 6

      .= Gik by A45, JORDAN3: 24;

      reconsider m = (( len go) - 1) as Nat by A59, FINSEQ_3: 26;

      

       A61: (m + 1) = ( len go);

      then

       A62: (( len go) -' 1) = m by NAT_D: 34;

      m >= 1 by A56, XREAL_1: 19;

      then

       A63: ( LSeg (go,m)) = ( LSeg ((go /. m),Gik)) by A60, A61, TOPREAL1:def 3;

      

       A64: {Gik} c= (( LSeg (go,m)) /\ ( L~ <*Gik, Gi1k, Gij*>))

      proof

        let x be object;

        assume x in {Gik};

        then

         A65: x = Gik by TARSKI:def 1;

        Gik in ( LSeg (Gik,Gi1k)) by RLTOPSP1: 68;

        then Gik in (( LSeg (Gik,Gi1k)) \/ ( LSeg (Gi1k,Gij))) by XBOOLE_0:def 3;

        then

         A66: Gik in ( L~ <*Gik, Gi1k, Gij*>) by SPRECT_1: 8;

        Gik in ( LSeg (go,m)) by A63, RLTOPSP1: 68;

        hence thesis by A65, A66, XBOOLE_0:def 4;

      end;

      ( LSeg (go,m)) c= ( L~ go) by TOPREAL3: 19;

      then ( LSeg (go,m)) c= ( L~ US) by A57;

      then (( LSeg (go,m)) /\ ( L~ <*Gik, Gi1k, Gij*>)) c= {Gik} by A7, A13, XBOOLE_1: 26;

      then

       A67: (( LSeg (go,m)) /\ ( L~ <*Gik, Gi1k, Gij*>)) = {Gik} by A64;

      

       A68: (go /. 1) = (US /. 1) by A45, SPRECT_3: 22

      .= Wmin by JORDAN1F: 5;

      then

       A69: Wmin in ( rng go) by FINSEQ_6: 42;

      

       A70: (LS . 1) = (LS /. 1) by A27, PARTFUN1:def 6

      .= Emax by JORDAN1F: 6;

      

       A71: (( L~ go) /\ ( L~ co)) c= {(go /. 1)}

      proof

        let x be object;

        assume

         A72: x in (( L~ go) /\ ( L~ co));

        then

         A73: x in ( L~ co) by XBOOLE_0:def 4;

         A74:

        now

          assume x = Emax;

          then

           A75: Emax = Gij by A30, A70, A73, JORDAN1E: 7;

          ((G * (( len G),j)) `1 ) = Ebo by A4, A10, A24, JORDAN1A: 71;

          then (Emax `1 ) <> Ebo by A2, A3, A23, A15, A75, JORDAN1G: 7;

          hence contradiction by EUCLID: 52;

        end;

        x in ( L~ go) by A72, XBOOLE_0:def 4;

        then x in (( L~ US) /\ ( L~ LS)) by A57, A36, A73, XBOOLE_0:def 4;

        then x in {Wmin, Emax} by JORDAN1E: 16;

        then x = Wmin or x = Emax by TARSKI:def 2;

        hence thesis by A68, A74, TARSKI:def 1;

      end;

      set W2 = (go /. 2);

      

       A76: 2 in ( dom go) by A56, FINSEQ_3: 25;

      go = ( mid (US,1,(Gik .. US))) by A47, JORDAN1G: 49

      .= (US | (Gik .. US)) by A47, FINSEQ_4: 21, FINSEQ_6: 116;

      then

       A77: W2 = (US /. 2) by A76, FINSEQ_4: 70;

      

       A78: ( rng go) c= ( L~ go) by A56, SPPOL_2: 18;

      

       A79: (go /. 1) = (LS /. ( len LS)) by A68, JORDAN1F: 8

      .= (co /. ( len co)) by A30, JORDAN1J: 35;

       {(go /. 1)} c= (( L~ go) /\ ( L~ co))

      proof

        let x be object;

        assume x in {(go /. 1)};

        then

         A80: x = (go /. 1) by TARSKI:def 1;

        then

         A81: x in ( rng go) by FINSEQ_6: 42;

        x in ( rng co) by A79, A80, FINSEQ_6: 168;

        hence thesis by A78, A55, A81, XBOOLE_0:def 4;

      end;

      then

       A82: (( L~ go) /\ ( L~ co)) = {(go /. 1)} by A71;

      now

        per cases ;

          suppose (Gij `1 ) <> (Gik `1 ) & (Gij `2 ) <> (Gik `2 );

          then pion is being_S-Seq by A21, TOPREAL3: 35;

          then

          consider pion1 be FinSequence of ( TOP-REAL 2) such that

           A83: pion1 is_sequence_on G and

           A84: pion1 is being_S-Seq and

           A85: ( L~ pion) = ( L~ pion1) and

           A86: (pion /. 1) = (pion1 /. 1) and

           A87: (pion /. ( len pion)) = (pion1 /. ( len pion1)) and

           A88: ( len pion) <= ( len pion1) by A44, GOBOARD3: 2;

          reconsider pion1 as being_S-Seq FinSequence of ( TOP-REAL 2) by A84;

          

           A89: ((go ^' pion1) /. ( len (go ^' pion1))) = (pion /. ( len pion)) by A87, FINSEQ_6: 156

          .= (pion /. 3) by FINSEQ_1: 45

          .= (co /. 1) by A49, FINSEQ_4: 18;

          

           A90: (go /. ( len go)) = (pion1 /. 1) by A60, A86, FINSEQ_4: 18;

          

           A91: (( L~ go) /\ ( L~ pion1)) c= {(pion1 /. 1)}

          proof

            let x be object;

            assume

             A92: x in (( L~ go) /\ ( L~ pion1));

            then

             A93: x in ( L~ pion1) by XBOOLE_0:def 4;

            x in ( L~ go) by A92, XBOOLE_0:def 4;

            hence thesis by A7, A13, A60, A57, A85, A90, A93, XBOOLE_0:def 4;

          end;

          ( len pion1) >= (2 + 1) by A88, FINSEQ_1: 45;

          then

           A94: ( len pion1) > (1 + 1) by NAT_1: 13;

          then

           A95: ( rng pion1) c= ( L~ pion1) by SPPOL_2: 18;

           {(pion1 /. 1)} c= (( L~ go) /\ ( L~ pion1))

          proof

            let x be object;

            assume x in {(pion1 /. 1)};

            then

             A96: x = (pion1 /. 1) by TARSKI:def 1;

            then

             A97: x in ( rng pion1) by FINSEQ_6: 42;

            x in ( rng go) by A90, A96, FINSEQ_6: 168;

            hence thesis by A78, A95, A97, XBOOLE_0:def 4;

          end;

          then

           A98: (( L~ go) /\ ( L~ pion1)) = {(pion1 /. 1)} by A91;

          then

           A99: (go ^' pion1) is s.n.c. by A90, JORDAN1J: 54;

          

           A100: (pion /. ( len pion)) = (pion /. 3) by FINSEQ_1: 45

          .= (co /. 1) by A49, FINSEQ_4: 18;

          

           A101: {(pion1 /. ( len pion1))} c= (( L~ co) /\ ( L~ pion1))

          proof

            let x be object;

            assume x in {(pion1 /. ( len pion1))};

            then

             A102: x = (pion1 /. ( len pion1)) by TARSKI:def 1;

            then

             A103: x in ( rng pion1) by FINSEQ_6: 168;

            x in ( rng co) by A87, A100, A102, FINSEQ_6: 42;

            hence thesis by A55, A95, A103, XBOOLE_0:def 4;

          end;

          (( L~ co) /\ ( L~ pion1)) c= {(pion1 /. ( len pion1))}

          proof

            let x be object;

            assume

             A104: x in (( L~ co) /\ ( L~ pion1));

            then

             A105: x in ( L~ pion1) by XBOOLE_0:def 4;

            x in ( L~ co) by A104, XBOOLE_0:def 4;

            hence thesis by A8, A13, A49, A36, A85, A87, A100, A105, XBOOLE_0:def 4;

          end;

          then

           A106: (( L~ co) /\ ( L~ pion1)) = {(pion1 /. ( len pion1))} by A101;

          

           A107: (( L~ (go ^' pion1)) /\ ( L~ co)) = ((( L~ go) \/ ( L~ pion1)) /\ ( L~ co)) by A90, TOPREAL8: 35

          .= ( {(go /. 1)} \/ {(co /. 1)}) by A82, A87, A100, A106, XBOOLE_1: 23

          .= ( {((go ^' pion1) /. 1)} \/ {(co /. 1)}) by FINSEQ_6: 155

          .= {((go ^' pion1) /. 1), (co /. 1)} by ENUMSET1: 1;

          

           A108: LA is_an_arc_of (( E-max C),( W-min C)) by JORDAN6:def 9;

          then

           A109: LA is connected by JORDAN6: 10;

          set godo = ((go ^' pion1) ^' co);

          

           A110: (co /. ( len co)) = ((go ^' pion1) /. 1) by A79, FINSEQ_6: 155;

          

           A111: (go ^' pion1) is_sequence_on G by A48, A83, A90, TOPREAL8: 12;

          then

           A112: godo is_sequence_on G by A33, A89, TOPREAL8: 12;

          

           A113: (( len pion1) - 1) >= 1 by A94, XREAL_1: 19;

          then

           A114: (( len pion1) -' 1) = (( len pion1) - 1) by XREAL_0:def 2;

          

           A115: {Gij} c= (( LSeg (pion1,(( len pion1) -' 1))) /\ ( LSeg (co,1)))

          proof

            let x be object;

            assume x in {Gij};

            then

             A116: x = Gij by TARSKI:def 1;

            (pion1 /. ((( len pion1) -' 1) + 1)) = (pion /. 3) by A87, A114, FINSEQ_1: 45

            .= Gij by FINSEQ_4: 18;

            then

             A117: Gij in ( LSeg (pion1,(( len pion1) -' 1))) by A113, A114, TOPREAL1: 21;

            Gij in ( LSeg (co,1)) by A50, RLTOPSP1: 68;

            hence thesis by A116, A117, XBOOLE_0:def 4;

          end;

          ( LSeg (pion1,(( len pion1) -' 1))) c= ( L~ pion) by A85, TOPREAL3: 19;

          then (( LSeg (pion1,(( len pion1) -' 1))) /\ ( LSeg (co,1))) c= {Gij} by A54, XBOOLE_1: 27;

          then

           A118: (( LSeg (pion1,(( len pion1) -' 1))) /\ ( LSeg (co,1))) = {Gij} by A115;

          ((( len pion1) - 1) + 1) <= ( len pion1);

          then

           A119: (( len pion1) -' 1) < ( len pion1) by A114, NAT_1: 13;

          ( len pion1) >= (2 + 1) by A88, FINSEQ_1: 45;

          then

           A120: (( len pion1) - 2) >= 0 by XREAL_1: 19;

          

          then ((( len pion1) -' 2) + 1) = ((( len pion1) - 2) + 1) by XREAL_0:def 2

          .= (( len pion1) -' 1) by A113, XREAL_0:def 2;

          then

           A121: (( LSeg ((go ^' pion1),(( len go) + (( len pion1) -' 2)))) /\ ( LSeg (co,1))) = {((go ^' pion1) /. ( len (go ^' pion1)))} by A49, A90, A89, A119, A118, TOPREAL8: 31;

          (( rng go) /\ ( rng pion1)) c= {(pion1 /. 1)} by A78, A95, A98, XBOOLE_1: 27;

          then

           A122: (go ^' pion1) is one-to-one by JORDAN1J: 55;

          ((( len (go ^' pion1)) + 1) - 1) = ((( len go) + ( len pion1)) - 1) by FINSEQ_6: 139;

          

          then (( len (go ^' pion1)) - 1) = (( len go) + (( len pion1) - 2))

          .= (( len go) + (( len pion1) -' 2)) by A120, XREAL_0:def 2;

          then

           A123: (( len (go ^' pion1)) -' 1) = (( len go) + (( len pion1) -' 2)) by XREAL_0:def 2;

          

           A124: ( L~ ( Cage (C,n))) = (( L~ US) \/ ( L~ LS)) by JORDAN1E: 13;

          then

           A125: ( L~ US) c= ( L~ ( Cage (C,n))) by XBOOLE_1: 7;

          then

           A126: ( L~ go) c= ( L~ ( Cage (C,n))) by A57;

          

           A127: {Gik} c= (( LSeg (go,m)) /\ ( LSeg (pion1,1)))

          proof

            let x be object;

            assume x in {Gik};

            then

             A128: x = Gik by TARSKI:def 1;

            

             A129: Gik in ( LSeg (go,m)) by A63, RLTOPSP1: 68;

            Gik in ( LSeg (pion1,1)) by A60, A90, A94, TOPREAL1: 21;

            hence thesis by A128, A129, XBOOLE_0:def 4;

          end;

          ( LSeg (pion1,1)) c= ( L~ pion) by A85, TOPREAL3: 19;

          then (( LSeg (go,(( len go) -' 1))) /\ ( LSeg (pion1,1))) c= {Gik} by A62, A67, XBOOLE_1: 27;

          then (( LSeg (go,(( len go) -' 1))) /\ ( LSeg (pion1,1))) = {(go /. ( len go))} by A60, A62, A127;

          then

           A130: (go ^' pion1) is unfolded by A90, TOPREAL8: 34;

          ( len (go ^' pion1)) >= ( len go) by TOPREAL8: 7;

          then

           A131: ( len (go ^' pion1)) >= (1 + 1) by A56, XXREAL_0: 2;

          then

           A132: ( len (go ^' pion1)) > (1 + 0 ) by NAT_1: 13;

           A133:

          now

            assume

             A134: (Gik .. US) <= 1;

            (Gik .. US) >= 1 by A47, FINSEQ_4: 21;

            then (Gik .. US) = 1 by A134, XXREAL_0: 1;

            then Gik = (US /. 1) by A47, FINSEQ_5: 38;

            hence contradiction by A42, A46, JORDAN1F: 5;

          end;

          

           A135: US is_sequence_on G by JORDAN1G: 4;

          

           A136: (Wmin `1 ) = Wbo by EUCLID: 52;

          set ff = ( Rotate (( Cage (C,n)),Wmin));

          

           A137: (1 + 1) <= ( len ( Cage (C,n))) by GOBOARD7: 34, XXREAL_0: 2;

          

           A138: ( len godo) >= ( len (go ^' pion1)) by TOPREAL8: 7;

          then

           A139: (1 + 1) <= ( len godo) by A131, XXREAL_0: 2;

          (go ^' pion1) is non trivial by A131, NAT_D: 60;

          then

          reconsider godo as non constant standard special_circular_sequence by A139, A89, A112, A130, A123, A121, A99, A122, A107, A110, JORDAN8: 4, JORDAN8: 5, TOPREAL8: 11, TOPREAL8: 33, TOPREAL8: 34;

          

           A140: ( L~ godo) = (( L~ (go ^' pion1)) \/ ( L~ co)) by A89, TOPREAL8: 35

          .= ((( L~ go) \/ ( L~ pion1)) \/ ( L~ co)) by A90, TOPREAL8: 35;

          

           A141: (( right_cell (godo,1,G)) \ ( L~ godo)) c= ( RightComp godo) by A139, A112, JORDAN9: 27;

          2 in ( dom godo) by A139, FINSEQ_3: 25;

          then

           A142: (godo /. 2) in ( rng godo) by PARTFUN2: 2;

          

           A143: ( W-min C) in LA by A108, TOPREAL1: 1;

          Wmin in ( rng ( Cage (C,n))) by SPRECT_2: 43;

          then

           A144: (ff /. 1) = Wmin by FINSEQ_6: 92;

          

           A145: ( L~ ff) = ( L~ ( Cage (C,n))) by REVROT_1: 33;

          then (( W-max ( L~ ff)) .. ff) > 1 by A144, SPRECT_5: 22;

          then (( N-min ( L~ ff)) .. ff) > 1 by A144, A145, SPRECT_5: 23, XXREAL_0: 2;

          then (( N-max ( L~ ff)) .. ff) > 1 by A144, A145, SPRECT_5: 24, XXREAL_0: 2;

          then

           A146: (Emax .. ff) > 1 by A144, A145, SPRECT_5: 25, XXREAL_0: 2;

          

           A147: ( Cage (C,n)) is_sequence_on G by JORDAN9:def 1;

          then

           A148: ff is_sequence_on G by REVROT_1: 34;

          

           A149: (Gi1k `1 ) = ((G * (i1,1)) `1 ) by A1, A6, A11, A16, GOBOARD5: 2

          .= (Gij `1 ) by A1, A4, A11, A10, GOBOARD5: 2;

          then

           A150: ( W-bound pio) = (Gij `1 ) by SPRECT_1: 54;

          

           A151: ( L~ LS) c= ( L~ ( Cage (C,n))) by A124, XBOOLE_1: 7;

          then

           A152: ( L~ co) c= ( L~ ( Cage (C,n))) by A36;

          

           A153: ( W-min C) in C by SPRECT_1: 13;

           A154:

          now

            assume ( W-min C) in ( L~ godo);

            then

             A155: ( W-min C) in (( L~ go) \/ ( L~ pion1)) or ( W-min C) in ( L~ co) by A140, XBOOLE_0:def 3;

            per cases by A155, XBOOLE_0:def 3;

              suppose ( W-min C) in ( L~ go);

              then C meets ( L~ ( Cage (C,n))) by A126, A153, XBOOLE_0: 3;

              hence contradiction by JORDAN10: 5;

            end;

              suppose ( W-min C) in ( L~ pion1);

              hence contradiction by A9, A13, A85, A143, XBOOLE_0: 3;

            end;

              suppose ( W-min C) in ( L~ co);

              then C meets ( L~ ( Cage (C,n))) by A152, A153, XBOOLE_0: 3;

              hence contradiction by JORDAN10: 5;

            end;

          end;

          

           A156: ( len US) >= 2 by A41, XXREAL_0: 2;

          

           A157: (( L~ go) \/ ( L~ co)) is compact by COMPTS_1: 10;

          (1 + 1) <= ( len ( Rotate (( Cage (C,n)),Wmin))) by GOBOARD7: 34, XXREAL_0: 2;

          

          then ( right_cell (( Rotate (( Cage (C,n)),Wmin)),1)) = ( right_cell (ff,1,( GoB ff))) by JORDAN1H: 23

          .= ( right_cell (ff,1,( GoB ( Cage (C,n))))) by REVROT_1: 28

          .= ( right_cell (ff,1,G)) by JORDAN1H: 44

          .= ( right_cell ((ff -: Emax),1,G)) by A146, A148, JORDAN1J: 53

          .= ( right_cell (US,1,G)) by JORDAN1E:def 1

          .= ( right_cell (( R_Cut (US,Gik)),1,G)) by A47, A135, A133, JORDAN1J: 52

          .= ( right_cell ((go ^' pion1),1,G)) by A58, A111, JORDAN1J: 51

          .= ( right_cell (godo,1,G)) by A132, A112, JORDAN1J: 51;

          then ( W-min C) in ( right_cell (godo,1,G)) by JORDAN1I: 6;

          then

           A158: ( W-min C) in (( right_cell (godo,1,G)) \ ( L~ godo)) by A154, XBOOLE_0:def 5;

          

           A159: ( rng godo) c= ( L~ godo) by A131, A138, SPPOL_2: 18, XXREAL_0: 2;

          

           A160: (godo /. 1) = ((go ^' pion1) /. 1) by FINSEQ_6: 155

          .= Wmin by A68, FINSEQ_6: 155;

          

           A161: (Gi1k `1 ) <= (Gik `1 ) by A1, A2, A3, A6, A16, JORDAN1A: 18;

          then

           A162: ( W-bound poz) = (Gi1k `1 ) by SPRECT_1: 54;

          ( W-bound (poz \/ pio)) = ( min (( W-bound poz),( W-bound pio))) by SPRECT_1: 47

          .= (Gij `1 ) by A149, A162, A150;

          then

           A163: ( W-bound ( L~ pion1)) = (Gij `1 ) by A85, TOPREAL3: 16;

          

           A164: LA c= C by JORDAN6: 61;

          (Gij `1 ) >= Wbo by A30, A151, PSCOMP_1: 24;

          then

           A165: (Gij `1 ) > Wbo by A38, XXREAL_0: 1;

          

           A166: ( E-max C) in LA by A108, TOPREAL1: 1;

          Wmin in (( L~ go) \/ ( L~ co)) by A78, A69, XBOOLE_0:def 3;

          then

           A167: ( W-min (( L~ go) \/ ( L~ co))) = Wmin by A126, A152, A157, JORDAN1J: 21, XBOOLE_1: 8;

          (( W-min (( L~ go) \/ ( L~ co))) `1 ) = ( W-bound (( L~ go) \/ ( L~ co))) by EUCLID: 52;

          then ( W-min ((( L~ go) \/ ( L~ co)) \/ ( L~ pion1))) = ( W-min (( L~ go) \/ ( L~ co))) by A163, A157, A167, A136, A165, JORDAN1J: 33;

          then

           A168: ( W-min ( L~ godo)) = Wmin by A140, A167, XBOOLE_1: 4;

          (godo /. 2) = ((go ^' pion1) /. 2) by A131, FINSEQ_6: 159

          .= (US /. 2) by A56, A77, FINSEQ_6: 159

          .= ((US ^' LS) /. 2) by A156, FINSEQ_6: 159

          .= (( Rotate (( Cage (C,n)),Wmin)) /. 2) by JORDAN1E: 11;

          then (godo /. 2) in ( W-most ( L~ ( Cage (C,n)))) by JORDAN1I: 25;

          

          then ((godo /. 2) `1 ) = (( W-min ( L~ godo)) `1 ) by A168, PSCOMP_1: 31

          .= ( W-bound ( L~ godo)) by EUCLID: 52;

          then (godo /. 2) in ( W-most ( L~ godo)) by A159, A142, SPRECT_2: 12;

          then (( Rotate (godo,( W-min ( L~ godo)))) /. 2) in ( W-most ( L~ godo)) by A160, A168, FINSEQ_6: 89;

          then

          reconsider godo as clockwise_oriented non constant standard special_circular_sequence by JORDAN1I: 25;

          ( len US) in ( dom US) by FINSEQ_5: 6;

          

          then

           A169: (US . ( len US)) = (US /. ( len US)) by PARTFUN1:def 6

          .= Emax by JORDAN1F: 7;

          

           A170: ( east_halfline ( E-max C)) misses ( L~ go)

          proof

            assume ( east_halfline ( E-max C)) meets ( L~ go);

            then

            consider p be object such that

             A171: p in ( east_halfline ( E-max C)) and

             A172: p in ( L~ go) by XBOOLE_0: 3;

            reconsider p as Point of ( TOP-REAL 2) by A171;

            p in ( L~ US) by A57, A172;

            then p in (( east_halfline ( E-max C)) /\ ( L~ ( Cage (C,n)))) by A125, A171, XBOOLE_0:def 4;

            then

             A173: (p `1 ) = Ebo by JORDAN1A: 83, PSCOMP_1: 50;

            then

             A174: p = Emax by A57, A172, JORDAN1J: 46;

            then Emax = Gik by A45, A169, A172, JORDAN1J: 43;

            then (Gik `1 ) = ((G * (( len G),k)) `1 ) by A6, A16, A24, A173, A174, JORDAN1A: 71;

            hence contradiction by A3, A19, A22, JORDAN1G: 7;

          end;

          now

            assume ( east_halfline ( E-max C)) meets ( L~ godo);

            then

             A175: ( east_halfline ( E-max C)) meets (( L~ go) \/ ( L~ pion1)) or ( east_halfline ( E-max C)) meets ( L~ co) by A140, XBOOLE_1: 70;

            per cases by A175, XBOOLE_1: 70;

              suppose ( east_halfline ( E-max C)) meets ( L~ go);

              hence contradiction by A170;

            end;

              suppose ( east_halfline ( E-max C)) meets ( L~ pion1);

              then

              consider p be object such that

               A176: p in ( east_halfline ( E-max C)) and

               A177: p in ( L~ pion1) by XBOOLE_0: 3;

              reconsider p as Point of ( TOP-REAL 2) by A176;

              

               A178: (p `2 ) = (( E-max C) `2 ) by A176, TOPREAL1:def 11;

               A179:

              now

                per cases by A13, A85, A177, XBOOLE_0:def 3;

                  suppose p in poz;

                  hence (p `1 ) <= (Gik `1 ) by A161, TOPREAL1: 3;

                end;

                  suppose p in pio;

                  hence (p `1 ) <= (Gik `1 ) by A149, A161, GOBOARD7: 5;

                end;

              end;

              (i2 + 1) <= ( len G) by A3, NAT_1: 13;

              then ((i2 + 1) - 1) <= (( len G) - 1) by XREAL_1: 9;

              then

               A180: i2 <= (( len G) -' 1) by XREAL_0:def 2;

              (( len G) -' 1) <= ( len G) by NAT_D: 35;

              then (Gik `1 ) <= ((G * ((( len G) -' 1),1)) `1 ) by A6, A18, A16, A24, A14, A180, JORDAN1A: 18;

              then (p `1 ) <= ((G * ((( len G) -' 1),1)) `1 ) by A179, XXREAL_0: 2;

              then (p `1 ) <= ( E-bound C) by A14, JORDAN8: 12;

              then

               A181: (p `1 ) <= (( E-max C) `1 ) by EUCLID: 52;

              (p `1 ) >= (( E-max C) `1 ) by A176, TOPREAL1:def 11;

              then (p `1 ) = (( E-max C) `1 ) by A181, XXREAL_0: 1;

              then p = ( E-max C) by A178, TOPREAL3: 6;

              hence contradiction by A9, A13, A85, A166, A177, XBOOLE_0: 3;

            end;

              suppose ( east_halfline ( E-max C)) meets ( L~ co);

              then

              consider p be object such that

               A182: p in ( east_halfline ( E-max C)) and

               A183: p in ( L~ co) by XBOOLE_0: 3;

              reconsider p as Point of ( TOP-REAL 2) by A182;

              

               A184: p in ( LSeg (co,( Index (p,co)))) by A183, JORDAN3: 9;

              consider t be Nat such that

               A185: t in ( dom LS) and

               A186: (LS . t) = Gij by A32, FINSEQ_2: 10;

              1 <= t by A185, FINSEQ_3: 25;

              then

               A187: 1 < t by A34, A186, XXREAL_0: 1;

              t <= ( len LS) by A185, FINSEQ_3: 25;

              then (( Index (Gij,LS)) + 1) = t by A186, A187, JORDAN3: 12;

              then

               A188: ( len ( L_Cut (LS,Gij))) = (( len LS) - ( Index (Gij,LS))) by A30, A186, JORDAN3: 26;

              ( Index (p,co)) < ( len co) by A183, JORDAN3: 8;

              then ( Index (p,co)) < (( len LS) -' ( Index (Gij,LS))) by A188, XREAL_0:def 2;

              then (( Index (p,co)) + 1) <= (( len LS) -' ( Index (Gij,LS))) by NAT_1: 13;

              then

               A189: ( Index (p,co)) <= ((( len LS) -' ( Index (Gij,LS))) - 1) by XREAL_1: 19;

              

               A190: co = ( mid (LS,(Gij .. LS),( len LS))) by A32, JORDAN1J: 37;

              p in ( L~ LS) by A36, A183;

              then p in (( east_halfline ( E-max C)) /\ ( L~ ( Cage (C,n)))) by A151, A182, XBOOLE_0:def 4;

              then

               A191: (p `1 ) = Ebo by JORDAN1A: 83, PSCOMP_1: 50;

              

               A192: (( Index (Gij,LS)) + 1) = (Gij .. LS) by A34, A32, JORDAN1J: 56;

              ( 0 + ( Index (Gij,LS))) < ( len LS) by A30, JORDAN3: 8;

              then (( len LS) - ( Index (Gij,LS))) > 0 by XREAL_1: 20;

              then ( Index (p,co)) <= ((( len LS) - ( Index (Gij,LS))) - 1) by A189, XREAL_0:def 2;

              then ( Index (p,co)) <= (( len LS) - (Gij .. LS)) by A192;

              then ( Index (p,co)) <= (( len LS) -' (Gij .. LS)) by XREAL_0:def 2;

              then

               A193: ( Index (p,co)) < ((( len LS) -' (Gij .. LS)) + 1) by NAT_1: 13;

              

               A194: 1 <= ( Index (p,co)) by A183, JORDAN3: 8;

              

               A195: (Gij .. LS) <= ( len LS) by A32, FINSEQ_4: 21;

              (Gij .. LS) <> ( len LS) by A31, A32, FINSEQ_4: 19;

              then

               A196: (Gij .. LS) < ( len LS) by A195, XXREAL_0: 1;

              

               A197: (1 + 1) <= ( len LS) by A25, XXREAL_0: 2;

              then

               A198: 2 in ( dom LS) by FINSEQ_3: 25;

              set tt = ((( Index (p,co)) + (Gij .. LS)) -' 1);

              set RC = ( Rotate (( Cage (C,n)),Emax));

              

               A199: ( E-max C) in ( right_cell (RC,1)) by JORDAN1I: 7;

              

               A200: ( GoB RC) = ( GoB ( Cage (C,n))) by REVROT_1: 28

              .= G by JORDAN1H: 44;

              

               A201: ( L~ RC) = ( L~ ( Cage (C,n))) by REVROT_1: 33;

              consider jj2 be Nat such that

               A202: 1 <= jj2 and

               A203: jj2 <= ( width G) and

               A204: Emax = (G * (( len G),jj2)) by JORDAN1D: 25;

              

               A205: ( len G) >= 4 by JORDAN8: 10;

              then ( len G) >= 1 by XXREAL_0: 2;

              then

               A206: [( len G), jj2] in ( Indices G) by A202, A203, MATRIX_0: 30;

              

               A207: ( len RC) = ( len ( Cage (C,n))) by FINSEQ_6: 179;

              LS = (RC -: Wmin) by JORDAN1G: 18;

              then

               A208: ( LSeg (LS,1)) = ( LSeg (RC,1)) by A197, SPPOL_2: 9;

              

               A209: Emax in ( rng ( Cage (C,n))) by SPRECT_2: 46;

              RC is_sequence_on G by A147, REVROT_1: 34;

              then

              consider ii,jj be Nat such that

               A210: [ii, (jj + 1)] in ( Indices G) and

               A211: [ii, jj] in ( Indices G) and

               A212: (RC /. 1) = (G * (ii,(jj + 1))) and

               A213: (RC /. (1 + 1)) = (G * (ii,jj)) by A137, A201, A207, A209, FINSEQ_6: 92, JORDAN1I: 23;

              

               A214: ((jj + 1) + 1) <> jj;

              

               A215: 1 <= jj by A211, MATRIX_0: 32;

              (RC /. 1) = ( E-max ( L~ RC)) by A201, A209, FINSEQ_6: 92;

              then

               A216: ii = ( len G) by A201, A210, A212, A204, A206, GOBOARD1: 5;

              then (ii - 1) >= (4 - 1) by A205, XREAL_1: 9;

              then

               A217: (ii - 1) >= 1 by XXREAL_0: 2;

              then

               A218: 1 <= (ii -' 1) by XREAL_0:def 2;

              

               A219: jj <= ( width G) by A211, MATRIX_0: 32;

              then

               A220: ((G * (( len G),jj)) `1 ) = Ebo by A24, A215, JORDAN1A: 71;

              

               A221: (jj + 1) <= ( width G) by A210, MATRIX_0: 32;

              (ii + 1) <> ii;

              then

               A222: ( right_cell (RC,1)) = ( cell (G,(ii -' 1),jj)) by A137, A207, A200, A210, A211, A212, A213, A214, GOBOARD5:def 6;

              

               A223: ii <= ( len G) by A211, MATRIX_0: 32;

              

               A224: 1 <= ii by A211, MATRIX_0: 32;

              

               A225: ii <= ( len G) by A210, MATRIX_0: 32;

              

               A226: 1 <= (jj + 1) by A210, MATRIX_0: 32;

              then

               A227: Ebo = ((G * (( len G),(jj + 1))) `1 ) by A24, A221, JORDAN1A: 71;

              

               A228: 1 <= ii by A210, MATRIX_0: 32;

              then

               A229: ((ii -' 1) + 1) = ii by XREAL_1: 235;

              then

               A230: (ii -' 1) < ( len G) by A225, NAT_1: 13;

              

              then

               A231: ((G * ((ii -' 1),(jj + 1))) `2 ) = ((G * (1,(jj + 1))) `2 ) by A226, A221, A218, GOBOARD5: 1

              .= ((G * (ii,(jj + 1))) `2 ) by A228, A225, A226, A221, GOBOARD5: 1;

              

               A232: (( E-max C) `2 ) = (p `2 ) by A182, TOPREAL1:def 11;

              then

               A233: (p `2 ) <= ((G * ((ii -' 1),(jj + 1))) `2 ) by A199, A225, A221, A215, A222, A229, A217, JORDAN9: 17;

              

               A234: ((G * ((ii -' 1),jj)) `2 ) = ((G * (1,jj)) `2 ) by A215, A219, A218, A230, GOBOARD5: 1

              .= ((G * (ii,jj)) `2 ) by A224, A223, A215, A219, GOBOARD5: 1;

              ((G * ((ii -' 1),jj)) `2 ) <= (p `2 ) by A232, A199, A225, A221, A215, A222, A229, A217, JORDAN9: 17;

              then p in ( LSeg ((RC /. 1),(RC /. (1 + 1)))) by A191, A212, A213, A216, A233, A234, A231, A220, A227, GOBOARD7: 7;

              then

               A235: p in ( LSeg (LS,1)) by A137, A208, A207, TOPREAL1:def 3;

              1 <= (Gij .. LS) by A32, FINSEQ_4: 21;

              then

               A236: ( LSeg (( mid (LS,(Gij .. LS),( len LS))),( Index (p,co)))) = ( LSeg (LS,((( Index (p,co)) + (Gij .. LS)) -' 1))) by A196, A194, A193, JORDAN4: 19;

              1 <= ( Index (Gij,LS)) by A30, JORDAN3: 8;

              then

               A237: (1 + 1) <= (Gij .. LS) by A192, XREAL_1: 7;

              then (( Index (p,co)) + (Gij .. LS)) >= ((1 + 1) + 1) by A194, XREAL_1: 7;

              then ((( Index (p,co)) + (Gij .. LS)) - 1) >= (((1 + 1) + 1) - 1) by XREAL_1: 9;

              then

               A238: tt >= (1 + 1) by XREAL_0:def 2;

              now

                per cases by A238, XXREAL_0: 1;

                  suppose tt > (1 + 1);

                  then ( LSeg (LS,1)) misses ( LSeg (LS,tt)) by TOPREAL1:def 7;

                  hence contradiction by A235, A184, A190, A236, XBOOLE_0: 3;

                end;

                  suppose

                   A239: tt = (1 + 1);

                  then (1 + 1) = ((( Index (p,co)) + (Gij .. LS)) - 1) by XREAL_0:def 2;

                  then ((1 + 1) + 1) = (( Index (p,co)) + (Gij .. LS));

                  then

                   A240: (Gij .. LS) = 2 by A194, A237, JORDAN1E: 6;

                  (( LSeg (LS,1)) /\ ( LSeg (LS,tt))) = {(LS /. 2)} by A25, A239, TOPREAL1:def 6;

                  then p in {(LS /. 2)} by A235, A184, A190, A236, XBOOLE_0:def 4;

                  then

                   A241: p = (LS /. 2) by TARSKI:def 1;

                  then

                   A242: p in ( rng LS) by A198, PARTFUN2: 2;

                  (p .. LS) = 2 by A198, A241, FINSEQ_5: 41;

                  then p = Gij by A32, A240, A242, FINSEQ_5: 9;

                  then (Gij `1 ) = Ebo by A241, JORDAN1G: 32;

                  then (Gij `1 ) = ((G * (( len G),j)) `1 ) by A4, A10, A24, JORDAN1A: 71;

                  hence contradiction by A2, A3, A23, A15, JORDAN1G: 7;

                end;

              end;

              hence contradiction;

            end;

          end;

          then ( east_halfline ( E-max C)) c= (( L~ godo) ` ) by SUBSET_1: 23;

          then

          consider W be Subset of ( TOP-REAL 2) such that

           A243: W is_a_component_of (( L~ godo) ` ) and

           A244: ( east_halfline ( E-max C)) c= W by GOBOARD9: 3;

           not W is bounded by A244, JORDAN2C: 121, RLTOPSP1: 42;

          then W is_outside_component_of ( L~ godo) by A243, JORDAN2C:def 3;

          then W c= ( UBD ( L~ godo)) by JORDAN2C: 23;

          then

           A245: ( east_halfline ( E-max C)) c= ( UBD ( L~ godo)) by A244;

          ( E-max C) in ( east_halfline ( E-max C)) by TOPREAL1: 38;

          then ( E-max C) in ( UBD ( L~ godo)) by A245;

          then ( E-max C) in ( LeftComp godo) by GOBRD14: 36;

          then LA meets ( L~ godo) by A109, A143, A166, A141, A158, JORDAN1J: 36;

          then

           A246: LA meets (( L~ go) \/ ( L~ pion1)) or LA meets ( L~ co) by A140, XBOOLE_1: 70;

          now

            per cases by A246, XBOOLE_1: 70;

              suppose LA meets ( L~ go);

              then LA meets ( L~ ( Cage (C,n))) by A57, A125, XBOOLE_1: 1, XBOOLE_1: 63;

              hence contradiction by A164, JORDAN10: 5, XBOOLE_1: 63;

            end;

              suppose LA meets ( L~ pion1);

              hence contradiction by A9, A13, A85;

            end;

              suppose LA meets ( L~ co);

              then LA meets ( L~ ( Cage (C,n))) by A36, A151, XBOOLE_1: 1, XBOOLE_1: 63;

              hence contradiction by A164, JORDAN10: 5, XBOOLE_1: 63;

            end;

          end;

          hence contradiction;

        end;

          suppose (Gij `1 ) = (Gik `1 );

          then

           A247: i1 = i2 by A23, A19, JORDAN1G: 7;

          then poz = {Gi1k} by RLTOPSP1: 70;

          then poz c= pio by A40, ZFMISC_1: 31;

          then (pio \/ poz) = pio by XBOOLE_1: 12;

          hence contradiction by A1, A3, A4, A5, A6, A7, A8, A9, A247, JORDAN1J: 58;

        end;

          suppose (Gij `2 ) = (Gik `2 );

          then

           A248: j = k by A23, A19, JORDAN1G: 6;

          then pio = {Gi1k} by RLTOPSP1: 70;

          then pio c= poz by A39, ZFMISC_1: 31;

          then (pio \/ poz) = poz by XBOOLE_1: 12;

          hence contradiction by A1, A2, A3, A4, A6, A7, A8, A9, A248, Th28;

        end;

      end;

      hence contradiction;

    end;

    theorem :: JORDAN15:46

    

     Th46: for C be Simple_closed_curve holds for i1,i2,j,k be Nat st 1 < i2 & i2 <= i1 & i1 < ( len ( Gauge (C,n))) & 1 <= j & j <= k & k <= ( width ( Gauge (C,n))) & ((( LSeg ((( Gauge (C,n)) * (i1,j)),(( Gauge (C,n)) * (i1,k)))) \/ ( LSeg ((( Gauge (C,n)) * (i1,k)),(( Gauge (C,n)) * (i2,k))))) /\ ( L~ ( Upper_Seq (C,n)))) = {(( Gauge (C,n)) * (i2,k))} & ((( LSeg ((( Gauge (C,n)) * (i1,j)),(( Gauge (C,n)) * (i1,k)))) \/ ( LSeg ((( Gauge (C,n)) * (i1,k)),(( Gauge (C,n)) * (i2,k))))) /\ ( L~ ( Lower_Seq (C,n)))) = {(( Gauge (C,n)) * (i1,j))} holds (( LSeg ((( Gauge (C,n)) * (i1,j)),(( Gauge (C,n)) * (i1,k)))) \/ ( LSeg ((( Gauge (C,n)) * (i1,k)),(( Gauge (C,n)) * (i2,k))))) meets ( Upper_Arc C)

    proof

      let C be Simple_closed_curve;

      let i1,i2,j,k be Nat;

      set G = ( Gauge (C,n));

      set pio = ( LSeg ((G * (i1,j)),(G * (i1,k))));

      set poz = ( LSeg ((G * (i1,k)),(G * (i2,k))));

      set US = ( Upper_Seq (C,n));

      set LS = ( Lower_Seq (C,n));

      assume that

       A1: 1 < i2 and

       A2: i2 <= i1 and

       A3: i1 < ( len G) and

       A4: 1 <= j and

       A5: j <= k and

       A6: k <= ( width G) and

       A7: ((pio \/ poz) /\ ( L~ US)) = {(G * (i2,k))} and

       A8: ((pio \/ poz) /\ ( L~ LS)) = {(G * (i1,j))} and

       A9: (pio \/ poz) misses ( Upper_Arc C);

      set Gi1k = (G * (i1,k));

      set Gik = (G * (i2,k));

      

       A10: 1 <= k by A4, A5, XXREAL_0: 2;

      

       A11: i2 < ( len G) by A2, A3, XXREAL_0: 2;

      then

       A12: [i2, k] in ( Indices G) by A1, A6, A10, MATRIX_0: 30;

      set Wmin = ( W-min ( L~ ( Cage (C,n))));

      set Wbo = ( W-bound ( L~ ( Cage (C,n))));

      

       A13: ( len G) = ( width G) by JORDAN8:def 1;

      set go = ( R_Cut (US,Gik));

      

       A14: ( len US) >= 3 by JORDAN1E: 15;

      then ( len US) >= 1 by XXREAL_0: 2;

      then 1 in ( dom US) by FINSEQ_3: 25;

      

      then

       A15: (US . 1) = (US /. 1) by PARTFUN1:def 6

      .= Wmin by JORDAN1F: 5;

      set Gij = (G * (i1,j));

      set co = ( L_Cut (LS,Gij));

      Gij in {Gij} by TARSKI:def 1;

      then

       A16: Gij in ( L~ LS) by A8, XBOOLE_0:def 4;

      

       A17: 1 < i1 by A1, A2, XXREAL_0: 2;

      

      then

       A18: (Gi1k `2 ) = ((G * (1,k)) `2 ) by A3, A6, A10, GOBOARD5: 1

      .= (Gik `2 ) by A1, A6, A11, A10, GOBOARD5: 1;

      

       A19: j <= ( width G) by A5, A6, XXREAL_0: 2;

      then

       A20: [i1, j] in ( Indices G) by A3, A4, A17, MATRIX_0: 30;

      ( len G) >= 4 by JORDAN8: 10;

      then

       A21: ( len G) >= 1 by XXREAL_0: 2;

      then

       A22: [( len G), j] in ( Indices G) by A4, A19, MATRIX_0: 30;

      

       A23: [1, k] in ( Indices G) by A6, A10, A21, MATRIX_0: 30;

       A24:

      now

        assume (Gik `1 ) = Wbo;

        then ((G * (1,k)) `1 ) = ((G * (i2,k)) `1 ) by A6, A10, A13, JORDAN1A: 73;

        hence contradiction by A1, A12, A23, JORDAN1G: 7;

      end;

      

       A25: [i1, j] in ( Indices G) by A3, A4, A17, A19, MATRIX_0: 30;

      set pion = <*Gik, Gi1k, Gij*>;

      

       A26: Gi1k in poz by RLTOPSP1: 68;

      set UA = ( Upper_Arc C);

      

       A27: Gi1k in pio by RLTOPSP1: 68;

      

       A28: [i1, k] in ( Indices G) by A3, A6, A17, A10, MATRIX_0: 30;

       A29:

      now

        let n be Nat;

        assume n in ( dom pion);

        then n in {1, 2, 3} by FINSEQ_1: 89, FINSEQ_3: 1;

        then n = 1 or n = 2 or n = 3 by ENUMSET1:def 1;

        hence ex i,j be Nat st [i, j] in ( Indices G) & (pion /. n) = (G * (i,j)) by A25, A12, A28, FINSEQ_4: 18;

      end;

      Gik in {Gik} by TARSKI:def 1;

      then

       A30: Gik in ( L~ US) by A7, XBOOLE_0:def 4;

      set Emax = ( E-max ( L~ ( Cage (C,n))));

      

       A31: ( len LS) >= (1 + 2) by JORDAN1E: 15;

      then

       A32: ( len LS) >= 1 by XXREAL_0: 2;

      then

       A33: 1 in ( dom LS) by FINSEQ_3: 25;

      

      then

       A34: (LS . 1) = (LS /. 1) by PARTFUN1:def 6

      .= Emax by JORDAN1F: 6;

      ( len LS) in ( dom LS) by A32, FINSEQ_3: 25;

      

      then

       A35: (LS . ( len LS)) = (LS /. ( len LS)) by PARTFUN1:def 6

      .= Wmin by JORDAN1F: 8;

      set Ebo = ( E-bound ( L~ ( Cage (C,n))));

      

       A36: ( L~ <*Gik, Gi1k, Gij*>) = (poz \/ pio) by TOPREAL3: 16;

      (Wmin `1 ) = Wbo by EUCLID: 52

      .= ((G * (1,k)) `1 ) by A6, A10, A13, JORDAN1A: 73;

      then

       A37: Gik <> (US . 1) by A1, A12, A15, A23, JORDAN1G: 7;

      then

      reconsider go as being_S-Seq FinSequence of ( TOP-REAL 2) by A30, JORDAN3: 35;

      

       A38: Gik in ( rng US) by A1, A6, A11, A30, A10, JORDAN1G: 4, JORDAN1J: 40;

      then

       A39: go is_sequence_on G by JORDAN1G: 4, JORDAN1J: 38;

      (Gi1k `1 ) = ((G * (i1,1)) `1 ) by A3, A6, A17, A10, GOBOARD5: 2

      .= (Gij `1 ) by A3, A4, A17, A19, GOBOARD5: 2;

      then

       A40: Gi1k = |[(Gij `1 ), (Gik `2 )]| by A18, EUCLID: 53;

      

       A41: [( len G), k] in ( Indices G) by A6, A10, A21, MATRIX_0: 30;

      

       A42: ( len go) >= (1 + 1) by TOPREAL1:def 8;

      (Wmin `1 ) = Wbo by EUCLID: 52

      .= ((G * (1,k)) `1 ) by A6, A10, A13, JORDAN1A: 73;

      then

       A43: Gij <> (LS . ( len LS)) by A1, A2, A23, A35, A20, JORDAN1G: 7;

      then

      reconsider co as being_S-Seq FinSequence of ( TOP-REAL 2) by A16, JORDAN3: 34;

      

       A44: Gij in ( rng LS) by A3, A4, A17, A16, A19, JORDAN1G: 5, JORDAN1J: 40;

      then

       A45: co is_sequence_on G by JORDAN1G: 5, JORDAN1J: 39;

      (Emax `1 ) = Ebo by EUCLID: 52

      .= ((G * (( len G),k)) `1 ) by A6, A10, A13, JORDAN1A: 71;

      then

       A46: Gij <> (LS . 1) by A3, A20, A41, A34, JORDAN1G: 7;

      

       A47: ( len co) >= (1 + 1) by TOPREAL1:def 8;

      then

      reconsider co as non constant s.c.c. being_S-Seq FinSequence of ( TOP-REAL 2) by A45, JGRAPH_1: 12, JORDAN8: 5;

      

       A48: ( L~ co) c= ( L~ LS) by A16, JORDAN3: 42;

      ( len co) >= 1 by A47, XXREAL_0: 2;

      then 1 in ( dom co) by FINSEQ_3: 25;

      

      then

       A49: (co /. 1) = (co . 1) by PARTFUN1:def 6

      .= Gij by A16, JORDAN3: 23;

      then

       A50: ( LSeg (co,1)) = ( LSeg (Gij,(co /. (1 + 1)))) by A47, TOPREAL1:def 3;

      

       A51: {Gij} c= (( LSeg (co,1)) /\ ( L~ <*Gik, Gi1k, Gij*>))

      proof

        let x be object;

        assume x in {Gij};

        then

         A52: x = Gij by TARSKI:def 1;

        Gij in ( LSeg (Gi1k,Gij)) by RLTOPSP1: 68;

        then Gij in (( LSeg (Gik,Gi1k)) \/ ( LSeg (Gi1k,Gij))) by XBOOLE_0:def 3;

        then

         A53: Gij in ( L~ <*Gik, Gi1k, Gij*>) by SPRECT_1: 8;

        Gij in ( LSeg (co,1)) by A50, RLTOPSP1: 68;

        hence thesis by A52, A53, XBOOLE_0:def 4;

      end;

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

      then ( LSeg (co,1)) c= ( L~ LS) by A48;

      then (( LSeg (co,1)) /\ ( L~ <*Gik, Gi1k, Gij*>)) c= {Gij} by A8, A36, XBOOLE_1: 26;

      then

       A54: (( L~ <*Gik, Gi1k, Gij*>) /\ ( LSeg (co,1))) = {Gij} by A51;

      

       A55: ( rng co) c= ( L~ co) by A47, SPPOL_2: 18;

      reconsider go as non constant s.c.c. being_S-Seq FinSequence of ( TOP-REAL 2) by A42, A39, JGRAPH_1: 12, JORDAN8: 5;

      

       A56: ( L~ go) c= ( L~ US) by A30, JORDAN3: 41;

      

       A57: ( len go) > 1 by A42, NAT_1: 13;

      then

       A58: ( len go) in ( dom go) by FINSEQ_3: 25;

      

      then

       A59: (go /. ( len go)) = (go . ( len go)) by PARTFUN1:def 6

      .= Gik by A30, JORDAN3: 24;

      reconsider m = (( len go) - 1) as Nat by A58, FINSEQ_3: 26;

      

       A60: (m + 1) = ( len go);

      then

       A61: (( len go) -' 1) = m by NAT_D: 34;

      m >= 1 by A42, XREAL_1: 19;

      then

       A62: ( LSeg (go,m)) = ( LSeg ((go /. m),Gik)) by A59, A60, TOPREAL1:def 3;

      

       A63: {Gik} c= (( LSeg (go,m)) /\ ( L~ <*Gik, Gi1k, Gij*>))

      proof

        let x be object;

        assume x in {Gik};

        then

         A64: x = Gik by TARSKI:def 1;

        Gik in ( LSeg (Gik,Gi1k)) by RLTOPSP1: 68;

        then Gik in (( LSeg (Gik,Gi1k)) \/ ( LSeg (Gi1k,Gij))) by XBOOLE_0:def 3;

        then

         A65: Gik in ( L~ <*Gik, Gi1k, Gij*>) by SPRECT_1: 8;

        Gik in ( LSeg (go,m)) by A62, RLTOPSP1: 68;

        hence thesis by A64, A65, XBOOLE_0:def 4;

      end;

      ( LSeg (go,m)) c= ( L~ go) by TOPREAL3: 19;

      then ( LSeg (go,m)) c= ( L~ US) by A56;

      then (( LSeg (go,m)) /\ ( L~ <*Gik, Gi1k, Gij*>)) c= {Gik} by A7, A36, XBOOLE_1: 26;

      then

       A66: (( LSeg (go,m)) /\ ( L~ <*Gik, Gi1k, Gij*>)) = {Gik} by A63;

      

       A67: (go /. 1) = (US /. 1) by A30, SPRECT_3: 22

      .= Wmin by JORDAN1F: 5;

      then

       A68: Wmin in ( rng go) by FINSEQ_6: 42;

      

       A69: (LS . 1) = (LS /. 1) by A33, PARTFUN1:def 6

      .= Emax by JORDAN1F: 6;

      

       A70: (( L~ go) /\ ( L~ co)) c= {(go /. 1)}

      proof

        let x be object;

        assume

         A71: x in (( L~ go) /\ ( L~ co));

        then

         A72: x in ( L~ co) by XBOOLE_0:def 4;

         A73:

        now

          assume x = Emax;

          then

           A74: Emax = Gij by A16, A69, A72, JORDAN1E: 7;

          ((G * (( len G),j)) `1 ) = Ebo by A4, A19, A13, JORDAN1A: 71;

          then (Emax `1 ) <> Ebo by A3, A25, A22, A74, JORDAN1G: 7;

          hence contradiction by EUCLID: 52;

        end;

        x in ( L~ go) by A71, XBOOLE_0:def 4;

        then x in (( L~ US) /\ ( L~ LS)) by A56, A48, A72, XBOOLE_0:def 4;

        then x in {Wmin, Emax} by JORDAN1E: 16;

        then x = Wmin or x = Emax by TARSKI:def 2;

        hence thesis by A67, A73, TARSKI:def 1;

      end;

      set W2 = (go /. 2);

      

       A75: 2 in ( dom go) by A42, FINSEQ_3: 25;

      go = ( mid (US,1,(Gik .. US))) by A38, JORDAN1G: 49

      .= (US | (Gik .. US)) by A38, FINSEQ_4: 21, FINSEQ_6: 116;

      then

       A76: W2 = (US /. 2) by A75, FINSEQ_4: 70;

      

       A77: ( rng go) c= ( L~ go) by A42, SPPOL_2: 18;

      

       A78: (go /. 1) = (LS /. ( len LS)) by A67, JORDAN1F: 8

      .= (co /. ( len co)) by A16, JORDAN1J: 35;

       {(go /. 1)} c= (( L~ go) /\ ( L~ co))

      proof

        let x be object;

        assume x in {(go /. 1)};

        then

         A79: x = (go /. 1) by TARSKI:def 1;

        then

         A80: x in ( rng go) by FINSEQ_6: 42;

        x in ( rng co) by A78, A79, FINSEQ_6: 168;

        hence thesis by A77, A55, A80, XBOOLE_0:def 4;

      end;

      then

       A81: (( L~ go) /\ ( L~ co)) = {(go /. 1)} by A70;

      now

        per cases ;

          suppose (Gij `1 ) <> (Gik `1 ) & (Gij `2 ) <> (Gik `2 );

          then pion is being_S-Seq by A40, TOPREAL3: 35;

          then

          consider pion1 be FinSequence of ( TOP-REAL 2) such that

           A82: pion1 is_sequence_on G and

           A83: pion1 is being_S-Seq and

           A84: ( L~ pion) = ( L~ pion1) and

           A85: (pion /. 1) = (pion1 /. 1) and

           A86: (pion /. ( len pion)) = (pion1 /. ( len pion1)) and

           A87: ( len pion) <= ( len pion1) by A29, GOBOARD3: 2;

          reconsider pion1 as being_S-Seq FinSequence of ( TOP-REAL 2) by A83;

          

           A88: ((go ^' pion1) /. ( len (go ^' pion1))) = (pion /. ( len pion)) by A86, FINSEQ_6: 156

          .= (pion /. 3) by FINSEQ_1: 45

          .= (co /. 1) by A49, FINSEQ_4: 18;

          

           A89: (go /. ( len go)) = (pion1 /. 1) by A59, A85, FINSEQ_4: 18;

          

           A90: (( L~ go) /\ ( L~ pion1)) c= {(pion1 /. 1)}

          proof

            let x be object;

            assume

             A91: x in (( L~ go) /\ ( L~ pion1));

            then

             A92: x in ( L~ pion1) by XBOOLE_0:def 4;

            x in ( L~ go) by A91, XBOOLE_0:def 4;

            hence thesis by A7, A36, A59, A56, A84, A89, A92, XBOOLE_0:def 4;

          end;

          ( len pion1) >= (2 + 1) by A87, FINSEQ_1: 45;

          then

           A93: ( len pion1) > (1 + 1) by NAT_1: 13;

          then

           A94: ( rng pion1) c= ( L~ pion1) by SPPOL_2: 18;

           {(pion1 /. 1)} c= (( L~ go) /\ ( L~ pion1))

          proof

            let x be object;

            assume x in {(pion1 /. 1)};

            then

             A95: x = (pion1 /. 1) by TARSKI:def 1;

            then

             A96: x in ( rng pion1) by FINSEQ_6: 42;

            x in ( rng go) by A89, A95, FINSEQ_6: 168;

            hence thesis by A77, A94, A96, XBOOLE_0:def 4;

          end;

          then

           A97: (( L~ go) /\ ( L~ pion1)) = {(pion1 /. 1)} by A90;

          then

           A98: (go ^' pion1) is s.n.c. by A89, JORDAN1J: 54;

          

           A99: {Gik} c= (( LSeg (go,m)) /\ ( LSeg (pion1,1)))

          proof

            let x be object;

            assume x in {Gik};

            then

             A100: x = Gik by TARSKI:def 1;

            

             A101: Gik in ( LSeg (go,m)) by A62, RLTOPSP1: 68;

            Gik in ( LSeg (pion1,1)) by A59, A89, A93, TOPREAL1: 21;

            hence thesis by A100, A101, XBOOLE_0:def 4;

          end;

          ( LSeg (pion1,1)) c= ( L~ pion) by A84, TOPREAL3: 19;

          then (( LSeg (go,(( len go) -' 1))) /\ ( LSeg (pion1,1))) c= {Gik} by A61, A66, XBOOLE_1: 27;

          then (( LSeg (go,(( len go) -' 1))) /\ ( LSeg (pion1,1))) = {(go /. ( len go))} by A59, A61, A99;

          then

           A102: (go ^' pion1) is unfolded by A89, TOPREAL8: 34;

          ( len (go ^' pion1)) >= ( len go) by TOPREAL8: 7;

          then

           A103: ( len (go ^' pion1)) >= (1 + 1) by A42, XXREAL_0: 2;

          then

           A104: ( len (go ^' pion1)) > (1 + 0 ) by NAT_1: 13;

          

           A105: (pion /. ( len pion)) = (pion /. 3) by FINSEQ_1: 45

          .= (co /. 1) by A49, FINSEQ_4: 18;

          

           A106: {(pion1 /. ( len pion1))} c= (( L~ co) /\ ( L~ pion1))

          proof

            let x be object;

            assume x in {(pion1 /. ( len pion1))};

            then

             A107: x = (pion1 /. ( len pion1)) by TARSKI:def 1;

            then

             A108: x in ( rng pion1) by FINSEQ_6: 168;

            x in ( rng co) by A86, A105, A107, FINSEQ_6: 42;

            hence thesis by A55, A94, A108, XBOOLE_0:def 4;

          end;

          (( L~ co) /\ ( L~ pion1)) c= {(pion1 /. ( len pion1))}

          proof

            let x be object;

            assume

             A109: x in (( L~ co) /\ ( L~ pion1));

            then

             A110: x in ( L~ pion1) by XBOOLE_0:def 4;

            x in ( L~ co) by A109, XBOOLE_0:def 4;

            hence thesis by A8, A36, A49, A48, A84, A86, A105, A110, XBOOLE_0:def 4;

          end;

          then

           A111: (( L~ co) /\ ( L~ pion1)) = {(pion1 /. ( len pion1))} by A106;

          

           A112: (( L~ (go ^' pion1)) /\ ( L~ co)) = ((( L~ go) \/ ( L~ pion1)) /\ ( L~ co)) by A89, TOPREAL8: 35

          .= ( {(go /. 1)} \/ {(co /. 1)}) by A81, A86, A105, A111, XBOOLE_1: 23

          .= ( {((go ^' pion1) /. 1)} \/ {(co /. 1)}) by FINSEQ_6: 155

          .= {((go ^' pion1) /. 1), (co /. 1)} by ENUMSET1: 1;

          

           A113: UA is_an_arc_of (( W-min C),( E-max C)) by JORDAN6:def 8;

          then

           A114: UA is connected by JORDAN6: 10;

          set godo = ((go ^' pion1) ^' co);

          

           A115: (co /. ( len co)) = ((go ^' pion1) /. 1) by A78, FINSEQ_6: 155;

          

           A116: (go ^' pion1) is_sequence_on G by A39, A82, A89, TOPREAL8: 12;

          then

           A117: godo is_sequence_on G by A45, A88, TOPREAL8: 12;

          

           A118: (( len pion1) - 1) >= 1 by A93, XREAL_1: 19;

          then

           A119: (( len pion1) -' 1) = (( len pion1) - 1) by XREAL_0:def 2;

          

           A120: {Gij} c= (( LSeg (pion1,(( len pion1) -' 1))) /\ ( LSeg (co,1)))

          proof

            let x be object;

            assume x in {Gij};

            then

             A121: x = Gij by TARSKI:def 1;

            (pion1 /. ((( len pion1) -' 1) + 1)) = (pion /. 3) by A86, A119, FINSEQ_1: 45

            .= Gij by FINSEQ_4: 18;

            then

             A122: Gij in ( LSeg (pion1,(( len pion1) -' 1))) by A118, A119, TOPREAL1: 21;

            Gij in ( LSeg (co,1)) by A50, RLTOPSP1: 68;

            hence thesis by A121, A122, XBOOLE_0:def 4;

          end;

          ( LSeg (pion1,(( len pion1) -' 1))) c= ( L~ pion) by A84, TOPREAL3: 19;

          then (( LSeg (pion1,(( len pion1) -' 1))) /\ ( LSeg (co,1))) c= {Gij} by A54, XBOOLE_1: 27;

          then

           A123: (( LSeg (pion1,(( len pion1) -' 1))) /\ ( LSeg (co,1))) = {Gij} by A120;

          ((( len pion1) - 1) + 1) <= ( len pion1);

          then

           A124: (( len pion1) -' 1) < ( len pion1) by A119, NAT_1: 13;

          ( len pion1) >= (2 + 1) by A87, FINSEQ_1: 45;

          then

           A125: (( len pion1) - 2) >= 0 by XREAL_1: 19;

          

          then ((( len pion1) -' 2) + 1) = ((( len pion1) - 2) + 1) by XREAL_0:def 2

          .= (( len pion1) -' 1) by A118, XREAL_0:def 2;

          then

           A126: (( LSeg ((go ^' pion1),(( len go) + (( len pion1) -' 2)))) /\ ( LSeg (co,1))) = {((go ^' pion1) /. ( len (go ^' pion1)))} by A49, A89, A88, A124, A123, TOPREAL8: 31;

          (( rng go) /\ ( rng pion1)) c= {(pion1 /. 1)} by A77, A94, A97, XBOOLE_1: 27;

          then

           A127: (go ^' pion1) is one-to-one by JORDAN1J: 55;

          ((( len (go ^' pion1)) + 1) - 1) = ((( len go) + ( len pion1)) - 1) by FINSEQ_6: 139;

          

          then (( len (go ^' pion1)) - 1) = (( len go) + (( len pion1) - 2))

          .= (( len go) + (( len pion1) -' 2)) by A125, XREAL_0:def 2;

          then

           A128: (( len (go ^' pion1)) -' 1) = (( len go) + (( len pion1) -' 2)) by XREAL_0:def 2;

          

           A129: ( L~ ( Cage (C,n))) = (( L~ US) \/ ( L~ LS)) by JORDAN1E: 13;

          then

           A130: ( L~ US) c= ( L~ ( Cage (C,n))) by XBOOLE_1: 7;

          then

           A131: ( L~ go) c= ( L~ ( Cage (C,n))) by A56;

          

           A132: ( len godo) >= ( len (go ^' pion1)) by TOPREAL8: 7;

          then

           A133: (1 + 1) <= ( len godo) by A103, XXREAL_0: 2;

          (go ^' pion1) is non trivial by A103, NAT_D: 60;

          then

          reconsider godo as non constant standard special_circular_sequence by A133, A88, A117, A102, A128, A126, A98, A127, A112, A115, JORDAN8: 4, JORDAN8: 5, TOPREAL8: 11, TOPREAL8: 33, TOPREAL8: 34;

          

           A134: ( L~ godo) = (( L~ (go ^' pion1)) \/ ( L~ co)) by A88, TOPREAL8: 35

          .= ((( L~ go) \/ ( L~ pion1)) \/ ( L~ co)) by A89, TOPREAL8: 35;

           A135:

          now

            assume

             A136: (Gik .. US) <= 1;

            (Gik .. US) >= 1 by A38, FINSEQ_4: 21;

            then (Gik .. US) = 1 by A136, XXREAL_0: 1;

            then Gik = (US /. 1) by A38, FINSEQ_5: 38;

            hence contradiction by A15, A37, JORDAN1F: 5;

          end;

          

           A137: US is_sequence_on G by JORDAN1G: 4;

          

           A138: (Gik `1 ) <= (Gi1k `1 ) by A1, A2, A3, A6, A10, JORDAN1A: 18;

          then

           A139: ( W-bound poz) = (Gik `1 ) by SPRECT_1: 54;

          

           A140: (Gi1k `1 ) = ((G * (i1,1)) `1 ) by A3, A6, A17, A10, GOBOARD5: 2

          .= (Gij `1 ) by A3, A4, A17, A19, GOBOARD5: 2;

          then

           A141: ( W-bound pio) = (Gij `1 ) by SPRECT_1: 54;

          ( W-bound (poz \/ pio)) = ( min (( W-bound poz),( W-bound pio))) by SPRECT_1: 47

          .= (Gik `1 ) by A140, A138, A139, A141, XXREAL_0:def 9;

          then

           A142: ( W-bound ( L~ pion1)) = (Gik `1 ) by A84, TOPREAL3: 16;

          

           A143: UA c= C by JORDAN6: 61;

          (Gik `1 ) >= Wbo by A30, A130, PSCOMP_1: 24;

          then

           A144: (Gik `1 ) > Wbo by A24, XXREAL_0: 1;

          

           A145: ( len US) >= 2 by A14, XXREAL_0: 2;

          

           A146: (( L~ go) \/ ( L~ co)) is compact by COMPTS_1: 10;

          

           A147: ( L~ LS) c= ( L~ ( Cage (C,n))) by A129, XBOOLE_1: 7;

          then

           A148: ( L~ co) c= ( L~ ( Cage (C,n))) by A48;

          

           A149: (( right_cell (godo,1,G)) \ ( L~ godo)) c= ( RightComp godo) by A133, A117, JORDAN9: 27;

          2 in ( dom godo) by A133, FINSEQ_3: 25;

          then

           A150: (godo /. 2) in ( rng godo) by PARTFUN2: 2;

          

           A151: ( rng godo) c= ( L~ godo) by A103, A132, SPPOL_2: 18, XXREAL_0: 2;

          

           A152: (godo /. 1) = ((go ^' pion1) /. 1) by FINSEQ_6: 155

          .= Wmin by A67, FINSEQ_6: 155;

          

           A153: ( W-min C) in UA by A113, TOPREAL1: 1;

          

           A154: ( W-min C) in C by SPRECT_1: 13;

           A155:

          now

            assume ( W-min C) in ( L~ godo);

            then

             A156: ( W-min C) in (( L~ go) \/ ( L~ pion1)) or ( W-min C) in ( L~ co) by A134, XBOOLE_0:def 3;

            per cases by A156, XBOOLE_0:def 3;

              suppose ( W-min C) in ( L~ go);

              then C meets ( L~ ( Cage (C,n))) by A131, A154, XBOOLE_0: 3;

              hence contradiction by JORDAN10: 5;

            end;

              suppose ( W-min C) in ( L~ pion1);

              hence contradiction by A9, A36, A84, A153, XBOOLE_0: 3;

            end;

              suppose ( W-min C) in ( L~ co);

              then C meets ( L~ ( Cage (C,n))) by A148, A154, XBOOLE_0: 3;

              hence contradiction by JORDAN10: 5;

            end;

          end;

          

           A157: (Wmin `1 ) = Wbo by EUCLID: 52;

          set ff = ( Rotate (( Cage (C,n)),Wmin));

          

           A158: (1 + 1) <= ( len ( Cage (C,n))) by GOBOARD7: 34, XXREAL_0: 2;

          Wmin in ( rng ( Cage (C,n))) by SPRECT_2: 43;

          then

           A159: (ff /. 1) = Wmin by FINSEQ_6: 92;

          

           A160: ( L~ ff) = ( L~ ( Cage (C,n))) by REVROT_1: 33;

          then (( W-max ( L~ ff)) .. ff) > 1 by A159, SPRECT_5: 22;

          then (( N-min ( L~ ff)) .. ff) > 1 by A159, A160, SPRECT_5: 23, XXREAL_0: 2;

          then (( N-max ( L~ ff)) .. ff) > 1 by A159, A160, SPRECT_5: 24, XXREAL_0: 2;

          then

           A161: (Emax .. ff) > 1 by A159, A160, SPRECT_5: 25, XXREAL_0: 2;

          

           A162: ( Cage (C,n)) is_sequence_on G by JORDAN9:def 1;

          then

           A163: ff is_sequence_on G by REVROT_1: 34;

          (1 + 1) <= ( len ( Rotate (( Cage (C,n)),Wmin))) by GOBOARD7: 34, XXREAL_0: 2;

          

          then ( right_cell (( Rotate (( Cage (C,n)),Wmin)),1)) = ( right_cell (ff,1,( GoB ff))) by JORDAN1H: 23

          .= ( right_cell (ff,1,( GoB ( Cage (C,n))))) by REVROT_1: 28

          .= ( right_cell (ff,1,G)) by JORDAN1H: 44

          .= ( right_cell ((ff -: Emax),1,G)) by A161, A163, JORDAN1J: 53

          .= ( right_cell (US,1,G)) by JORDAN1E:def 1

          .= ( right_cell (( R_Cut (US,Gik)),1,G)) by A38, A137, A135, JORDAN1J: 52

          .= ( right_cell ((go ^' pion1),1,G)) by A57, A116, JORDAN1J: 51

          .= ( right_cell (godo,1,G)) by A104, A117, JORDAN1J: 51;

          then ( W-min C) in ( right_cell (godo,1,G)) by JORDAN1I: 6;

          then

           A164: ( W-min C) in (( right_cell (godo,1,G)) \ ( L~ godo)) by A155, XBOOLE_0:def 5;

          

           A165: ( E-max C) in UA by A113, TOPREAL1: 1;

          Wmin in (( L~ go) \/ ( L~ co)) by A77, A68, XBOOLE_0:def 3;

          then

           A166: ( W-min (( L~ go) \/ ( L~ co))) = Wmin by A131, A148, A146, JORDAN1J: 21, XBOOLE_1: 8;

          (( W-min (( L~ go) \/ ( L~ co))) `1 ) = ( W-bound (( L~ go) \/ ( L~ co))) by EUCLID: 52;

          then ( W-min ((( L~ go) \/ ( L~ co)) \/ ( L~ pion1))) = ( W-min (( L~ go) \/ ( L~ co))) by A142, A146, A166, A157, A144, JORDAN1J: 33;

          then

           A167: ( W-min ( L~ godo)) = Wmin by A134, A166, XBOOLE_1: 4;

          (godo /. 2) = ((go ^' pion1) /. 2) by A103, FINSEQ_6: 159

          .= (US /. 2) by A42, A76, FINSEQ_6: 159

          .= ((US ^' LS) /. 2) by A145, FINSEQ_6: 159

          .= (( Rotate (( Cage (C,n)),Wmin)) /. 2) by JORDAN1E: 11;

          then (godo /. 2) in ( W-most ( L~ ( Cage (C,n)))) by JORDAN1I: 25;

          

          then ((godo /. 2) `1 ) = (( W-min ( L~ godo)) `1 ) by A167, PSCOMP_1: 31

          .= ( W-bound ( L~ godo)) by EUCLID: 52;

          then (godo /. 2) in ( W-most ( L~ godo)) by A151, A150, SPRECT_2: 12;

          then (( Rotate (godo,( W-min ( L~ godo)))) /. 2) in ( W-most ( L~ godo)) by A152, A167, FINSEQ_6: 89;

          then

          reconsider godo as clockwise_oriented non constant standard special_circular_sequence by JORDAN1I: 25;

          ( len US) in ( dom US) by FINSEQ_5: 6;

          

          then

           A168: (US . ( len US)) = (US /. ( len US)) by PARTFUN1:def 6

          .= Emax by JORDAN1F: 7;

          

           A169: ( east_halfline ( E-max C)) misses ( L~ go)

          proof

            assume ( east_halfline ( E-max C)) meets ( L~ go);

            then

            consider p be object such that

             A170: p in ( east_halfline ( E-max C)) and

             A171: p in ( L~ go) by XBOOLE_0: 3;

            reconsider p as Point of ( TOP-REAL 2) by A170;

            p in ( L~ US) by A56, A171;

            then p in (( east_halfline ( E-max C)) /\ ( L~ ( Cage (C,n)))) by A130, A170, XBOOLE_0:def 4;

            then

             A172: (p `1 ) = Ebo by JORDAN1A: 83, PSCOMP_1: 50;

            then

             A173: p = Emax by A56, A171, JORDAN1J: 46;

            then Emax = Gik by A30, A168, A171, JORDAN1J: 43;

            then (Gik `1 ) = ((G * (( len G),k)) `1 ) by A6, A10, A13, A172, A173, JORDAN1A: 71;

            hence contradiction by A2, A3, A12, A41, JORDAN1G: 7;

          end;

          now

            assume ( east_halfline ( E-max C)) meets ( L~ godo);

            then

             A174: ( east_halfline ( E-max C)) meets (( L~ go) \/ ( L~ pion1)) or ( east_halfline ( E-max C)) meets ( L~ co) by A134, XBOOLE_1: 70;

            per cases by A174, XBOOLE_1: 70;

              suppose ( east_halfline ( E-max C)) meets ( L~ go);

              hence contradiction by A169;

            end;

              suppose ( east_halfline ( E-max C)) meets ( L~ pion1);

              then

              consider p be object such that

               A175: p in ( east_halfline ( E-max C)) and

               A176: p in ( L~ pion1) by XBOOLE_0: 3;

              reconsider p as Point of ( TOP-REAL 2) by A175;

              

               A177: (p `2 ) = (( E-max C) `2 ) by A175, TOPREAL1:def 11;

               A178:

              now

                per cases by A36, A84, A176, XBOOLE_0:def 3;

                  suppose p in poz;

                  hence (p `1 ) <= (Gi1k `1 ) by A138, TOPREAL1: 3;

                end;

                  suppose p in pio;

                  hence (p `1 ) <= (Gi1k `1 ) by A140, GOBOARD7: 5;

                end;

              end;

              (i1 + 1) <= ( len G) by A3, NAT_1: 13;

              then ((i1 + 1) - 1) <= (( len G) - 1) by XREAL_1: 9;

              then

               A179: i1 <= (( len G) -' 1) by XREAL_0:def 2;

              (( len G) -' 1) <= ( len G) by NAT_D: 35;

              then (Gi1k `1 ) <= ((G * ((( len G) -' 1),1)) `1 ) by A6, A17, A10, A13, A21, A179, JORDAN1A: 18;

              then (p `1 ) <= ((G * ((( len G) -' 1),1)) `1 ) by A178, XXREAL_0: 2;

              then (p `1 ) <= ( E-bound C) by A21, JORDAN8: 12;

              then

               A180: (p `1 ) <= (( E-max C) `1 ) by EUCLID: 52;

              (p `1 ) >= (( E-max C) `1 ) by A175, TOPREAL1:def 11;

              then (p `1 ) = (( E-max C) `1 ) by A180, XXREAL_0: 1;

              then p = ( E-max C) by A177, TOPREAL3: 6;

              hence contradiction by A9, A36, A84, A165, A176, XBOOLE_0: 3;

            end;

              suppose ( east_halfline ( E-max C)) meets ( L~ co);

              then

              consider p be object such that

               A181: p in ( east_halfline ( E-max C)) and

               A182: p in ( L~ co) by XBOOLE_0: 3;

              reconsider p as Point of ( TOP-REAL 2) by A181;

              

               A183: p in ( LSeg (co,( Index (p,co)))) by A182, JORDAN3: 9;

              consider t be Nat such that

               A184: t in ( dom LS) and

               A185: (LS . t) = Gij by A44, FINSEQ_2: 10;

              1 <= t by A184, FINSEQ_3: 25;

              then

               A186: 1 < t by A46, A185, XXREAL_0: 1;

              t <= ( len LS) by A184, FINSEQ_3: 25;

              then (( Index (Gij,LS)) + 1) = t by A185, A186, JORDAN3: 12;

              then

               A187: ( len ( L_Cut (LS,Gij))) = (( len LS) - ( Index (Gij,LS))) by A16, A185, JORDAN3: 26;

              ( Index (p,co)) < ( len co) by A182, JORDAN3: 8;

              then ( Index (p,co)) < (( len LS) -' ( Index (Gij,LS))) by A187, XREAL_0:def 2;

              then (( Index (p,co)) + 1) <= (( len LS) -' ( Index (Gij,LS))) by NAT_1: 13;

              then

               A188: ( Index (p,co)) <= ((( len LS) -' ( Index (Gij,LS))) - 1) by XREAL_1: 19;

              

               A189: co = ( mid (LS,(Gij .. LS),( len LS))) by A44, JORDAN1J: 37;

              p in ( L~ LS) by A48, A182;

              then p in (( east_halfline ( E-max C)) /\ ( L~ ( Cage (C,n)))) by A147, A181, XBOOLE_0:def 4;

              then

               A190: (p `1 ) = Ebo by JORDAN1A: 83, PSCOMP_1: 50;

              

               A191: (( Index (Gij,LS)) + 1) = (Gij .. LS) by A46, A44, JORDAN1J: 56;

              ( 0 + ( Index (Gij,LS))) < ( len LS) by A16, JORDAN3: 8;

              then (( len LS) - ( Index (Gij,LS))) > 0 by XREAL_1: 20;

              then ( Index (p,co)) <= ((( len LS) - ( Index (Gij,LS))) - 1) by A188, XREAL_0:def 2;

              then ( Index (p,co)) <= (( len LS) - (Gij .. LS)) by A191;

              then ( Index (p,co)) <= (( len LS) -' (Gij .. LS)) by XREAL_0:def 2;

              then

               A192: ( Index (p,co)) < ((( len LS) -' (Gij .. LS)) + 1) by NAT_1: 13;

              

               A193: 1 <= ( Index (p,co)) by A182, JORDAN3: 8;

              

               A194: (Gij .. LS) <= ( len LS) by A44, FINSEQ_4: 21;

              (Gij .. LS) <> ( len LS) by A43, A44, FINSEQ_4: 19;

              then

               A195: (Gij .. LS) < ( len LS) by A194, XXREAL_0: 1;

              

               A196: (1 + 1) <= ( len LS) by A31, XXREAL_0: 2;

              then

               A197: 2 in ( dom LS) by FINSEQ_3: 25;

              set tt = ((( Index (p,co)) + (Gij .. LS)) -' 1);

              set RC = ( Rotate (( Cage (C,n)),Emax));

              

               A198: ( E-max C) in ( right_cell (RC,1)) by JORDAN1I: 7;

              

               A199: ( GoB RC) = ( GoB ( Cage (C,n))) by REVROT_1: 28

              .= G by JORDAN1H: 44;

              

               A200: ( L~ RC) = ( L~ ( Cage (C,n))) by REVROT_1: 33;

              consider jj2 be Nat such that

               A201: 1 <= jj2 and

               A202: jj2 <= ( width G) and

               A203: Emax = (G * (( len G),jj2)) by JORDAN1D: 25;

              

               A204: ( len G) >= 4 by JORDAN8: 10;

              then ( len G) >= 1 by XXREAL_0: 2;

              then

               A205: [( len G), jj2] in ( Indices G) by A201, A202, MATRIX_0: 30;

              

               A206: ( len RC) = ( len ( Cage (C,n))) by FINSEQ_6: 179;

              LS = (RC -: Wmin) by JORDAN1G: 18;

              then

               A207: ( LSeg (LS,1)) = ( LSeg (RC,1)) by A196, SPPOL_2: 9;

              

               A208: Emax in ( rng ( Cage (C,n))) by SPRECT_2: 46;

              RC is_sequence_on G by A162, REVROT_1: 34;

              then

              consider ii,jj be Nat such that

               A209: [ii, (jj + 1)] in ( Indices G) and

               A210: [ii, jj] in ( Indices G) and

               A211: (RC /. 1) = (G * (ii,(jj + 1))) and

               A212: (RC /. (1 + 1)) = (G * (ii,jj)) by A158, A200, A206, A208, FINSEQ_6: 92, JORDAN1I: 23;

              

               A213: ((jj + 1) + 1) <> jj;

              

               A214: 1 <= jj by A210, MATRIX_0: 32;

              (RC /. 1) = ( E-max ( L~ RC)) by A200, A208, FINSEQ_6: 92;

              then

               A215: ii = ( len G) by A200, A209, A211, A203, A205, GOBOARD1: 5;

              then (ii - 1) >= (4 - 1) by A204, XREAL_1: 9;

              then

               A216: (ii - 1) >= 1 by XXREAL_0: 2;

              then

               A217: 1 <= (ii -' 1) by XREAL_0:def 2;

              

               A218: jj <= ( width G) by A210, MATRIX_0: 32;

              then

               A219: ((G * (( len G),jj)) `1 ) = Ebo by A13, A214, JORDAN1A: 71;

              

               A220: (jj + 1) <= ( width G) by A209, MATRIX_0: 32;

              (ii + 1) <> ii;

              then

               A221: ( right_cell (RC,1)) = ( cell (G,(ii -' 1),jj)) by A158, A206, A199, A209, A210, A211, A212, A213, GOBOARD5:def 6;

              

               A222: ii <= ( len G) by A210, MATRIX_0: 32;

              

               A223: 1 <= ii by A210, MATRIX_0: 32;

              

               A224: ii <= ( len G) by A209, MATRIX_0: 32;

              

               A225: 1 <= (jj + 1) by A209, MATRIX_0: 32;

              then

               A226: Ebo = ((G * (( len G),(jj + 1))) `1 ) by A13, A220, JORDAN1A: 71;

              

               A227: 1 <= ii by A209, MATRIX_0: 32;

              then

               A228: ((ii -' 1) + 1) = ii by XREAL_1: 235;

              then

               A229: (ii -' 1) < ( len G) by A224, NAT_1: 13;

              

              then

               A230: ((G * ((ii -' 1),(jj + 1))) `2 ) = ((G * (1,(jj + 1))) `2 ) by A225, A220, A217, GOBOARD5: 1

              .= ((G * (ii,(jj + 1))) `2 ) by A227, A224, A225, A220, GOBOARD5: 1;

              

               A231: (( E-max C) `2 ) = (p `2 ) by A181, TOPREAL1:def 11;

              then

               A232: (p `2 ) <= ((G * ((ii -' 1),(jj + 1))) `2 ) by A198, A224, A220, A214, A221, A228, A216, JORDAN9: 17;

              

               A233: ((G * ((ii -' 1),jj)) `2 ) = ((G * (1,jj)) `2 ) by A214, A218, A217, A229, GOBOARD5: 1

              .= ((G * (ii,jj)) `2 ) by A223, A222, A214, A218, GOBOARD5: 1;

              ((G * ((ii -' 1),jj)) `2 ) <= (p `2 ) by A231, A198, A224, A220, A214, A221, A228, A216, JORDAN9: 17;

              then p in ( LSeg ((RC /. 1),(RC /. (1 + 1)))) by A190, A211, A212, A215, A232, A233, A230, A219, A226, GOBOARD7: 7;

              then

               A234: p in ( LSeg (LS,1)) by A158, A207, A206, TOPREAL1:def 3;

              1 <= (Gij .. LS) by A44, FINSEQ_4: 21;

              then

               A235: ( LSeg (( mid (LS,(Gij .. LS),( len LS))),( Index (p,co)))) = ( LSeg (LS,((( Index (p,co)) + (Gij .. LS)) -' 1))) by A195, A193, A192, JORDAN4: 19;

              1 <= ( Index (Gij,LS)) by A16, JORDAN3: 8;

              then

               A236: (1 + 1) <= (Gij .. LS) by A191, XREAL_1: 7;

              then (( Index (p,co)) + (Gij .. LS)) >= ((1 + 1) + 1) by A193, XREAL_1: 7;

              then ((( Index (p,co)) + (Gij .. LS)) - 1) >= (((1 + 1) + 1) - 1) by XREAL_1: 9;

              then

               A237: tt >= (1 + 1) by XREAL_0:def 2;

              now

                per cases by A237, XXREAL_0: 1;

                  suppose tt > (1 + 1);

                  then ( LSeg (LS,1)) misses ( LSeg (LS,tt)) by TOPREAL1:def 7;

                  hence contradiction by A234, A183, A189, A235, XBOOLE_0: 3;

                end;

                  suppose

                   A238: tt = (1 + 1);

                  then (1 + 1) = ((( Index (p,co)) + (Gij .. LS)) - 1) by XREAL_0:def 2;

                  then ((1 + 1) + 1) = (( Index (p,co)) + (Gij .. LS));

                  then

                   A239: (Gij .. LS) = 2 by A193, A236, JORDAN1E: 6;

                  (( LSeg (LS,1)) /\ ( LSeg (LS,tt))) = {(LS /. 2)} by A31, A238, TOPREAL1:def 6;

                  then p in {(LS /. 2)} by A234, A183, A189, A235, XBOOLE_0:def 4;

                  then

                   A240: p = (LS /. 2) by TARSKI:def 1;

                  then

                   A241: p in ( rng LS) by A197, PARTFUN2: 2;

                  (p .. LS) = 2 by A197, A240, FINSEQ_5: 41;

                  then p = Gij by A44, A239, A241, FINSEQ_5: 9;

                  then (Gij `1 ) = Ebo by A240, JORDAN1G: 32;

                  then (Gij `1 ) = ((G * (( len G),j)) `1 ) by A4, A19, A13, JORDAN1A: 71;

                  hence contradiction by A3, A25, A22, JORDAN1G: 7;

                end;

              end;

              hence contradiction;

            end;

          end;

          then ( east_halfline ( E-max C)) c= (( L~ godo) ` ) by SUBSET_1: 23;

          then

          consider W be Subset of ( TOP-REAL 2) such that

           A242: W is_a_component_of (( L~ godo) ` ) and

           A243: ( east_halfline ( E-max C)) c= W by GOBOARD9: 3;

           not W is bounded by A243, JORDAN2C: 121, RLTOPSP1: 42;

          then W is_outside_component_of ( L~ godo) by A242, JORDAN2C:def 3;

          then W c= ( UBD ( L~ godo)) by JORDAN2C: 23;

          then

           A244: ( east_halfline ( E-max C)) c= ( UBD ( L~ godo)) by A243;

          ( E-max C) in ( east_halfline ( E-max C)) by TOPREAL1: 38;

          then ( E-max C) in ( UBD ( L~ godo)) by A244;

          then ( E-max C) in ( LeftComp godo) by GOBRD14: 36;

          then UA meets ( L~ godo) by A114, A153, A165, A149, A164, JORDAN1J: 36;

          then

           A245: UA meets (( L~ go) \/ ( L~ pion1)) or UA meets ( L~ co) by A134, XBOOLE_1: 70;

          now

            per cases by A245, XBOOLE_1: 70;

              suppose UA meets ( L~ go);

              then UA meets ( L~ ( Cage (C,n))) by A56, A130, XBOOLE_1: 1, XBOOLE_1: 63;

              hence contradiction by A143, JORDAN10: 5, XBOOLE_1: 63;

            end;

              suppose UA meets ( L~ pion1);

              hence contradiction by A9, A36, A84;

            end;

              suppose UA meets ( L~ co);

              then UA meets ( L~ ( Cage (C,n))) by A48, A147, XBOOLE_1: 1, XBOOLE_1: 63;

              hence contradiction by A143, JORDAN10: 5, XBOOLE_1: 63;

            end;

          end;

          hence contradiction;

        end;

          suppose (Gij `1 ) = (Gik `1 );

          then

           A246: i1 = i2 by A25, A12, JORDAN1G: 7;

          then poz = {Gi1k} by RLTOPSP1: 70;

          then poz c= pio by A27, ZFMISC_1: 31;

          then (pio \/ poz) = pio by XBOOLE_1: 12;

          hence contradiction by A1, A3, A4, A5, A6, A7, A8, A9, A246, JORDAN1J: 59;

        end;

          suppose (Gij `2 ) = (Gik `2 );

          then

           A247: j = k by A25, A12, JORDAN1G: 6;

          then pio = {Gi1k} by RLTOPSP1: 70;

          then pio c= poz by A26, ZFMISC_1: 31;

          then (pio \/ poz) = poz by XBOOLE_1: 12;

          hence contradiction by A1, A2, A3, A4, A6, A7, A8, A9, A247, Th37;

        end;

      end;

      hence contradiction;

    end;

    theorem :: JORDAN15:47

    

     Th47: for C be Simple_closed_curve holds for i1,i2,j,k be Nat st 1 < i2 & i2 <= i1 & i1 < ( len ( Gauge (C,n))) & 1 <= j & j <= k & k <= ( width ( Gauge (C,n))) & ((( LSeg ((( Gauge (C,n)) * (i1,j)),(( Gauge (C,n)) * (i1,k)))) \/ ( LSeg ((( Gauge (C,n)) * (i1,k)),(( Gauge (C,n)) * (i2,k))))) /\ ( L~ ( Upper_Seq (C,n)))) = {(( Gauge (C,n)) * (i2,k))} & ((( LSeg ((( Gauge (C,n)) * (i1,j)),(( Gauge (C,n)) * (i1,k)))) \/ ( LSeg ((( Gauge (C,n)) * (i1,k)),(( Gauge (C,n)) * (i2,k))))) /\ ( L~ ( Lower_Seq (C,n)))) = {(( Gauge (C,n)) * (i1,j))} holds (( LSeg ((( Gauge (C,n)) * (i1,j)),(( Gauge (C,n)) * (i1,k)))) \/ ( LSeg ((( Gauge (C,n)) * (i1,k)),(( Gauge (C,n)) * (i2,k))))) meets ( Lower_Arc C)

    proof

      let C be Simple_closed_curve;

      let i1,i2,j,k be Nat;

      set G = ( Gauge (C,n));

      set pio = ( LSeg ((G * (i1,j)),(G * (i1,k))));

      set poz = ( LSeg ((G * (i1,k)),(G * (i2,k))));

      set US = ( Upper_Seq (C,n));

      set LS = ( Lower_Seq (C,n));

      assume that

       A1: 1 < i2 and

       A2: i2 <= i1 and

       A3: i1 < ( len G) and

       A4: 1 <= j and

       A5: j <= k and

       A6: k <= ( width G) and

       A7: ((pio \/ poz) /\ ( L~ US)) = {(G * (i2,k))} and

       A8: ((pio \/ poz) /\ ( L~ LS)) = {(G * (i1,j))} and

       A9: (pio \/ poz) misses ( Lower_Arc C);

      set Gi1k = (G * (i1,k));

      set Gik = (G * (i2,k));

      

       A10: 1 <= k by A4, A5, XXREAL_0: 2;

      

       A11: i2 < ( len G) by A2, A3, XXREAL_0: 2;

      then

       A12: [i2, k] in ( Indices G) by A1, A6, A10, MATRIX_0: 30;

      set Wmin = ( W-min ( L~ ( Cage (C,n))));

      set Wbo = ( W-bound ( L~ ( Cage (C,n))));

      

       A13: ( len G) = ( width G) by JORDAN8:def 1;

      set go = ( R_Cut (US,Gik));

      

       A14: ( len US) >= 3 by JORDAN1E: 15;

      then ( len US) >= 1 by XXREAL_0: 2;

      then 1 in ( dom US) by FINSEQ_3: 25;

      

      then

       A15: (US . 1) = (US /. 1) by PARTFUN1:def 6

      .= Wmin by JORDAN1F: 5;

      set Gij = (G * (i1,j));

      set co = ( L_Cut (LS,Gij));

      Gij in {Gij} by TARSKI:def 1;

      then

       A16: Gij in ( L~ LS) by A8, XBOOLE_0:def 4;

      

       A17: 1 < i1 by A1, A2, XXREAL_0: 2;

      

      then

       A18: (Gi1k `2 ) = ((G * (1,k)) `2 ) by A3, A6, A10, GOBOARD5: 1

      .= (Gik `2 ) by A1, A6, A11, A10, GOBOARD5: 1;

      

       A19: j <= ( width G) by A5, A6, XXREAL_0: 2;

      then

       A20: [i1, j] in ( Indices G) by A3, A4, A17, MATRIX_0: 30;

      ( len G) >= 4 by JORDAN8: 10;

      then

       A21: ( len G) >= 1 by XXREAL_0: 2;

      then

       A22: [( len G), j] in ( Indices G) by A4, A19, MATRIX_0: 30;

      

       A23: [1, k] in ( Indices G) by A6, A10, A21, MATRIX_0: 30;

       A24:

      now

        assume (Gik `1 ) = Wbo;

        then ((G * (1,k)) `1 ) = ((G * (i2,k)) `1 ) by A6, A10, A13, JORDAN1A: 73;

        hence contradiction by A1, A12, A23, JORDAN1G: 7;

      end;

      

       A25: [i1, j] in ( Indices G) by A3, A4, A17, A19, MATRIX_0: 30;

      set pion = <*Gik, Gi1k, Gij*>;

      

       A26: Gi1k in poz by RLTOPSP1: 68;

      set LA = ( Lower_Arc C);

      

       A27: Gi1k in pio by RLTOPSP1: 68;

      

       A28: [i1, k] in ( Indices G) by A3, A6, A17, A10, MATRIX_0: 30;

       A29:

      now

        let n be Nat;

        assume n in ( dom pion);

        then n in {1, 2, 3} by FINSEQ_1: 89, FINSEQ_3: 1;

        then n = 1 or n = 2 or n = 3 by ENUMSET1:def 1;

        hence ex i,j be Nat st [i, j] in ( Indices G) & (pion /. n) = (G * (i,j)) by A25, A12, A28, FINSEQ_4: 18;

      end;

      Gik in {Gik} by TARSKI:def 1;

      then

       A30: Gik in ( L~ US) by A7, XBOOLE_0:def 4;

      set Emax = ( E-max ( L~ ( Cage (C,n))));

      

       A31: ( len LS) >= (1 + 2) by JORDAN1E: 15;

      then

       A32: ( len LS) >= 1 by XXREAL_0: 2;

      then

       A33: 1 in ( dom LS) by FINSEQ_3: 25;

      

      then

       A34: (LS . 1) = (LS /. 1) by PARTFUN1:def 6

      .= Emax by JORDAN1F: 6;

      ( len LS) in ( dom LS) by A32, FINSEQ_3: 25;

      

      then

       A35: (LS . ( len LS)) = (LS /. ( len LS)) by PARTFUN1:def 6

      .= Wmin by JORDAN1F: 8;

      set Ebo = ( E-bound ( L~ ( Cage (C,n))));

      

       A36: ( L~ <*Gik, Gi1k, Gij*>) = (poz \/ pio) by TOPREAL3: 16;

      (Wmin `1 ) = Wbo by EUCLID: 52

      .= ((G * (1,k)) `1 ) by A6, A10, A13, JORDAN1A: 73;

      then

       A37: Gik <> (US . 1) by A1, A12, A15, A23, JORDAN1G: 7;

      then

      reconsider go as being_S-Seq FinSequence of ( TOP-REAL 2) by A30, JORDAN3: 35;

      

       A38: Gik in ( rng US) by A1, A6, A11, A30, A10, JORDAN1G: 4, JORDAN1J: 40;

      then

       A39: go is_sequence_on G by JORDAN1G: 4, JORDAN1J: 38;

      (Gi1k `1 ) = ((G * (i1,1)) `1 ) by A3, A6, A17, A10, GOBOARD5: 2

      .= (Gij `1 ) by A3, A4, A17, A19, GOBOARD5: 2;

      then

       A40: Gi1k = |[(Gij `1 ), (Gik `2 )]| by A18, EUCLID: 53;

      

       A41: [( len G), k] in ( Indices G) by A6, A10, A21, MATRIX_0: 30;

      

       A42: ( len go) >= (1 + 1) by TOPREAL1:def 8;

      (Wmin `1 ) = Wbo by EUCLID: 52

      .= ((G * (1,k)) `1 ) by A6, A10, A13, JORDAN1A: 73;

      then

       A43: Gij <> (LS . ( len LS)) by A1, A2, A23, A35, A20, JORDAN1G: 7;

      then

      reconsider co as being_S-Seq FinSequence of ( TOP-REAL 2) by A16, JORDAN3: 34;

      

       A44: Gij in ( rng LS) by A3, A4, A17, A16, A19, JORDAN1G: 5, JORDAN1J: 40;

      then

       A45: co is_sequence_on G by JORDAN1G: 5, JORDAN1J: 39;

      (Emax `1 ) = Ebo by EUCLID: 52

      .= ((G * (( len G),k)) `1 ) by A6, A10, A13, JORDAN1A: 71;

      then

       A46: Gij <> (LS . 1) by A3, A20, A41, A34, JORDAN1G: 7;

      

       A47: ( len co) >= (1 + 1) by TOPREAL1:def 8;

      then

      reconsider co as non constant s.c.c. being_S-Seq FinSequence of ( TOP-REAL 2) by A45, JGRAPH_1: 12, JORDAN8: 5;

      

       A48: ( L~ co) c= ( L~ LS) by A16, JORDAN3: 42;

      ( len co) >= 1 by A47, XXREAL_0: 2;

      then 1 in ( dom co) by FINSEQ_3: 25;

      

      then

       A49: (co /. 1) = (co . 1) by PARTFUN1:def 6

      .= Gij by A16, JORDAN3: 23;

      then

       A50: ( LSeg (co,1)) = ( LSeg (Gij,(co /. (1 + 1)))) by A47, TOPREAL1:def 3;

      

       A51: {Gij} c= (( LSeg (co,1)) /\ ( L~ <*Gik, Gi1k, Gij*>))

      proof

        let x be object;

        assume x in {Gij};

        then

         A52: x = Gij by TARSKI:def 1;

        Gij in ( LSeg (Gi1k,Gij)) by RLTOPSP1: 68;

        then Gij in (( LSeg (Gik,Gi1k)) \/ ( LSeg (Gi1k,Gij))) by XBOOLE_0:def 3;

        then

         A53: Gij in ( L~ <*Gik, Gi1k, Gij*>) by SPRECT_1: 8;

        Gij in ( LSeg (co,1)) by A50, RLTOPSP1: 68;

        hence thesis by A52, A53, XBOOLE_0:def 4;

      end;

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

      then ( LSeg (co,1)) c= ( L~ LS) by A48;

      then (( LSeg (co,1)) /\ ( L~ <*Gik, Gi1k, Gij*>)) c= {Gij} by A8, A36, XBOOLE_1: 26;

      then

       A54: (( L~ <*Gik, Gi1k, Gij*>) /\ ( LSeg (co,1))) = {Gij} by A51;

      

       A55: ( rng co) c= ( L~ co) by A47, SPPOL_2: 18;

      reconsider go as non constant s.c.c. being_S-Seq FinSequence of ( TOP-REAL 2) by A42, A39, JGRAPH_1: 12, JORDAN8: 5;

      

       A56: ( L~ go) c= ( L~ US) by A30, JORDAN3: 41;

      

       A57: ( len go) > 1 by A42, NAT_1: 13;

      then

       A58: ( len go) in ( dom go) by FINSEQ_3: 25;

      

      then

       A59: (go /. ( len go)) = (go . ( len go)) by PARTFUN1:def 6

      .= Gik by A30, JORDAN3: 24;

      reconsider m = (( len go) - 1) as Nat by A58, FINSEQ_3: 26;

      

       A60: (m + 1) = ( len go);

      then

       A61: (( len go) -' 1) = m by NAT_D: 34;

      m >= 1 by A42, XREAL_1: 19;

      then

       A62: ( LSeg (go,m)) = ( LSeg ((go /. m),Gik)) by A59, A60, TOPREAL1:def 3;

      

       A63: {Gik} c= (( LSeg (go,m)) /\ ( L~ <*Gik, Gi1k, Gij*>))

      proof

        let x be object;

        assume x in {Gik};

        then

         A64: x = Gik by TARSKI:def 1;

        Gik in ( LSeg (Gik,Gi1k)) by RLTOPSP1: 68;

        then Gik in (( LSeg (Gik,Gi1k)) \/ ( LSeg (Gi1k,Gij))) by XBOOLE_0:def 3;

        then

         A65: Gik in ( L~ <*Gik, Gi1k, Gij*>) by SPRECT_1: 8;

        Gik in ( LSeg (go,m)) by A62, RLTOPSP1: 68;

        hence thesis by A64, A65, XBOOLE_0:def 4;

      end;

      ( LSeg (go,m)) c= ( L~ go) by TOPREAL3: 19;

      then ( LSeg (go,m)) c= ( L~ US) by A56;

      then (( LSeg (go,m)) /\ ( L~ <*Gik, Gi1k, Gij*>)) c= {Gik} by A7, A36, XBOOLE_1: 26;

      then

       A66: (( LSeg (go,m)) /\ ( L~ <*Gik, Gi1k, Gij*>)) = {Gik} by A63;

      

       A67: (go /. 1) = (US /. 1) by A30, SPRECT_3: 22

      .= Wmin by JORDAN1F: 5;

      then

       A68: Wmin in ( rng go) by FINSEQ_6: 42;

      

       A69: (LS . 1) = (LS /. 1) by A33, PARTFUN1:def 6

      .= Emax by JORDAN1F: 6;

      

       A70: (( L~ go) /\ ( L~ co)) c= {(go /. 1)}

      proof

        let x be object;

        assume

         A71: x in (( L~ go) /\ ( L~ co));

        then

         A72: x in ( L~ co) by XBOOLE_0:def 4;

         A73:

        now

          assume x = Emax;

          then

           A74: Emax = Gij by A16, A69, A72, JORDAN1E: 7;

          ((G * (( len G),j)) `1 ) = Ebo by A4, A19, A13, JORDAN1A: 71;

          then (Emax `1 ) <> Ebo by A3, A25, A22, A74, JORDAN1G: 7;

          hence contradiction by EUCLID: 52;

        end;

        x in ( L~ go) by A71, XBOOLE_0:def 4;

        then x in (( L~ US) /\ ( L~ LS)) by A56, A48, A72, XBOOLE_0:def 4;

        then x in {Wmin, Emax} by JORDAN1E: 16;

        then x = Wmin or x = Emax by TARSKI:def 2;

        hence thesis by A67, A73, TARSKI:def 1;

      end;

      set W2 = (go /. 2);

      

       A75: 2 in ( dom go) by A42, FINSEQ_3: 25;

      go = ( mid (US,1,(Gik .. US))) by A38, JORDAN1G: 49

      .= (US | (Gik .. US)) by A38, FINSEQ_4: 21, FINSEQ_6: 116;

      then

       A76: W2 = (US /. 2) by A75, FINSEQ_4: 70;

      

       A77: ( rng go) c= ( L~ go) by A42, SPPOL_2: 18;

      

       A78: (go /. 1) = (LS /. ( len LS)) by A67, JORDAN1F: 8

      .= (co /. ( len co)) by A16, JORDAN1J: 35;

       {(go /. 1)} c= (( L~ go) /\ ( L~ co))

      proof

        let x be object;

        assume x in {(go /. 1)};

        then

         A79: x = (go /. 1) by TARSKI:def 1;

        then

         A80: x in ( rng go) by FINSEQ_6: 42;

        x in ( rng co) by A78, A79, FINSEQ_6: 168;

        hence thesis by A77, A55, A80, XBOOLE_0:def 4;

      end;

      then

       A81: (( L~ go) /\ ( L~ co)) = {(go /. 1)} by A70;

      now

        per cases ;

          suppose (Gij `1 ) <> (Gik `1 ) & (Gij `2 ) <> (Gik `2 );

          then pion is being_S-Seq by A40, TOPREAL3: 35;

          then

          consider pion1 be FinSequence of ( TOP-REAL 2) such that

           A82: pion1 is_sequence_on G and

           A83: pion1 is being_S-Seq and

           A84: ( L~ pion) = ( L~ pion1) and

           A85: (pion /. 1) = (pion1 /. 1) and

           A86: (pion /. ( len pion)) = (pion1 /. ( len pion1)) and

           A87: ( len pion) <= ( len pion1) by A29, GOBOARD3: 2;

          reconsider pion1 as being_S-Seq FinSequence of ( TOP-REAL 2) by A83;

          

           A88: ((go ^' pion1) /. ( len (go ^' pion1))) = (pion /. ( len pion)) by A86, FINSEQ_6: 156

          .= (pion /. 3) by FINSEQ_1: 45

          .= (co /. 1) by A49, FINSEQ_4: 18;

          

           A89: (go /. ( len go)) = (pion1 /. 1) by A59, A85, FINSEQ_4: 18;

          

           A90: (( L~ go) /\ ( L~ pion1)) c= {(pion1 /. 1)}

          proof

            let x be object;

            assume

             A91: x in (( L~ go) /\ ( L~ pion1));

            then

             A92: x in ( L~ pion1) by XBOOLE_0:def 4;

            x in ( L~ go) by A91, XBOOLE_0:def 4;

            hence thesis by A7, A36, A59, A56, A84, A89, A92, XBOOLE_0:def 4;

          end;

          ( len pion1) >= (2 + 1) by A87, FINSEQ_1: 45;

          then

           A93: ( len pion1) > (1 + 1) by NAT_1: 13;

          then

           A94: ( rng pion1) c= ( L~ pion1) by SPPOL_2: 18;

           {(pion1 /. 1)} c= (( L~ go) /\ ( L~ pion1))

          proof

            let x be object;

            assume x in {(pion1 /. 1)};

            then

             A95: x = (pion1 /. 1) by TARSKI:def 1;

            then

             A96: x in ( rng pion1) by FINSEQ_6: 42;

            x in ( rng go) by A89, A95, FINSEQ_6: 168;

            hence thesis by A77, A94, A96, XBOOLE_0:def 4;

          end;

          then

           A97: (( L~ go) /\ ( L~ pion1)) = {(pion1 /. 1)} by A90;

          then

           A98: (go ^' pion1) is s.n.c. by A89, JORDAN1J: 54;

          

           A99: {Gik} c= (( LSeg (go,m)) /\ ( LSeg (pion1,1)))

          proof

            let x be object;

            assume x in {Gik};

            then

             A100: x = Gik by TARSKI:def 1;

            

             A101: Gik in ( LSeg (go,m)) by A62, RLTOPSP1: 68;

            Gik in ( LSeg (pion1,1)) by A59, A89, A93, TOPREAL1: 21;

            hence thesis by A100, A101, XBOOLE_0:def 4;

          end;

          ( LSeg (pion1,1)) c= ( L~ pion) by A84, TOPREAL3: 19;

          then (( LSeg (go,(( len go) -' 1))) /\ ( LSeg (pion1,1))) c= {Gik} by A61, A66, XBOOLE_1: 27;

          then (( LSeg (go,(( len go) -' 1))) /\ ( LSeg (pion1,1))) = {(go /. ( len go))} by A59, A61, A99;

          then

           A102: (go ^' pion1) is unfolded by A89, TOPREAL8: 34;

          ( len (go ^' pion1)) >= ( len go) by TOPREAL8: 7;

          then

           A103: ( len (go ^' pion1)) >= (1 + 1) by A42, XXREAL_0: 2;

          then

           A104: ( len (go ^' pion1)) > (1 + 0 ) by NAT_1: 13;

          

           A105: (pion /. ( len pion)) = (pion /. 3) by FINSEQ_1: 45

          .= (co /. 1) by A49, FINSEQ_4: 18;

          

           A106: {(pion1 /. ( len pion1))} c= (( L~ co) /\ ( L~ pion1))

          proof

            let x be object;

            assume x in {(pion1 /. ( len pion1))};

            then

             A107: x = (pion1 /. ( len pion1)) by TARSKI:def 1;

            then

             A108: x in ( rng pion1) by FINSEQ_6: 168;

            x in ( rng co) by A86, A105, A107, FINSEQ_6: 42;

            hence thesis by A55, A94, A108, XBOOLE_0:def 4;

          end;

          (( L~ co) /\ ( L~ pion1)) c= {(pion1 /. ( len pion1))}

          proof

            let x be object;

            assume

             A109: x in (( L~ co) /\ ( L~ pion1));

            then

             A110: x in ( L~ pion1) by XBOOLE_0:def 4;

            x in ( L~ co) by A109, XBOOLE_0:def 4;

            hence thesis by A8, A36, A49, A48, A84, A86, A105, A110, XBOOLE_0:def 4;

          end;

          then

           A111: (( L~ co) /\ ( L~ pion1)) = {(pion1 /. ( len pion1))} by A106;

          

           A112: (( L~ (go ^' pion1)) /\ ( L~ co)) = ((( L~ go) \/ ( L~ pion1)) /\ ( L~ co)) by A89, TOPREAL8: 35

          .= ( {(go /. 1)} \/ {(co /. 1)}) by A81, A86, A105, A111, XBOOLE_1: 23

          .= ( {((go ^' pion1) /. 1)} \/ {(co /. 1)}) by FINSEQ_6: 155

          .= {((go ^' pion1) /. 1), (co /. 1)} by ENUMSET1: 1;

          

           A113: LA is_an_arc_of (( E-max C),( W-min C)) by JORDAN6:def 9;

          then

           A114: LA is connected by JORDAN6: 10;

          set godo = ((go ^' pion1) ^' co);

          

           A115: (co /. ( len co)) = ((go ^' pion1) /. 1) by A78, FINSEQ_6: 155;

          

           A116: (go ^' pion1) is_sequence_on G by A39, A82, A89, TOPREAL8: 12;

          then

           A117: godo is_sequence_on G by A45, A88, TOPREAL8: 12;

          

           A118: (( len pion1) - 1) >= 1 by A93, XREAL_1: 19;

          then

           A119: (( len pion1) -' 1) = (( len pion1) - 1) by XREAL_0:def 2;

          

           A120: {Gij} c= (( LSeg (pion1,(( len pion1) -' 1))) /\ ( LSeg (co,1)))

          proof

            let x be object;

            assume x in {Gij};

            then

             A121: x = Gij by TARSKI:def 1;

            (pion1 /. ((( len pion1) -' 1) + 1)) = (pion /. 3) by A86, A119, FINSEQ_1: 45

            .= Gij by FINSEQ_4: 18;

            then

             A122: Gij in ( LSeg (pion1,(( len pion1) -' 1))) by A118, A119, TOPREAL1: 21;

            Gij in ( LSeg (co,1)) by A50, RLTOPSP1: 68;

            hence thesis by A121, A122, XBOOLE_0:def 4;

          end;

          ( LSeg (pion1,(( len pion1) -' 1))) c= ( L~ pion) by A84, TOPREAL3: 19;

          then (( LSeg (pion1,(( len pion1) -' 1))) /\ ( LSeg (co,1))) c= {Gij} by A54, XBOOLE_1: 27;

          then

           A123: (( LSeg (pion1,(( len pion1) -' 1))) /\ ( LSeg (co,1))) = {Gij} by A120;

          ((( len pion1) - 1) + 1) <= ( len pion1);

          then

           A124: (( len pion1) -' 1) < ( len pion1) by A119, NAT_1: 13;

          ( len pion1) >= (2 + 1) by A87, FINSEQ_1: 45;

          then

           A125: (( len pion1) - 2) >= 0 by XREAL_1: 19;

          

          then ((( len pion1) -' 2) + 1) = ((( len pion1) - 2) + 1) by XREAL_0:def 2

          .= (( len pion1) -' 1) by A118, XREAL_0:def 2;

          then

           A126: (( LSeg ((go ^' pion1),(( len go) + (( len pion1) -' 2)))) /\ ( LSeg (co,1))) = {((go ^' pion1) /. ( len (go ^' pion1)))} by A49, A89, A88, A124, A123, TOPREAL8: 31;

          (( rng go) /\ ( rng pion1)) c= {(pion1 /. 1)} by A77, A94, A97, XBOOLE_1: 27;

          then

           A127: (go ^' pion1) is one-to-one by JORDAN1J: 55;

          ((( len (go ^' pion1)) + 1) - 1) = ((( len go) + ( len pion1)) - 1) by FINSEQ_6: 139;

          

          then (( len (go ^' pion1)) - 1) = (( len go) + (( len pion1) - 2))

          .= (( len go) + (( len pion1) -' 2)) by A125, XREAL_0:def 2;

          then

           A128: (( len (go ^' pion1)) -' 1) = (( len go) + (( len pion1) -' 2)) by XREAL_0:def 2;

          

           A129: ( L~ ( Cage (C,n))) = (( L~ US) \/ ( L~ LS)) by JORDAN1E: 13;

          then

           A130: ( L~ US) c= ( L~ ( Cage (C,n))) by XBOOLE_1: 7;

          then

           A131: ( L~ go) c= ( L~ ( Cage (C,n))) by A56;

          

           A132: ( len godo) >= ( len (go ^' pion1)) by TOPREAL8: 7;

          then

           A133: (1 + 1) <= ( len godo) by A103, XXREAL_0: 2;

          (go ^' pion1) is non trivial by A103, NAT_D: 60;

          then

          reconsider godo as non constant standard special_circular_sequence by A133, A88, A117, A102, A128, A126, A98, A127, A112, A115, JORDAN8: 4, JORDAN8: 5, TOPREAL8: 11, TOPREAL8: 33, TOPREAL8: 34;

          

           A134: ( L~ godo) = (( L~ (go ^' pion1)) \/ ( L~ co)) by A88, TOPREAL8: 35

          .= ((( L~ go) \/ ( L~ pion1)) \/ ( L~ co)) by A89, TOPREAL8: 35;

           A135:

          now

            assume

             A136: (Gik .. US) <= 1;

            (Gik .. US) >= 1 by A38, FINSEQ_4: 21;

            then (Gik .. US) = 1 by A136, XXREAL_0: 1;

            then Gik = (US /. 1) by A38, FINSEQ_5: 38;

            hence contradiction by A15, A37, JORDAN1F: 5;

          end;

          

           A137: US is_sequence_on G by JORDAN1G: 4;

          

           A138: (Gik `1 ) <= (Gi1k `1 ) by A1, A2, A3, A6, A10, JORDAN1A: 18;

          then

           A139: ( W-bound poz) = (Gik `1 ) by SPRECT_1: 54;

          

           A140: (Gi1k `1 ) = ((G * (i1,1)) `1 ) by A3, A6, A17, A10, GOBOARD5: 2

          .= (Gij `1 ) by A3, A4, A17, A19, GOBOARD5: 2;

          then

           A141: ( W-bound pio) = (Gij `1 ) by SPRECT_1: 54;

          ( W-bound (poz \/ pio)) = ( min (( W-bound poz),( W-bound pio))) by SPRECT_1: 47

          .= (Gik `1 ) by A140, A138, A139, A141, XXREAL_0:def 9;

          then

           A142: ( W-bound ( L~ pion1)) = (Gik `1 ) by A84, TOPREAL3: 16;

          

           A143: LA c= C by JORDAN6: 61;

          (Gik `1 ) >= Wbo by A30, A130, PSCOMP_1: 24;

          then

           A144: (Gik `1 ) > Wbo by A24, XXREAL_0: 1;

          

           A145: ( len US) >= 2 by A14, XXREAL_0: 2;

          

           A146: (( L~ go) \/ ( L~ co)) is compact by COMPTS_1: 10;

          

           A147: ( L~ LS) c= ( L~ ( Cage (C,n))) by A129, XBOOLE_1: 7;

          then

           A148: ( L~ co) c= ( L~ ( Cage (C,n))) by A48;

          

           A149: (( right_cell (godo,1,G)) \ ( L~ godo)) c= ( RightComp godo) by A133, A117, JORDAN9: 27;

          2 in ( dom godo) by A133, FINSEQ_3: 25;

          then

           A150: (godo /. 2) in ( rng godo) by PARTFUN2: 2;

          

           A151: ( rng godo) c= ( L~ godo) by A103, A132, SPPOL_2: 18, XXREAL_0: 2;

          

           A152: (godo /. 1) = ((go ^' pion1) /. 1) by FINSEQ_6: 155

          .= Wmin by A67, FINSEQ_6: 155;

          

           A153: ( W-min C) in LA by A113, TOPREAL1: 1;

          

           A154: ( W-min C) in C by SPRECT_1: 13;

           A155:

          now

            assume ( W-min C) in ( L~ godo);

            then

             A156: ( W-min C) in (( L~ go) \/ ( L~ pion1)) or ( W-min C) in ( L~ co) by A134, XBOOLE_0:def 3;

            per cases by A156, XBOOLE_0:def 3;

              suppose ( W-min C) in ( L~ go);

              then C meets ( L~ ( Cage (C,n))) by A131, A154, XBOOLE_0: 3;

              hence contradiction by JORDAN10: 5;

            end;

              suppose ( W-min C) in ( L~ pion1);

              hence contradiction by A9, A36, A84, A153, XBOOLE_0: 3;

            end;

              suppose ( W-min C) in ( L~ co);

              then C meets ( L~ ( Cage (C,n))) by A148, A154, XBOOLE_0: 3;

              hence contradiction by JORDAN10: 5;

            end;

          end;

          

           A157: (Wmin `1 ) = Wbo by EUCLID: 52;

          set ff = ( Rotate (( Cage (C,n)),Wmin));

          

           A158: (1 + 1) <= ( len ( Cage (C,n))) by GOBOARD7: 34, XXREAL_0: 2;

          Wmin in ( rng ( Cage (C,n))) by SPRECT_2: 43;

          then

           A159: (ff /. 1) = Wmin by FINSEQ_6: 92;

          

           A160: ( L~ ff) = ( L~ ( Cage (C,n))) by REVROT_1: 33;

          then (( W-max ( L~ ff)) .. ff) > 1 by A159, SPRECT_5: 22;

          then (( N-min ( L~ ff)) .. ff) > 1 by A159, A160, SPRECT_5: 23, XXREAL_0: 2;

          then (( N-max ( L~ ff)) .. ff) > 1 by A159, A160, SPRECT_5: 24, XXREAL_0: 2;

          then

           A161: (Emax .. ff) > 1 by A159, A160, SPRECT_5: 25, XXREAL_0: 2;

          

           A162: ( Cage (C,n)) is_sequence_on G by JORDAN9:def 1;

          then

           A163: ff is_sequence_on G by REVROT_1: 34;

          (1 + 1) <= ( len ( Rotate (( Cage (C,n)),Wmin))) by GOBOARD7: 34, XXREAL_0: 2;

          

          then ( right_cell (( Rotate (( Cage (C,n)),Wmin)),1)) = ( right_cell (ff,1,( GoB ff))) by JORDAN1H: 23

          .= ( right_cell (ff,1,( GoB ( Cage (C,n))))) by REVROT_1: 28

          .= ( right_cell (ff,1,G)) by JORDAN1H: 44

          .= ( right_cell ((ff -: Emax),1,G)) by A161, A163, JORDAN1J: 53

          .= ( right_cell (US,1,G)) by JORDAN1E:def 1

          .= ( right_cell (( R_Cut (US,Gik)),1,G)) by A38, A137, A135, JORDAN1J: 52

          .= ( right_cell ((go ^' pion1),1,G)) by A57, A116, JORDAN1J: 51

          .= ( right_cell (godo,1,G)) by A104, A117, JORDAN1J: 51;

          then ( W-min C) in ( right_cell (godo,1,G)) by JORDAN1I: 6;

          then

           A164: ( W-min C) in (( right_cell (godo,1,G)) \ ( L~ godo)) by A155, XBOOLE_0:def 5;

          

           A165: ( E-max C) in LA by A113, TOPREAL1: 1;

          Wmin in (( L~ go) \/ ( L~ co)) by A77, A68, XBOOLE_0:def 3;

          then

           A166: ( W-min (( L~ go) \/ ( L~ co))) = Wmin by A131, A148, A146, JORDAN1J: 21, XBOOLE_1: 8;

          (( W-min (( L~ go) \/ ( L~ co))) `1 ) = ( W-bound (( L~ go) \/ ( L~ co))) by EUCLID: 52;

          then ( W-min ((( L~ go) \/ ( L~ co)) \/ ( L~ pion1))) = ( W-min (( L~ go) \/ ( L~ co))) by A142, A146, A166, A157, A144, JORDAN1J: 33;

          then

           A167: ( W-min ( L~ godo)) = Wmin by A134, A166, XBOOLE_1: 4;

          (godo /. 2) = ((go ^' pion1) /. 2) by A103, FINSEQ_6: 159

          .= (US /. 2) by A42, A76, FINSEQ_6: 159

          .= ((US ^' LS) /. 2) by A145, FINSEQ_6: 159

          .= (( Rotate (( Cage (C,n)),Wmin)) /. 2) by JORDAN1E: 11;

          then (godo /. 2) in ( W-most ( L~ ( Cage (C,n)))) by JORDAN1I: 25;

          

          then ((godo /. 2) `1 ) = (( W-min ( L~ godo)) `1 ) by A167, PSCOMP_1: 31

          .= ( W-bound ( L~ godo)) by EUCLID: 52;

          then (godo /. 2) in ( W-most ( L~ godo)) by A151, A150, SPRECT_2: 12;

          then (( Rotate (godo,( W-min ( L~ godo)))) /. 2) in ( W-most ( L~ godo)) by A152, A167, FINSEQ_6: 89;

          then

          reconsider godo as clockwise_oriented non constant standard special_circular_sequence by JORDAN1I: 25;

          ( len US) in ( dom US) by FINSEQ_5: 6;

          

          then

           A168: (US . ( len US)) = (US /. ( len US)) by PARTFUN1:def 6

          .= Emax by JORDAN1F: 7;

          

           A169: ( east_halfline ( E-max C)) misses ( L~ go)

          proof

            assume ( east_halfline ( E-max C)) meets ( L~ go);

            then

            consider p be object such that

             A170: p in ( east_halfline ( E-max C)) and

             A171: p in ( L~ go) by XBOOLE_0: 3;

            reconsider p as Point of ( TOP-REAL 2) by A170;

            p in ( L~ US) by A56, A171;

            then p in (( east_halfline ( E-max C)) /\ ( L~ ( Cage (C,n)))) by A130, A170, XBOOLE_0:def 4;

            then

             A172: (p `1 ) = Ebo by JORDAN1A: 83, PSCOMP_1: 50;

            then

             A173: p = Emax by A56, A171, JORDAN1J: 46;

            then Emax = Gik by A30, A168, A171, JORDAN1J: 43;

            then (Gik `1 ) = ((G * (( len G),k)) `1 ) by A6, A10, A13, A172, A173, JORDAN1A: 71;

            hence contradiction by A2, A3, A12, A41, JORDAN1G: 7;

          end;

          now

            assume ( east_halfline ( E-max C)) meets ( L~ godo);

            then

             A174: ( east_halfline ( E-max C)) meets (( L~ go) \/ ( L~ pion1)) or ( east_halfline ( E-max C)) meets ( L~ co) by A134, XBOOLE_1: 70;

            per cases by A174, XBOOLE_1: 70;

              suppose ( east_halfline ( E-max C)) meets ( L~ go);

              hence contradiction by A169;

            end;

              suppose ( east_halfline ( E-max C)) meets ( L~ pion1);

              then

              consider p be object such that

               A175: p in ( east_halfline ( E-max C)) and

               A176: p in ( L~ pion1) by XBOOLE_0: 3;

              reconsider p as Point of ( TOP-REAL 2) by A175;

              

               A177: (p `2 ) = (( E-max C) `2 ) by A175, TOPREAL1:def 11;

               A178:

              now

                per cases by A36, A84, A176, XBOOLE_0:def 3;

                  suppose p in poz;

                  hence (p `1 ) <= (Gi1k `1 ) by A138, TOPREAL1: 3;

                end;

                  suppose p in pio;

                  hence (p `1 ) <= (Gi1k `1 ) by A140, GOBOARD7: 5;

                end;

              end;

              (i1 + 1) <= ( len G) by A3, NAT_1: 13;

              then ((i1 + 1) - 1) <= (( len G) - 1) by XREAL_1: 9;

              then

               A179: i1 <= (( len G) -' 1) by XREAL_0:def 2;

              (( len G) -' 1) <= ( len G) by NAT_D: 35;

              then (Gi1k `1 ) <= ((G * ((( len G) -' 1),1)) `1 ) by A6, A17, A10, A13, A21, A179, JORDAN1A: 18;

              then (p `1 ) <= ((G * ((( len G) -' 1),1)) `1 ) by A178, XXREAL_0: 2;

              then (p `1 ) <= ( E-bound C) by A21, JORDAN8: 12;

              then

               A180: (p `1 ) <= (( E-max C) `1 ) by EUCLID: 52;

              (p `1 ) >= (( E-max C) `1 ) by A175, TOPREAL1:def 11;

              then (p `1 ) = (( E-max C) `1 ) by A180, XXREAL_0: 1;

              then p = ( E-max C) by A177, TOPREAL3: 6;

              hence contradiction by A9, A36, A84, A165, A176, XBOOLE_0: 3;

            end;

              suppose ( east_halfline ( E-max C)) meets ( L~ co);

              then

              consider p be object such that

               A181: p in ( east_halfline ( E-max C)) and

               A182: p in ( L~ co) by XBOOLE_0: 3;

              reconsider p as Point of ( TOP-REAL 2) by A181;

              

               A183: p in ( LSeg (co,( Index (p,co)))) by A182, JORDAN3: 9;

              consider t be Nat such that

               A184: t in ( dom LS) and

               A185: (LS . t) = Gij by A44, FINSEQ_2: 10;

              1 <= t by A184, FINSEQ_3: 25;

              then

               A186: 1 < t by A46, A185, XXREAL_0: 1;

              t <= ( len LS) by A184, FINSEQ_3: 25;

              then (( Index (Gij,LS)) + 1) = t by A185, A186, JORDAN3: 12;

              then

               A187: ( len ( L_Cut (LS,Gij))) = (( len LS) - ( Index (Gij,LS))) by A16, A185, JORDAN3: 26;

              ( Index (p,co)) < ( len co) by A182, JORDAN3: 8;

              then ( Index (p,co)) < (( len LS) -' ( Index (Gij,LS))) by A187, XREAL_0:def 2;

              then (( Index (p,co)) + 1) <= (( len LS) -' ( Index (Gij,LS))) by NAT_1: 13;

              then

               A188: ( Index (p,co)) <= ((( len LS) -' ( Index (Gij,LS))) - 1) by XREAL_1: 19;

              

               A189: co = ( mid (LS,(Gij .. LS),( len LS))) by A44, JORDAN1J: 37;

              p in ( L~ LS) by A48, A182;

              then p in (( east_halfline ( E-max C)) /\ ( L~ ( Cage (C,n)))) by A147, A181, XBOOLE_0:def 4;

              then

               A190: (p `1 ) = Ebo by JORDAN1A: 83, PSCOMP_1: 50;

              

               A191: (( Index (Gij,LS)) + 1) = (Gij .. LS) by A46, A44, JORDAN1J: 56;

              ( 0 + ( Index (Gij,LS))) < ( len LS) by A16, JORDAN3: 8;

              then (( len LS) - ( Index (Gij,LS))) > 0 by XREAL_1: 20;

              then ( Index (p,co)) <= ((( len LS) - ( Index (Gij,LS))) - 1) by A188, XREAL_0:def 2;

              then ( Index (p,co)) <= (( len LS) - (Gij .. LS)) by A191;

              then ( Index (p,co)) <= (( len LS) -' (Gij .. LS)) by XREAL_0:def 2;

              then

               A192: ( Index (p,co)) < ((( len LS) -' (Gij .. LS)) + 1) by NAT_1: 13;

              

               A193: 1 <= ( Index (p,co)) by A182, JORDAN3: 8;

              

               A194: (Gij .. LS) <= ( len LS) by A44, FINSEQ_4: 21;

              (Gij .. LS) <> ( len LS) by A43, A44, FINSEQ_4: 19;

              then

               A195: (Gij .. LS) < ( len LS) by A194, XXREAL_0: 1;

              

               A196: (1 + 1) <= ( len LS) by A31, XXREAL_0: 2;

              then

               A197: 2 in ( dom LS) by FINSEQ_3: 25;

              set tt = ((( Index (p,co)) + (Gij .. LS)) -' 1);

              set RC = ( Rotate (( Cage (C,n)),Emax));

              

               A198: ( E-max C) in ( right_cell (RC,1)) by JORDAN1I: 7;

              

               A199: ( GoB RC) = ( GoB ( Cage (C,n))) by REVROT_1: 28

              .= G by JORDAN1H: 44;

              

               A200: ( L~ RC) = ( L~ ( Cage (C,n))) by REVROT_1: 33;

              consider jj2 be Nat such that

               A201: 1 <= jj2 and

               A202: jj2 <= ( width G) and

               A203: Emax = (G * (( len G),jj2)) by JORDAN1D: 25;

              

               A204: ( len G) >= 4 by JORDAN8: 10;

              then ( len G) >= 1 by XXREAL_0: 2;

              then

               A205: [( len G), jj2] in ( Indices G) by A201, A202, MATRIX_0: 30;

              

               A206: ( len RC) = ( len ( Cage (C,n))) by FINSEQ_6: 179;

              LS = (RC -: Wmin) by JORDAN1G: 18;

              then

               A207: ( LSeg (LS,1)) = ( LSeg (RC,1)) by A196, SPPOL_2: 9;

              

               A208: Emax in ( rng ( Cage (C,n))) by SPRECT_2: 46;

              RC is_sequence_on G by A162, REVROT_1: 34;

              then

              consider ii,jj be Nat such that

               A209: [ii, (jj + 1)] in ( Indices G) and

               A210: [ii, jj] in ( Indices G) and

               A211: (RC /. 1) = (G * (ii,(jj + 1))) and

               A212: (RC /. (1 + 1)) = (G * (ii,jj)) by A158, A200, A206, A208, FINSEQ_6: 92, JORDAN1I: 23;

              

               A213: ((jj + 1) + 1) <> jj;

              

               A214: 1 <= jj by A210, MATRIX_0: 32;

              (RC /. 1) = ( E-max ( L~ RC)) by A200, A208, FINSEQ_6: 92;

              then

               A215: ii = ( len G) by A200, A209, A211, A203, A205, GOBOARD1: 5;

              then (ii - 1) >= (4 - 1) by A204, XREAL_1: 9;

              then

               A216: (ii - 1) >= 1 by XXREAL_0: 2;

              then

               A217: 1 <= (ii -' 1) by XREAL_0:def 2;

              

               A218: jj <= ( width G) by A210, MATRIX_0: 32;

              then

               A219: ((G * (( len G),jj)) `1 ) = Ebo by A13, A214, JORDAN1A: 71;

              

               A220: (jj + 1) <= ( width G) by A209, MATRIX_0: 32;

              (ii + 1) <> ii;

              then

               A221: ( right_cell (RC,1)) = ( cell (G,(ii -' 1),jj)) by A158, A206, A199, A209, A210, A211, A212, A213, GOBOARD5:def 6;

              

               A222: ii <= ( len G) by A210, MATRIX_0: 32;

              

               A223: 1 <= ii by A210, MATRIX_0: 32;

              

               A224: ii <= ( len G) by A209, MATRIX_0: 32;

              

               A225: 1 <= (jj + 1) by A209, MATRIX_0: 32;

              then

               A226: Ebo = ((G * (( len G),(jj + 1))) `1 ) by A13, A220, JORDAN1A: 71;

              

               A227: 1 <= ii by A209, MATRIX_0: 32;

              then

               A228: ((ii -' 1) + 1) = ii by XREAL_1: 235;

              then

               A229: (ii -' 1) < ( len G) by A224, NAT_1: 13;

              

              then

               A230: ((G * ((ii -' 1),(jj + 1))) `2 ) = ((G * (1,(jj + 1))) `2 ) by A225, A220, A217, GOBOARD5: 1

              .= ((G * (ii,(jj + 1))) `2 ) by A227, A224, A225, A220, GOBOARD5: 1;

              

               A231: (( E-max C) `2 ) = (p `2 ) by A181, TOPREAL1:def 11;

              then

               A232: (p `2 ) <= ((G * ((ii -' 1),(jj + 1))) `2 ) by A198, A224, A220, A214, A221, A228, A216, JORDAN9: 17;

              

               A233: ((G * ((ii -' 1),jj)) `2 ) = ((G * (1,jj)) `2 ) by A214, A218, A217, A229, GOBOARD5: 1

              .= ((G * (ii,jj)) `2 ) by A223, A222, A214, A218, GOBOARD5: 1;

              ((G * ((ii -' 1),jj)) `2 ) <= (p `2 ) by A231, A198, A224, A220, A214, A221, A228, A216, JORDAN9: 17;

              then p in ( LSeg ((RC /. 1),(RC /. (1 + 1)))) by A190, A211, A212, A215, A232, A233, A230, A219, A226, GOBOARD7: 7;

              then

               A234: p in ( LSeg (LS,1)) by A158, A207, A206, TOPREAL1:def 3;

              1 <= (Gij .. LS) by A44, FINSEQ_4: 21;

              then

               A235: ( LSeg (( mid (LS,(Gij .. LS),( len LS))),( Index (p,co)))) = ( LSeg (LS,((( Index (p,co)) + (Gij .. LS)) -' 1))) by A195, A193, A192, JORDAN4: 19;

              1 <= ( Index (Gij,LS)) by A16, JORDAN3: 8;

              then

               A236: (1 + 1) <= (Gij .. LS) by A191, XREAL_1: 7;

              then (( Index (p,co)) + (Gij .. LS)) >= ((1 + 1) + 1) by A193, XREAL_1: 7;

              then ((( Index (p,co)) + (Gij .. LS)) - 1) >= (((1 + 1) + 1) - 1) by XREAL_1: 9;

              then

               A237: tt >= (1 + 1) by XREAL_0:def 2;

              now

                per cases by A237, XXREAL_0: 1;

                  suppose tt > (1 + 1);

                  then ( LSeg (LS,1)) misses ( LSeg (LS,tt)) by TOPREAL1:def 7;

                  hence contradiction by A234, A183, A189, A235, XBOOLE_0: 3;

                end;

                  suppose

                   A238: tt = (1 + 1);

                  then (1 + 1) = ((( Index (p,co)) + (Gij .. LS)) - 1) by XREAL_0:def 2;

                  then ((1 + 1) + 1) = (( Index (p,co)) + (Gij .. LS));

                  then

                   A239: (Gij .. LS) = 2 by A193, A236, JORDAN1E: 6;

                  (( LSeg (LS,1)) /\ ( LSeg (LS,tt))) = {(LS /. 2)} by A31, A238, TOPREAL1:def 6;

                  then p in {(LS /. 2)} by A234, A183, A189, A235, XBOOLE_0:def 4;

                  then

                   A240: p = (LS /. 2) by TARSKI:def 1;

                  then

                   A241: p in ( rng LS) by A197, PARTFUN2: 2;

                  (p .. LS) = 2 by A197, A240, FINSEQ_5: 41;

                  then p = Gij by A44, A239, A241, FINSEQ_5: 9;

                  then (Gij `1 ) = Ebo by A240, JORDAN1G: 32;

                  then (Gij `1 ) = ((G * (( len G),j)) `1 ) by A4, A19, A13, JORDAN1A: 71;

                  hence contradiction by A3, A25, A22, JORDAN1G: 7;

                end;

              end;

              hence contradiction;

            end;

          end;

          then ( east_halfline ( E-max C)) c= (( L~ godo) ` ) by SUBSET_1: 23;

          then

          consider W be Subset of ( TOP-REAL 2) such that

           A242: W is_a_component_of (( L~ godo) ` ) and

           A243: ( east_halfline ( E-max C)) c= W by GOBOARD9: 3;

           not W is bounded by A243, JORDAN2C: 121, RLTOPSP1: 42;

          then W is_outside_component_of ( L~ godo) by A242, JORDAN2C:def 3;

          then W c= ( UBD ( L~ godo)) by JORDAN2C: 23;

          then

           A244: ( east_halfline ( E-max C)) c= ( UBD ( L~ godo)) by A243;

          ( E-max C) in ( east_halfline ( E-max C)) by TOPREAL1: 38;

          then ( E-max C) in ( UBD ( L~ godo)) by A244;

          then ( E-max C) in ( LeftComp godo) by GOBRD14: 36;

          then LA meets ( L~ godo) by A114, A153, A165, A149, A164, JORDAN1J: 36;

          then

           A245: LA meets (( L~ go) \/ ( L~ pion1)) or LA meets ( L~ co) by A134, XBOOLE_1: 70;

          now

            per cases by A245, XBOOLE_1: 70;

              suppose LA meets ( L~ go);

              then LA meets ( L~ ( Cage (C,n))) by A56, A130, XBOOLE_1: 1, XBOOLE_1: 63;

              hence contradiction by A143, JORDAN10: 5, XBOOLE_1: 63;

            end;

              suppose LA meets ( L~ pion1);

              hence contradiction by A9, A36, A84;

            end;

              suppose LA meets ( L~ co);

              then LA meets ( L~ ( Cage (C,n))) by A48, A147, XBOOLE_1: 1, XBOOLE_1: 63;

              hence contradiction by A143, JORDAN10: 5, XBOOLE_1: 63;

            end;

          end;

          hence contradiction;

        end;

          suppose (Gij `1 ) = (Gik `1 );

          then

           A246: i1 = i2 by A25, A12, JORDAN1G: 7;

          then poz = {Gi1k} by RLTOPSP1: 70;

          then poz c= pio by A27, ZFMISC_1: 31;

          then (pio \/ poz) = pio by XBOOLE_1: 12;

          hence contradiction by A1, A3, A4, A5, A6, A7, A8, A9, A246, JORDAN1J: 58;

        end;

          suppose (Gij `2 ) = (Gik `2 );

          then

           A247: j = k by A25, A12, JORDAN1G: 6;

          then pio = {Gi1k} by RLTOPSP1: 70;

          then pio c= poz by A26, ZFMISC_1: 31;

          then (pio \/ poz) = poz by XBOOLE_1: 12;

          hence contradiction by A1, A2, A3, A4, A6, A7, A8, A9, A247, Th36;

        end;

      end;

      hence contradiction;

    end;

    theorem :: JORDAN15:48

    

     Th48: for C be Simple_closed_curve holds for i1,i2,j,k be Nat holds 1 < i1 & i1 < ( len ( Gauge (C,(n + 1)))) & 1 < i2 & i2 < ( len ( Gauge (C,(n + 1)))) & 1 <= j & j <= k & k <= ( width ( Gauge (C,(n + 1)))) & (( Gauge (C,(n + 1))) * (i1,k)) in ( Upper_Arc ( L~ ( Cage (C,(n + 1))))) & (( Gauge (C,(n + 1))) * (i2,j)) in ( Lower_Arc ( L~ ( Cage (C,(n + 1))))) implies (( LSeg ((( Gauge (C,(n + 1))) * (i2,j)),(( Gauge (C,(n + 1))) * (i2,k)))) \/ ( LSeg ((( Gauge (C,(n + 1))) * (i2,k)),(( Gauge (C,(n + 1))) * (i1,k))))) meets ( Upper_Arc C)

    proof

      let C be Simple_closed_curve;

      let i1,i2,j,k be Nat;

      set G = ( Gauge (C,(n + 1)));

      assume that

       A1: 1 < i1 and

       A2: i1 < ( len G) and

       A3: 1 < i2 and

       A4: i2 < ( len G) and

       A5: 1 <= j and

       A6: j <= k and

       A7: k <= ( width G) and

       A8: (G * (i1,k)) in ( Upper_Arc ( L~ ( Cage (C,(n + 1))))) and

       A9: (G * (i2,j)) in ( Lower_Arc ( L~ ( Cage (C,(n + 1)))));

      

       A10: 1 <= k by A5, A6, XXREAL_0: 2;

      then

       A11: [i2, k] in ( Indices G) by A3, A4, A7, MATRIX_0: 30;

      

       A12: [i1, k] in ( Indices G) by A1, A2, A7, A10, MATRIX_0: 30;

      ((G * (i2,k)) `2 ) = ((G * (1,k)) `2 ) by A3, A4, A7, A10, GOBOARD5: 1

      .= ((G * (i1,k)) `2 ) by A1, A2, A7, A10, GOBOARD5: 1;

      then

       A13: ( LSeg ((G * (i2,k)),(G * (i1,k)))) is horizontal by SPPOL_1: 15;

      

       A14: ( Lower_Arc ( L~ ( Cage (C,(n + 1))))) = ( L~ ( Lower_Seq (C,(n + 1)))) by JORDAN1G: 56;

      

       A15: j <= ( width G) by A6, A7, XXREAL_0: 2;

      then

       A16: [i2, j] in ( Indices G) by A3, A4, A5, MATRIX_0: 30;

      ((G * (i2,j)) `1 ) = ((G * (i2,1)) `1 ) by A3, A4, A5, A15, GOBOARD5: 2

      .= ((G * (i2,k)) `1 ) by A3, A4, A7, A10, GOBOARD5: 2;

      then

       A17: ( LSeg ((G * (i2,j)),(G * (i2,k)))) is vertical by SPPOL_1: 16;

      

       A18: ( Upper_Arc ( L~ ( Cage (C,(n + 1))))) = ( L~ ( Upper_Seq (C,(n + 1)))) by JORDAN1G: 55;

      

       A19: [i2, k] in ( Indices G) by A3, A4, A7, A10, MATRIX_0: 30;

      now

        per cases ;

          suppose

           A20: ( LSeg ((G * (i2,j)),(G * (i2,k)))) meets ( Upper_Arc ( L~ ( Cage (C,(n + 1)))));

          set X = (( LSeg ((G * (i2,j)),(G * (i2,k)))) /\ ( L~ ( Upper_Seq (C,(n + 1)))));

          ex x be object st x in ( LSeg ((G * (i2,j)),(G * (i2,k)))) & x in ( L~ ( Upper_Seq (C,(n + 1)))) by A18, A20, XBOOLE_0: 3;

          then

          reconsider X1 = X as non empty compact Subset of ( TOP-REAL 2) by XBOOLE_0:def 4;

          consider pp be object such that

           A21: pp in ( S-most X1) by XBOOLE_0:def 1;

          reconsider pp as Point of ( TOP-REAL 2) by A21;

          

           A22: pp in X by A21, XBOOLE_0:def 4;

          then

           A23: pp in ( L~ ( Upper_Seq (C,(n + 1)))) by XBOOLE_0:def 4;

          

           A24: pp in ( LSeg ((G * (i2,j)),(G * (i2,k)))) by A22, XBOOLE_0:def 4;

          consider m be Nat such that

           A25: j <= m and

           A26: m <= k and

           A27: ((G * (i2,m)) `2 ) = ( lower_bound ( proj2 .: (( LSeg ((G * (i2,j)),(G * (i2,k)))) /\ ( L~ ( Upper_Seq (C,(n + 1))))))) by A6, A18, A16, A19, A20, JORDAN1F: 1, JORDAN1G: 4;

          

           A28: m <= ( width G) by A7, A26, XXREAL_0: 2;

          1 <= m by A5, A25, XXREAL_0: 2;

          then

           A29: ((G * (i2,m)) `1 ) = ((G * (i2,1)) `1 ) by A3, A4, A28, GOBOARD5: 2;

          then

           A30: |[((G * (i2,1)) `1 ), ( lower_bound ( proj2 .: X))]| = (G * (i2,m)) by A27, EUCLID: 53;

          then ((G * (i2,j)) `1 ) = ( |[((G * (i2,1)) `1 ), ( lower_bound ( proj2 .: X))]| `1 ) by A3, A4, A5, A15, A29, GOBOARD5: 2;

          then

           A31: (pp `1 ) = ( |[((G * (i2,1)) `1 ), ( lower_bound ( proj2 .: X))]| `1 ) by A17, A24, SPPOL_1: 41;

          ( |[((G * (i2,1)) `1 ), ( lower_bound ( proj2 .: X))]| `2 ) = ( S-bound X) by A27, A30, SPRECT_1: 44

          .= (( S-min X) `2 ) by EUCLID: 52

          .= (pp `2 ) by A21, PSCOMP_1: 55;

          then (G * (i2,m)) in ( Upper_Arc ( L~ ( Cage (C,(n + 1))))) by A18, A30, A23, A31, TOPREAL3: 6;

          then ( LSeg ((G * (i2,j)),(G * (i2,m)))) meets ( Upper_Arc C) by A3, A4, A5, A9, A25, A28, Th24;

          then ( LSeg ((G * (i2,j)),(G * (i2,k)))) meets ( Upper_Arc C) by A3, A4, A5, A7, A25, A26, Th5, XBOOLE_1: 63;

          hence thesis by XBOOLE_1: 70;

        end;

          suppose

           A32: ( LSeg ((G * (i2,k)),(G * (i1,k)))) meets ( Lower_Arc ( L~ ( Cage (C,(n + 1))))) & i2 <= i1;

          set X = (( LSeg ((G * (i2,k)),(G * (i1,k)))) /\ ( L~ ( Lower_Seq (C,(n + 1)))));

          ex x be object st x in ( LSeg ((G * (i2,k)),(G * (i1,k)))) & x in ( L~ ( Lower_Seq (C,(n + 1)))) by A14, A32, XBOOLE_0: 3;

          then

          reconsider X1 = X as non empty compact Subset of ( TOP-REAL 2) by XBOOLE_0:def 4;

          consider pp be object such that

           A33: pp in ( E-most X1) by XBOOLE_0:def 1;

          reconsider pp as Point of ( TOP-REAL 2) by A33;

          

           A34: pp in X by A33, XBOOLE_0:def 4;

          then

           A35: pp in ( L~ ( Lower_Seq (C,(n + 1)))) by XBOOLE_0:def 4;

          

           A36: pp in ( LSeg ((G * (i2,k)),(G * (i1,k)))) by A34, XBOOLE_0:def 4;

          consider m be Nat such that

           A37: i2 <= m and

           A38: m <= i1 and

           A39: ((G * (m,k)) `1 ) = ( upper_bound ( proj1 .: (( LSeg ((G * (i2,k)),(G * (i1,k)))) /\ ( L~ ( Lower_Seq (C,(n + 1))))))) by A14, A11, A12, A32, JORDAN1F: 4, JORDAN1G: 5;

          

           A40: 1 < m by A3, A37, XXREAL_0: 2;

          m < ( len G) by A2, A38, XXREAL_0: 2;

          then

           A41: ((G * (m,k)) `2 ) = ((G * (1,k)) `2 ) by A7, A10, A40, GOBOARD5: 1;

          then

           A42: |[( upper_bound ( proj1 .: X)), ((G * (1,k)) `2 )]| = (G * (m,k)) by A39, EUCLID: 53;

          then ((G * (i2,k)) `2 ) = ( |[( upper_bound ( proj1 .: X)), ((G * (1,k)) `2 )]| `2 ) by A3, A4, A7, A10, A41, GOBOARD5: 1;

          then

           A43: (pp `2 ) = ( |[( upper_bound ( proj1 .: X)), ((G * (1,k)) `2 )]| `2 ) by A13, A36, SPPOL_1: 40;

          ( |[( upper_bound ( proj1 .: X)), ((G * (1,k)) `2 )]| `1 ) = ( E-bound X) by A39, A42, SPRECT_1: 46

          .= (( E-min X) `1 ) by EUCLID: 52

          .= (pp `1 ) by A33, PSCOMP_1: 47;

          then (G * (m,k)) in ( Lower_Arc ( L~ ( Cage (C,(n + 1))))) by A14, A42, A35, A43, TOPREAL3: 6;

          then ( LSeg ((G * (m,k)),(G * (i1,k)))) meets ( Upper_Arc C) by A2, A7, A8, A10, A38, A40, Th33;

          then ( LSeg ((G * (i2,k)),(G * (i1,k)))) meets ( Upper_Arc C) by A2, A3, A7, A10, A37, A38, Th6, XBOOLE_1: 63;

          hence thesis by XBOOLE_1: 70;

        end;

          suppose

           A44: ( LSeg ((G * (i2,k)),(G * (i1,k)))) meets ( Lower_Arc ( L~ ( Cage (C,(n + 1))))) & i1 < i2;

          set X = (( LSeg ((G * (i1,k)),(G * (i2,k)))) /\ ( L~ ( Lower_Seq (C,(n + 1)))));

          ex x be object st x in ( LSeg ((G * (i1,k)),(G * (i2,k)))) & x in ( L~ ( Lower_Seq (C,(n + 1)))) by A14, A44, XBOOLE_0: 3;

          then

          reconsider X1 = X as non empty compact Subset of ( TOP-REAL 2) by XBOOLE_0:def 4;

          consider pp be object such that

           A45: pp in ( W-most X1) by XBOOLE_0:def 1;

          reconsider pp as Point of ( TOP-REAL 2) by A45;

          

           A46: pp in X by A45, XBOOLE_0:def 4;

          then

           A47: pp in ( L~ ( Lower_Seq (C,(n + 1)))) by XBOOLE_0:def 4;

          

           A48: pp in ( LSeg ((G * (i1,k)),(G * (i2,k)))) by A46, XBOOLE_0:def 4;

          consider m be Nat such that

           A49: i1 <= m and

           A50: m <= i2 and

           A51: ((G * (m,k)) `1 ) = ( lower_bound ( proj1 .: (( LSeg ((G * (i1,k)),(G * (i2,k)))) /\ ( L~ ( Lower_Seq (C,(n + 1))))))) by A14, A11, A12, A44, JORDAN1F: 3, JORDAN1G: 5;

          

           A52: m < ( len G) by A4, A50, XXREAL_0: 2;

          1 < m by A1, A49, XXREAL_0: 2;

          then

           A53: ((G * (m,k)) `2 ) = ((G * (1,k)) `2 ) by A7, A10, A52, GOBOARD5: 1;

          then

           A54: |[( lower_bound ( proj1 .: X)), ((G * (1,k)) `2 )]| = (G * (m,k)) by A51, EUCLID: 53;

          then ((G * (i1,k)) `2 ) = ( |[( lower_bound ( proj1 .: X)), ((G * (1,k)) `2 )]| `2 ) by A1, A2, A7, A10, A53, GOBOARD5: 1;

          then

           A55: (pp `2 ) = ( |[( lower_bound ( proj1 .: X)), ((G * (1,k)) `2 )]| `2 ) by A13, A48, SPPOL_1: 40;

          ( |[( lower_bound ( proj1 .: X)), ((G * (1,k)) `2 )]| `1 ) = ( W-bound X) by A51, A54, SPRECT_1: 43

          .= (( W-min X) `1 ) by EUCLID: 52

          .= (pp `1 ) by A45, PSCOMP_1: 31;

          then (G * (m,k)) in ( Lower_Arc ( L~ ( Cage (C,(n + 1))))) by A14, A54, A47, A55, TOPREAL3: 6;

          then ( LSeg ((G * (i1,k)),(G * (m,k)))) meets ( Upper_Arc C) by A1, A7, A8, A10, A49, A52, Th41;

          then ( LSeg ((G * (i1,k)),(G * (i2,k)))) meets ( Upper_Arc C) by A1, A4, A7, A10, A49, A50, Th6, XBOOLE_1: 63;

          hence thesis by XBOOLE_1: 70;

        end;

          suppose

           A56: ( LSeg ((G * (i2,j)),(G * (i2,k)))) misses ( Upper_Arc ( L~ ( Cage (C,(n + 1))))) & ( LSeg ((( Gauge (C,(n + 1))) * (i2,k)),(( Gauge (C,(n + 1))) * (i1,k)))) misses ( Lower_Arc ( L~ ( Cage (C,(n + 1)))));

          consider j1 be Nat such that

           A57: j <= j1 and

           A58: j1 <= k and

           A59: (( LSeg ((G * (i2,j1)),(G * (i2,k)))) /\ ( L~ ( Lower_Seq (C,(n + 1))))) = {(G * (i2,j1))} by A3, A4, A5, A6, A7, A9, A14, Th9;

          (G * (i2,j1)) in (( LSeg ((G * (i2,j1)),(G * (i2,k)))) /\ ( L~ ( Lower_Seq (C,(n + 1))))) by A59, TARSKI:def 1;

          then

           A60: (G * (i2,j1)) in ( L~ ( Lower_Seq (C,(n + 1)))) by XBOOLE_0:def 4;

          

           A61: 1 <= j1 by A5, A57, XXREAL_0: 2;

          now

            per cases ;

              suppose

               A62: i2 <= i1;

              

               A63: ( LSeg ((G * (i2,j1)),(G * (i2,k)))) c= ( LSeg ((G * (i2,j)),(G * (i2,k)))) by A3, A4, A5, A7, A57, A58, Th5;

              consider i3 be Nat such that

               A64: i2 <= i3 and

               A65: i3 <= i1 and

               A66: (( LSeg ((G * (i2,k)),(G * (i3,k)))) /\ ( L~ ( Upper_Seq (C,(n + 1))))) = {(G * (i3,k))} by A2, A3, A7, A8, A18, A10, A62, Th13;

              

               A67: ( LSeg ((G * (i2,k)),(G * (i3,k)))) c= ( LSeg ((G * (i2,k)),(G * (i1,k)))) by A2, A3, A7, A10, A64, A65, Th6;

              then

               A68: (( LSeg ((G * (i2,j1)),(G * (i2,k)))) \/ ( LSeg ((G * (i2,k)),(G * (i3,k))))) c= (( LSeg ((G * (i2,j)),(G * (i2,k)))) \/ ( LSeg ((G * (i2,k)),(G * (i1,k))))) by A63, XBOOLE_1: 13;

              (G * (i3,k)) in (( LSeg ((G * (i2,k)),(G * (i3,k)))) /\ ( L~ ( Upper_Seq (C,(n + 1))))) by A66, TARSKI:def 1;

              then

               A69: (G * (i3,k)) in ( L~ ( Upper_Seq (C,(n + 1)))) by XBOOLE_0:def 4;

              

               A70: ((( LSeg ((G * (i2,j1)),(G * (i2,k)))) \/ ( LSeg ((G * (i2,k)),(G * (i3,k))))) /\ ( L~ ( Upper_Seq (C,(n + 1))))) = {(G * (i3,k))}

              proof

                thus ((( LSeg ((G * (i2,j1)),(G * (i2,k)))) \/ ( LSeg ((G * (i2,k)),(G * (i3,k))))) /\ ( L~ ( Upper_Seq (C,(n + 1))))) c= {(G * (i3,k))}

                proof

                  let x be object;

                  assume

                   A71: x in ((( LSeg ((G * (i2,j1)),(G * (i2,k)))) \/ ( LSeg ((G * (i2,k)),(G * (i3,k))))) /\ ( L~ ( Upper_Seq (C,(n + 1)))));

                  then x in (( LSeg ((G * (i2,j1)),(G * (i2,k)))) \/ ( LSeg ((G * (i2,k)),(G * (i3,k))))) by XBOOLE_0:def 4;

                  then

                   A72: x in ( LSeg ((G * (i2,j1)),(G * (i2,k)))) or x in ( LSeg ((G * (i2,k)),(G * (i3,k)))) by XBOOLE_0:def 3;

                  x in ( L~ ( Upper_Seq (C,(n + 1)))) by A71, XBOOLE_0:def 4;

                  hence thesis by A18, A56, A66, A63, A72, XBOOLE_0:def 4;

                end;

                let x be object;

                (G * (i3,k)) in ( LSeg ((G * (i2,k)),(G * (i3,k)))) by RLTOPSP1: 68;

                then

                 A73: (G * (i3,k)) in (( LSeg ((G * (i2,j1)),(G * (i2,k)))) \/ ( LSeg ((G * (i2,k)),(G * (i3,k))))) by XBOOLE_0:def 3;

                assume x in {(G * (i3,k))};

                then x = (G * (i3,k)) by TARSKI:def 1;

                hence thesis by A69, A73, XBOOLE_0:def 4;

              end;

              

               A74: ((( LSeg ((G * (i2,j1)),(G * (i2,k)))) \/ ( LSeg ((G * (i2,k)),(G * (i3,k))))) /\ ( L~ ( Lower_Seq (C,(n + 1))))) = {(G * (i2,j1))}

              proof

                thus ((( LSeg ((G * (i2,j1)),(G * (i2,k)))) \/ ( LSeg ((G * (i2,k)),(G * (i3,k))))) /\ ( L~ ( Lower_Seq (C,(n + 1))))) c= {(G * (i2,j1))}

                proof

                  let x be object;

                  assume

                   A75: x in ((( LSeg ((G * (i2,j1)),(G * (i2,k)))) \/ ( LSeg ((G * (i2,k)),(G * (i3,k))))) /\ ( L~ ( Lower_Seq (C,(n + 1)))));

                  then x in (( LSeg ((G * (i2,j1)),(G * (i2,k)))) \/ ( LSeg ((G * (i2,k)),(G * (i3,k))))) by XBOOLE_0:def 4;

                  then

                   A76: x in ( LSeg ((G * (i2,j1)),(G * (i2,k)))) or x in ( LSeg ((G * (i2,k)),(G * (i3,k)))) by XBOOLE_0:def 3;

                  x in ( L~ ( Lower_Seq (C,(n + 1)))) by A75, XBOOLE_0:def 4;

                  hence thesis by A14, A56, A59, A67, A76, XBOOLE_0:def 4;

                end;

                let x be object;

                (G * (i2,j1)) in ( LSeg ((G * (i2,j1)),(G * (i2,k)))) by RLTOPSP1: 68;

                then

                 A77: (G * (i2,j1)) in (( LSeg ((G * (i2,j1)),(G * (i2,k)))) \/ ( LSeg ((G * (i2,k)),(G * (i3,k))))) by XBOOLE_0:def 3;

                assume x in {(G * (i2,j1))};

                then x = (G * (i2,j1)) by TARSKI:def 1;

                hence thesis by A60, A77, XBOOLE_0:def 4;

              end;

              i3 < ( len G) by A2, A65, XXREAL_0: 2;

              hence thesis by A3, A7, A58, A61, A64, A68, A70, A74, Th44, XBOOLE_1: 63;

            end;

              suppose

               A78: i1 < i2;

              

               A79: ( LSeg ((G * (i2,j1)),(G * (i2,k)))) c= ( LSeg ((G * (i2,j)),(G * (i2,k)))) by A3, A4, A5, A7, A57, A58, Th5;

              consider i3 be Nat such that

               A80: i1 <= i3 and

               A81: i3 <= i2 and

               A82: (( LSeg ((G * (i3,k)),(G * (i2,k)))) /\ ( L~ ( Upper_Seq (C,(n + 1))))) = {(G * (i3,k))} by A1, A4, A7, A8, A18, A10, A78, Th18;

              

               A83: ( LSeg ((G * (i2,k)),(G * (i3,k)))) c= ( LSeg ((G * (i2,k)),(G * (i1,k)))) by A1, A4, A7, A10, A80, A81, Th6;

              then

               A84: (( LSeg ((G * (i2,j1)),(G * (i2,k)))) \/ ( LSeg ((G * (i2,k)),(G * (i3,k))))) c= (( LSeg ((G * (i2,j)),(G * (i2,k)))) \/ ( LSeg ((G * (i2,k)),(G * (i1,k))))) by A79, XBOOLE_1: 13;

              (G * (i3,k)) in (( LSeg ((G * (i2,k)),(G * (i3,k)))) /\ ( L~ ( Upper_Seq (C,(n + 1))))) by A82, TARSKI:def 1;

              then

               A85: (G * (i3,k)) in ( L~ ( Upper_Seq (C,(n + 1)))) by XBOOLE_0:def 4;

              

               A86: ((( LSeg ((G * (i2,j1)),(G * (i2,k)))) \/ ( LSeg ((G * (i2,k)),(G * (i3,k))))) /\ ( L~ ( Upper_Seq (C,(n + 1))))) = {(G * (i3,k))}

              proof

                thus ((( LSeg ((G * (i2,j1)),(G * (i2,k)))) \/ ( LSeg ((G * (i2,k)),(G * (i3,k))))) /\ ( L~ ( Upper_Seq (C,(n + 1))))) c= {(G * (i3,k))}

                proof

                  let x be object;

                  assume

                   A87: x in ((( LSeg ((G * (i2,j1)),(G * (i2,k)))) \/ ( LSeg ((G * (i2,k)),(G * (i3,k))))) /\ ( L~ ( Upper_Seq (C,(n + 1)))));

                  then x in (( LSeg ((G * (i2,j1)),(G * (i2,k)))) \/ ( LSeg ((G * (i2,k)),(G * (i3,k))))) by XBOOLE_0:def 4;

                  then

                   A88: x in ( LSeg ((G * (i2,j1)),(G * (i2,k)))) or x in ( LSeg ((G * (i2,k)),(G * (i3,k)))) by XBOOLE_0:def 3;

                  x in ( L~ ( Upper_Seq (C,(n + 1)))) by A87, XBOOLE_0:def 4;

                  hence thesis by A18, A56, A82, A79, A88, XBOOLE_0:def 4;

                end;

                let x be object;

                (G * (i3,k)) in ( LSeg ((G * (i2,k)),(G * (i3,k)))) by RLTOPSP1: 68;

                then

                 A89: (G * (i3,k)) in (( LSeg ((G * (i2,j1)),(G * (i2,k)))) \/ ( LSeg ((G * (i2,k)),(G * (i3,k))))) by XBOOLE_0:def 3;

                assume x in {(G * (i3,k))};

                then x = (G * (i3,k)) by TARSKI:def 1;

                hence thesis by A85, A89, XBOOLE_0:def 4;

              end;

              

               A90: ((( LSeg ((G * (i2,j1)),(G * (i2,k)))) \/ ( LSeg ((G * (i2,k)),(G * (i3,k))))) /\ ( L~ ( Lower_Seq (C,(n + 1))))) = {(G * (i2,j1))}

              proof

                thus ((( LSeg ((G * (i2,j1)),(G * (i2,k)))) \/ ( LSeg ((G * (i2,k)),(G * (i3,k))))) /\ ( L~ ( Lower_Seq (C,(n + 1))))) c= {(G * (i2,j1))}

                proof

                  let x be object;

                  assume

                   A91: x in ((( LSeg ((G * (i2,j1)),(G * (i2,k)))) \/ ( LSeg ((G * (i2,k)),(G * (i3,k))))) /\ ( L~ ( Lower_Seq (C,(n + 1)))));

                  then x in (( LSeg ((G * (i2,j1)),(G * (i2,k)))) \/ ( LSeg ((G * (i2,k)),(G * (i3,k))))) by XBOOLE_0:def 4;

                  then

                   A92: x in ( LSeg ((G * (i2,j1)),(G * (i2,k)))) or x in ( LSeg ((G * (i2,k)),(G * (i3,k)))) by XBOOLE_0:def 3;

                  x in ( L~ ( Lower_Seq (C,(n + 1)))) by A91, XBOOLE_0:def 4;

                  hence thesis by A14, A56, A59, A83, A92, XBOOLE_0:def 4;

                end;

                let x be object;

                (G * (i2,j1)) in ( LSeg ((G * (i2,j1)),(G * (i2,k)))) by RLTOPSP1: 68;

                then

                 A93: (G * (i2,j1)) in (( LSeg ((G * (i2,j1)),(G * (i2,k)))) \/ ( LSeg ((G * (i2,k)),(G * (i3,k))))) by XBOOLE_0:def 3;

                assume x in {(G * (i2,j1))};

                then x = (G * (i2,j1)) by TARSKI:def 1;

                hence thesis by A60, A93, XBOOLE_0:def 4;

              end;

              1 < i3 by A1, A80, XXREAL_0: 2;

              hence thesis by A4, A7, A58, A61, A81, A84, A86, A90, Th46, XBOOLE_1: 63;

            end;

          end;

          hence thesis;

        end;

      end;

      hence thesis;

    end;

    theorem :: JORDAN15:49

    

     Th49: for C be Simple_closed_curve holds for i1,i2,j,k be Nat holds 1 < i1 & i1 < ( len ( Gauge (C,(n + 1)))) & 1 < i2 & i2 < ( len ( Gauge (C,(n + 1)))) & 1 <= j & j <= k & k <= ( width ( Gauge (C,(n + 1)))) & (( Gauge (C,(n + 1))) * (i1,k)) in ( Upper_Arc ( L~ ( Cage (C,(n + 1))))) & (( Gauge (C,(n + 1))) * (i2,j)) in ( Lower_Arc ( L~ ( Cage (C,(n + 1))))) implies (( LSeg ((( Gauge (C,(n + 1))) * (i2,j)),(( Gauge (C,(n + 1))) * (i2,k)))) \/ ( LSeg ((( Gauge (C,(n + 1))) * (i2,k)),(( Gauge (C,(n + 1))) * (i1,k))))) meets ( Lower_Arc C)

    proof

      let C be Simple_closed_curve;

      let i1,i2,j,k be Nat;

      set G = ( Gauge (C,(n + 1)));

      assume that

       A1: 1 < i1 and

       A2: i1 < ( len G) and

       A3: 1 < i2 and

       A4: i2 < ( len G) and

       A5: 1 <= j and

       A6: j <= k and

       A7: k <= ( width G) and

       A8: (G * (i1,k)) in ( Upper_Arc ( L~ ( Cage (C,(n + 1))))) and

       A9: (G * (i2,j)) in ( Lower_Arc ( L~ ( Cage (C,(n + 1)))));

      

       A10: 1 <= k by A5, A6, XXREAL_0: 2;

      then

       A11: [i2, k] in ( Indices G) by A3, A4, A7, MATRIX_0: 30;

      

       A12: [i1, k] in ( Indices G) by A1, A2, A7, A10, MATRIX_0: 30;

      ((G * (i2,k)) `2 ) = ((G * (1,k)) `2 ) by A3, A4, A7, A10, GOBOARD5: 1

      .= ((G * (i1,k)) `2 ) by A1, A2, A7, A10, GOBOARD5: 1;

      then

       A13: ( LSeg ((G * (i2,k)),(G * (i1,k)))) is horizontal by SPPOL_1: 15;

      

       A14: ( Lower_Arc ( L~ ( Cage (C,(n + 1))))) = ( L~ ( Lower_Seq (C,(n + 1)))) by JORDAN1G: 56;

      

       A15: j <= ( width G) by A6, A7, XXREAL_0: 2;

      then

       A16: [i2, j] in ( Indices G) by A3, A4, A5, MATRIX_0: 30;

      ((G * (i2,j)) `1 ) = ((G * (i2,1)) `1 ) by A3, A4, A5, A15, GOBOARD5: 2

      .= ((G * (i2,k)) `1 ) by A3, A4, A7, A10, GOBOARD5: 2;

      then

       A17: ( LSeg ((G * (i2,j)),(G * (i2,k)))) is vertical by SPPOL_1: 16;

      

       A18: ( Upper_Arc ( L~ ( Cage (C,(n + 1))))) = ( L~ ( Upper_Seq (C,(n + 1)))) by JORDAN1G: 55;

      

       A19: [i2, k] in ( Indices G) by A3, A4, A7, A10, MATRIX_0: 30;

      now

        per cases ;

          suppose

           A20: ( LSeg ((G * (i2,j)),(G * (i2,k)))) meets ( Upper_Arc ( L~ ( Cage (C,(n + 1)))));

          set X = (( LSeg ((G * (i2,j)),(G * (i2,k)))) /\ ( L~ ( Upper_Seq (C,(n + 1)))));

          ex x be object st x in ( LSeg ((G * (i2,j)),(G * (i2,k)))) & x in ( L~ ( Upper_Seq (C,(n + 1)))) by A18, A20, XBOOLE_0: 3;

          then

          reconsider X1 = X as non empty compact Subset of ( TOP-REAL 2) by XBOOLE_0:def 4;

          consider pp be object such that

           A21: pp in ( S-most X1) by XBOOLE_0:def 1;

          reconsider pp as Point of ( TOP-REAL 2) by A21;

          

           A22: pp in X by A21, XBOOLE_0:def 4;

          then

           A23: pp in ( L~ ( Upper_Seq (C,(n + 1)))) by XBOOLE_0:def 4;

          

           A24: pp in ( LSeg ((G * (i2,j)),(G * (i2,k)))) by A22, XBOOLE_0:def 4;

          consider m be Nat such that

           A25: j <= m and

           A26: m <= k and

           A27: ((G * (i2,m)) `2 ) = ( lower_bound ( proj2 .: (( LSeg ((G * (i2,j)),(G * (i2,k)))) /\ ( L~ ( Upper_Seq (C,(n + 1))))))) by A6, A18, A16, A19, A20, JORDAN1F: 1, JORDAN1G: 4;

          

           A28: m <= ( width G) by A7, A26, XXREAL_0: 2;

          1 <= m by A5, A25, XXREAL_0: 2;

          then

           A29: ((G * (i2,m)) `1 ) = ((G * (i2,1)) `1 ) by A3, A4, A28, GOBOARD5: 2;

          then

           A30: |[((G * (i2,1)) `1 ), ( lower_bound ( proj2 .: X))]| = (G * (i2,m)) by A27, EUCLID: 53;

          then ((G * (i2,j)) `1 ) = ( |[((G * (i2,1)) `1 ), ( lower_bound ( proj2 .: X))]| `1 ) by A3, A4, A5, A15, A29, GOBOARD5: 2;

          then

           A31: (pp `1 ) = ( |[((G * (i2,1)) `1 ), ( lower_bound ( proj2 .: X))]| `1 ) by A17, A24, SPPOL_1: 41;

          ( |[((G * (i2,1)) `1 ), ( lower_bound ( proj2 .: X))]| `2 ) = ( S-bound X) by A27, A30, SPRECT_1: 44

          .= (( S-min X) `2 ) by EUCLID: 52

          .= (pp `2 ) by A21, PSCOMP_1: 55;

          then (G * (i2,m)) in ( Upper_Arc ( L~ ( Cage (C,(n + 1))))) by A18, A30, A23, A31, TOPREAL3: 6;

          then ( LSeg ((G * (i2,j)),(G * (i2,m)))) meets ( Lower_Arc C) by A3, A4, A5, A9, A25, A28, Th23;

          then ( LSeg ((G * (i2,j)),(G * (i2,k)))) meets ( Lower_Arc C) by A3, A4, A5, A7, A25, A26, Th5, XBOOLE_1: 63;

          hence thesis by XBOOLE_1: 70;

        end;

          suppose

           A32: ( LSeg ((G * (i2,k)),(G * (i1,k)))) meets ( Lower_Arc ( L~ ( Cage (C,(n + 1))))) & i2 <= i1;

          set X = (( LSeg ((G * (i2,k)),(G * (i1,k)))) /\ ( L~ ( Lower_Seq (C,(n + 1)))));

          ex x be object st x in ( LSeg ((G * (i2,k)),(G * (i1,k)))) & x in ( L~ ( Lower_Seq (C,(n + 1)))) by A14, A32, XBOOLE_0: 3;

          then

          reconsider X1 = X as non empty compact Subset of ( TOP-REAL 2) by XBOOLE_0:def 4;

          consider pp be object such that

           A33: pp in ( E-most X1) by XBOOLE_0:def 1;

          reconsider pp as Point of ( TOP-REAL 2) by A33;

          

           A34: pp in X by A33, XBOOLE_0:def 4;

          then

           A35: pp in ( L~ ( Lower_Seq (C,(n + 1)))) by XBOOLE_0:def 4;

          

           A36: pp in ( LSeg ((G * (i2,k)),(G * (i1,k)))) by A34, XBOOLE_0:def 4;

          consider m be Nat such that

           A37: i2 <= m and

           A38: m <= i1 and

           A39: ((G * (m,k)) `1 ) = ( upper_bound ( proj1 .: (( LSeg ((G * (i2,k)),(G * (i1,k)))) /\ ( L~ ( Lower_Seq (C,(n + 1))))))) by A14, A11, A12, A32, JORDAN1F: 4, JORDAN1G: 5;

          

           A40: 1 < m by A3, A37, XXREAL_0: 2;

          m < ( len G) by A2, A38, XXREAL_0: 2;

          then

           A41: ((G * (m,k)) `2 ) = ((G * (1,k)) `2 ) by A7, A10, A40, GOBOARD5: 1;

          then

           A42: |[( upper_bound ( proj1 .: X)), ((G * (1,k)) `2 )]| = (G * (m,k)) by A39, EUCLID: 53;

          then ((G * (i2,k)) `2 ) = ( |[( upper_bound ( proj1 .: X)), ((G * (1,k)) `2 )]| `2 ) by A3, A4, A7, A10, A41, GOBOARD5: 1;

          then

           A43: (pp `2 ) = ( |[( upper_bound ( proj1 .: X)), ((G * (1,k)) `2 )]| `2 ) by A13, A36, SPPOL_1: 40;

          ( |[( upper_bound ( proj1 .: X)), ((G * (1,k)) `2 )]| `1 ) = ( E-bound X) by A39, A42, SPRECT_1: 46

          .= (( E-min X) `1 ) by EUCLID: 52

          .= (pp `1 ) by A33, PSCOMP_1: 47;

          then (G * (m,k)) in ( Lower_Arc ( L~ ( Cage (C,(n + 1))))) by A14, A42, A35, A43, TOPREAL3: 6;

          then ( LSeg ((G * (m,k)),(G * (i1,k)))) meets ( Lower_Arc C) by A2, A7, A8, A10, A38, A40, Th32;

          then ( LSeg ((G * (i2,k)),(G * (i1,k)))) meets ( Lower_Arc C) by A2, A3, A7, A10, A37, A38, Th6, XBOOLE_1: 63;

          hence thesis by XBOOLE_1: 70;

        end;

          suppose

           A44: ( LSeg ((G * (i2,k)),(G * (i1,k)))) meets ( Lower_Arc ( L~ ( Cage (C,(n + 1))))) & i1 < i2;

          set X = (( LSeg ((G * (i1,k)),(G * (i2,k)))) /\ ( L~ ( Lower_Seq (C,(n + 1)))));

          ex x be object st x in ( LSeg ((G * (i1,k)),(G * (i2,k)))) & x in ( L~ ( Lower_Seq (C,(n + 1)))) by A14, A44, XBOOLE_0: 3;

          then

          reconsider X1 = X as non empty compact Subset of ( TOP-REAL 2) by XBOOLE_0:def 4;

          consider pp be object such that

           A45: pp in ( W-most X1) by XBOOLE_0:def 1;

          reconsider pp as Point of ( TOP-REAL 2) by A45;

          

           A46: pp in X by A45, XBOOLE_0:def 4;

          then

           A47: pp in ( L~ ( Lower_Seq (C,(n + 1)))) by XBOOLE_0:def 4;

          

           A48: pp in ( LSeg ((G * (i1,k)),(G * (i2,k)))) by A46, XBOOLE_0:def 4;

          consider m be Nat such that

           A49: i1 <= m and

           A50: m <= i2 and

           A51: ((G * (m,k)) `1 ) = ( lower_bound ( proj1 .: (( LSeg ((G * (i1,k)),(G * (i2,k)))) /\ ( L~ ( Lower_Seq (C,(n + 1))))))) by A14, A11, A12, A44, JORDAN1F: 3, JORDAN1G: 5;

          

           A52: m < ( len G) by A4, A50, XXREAL_0: 2;

          1 < m by A1, A49, XXREAL_0: 2;

          then

           A53: ((G * (m,k)) `2 ) = ((G * (1,k)) `2 ) by A7, A10, A52, GOBOARD5: 1;

          then

           A54: |[( lower_bound ( proj1 .: X)), ((G * (1,k)) `2 )]| = (G * (m,k)) by A51, EUCLID: 53;

          then ((G * (i1,k)) `2 ) = ( |[( lower_bound ( proj1 .: X)), ((G * (1,k)) `2 )]| `2 ) by A1, A2, A7, A10, A53, GOBOARD5: 1;

          then

           A55: (pp `2 ) = ( |[( lower_bound ( proj1 .: X)), ((G * (1,k)) `2 )]| `2 ) by A13, A48, SPPOL_1: 40;

          ( |[( lower_bound ( proj1 .: X)), ((G * (1,k)) `2 )]| `1 ) = ( W-bound X) by A51, A54, SPRECT_1: 43

          .= (( W-min X) `1 ) by EUCLID: 52

          .= (pp `1 ) by A45, PSCOMP_1: 31;

          then (G * (m,k)) in ( Lower_Arc ( L~ ( Cage (C,(n + 1))))) by A14, A54, A47, A55, TOPREAL3: 6;

          then ( LSeg ((G * (i1,k)),(G * (m,k)))) meets ( Lower_Arc C) by A1, A7, A8, A10, A49, A52, Th40;

          then ( LSeg ((G * (i1,k)),(G * (i2,k)))) meets ( Lower_Arc C) by A1, A4, A7, A10, A49, A50, Th6, XBOOLE_1: 63;

          hence thesis by XBOOLE_1: 70;

        end;

          suppose

           A56: ( LSeg ((G * (i2,j)),(G * (i2,k)))) misses ( Upper_Arc ( L~ ( Cage (C,(n + 1))))) & ( LSeg ((( Gauge (C,(n + 1))) * (i2,k)),(( Gauge (C,(n + 1))) * (i1,k)))) misses ( Lower_Arc ( L~ ( Cage (C,(n + 1)))));

          consider j1 be Nat such that

           A57: j <= j1 and

           A58: j1 <= k and

           A59: (( LSeg ((G * (i2,j1)),(G * (i2,k)))) /\ ( L~ ( Lower_Seq (C,(n + 1))))) = {(G * (i2,j1))} by A3, A4, A5, A6, A7, A9, A14, Th9;

          (G * (i2,j1)) in (( LSeg ((G * (i2,j1)),(G * (i2,k)))) /\ ( L~ ( Lower_Seq (C,(n + 1))))) by A59, TARSKI:def 1;

          then

           A60: (G * (i2,j1)) in ( L~ ( Lower_Seq (C,(n + 1)))) by XBOOLE_0:def 4;

          

           A61: 1 <= j1 by A5, A57, XXREAL_0: 2;

          now

            per cases ;

              suppose

               A62: i2 <= i1;

              

               A63: ( LSeg ((G * (i2,j1)),(G * (i2,k)))) c= ( LSeg ((G * (i2,j)),(G * (i2,k)))) by A3, A4, A5, A7, A57, A58, Th5;

              consider i3 be Nat such that

               A64: i2 <= i3 and

               A65: i3 <= i1 and

               A66: (( LSeg ((G * (i2,k)),(G * (i3,k)))) /\ ( L~ ( Upper_Seq (C,(n + 1))))) = {(G * (i3,k))} by A2, A3, A7, A8, A18, A10, A62, Th13;

              

               A67: ( LSeg ((G * (i2,k)),(G * (i3,k)))) c= ( LSeg ((G * (i2,k)),(G * (i1,k)))) by A2, A3, A7, A10, A64, A65, Th6;

              then

               A68: (( LSeg ((G * (i2,j1)),(G * (i2,k)))) \/ ( LSeg ((G * (i2,k)),(G * (i3,k))))) c= (( LSeg ((G * (i2,j)),(G * (i2,k)))) \/ ( LSeg ((G * (i2,k)),(G * (i1,k))))) by A63, XBOOLE_1: 13;

              (G * (i3,k)) in (( LSeg ((G * (i2,k)),(G * (i3,k)))) /\ ( L~ ( Upper_Seq (C,(n + 1))))) by A66, TARSKI:def 1;

              then

               A69: (G * (i3,k)) in ( L~ ( Upper_Seq (C,(n + 1)))) by XBOOLE_0:def 4;

              

               A70: ((( LSeg ((G * (i2,j1)),(G * (i2,k)))) \/ ( LSeg ((G * (i2,k)),(G * (i3,k))))) /\ ( L~ ( Upper_Seq (C,(n + 1))))) = {(G * (i3,k))}

              proof

                thus ((( LSeg ((G * (i2,j1)),(G * (i2,k)))) \/ ( LSeg ((G * (i2,k)),(G * (i3,k))))) /\ ( L~ ( Upper_Seq (C,(n + 1))))) c= {(G * (i3,k))}

                proof

                  let x be object;

                  assume

                   A71: x in ((( LSeg ((G * (i2,j1)),(G * (i2,k)))) \/ ( LSeg ((G * (i2,k)),(G * (i3,k))))) /\ ( L~ ( Upper_Seq (C,(n + 1)))));

                  then x in (( LSeg ((G * (i2,j1)),(G * (i2,k)))) \/ ( LSeg ((G * (i2,k)),(G * (i3,k))))) by XBOOLE_0:def 4;

                  then

                   A72: x in ( LSeg ((G * (i2,j1)),(G * (i2,k)))) or x in ( LSeg ((G * (i2,k)),(G * (i3,k)))) by XBOOLE_0:def 3;

                  x in ( L~ ( Upper_Seq (C,(n + 1)))) by A71, XBOOLE_0:def 4;

                  hence thesis by A18, A56, A66, A63, A72, XBOOLE_0:def 4;

                end;

                let x be object;

                (G * (i3,k)) in ( LSeg ((G * (i2,k)),(G * (i3,k)))) by RLTOPSP1: 68;

                then

                 A73: (G * (i3,k)) in (( LSeg ((G * (i2,j1)),(G * (i2,k)))) \/ ( LSeg ((G * (i2,k)),(G * (i3,k))))) by XBOOLE_0:def 3;

                assume x in {(G * (i3,k))};

                then x = (G * (i3,k)) by TARSKI:def 1;

                hence thesis by A69, A73, XBOOLE_0:def 4;

              end;

              

               A74: ((( LSeg ((G * (i2,j1)),(G * (i2,k)))) \/ ( LSeg ((G * (i2,k)),(G * (i3,k))))) /\ ( L~ ( Lower_Seq (C,(n + 1))))) = {(G * (i2,j1))}

              proof

                thus ((( LSeg ((G * (i2,j1)),(G * (i2,k)))) \/ ( LSeg ((G * (i2,k)),(G * (i3,k))))) /\ ( L~ ( Lower_Seq (C,(n + 1))))) c= {(G * (i2,j1))}

                proof

                  let x be object;

                  assume

                   A75: x in ((( LSeg ((G * (i2,j1)),(G * (i2,k)))) \/ ( LSeg ((G * (i2,k)),(G * (i3,k))))) /\ ( L~ ( Lower_Seq (C,(n + 1)))));

                  then x in (( LSeg ((G * (i2,j1)),(G * (i2,k)))) \/ ( LSeg ((G * (i2,k)),(G * (i3,k))))) by XBOOLE_0:def 4;

                  then

                   A76: x in ( LSeg ((G * (i2,j1)),(G * (i2,k)))) or x in ( LSeg ((G * (i2,k)),(G * (i3,k)))) by XBOOLE_0:def 3;

                  x in ( L~ ( Lower_Seq (C,(n + 1)))) by A75, XBOOLE_0:def 4;

                  hence thesis by A14, A56, A59, A67, A76, XBOOLE_0:def 4;

                end;

                let x be object;

                (G * (i2,j1)) in ( LSeg ((G * (i2,j1)),(G * (i2,k)))) by RLTOPSP1: 68;

                then

                 A77: (G * (i2,j1)) in (( LSeg ((G * (i2,j1)),(G * (i2,k)))) \/ ( LSeg ((G * (i2,k)),(G * (i3,k))))) by XBOOLE_0:def 3;

                assume x in {(G * (i2,j1))};

                then x = (G * (i2,j1)) by TARSKI:def 1;

                hence thesis by A60, A77, XBOOLE_0:def 4;

              end;

              i3 < ( len G) by A2, A65, XXREAL_0: 2;

              hence thesis by A3, A7, A58, A61, A64, A68, A70, A74, Th45, XBOOLE_1: 63;

            end;

              suppose

               A78: i1 < i2;

              

               A79: ( LSeg ((G * (i2,j1)),(G * (i2,k)))) c= ( LSeg ((G * (i2,j)),(G * (i2,k)))) by A3, A4, A5, A7, A57, A58, Th5;

              consider i3 be Nat such that

               A80: i1 <= i3 and

               A81: i3 <= i2 and

               A82: (( LSeg ((G * (i3,k)),(G * (i2,k)))) /\ ( L~ ( Upper_Seq (C,(n + 1))))) = {(G * (i3,k))} by A1, A4, A7, A8, A18, A10, A78, Th18;

              

               A83: ( LSeg ((G * (i2,k)),(G * (i3,k)))) c= ( LSeg ((G * (i2,k)),(G * (i1,k)))) by A1, A4, A7, A10, A80, A81, Th6;

              then

               A84: (( LSeg ((G * (i2,j1)),(G * (i2,k)))) \/ ( LSeg ((G * (i2,k)),(G * (i3,k))))) c= (( LSeg ((G * (i2,j)),(G * (i2,k)))) \/ ( LSeg ((G * (i2,k)),(G * (i1,k))))) by A79, XBOOLE_1: 13;

              (G * (i3,k)) in (( LSeg ((G * (i2,k)),(G * (i3,k)))) /\ ( L~ ( Upper_Seq (C,(n + 1))))) by A82, TARSKI:def 1;

              then

               A85: (G * (i3,k)) in ( L~ ( Upper_Seq (C,(n + 1)))) by XBOOLE_0:def 4;

              

               A86: ((( LSeg ((G * (i2,j1)),(G * (i2,k)))) \/ ( LSeg ((G * (i2,k)),(G * (i3,k))))) /\ ( L~ ( Upper_Seq (C,(n + 1))))) = {(G * (i3,k))}

              proof

                thus ((( LSeg ((G * (i2,j1)),(G * (i2,k)))) \/ ( LSeg ((G * (i2,k)),(G * (i3,k))))) /\ ( L~ ( Upper_Seq (C,(n + 1))))) c= {(G * (i3,k))}

                proof

                  let x be object;

                  assume

                   A87: x in ((( LSeg ((G * (i2,j1)),(G * (i2,k)))) \/ ( LSeg ((G * (i2,k)),(G * (i3,k))))) /\ ( L~ ( Upper_Seq (C,(n + 1)))));

                  then x in (( LSeg ((G * (i2,j1)),(G * (i2,k)))) \/ ( LSeg ((G * (i2,k)),(G * (i3,k))))) by XBOOLE_0:def 4;

                  then

                   A88: x in ( LSeg ((G * (i2,j1)),(G * (i2,k)))) or x in ( LSeg ((G * (i2,k)),(G * (i3,k)))) by XBOOLE_0:def 3;

                  x in ( L~ ( Upper_Seq (C,(n + 1)))) by A87, XBOOLE_0:def 4;

                  hence thesis by A18, A56, A82, A79, A88, XBOOLE_0:def 4;

                end;

                let x be object;

                (G * (i3,k)) in ( LSeg ((G * (i2,k)),(G * (i3,k)))) by RLTOPSP1: 68;

                then

                 A89: (G * (i3,k)) in (( LSeg ((G * (i2,j1)),(G * (i2,k)))) \/ ( LSeg ((G * (i2,k)),(G * (i3,k))))) by XBOOLE_0:def 3;

                assume x in {(G * (i3,k))};

                then x = (G * (i3,k)) by TARSKI:def 1;

                hence thesis by A85, A89, XBOOLE_0:def 4;

              end;

              

               A90: ((( LSeg ((G * (i2,j1)),(G * (i2,k)))) \/ ( LSeg ((G * (i2,k)),(G * (i3,k))))) /\ ( L~ ( Lower_Seq (C,(n + 1))))) = {(G * (i2,j1))}

              proof

                thus ((( LSeg ((G * (i2,j1)),(G * (i2,k)))) \/ ( LSeg ((G * (i2,k)),(G * (i3,k))))) /\ ( L~ ( Lower_Seq (C,(n + 1))))) c= {(G * (i2,j1))}

                proof

                  let x be object;

                  assume

                   A91: x in ((( LSeg ((G * (i2,j1)),(G * (i2,k)))) \/ ( LSeg ((G * (i2,k)),(G * (i3,k))))) /\ ( L~ ( Lower_Seq (C,(n + 1)))));

                  then x in (( LSeg ((G * (i2,j1)),(G * (i2,k)))) \/ ( LSeg ((G * (i2,k)),(G * (i3,k))))) by XBOOLE_0:def 4;

                  then

                   A92: x in ( LSeg ((G * (i2,j1)),(G * (i2,k)))) or x in ( LSeg ((G * (i2,k)),(G * (i3,k)))) by XBOOLE_0:def 3;

                  x in ( L~ ( Lower_Seq (C,(n + 1)))) by A91, XBOOLE_0:def 4;

                  hence thesis by A14, A56, A59, A83, A92, XBOOLE_0:def 4;

                end;

                let x be object;

                (G * (i2,j1)) in ( LSeg ((G * (i2,j1)),(G * (i2,k)))) by RLTOPSP1: 68;

                then

                 A93: (G * (i2,j1)) in (( LSeg ((G * (i2,j1)),(G * (i2,k)))) \/ ( LSeg ((G * (i2,k)),(G * (i3,k))))) by XBOOLE_0:def 3;

                assume x in {(G * (i2,j1))};

                then x = (G * (i2,j1)) by TARSKI:def 1;

                hence thesis by A60, A93, XBOOLE_0:def 4;

              end;

              1 < i3 by A1, A80, XXREAL_0: 2;

              hence thesis by A4, A7, A58, A61, A81, A84, A86, A90, Th47, XBOOLE_1: 63;

            end;

          end;

          hence thesis;

        end;

      end;

      hence thesis;

    end;

    theorem :: JORDAN15:50

    for C be Simple_closed_curve holds for i,j,k be Nat holds 1 < i & i < ( len ( Gauge (C,(n + 1)))) & 1 <= j & j <= k & k <= ( width ( Gauge (C,(n + 1)))) & (( Gauge (C,(n + 1))) * (i,k)) in ( Upper_Arc ( L~ ( Cage (C,(n + 1))))) & (( Gauge (C,(n + 1))) * (( Center ( Gauge (C,(n + 1)))),j)) in ( Lower_Arc ( L~ ( Cage (C,(n + 1))))) implies (( LSeg ((( Gauge (C,(n + 1))) * (( Center ( Gauge (C,(n + 1)))),j)),(( Gauge (C,(n + 1))) * (( Center ( Gauge (C,(n + 1)))),k)))) \/ ( LSeg ((( Gauge (C,(n + 1))) * (( Center ( Gauge (C,(n + 1)))),k)),(( Gauge (C,(n + 1))) * (i,k))))) meets ( Upper_Arc C)

    proof

      let C be Simple_closed_curve;

      let i,j,k be Nat;

      assume that

       A1: 1 < i and

       A2: i < ( len ( Gauge (C,(n + 1)))) and

       A3: 1 <= j and

       A4: j <= k and

       A5: k <= ( width ( Gauge (C,(n + 1)))) and

       A6: (( Gauge (C,(n + 1))) * (i,k)) in ( Upper_Arc ( L~ ( Cage (C,(n + 1))))) and

       A7: (( Gauge (C,(n + 1))) * (( Center ( Gauge (C,(n + 1)))),j)) in ( Lower_Arc ( L~ ( Cage (C,(n + 1)))));

      

       A8: ( len ( Gauge (C,(n + 1)))) >= 4 by JORDAN8: 10;

      then ( len ( Gauge (C,(n + 1)))) >= 3 by XXREAL_0: 2;

      then

       A9: ( Center ( Gauge (C,(n + 1)))) < ( len ( Gauge (C,(n + 1)))) by JORDAN1B: 15;

      ( len ( Gauge (C,(n + 1)))) >= 2 by A8, XXREAL_0: 2;

      then 1 < ( Center ( Gauge (C,(n + 1)))) by JORDAN1B: 14;

      hence thesis by A1, A2, A3, A4, A5, A6, A7, A9, Th48;

    end;

    theorem :: JORDAN15:51

    for C be Simple_closed_curve holds for i,j,k be Nat holds 1 < i & i < ( len ( Gauge (C,(n + 1)))) & 1 <= j & j <= k & k <= ( width ( Gauge (C,(n + 1)))) & (( Gauge (C,(n + 1))) * (i,k)) in ( Upper_Arc ( L~ ( Cage (C,(n + 1))))) & (( Gauge (C,(n + 1))) * (( Center ( Gauge (C,(n + 1)))),j)) in ( Lower_Arc ( L~ ( Cage (C,(n + 1))))) implies (( LSeg ((( Gauge (C,(n + 1))) * (( Center ( Gauge (C,(n + 1)))),j)),(( Gauge (C,(n + 1))) * (( Center ( Gauge (C,(n + 1)))),k)))) \/ ( LSeg ((( Gauge (C,(n + 1))) * (( Center ( Gauge (C,(n + 1)))),k)),(( Gauge (C,(n + 1))) * (i,k))))) meets ( Lower_Arc C)

    proof

      let C be Simple_closed_curve;

      let i,j,k be Nat;

      assume that

       A1: 1 < i and

       A2: i < ( len ( Gauge (C,(n + 1)))) and

       A3: 1 <= j and

       A4: j <= k and

       A5: k <= ( width ( Gauge (C,(n + 1)))) and

       A6: (( Gauge (C,(n + 1))) * (i,k)) in ( Upper_Arc ( L~ ( Cage (C,(n + 1))))) and

       A7: (( Gauge (C,(n + 1))) * (( Center ( Gauge (C,(n + 1)))),j)) in ( Lower_Arc ( L~ ( Cage (C,(n + 1)))));

      

       A8: ( len ( Gauge (C,(n + 1)))) >= 4 by JORDAN8: 10;

      then ( len ( Gauge (C,(n + 1)))) >= 3 by XXREAL_0: 2;

      then

       A9: ( Center ( Gauge (C,(n + 1)))) < ( len ( Gauge (C,(n + 1)))) by JORDAN1B: 15;

      ( len ( Gauge (C,(n + 1)))) >= 2 by A8, XXREAL_0: 2;

      then 1 < ( Center ( Gauge (C,(n + 1)))) by JORDAN1B: 14;

      hence thesis by A1, A2, A3, A4, A5, A6, A7, A9, Th49;

    end;