jordan19.miz



    begin

    reserve n for Nat;

    definition

      let C be Simple_closed_curve;

      :: JORDAN19:def1

      func Upper_Appr C -> SetSequence of the carrier of ( TOP-REAL 2) means

      : Def1: for i be Nat holds (it . i) = ( Upper_Arc ( L~ ( Cage (C,i))));

      existence

      proof

        deffunc O( Nat) = ( Upper_Arc ( L~ ( Cage (C,$1))));

        consider S be SetSequence of the carrier of ( TOP-REAL 2) such that

         A1: for i be Element of NAT holds (S . i) = O(i) from FUNCT_2:sch 4;

        take S;

        let i be Nat;

        i in NAT by ORDINAL1:def 12;

        hence thesis by A1;

      end;

      uniqueness

      proof

        let s1,s2 be SetSequence of the carrier of ( TOP-REAL 2) such that

         A2: for i be Nat holds (s1 . i) = ( Upper_Arc ( L~ ( Cage (C,i)))) and

         A3: for i be Nat holds (s2 . i) = ( Upper_Arc ( L~ ( Cage (C,i))));

        let i be Element of NAT ;

        

        thus (s1 . i) = ( Upper_Arc ( L~ ( Cage (C,i)))) by A2

        .= (s2 . i) by A3;

      end;

      :: JORDAN19:def2

      func Lower_Appr C -> SetSequence of the carrier of ( TOP-REAL 2) means

      : Def2: for i be Nat holds (it . i) = ( Lower_Arc ( L~ ( Cage (C,i))));

      existence

      proof

        deffunc O( Nat) = ( Lower_Arc ( L~ ( Cage (C,$1))));

        consider S be SetSequence of the carrier of ( TOP-REAL 2) such that

         A4: for i be Element of NAT holds (S . i) = O(i) from FUNCT_2:sch 4;

        take S;

        let i be Nat;

        i in NAT by ORDINAL1:def 12;

        hence thesis by A4;

      end;

      uniqueness

      proof

        deffunc O( Nat) = ( Lower_Arc ( L~ ( Cage (C,$1))));

        let s1,s2 be SetSequence of the carrier of ( TOP-REAL 2) such that

         A5: for i be Nat holds (s1 . i) = ( Lower_Arc ( L~ ( Cage (C,i)))) and

         A6: for i be Nat holds (s2 . i) = ( Lower_Arc ( L~ ( Cage (C,i))));

        let i be Element of NAT ;

        

        thus (s1 . i) = ( Lower_Arc ( L~ ( Cage (C,i)))) by A5

        .= (s2 . i) by A6;

      end;

    end

    definition

      let C be Simple_closed_curve;

      :: JORDAN19:def3

      func North_Arc C -> Subset of ( TOP-REAL 2) equals ( Lim_inf ( Upper_Appr C));

      coherence ;

      :: JORDAN19:def4

      func South_Arc C -> Subset of ( TOP-REAL 2) equals ( Lim_inf ( Lower_Appr C));

      coherence ;

    end

     Lm1:

    now

      let G be Go-board;

      let j be Nat;

      assume that

       A1: 1 <= j and

       A2: j <= ( width G);

      ( 0 + 1) <= ((( len G) div 2) + 1) by XREAL_1: 6;

      then

       A3: ( 0 + 1) <= ( Center G) by JORDAN1A:def 1;

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

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

    end;

     Lm2:

    now

      let D be non empty Subset of ( TOP-REAL 2);

      let n,i be Nat;

      set a = ( N-bound D), s = ( S-bound D), w = ( W-bound D), e = ( E-bound D);

      set G = ( Gauge (D,n));

      assume [i, ( width G)] in ( Indices G);

      

      hence ((G * (i,( width G))) `2 ) = ( |[(w + (((e - w) / (2 |^ n)) * (i - 2))), (s + (((a - s) / (2 |^ n)) * (( width G) - 2)))]| `2 ) by JORDAN8:def 1

      .= (s + (((a - s) / (2 |^ n)) * (( width G) - 2))) by EUCLID: 52;

    end;

    theorem :: JORDAN19:1

    

     Th1: for n,m be Nat holds n <= m & n <> 0 implies ((n + 1) / n) >= ((m + 1) / m)

    proof

      let n,m be Nat;

      assume that

       A1: n <= m and

       A2: n <> 0 ;

      

       A3: n > 0 by A2;

      

       A4: (1 / n) >= (1 / m) by A1, A2, XREAL_1: 85;

      

       A5: ((n + 1) / n) = ((n / n) + (1 / n))

      .= (1 + (1 / n)) by A2, XCMPLX_1: 60;

      ((m + 1) / m) = ((m / m) + (1 / m))

      .= (1 + (1 / m)) by A1, A3, XCMPLX_1: 60;

      hence thesis by A4, A5, XREAL_1: 7;

    end;

    theorem :: JORDAN19:2

    

     Th2: for E be compact non vertical non horizontal Subset of ( TOP-REAL 2) holds for m,j be Nat st 1 <= m & m <= n & 1 <= j & j <= ( width ( Gauge (E,n))) holds ( LSeg ((( Gauge (E,n)) * (( Center ( Gauge (E,n))),( width ( Gauge (E,n))))),(( Gauge (E,n)) * (( Center ( Gauge (E,n))),j)))) c= ( LSeg ((( Gauge (E,m)) * (( Center ( Gauge (E,m))),( width ( Gauge (E,m))))),(( Gauge (E,n)) * (( Center ( Gauge (E,n))),j))))

    proof

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

      let m,j be Nat;

      set a = ( N-bound E), s = ( S-bound E), w = ( W-bound E), e = ( E-bound E), G = ( Gauge (E,n)), M = ( Gauge (E,m)), sn = ( Center G), sm = ( Center M);

      assume that

       A1: 1 <= m and

       A2: m <= n and

       A3: 1 <= j and

       A4: j <= ( width G);

      

       A5: ( width M) = ( len M) by JORDAN8:def 1

      .= ((2 |^ m) + 3) by JORDAN8:def 1;

      

       A6: ( width G) = ( len G) by JORDAN8:def 1

      .= ((2 |^ n) + 3) by JORDAN8:def 1;

       A7:

      now

        let t be Nat;

        assume that

         A8: ( width G) >= t and

         A9: t >= j;

        

         A10: ( len M) = ( width M) by JORDAN8:def 1;

        

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

        

         A12: 0 < (a - s) by SPRECT_1: 32, XREAL_1: 50;

        

         A13: t >= 1 by A3, A9, XXREAL_0: 2;

        

         A14: 0 < (2 |^ m) by NEWTON: 83;

        

         A15: 1 <= ( len M) by GOBRD11: 34;

        then

         A16: ((M * (sm,( width M))) `1 ) = ((G * (sn,t)) `1 ) by A1, A2, A8, A10, A11, A13, JORDAN1A: 36;

        

         A17: ((G * (sn,t)) `1 ) = ((G * (sn,j)) `1 ) by A1, A2, A3, A4, A8, A11, A13, JORDAN1A: 36;

         [sn, t] in ( Indices G) by A8, A13, Lm1;

        

        then

         A18: ((G * (sn,t)) `2 ) = ( |[(w + (((e - w) / (2 |^ n)) * (sn - 2))), (s + (((a - s) / (2 |^ n)) * (t - 2)))]| `2 ) by JORDAN8:def 1

        .= (s + (((a - s) / (2 |^ n)) * (t - 2))) by EUCLID: 52;

         [sm, ( width M)] in ( Indices M) by A10, A15, Lm1;

        then

         A19: ((M * (sm,( width M))) `2 ) = (s + (((a - s) / (2 |^ m)) * (( width M) - 2))) by Lm2;

        

         A20: (((2 |^ m) + 1) / (2 |^ m)) >= (((2 |^ n) + 1) / (2 |^ n)) by A2, A14, Th1, PREPOWER: 93;

        (t - 2) <= (((2 |^ n) + 3) - 2) by A6, A8, XREAL_1: 9;

        then ((t - 2) / (2 |^ n)) <= (((2 |^ n) + 1) / (2 |^ n)) by XREAL_1: 72;

        then ((t - 2) / (2 |^ n)) <= ((( width M) - 2) / (2 |^ m)) by A5, A20, XXREAL_0: 2;

        then ((a - s) * ((t - 2) / (2 |^ n))) <= ((a - s) * ((( width M) - 2) / (2 |^ m))) by A12, XREAL_1: 64;

        then

         A21: (s + (((a - s) / (2 |^ m)) * (( width M) - 2))) >= (s + (((a - s) / (2 |^ n)) * (t - 2))) by XREAL_1: 6;

        

         A22: 1 <= sn by JORDAN1B: 11;

        sn <= ( len G) by JORDAN1B: 13;

        then ((G * (sn,t)) `2 ) >= ((G * (sn,j)) `2 ) by A3, A8, A9, A22, SPRECT_3: 12;

        hence (G * (sn,t)) in ( LSeg ((M * (sm,( width M))),(G * (sn,j)))) by A16, A17, A18, A19, A21, GOBOARD7: 7;

      end;

      then

       A23: (G * (sn,( width G))) in ( LSeg ((M * (sm,( width M))),(G * (sn,j)))) by A4;

      (G * (sn,j)) in ( LSeg ((M * (sm,( width M))),(G * (sn,j)))) by A4, A7;

      hence thesis by A23, TOPREAL1: 6;

    end;

    theorem :: JORDAN19:3

    

     Th3: for C be compact connected non vertical non horizontal Subset of ( TOP-REAL 2) holds for i,j be Nat st 1 <= i & i <= ( len ( Gauge (C,n))) & 1 <= j & j <= ( width ( Gauge (C,n))) & (( Gauge (C,n)) * (i,j)) in ( L~ ( Cage (C,n))) holds ( LSeg ((( Gauge (C,n)) * (i,( width ( Gauge (C,n))))),(( Gauge (C,n)) * (i,j)))) meets ( L~ ( Upper_Seq (C,n)))

    proof

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

      let i,j be Nat;

      set Gij = (( Gauge (C,n)) * (i,j));

      assume that

       A1: 1 <= i and

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

       A3: 1 <= j and

       A4: j <= ( width ( Gauge (C,n))) and

       A5: Gij in ( L~ ( Cage (C,n)));

      set NE = ( SW-corner ( L~ ( Cage (C,n))));

      set v1 = ( L_Cut (( Lower_Seq (C,n)),Gij));

      set wG = ( width ( Gauge (C,n)));

      set lG = ( len ( Gauge (C,n)));

      set Gv1 = ( <*(( Gauge (C,n)) * (i,wG))*> ^ v1);

      set v = (Gv1 ^ <*NE*>);

      set h = ( mid (( Upper_Seq (C,n)),2,( len ( Upper_Seq (C,n)))));

      

       A6: ( L~ ( Cage (C,n))) = (( L~ ( Lower_Seq (C,n))) \/ ( L~ ( Upper_Seq (C,n)))) by JORDAN1E: 13;

      

       A7: ( len ( Upper_Seq (C,n))) >= 3 by JORDAN1E: 15;

      

       A8: ( len ( Lower_Seq (C,n))) >= 3 by JORDAN1E: 15;

      

       A9: ( len ( Upper_Seq (C,n))) >= 2 by A7, XXREAL_0: 2;

      

       A10: ( len ( Upper_Seq (C,n))) >= 1 by A7, XXREAL_0: 2;

      

       A11: ( len ( Lower_Seq (C,n))) >= 1 by A8, XXREAL_0: 2;

      

       A12: ( len ( Gauge (C,n))) = ( width ( Gauge (C,n))) by JORDAN8:def 1;

      then ( width ( Gauge (C,n))) >= 4 by JORDAN8: 10;

      then

       A13: 1 <= ( width ( Gauge (C,n))) by XXREAL_0: 2;

      

       A14: ((( Gauge (C,n)) * (i,wG)) `2 ) = ( N-bound ( L~ ( Cage (C,n)))) by A1, A2, A12, JORDAN1A: 70;

      set Ema = ( E-max ( L~ ( Cage (C,n))));

      now

        per cases by A2, A5, A6, XBOOLE_0:def 3, XXREAL_0: 1;

          suppose

           A15: Gij in ( L~ ( Lower_Seq (C,n))) & i = lG;

          set G11 = (( Gauge (C,n)) * (lG,wG));

          

           A16: (G11 `1 ) = ( E-bound ( L~ ( Cage (C,n)))) by A1, A12, A15, JORDAN1A: 71;

          

           A17: (Ema `1 ) = ( E-bound ( L~ ( Cage (C,n)))) by EUCLID: 52;

          

           A18: ( N-bound ( L~ ( Cage (C,n)))) = (G11 `2 ) by A1, A12, A15, JORDAN1A: 70;

          Ema in ( L~ ( Cage (C,n))) by SPRECT_1: 14;

          then

           A19: (G11 `2 ) >= (Ema `2 ) by A18, PSCOMP_1: 24;

          

           A20: (Gij `1 ) = ( E-bound ( L~ ( Cage (C,n)))) by A3, A4, A12, A15, JORDAN1A: 71;

          then Gij in ( E-most ( L~ ( Cage (C,n)))) by A5, SPRECT_2: 13;

          then (Ema `2 ) >= (Gij `2 ) by PSCOMP_1: 47;

          then

           A21: Ema in ( LSeg ((( Gauge (C,n)) * (lG,wG)),(( Gauge (C,n)) * (lG,j)))) by A15, A16, A17, A19, A20, GOBOARD7: 7;

          

           A22: ( rng ( Upper_Seq (C,n))) c= ( L~ ( Upper_Seq (C,n))) by A7, SPPOL_2: 18, XXREAL_0: 2;

          (( Upper_Seq (C,n)) /. ( len ( Upper_Seq (C,n)))) = Ema by JORDAN1F: 7;

          then Ema in ( rng ( Upper_Seq (C,n))) by FINSEQ_6: 168;

          hence thesis by A15, A21, A22, XBOOLE_0: 3;

        end;

          suppose

           A23: Gij in ( L~ ( Lower_Seq (C,n))) & Gij <> (( Lower_Seq (C,n)) . ( len ( Lower_Seq (C,n)))) & ( W-min ( L~ ( Cage (C,n)))) <> NE & i < lG;

          then

           A24: v1 is non empty by JORDAN1E: 3;

          then

           A25: ( 0 + 1) <= ( len v1) by NAT_1: 13;

          then

           A26: 1 in ( dom v1) by FINSEQ_3: 25;

          

           A27: ( len v1) in ( dom v1) by A25, FINSEQ_3: 25;

          

           A28: ( len ( Lower_Seq (C,n))) in ( dom ( Lower_Seq (C,n))) by A11, FINSEQ_3: 25;

          

           A29: (v1 /. ( len v1)) = (v1 . ( len v1)) by A27, PARTFUN1:def 6

          .= (( Lower_Seq (C,n)) . ( len ( Lower_Seq (C,n)))) by A23, JORDAN1B: 4

          .= (( Lower_Seq (C,n)) /. ( len ( Lower_Seq (C,n)))) by A28, PARTFUN1:def 6

          .= ( W-min ( L~ ( Cage (C,n)))) by JORDAN1F: 8;

          then

           A30: (Gv1 /. ( len Gv1)) = ( W-min ( L~ ( Cage (C,n)))) by A24, SPRECT_3: 1;

          

           A31: (v1 /. 1) = (v1 . 1) by A26, PARTFUN1:def 6

          .= Gij by A23, JORDAN3: 23;

          then

           A32: ((v1 ^ <*NE*>) /. 1) = Gij by A25, BOOLMARK: 7;

          

           A33: (1 + ( len v1)) >= (1 + 1) by A25, XREAL_1: 7;

          ( len v) = (( len Gv1) + 1) by FINSEQ_2: 16

          .= ((1 + ( len v1)) + 1) by FINSEQ_5: 8;

          then 2 < ( len v) by A33, NAT_1: 13;

          then

           A34: 2 < ( len ( Rev v)) by FINSEQ_5:def 3;

          ( S-bound ( L~ ( Cage (C,n)))) < ( N-bound ( L~ ( Cage (C,n)))) by SPRECT_1: 32;

          then NE <> (( Gauge (C,n)) * (i,wG)) by A14, EUCLID: 52;

          then not NE in {(( Gauge (C,n)) * (i,wG))} by TARSKI:def 1;

          then

           A35: not NE in ( rng <*(( Gauge (C,n)) * (i,wG))*>) by FINSEQ_1: 39;

          ( len ( Cage (C,n))) > 4 by GOBOARD7: 34;

          then

           A36: ( rng ( Cage (C,n))) c= ( L~ ( Cage (C,n))) by SPPOL_2: 18, XXREAL_0: 2;

          

           A37: not NE in ( rng ( Cage (C,n)))

          proof

            assume

             A38: NE in ( rng ( Cage (C,n)));

            

             A39: (NE `1 ) = ( W-bound ( L~ ( Cage (C,n)))) by EUCLID: 52;

            

             A40: (NE `2 ) = ( S-bound ( L~ ( Cage (C,n)))) by EUCLID: 52;

            then (NE `2 ) <= ( N-bound ( L~ ( Cage (C,n)))) by SPRECT_1: 22;

            then NE in { p where p be Point of ( TOP-REAL 2) : (p `1 ) = ( W-bound ( L~ ( Cage (C,n)))) & (p `2 ) <= ( N-bound ( L~ ( Cage (C,n)))) & (p `2 ) >= ( S-bound ( L~ ( Cage (C,n)))) } by A39, A40;

            then NE in ( LSeg (( SW-corner ( L~ ( Cage (C,n)))),( NW-corner ( L~ ( Cage (C,n)))))) by SPRECT_1: 26;

            then NE in (( LSeg (( SW-corner ( L~ ( Cage (C,n)))),( NW-corner ( L~ ( Cage (C,n)))))) /\ ( L~ ( Cage (C,n)))) by A36, A38, XBOOLE_0:def 4;

            then

             A41: (NE `2 ) >= (( W-min ( L~ ( Cage (C,n)))) `2 ) by PSCOMP_1: 31;

            (( W-min ( L~ ( Cage (C,n)))) `2 ) >= (NE `2 ) by PSCOMP_1: 30;

            then

             A42: (( W-min ( L~ ( Cage (C,n)))) `2 ) = (NE `2 ) by A41, XXREAL_0: 1;

            (( W-min ( L~ ( Cage (C,n)))) `1 ) = (NE `1 ) by PSCOMP_1: 29;

            hence contradiction by A23, A42, TOPREAL3: 6;

          end;

          now

            per cases ;

              suppose Gij <> (( Lower_Seq (C,n)) . (( Index (Gij,( Lower_Seq (C,n)))) + 1));

              then v1 = ( <*Gij*> ^ ( mid (( Lower_Seq (C,n)),(( Index (Gij,( Lower_Seq (C,n)))) + 1),( len ( Lower_Seq (C,n)))))) by JORDAN3:def 3;

              then ( rng v1) = (( rng <*Gij*>) \/ ( rng ( mid (( Lower_Seq (C,n)),(( Index (Gij,( Lower_Seq (C,n)))) + 1),( len ( Lower_Seq (C,n))))))) by FINSEQ_1: 31;

              then

               A43: ( rng v1) = ( {Gij} \/ ( rng ( mid (( Lower_Seq (C,n)),(( Index (Gij,( Lower_Seq (C,n)))) + 1),( len ( Lower_Seq (C,n))))))) by FINSEQ_1: 38;

               not NE in ( L~ ( Cage (C,n)))

              proof

                assume NE in ( L~ ( Cage (C,n)));

                then

                consider i be Nat such that

                 A44: 1 <= i and

                 A45: (i + 1) <= ( len ( Cage (C,n))) and

                 A46: NE in ( LSeg ((( Cage (C,n)) /. i),(( Cage (C,n)) /. (i + 1)))) by SPPOL_2: 14;

                per cases by A44, A45, TOPREAL1:def 5;

                  suppose

                   A47: ((( Cage (C,n)) /. i) `1 ) = ((( Cage (C,n)) /. (i + 1)) `1 );

                  then

                   A48: (NE `1 ) = ((( Cage (C,n)) /. i) `1 ) by A46, GOBOARD7: 5;

                  

                   A49: (NE `2 ) = ( S-bound ( L~ ( Cage (C,n)))) by EUCLID: 52;

                  

                   A50: i < ( len ( Cage (C,n))) by A45, NAT_1: 13;

                  then

                   A51: ((( Cage (C,n)) /. i) `2 ) >= (NE `2 ) by A44, A49, JORDAN5D: 11;

                  

                   A52: 1 <= (i + 1) by NAT_1: 11;

                  then

                   A53: ((( Cage (C,n)) /. (i + 1)) `2 ) >= (NE `2 ) by A45, A49, JORDAN5D: 11;

                  

                   A54: i in ( dom ( Cage (C,n))) by A44, A50, FINSEQ_3: 25;

                  

                   A55: (i + 1) in ( dom ( Cage (C,n))) by A45, A52, FINSEQ_3: 25;

                  ((( Cage (C,n)) /. i) `2 ) <= ((( Cage (C,n)) /. (i + 1)) `2 ) or ((( Cage (C,n)) /. i) `2 ) >= ((( Cage (C,n)) /. (i + 1)) `2 );

                  then (NE `2 ) >= ((( Cage (C,n)) /. (i + 1)) `2 ) or (NE `2 ) >= ((( Cage (C,n)) /. i) `2 ) by A46, TOPREAL1: 4;

                  then (NE `2 ) = ((( Cage (C,n)) /. (i + 1)) `2 ) or (NE `2 ) = ((( Cage (C,n)) /. i) `2 ) by A51, A53, XXREAL_0: 1;

                  then NE = (( Cage (C,n)) /. (i + 1)) or NE = (( Cage (C,n)) /. i) by A47, A48, TOPREAL3: 6;

                  hence contradiction by A37, A54, A55, PARTFUN2: 2;

                end;

                  suppose

                   A56: ((( Cage (C,n)) /. i) `2 ) = ((( Cage (C,n)) /. (i + 1)) `2 );

                  then

                   A57: (NE `2 ) = ((( Cage (C,n)) /. i) `2 ) by A46, GOBOARD7: 6;

                  

                   A58: (NE `1 ) = ( W-bound ( L~ ( Cage (C,n)))) by EUCLID: 52;

                  

                   A59: i < ( len ( Cage (C,n))) by A45, NAT_1: 13;

                  then

                   A60: ((( Cage (C,n)) /. i) `1 ) >= (NE `1 ) by A44, A58, JORDAN5D: 12;

                  

                   A61: 1 <= (i + 1) by NAT_1: 11;

                  then

                   A62: ((( Cage (C,n)) /. (i + 1)) `1 ) >= (NE `1 ) by A45, A58, JORDAN5D: 12;

                  

                   A63: i in ( dom ( Cage (C,n))) by A44, A59, FINSEQ_3: 25;

                  

                   A64: (i + 1) in ( dom ( Cage (C,n))) by A45, A61, FINSEQ_3: 25;

                  ((( Cage (C,n)) /. i) `1 ) <= ((( Cage (C,n)) /. (i + 1)) `1 ) or ((( Cage (C,n)) /. i) `1 ) >= ((( Cage (C,n)) /. (i + 1)) `1 );

                  then (NE `1 ) >= ((( Cage (C,n)) /. (i + 1)) `1 ) or (NE `1 ) >= ((( Cage (C,n)) /. i) `1 ) by A46, TOPREAL1: 3;

                  then (NE `1 ) = ((( Cage (C,n)) /. (i + 1)) `1 ) or (NE `1 ) = ((( Cage (C,n)) /. i) `1 ) by A60, A62, XXREAL_0: 1;

                  then NE = (( Cage (C,n)) /. (i + 1)) or NE = (( Cage (C,n)) /. i) by A56, A57, TOPREAL3: 6;

                  hence contradiction by A37, A63, A64, PARTFUN2: 2;

                end;

              end;

              then

               A65: not NE in {Gij} by A5, TARSKI:def 1;

              

               A66: ( rng ( mid (( Lower_Seq (C,n)),(( Index (Gij,( Lower_Seq (C,n)))) + 1),( len ( Lower_Seq (C,n)))))) c= ( rng ( Lower_Seq (C,n))) by FINSEQ_6: 119;

              ( rng ( Lower_Seq (C,n))) c= ( rng ( Cage (C,n))) by JORDAN1G: 39;

              then ( rng ( mid (( Lower_Seq (C,n)),(( Index (Gij,( Lower_Seq (C,n)))) + 1),( len ( Lower_Seq (C,n)))))) c= ( rng ( Cage (C,n))) by A66;

              then not NE in ( rng ( mid (( Lower_Seq (C,n)),(( Index (Gij,( Lower_Seq (C,n)))) + 1),( len ( Lower_Seq (C,n)))))) by A37;

              hence not NE in ( rng v1) by A43, A65, XBOOLE_0:def 3;

            end;

              suppose Gij = (( Lower_Seq (C,n)) . (( Index (Gij,( Lower_Seq (C,n)))) + 1));

              then v1 = ( mid (( Lower_Seq (C,n)),(( Index (Gij,( Lower_Seq (C,n)))) + 1),( len ( Lower_Seq (C,n))))) by JORDAN3:def 3;

              then

               A67: ( rng v1) c= ( rng ( Lower_Seq (C,n))) by FINSEQ_6: 119;

              ( rng ( Lower_Seq (C,n))) c= ( rng ( Cage (C,n))) by JORDAN1G: 39;

              then ( rng v1) c= ( rng ( Cage (C,n))) by A67;

              hence not NE in ( rng v1) by A37;

            end;

          end;

          then not NE in (( rng <*(( Gauge (C,n)) * (i,wG))*>) \/ ( rng v1)) by A35, XBOOLE_0:def 3;

          then not NE in ( rng Gv1) by FINSEQ_1: 31;

          then ( rng Gv1) misses {NE} by ZFMISC_1: 50;

          then

           A68: ( rng Gv1) misses ( rng <*NE*>) by FINSEQ_1: 38;

          

           A69: not (( Gauge (C,n)) * (i,wG)) in ( L~ ( Lower_Seq (C,n))) by A1, A23, JORDAN1G: 45;

          ( rng ( Lower_Seq (C,n))) c= ( L~ ( Lower_Seq (C,n))) by A8, SPPOL_2: 18, XXREAL_0: 2;

          then

           A70: not (( Gauge (C,n)) * (i,wG)) in ( rng ( Lower_Seq (C,n))) by A1, A23, JORDAN1G: 45;

           not (( Gauge (C,n)) * (i,wG)) in {Gij} by A23, A69, TARSKI:def 1;

          then

           A71: not (( Gauge (C,n)) * (i,wG)) in ( rng <*Gij*>) by FINSEQ_1: 38;

          set ci = ( mid (( Lower_Seq (C,n)),(( Index (Gij,( Lower_Seq (C,n)))) + 1),( len ( Lower_Seq (C,n)))));

          now

            per cases ;

              suppose

               A72: Gij <> (( Lower_Seq (C,n)) . (( Index (Gij,( Lower_Seq (C,n)))) + 1));

              ( rng ci) c= ( rng ( Lower_Seq (C,n))) by FINSEQ_6: 119;

              then not (( Gauge (C,n)) * (i,wG)) in ( rng ci) by A70;

              then not (( Gauge (C,n)) * (i,wG)) in (( rng <*Gij*>) \/ ( rng ci)) by A71, XBOOLE_0:def 3;

              then not (( Gauge (C,n)) * (i,wG)) in ( rng ( <*Gij*> ^ ci)) by FINSEQ_1: 31;

              hence not (( Gauge (C,n)) * (i,wG)) in ( rng v1) by A72, JORDAN3:def 3;

            end;

              suppose Gij = (( Lower_Seq (C,n)) . (( Index (Gij,( Lower_Seq (C,n)))) + 1));

              then v1 = ci by JORDAN3:def 3;

              then ( rng v1) c= ( rng ( Lower_Seq (C,n))) by FINSEQ_6: 119;

              hence not (( Gauge (C,n)) * (i,wG)) in ( rng v1) by A70;

            end;

          end;

          then {(( Gauge (C,n)) * (i,wG))} misses ( rng v1) by ZFMISC_1: 50;

          then

           A73: ( rng <*(( Gauge (C,n)) * (i,wG))*>) misses ( rng v1) by FINSEQ_1: 38;

          

           A74: <*(( Gauge (C,n)) * (i,wG))*> is one-to-one by FINSEQ_3: 93;

          

           A75: v1 is being_S-Seq by A23, JORDAN3: 34;

          then

           A76: Gv1 is one-to-one by A73, A74, FINSEQ_3: 91;

           <*NE*> is one-to-one by FINSEQ_3: 93;

          then

           A77: v is one-to-one by A68, A76, FINSEQ_3: 91;

          (( <*(( Gauge (C,n)) * (i,wG))*> /. ( len <*(( Gauge (C,n)) * (i,wG))*>)) `1 ) = (( <*(( Gauge (C,n)) * (i,wG))*> /. 1) `1 ) by FINSEQ_1: 39

          .= ((( Gauge (C,n)) * (i,wG)) `1 ) by FINSEQ_4: 16

          .= ((( Gauge (C,n)) * (i,1)) `1 ) by A1, A2, A13, GOBOARD5: 2

          .= ((v1 /. 1) `1 ) by A1, A2, A3, A4, A31, GOBOARD5: 2;

          then

           A78: Gv1 is special by A75, GOBOARD2: 8;

          ((Gv1 /. ( len Gv1)) `1 ) = (NE `1 ) by A30, PSCOMP_1: 29

          .= (( <*NE*> /. 1) `1 ) by FINSEQ_4: 16;

          then v is special by A78, GOBOARD2: 8;

          then

           A79: ( Rev v) is special by SPPOL_2: 40;

          

           A80: ( len ( Upper_Seq (C,n))) >= (2 + 1) by JORDAN1E: 15;

          then

           A81: ( len ( Upper_Seq (C,n))) > 2 by NAT_1: 13;

          ( len ( Upper_Seq (C,n))) > 1 by A80, XXREAL_0: 2;

          then

           A82: h is S-Sequence_in_R2 by A81, JORDAN3: 6;

          then

           A83: 2 <= ( len h) by TOPREAL1:def 8;

          3 <= ( len ( Upper_Seq (C,n))) by JORDAN1E: 15;

          then 2 <= ( len ( Upper_Seq (C,n))) by XXREAL_0: 2;

          then

           A84: 2 in ( dom ( Upper_Seq (C,n))) by FINSEQ_3: 25;

          

           A85: ( len ( Upper_Seq (C,n))) in ( dom ( Upper_Seq (C,n))) by FINSEQ_5: 6;

          then

           A86: h is_in_the_area_of ( Cage (C,n)) by A84, JORDAN1E: 17, SPRECT_2: 22;

          (( Upper_Seq (C,n)) /. ( len ( Upper_Seq (C,n)))) = ( E-max ( L~ ( Cage (C,n)))) by JORDAN1F: 7;

          then ((( Upper_Seq (C,n)) /. ( len ( Upper_Seq (C,n)))) `1 ) = ( E-bound ( L~ ( Cage (C,n)))) by EUCLID: 52;

          then

           A87: ((h /. ( len h)) `1 ) = ( E-bound ( L~ ( Cage (C,n)))) by A84, A85, SPRECT_2: 9;

          ((( Upper_Seq (C,n)) /. (1 + 1)) `1 ) = ( W-bound ( L~ ( Cage (C,n)))) by JORDAN1G: 31;

          then ((h /. 1) `1 ) = ( W-bound ( L~ ( Cage (C,n)))) by A84, A85, SPRECT_2: 8;

          then

           A88: h is_a_h.c._for ( Cage (C,n)) by A86, A87, SPRECT_2:def 2;

          now

            let m be Nat;

            assume

             A89: m in ( dom <*(( Gauge (C,n)) * (i,wG))*>);

            then m in ( Seg 1) by FINSEQ_1: 38;

            then m = 1 by FINSEQ_1: 2, TARSKI:def 1;

            then ( <*(( Gauge (C,n)) * (i,wG))*> . m) = (( Gauge (C,n)) * (i,wG)) by FINSEQ_1: 40;

            then

             A90: ( <*(( Gauge (C,n)) * (i,wG))*> /. m) = (( Gauge (C,n)) * (i,wG)) by A89, PARTFUN1:def 6;

            ((( Gauge (C,n)) * (1,wG)) `1 ) <= ((( Gauge (C,n)) * (i,wG)) `1 ) by A1, A2, A13, SPRECT_3: 13;

            hence ( W-bound ( L~ ( Cage (C,n)))) <= (( <*(( Gauge (C,n)) * (i,wG))*> /. m) `1 ) by A12, A13, A90, JORDAN1A: 73;

            ((( Gauge (C,n)) * (i,wG)) `1 ) <= ((( Gauge (C,n)) * (( len ( Gauge (C,n))),wG)) `1 ) by A1, A2, A13, SPRECT_3: 13;

            hence (( <*(( Gauge (C,n)) * (i,wG))*> /. m) `1 ) <= ( E-bound ( L~ ( Cage (C,n)))) by A12, A13, A90, JORDAN1A: 71;

            (( <*(( Gauge (C,n)) * (i,wG))*> /. m) `2 ) = ( N-bound ( L~ ( Cage (C,n)))) by A1, A2, A12, A90, JORDAN1A: 70;

            hence ( S-bound ( L~ ( Cage (C,n)))) <= (( <*(( Gauge (C,n)) * (i,wG))*> /. m) `2 ) by SPRECT_1: 22;

            thus (( <*(( Gauge (C,n)) * (i,wG))*> /. m) `2 ) <= ( N-bound ( L~ ( Cage (C,n)))) by A1, A2, A12, A90, JORDAN1A: 70;

          end;

          then

           A91: <*(( Gauge (C,n)) * (i,wG))*> is_in_the_area_of ( Cage (C,n)) by SPRECT_2:def 1;

           <*Gij*> is_in_the_area_of ( Cage (C,n)) by A23, JORDAN1E: 18, SPRECT_3: 46;

          then v1 is_in_the_area_of ( Cage (C,n)) by A23, JORDAN1E: 18, SPRECT_3: 56;

          then

           A92: Gv1 is_in_the_area_of ( Cage (C,n)) by A91, SPRECT_2: 24;

           <*NE*> is_in_the_area_of ( Cage (C,n)) by SPRECT_2: 28;

          then v is_in_the_area_of ( Cage (C,n)) by A92, SPRECT_2: 24;

          then

           A93: ( Rev v) is_in_the_area_of ( Cage (C,n)) by SPRECT_3: 51;

          v = ( <*(( Gauge (C,n)) * (i,wG))*> ^ (v1 ^ <*NE*>)) by FINSEQ_1: 32;

          then (v /. 1) = (( Gauge (C,n)) * (i,wG)) by FINSEQ_5: 15;

          then ((v /. 1) `2 ) = ( N-bound ( L~ ( Cage (C,n)))) by A1, A2, A12, JORDAN1A: 70;

          then ((( Rev v) /. ( len v)) `2 ) = ( N-bound ( L~ ( Cage (C,n)))) by FINSEQ_5: 65;

          then

           A94: ((( Rev v) /. ( len ( Rev v))) `2 ) = ( N-bound ( L~ ( Cage (C,n)))) by FINSEQ_5:def 3;

          ( len v) = (( len Gv1) + 1) by FINSEQ_2: 16;

          then (v /. ( len v)) = NE by FINSEQ_4: 67;

          then ((v /. ( len v)) `2 ) = ( S-bound ( L~ ( Cage (C,n)))) by EUCLID: 52;

          then ((( Rev v) /. 1) `2 ) = ( S-bound ( L~ ( Cage (C,n)))) by FINSEQ_5: 65;

          then ( Rev v) is_a_v.c._for ( Cage (C,n)) by A93, A94, SPRECT_2:def 3;

          then ( L~ h) meets ( L~ ( Rev v)) by A34, A77, A79, A82, A83, A88, SPRECT_2: 29;

          then ( L~ h) meets ( L~ v) by SPPOL_2: 22;

          then

          consider x be object such that

           A95: x in ( L~ h) and

           A96: x in ( L~ v) by XBOOLE_0: 3;

          

           A97: ( L~ h) c= ( L~ ( Upper_Seq (C,n))) by A9, A10, JORDAN4: 35;

          

           A98: ( L~ v1) c= ( L~ ( Lower_Seq (C,n))) by A23, JORDAN3: 42;

          ( L~ v) = ( L~ ( <*(( Gauge (C,n)) * (i,wG))*> ^ (v1 ^ <*NE*>))) by FINSEQ_1: 32

          .= (( LSeg ((( Gauge (C,n)) * (i,wG)),((v1 ^ <*NE*>) /. 1))) \/ ( L~ (v1 ^ <*NE*>))) by SPPOL_2: 20

          .= (( LSeg ((( Gauge (C,n)) * (i,wG)),((v1 ^ <*NE*>) /. 1))) \/ (( L~ v1) \/ ( LSeg ((v1 /. ( len v1)),NE)))) by A24, SPPOL_2: 19;

          then

           A99: x in ( LSeg ((( Gauge (C,n)) * (i,wG)),((v1 ^ <*NE*>) /. 1))) or x in (( L~ v1) \/ ( LSeg ((v1 /. ( len v1)),NE))) by A96, XBOOLE_0:def 3;

          (( Upper_Seq (C,n)) /. 1) = ( W-min ( L~ ( Cage (C,n)))) by JORDAN1F: 5;

          then

           A100: not ( W-min ( L~ ( Cage (C,n)))) in ( L~ h) by A81, JORDAN5B: 16;

          now

            per cases by A99, XBOOLE_0:def 3;

              suppose x in ( LSeg ((( Gauge (C,n)) * (i,wG)),((v1 ^ <*NE*>) /. 1)));

              then x in ( L~ <*(( Gauge (C,n)) * (i,wG)), Gij*>) by A32, SPPOL_2: 21;

              hence ( L~ ( Upper_Seq (C,n))) meets ( L~ <*(( Gauge (C,n)) * (i,wG)), Gij*>) by A95, A97, XBOOLE_0: 3;

            end;

              suppose

               A101: x in ( L~ v1);

              then x in (( L~ ( Lower_Seq (C,n))) /\ ( L~ ( Upper_Seq (C,n)))) by A95, A97, A98, XBOOLE_0:def 4;

              then x in {( W-min ( L~ ( Cage (C,n)))), ( E-max ( L~ ( Cage (C,n))))} by JORDAN1E: 16;

              then

               A102: x = ( E-max ( L~ ( Cage (C,n)))) by A95, A100, TARSKI:def 2;

              1 in ( dom ( Lower_Seq (C,n))) by A11, FINSEQ_3: 25;

              

              then (( Lower_Seq (C,n)) . 1) = (( Lower_Seq (C,n)) /. 1) by PARTFUN1:def 6

              .= ( E-max ( L~ ( Cage (C,n)))) by JORDAN1F: 6;

              then x = Gij by A23, A101, A102, JORDAN1E: 7;

              then x in ( LSeg ((( Gauge (C,n)) * (i,wG)),Gij)) by RLTOPSP1: 68;

              then x in ( L~ <*(( Gauge (C,n)) * (i,wG)), Gij*>) by SPPOL_2: 21;

              hence ( L~ ( Upper_Seq (C,n))) meets ( L~ <*(( Gauge (C,n)) * (i,wG)), Gij*>) by A95, A97, XBOOLE_0: 3;

            end;

              suppose

               A103: x in ( LSeg ((v1 /. ( len v1)),NE));

              x in ( L~ ( Cage (C,n))) by A6, A95, A97, XBOOLE_0:def 3;

              then x in (( LSeg (( W-min ( L~ ( Cage (C,n)))),NE)) /\ ( L~ ( Cage (C,n)))) by A29, A103, XBOOLE_0:def 4;

              then x in {( W-min ( L~ ( Cage (C,n))))} by PSCOMP_1: 35;

              hence ( L~ ( Upper_Seq (C,n))) meets ( L~ <*(( Gauge (C,n)) * (i,wG)), Gij*>) by A95, A100, TARSKI:def 1;

            end;

          end;

          then ( L~ <*(( Gauge (C,n)) * (i,wG)), Gij*>) meets ( L~ ( Upper_Seq (C,n)));

          hence thesis by SPPOL_2: 21;

        end;

          suppose

           A104: Gij in ( L~ ( Lower_Seq (C,n))) & Gij <> (( Lower_Seq (C,n)) . ( len ( Lower_Seq (C,n)))) & ( W-min ( L~ ( Cage (C,n)))) = NE & i < lG;

          then

           A105: v1 is non empty by JORDAN1E: 3;

          then

           A106: ( 0 + 1) <= ( len v1) by NAT_1: 13;

          then

           A107: 1 in ( dom v1) by FINSEQ_3: 25;

          set v = Gv1;

          

           A108: ( len v1) in ( dom v1) by A106, FINSEQ_3: 25;

          

           A109: ( len ( Lower_Seq (C,n))) in ( dom ( Lower_Seq (C,n))) by A11, FINSEQ_3: 25;

          (v1 /. ( len v1)) = (v1 . ( len v1)) by A108, PARTFUN1:def 6

          .= (( Lower_Seq (C,n)) . ( len ( Lower_Seq (C,n)))) by A104, JORDAN1B: 4

          .= (( Lower_Seq (C,n)) /. ( len ( Lower_Seq (C,n)))) by A109, PARTFUN1:def 6

          .= ( W-min ( L~ ( Cage (C,n)))) by JORDAN1F: 8;

          then

           A110: (Gv1 /. ( len Gv1)) = ( W-min ( L~ ( Cage (C,n)))) by A105, SPRECT_3: 1;

          

           A111: (v1 /. 1) = (v1 . 1) by A107, PARTFUN1:def 6

          .= Gij by A104, JORDAN3: 23;

          (1 + ( len v1)) >= (1 + 1) by A106, XREAL_1: 7;

          then 2 <= ( len v) by FINSEQ_5: 8;

          then

           A112: 2 <= ( len ( Rev v)) by FINSEQ_5:def 3;

          

           A113: not (( Gauge (C,n)) * (i,wG)) in ( L~ ( Lower_Seq (C,n))) by A1, A104, JORDAN1G: 45;

          ( rng ( Lower_Seq (C,n))) c= ( L~ ( Lower_Seq (C,n))) by A8, SPPOL_2: 18, XXREAL_0: 2;

          then

           A114: not (( Gauge (C,n)) * (i,wG)) in ( rng ( Lower_Seq (C,n))) by A1, A104, JORDAN1G: 45;

           not (( Gauge (C,n)) * (i,wG)) in {Gij} by A104, A113, TARSKI:def 1;

          then

           A115: not (( Gauge (C,n)) * (i,wG)) in ( rng <*Gij*>) by FINSEQ_1: 38;

          set ci = ( mid (( Lower_Seq (C,n)),(( Index (Gij,( Lower_Seq (C,n)))) + 1),( len ( Lower_Seq (C,n)))));

          now

            per cases ;

              suppose

               A116: Gij <> (( Lower_Seq (C,n)) . (( Index (Gij,( Lower_Seq (C,n)))) + 1));

              ( rng ci) c= ( rng ( Lower_Seq (C,n))) by FINSEQ_6: 119;

              then not (( Gauge (C,n)) * (i,wG)) in ( rng ci) by A114;

              then not (( Gauge (C,n)) * (i,wG)) in (( rng <*Gij*>) \/ ( rng ci)) by A115, XBOOLE_0:def 3;

              then not (( Gauge (C,n)) * (i,wG)) in ( rng ( <*Gij*> ^ ci)) by FINSEQ_1: 31;

              hence not (( Gauge (C,n)) * (i,wG)) in ( rng v1) by A116, JORDAN3:def 3;

            end;

              suppose Gij = (( Lower_Seq (C,n)) . (( Index (Gij,( Lower_Seq (C,n)))) + 1));

              then v1 = ci by JORDAN3:def 3;

              then ( rng v1) c= ( rng ( Lower_Seq (C,n))) by FINSEQ_6: 119;

              hence not (( Gauge (C,n)) * (i,wG)) in ( rng v1) by A114;

            end;

          end;

          then {(( Gauge (C,n)) * (i,wG))} misses ( rng v1) by ZFMISC_1: 50;

          then

           A117: ( rng <*(( Gauge (C,n)) * (i,wG))*>) misses ( rng v1) by FINSEQ_1: 38;

          

           A118: <*(( Gauge (C,n)) * (i,wG))*> is one-to-one by FINSEQ_3: 93;

          

           A119: v1 is being_S-Seq by A104, JORDAN3: 34;

          then

           A120: Gv1 is one-to-one by A117, A118, FINSEQ_3: 91;

          (( <*(( Gauge (C,n)) * (i,wG))*> /. ( len <*(( Gauge (C,n)) * (i,wG))*>)) `1 ) = (( <*(( Gauge (C,n)) * (i,wG))*> /. 1) `1 ) by FINSEQ_1: 39

          .= ((( Gauge (C,n)) * (i,wG)) `1 ) by FINSEQ_4: 16

          .= ((( Gauge (C,n)) * (i,1)) `1 ) by A1, A2, A13, GOBOARD5: 2

          .= ((v1 /. 1) `1 ) by A1, A2, A3, A4, A111, GOBOARD5: 2;

          then Gv1 is special by A119, GOBOARD2: 8;

          then

           A121: ( Rev v) is special by SPPOL_2: 40;

          

           A122: ( len ( Upper_Seq (C,n))) >= (2 + 1) by JORDAN1E: 15;

          then

           A123: ( len ( Upper_Seq (C,n))) > 2 by NAT_1: 13;

          ( len ( Upper_Seq (C,n))) > 1 by A122, XXREAL_0: 2;

          then

           A124: h is S-Sequence_in_R2 by A123, JORDAN3: 6;

          then

           A125: 2 <= ( len h) by TOPREAL1:def 8;

          3 <= ( len ( Upper_Seq (C,n))) by JORDAN1E: 15;

          then 2 <= ( len ( Upper_Seq (C,n))) by XXREAL_0: 2;

          then

           A126: 2 in ( dom ( Upper_Seq (C,n))) by FINSEQ_3: 25;

          

           A127: ( len ( Upper_Seq (C,n))) in ( dom ( Upper_Seq (C,n))) by FINSEQ_5: 6;

          then

           A128: h is_in_the_area_of ( Cage (C,n)) by A126, JORDAN1E: 17, SPRECT_2: 22;

          (( Upper_Seq (C,n)) /. ( len ( Upper_Seq (C,n)))) = ( E-max ( L~ ( Cage (C,n)))) by JORDAN1F: 7;

          then ((( Upper_Seq (C,n)) /. ( len ( Upper_Seq (C,n)))) `1 ) = ( E-bound ( L~ ( Cage (C,n)))) by EUCLID: 52;

          then

           A129: ((h /. ( len h)) `1 ) = ( E-bound ( L~ ( Cage (C,n)))) by A126, A127, SPRECT_2: 9;

          ((( Upper_Seq (C,n)) /. (1 + 1)) `1 ) = ( W-bound ( L~ ( Cage (C,n)))) by JORDAN1G: 31;

          then ((h /. 1) `1 ) = ( W-bound ( L~ ( Cage (C,n)))) by A126, A127, SPRECT_2: 8;

          then

           A130: h is_a_h.c._for ( Cage (C,n)) by A128, A129, SPRECT_2:def 2;

          now

            let m be Nat;

            assume

             A131: m in ( dom <*(( Gauge (C,n)) * (i,wG))*>);

            then m in ( Seg 1) by FINSEQ_1: 38;

            then m = 1 by FINSEQ_1: 2, TARSKI:def 1;

            then ( <*(( Gauge (C,n)) * (i,wG))*> . m) = (( Gauge (C,n)) * (i,wG)) by FINSEQ_1: 40;

            then

             A132: ( <*(( Gauge (C,n)) * (i,wG))*> /. m) = (( Gauge (C,n)) * (i,wG)) by A131, PARTFUN1:def 6;

            ((( Gauge (C,n)) * (1,wG)) `1 ) <= ((( Gauge (C,n)) * (i,wG)) `1 ) by A1, A2, A13, SPRECT_3: 13;

            hence ( W-bound ( L~ ( Cage (C,n)))) <= (( <*(( Gauge (C,n)) * (i,wG))*> /. m) `1 ) by A12, A13, A132, JORDAN1A: 73;

            ((( Gauge (C,n)) * (i,wG)) `1 ) <= ((( Gauge (C,n)) * (( len ( Gauge (C,n))),wG)) `1 ) by A1, A2, A13, SPRECT_3: 13;

            hence (( <*(( Gauge (C,n)) * (i,wG))*> /. m) `1 ) <= ( E-bound ( L~ ( Cage (C,n)))) by A12, A13, A132, JORDAN1A: 71;

            (( <*(( Gauge (C,n)) * (i,wG))*> /. m) `2 ) = ( N-bound ( L~ ( Cage (C,n)))) by A1, A2, A12, A132, JORDAN1A: 70;

            hence ( S-bound ( L~ ( Cage (C,n)))) <= (( <*(( Gauge (C,n)) * (i,wG))*> /. m) `2 ) by SPRECT_1: 22;

            thus (( <*(( Gauge (C,n)) * (i,wG))*> /. m) `2 ) <= ( N-bound ( L~ ( Cage (C,n)))) by A1, A2, A12, A132, JORDAN1A: 70;

          end;

          then

           A133: <*(( Gauge (C,n)) * (i,wG))*> is_in_the_area_of ( Cage (C,n)) by SPRECT_2:def 1;

           <*Gij*> is_in_the_area_of ( Cage (C,n)) by A104, JORDAN1E: 18, SPRECT_3: 46;

          then v1 is_in_the_area_of ( Cage (C,n)) by A104, JORDAN1E: 18, SPRECT_3: 56;

          then Gv1 is_in_the_area_of ( Cage (C,n)) by A133, SPRECT_2: 24;

          then

           A134: ( Rev v) is_in_the_area_of ( Cage (C,n)) by SPRECT_3: 51;

          (v /. 1) = (( Gauge (C,n)) * (i,wG)) by FINSEQ_5: 15;

          then ((v /. 1) `2 ) = ( N-bound ( L~ ( Cage (C,n)))) by A1, A2, A12, JORDAN1A: 70;

          then ((( Rev v) /. ( len v)) `2 ) = ( N-bound ( L~ ( Cage (C,n)))) by FINSEQ_5: 65;

          then

           A135: ((( Rev v) /. ( len ( Rev v))) `2 ) = ( N-bound ( L~ ( Cage (C,n)))) by FINSEQ_5:def 3;

          ((v /. ( len v)) `2 ) = ( S-bound ( L~ ( Cage (C,n)))) by A104, A110, EUCLID: 52;

          then ((( Rev v) /. 1) `2 ) = ( S-bound ( L~ ( Cage (C,n)))) by FINSEQ_5: 65;

          then ( Rev v) is_a_v.c._for ( Cage (C,n)) by A134, A135, SPRECT_2:def 3;

          then ( L~ h) meets ( L~ ( Rev v)) by A112, A120, A121, A124, A125, A130, SPRECT_2: 29;

          then ( L~ h) meets ( L~ v) by SPPOL_2: 22;

          then

          consider x be object such that

           A136: x in ( L~ h) and

           A137: x in ( L~ v) by XBOOLE_0: 3;

          

           A138: ( L~ h) c= ( L~ ( Upper_Seq (C,n))) by A9, A10, JORDAN4: 35;

          

           A139: ( L~ v1) c= ( L~ ( Lower_Seq (C,n))) by A104, JORDAN3: 42;

          

           A140: ( L~ v) = (( LSeg ((( Gauge (C,n)) * (i,wG)),(v1 /. 1))) \/ ( L~ v1)) by A105, SPPOL_2: 20;

          (( Upper_Seq (C,n)) /. 1) = ( W-min ( L~ ( Cage (C,n)))) by JORDAN1F: 5;

          then

           A141: not ( W-min ( L~ ( Cage (C,n)))) in ( L~ h) by A123, JORDAN5B: 16;

          now

            per cases by A137, A140, XBOOLE_0:def 3;

              suppose x in ( LSeg ((( Gauge (C,n)) * (i,wG)),(v1 /. 1)));

              then x in ( L~ <*(( Gauge (C,n)) * (i,wG)), Gij*>) by A111, SPPOL_2: 21;

              hence ( L~ ( Upper_Seq (C,n))) meets ( L~ <*(( Gauge (C,n)) * (i,wG)), Gij*>) by A136, A138, XBOOLE_0: 3;

            end;

              suppose

               A142: x in ( L~ v1);

              then x in (( L~ ( Lower_Seq (C,n))) /\ ( L~ ( Upper_Seq (C,n)))) by A136, A138, A139, XBOOLE_0:def 4;

              then x in {( W-min ( L~ ( Cage (C,n)))), ( E-max ( L~ ( Cage (C,n))))} by JORDAN1E: 16;

              then

               A143: x = ( E-max ( L~ ( Cage (C,n)))) by A136, A141, TARSKI:def 2;

              1 in ( dom ( Lower_Seq (C,n))) by A11, FINSEQ_3: 25;

              

              then (( Lower_Seq (C,n)) . 1) = (( Lower_Seq (C,n)) /. 1) by PARTFUN1:def 6

              .= ( E-max ( L~ ( Cage (C,n)))) by JORDAN1F: 6;

              then x = Gij by A104, A142, A143, JORDAN1E: 7;

              then x in ( LSeg ((( Gauge (C,n)) * (i,wG)),Gij)) by RLTOPSP1: 68;

              then x in ( L~ <*(( Gauge (C,n)) * (i,wG)), Gij*>) by SPPOL_2: 21;

              hence ( L~ ( Upper_Seq (C,n))) meets ( L~ <*(( Gauge (C,n)) * (i,wG)), Gij*>) by A136, A138, XBOOLE_0: 3;

            end;

          end;

          then ( L~ <*(( Gauge (C,n)) * (i,wG)), Gij*>) meets ( L~ ( Upper_Seq (C,n)));

          hence thesis by SPPOL_2: 21;

        end;

          suppose

           A144: Gij in ( L~ ( Upper_Seq (C,n)));

          Gij in ( LSeg ((( Gauge (C,n)) * (i,wG)),Gij)) by RLTOPSP1: 68;

          hence thesis by A144, XBOOLE_0: 3;

        end;

          suppose

           A145: Gij in ( L~ ( Lower_Seq (C,n))) & Gij = (( Lower_Seq (C,n)) . ( len ( Lower_Seq (C,n))));

          ( len ( Lower_Seq (C,n))) in ( dom ( Lower_Seq (C,n))) by A11, FINSEQ_3: 25;

          

          then

           A146: (( Lower_Seq (C,n)) . ( len ( Lower_Seq (C,n)))) = (( Lower_Seq (C,n)) /. ( len ( Lower_Seq (C,n)))) by PARTFUN1:def 6

          .= ( W-min ( L~ ( Cage (C,n)))) by JORDAN1F: 8;

          

           A147: ( rng ( Upper_Seq (C,n))) c= ( L~ ( Upper_Seq (C,n))) by A7, SPPOL_2: 18, XXREAL_0: 2;

          

           A148: ( W-min ( L~ ( Cage (C,n)))) in ( rng ( Upper_Seq (C,n))) by JORDAN1J: 5;

          Gij in ( LSeg ((( Gauge (C,n)) * (i,wG)),Gij)) by RLTOPSP1: 68;

          hence thesis by A145, A146, A147, A148, XBOOLE_0: 3;

        end;

      end;

      hence thesis;

    end;

    theorem :: JORDAN19:4

    

     Th4: for C be compact connected non vertical non horizontal Subset of ( TOP-REAL 2) holds for n be Nat st n > 0 holds for i,j be Nat st 1 <= i & i <= ( len ( Gauge (C,n))) & 1 <= j & j <= ( width ( Gauge (C,n))) & (( Gauge (C,n)) * (i,j)) in ( L~ ( Cage (C,n))) holds ( LSeg ((( Gauge (C,n)) * (i,( width ( Gauge (C,n))))),(( Gauge (C,n)) * (i,j)))) meets ( Upper_Arc ( L~ ( Cage (C,n))))

    proof

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

      let n be Nat;

      assume

       A1: n > 0 ;

      let i,j be Nat;

      assume that

       A2: 1 <= i and

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

       A4: 1 <= j and

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

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

      ( L~ ( Upper_Seq (C,n))) = ( Upper_Arc ( L~ ( Cage (C,n)))) by A1, JORDAN1G: 55;

      hence thesis by A2, A3, A4, A5, A6, Th3;

    end;

    theorem :: JORDAN19:5

    for C be compact connected non vertical non horizontal Subset of ( TOP-REAL 2) holds for j be Nat holds (( Gauge (C,(n + 1))) * (( Center ( Gauge (C,(n + 1)))),j)) in ( Lower_Arc ( L~ ( Cage (C,(n + 1))))) & 1 <= j & j <= ( width ( Gauge (C,(n + 1)))) implies ( LSeg ((( Gauge (C,1)) * (( Center ( Gauge (C,1))),( width ( Gauge (C,1))))),(( Gauge (C,(n + 1))) * (( Center ( Gauge (C,(n + 1)))),j)))) meets ( Upper_Arc ( L~ ( Cage (C,(n + 1)))))

    proof

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

      let j be Nat;

      assume that

       A1: (( Gauge (C,(n + 1))) * (( Center ( Gauge (C,(n + 1)))),j)) in ( Lower_Arc ( L~ ( Cage (C,(n + 1))))) and

       A2: 1 <= j and

       A3: j <= ( width ( Gauge (C,(n + 1))));

      set in1 = ( Center ( Gauge (C,(n + 1))));

      

       A4: (n + 1) >= ( 0 + 1) by NAT_1: 11;

      

       A5: 1 <= in1 by JORDAN1B: 11;

      

       A6: in1 <= ( len ( Gauge (C,(n + 1)))) by JORDAN1B: 13;

      

       A7: ( LSeg ((( Gauge (C,(n + 1))) * (( Center ( Gauge (C,(n + 1)))),( width ( Gauge (C,(n + 1)))))),(( Gauge (C,(n + 1))) * (( Center ( Gauge (C,(n + 1)))),j)))) c= ( LSeg ((( Gauge (C,1)) * (( Center ( Gauge (C,1))),( width ( Gauge (C,1))))),(( Gauge (C,(n + 1))) * (( Center ( Gauge (C,(n + 1)))),j)))) by A2, A3, A4, Th2;

      ( Lower_Arc ( L~ ( Cage (C,(n + 1))))) c= ( L~ ( Cage (C,(n + 1)))) by JORDAN6: 61;

      hence thesis by A1, A2, A3, A5, A6, A7, Th4, XBOOLE_1: 63;

    end;

    theorem :: JORDAN19:6

    

     Th6: for C be compact connected non vertical non horizontal Subset of ( TOP-REAL 2) holds for f be FinSequence of ( TOP-REAL 2) holds for k be Nat st 1 <= k & (k + 1) <= ( len f) & f is_sequence_on ( Gauge (C,n)) holds ( dist ((f /. k),(f /. (k + 1)))) = ((( N-bound C) - ( S-bound C)) / (2 |^ n)) or ( dist ((f /. k),(f /. (k + 1)))) = ((( E-bound C) - ( W-bound C)) / (2 |^ n))

    proof

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

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

      let k be Nat;

      assume that

       A1: 1 <= k and

       A2: (k + 1) <= ( len f);

      assume f is_sequence_on ( Gauge (C,n));

      then

      consider i1,j1,i2,j2 be Nat such that

       A3: [i1, j1] in ( Indices ( Gauge (C,n))) and

       A4: (f /. k) = (( Gauge (C,n)) * (i1,j1)) and

       A5: [i2, j2] in ( Indices ( Gauge (C,n))) and

       A6: (f /. (k + 1)) = (( Gauge (C,n)) * (i2,j2)) and

       A7: i1 = i2 & (j1 + 1) = j2 or (i1 + 1) = i2 & j1 = j2 or i1 = (i2 + 1) & j1 = j2 or i1 = i2 & j1 = (j2 + 1) by A1, A2, JORDAN8: 3;

      per cases by A7;

        suppose i1 = i2 & (j1 + 1) = j2;

        hence thesis by A3, A4, A5, A6, GOBRD14: 9;

      end;

        suppose (i1 + 1) = i2 & j1 = j2;

        hence thesis by A3, A4, A5, A6, GOBRD14: 10;

      end;

        suppose i1 = (i2 + 1) & j1 = j2;

        hence thesis by A3, A4, A5, A6, GOBRD14: 10;

      end;

        suppose i1 = i2 & j1 = (j2 + 1);

        hence thesis by A3, A4, A5, A6, GOBRD14: 9;

      end;

    end;

    theorem :: JORDAN19:7

    for M be symmetric triangle MetrStruct holds for r be Real holds for p,q,x be Element of M st p in ( Ball (x,r)) & q in ( Ball (x,r)) holds ( dist (p,q)) < (2 * r)

    proof

      let M be symmetric triangle MetrStruct;

      let r be Real;

      let p,q,x be Element of M;

      assume that

       A1: p in ( Ball (x,r)) and

       A2: q in ( Ball (x,r));

      

       A3: ( dist (p,x)) < r by A1, METRIC_1: 11;

      

       A4: ( dist (x,q)) < r by A2, METRIC_1: 11;

      

       A5: ( dist (p,q)) <= (( dist (p,x)) + ( dist (x,q))) by METRIC_1: 4;

      (( dist (p,x)) + ( dist (x,q))) < (r + r) by A3, A4, XREAL_1: 8;

      hence thesis by A5, XXREAL_0: 2;

    end;

    theorem :: JORDAN19:8

    for C be compact connected non vertical non horizontal Subset of ( TOP-REAL 2) holds ( N-bound C) < ( N-bound ( L~ ( Cage (C,n))))

    proof

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

      

       A1: (2 |^ n) > 0 by NEWTON: 83;

      ( N-bound C) > (( S-bound C) + 0 ) by SPRECT_1: 32;

      then (( N-bound C) - ( S-bound C)) > 0 by XREAL_1: 20;

      then

       A2: ((( N-bound C) - ( S-bound C)) / (2 |^ n)) > (( N-bound C) - ( N-bound C)) by A1, XREAL_1: 139;

      ( N-bound ( L~ ( Cage (C,n)))) = (( N-bound C) + ((( N-bound C) - ( S-bound C)) / (2 |^ n))) by JORDAN10: 6;

      hence thesis by A2, XREAL_1: 19;

    end;

    theorem :: JORDAN19:9

    

     Th9: for C be compact connected non vertical non horizontal Subset of ( TOP-REAL 2) holds ( E-bound C) < ( E-bound ( L~ ( Cage (C,n))))

    proof

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

      

       A1: (2 |^ n) > 0 by NEWTON: 83;

      ( E-bound C) > (( W-bound C) + 0 ) by SPRECT_1: 31;

      then (( E-bound C) - ( W-bound C)) > 0 by XREAL_1: 20;

      then

       A2: ((( E-bound C) - ( W-bound C)) / (2 |^ n)) > (( E-bound C) - ( E-bound C)) by A1, XREAL_1: 139;

      ( E-bound ( L~ ( Cage (C,n)))) = (( E-bound C) + ((( E-bound C) - ( W-bound C)) / (2 |^ n))) by JORDAN1A: 64;

      hence thesis by A2, XREAL_1: 19;

    end;

    theorem :: JORDAN19:10

    for C be compact connected non vertical non horizontal Subset of ( TOP-REAL 2) holds ( S-bound ( L~ ( Cage (C,n)))) < ( S-bound C)

    proof

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

      

       A1: (2 |^ n) > 0 by NEWTON: 83;

      ( N-bound C) > (( S-bound C) + 0 ) by SPRECT_1: 32;

      then (( N-bound C) - ( S-bound C)) > 0 by XREAL_1: 20;

      then

       A2: ((( N-bound C) - ( S-bound C)) / (2 |^ n)) > (( S-bound C) - ( S-bound C)) by A1, XREAL_1: 139;

      ( S-bound ( L~ ( Cage (C,n)))) = (( S-bound C) - ((( N-bound C) - ( S-bound C)) / (2 |^ n))) by JORDAN1A: 63;

      hence thesis by A2, XREAL_1: 11;

    end;

    theorem :: JORDAN19:11

    

     Th11: for C be compact connected non vertical non horizontal Subset of ( TOP-REAL 2) holds ( W-bound ( L~ ( Cage (C,n)))) < ( W-bound C)

    proof

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

      

       A1: (2 |^ n) > 0 by NEWTON: 83;

      ( E-bound C) > (( W-bound C) + 0 ) by SPRECT_1: 31;

      then (( E-bound C) - ( W-bound C)) > 0 by XREAL_1: 20;

      then

       A2: ((( E-bound C) - ( W-bound C)) / (2 |^ n)) > (( W-bound C) - ( W-bound C)) by A1, XREAL_1: 139;

      ( W-bound ( L~ ( Cage (C,n)))) = (( W-bound C) - ((( E-bound C) - ( W-bound C)) / (2 |^ n))) by JORDAN1A: 62;

      hence thesis by A2, XREAL_1: 11;

    end;

    theorem :: JORDAN19:12

    

     Th12: for C be Simple_closed_curve holds for i,j,k be Nat st 1 < i & i < ( len ( Gauge (C,n))) & 1 <= k & k <= j & j <= ( width ( Gauge (C,n))) & (( LSeg ((( Gauge (C,n)) * (i,k)),(( Gauge (C,n)) * (i,j)))) /\ ( L~ ( Upper_Seq (C,n)))) = {(( Gauge (C,n)) * (i,k))} & (( LSeg ((( Gauge (C,n)) * (i,k)),(( Gauge (C,n)) * (i,j)))) /\ ( L~ ( Lower_Seq (C,n)))) = {(( Gauge (C,n)) * (i,j))} holds ( LSeg ((( Gauge (C,n)) * (i,k)),(( Gauge (C,n)) * (i,j)))) 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 Gik = (Ga * (i,k));

      set Gij = (Ga * (i,j));

      assume that

       A1: 1 < i and

       A2: i < ( len Ga) and

       A3: 1 <= k and

       A4: k <= j and

       A5: j <= ( width Ga) and

       A6: (( LSeg (Gik,Gij)) /\ ( L~ US)) = {Gik} and

       A7: (( LSeg (Gik,Gij)) /\ ( L~ LS)) = {Gij} and

       A8: ( LSeg (Gik,Gij)) 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;

      then

       A11: j <> k by A1, A2, A3, A5, A9, JORDAN1J: 57;

      

       A12: 1 <= j by A3, A4, XXREAL_0: 2;

      

       A13: k <= ( width Ga) by A4, A5, XXREAL_0: 2;

      

       A14: [i, j] in ( Indices Ga) by A1, A2, A5, A12, MATRIX_0: 30;

      

       A15: [i, k] in ( Indices Ga) by A1, A2, A3, A13, MATRIX_0: 30;

      set co = ( L_Cut (LS,Gij));

      set go = ( R_Cut (US,Gik));

      

       A16: ( len Ga) = ( width Ga) by JORDAN8:def 1;

      

       A17: ( len US) >= 3 by JORDAN1E: 15;

      then ( len US) >= 1 by XXREAL_0: 2;

      then 1 in ( dom US) by FINSEQ_3: 25;

      

      then

       A18: (US . 1) = (US /. 1) by PARTFUN1:def 6

      .= Wmin by JORDAN1F: 5;

      

       A19: (Wmin `1 ) = Wbo by EUCLID: 52

      .= ((Ga * (1,k)) `1 ) by A3, A13, A16, JORDAN1A: 73;

      ( len Ga) >= 4 by JORDAN8: 10;

      then

       A20: ( len Ga) >= 1 by XXREAL_0: 2;

      then

       A21: [1, k] in ( Indices Ga) by A3, A13, MATRIX_0: 30;

      then

       A22: Gik <> (US . 1) by A1, A15, A18, A19, JORDAN1G: 7;

      then

      reconsider go as being_S-Seq FinSequence of ( TOP-REAL 2) by A10, JORDAN3: 35;

      

       A23: ( len LS) >= (1 + 2) by JORDAN1E: 15;

      then

       A24: ( len LS) >= 1 by XXREAL_0: 2;

      then

       A25: 1 in ( dom LS) by FINSEQ_3: 25;

      ( len LS) in ( dom LS) by A24, FINSEQ_3: 25;

      

      then

       A26: (LS . ( len LS)) = (LS /. ( len LS)) by PARTFUN1:def 6

      .= Wmin by JORDAN1F: 8;

      

       A27: (Wmin `1 ) = Wbo by EUCLID: 52

      .= ((Ga * (1,k)) `1 ) by A3, A13, A16, JORDAN1A: 73;

      

       A28: [i, j] in ( Indices Ga) by A1, A2, A5, A12, MATRIX_0: 30;

      then

       A29: Gij <> (LS . ( len LS)) by A1, A21, A26, A27, 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 A3, A13, A20, MATRIX_0: 30;

      

       A31: (LS . 1) = (LS /. 1) by A25, PARTFUN1:def 6

      .= Emax by JORDAN1F: 6;

      (Emax `1 ) = Ebo by EUCLID: 52

      .= ((Ga * (( len Ga),k)) `1 ) by A3, A13, A16, JORDAN1A: 71;

      then

       A32: Gij <> (LS . 1) by A2, A28, A30, A31, JORDAN1G: 7;

      

       A33: ( len go) >= (1 + 1) by TOPREAL1:def 8;

      

       A34: Gik in ( rng US) by A1, A2, A3, A10, 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: Gij in ( rng LS) by A1, A2, A5, A9, A12, 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

      .= Gik 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

      .= Gij 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= {Gik} by A6, XBOOLE_1: 26;

      m >= 1 by A33, XREAL_1: 19;

      then

       A48: ( LSeg (go,m)) = ( LSeg ((go /. m),Gik)) by A41, A43, TOPREAL1:def 3;

       {Gik} c= (( LSeg (go,m)) /\ ( LSeg (Gik,Gij)))

      proof

        let x be object;

        assume x in {Gik};

        then

         A49: x = Gik by TARSKI:def 1;

        

         A50: Gik in ( LSeg (go,m)) by A48, RLTOPSP1: 68;

        Gik in ( LSeg (Gik,Gij)) by RLTOPSP1: 68;

        hence thesis by A49, A50, XBOOLE_0:def 4;

      end;

      then

       A51: (( LSeg (go,m)) /\ ( LSeg (Gik,Gij))) = {Gik} 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= {Gij} by A7, XBOOLE_1: 26;

      

       A55: ( LSeg (co,1)) = ( LSeg (Gij,(co /. (1 + 1)))) by A36, A42, TOPREAL1:def 3;

       {Gij} c= (( LSeg (co,1)) /\ ( LSeg (Gik,Gij)))

      proof

        let x be object;

        assume x in {Gij};

        then

         A56: x = Gij by TARSKI:def 1;

        

         A57: Gij in ( LSeg (co,1)) by A55, RLTOPSP1: 68;

        Gij in ( LSeg (Gik,Gij)) by RLTOPSP1: 68;

        hence thesis by A56, A57, XBOOLE_0:def 4;

      end;

      then

       A58: (( LSeg (Gik,Gij)) /\ ( LSeg (co,1))) = {Gij} 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 A25, PARTFUN1:def 6

      .= Emax by JORDAN1F: 6;

      

       A67: [( len Ga), j] in ( Indices Ga) by A5, A12, A20, 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~ go) by XBOOLE_0:def 4;

        

         A70: x in ( L~ co) 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

         A71: x = Wmin or x = Emax by TARSKI:def 2;

        now

          assume x = Emax;

          then

           A72: Emax = Gij by A9, A66, A70, JORDAN1E: 7;

          ((Ga * (( len Ga),j)) `1 ) = Ebo by A5, A12, A16, JORDAN1A: 71;

          then (Emax `1 ) <> Ebo by A2, A14, A67, A72, JORDAN1G: 7;

          hence contradiction by EUCLID: 52;

        end;

        hence thesis by A59, A71, TARSKI:def 1;

      end;

      then

       A73: (( L~ go) /\ ( L~ co)) = {(go /. 1)} by A63;

      set W2 = (go /. 2);

      

       A74: 2 in ( dom go) by A33, FINSEQ_3: 25;

       A75:

      now

        assume (Gik `1 ) = Wbo;

        then ((Ga * (1,k)) `1 ) = ((Ga * (i,k)) `1 ) by A3, A13, A16, JORDAN1A: 73;

        hence contradiction by A1, A15, A21, JORDAN1G: 7;

      end;

      go = ( mid (US,1,(Gik .. US))) by A34, JORDAN1G: 49

      .= (US | (Gik .. US)) by A34, FINSEQ_4: 21, FINSEQ_6: 116;

      then

       A76: W2 = (US /. 2) by A74, FINSEQ_4: 70;

      

       A77: Wmin in ( rng go) by A59, FINSEQ_6: 42;

      set pion = <*Gik, Gij*>;

       A78:

      now

        let n be Nat;

        assume n in ( dom pion);

        then n in ( Seg 2) by FINSEQ_1: 89;

        then n = 1 or n = 2 by FINSEQ_1: 2, TARSKI:def 2;

        hence ex i,j be Nat st [i, j] in ( Indices Ga) & (pion /. n) = (Ga * (i,j)) by A14, A15, FINSEQ_4: 17;

      end;

      

       A79: Gik <> Gij by A11, A14, A15, GOBOARD1: 5;

      

       A80: (Gik `1 ) = ((Ga * (i,1)) `1 ) by A1, A2, A3, A13, GOBOARD5: 2

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

      then ( LSeg (Gik,Gij)) is vertical by SPPOL_1: 16;

      then pion is being_S-Seq by A79, JORDAN1B: 7;

      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 A78, 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 A33, 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 A41, A84, FINSEQ_4: 17;

      then

       A95: (go ^' pion1) is_sequence_on Ga by A35, 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 A42, FINSEQ_4: 17;

      then

       A97: godo is_sequence_on Ga by A38, 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 A44, A51, 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 A48, RLTOPSP1: 68;

        Gik in ( LSeg (pion1,1)) by A41, 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 A41, A44, 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 A58, 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;

        

         A111: Gij in ( LSeg (co,1)) by A55, RLTOPSP1: 68;

        (pion1 /. ((( len pion1) -' 1) + 1)) = (pion /. 2) by A85, A106, FINSEQ_1: 44

        .= Gij by FINSEQ_4: 17;

        then Gij in ( LSeg (pion1,(( len pion1) -' 1))) by A105, A106, TOPREAL1: 21;

        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 A42, 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 go) by A94, FINSEQ_6: 168;

        x in ( rng pion1) by A116, FINSEQ_6: 42;

        hence thesis by A61, 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~ go) by XBOOLE_0:def 4;

        x in ( L~ pion1) by A118, XBOOLE_0:def 4;

        then x in (( L~ pion1) /\ ( L~ US)) by A46, A119, XBOOLE_0:def 4;

        hence thesis by A6, A41, 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 A61, 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 A42, 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 co) by A85, A123, FINSEQ_6: 42;

        x in ( rng pion1) by A125, FINSEQ_6: 168;

        hence thesis by A62, 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~ co) by XBOOLE_0:def 4;

        x in ( L~ pion1) by A127, XBOOLE_0:def 4;

        then x in (( L~ pion1) /\ ( L~ LS)) by A53, A128, XBOOLE_0:def 4;

        hence thesis by A7, A42, 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 A73, 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 A60, 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 A34, FINSEQ_4: 21;

        then (Gik .. US) = 1 by A139, XXREAL_0: 1;

        then Gik = (US /. 1) by A34, FINSEQ_5: 38;

        hence contradiction by A18, A22, 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 A46, A145;

      

       A148: ( L~ co) c= ( L~ ( Cage (C,n))) by A53, 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 A34, A93, A138, JORDAN1J: 52

      .= ( right_cell ((go ^' pion1),1,Ga)) by A39, 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 A59, FINSEQ_6: 155;

      

       A155: ( len US) >= 2 by A17, XXREAL_0: 2;

      

       A156: (godo /. 2) = ((go ^' pion1) /. 2) by A89, FINSEQ_6: 159

      .= (US /. 2) by A33, A76, 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 A61, A77, 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;

      ( W-bound ( LSeg (Gik,Gij))) = (Gik `1 ) by A80, SPRECT_1: 54;

      then

       A161: ( W-bound ( L~ pion1)) = (Gik `1 ) by A83, SPPOL_2: 21;

      (Gik `1 ) >= Wbo by A10, A145, PSCOMP_1: 24;

      then (Gik `1 ) > Wbo by A75, XXREAL_0: 1;

      then ( W-min ((( L~ go) \/ ( L~ co)) \/ ( L~ pion1))) = ( W-min (( L~ go) \/ ( L~ co))) by A157, A158, A159, A160, A161, JORDAN1J: 33;

      then

       A162: ( W-min ( L~ godo)) = Wmin by A143, A158, XBOOLE_1: 4;

      

       A163: ( rng godo) c= ( L~ godo) by A89, A91, SPPOL_2: 18, XXREAL_0: 2;

      2 in ( dom godo) by A92, FINSEQ_3: 25;

      then

       A164: (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 A162, PSCOMP_1: 31

      .= ( W-bound ( L~ godo)) by EUCLID: 52;

      then (godo /. 2) in ( W-most ( L~ godo)) by A163, A164, SPRECT_2: 12;

      then (( Rotate (godo,( W-min ( L~ godo)))) /. 2) in ( W-most ( L~ godo)) by A154, A162, 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

       A165: (US . ( len US)) = (US /. ( len US)) by PARTFUN1:def 6

      .= Emax by JORDAN1F: 7;

      

       A166: ( 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

         A167: p in ( east_halfline ( E-max C)) and

         A168: p in ( L~ go) by XBOOLE_0: 3;

        reconsider p as Point of ( TOP-REAL 2) by A167;

        p in ( L~ US) by A46, A168;

        then p in (( east_halfline ( E-max C)) /\ ( L~ ( Cage (C,n)))) by A145, A167, XBOOLE_0:def 4;

        then

         A169: (p `1 ) = Ebo by JORDAN1A: 83, PSCOMP_1: 50;

        then

         A170: p = Emax by A46, A168, JORDAN1J: 46;

        then Emax = Gik by A10, A165, A168, JORDAN1J: 43;

        then (Gik `1 ) = ((Ga * (( len Ga),k)) `1 ) by A3, A13, A16, A169, A170, JORDAN1A: 71;

        hence contradiction by A2, A15, A30, JORDAN1G: 7;

      end;

      now

        assume ( east_halfline ( E-max C)) meets ( L~ godo);

        then

         A171: ( 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 A171, XBOOLE_1: 70;

          suppose ( east_halfline ( E-max C)) meets ( L~ go);

          hence contradiction by A166;

        end;

          suppose ( east_halfline ( E-max C)) meets ( L~ pion1);

          then

          consider p be object such that

           A172: p in ( east_halfline ( E-max C)) and

           A173: p in ( L~ pion1) by XBOOLE_0: 3;

          reconsider p as Point of ( TOP-REAL 2) by A172;

          

           A174: (p `1 ) = (Gik `1 ) by A80, A83, A150, A173, GOBOARD7: 5;

          (i + 1) <= ( len Ga) by A2, NAT_1: 13;

          then ((i + 1) - 1) <= (( len Ga) - 1) by XREAL_1: 9;

          then

           A175: i <= (( len Ga) -' 1) by XREAL_0:def 2;

          (( len Ga) -' 1) <= ( len Ga) by NAT_D: 35;

          then (p `1 ) <= ((Ga * ((( len Ga) -' 1),1)) `1 ) by A1, A3, A13, A16, A20, A174, A175, JORDAN1A: 18;

          then (p `1 ) <= ( E-bound C) by A20, JORDAN8: 12;

          then

           A176: (p `1 ) <= (( E-max C) `1 ) by EUCLID: 52;

          (p `1 ) >= (( E-max C) `1 ) by A172, TOPREAL1:def 11;

          then

           A177: (p `1 ) = (( E-max C) `1 ) by A176, XXREAL_0: 1;

          (p `2 ) = (( E-max C) `2 ) by A172, TOPREAL1:def 11;

          then p = ( E-max C) by A177, TOPREAL3: 6;

          hence contradiction by A8, A83, A134, A150, A173, XBOOLE_0: 3;

        end;

          suppose ( east_halfline ( E-max C)) meets ( L~ co);

          then

          consider p be object such that

           A178: p in ( east_halfline ( E-max C)) and

           A179: p in ( L~ co) by XBOOLE_0: 3;

          reconsider p as Point of ( TOP-REAL 2) by A178;

          p in ( L~ LS) by A53, A179;

          then p in (( east_halfline ( E-max C)) /\ ( L~ ( Cage (C,n)))) by A146, A178, XBOOLE_0:def 4;

          then

           A180: (p `1 ) = Ebo by JORDAN1A: 83, PSCOMP_1: 50;

          

           A181: (( E-max C) `2 ) = (p `2 ) by A178, TOPREAL1:def 11;

          set RC = ( Rotate (( Cage (C,n)),Emax));

          

           A182: ( E-max C) in ( right_cell (RC,1)) by JORDAN1I: 7;

          

           A183: (1 + 1) <= ( len LS) by A23, XXREAL_0: 2;

          LS = (RC -: Wmin) by JORDAN1G: 18;

          then

           A184: ( LSeg (LS,1)) = ( LSeg (RC,1)) by A183, SPPOL_2: 9;

          

           A185: ( L~ RC) = ( L~ ( Cage (C,n))) by REVROT_1: 33;

          

           A186: ( len RC) = ( len ( Cage (C,n))) by FINSEQ_6: 179;

          

           A187: ( GoB RC) = ( GoB ( Cage (C,n))) by REVROT_1: 28

          .= Ga by JORDAN1H: 44;

          

           A188: Emax in ( rng ( Cage (C,n))) by SPRECT_2: 46;

          

           A189: RC is_sequence_on Ga by A140, REVROT_1: 34;

          

           A190: (RC /. 1) = ( E-max ( L~ RC)) by A185, A188, FINSEQ_6: 92;

          consider ii,jj be Nat such that

           A191: [ii, (jj + 1)] in ( Indices Ga) and

           A192: [ii, jj] in ( Indices Ga) and

           A193: (RC /. 1) = (Ga * (ii,(jj + 1))) and

           A194: (RC /. (1 + 1)) = (Ga * (ii,jj)) by A87, A185, A186, A188, A189, FINSEQ_6: 92, JORDAN1I: 23;

          consider jj2 be Nat such that

           A195: 1 <= jj2 and

           A196: jj2 <= ( width Ga) and

           A197: Emax = (Ga * (( len Ga),jj2)) by JORDAN1D: 25;

          

           A198: ( len Ga) >= 4 by JORDAN8: 10;

          then ( len Ga) >= 1 by XXREAL_0: 2;

          then [( len Ga), jj2] in ( Indices Ga) by A195, A196, MATRIX_0: 30;

          then

           A199: ii = ( len Ga) by A185, A190, A191, A193, A197, GOBOARD1: 5;

          

           A200: 1 <= ii by A191, MATRIX_0: 32;

          

           A201: ii <= ( len Ga) by A191, MATRIX_0: 32;

          

           A202: 1 <= (jj + 1) by A191, MATRIX_0: 32;

          

           A203: (jj + 1) <= ( width Ga) by A191, MATRIX_0: 32;

          

           A204: 1 <= ii by A192, MATRIX_0: 32;

          

           A205: ii <= ( len Ga) by A192, MATRIX_0: 32;

          

           A206: 1 <= jj by A192, MATRIX_0: 32;

          

           A207: jj <= ( width Ga) by A192, MATRIX_0: 32;

          

           A208: (ii + 1) <> ii;

          ((jj + 1) + 1) <> jj;

          then

           A209: ( right_cell (RC,1)) = ( cell (Ga,(ii -' 1),jj)) by A87, A186, A187, A191, A192, A193, A194, A208, GOBOARD5:def 6;

          

           A210: ((ii -' 1) + 1) = ii by A200, XREAL_1: 235;

          (ii - 1) >= (4 - 1) by A198, A199, XREAL_1: 9;

          then

           A211: (ii - 1) >= 1 by XXREAL_0: 2;

          then

           A212: 1 <= (ii -' 1) by XREAL_0:def 2;

          

           A213: ((Ga * ((ii -' 1),jj)) `2 ) <= (p `2 ) by A181, A182, A201, A203, A206, A209, A210, A211, JORDAN9: 17;

          

           A214: (p `2 ) <= ((Ga * ((ii -' 1),(jj + 1))) `2 ) by A181, A182, A201, A203, A206, A209, A210, A211, JORDAN9: 17;

          

           A215: (ii -' 1) < ( len Ga) by A201, A210, NAT_1: 13;

          

          then

           A216: ((Ga * ((ii -' 1),jj)) `2 ) = ((Ga * (1,jj)) `2 ) by A206, A207, A212, GOBOARD5: 1

          .= ((Ga * (ii,jj)) `2 ) by A204, A205, A206, A207, GOBOARD5: 1;

          

           A217: ((Ga * ((ii -' 1),(jj + 1))) `2 ) = ((Ga * (1,(jj + 1))) `2 ) by A202, A203, A212, A215, GOBOARD5: 1

          .= ((Ga * (ii,(jj + 1))) `2 ) by A200, A201, A202, A203, GOBOARD5: 1;

          

           A218: ((Ga * (( len Ga),jj)) `1 ) = Ebo by A16, A206, A207, JORDAN1A: 71;

          Ebo = ((Ga * (( len Ga),(jj + 1))) `1 ) by A16, A202, A203, JORDAN1A: 71;

          then p in ( LSeg ((RC /. 1),(RC /. (1 + 1)))) by A180, A193, A194, A199, A213, A214, A216, A217, A218, GOBOARD7: 7;

          then

           A219: p in ( LSeg (LS,1)) by A87, A184, A186, TOPREAL1:def 3;

          

           A220: p in ( LSeg (co,( Index (p,co)))) by A179, JORDAN3: 9;

          

           A221: co = ( mid (LS,(Gij .. LS),( len LS))) by A37, JORDAN1J: 37;

          

           A222: 1 <= (Gij .. LS) by A37, FINSEQ_4: 21;

          

           A223: (Gij .. LS) <= ( len LS) by A37, FINSEQ_4: 21;

          (Gij .. LS) <> ( len LS) by A29, A37, FINSEQ_4: 19;

          then

           A224: (Gij .. LS) < ( len LS) by A223, XXREAL_0: 1;

          

           A225: 1 <= ( Index (p,co)) by A179, JORDAN3: 8;

          

           A226: ( Index (p,co)) < ( len co) by A179, JORDAN3: 8;

          

           A227: (( Index (Gij,LS)) + 1) = (Gij .. LS) by A32, A37, JORDAN1J: 56;

          consider t be Nat such that

           A228: t in ( dom LS) and

           A229: (LS . t) = Gij by A37, FINSEQ_2: 10;

          

           A230: 1 <= t by A228, FINSEQ_3: 25;

          

           A231: t <= ( len LS) by A228, FINSEQ_3: 25;

          1 < t by A32, A229, A230, XXREAL_0: 1;

          then (( Index (Gij,LS)) + 1) = t by A229, A231, JORDAN3: 12;

          then

           A232: ( len ( L_Cut (LS,Gij))) = (( len LS) - ( Index (Gij,LS))) by A9, A229, JORDAN3: 26;

          set tt = ((( Index (p,co)) + (Gij .. LS)) -' 1);

          

           A233: 1 <= ( Index (Gij,LS)) by A9, JORDAN3: 8;

          ( 0 + ( Index (Gij,LS))) < ( len LS) by A9, JORDAN3: 8;

          then

           A234: (( len LS) - ( Index (Gij,LS))) > 0 by XREAL_1: 20;

          ( Index (p,co)) < (( len LS) -' ( Index (Gij,LS))) by A226, A232, XREAL_0:def 2;

          then (( Index (p,co)) + 1) <= (( len LS) -' ( Index (Gij,LS))) by NAT_1: 13;

          then ( Index (p,co)) <= ((( len LS) -' ( Index (Gij,LS))) - 1) by XREAL_1: 19;

          then ( Index (p,co)) <= ((( len LS) - ( Index (Gij,LS))) - 1) by A234, XREAL_0:def 2;

          then ( Index (p,co)) <= (( len LS) - (Gij .. LS)) by A227;

          then ( Index (p,co)) <= (( len LS) -' (Gij .. LS)) by XREAL_0:def 2;

          then ( Index (p,co)) < ((( len LS) -' (Gij .. LS)) + 1) by NAT_1: 13;

          then

           A235: ( LSeg (( mid (LS,(Gij .. LS),( len LS))),( Index (p,co)))) = ( LSeg (LS,((( Index (p,co)) + (Gij .. LS)) -' 1))) by A222, A224, A225, JORDAN4: 19;

          

           A236: (1 + 1) <= (Gij .. LS) by A227, A233, XREAL_1: 7;

          then (( Index (p,co)) + (Gij .. LS)) >= ((1 + 1) + 1) by A225, 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;

          

           A238: 2 in ( dom LS) by A183, FINSEQ_3: 25;

          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 A219, A220, A221, A235, XBOOLE_0: 3;

            end;

              suppose

               A239: tt = (1 + 1);

              then (( LSeg (LS,1)) /\ ( LSeg (LS,tt))) = {(LS /. 2)} by A23, TOPREAL1:def 6;

              then p in {(LS /. 2)} by A219, A220, A221, A235, XBOOLE_0:def 4;

              then

               A240: p = (LS /. 2) by TARSKI:def 1;

              then

               A241: (p .. LS) = 2 by A238, FINSEQ_5: 41;

              (1 + 1) = ((( Index (p,co)) + (Gij .. LS)) - 1) by A239, XREAL_0:def 2;

              then ((1 + 1) + 1) = (( Index (p,co)) + (Gij .. LS));

              then

               A242: (Gij .. LS) = 2 by A225, A236, JORDAN1E: 6;

              p in ( rng LS) by A238, A240, PARTFUN2: 2;

              then p = Gij by A37, A241, A242, FINSEQ_5: 9;

              then (Gij `1 ) = Ebo by A240, JORDAN1G: 32;

              then (Gij `1 ) = ((Ga * (( len Ga),j)) `1 ) by A5, A12, A16, JORDAN1A: 71;

              hence contradiction by A2, A14, 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

       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 UA meets ( L~ godo) by A132, A133, A134, A142, A153, JORDAN1J: 36;

      then

       A246: UA meets (( L~ go) \/ ( L~ pion1)) or UA meets ( L~ co) by A143, XBOOLE_1: 70;

      

       A247: UA c= C by JORDAN6: 61;

      per cases by A246, XBOOLE_1: 70;

        suppose UA meets ( L~ go);

        then UA meets ( L~ ( Cage (C,n))) by A46, A145, XBOOLE_1: 1, XBOOLE_1: 63;

        hence contradiction by A247, 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 A53, A146, XBOOLE_1: 1, XBOOLE_1: 63;

        hence contradiction by A247, JORDAN10: 5, XBOOLE_1: 63;

      end;

    end;

    theorem :: JORDAN19:13

    

     Th13: for C be Simple_closed_curve holds for i,j,k be Nat st 1 < i & i < ( len ( Gauge (C,n))) & 1 <= k & k <= j & j <= ( width ( Gauge (C,n))) & (( LSeg ((( Gauge (C,n)) * (i,k)),(( Gauge (C,n)) * (i,j)))) /\ ( L~ ( Upper_Seq (C,n)))) = {(( Gauge (C,n)) * (i,k))} & (( LSeg ((( Gauge (C,n)) * (i,k)),(( Gauge (C,n)) * (i,j)))) /\ ( L~ ( Lower_Seq (C,n)))) = {(( Gauge (C,n)) * (i,j))} holds ( LSeg ((( Gauge (C,n)) * (i,k)),(( Gauge (C,n)) * (i,j)))) 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 Gik = (Ga * (i,k));

      set Gij = (Ga * (i,j));

      assume that

       A1: 1 < i and

       A2: i < ( len Ga) and

       A3: 1 <= k and

       A4: k <= j and

       A5: j <= ( width Ga) and

       A6: (( LSeg (Gik,Gij)) /\ ( L~ US)) = {Gik} and

       A7: (( LSeg (Gik,Gij)) /\ ( L~ LS)) = {Gij} and

       A8: ( LSeg (Gik,Gij)) 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;

      then

       A11: j <> k by A1, A2, A3, A5, A9, JORDAN1J: 57;

      

       A12: 1 <= j by A3, A4, XXREAL_0: 2;

      

       A13: k <= ( width Ga) by A4, A5, XXREAL_0: 2;

      

       A14: [i, j] in ( Indices Ga) by A1, A2, A5, A12, MATRIX_0: 30;

      

       A15: [i, k] in ( Indices Ga) by A1, A2, A3, A13, MATRIX_0: 30;

      set co = ( L_Cut (LS,Gij));

      set go = ( R_Cut (US,Gik));

      

       A16: ( len Ga) = ( width Ga) by JORDAN8:def 1;

      

       A17: ( len US) >= 3 by JORDAN1E: 15;

      then ( len US) >= 1 by XXREAL_0: 2;

      then 1 in ( dom US) by FINSEQ_3: 25;

      

      then

       A18: (US . 1) = (US /. 1) by PARTFUN1:def 6

      .= Wmin by JORDAN1F: 5;

      

       A19: (Wmin `1 ) = Wbo by EUCLID: 52

      .= ((Ga * (1,k)) `1 ) by A3, A13, A16, JORDAN1A: 73;

      ( len Ga) >= 4 by JORDAN8: 10;

      then

       A20: ( len Ga) >= 1 by XXREAL_0: 2;

      then

       A21: [1, k] in ( Indices Ga) by A3, A13, MATRIX_0: 30;

      then

       A22: Gik <> (US . 1) by A1, A15, A18, A19, JORDAN1G: 7;

      then

      reconsider go as being_S-Seq FinSequence of ( TOP-REAL 2) by A10, JORDAN3: 35;

      

       A23: ( len LS) >= (1 + 2) by JORDAN1E: 15;

      then

       A24: ( len LS) >= 1 by XXREAL_0: 2;

      then

       A25: 1 in ( dom LS) by FINSEQ_3: 25;

      ( len LS) in ( dom LS) by A24, FINSEQ_3: 25;

      

      then

       A26: (LS . ( len LS)) = (LS /. ( len LS)) by PARTFUN1:def 6

      .= Wmin by JORDAN1F: 8;

      

       A27: (Wmin `1 ) = Wbo by EUCLID: 52

      .= ((Ga * (1,k)) `1 ) by A3, A13, A16, JORDAN1A: 73;

      

       A28: [i, j] in ( Indices Ga) by A1, A2, A5, A12, MATRIX_0: 30;

      then

       A29: Gij <> (LS . ( len LS)) by A1, A21, A26, A27, 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 A3, A13, A20, MATRIX_0: 30;

      

       A31: (LS . 1) = (LS /. 1) by A25, PARTFUN1:def 6

      .= Emax by JORDAN1F: 6;

      (Emax `1 ) = Ebo by EUCLID: 52

      .= ((Ga * (( len Ga),k)) `1 ) by A3, A13, A16, JORDAN1A: 71;

      then

       A32: Gij <> (LS . 1) by A2, A28, A30, A31, JORDAN1G: 7;

      

       A33: ( len go) >= (1 + 1) by TOPREAL1:def 8;

      

       A34: Gik in ( rng US) by A1, A2, A3, A10, 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: Gij in ( rng LS) by A1, A2, A5, A9, A12, 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

      .= Gik 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

      .= Gij 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= {Gik} by A6, XBOOLE_1: 26;

      m >= 1 by A33, XREAL_1: 19;

      then

       A48: ( LSeg (go,m)) = ( LSeg ((go /. m),Gik)) by A41, A43, TOPREAL1:def 3;

       {Gik} c= (( LSeg (go,m)) /\ ( LSeg (Gik,Gij)))

      proof

        let x be object;

        assume x in {Gik};

        then

         A49: x = Gik by TARSKI:def 1;

        

         A50: Gik in ( LSeg (go,m)) by A48, RLTOPSP1: 68;

        Gik in ( LSeg (Gik,Gij)) by RLTOPSP1: 68;

        hence thesis by A49, A50, XBOOLE_0:def 4;

      end;

      then

       A51: (( LSeg (go,m)) /\ ( LSeg (Gik,Gij))) = {Gik} 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= {Gij} by A7, XBOOLE_1: 26;

      

       A55: ( LSeg (co,1)) = ( LSeg (Gij,(co /. (1 + 1)))) by A36, A42, TOPREAL1:def 3;

       {Gij} c= (( LSeg (co,1)) /\ ( LSeg (Gik,Gij)))

      proof

        let x be object;

        assume x in {Gij};

        then

         A56: x = Gij by TARSKI:def 1;

        

         A57: Gij in ( LSeg (co,1)) by A55, RLTOPSP1: 68;

        Gij in ( LSeg (Gik,Gij)) by RLTOPSP1: 68;

        hence thesis by A56, A57, XBOOLE_0:def 4;

      end;

      then

       A58: (( LSeg (Gik,Gij)) /\ ( LSeg (co,1))) = {Gij} 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 A25, PARTFUN1:def 6

      .= Emax by JORDAN1F: 6;

      

       A67: [( len Ga), j] in ( Indices Ga) by A5, A12, A20, 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~ go) by XBOOLE_0:def 4;

        

         A70: x in ( L~ co) 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

         A71: x = Wmin or x = Emax by TARSKI:def 2;

        now

          assume x = Emax;

          then

           A72: Emax = Gij by A9, A66, A70, JORDAN1E: 7;

          ((Ga * (( len Ga),j)) `1 ) = Ebo by A5, A12, A16, JORDAN1A: 71;

          then (Emax `1 ) <> Ebo by A2, A14, A67, A72, JORDAN1G: 7;

          hence contradiction by EUCLID: 52;

        end;

        hence thesis by A59, A71, TARSKI:def 1;

      end;

      then

       A73: (( L~ go) /\ ( L~ co)) = {(go /. 1)} by A63;

      set W2 = (go /. 2);

      

       A74: 2 in ( dom go) by A33, FINSEQ_3: 25;

       A75:

      now

        assume (Gik `1 ) = Wbo;

        then ((Ga * (1,k)) `1 ) = ((Ga * (i,k)) `1 ) by A3, A13, A16, JORDAN1A: 73;

        hence contradiction by A1, A15, A21, JORDAN1G: 7;

      end;

      go = ( mid (US,1,(Gik .. US))) by A34, JORDAN1G: 49

      .= (US | (Gik .. US)) by A34, FINSEQ_4: 21, FINSEQ_6: 116;

      then

       A76: W2 = (US /. 2) by A74, FINSEQ_4: 70;

      

       A77: Wmin in ( rng go) by A59, FINSEQ_6: 42;

      set pion = <*Gik, Gij*>;

       A78:

      now

        let n be Nat;

        assume n in ( dom pion);

        then n in ( Seg 2) by FINSEQ_1: 89;

        then n = 1 or n = 2 by FINSEQ_1: 2, TARSKI:def 2;

        hence ex i,j be Nat st [i, j] in ( Indices Ga) & (pion /. n) = (Ga * (i,j)) by A14, A15, FINSEQ_4: 17;

      end;

      

       A79: Gik <> Gij by A11, A14, A15, GOBOARD1: 5;

      

       A80: (Gik `1 ) = ((Ga * (i,1)) `1 ) by A1, A2, A3, A13, GOBOARD5: 2

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

      then ( LSeg (Gik,Gij)) is vertical by SPPOL_1: 16;

      then pion is being_S-Seq by A79, JORDAN1B: 7;

      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 A78, 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 A33, 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 A41, A84, FINSEQ_4: 17;

      then

       A95: (go ^' pion1) is_sequence_on Ga by A35, 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 A42, FINSEQ_4: 17;

      then

       A97: godo is_sequence_on Ga by A38, 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 A44, A51, 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 A48, RLTOPSP1: 68;

        Gik in ( LSeg (pion1,1)) by A41, 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 A41, A44, 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 A58, 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;

        

         A111: Gij in ( LSeg (co,1)) by A55, RLTOPSP1: 68;

        (pion1 /. ((( len pion1) -' 1) + 1)) = (pion /. 2) by A85, A106, FINSEQ_1: 44

        .= Gij by FINSEQ_4: 17;

        then Gij in ( LSeg (pion1,(( len pion1) -' 1))) by A105, A106, TOPREAL1: 21;

        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 A42, 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 go) by A94, FINSEQ_6: 168;

        x in ( rng pion1) by A116, FINSEQ_6: 42;

        hence thesis by A61, 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~ go) by XBOOLE_0:def 4;

        x in ( L~ pion1) by A118, XBOOLE_0:def 4;

        then x in (( L~ pion1) /\ ( L~ US)) by A46, A119, XBOOLE_0:def 4;

        hence thesis by A6, A41, 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 A61, 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 A42, 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 co) by A85, A123, FINSEQ_6: 42;

        x in ( rng pion1) by A125, FINSEQ_6: 168;

        hence thesis by A62, 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~ co) by XBOOLE_0:def 4;

        x in ( L~ pion1) by A127, XBOOLE_0:def 4;

        then x in (( L~ pion1) /\ ( L~ LS)) by A53, A128, XBOOLE_0:def 4;

        hence thesis by A7, A42, 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 A73, 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 A60, 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 A34, FINSEQ_4: 21;

        then (Gik .. US) = 1 by A139, XXREAL_0: 1;

        then Gik = (US /. 1) by A34, FINSEQ_5: 38;

        hence contradiction by A18, A22, 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 A46, A145;

      

       A148: ( L~ co) c= ( L~ ( Cage (C,n))) by A53, 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 A34, A93, A138, JORDAN1J: 52

      .= ( right_cell ((go ^' pion1),1,Ga)) by A39, 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 A59, FINSEQ_6: 155;

      

       A155: ( len US) >= 2 by A17, XXREAL_0: 2;

      

       A156: (godo /. 2) = ((go ^' pion1) /. 2) by A89, FINSEQ_6: 159

      .= (US /. 2) by A33, A76, 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 A61, A77, 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;

      ( W-bound ( LSeg (Gik,Gij))) = (Gik `1 ) by A80, SPRECT_1: 54;

      then

       A161: ( W-bound ( L~ pion1)) = (Gik `1 ) by A83, SPPOL_2: 21;

      (Gik `1 ) >= Wbo by A10, A145, PSCOMP_1: 24;

      then (Gik `1 ) > Wbo by A75, XXREAL_0: 1;

      then ( W-min ((( L~ go) \/ ( L~ co)) \/ ( L~ pion1))) = ( W-min (( L~ go) \/ ( L~ co))) by A157, A158, A159, A160, A161, JORDAN1J: 33;

      then

       A162: ( W-min ( L~ godo)) = Wmin by A143, A158, XBOOLE_1: 4;

      

       A163: ( rng godo) c= ( L~ godo) by A89, A91, SPPOL_2: 18, XXREAL_0: 2;

      2 in ( dom godo) by A92, FINSEQ_3: 25;

      then

       A164: (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 A162, PSCOMP_1: 31

      .= ( W-bound ( L~ godo)) by EUCLID: 52;

      then (godo /. 2) in ( W-most ( L~ godo)) by A163, A164, SPRECT_2: 12;

      then (( Rotate (godo,( W-min ( L~ godo)))) /. 2) in ( W-most ( L~ godo)) by A154, A162, 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

       A165: (US . ( len US)) = (US /. ( len US)) by PARTFUN1:def 6

      .= Emax by JORDAN1F: 7;

      

       A166: ( 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

         A167: p in ( east_halfline ( E-max C)) and

         A168: p in ( L~ go) by XBOOLE_0: 3;

        reconsider p as Point of ( TOP-REAL 2) by A167;

        p in ( L~ US) by A46, A168;

        then p in (( east_halfline ( E-max C)) /\ ( L~ ( Cage (C,n)))) by A145, A167, XBOOLE_0:def 4;

        then

         A169: (p `1 ) = Ebo by JORDAN1A: 83, PSCOMP_1: 50;

        then

         A170: p = Emax by A46, A168, JORDAN1J: 46;

        then Emax = Gik by A10, A165, A168, JORDAN1J: 43;

        then (Gik `1 ) = ((Ga * (( len Ga),k)) `1 ) by A3, A13, A16, A169, A170, JORDAN1A: 71;

        hence contradiction by A2, A15, A30, JORDAN1G: 7;

      end;

      now

        assume ( east_halfline ( E-max C)) meets ( L~ godo);

        then

         A171: ( 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 A171, XBOOLE_1: 70;

          suppose ( east_halfline ( E-max C)) meets ( L~ go);

          hence contradiction by A166;

        end;

          suppose ( east_halfline ( E-max C)) meets ( L~ pion1);

          then

          consider p be object such that

           A172: p in ( east_halfline ( E-max C)) and

           A173: p in ( L~ pion1) by XBOOLE_0: 3;

          reconsider p as Point of ( TOP-REAL 2) by A172;

          

           A174: (p `1 ) = (Gik `1 ) by A80, A83, A150, A173, GOBOARD7: 5;

          (i + 1) <= ( len Ga) by A2, NAT_1: 13;

          then ((i + 1) - 1) <= (( len Ga) - 1) by XREAL_1: 9;

          then

           A175: i <= (( len Ga) -' 1) by XREAL_0:def 2;

          (( len Ga) -' 1) <= ( len Ga) by NAT_D: 35;

          then (p `1 ) <= ((Ga * ((( len Ga) -' 1),1)) `1 ) by A1, A3, A13, A16, A20, A174, A175, JORDAN1A: 18;

          then (p `1 ) <= ( E-bound C) by A20, JORDAN8: 12;

          then

           A176: (p `1 ) <= (( E-max C) `1 ) by EUCLID: 52;

          (p `1 ) >= (( E-max C) `1 ) by A172, TOPREAL1:def 11;

          then

           A177: (p `1 ) = (( E-max C) `1 ) by A176, XXREAL_0: 1;

          (p `2 ) = (( E-max C) `2 ) by A172, TOPREAL1:def 11;

          then p = ( E-max C) by A177, TOPREAL3: 6;

          hence contradiction by A8, A83, A134, A150, A173, XBOOLE_0: 3;

        end;

          suppose ( east_halfline ( E-max C)) meets ( L~ co);

          then

          consider p be object such that

           A178: p in ( east_halfline ( E-max C)) and

           A179: p in ( L~ co) by XBOOLE_0: 3;

          reconsider p as Point of ( TOP-REAL 2) by A178;

          p in ( L~ LS) by A53, A179;

          then p in (( east_halfline ( E-max C)) /\ ( L~ ( Cage (C,n)))) by A146, A178, XBOOLE_0:def 4;

          then

           A180: (p `1 ) = Ebo by JORDAN1A: 83, PSCOMP_1: 50;

          

           A181: (( E-max C) `2 ) = (p `2 ) by A178, TOPREAL1:def 11;

          set RC = ( Rotate (( Cage (C,n)),Emax));

          

           A182: ( E-max C) in ( right_cell (RC,1)) by JORDAN1I: 7;

          

           A183: (1 + 1) <= ( len LS) by A23, XXREAL_0: 2;

          LS = (RC -: Wmin) by JORDAN1G: 18;

          then

           A184: ( LSeg (LS,1)) = ( LSeg (RC,1)) by A183, SPPOL_2: 9;

          

           A185: ( L~ RC) = ( L~ ( Cage (C,n))) by REVROT_1: 33;

          

           A186: ( len RC) = ( len ( Cage (C,n))) by FINSEQ_6: 179;

          

           A187: ( GoB RC) = ( GoB ( Cage (C,n))) by REVROT_1: 28

          .= Ga by JORDAN1H: 44;

          

           A188: Emax in ( rng ( Cage (C,n))) by SPRECT_2: 46;

          

           A189: RC is_sequence_on Ga by A140, REVROT_1: 34;

          

           A190: (RC /. 1) = ( E-max ( L~ RC)) by A185, A188, FINSEQ_6: 92;

          consider ii,jj be Nat such that

           A191: [ii, (jj + 1)] in ( Indices Ga) and

           A192: [ii, jj] in ( Indices Ga) and

           A193: (RC /. 1) = (Ga * (ii,(jj + 1))) and

           A194: (RC /. (1 + 1)) = (Ga * (ii,jj)) by A87, A185, A186, A188, A189, FINSEQ_6: 92, JORDAN1I: 23;

          consider jj2 be Nat such that

           A195: 1 <= jj2 and

           A196: jj2 <= ( width Ga) and

           A197: Emax = (Ga * (( len Ga),jj2)) by JORDAN1D: 25;

          

           A198: ( len Ga) >= 4 by JORDAN8: 10;

          then ( len Ga) >= 1 by XXREAL_0: 2;

          then [( len Ga), jj2] in ( Indices Ga) by A195, A196, MATRIX_0: 30;

          then

           A199: ii = ( len Ga) by A185, A190, A191, A193, A197, GOBOARD1: 5;

          

           A200: 1 <= ii by A191, MATRIX_0: 32;

          

           A201: ii <= ( len Ga) by A191, MATRIX_0: 32;

          

           A202: 1 <= (jj + 1) by A191, MATRIX_0: 32;

          

           A203: (jj + 1) <= ( width Ga) by A191, MATRIX_0: 32;

          

           A204: 1 <= ii by A192, MATRIX_0: 32;

          

           A205: ii <= ( len Ga) by A192, MATRIX_0: 32;

          

           A206: 1 <= jj by A192, MATRIX_0: 32;

          

           A207: jj <= ( width Ga) by A192, MATRIX_0: 32;

          

           A208: (ii + 1) <> ii;

          ((jj + 1) + 1) <> jj;

          then

           A209: ( right_cell (RC,1)) = ( cell (Ga,(ii -' 1),jj)) by A87, A186, A187, A191, A192, A193, A194, A208, GOBOARD5:def 6;

          

           A210: ((ii -' 1) + 1) = ii by A200, XREAL_1: 235;

          (ii - 1) >= (4 - 1) by A198, A199, XREAL_1: 9;

          then

           A211: (ii - 1) >= 1 by XXREAL_0: 2;

          then

           A212: 1 <= (ii -' 1) by XREAL_0:def 2;

          

           A213: ((Ga * ((ii -' 1),jj)) `2 ) <= (p `2 ) by A181, A182, A201, A203, A206, A209, A210, A211, JORDAN9: 17;

          

           A214: (p `2 ) <= ((Ga * ((ii -' 1),(jj + 1))) `2 ) by A181, A182, A201, A203, A206, A209, A210, A211, JORDAN9: 17;

          

           A215: (ii -' 1) < ( len Ga) by A201, A210, NAT_1: 13;

          

          then

           A216: ((Ga * ((ii -' 1),jj)) `2 ) = ((Ga * (1,jj)) `2 ) by A206, A207, A212, GOBOARD5: 1

          .= ((Ga * (ii,jj)) `2 ) by A204, A205, A206, A207, GOBOARD5: 1;

          

           A217: ((Ga * ((ii -' 1),(jj + 1))) `2 ) = ((Ga * (1,(jj + 1))) `2 ) by A202, A203, A212, A215, GOBOARD5: 1

          .= ((Ga * (ii,(jj + 1))) `2 ) by A200, A201, A202, A203, GOBOARD5: 1;

          

           A218: ((Ga * (( len Ga),jj)) `1 ) = Ebo by A16, A206, A207, JORDAN1A: 71;

          Ebo = ((Ga * (( len Ga),(jj + 1))) `1 ) by A16, A202, A203, JORDAN1A: 71;

          then p in ( LSeg ((RC /. 1),(RC /. (1 + 1)))) by A180, A193, A194, A199, A213, A214, A216, A217, A218, GOBOARD7: 7;

          then

           A219: p in ( LSeg (LS,1)) by A87, A184, A186, TOPREAL1:def 3;

          

           A220: p in ( LSeg (co,( Index (p,co)))) by A179, JORDAN3: 9;

          

           A221: co = ( mid (LS,(Gij .. LS),( len LS))) by A37, JORDAN1J: 37;

          

           A222: 1 <= (Gij .. LS) by A37, FINSEQ_4: 21;

          

           A223: (Gij .. LS) <= ( len LS) by A37, FINSEQ_4: 21;

          (Gij .. LS) <> ( len LS) by A29, A37, FINSEQ_4: 19;

          then

           A224: (Gij .. LS) < ( len LS) by A223, XXREAL_0: 1;

          

           A225: 1 <= ( Index (p,co)) by A179, JORDAN3: 8;

          

           A226: ( Index (p,co)) < ( len co) by A179, JORDAN3: 8;

          

           A227: (( Index (Gij,LS)) + 1) = (Gij .. LS) by A32, A37, JORDAN1J: 56;

          consider t be Nat such that

           A228: t in ( dom LS) and

           A229: (LS . t) = Gij by A37, FINSEQ_2: 10;

          

           A230: 1 <= t by A228, FINSEQ_3: 25;

          

           A231: t <= ( len LS) by A228, FINSEQ_3: 25;

          1 < t by A32, A229, A230, XXREAL_0: 1;

          then (( Index (Gij,LS)) + 1) = t by A229, A231, JORDAN3: 12;

          then

           A232: ( len ( L_Cut (LS,Gij))) = (( len LS) - ( Index (Gij,LS))) by A9, A229, JORDAN3: 26;

          set tt = ((( Index (p,co)) + (Gij .. LS)) -' 1);

          

           A233: 1 <= ( Index (Gij,LS)) by A9, JORDAN3: 8;

          ( 0 + ( Index (Gij,LS))) < ( len LS) by A9, JORDAN3: 8;

          then

           A234: (( len LS) - ( Index (Gij,LS))) > 0 by XREAL_1: 20;

          ( Index (p,co)) < (( len LS) -' ( Index (Gij,LS))) by A226, A232, XREAL_0:def 2;

          then (( Index (p,co)) + 1) <= (( len LS) -' ( Index (Gij,LS))) by NAT_1: 13;

          then ( Index (p,co)) <= ((( len LS) -' ( Index (Gij,LS))) - 1) by XREAL_1: 19;

          then ( Index (p,co)) <= ((( len LS) - ( Index (Gij,LS))) - 1) by A234, XREAL_0:def 2;

          then ( Index (p,co)) <= (( len LS) - (Gij .. LS)) by A227;

          then ( Index (p,co)) <= (( len LS) -' (Gij .. LS)) by XREAL_0:def 2;

          then ( Index (p,co)) < ((( len LS) -' (Gij .. LS)) + 1) by NAT_1: 13;

          then

           A235: ( LSeg (( mid (LS,(Gij .. LS),( len LS))),( Index (p,co)))) = ( LSeg (LS,((( Index (p,co)) + (Gij .. LS)) -' 1))) by A222, A224, A225, JORDAN4: 19;

          

           A236: (1 + 1) <= (Gij .. LS) by A227, A233, XREAL_1: 7;

          then (( Index (p,co)) + (Gij .. LS)) >= ((1 + 1) + 1) by A225, 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;

          

           A238: 2 in ( dom LS) by A183, FINSEQ_3: 25;

          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 A219, A220, A221, A235, XBOOLE_0: 3;

            end;

              suppose

               A239: tt = (1 + 1);

              then (( LSeg (LS,1)) /\ ( LSeg (LS,tt))) = {(LS /. 2)} by A23, TOPREAL1:def 6;

              then p in {(LS /. 2)} by A219, A220, A221, A235, XBOOLE_0:def 4;

              then

               A240: p = (LS /. 2) by TARSKI:def 1;

              then

               A241: (p .. LS) = 2 by A238, FINSEQ_5: 41;

              (1 + 1) = ((( Index (p,co)) + (Gij .. LS)) - 1) by A239, XREAL_0:def 2;

              then ((1 + 1) + 1) = (( Index (p,co)) + (Gij .. LS));

              then

               A242: (Gij .. LS) = 2 by A225, A236, JORDAN1E: 6;

              p in ( rng LS) by A238, A240, PARTFUN2: 2;

              then p = Gij by A37, A241, A242, FINSEQ_5: 9;

              then (Gij `1 ) = Ebo by A240, JORDAN1G: 32;

              then (Gij `1 ) = ((Ga * (( len Ga),j)) `1 ) by A5, A12, A16, JORDAN1A: 71;

              hence contradiction by A2, A14, 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

       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 A132, A133, A134, A142, A153, JORDAN1J: 36;

      then

       A246: LA meets (( L~ go) \/ ( L~ pion1)) or LA meets ( L~ co) by A143, XBOOLE_1: 70;

      

       A247: LA c= C by JORDAN6: 61;

      per cases by A246, XBOOLE_1: 70;

        suppose LA meets ( L~ go);

        then LA meets ( L~ ( Cage (C,n))) by A46, A145, XBOOLE_1: 1, XBOOLE_1: 63;

        hence contradiction by A247, 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 A53, A146, XBOOLE_1: 1, XBOOLE_1: 63;

        hence contradiction by A247, JORDAN10: 5, XBOOLE_1: 63;

      end;

    end;

    theorem :: JORDAN19:14

    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 & (( LSeg ((( Gauge (C,n)) * (i,j)),(( Gauge (C,n)) * (i,k)))) /\ ( Lower_Arc ( L~ ( Cage (C,n))))) = {(( Gauge (C,n)) * (i,k))} & (( LSeg ((( Gauge (C,n)) * (i,j)),(( Gauge (C,n)) * (i,k)))) /\ ( Upper_Arc ( L~ ( Cage (C,n))))) = {(( Gauge (C,n)) * (i,j))} 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: (( LSeg ((( Gauge (C,n)) * (i,j)),(( Gauge (C,n)) * (i,k)))) /\ ( Lower_Arc ( L~ ( Cage (C,n))))) = {(( Gauge (C,n)) * (i,k))} and

       A8: (( LSeg ((( Gauge (C,n)) * (i,j)),(( Gauge (C,n)) * (i,k)))) /\ ( Upper_Arc ( L~ ( Cage (C,n))))) = {(( Gauge (C,n)) * (i,j))};

      

       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, Th12;

    end;

    theorem :: JORDAN19:15

    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 & (( LSeg ((( Gauge (C,n)) * (i,j)),(( Gauge (C,n)) * (i,k)))) /\ ( Lower_Arc ( L~ ( Cage (C,n))))) = {(( Gauge (C,n)) * (i,k))} & (( LSeg ((( Gauge (C,n)) * (i,j)),(( Gauge (C,n)) * (i,k)))) /\ ( Upper_Arc ( L~ ( Cage (C,n))))) = {(( Gauge (C,n)) * (i,j))} 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: (( LSeg ((( Gauge (C,n)) * (i,j)),(( Gauge (C,n)) * (i,k)))) /\ ( Lower_Arc ( L~ ( Cage (C,n))))) = {(( Gauge (C,n)) * (i,k))} and

       A8: (( LSeg ((( Gauge (C,n)) * (i,j)),(( Gauge (C,n)) * (i,k)))) /\ ( Upper_Arc ( L~ ( Cage (C,n))))) = {(( Gauge (C,n)) * (i,j))};

      

       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, Th13;

    end;

    theorem :: JORDAN19:16

    

     Th16: 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~ ( Lower_Seq (C,n))) & (( Gauge (C,n)) * (i,j)) in ( L~ ( Upper_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~ ( Lower_Seq (C,n))) and

       A7: (( Gauge (C,n)) * (i,j)) in ( L~ ( Upper_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~ ( Upper_Seq (C,n)))) = {(( Gauge (C,n)) * (i,j1))} and

       A12: (( LSeg ((( Gauge (C,n)) * (i,j1)),(( Gauge (C,n)) * (i,k1)))) /\ ( L~ ( Lower_Seq (C,n)))) = {(( Gauge (C,n)) * (i,k1))} by A1, A2, A3, A4, A5, A6, A7, JORDAN15: 17;

      

       A13: 1 <= j1 by A3, A8, XXREAL_0: 2;

      k1 <= ( width ( Gauge (C,n))) by A5, A10, 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, Th12;

      hence thesis by A1, A2, A3, A5, A8, A9, A10, JORDAN15: 5, XBOOLE_1: 63;

    end;

    theorem :: JORDAN19:17

    

     Th17: 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~ ( Lower_Seq (C,n))) & (( Gauge (C,n)) * (i,j)) in ( L~ ( Upper_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~ ( Lower_Seq (C,n))) and

       A7: (( Gauge (C,n)) * (i,j)) in ( L~ ( Upper_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~ ( Upper_Seq (C,n)))) = {(( Gauge (C,n)) * (i,j1))} and

       A12: (( LSeg ((( Gauge (C,n)) * (i,j1)),(( Gauge (C,n)) * (i,k1)))) /\ ( L~ ( Lower_Seq (C,n)))) = {(( Gauge (C,n)) * (i,k1))} by A1, A2, A3, A4, A5, A6, A7, JORDAN15: 17;

      

       A13: 1 <= j1 by A3, A8, XXREAL_0: 2;

      k1 <= ( width ( Gauge (C,n))) by A5, A10, 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, Th13;

      hence thesis by A1, A2, A3, A5, A8, A9, A10, JORDAN15: 5, XBOOLE_1: 63;

    end;

    theorem :: JORDAN19:18

    

     Th18: 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 ( Lower_Arc ( L~ ( Cage (C,n)))) & (( Gauge (C,n)) * (i,j)) in ( Upper_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 ( Lower_Arc ( L~ ( Cage (C,n)))) and

       A8: (( Gauge (C,n)) * (i,j)) in ( Upper_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, Th16;

    end;

    theorem :: JORDAN19:19

    

     Th19: 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 ( Lower_Arc ( L~ ( Cage (C,n)))) & (( Gauge (C,n)) * (i,j)) in ( Upper_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 ( Lower_Arc ( L~ ( Cage (C,n)))) and

       A8: (( Gauge (C,n)) * (i,j)) in ( Upper_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, Th17;

    end;

    theorem :: JORDAN19:20

    

     Th20: 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)) * (i1,j))} & ((( 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)) * (i2,k))} 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 * (i1,j))} and

       A8: ((pio \/ poz) /\ ( L~ LS)) = {(G * (i2,k))} and

       A9: (pio \/ poz) misses ( Upper_Arc C);

      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 Gik = (G * (i2,k));

      set Gij = (G * (i1,j));

      set Gi1k = (G * (i1,k));

      

       A10: i1 < ( len G) by A2, A3, XXREAL_0: 2;

      

       A11: 1 < i2 by A1, A2, XXREAL_0: 2;

      

       A12: ( L~ <*Gij, Gi1k, Gik*>) = (poz \/ pio) by TOPREAL3: 16;

      Gik in {Gik} by TARSKI:def 1;

      then

       A13: Gik in ( L~ LS) by A8, XBOOLE_0:def 4;

      Gij in {Gij} by TARSKI:def 1;

      then

       A14: Gij in ( L~ US) by A7, XBOOLE_0:def 4;

      

       A15: j <= ( width G) by A5, A6, XXREAL_0: 2;

      

       A16: 1 <= k by A4, A5, XXREAL_0: 2;

      

       A17: [i1, j] in ( Indices G) by A1, A4, A10, A15, MATRIX_0: 30;

      

       A18: [i2, k] in ( Indices G) by A3, A6, A11, A16, MATRIX_0: 30;

      

       A19: [i1, k] in ( Indices G) by A1, A6, A10, A16, MATRIX_0: 30;

      set go = ( R_Cut (US,Gij));

      set co = ( L_Cut (LS,Gik));

      

       A20: ( len G) = ( width G) by JORDAN8:def 1;

      

       A21: ( len US) >= 3 by JORDAN1E: 15;

      then ( len US) >= 1 by XXREAL_0: 2;

      then 1 in ( dom US) by FINSEQ_3: 25;

      

      then

       A22: (US . 1) = (US /. 1) by PARTFUN1:def 6

      .= Wmin by JORDAN1F: 5;

      

       A23: (Wmin `1 ) = Wbo by EUCLID: 52

      .= ((G * (1,k)) `1 ) by A6, A16, A20, JORDAN1A: 73;

      ( len G) >= 4 by JORDAN8: 10;

      then

       A24: ( len G) >= 1 by XXREAL_0: 2;

      then

       A25: [1, k] in ( Indices G) by A6, A16, MATRIX_0: 30;

      then

       A26: Gij <> (US . 1) by A1, A17, A22, A23, JORDAN1G: 7;

      then

      reconsider go as being_S-Seq FinSequence of ( TOP-REAL 2) by A14, JORDAN3: 35;

      

       A27: [1, j] in ( Indices G) by A4, A15, A24, MATRIX_0: 30;

      

       A28: ( len LS) >= (1 + 2) by JORDAN1E: 15;

      then

       A29: ( len LS) >= 1 by XXREAL_0: 2;

      then

       A30: 1 in ( dom LS) by FINSEQ_3: 25;

      ( len LS) in ( dom LS) by A29, FINSEQ_3: 25;

      

      then

       A31: (LS . ( len LS)) = (LS /. ( len LS)) by PARTFUN1:def 6

      .= Wmin by JORDAN1F: 8;

      (Wmin `1 ) = Wbo by EUCLID: 52

      .= ((G * (1,k)) `1 ) by A6, A16, A20, JORDAN1A: 73;

      then

       A32: Gik <> (LS . ( len LS)) by A1, A2, A18, A25, A31, JORDAN1G: 7;

      then

      reconsider co as being_S-Seq FinSequence of ( TOP-REAL 2) by A13, JORDAN3: 34;

      

       A33: [( len G), k] in ( Indices G) by A6, A16, A24, MATRIX_0: 30;

      

       A34: (LS . 1) = (LS /. 1) by A30, PARTFUN1:def 6

      .= Emax by JORDAN1F: 6;

      (Emax `1 ) = Ebo by EUCLID: 52

      .= ((G * (( len G),k)) `1 ) by A6, A16, A20, JORDAN1A: 71;

      then

       A35: Gik <> (LS . 1) by A3, A18, A33, A34, JORDAN1G: 7;

      

       A36: ( len go) >= (1 + 1) by TOPREAL1:def 8;

      

       A37: Gij in ( rng US) by A1, A4, A10, A14, A15, JORDAN1G: 4, JORDAN1J: 40;

      then

       A38: go is_sequence_on G by JORDAN1G: 4, JORDAN1J: 38;

      

       A39: ( len co) >= (1 + 1) by TOPREAL1:def 8;

      

       A40: Gik in ( rng LS) by A3, A6, A11, A13, A16, JORDAN1G: 5, JORDAN1J: 40;

      then

       A41: co is_sequence_on G by JORDAN1G: 5, JORDAN1J: 39;

      reconsider go as non constant s.c.c. being_S-Seq FinSequence of ( TOP-REAL 2) by A36, A38, JGRAPH_1: 12, JORDAN8: 5;

      reconsider co as non constant s.c.c. being_S-Seq FinSequence of ( TOP-REAL 2) by A39, A41, JGRAPH_1: 12, JORDAN8: 5;

      

       A42: ( len go) > 1 by A36, NAT_1: 13;

      then

       A43: ( len go) in ( dom go) by FINSEQ_3: 25;

      

      then

       A44: (go /. ( len go)) = (go . ( len go)) by PARTFUN1:def 6

      .= Gij by A14, JORDAN3: 24;

      ( len co) >= 1 by A39, XXREAL_0: 2;

      then 1 in ( dom co) by FINSEQ_3: 25;

      

      then

       A45: (co /. 1) = (co . 1) by PARTFUN1:def 6

      .= Gik by A13, JORDAN3: 23;

      reconsider m = (( len go) - 1) as Nat by A43, FINSEQ_3: 26;

      

       A46: (m + 1) = ( len go);

      then

       A47: (( len go) -' 1) = m by NAT_D: 34;

      

       A48: ( LSeg (go,m)) c= ( L~ go) by TOPREAL3: 19;

      

       A49: ( L~ go) c= ( L~ US) by A14, JORDAN3: 41;

      then ( LSeg (go,m)) c= ( L~ US) by A48;

      then

       A50: (( LSeg (go,m)) /\ ( L~ <*Gij, Gi1k, Gik*>)) c= {Gij} by A7, A12, XBOOLE_1: 26;

      m >= 1 by A36, XREAL_1: 19;

      then

       A51: ( LSeg (go,m)) = ( LSeg ((go /. m),Gij)) by A44, A46, TOPREAL1:def 3;

       {Gij} c= (( LSeg (go,m)) /\ ( L~ <*Gij, Gi1k, Gik*>))

      proof

        let x be object;

        assume x in {Gij};

        then

         A52: x = Gij by TARSKI:def 1;

        

         A53: Gij in ( LSeg (go,m)) by A51, RLTOPSP1: 68;

        Gij in ( LSeg (Gij,Gi1k)) by RLTOPSP1: 68;

        then Gij in (( LSeg (Gij,Gi1k)) \/ ( LSeg (Gi1k,Gik))) by XBOOLE_0:def 3;

        then Gij in ( L~ <*Gij, Gi1k, Gik*>) by SPRECT_1: 8;

        hence thesis by A52, A53, XBOOLE_0:def 4;

      end;

      then

       A54: (( LSeg (go,m)) /\ ( L~ <*Gij, Gi1k, Gik*>)) = {Gij} by A50;

      

       A55: ( LSeg (co,1)) c= ( L~ co) by TOPREAL3: 19;

      

       A56: ( L~ co) c= ( L~ LS) by A13, JORDAN3: 42;

      then ( LSeg (co,1)) c= ( L~ LS) by A55;

      then

       A57: (( LSeg (co,1)) /\ ( L~ <*Gij, Gi1k, Gik*>)) c= {Gik} by A8, A12, XBOOLE_1: 26;

      

       A58: ( LSeg (co,1)) = ( LSeg (Gik,(co /. (1 + 1)))) by A39, A45, TOPREAL1:def 3;

       {Gik} c= (( LSeg (co,1)) /\ ( L~ <*Gij, Gi1k, Gik*>))

      proof

        let x be object;

        assume x in {Gik};

        then

         A59: x = Gik by TARSKI:def 1;

        

         A60: Gik in ( LSeg (co,1)) by A58, RLTOPSP1: 68;

        Gik in ( LSeg (Gi1k,Gik)) by RLTOPSP1: 68;

        then Gik in (( LSeg (Gij,Gi1k)) \/ ( LSeg (Gi1k,Gik))) by XBOOLE_0:def 3;

        then Gik in ( L~ <*Gij, Gi1k, Gik*>) by SPRECT_1: 8;

        hence thesis by A59, A60, XBOOLE_0:def 4;

      end;

      then

       A61: (( L~ <*Gij, Gi1k, Gik*>) /\ ( LSeg (co,1))) = {Gik} by A57;

      

       A62: (go /. 1) = (US /. 1) by A14, SPRECT_3: 22

      .= Wmin by JORDAN1F: 5;

      

      then

       A63: (go /. 1) = (LS /. ( len LS)) by JORDAN1F: 8

      .= (co /. ( len co)) by A13, JORDAN1J: 35;

      

       A64: ( rng go) c= ( L~ go) by A36, SPPOL_2: 18;

      

       A65: ( rng co) c= ( L~ co) by A39, SPPOL_2: 18;

      

       A66: {(go /. 1)} c= (( L~ go) /\ ( L~ co))

      proof

        let x be object;

        assume x in {(go /. 1)};

        then

         A67: x = (go /. 1) by TARSKI:def 1;

        then

         A68: x in ( rng go) by FINSEQ_6: 42;

        x in ( rng co) by A63, A67, FINSEQ_6: 168;

        hence thesis by A64, A65, A68, XBOOLE_0:def 4;

      end;

      

       A69: (LS . 1) = (LS /. 1) by A30, PARTFUN1:def 6

      .= Emax by JORDAN1F: 6;

      

       A70: [( len G), j] in ( Indices G) by A4, A15, A24, MATRIX_0: 30;

      (( 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~ go) by XBOOLE_0:def 4;

        

         A73: x in ( L~ co) by A71, XBOOLE_0:def 4;

        then x in (( L~ US) /\ ( L~ LS)) by A49, A56, A72, XBOOLE_0:def 4;

        then x in {Wmin, Emax} by JORDAN1E: 16;

        then

         A74: x = Wmin or x = Emax by TARSKI:def 2;

        now

          assume x = Emax;

          then

           A75: Emax = Gik by A13, A69, A73, JORDAN1E: 7;

          ((G * (( len G),j)) `1 ) = Ebo by A4, A15, A20, JORDAN1A: 71;

          then (Emax `1 ) <> Ebo by A3, A18, A70, A75, JORDAN1G: 7;

          hence contradiction by EUCLID: 52;

        end;

        hence thesis by A62, A74, TARSKI:def 1;

      end;

      then

       A76: (( L~ go) /\ ( L~ co)) = {(go /. 1)} by A66;

      set W2 = (go /. 2);

      

       A77: 2 in ( dom go) by A36, FINSEQ_3: 25;

       A78:

      now

        assume (Gij `1 ) = Wbo;

        then ((G * (1,j)) `1 ) = ((G * (i1,j)) `1 ) by A4, A15, A20, JORDAN1A: 73;

        hence contradiction by A1, A17, A27, JORDAN1G: 7;

      end;

      go = ( mid (US,1,(Gij .. US))) by A37, JORDAN1G: 49

      .= (US | (Gij .. US)) by A37, FINSEQ_4: 21, FINSEQ_6: 116;

      then

       A79: W2 = (US /. 2) by A77, FINSEQ_4: 70;

      

       A80: Wmin in ( rng go) by A62, FINSEQ_6: 42;

      set pion = <*Gij, Gi1k, Gik*>;

       A81:

      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 A17, A18, A19, FINSEQ_4: 18;

      end;

      

       A82: (Gi1k `1 ) = ((G * (i1,1)) `1 ) by A1, A6, A10, A16, GOBOARD5: 2

      .= (Gij `1 ) by A1, A4, A10, A15, GOBOARD5: 2;

      (Gi1k `2 ) = ((G * (1,k)) `2 ) by A1, A6, A10, A16, GOBOARD5: 1

      .= (Gik `2 ) by A3, A6, A11, A16, GOBOARD5: 1;

      then

       A83: Gi1k = |[(Gij `1 ), (Gik `2 )]| by A82, EUCLID: 53;

      

       A84: Gi1k in pio by RLTOPSP1: 68;

      

       A85: Gi1k in poz by RLTOPSP1: 68;

      now

        per cases ;

          suppose (Gik `1 ) <> (Gij `1 ) & (Gik `2 ) <> (Gij `2 );

          then pion is being_S-Seq by A83, TOPREAL3: 34;

          then

          consider pion1 be FinSequence of ( TOP-REAL 2) such that

           A86: pion1 is_sequence_on G and

           A87: pion1 is being_S-Seq and

           A88: ( L~ pion) = ( L~ pion1) and

           A89: (pion /. 1) = (pion1 /. 1) and

           A90: (pion /. ( len pion)) = (pion1 /. ( len pion1)) and

           A91: ( len pion) <= ( len pion1) by A81, GOBOARD3: 2;

          reconsider pion1 as being_S-Seq FinSequence of ( TOP-REAL 2) by A87;

          set godo = ((go ^' pion1) ^' co);

          

           A92: (Gi1k `1 ) = ((G * (i1,1)) `1 ) by A1, A6, A10, A16, GOBOARD5: 2

          .= (Gij `1 ) by A1, A4, A10, A15, GOBOARD5: 2;

          

           A93: (Gi1k `1 ) <= (Gik `1 ) by A1, A2, A3, A6, A16, JORDAN1A: 18;

          then

           A94: ( W-bound poz) = (Gi1k `1 ) by SPRECT_1: 54;

          

           A95: ( W-bound pio) = (Gij `1 ) by A92, SPRECT_1: 54;

          ( W-bound (poz \/ pio)) = ( min (( W-bound poz),( W-bound pio))) by SPRECT_1: 47

          .= (Gij `1 ) by A92, A94, A95;

          then

           A96: ( W-bound ( L~ pion1)) = (Gij `1 ) by A88, TOPREAL3: 16;

          

           A97: (1 + 1) <= ( len ( Cage (C,n))) by GOBOARD7: 34, XXREAL_0: 2;

          

           A98: (1 + 1) <= ( len ( Rotate (( Cage (C,n)),Wmin))) by GOBOARD7: 34, XXREAL_0: 2;

          ( len (go ^' pion1)) >= ( len go) by TOPREAL8: 7;

          then

           A99: ( len (go ^' pion1)) >= (1 + 1) by A36, XXREAL_0: 2;

          then

           A100: ( len (go ^' pion1)) > (1 + 0 ) by NAT_1: 13;

          

           A101: ( len godo) >= ( len (go ^' pion1)) by TOPREAL8: 7;

          then

           A102: (1 + 1) <= ( len godo) by A99, XXREAL_0: 2;

          

           A103: US is_sequence_on G by JORDAN1G: 4;

          

           A104: (go /. ( len go)) = (pion1 /. 1) by A44, A89, FINSEQ_4: 18;

          then

           A105: (go ^' pion1) is_sequence_on G by A38, A86, TOPREAL8: 12;

          

           A106: ((go ^' pion1) /. ( len (go ^' pion1))) = (pion /. ( len pion)) by A90, FINSEQ_6: 156

          .= (pion /. 3) by FINSEQ_1: 45

          .= (co /. 1) by A45, FINSEQ_4: 18;

          then

           A107: godo is_sequence_on G by A41, A105, TOPREAL8: 12;

          ( LSeg (pion1,1)) c= ( L~ pion) by A88, TOPREAL3: 19;

          then

           A108: (( LSeg (go,(( len go) -' 1))) /\ ( LSeg (pion1,1))) c= {Gij} by A47, A54, XBOOLE_1: 27;

          ( len pion1) >= (2 + 1) by A91, FINSEQ_1: 45;

          then

           A109: ( len pion1) > (1 + 1) by NAT_1: 13;

           {Gij} c= (( LSeg (go,m)) /\ ( LSeg (pion1,1)))

          proof

            let x be object;

            assume x in {Gij};

            then

             A110: x = Gij by TARSKI:def 1;

            

             A111: Gij in ( LSeg (go,m)) by A51, RLTOPSP1: 68;

            Gij in ( LSeg (pion1,1)) by A44, A104, A109, TOPREAL1: 21;

            hence thesis by A110, A111, XBOOLE_0:def 4;

          end;

          then (( LSeg (go,(( len go) -' 1))) /\ ( LSeg (pion1,1))) = {(go /. ( len go))} by A44, A47, A108;

          then

           A112: (go ^' pion1) is unfolded by A104, TOPREAL8: 34;

          ( len pion1) >= (2 + 1) by A91, FINSEQ_1: 45;

          then

           A113: (( 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 A113, XREAL_0:def 2;

          then

           A114: (( len (go ^' pion1)) -' 1) = (( len go) + (( len pion1) -' 2)) by XREAL_0:def 2;

          

           A115: (( len pion1) - 1) >= 1 by A109, XREAL_1: 19;

          then

           A116: (( len pion1) -' 1) = (( len pion1) - 1) by XREAL_0:def 2;

          

           A117: ((( len pion1) -' 2) + 1) = ((( len pion1) - 2) + 1) by A113, XREAL_0:def 2

          .= (( len pion1) -' 1) by A115, XREAL_0:def 2;

          ((( len pion1) - 1) + 1) <= ( len pion1);

          then

           A118: (( len pion1) -' 1) < ( len pion1) by A116, NAT_1: 13;

          ( LSeg (pion1,(( len pion1) -' 1))) c= ( L~ pion) by A88, TOPREAL3: 19;

          then

           A119: (( LSeg (pion1,(( len pion1) -' 1))) /\ ( LSeg (co,1))) c= {Gik} by A61, XBOOLE_1: 27;

           {Gik} c= (( LSeg (pion1,(( len pion1) -' 1))) /\ ( LSeg (co,1)))

          proof

            let x be object;

            assume x in {Gik};

            then

             A120: x = Gik by TARSKI:def 1;

            

             A121: Gik in ( LSeg (co,1)) by A58, RLTOPSP1: 68;

            (pion1 /. ((( len pion1) -' 1) + 1)) = (pion /. 3) by A90, A116, FINSEQ_1: 45

            .= Gik by FINSEQ_4: 18;

            then Gik in ( LSeg (pion1,(( len pion1) -' 1))) by A115, A116, TOPREAL1: 21;

            hence thesis by A120, A121, XBOOLE_0:def 4;

          end;

          then (( LSeg (pion1,(( len pion1) -' 1))) /\ ( LSeg (co,1))) = {Gik} by A119;

          then

           A122: (( LSeg ((go ^' pion1),(( len go) + (( len pion1) -' 2)))) /\ ( LSeg (co,1))) = {((go ^' pion1) /. ( len (go ^' pion1)))} by A45, A104, A106, A117, A118, TOPREAL8: 31;

          

           A123: (go ^' pion1) is non trivial by A99, NAT_D: 60;

          

           A124: ( rng pion1) c= ( L~ pion1) by A109, SPPOL_2: 18;

          

           A125: {(pion1 /. 1)} c= (( L~ go) /\ ( L~ pion1))

          proof

            let x be object;

            assume x in {(pion1 /. 1)};

            then

             A126: x = (pion1 /. 1) by TARSKI:def 1;

            then

             A127: x in ( rng go) by A104, FINSEQ_6: 168;

            x in ( rng pion1) by A126, FINSEQ_6: 42;

            hence thesis by A64, A124, A127, XBOOLE_0:def 4;

          end;

          (( L~ go) /\ ( L~ pion1)) c= {(pion1 /. 1)}

          proof

            let x be object;

            assume

             A128: x in (( L~ go) /\ ( L~ pion1));

            then

             A129: x in ( L~ go) by XBOOLE_0:def 4;

            x in ( L~ pion1) by A128, XBOOLE_0:def 4;

            hence thesis by A7, A12, A44, A49, A88, A104, A129, XBOOLE_0:def 4;

          end;

          then

           A130: (( L~ go) /\ ( L~ pion1)) = {(pion1 /. 1)} by A125;

          then

           A131: (go ^' pion1) is s.n.c. by A104, JORDAN1J: 54;

          (( rng go) /\ ( rng pion1)) c= {(pion1 /. 1)} by A64, A124, A130, XBOOLE_1: 27;

          then

           A132: (go ^' pion1) is one-to-one by JORDAN1J: 55;

          

           A133: (pion /. ( len pion)) = (pion /. 3) by FINSEQ_1: 45

          .= (co /. 1) by A45, FINSEQ_4: 18;

          

           A134: {(pion1 /. ( len pion1))} c= (( L~ co) /\ ( L~ pion1))

          proof

            let x be object;

            assume x in {(pion1 /. ( len pion1))};

            then

             A135: x = (pion1 /. ( len pion1)) by TARSKI:def 1;

            then

             A136: x in ( rng co) by A90, A133, FINSEQ_6: 42;

            x in ( rng pion1) by A135, FINSEQ_6: 168;

            hence thesis by A65, A124, A136, XBOOLE_0:def 4;

          end;

          (( L~ co) /\ ( L~ pion1)) c= {(pion1 /. ( len pion1))}

          proof

            let x be object;

            assume

             A137: x in (( L~ co) /\ ( L~ pion1));

            then

             A138: x in ( L~ co) by XBOOLE_0:def 4;

            x in ( L~ pion1) by A137, XBOOLE_0:def 4;

            hence thesis by A8, A12, A45, A56, A88, A90, A133, A138, XBOOLE_0:def 4;

          end;

          then

           A139: (( L~ co) /\ ( L~ pion1)) = {(pion1 /. ( len pion1))} by A134;

          

           A140: (( L~ (go ^' pion1)) /\ ( L~ co)) = ((( L~ go) \/ ( L~ pion1)) /\ ( L~ co)) by A104, TOPREAL8: 35

          .= ( {(go /. 1)} \/ {(co /. 1)}) by A76, A90, A133, A139, 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 A63, FINSEQ_6: 155;

          then

          reconsider godo as non constant standard special_circular_sequence by A102, A106, A107, A112, A114, A122, A123, A131, A132, A140, JORDAN8: 4, JORDAN8: 5, TOPREAL8: 11, TOPREAL8: 33, TOPREAL8: 34;

          

           A141: UA is_an_arc_of (( W-min C),( E-max C)) by JORDAN6:def 8;

          then

           A142: UA is connected by JORDAN6: 10;

          

           A143: ( W-min C) in UA by A141, TOPREAL1: 1;

          

           A144: ( E-max C) in UA by A141, TOPREAL1: 1;

          set ff = ( Rotate (( Cage (C,n)),Wmin));

          Wmin in ( rng ( Cage (C,n))) by SPRECT_2: 43;

          then

           A145: (ff /. 1) = Wmin by FINSEQ_6: 92;

          

           A146: ( L~ ff) = ( L~ ( Cage (C,n))) by REVROT_1: 33;

          then (( W-max ( L~ ff)) .. ff) > 1 by A145, SPRECT_5: 22;

          then (( N-min ( L~ ff)) .. ff) > 1 by A145, A146, SPRECT_5: 23, XXREAL_0: 2;

          then (( N-max ( L~ ff)) .. ff) > 1 by A145, A146, SPRECT_5: 24, XXREAL_0: 2;

          then

           A147: (Emax .. ff) > 1 by A145, A146, SPRECT_5: 25, XXREAL_0: 2;

           A148:

          now

            assume

             A149: (Gij .. US) <= 1;

            (Gij .. US) >= 1 by A37, FINSEQ_4: 21;

            then (Gij .. US) = 1 by A149, XXREAL_0: 1;

            then Gij = (US /. 1) by A37, FINSEQ_5: 38;

            hence contradiction by A22, A26, JORDAN1F: 5;

          end;

          

           A150: ( Cage (C,n)) is_sequence_on G by JORDAN9:def 1;

          then

           A151: ff is_sequence_on G by REVROT_1: 34;

          

           A152: (( right_cell (godo,1,G)) \ ( L~ godo)) c= ( RightComp godo) by A102, A107, JORDAN9: 27;

          

           A153: ( L~ godo) = (( L~ (go ^' pion1)) \/ ( L~ co)) by A106, TOPREAL8: 35

          .= ((( L~ go) \/ ( L~ pion1)) \/ ( L~ co)) by A104, TOPREAL8: 35;

          

           A154: ( L~ ( Cage (C,n))) = (( L~ US) \/ ( L~ LS)) by JORDAN1E: 13;

          then

           A155: ( L~ US) c= ( L~ ( Cage (C,n))) by XBOOLE_1: 7;

          

           A156: ( L~ LS) c= ( L~ ( Cage (C,n))) by A154, XBOOLE_1: 7;

          

           A157: ( L~ go) c= ( L~ ( Cage (C,n))) by A49, A155;

          

           A158: ( L~ co) c= ( L~ ( Cage (C,n))) by A56, A156;

          

           A159: ( W-min C) in C by SPRECT_1: 13;

           A160:

          now

            assume ( W-min C) in ( L~ godo);

            then

             A161: ( W-min C) in (( L~ go) \/ ( L~ pion1)) or ( W-min C) in ( L~ co) by A153, XBOOLE_0:def 3;

            per cases by A161, XBOOLE_0:def 3;

              suppose ( W-min C) in ( L~ go);

              then C meets ( L~ ( Cage (C,n))) by A157, A159, XBOOLE_0: 3;

              hence contradiction by JORDAN10: 5;

            end;

              suppose ( W-min C) in ( L~ pion1);

              hence contradiction by A9, A12, A88, A143, XBOOLE_0: 3;

            end;

              suppose ( W-min C) in ( L~ co);

              then C meets ( L~ ( Cage (C,n))) by A158, A159, 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 A98, 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 A147, A151, JORDAN1J: 53

          .= ( right_cell (US,1,G)) by JORDAN1E:def 1

          .= ( right_cell (( R_Cut (US,Gij)),1,G)) by A37, A103, A148, JORDAN1J: 52

          .= ( right_cell ((go ^' pion1),1,G)) by A42, A105, JORDAN1J: 51

          .= ( right_cell (godo,1,G)) by A100, A107, JORDAN1J: 51;

          then ( W-min C) in ( right_cell (godo,1,G)) by JORDAN1I: 6;

          then

           A162: ( W-min C) in (( right_cell (godo,1,G)) \ ( L~ godo)) by A160, XBOOLE_0:def 5;

          

           A163: (godo /. 1) = ((go ^' pion1) /. 1) by FINSEQ_6: 155

          .= Wmin by A62, FINSEQ_6: 155;

          

           A164: ( len US) >= 2 by A21, XXREAL_0: 2;

          

           A165: (godo /. 2) = ((go ^' pion1) /. 2) by A99, FINSEQ_6: 159

          .= (US /. 2) by A36, A79, FINSEQ_6: 159

          .= ((US ^' LS) /. 2) by A164, FINSEQ_6: 159

          .= (( Rotate (( Cage (C,n)),Wmin)) /. 2) by JORDAN1E: 11;

          

           A166: (( L~ go) \/ ( L~ co)) is compact by COMPTS_1: 10;

          Wmin in (( L~ go) \/ ( L~ co)) by A64, A80, XBOOLE_0:def 3;

          then

           A167: ( W-min (( L~ go) \/ ( L~ co))) = Wmin by A157, A158, A166, JORDAN1J: 21, XBOOLE_1: 8;

          

           A168: (( W-min (( L~ go) \/ ( L~ co))) `1 ) = ( W-bound (( L~ go) \/ ( L~ co))) by EUCLID: 52;

          

           A169: (Wmin `1 ) = Wbo by EUCLID: 52;

          (Gij `1 ) >= Wbo by A14, A155, PSCOMP_1: 24;

          then (Gij `1 ) > Wbo by A78, XXREAL_0: 1;

          then ( W-min ((( L~ go) \/ ( L~ co)) \/ ( L~ pion1))) = ( W-min (( L~ go) \/ ( L~ co))) by A96, A166, A167, A168, A169, JORDAN1J: 33;

          then

           A170: ( W-min ( L~ godo)) = Wmin by A153, A167, XBOOLE_1: 4;

          

           A171: ( rng godo) c= ( L~ godo) by A99, A101, SPPOL_2: 18, XXREAL_0: 2;

          2 in ( dom godo) by A102, FINSEQ_3: 25;

          then

           A172: (godo /. 2) in ( rng godo) by PARTFUN2: 2;

          (godo /. 2) in ( W-most ( L~ ( Cage (C,n)))) by A165, JORDAN1I: 25;

          

          then ((godo /. 2) `1 ) = (( W-min ( L~ godo)) `1 ) by A170, PSCOMP_1: 31

          .= ( W-bound ( L~ godo)) by EUCLID: 52;

          then (godo /. 2) in ( W-most ( L~ godo)) by A171, A172, SPRECT_2: 12;

          then (( Rotate (godo,( W-min ( L~ godo)))) /. 2) in ( W-most ( L~ godo)) by A163, A170, 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

           A173: (US . ( len US)) = (US /. ( len US)) by PARTFUN1:def 6

          .= Emax by JORDAN1F: 7;

          

           A174: ( 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

             A175: p in ( east_halfline ( E-max C)) and

             A176: p in ( L~ go) by XBOOLE_0: 3;

            reconsider p as Point of ( TOP-REAL 2) by A175;

            p in ( L~ US) by A49, A176;

            then p in (( east_halfline ( E-max C)) /\ ( L~ ( Cage (C,n)))) by A155, A175, XBOOLE_0:def 4;

            then

             A177: (p `1 ) = Ebo by JORDAN1A: 83, PSCOMP_1: 50;

            then

             A178: p = Emax by A49, A176, JORDAN1J: 46;

            then Emax = Gij by A14, A173, A176, JORDAN1J: 43;

            then (Gij `1 ) = ((G * (( len G),k)) `1 ) by A6, A16, A20, A177, A178, JORDAN1A: 71;

            hence contradiction by A2, A3, A17, A33, JORDAN1G: 7;

          end;

          now

            assume ( east_halfline ( E-max C)) meets ( L~ godo);

            then

             A179: ( east_halfline ( E-max C)) meets (( L~ go) \/ ( L~ pion1)) or ( east_halfline ( E-max C)) meets ( L~ co) by A153, XBOOLE_1: 70;

            per cases by A179, XBOOLE_1: 70;

              suppose ( east_halfline ( E-max C)) meets ( L~ go);

              hence contradiction by A174;

            end;

              suppose ( east_halfline ( E-max C)) meets ( L~ pion1);

              then

              consider p be object such that

               A180: p in ( east_halfline ( E-max C)) and

               A181: p in ( L~ pion1) by XBOOLE_0: 3;

              reconsider p as Point of ( TOP-REAL 2) by A180;

               A182:

              now

                per cases by A12, A88, A181, XBOOLE_0:def 3;

                  suppose p in poz;

                  hence (p `1 ) <= (Gik `1 ) by A93, TOPREAL1: 3;

                end;

                  suppose p in pio;

                  hence (p `1 ) <= (Gik `1 ) by A92, A93, GOBOARD7: 5;

                end;

              end;

              (i2 + 1) <= ( len G) by A3, NAT_1: 13;

              then i2 <= (( len G) - 1) by XREAL_1: 19;

              then

               A183: 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, A11, A16, A20, A24, A183, JORDAN1A: 18;

              then (p `1 ) <= ((G * ((( len G) -' 1),1)) `1 ) by A182, XXREAL_0: 2;

              then (p `1 ) <= ( E-bound C) by A24, JORDAN8: 12;

              then

               A184: (p `1 ) <= (( E-max C) `1 ) by EUCLID: 52;

              (p `1 ) >= (( E-max C) `1 ) by A180, TOPREAL1:def 11;

              then

               A185: (p `1 ) = (( E-max C) `1 ) by A184, XXREAL_0: 1;

              (p `2 ) = (( E-max C) `2 ) by A180, TOPREAL1:def 11;

              then p = ( E-max C) by A185, TOPREAL3: 6;

              hence contradiction by A9, A12, A88, A144, A181, XBOOLE_0: 3;

            end;

              suppose ( east_halfline ( E-max C)) meets ( L~ co);

              then

              consider p be object such that

               A186: p in ( east_halfline ( E-max C)) and

               A187: p in ( L~ co) by XBOOLE_0: 3;

              reconsider p as Point of ( TOP-REAL 2) by A186;

              p in ( L~ LS) by A56, A187;

              then p in (( east_halfline ( E-max C)) /\ ( L~ ( Cage (C,n)))) by A156, A186, XBOOLE_0:def 4;

              then

               A188: (p `1 ) = Ebo by JORDAN1A: 83, PSCOMP_1: 50;

              

               A189: (( E-max C) `2 ) = (p `2 ) by A186, TOPREAL1:def 11;

              set RC = ( Rotate (( Cage (C,n)),Emax));

              

               A190: ( E-max C) in ( right_cell (RC,1)) by JORDAN1I: 7;

              

               A191: (1 + 1) <= ( len LS) by A28, XXREAL_0: 2;

              LS = (RC -: Wmin) by JORDAN1G: 18;

              then

               A192: ( LSeg (LS,1)) = ( LSeg (RC,1)) by A191, SPPOL_2: 9;

              

               A193: ( L~ RC) = ( L~ ( Cage (C,n))) by REVROT_1: 33;

              

               A194: ( len RC) = ( len ( Cage (C,n))) by FINSEQ_6: 179;

              

               A195: ( GoB RC) = ( GoB ( Cage (C,n))) by REVROT_1: 28

              .= G by JORDAN1H: 44;

              

               A196: Emax in ( rng ( Cage (C,n))) by SPRECT_2: 46;

              

               A197: RC is_sequence_on G by A150, REVROT_1: 34;

              

               A198: (RC /. 1) = ( E-max ( L~ RC)) by A193, A196, FINSEQ_6: 92;

              consider ii,jj be Nat such that

               A199: [ii, (jj + 1)] in ( Indices G) and

               A200: [ii, jj] in ( Indices G) and

               A201: (RC /. 1) = (G * (ii,(jj + 1))) and

               A202: (RC /. (1 + 1)) = (G * (ii,jj)) by A97, A193, A194, A196, A197, FINSEQ_6: 92, JORDAN1I: 23;

              consider jj2 be Nat such that

               A203: 1 <= jj2 and

               A204: jj2 <= ( width G) and

               A205: Emax = (G * (( len G),jj2)) by JORDAN1D: 25;

              

               A206: ( len G) >= 4 by JORDAN8: 10;

              then ( len G) >= 1 by XXREAL_0: 2;

              then [( len G), jj2] in ( Indices G) by A203, A204, MATRIX_0: 30;

              then

               A207: ii = ( len G) by A193, A198, A199, A201, A205, GOBOARD1: 5;

              

               A208: 1 <= ii by A199, MATRIX_0: 32;

              

               A209: ii <= ( len G) by A199, MATRIX_0: 32;

              

               A210: 1 <= (jj + 1) by A199, MATRIX_0: 32;

              

               A211: (jj + 1) <= ( width G) by A199, MATRIX_0: 32;

              

               A212: 1 <= ii by A200, MATRIX_0: 32;

              

               A213: ii <= ( len G) by A200, MATRIX_0: 32;

              

               A214: 1 <= jj by A200, MATRIX_0: 32;

              

               A215: jj <= ( width G) by A200, MATRIX_0: 32;

              

               A216: (ii + 1) <> ii;

              ((jj + 1) + 1) <> jj;

              then

               A217: ( right_cell (RC,1)) = ( cell (G,(ii -' 1),jj)) by A97, A194, A195, A199, A200, A201, A202, A216, GOBOARD5:def 6;

              

               A218: ((ii -' 1) + 1) = ii by A208, XREAL_1: 235;

              (ii - 1) >= (4 - 1) by A206, A207, XREAL_1: 9;

              then

               A219: (ii - 1) >= 1 by XXREAL_0: 2;

              then

               A220: 1 <= (ii -' 1) by XREAL_0:def 2;

              

               A221: ((G * ((ii -' 1),jj)) `2 ) <= (p `2 ) by A189, A190, A209, A211, A214, A217, A218, A219, JORDAN9: 17;

              

               A222: (p `2 ) <= ((G * ((ii -' 1),(jj + 1))) `2 ) by A189, A190, A209, A211, A214, A217, A218, A219, JORDAN9: 17;

              

               A223: (ii -' 1) < ( len G) by A209, A218, NAT_1: 13;

              

              then

               A224: ((G * ((ii -' 1),jj)) `2 ) = ((G * (1,jj)) `2 ) by A214, A215, A220, GOBOARD5: 1

              .= ((G * (ii,jj)) `2 ) by A212, A213, A214, A215, GOBOARD5: 1;

              

               A225: ((G * ((ii -' 1),(jj + 1))) `2 ) = ((G * (1,(jj + 1))) `2 ) by A210, A211, A220, A223, GOBOARD5: 1

              .= ((G * (ii,(jj + 1))) `2 ) by A208, A209, A210, A211, GOBOARD5: 1;

              

               A226: ((G * (( len G),jj)) `1 ) = Ebo by A20, A214, A215, JORDAN1A: 71;

              Ebo = ((G * (( len G),(jj + 1))) `1 ) by A20, A210, A211, JORDAN1A: 71;

              then p in ( LSeg ((RC /. 1),(RC /. (1 + 1)))) by A188, A201, A202, A207, A221, A222, A224, A225, A226, GOBOARD7: 7;

              then

               A227: p in ( LSeg (LS,1)) by A97, A192, A194, TOPREAL1:def 3;

              

               A228: p in ( LSeg (co,( Index (p,co)))) by A187, JORDAN3: 9;

              

               A229: co = ( mid (LS,(Gik .. LS),( len LS))) by A40, JORDAN1J: 37;

              

               A230: 1 <= (Gik .. LS) by A40, FINSEQ_4: 21;

              

               A231: (Gik .. LS) <= ( len LS) by A40, FINSEQ_4: 21;

              (Gik .. LS) <> ( len LS) by A32, A40, FINSEQ_4: 19;

              then

               A232: (Gik .. LS) < ( len LS) by A231, XXREAL_0: 1;

              

               A233: 1 <= ( Index (p,co)) by A187, JORDAN3: 8;

              

               A234: ( Index (p,co)) < ( len co) by A187, JORDAN3: 8;

              

               A235: (( Index (Gik,LS)) + 1) = (Gik .. LS) by A35, A40, JORDAN1J: 56;

              consider t be Nat such that

               A236: t in ( dom LS) and

               A237: (LS . t) = Gik by A40, FINSEQ_2: 10;

              

               A238: 1 <= t by A236, FINSEQ_3: 25;

              

               A239: t <= ( len LS) by A236, FINSEQ_3: 25;

              1 < t by A35, A237, A238, XXREAL_0: 1;

              then (( Index (Gik,LS)) + 1) = t by A237, A239, JORDAN3: 12;

              then

               A240: ( len ( L_Cut (LS,Gik))) = (( len LS) - ( Index (Gik,LS))) by A13, A237, JORDAN3: 26;

              set tt = ((( Index (p,co)) + (Gik .. LS)) -' 1);

              

               A241: 1 <= ( Index (Gik,LS)) by A13, JORDAN3: 8;

              ( 0 + ( Index (Gik,LS))) < ( len LS) by A13, JORDAN3: 8;

              then

               A242: (( len LS) - ( Index (Gik,LS))) > 0 by XREAL_1: 20;

              ( Index (p,co)) < (( len LS) -' ( Index (Gik,LS))) by A234, A240, XREAL_0:def 2;

              then (( Index (p,co)) + 1) <= (( len LS) -' ( Index (Gik,LS))) by NAT_1: 13;

              then ( Index (p,co)) <= ((( len LS) -' ( Index (Gik,LS))) - 1) by XREAL_1: 19;

              then ( Index (p,co)) <= ((( len LS) - ( Index (Gik,LS))) - 1) by A242, XREAL_0:def 2;

              then ( Index (p,co)) <= (( len LS) - (Gik .. LS)) by A235;

              then ( Index (p,co)) <= (( len LS) -' (Gik .. LS)) by XREAL_0:def 2;

              then ( Index (p,co)) < ((( len LS) -' (Gik .. LS)) + 1) by NAT_1: 13;

              then

               A243: ( LSeg (( mid (LS,(Gik .. LS),( len LS))),( Index (p,co)))) = ( LSeg (LS,((( Index (p,co)) + (Gik .. LS)) -' 1))) by A230, A232, A233, JORDAN4: 19;

              

               A244: (1 + 1) <= (Gik .. LS) by A235, A241, XREAL_1: 7;

              then (( Index (p,co)) + (Gik .. LS)) >= ((1 + 1) + 1) by A233, XREAL_1: 7;

              then ((( Index (p,co)) + (Gik .. LS)) - 1) >= (((1 + 1) + 1) - 1) by XREAL_1: 9;

              then

               A245: tt >= (1 + 1) by XREAL_0:def 2;

              

               A246: 2 in ( dom LS) by A191, FINSEQ_3: 25;

              now

                per cases by A245, XXREAL_0: 1;

                  suppose tt > (1 + 1);

                  then ( LSeg (LS,1)) misses ( LSeg (LS,tt)) by TOPREAL1:def 7;

                  hence contradiction by A227, A228, A229, A243, XBOOLE_0: 3;

                end;

                  suppose

                   A247: tt = (1 + 1);

                  then (( LSeg (LS,1)) /\ ( LSeg (LS,tt))) = {(LS /. 2)} by A28, TOPREAL1:def 6;

                  then p in {(LS /. 2)} by A227, A228, A229, A243, XBOOLE_0:def 4;

                  then

                   A248: p = (LS /. 2) by TARSKI:def 1;

                  then

                   A249: (p .. LS) = 2 by A246, FINSEQ_5: 41;

                  (1 + 1) = ((( Index (p,co)) + (Gik .. LS)) - 1) by A247, XREAL_0:def 2;

                  then ((1 + 1) + 1) = (( Index (p,co)) + (Gik .. LS));

                  then

                   A250: (Gik .. LS) = 2 by A233, A244, JORDAN1E: 6;

                  p in ( rng LS) by A246, A248, PARTFUN2: 2;

                  then p = Gik by A40, A249, A250, FINSEQ_5: 9;

                  then (Gik `1 ) = Ebo by A248, JORDAN1G: 32;

                  then (Gik `1 ) = ((G * (( len G),j)) `1 ) by A4, A15, A20, JORDAN1A: 71;

                  hence contradiction by A3, A18, A70, 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

           A251: W is_a_component_of (( L~ godo) ` ) and

           A252: ( east_halfline ( E-max C)) c= W by GOBOARD9: 3;

           not W is bounded by A252, JORDAN2C: 121, RLTOPSP1: 42;

          then W is_outside_component_of ( L~ godo) by A251, JORDAN2C:def 3;

          then W c= ( UBD ( L~ godo)) by JORDAN2C: 23;

          then

           A253: ( east_halfline ( E-max C)) c= ( UBD ( L~ godo)) by A252;

          ( E-max C) in ( east_halfline ( E-max C)) by TOPREAL1: 38;

          then ( E-max C) in ( UBD ( L~ godo)) by A253;

          then ( E-max C) in ( LeftComp godo) by GOBRD14: 36;

          then UA meets ( L~ godo) by A142, A143, A144, A152, A162, JORDAN1J: 36;

          then

           A254: UA meets (( L~ go) \/ ( L~ pion1)) or UA meets ( L~ co) by A153, XBOOLE_1: 70;

          

           A255: UA c= C by JORDAN6: 61;

          now

            per cases by A254, XBOOLE_1: 70;

              suppose UA meets ( L~ go);

              then UA meets ( L~ ( Cage (C,n))) by A49, A155, XBOOLE_1: 1, XBOOLE_1: 63;

              hence contradiction by A255, JORDAN10: 5, XBOOLE_1: 63;

            end;

              suppose UA meets ( L~ pion1);

              hence contradiction by A9, A12, A88;

            end;

              suppose UA meets ( L~ co);

              then UA meets ( L~ ( Cage (C,n))) by A56, A156, XBOOLE_1: 1, XBOOLE_1: 63;

              hence contradiction by A255, JORDAN10: 5, XBOOLE_1: 63;

            end;

          end;

          hence contradiction;

        end;

          suppose (Gik `1 ) = (Gij `1 );

          then

           A256: i1 = i2 by A17, A18, JORDAN1G: 7;

          then poz = {Gi1k} by RLTOPSP1: 70;

          then poz c= pio by A84, ZFMISC_1: 31;

          then (pio \/ poz) = pio by XBOOLE_1: 12;

          hence contradiction by A1, A3, A4, A5, A6, A7, A8, A9, A256, Th12;

        end;

          suppose (Gik `2 ) = (Gij `2 );

          then

           A257: j = k by A17, A18, JORDAN1G: 6;

          then pio = {Gi1k} by RLTOPSP1: 70;

          then pio c= poz by A85, ZFMISC_1: 31;

          then (pio \/ poz) = poz by XBOOLE_1: 12;

          hence contradiction by A1, A2, A3, A4, A6, A7, A8, A9, A257, JORDAN15: 37;

        end;

      end;

      hence contradiction;

    end;

    theorem :: JORDAN19:21

    

     Th21: 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)) * (i1,j))} & ((( 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)) * (i2,k))} 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 * (i1,j))} and

       A8: ((pio \/ poz) /\ ( L~ LS)) = {(G * (i2,k))} and

       A9: (pio \/ poz) misses ( Lower_Arc C);

      set UA = ( 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 Gik = (G * (i2,k));

      set Gij = (G * (i1,j));

      set Gi1k = (G * (i1,k));

      

       A10: i1 < ( len G) by A2, A3, XXREAL_0: 2;

      

       A11: 1 < i2 by A1, A2, XXREAL_0: 2;

      

       A12: ( L~ <*Gij, Gi1k, Gik*>) = (poz \/ pio) by TOPREAL3: 16;

      Gik in {Gik} by TARSKI:def 1;

      then

       A13: Gik in ( L~ LS) by A8, XBOOLE_0:def 4;

      Gij in {Gij} by TARSKI:def 1;

      then

       A14: Gij in ( L~ US) by A7, XBOOLE_0:def 4;

      

       A15: j <= ( width G) by A5, A6, XXREAL_0: 2;

      

       A16: 1 <= k by A4, A5, XXREAL_0: 2;

      

       A17: [i1, j] in ( Indices G) by A1, A4, A10, A15, MATRIX_0: 30;

      

       A18: [i2, k] in ( Indices G) by A3, A6, A11, A16, MATRIX_0: 30;

      

       A19: [i1, k] in ( Indices G) by A1, A6, A10, A16, MATRIX_0: 30;

      set go = ( R_Cut (US,Gij));

      set co = ( L_Cut (LS,Gik));

      

       A20: ( len G) = ( width G) by JORDAN8:def 1;

      

       A21: ( len US) >= 3 by JORDAN1E: 15;

      then ( len US) >= 1 by XXREAL_0: 2;

      then 1 in ( dom US) by FINSEQ_3: 25;

      

      then

       A22: (US . 1) = (US /. 1) by PARTFUN1:def 6

      .= Wmin by JORDAN1F: 5;

      

       A23: (Wmin `1 ) = Wbo by EUCLID: 52

      .= ((G * (1,k)) `1 ) by A6, A16, A20, JORDAN1A: 73;

      ( len G) >= 4 by JORDAN8: 10;

      then

       A24: ( len G) >= 1 by XXREAL_0: 2;

      then

       A25: [1, k] in ( Indices G) by A6, A16, MATRIX_0: 30;

      then

       A26: Gij <> (US . 1) by A1, A17, A22, A23, JORDAN1G: 7;

      then

      reconsider go as being_S-Seq FinSequence of ( TOP-REAL 2) by A14, JORDAN3: 35;

      

       A27: [1, j] in ( Indices G) by A4, A15, A24, MATRIX_0: 30;

      

       A28: ( len LS) >= (1 + 2) by JORDAN1E: 15;

      then

       A29: ( len LS) >= 1 by XXREAL_0: 2;

      then

       A30: 1 in ( dom LS) by FINSEQ_3: 25;

      ( len LS) in ( dom LS) by A29, FINSEQ_3: 25;

      

      then

       A31: (LS . ( len LS)) = (LS /. ( len LS)) by PARTFUN1:def 6

      .= Wmin by JORDAN1F: 8;

      (Wmin `1 ) = Wbo by EUCLID: 52

      .= ((G * (1,k)) `1 ) by A6, A16, A20, JORDAN1A: 73;

      then

       A32: Gik <> (LS . ( len LS)) by A1, A2, A18, A25, A31, JORDAN1G: 7;

      then

      reconsider co as being_S-Seq FinSequence of ( TOP-REAL 2) by A13, JORDAN3: 34;

      

       A33: [( len G), k] in ( Indices G) by A6, A16, A24, MATRIX_0: 30;

      

       A34: (LS . 1) = (LS /. 1) by A30, PARTFUN1:def 6

      .= Emax by JORDAN1F: 6;

      (Emax `1 ) = Ebo by EUCLID: 52

      .= ((G * (( len G),k)) `1 ) by A6, A16, A20, JORDAN1A: 71;

      then

       A35: Gik <> (LS . 1) by A3, A18, A33, A34, JORDAN1G: 7;

      

       A36: ( len go) >= (1 + 1) by TOPREAL1:def 8;

      

       A37: Gij in ( rng US) by A1, A4, A10, A14, A15, JORDAN1G: 4, JORDAN1J: 40;

      then

       A38: go is_sequence_on G by JORDAN1G: 4, JORDAN1J: 38;

      

       A39: ( len co) >= (1 + 1) by TOPREAL1:def 8;

      

       A40: Gik in ( rng LS) by A3, A6, A11, A13, A16, JORDAN1G: 5, JORDAN1J: 40;

      then

       A41: co is_sequence_on G by JORDAN1G: 5, JORDAN1J: 39;

      reconsider go as non constant s.c.c. being_S-Seq FinSequence of ( TOP-REAL 2) by A36, A38, JGRAPH_1: 12, JORDAN8: 5;

      reconsider co as non constant s.c.c. being_S-Seq FinSequence of ( TOP-REAL 2) by A39, A41, JGRAPH_1: 12, JORDAN8: 5;

      

       A42: ( len go) > 1 by A36, NAT_1: 13;

      then

       A43: ( len go) in ( dom go) by FINSEQ_3: 25;

      

      then

       A44: (go /. ( len go)) = (go . ( len go)) by PARTFUN1:def 6

      .= Gij by A14, JORDAN3: 24;

      ( len co) >= 1 by A39, XXREAL_0: 2;

      then 1 in ( dom co) by FINSEQ_3: 25;

      

      then

       A45: (co /. 1) = (co . 1) by PARTFUN1:def 6

      .= Gik by A13, JORDAN3: 23;

      reconsider m = (( len go) - 1) as Nat by A43, FINSEQ_3: 26;

      

       A46: (m + 1) = ( len go);

      then

       A47: (( len go) -' 1) = m by NAT_D: 34;

      

       A48: ( LSeg (go,m)) c= ( L~ go) by TOPREAL3: 19;

      

       A49: ( L~ go) c= ( L~ US) by A14, JORDAN3: 41;

      then ( LSeg (go,m)) c= ( L~ US) by A48;

      then

       A50: (( LSeg (go,m)) /\ ( L~ <*Gij, Gi1k, Gik*>)) c= {Gij} by A7, A12, XBOOLE_1: 26;

      m >= 1 by A36, XREAL_1: 19;

      then

       A51: ( LSeg (go,m)) = ( LSeg ((go /. m),Gij)) by A44, A46, TOPREAL1:def 3;

       {Gij} c= (( LSeg (go,m)) /\ ( L~ <*Gij, Gi1k, Gik*>))

      proof

        let x be object;

        assume x in {Gij};

        then

         A52: x = Gij by TARSKI:def 1;

        

         A53: Gij in ( LSeg (go,m)) by A51, RLTOPSP1: 68;

        Gij in ( LSeg (Gij,Gi1k)) by RLTOPSP1: 68;

        then Gij in (( LSeg (Gij,Gi1k)) \/ ( LSeg (Gi1k,Gik))) by XBOOLE_0:def 3;

        then Gij in ( L~ <*Gij, Gi1k, Gik*>) by SPRECT_1: 8;

        hence thesis by A52, A53, XBOOLE_0:def 4;

      end;

      then

       A54: (( LSeg (go,m)) /\ ( L~ <*Gij, Gi1k, Gik*>)) = {Gij} by A50;

      

       A55: ( LSeg (co,1)) c= ( L~ co) by TOPREAL3: 19;

      

       A56: ( L~ co) c= ( L~ LS) by A13, JORDAN3: 42;

      then ( LSeg (co,1)) c= ( L~ LS) by A55;

      then

       A57: (( LSeg (co,1)) /\ ( L~ <*Gij, Gi1k, Gik*>)) c= {Gik} by A8, A12, XBOOLE_1: 26;

      

       A58: ( LSeg (co,1)) = ( LSeg (Gik,(co /. (1 + 1)))) by A39, A45, TOPREAL1:def 3;

       {Gik} c= (( LSeg (co,1)) /\ ( L~ <*Gij, Gi1k, Gik*>))

      proof

        let x be object;

        assume x in {Gik};

        then

         A59: x = Gik by TARSKI:def 1;

        

         A60: Gik in ( LSeg (co,1)) by A58, RLTOPSP1: 68;

        Gik in ( LSeg (Gi1k,Gik)) by RLTOPSP1: 68;

        then Gik in (( LSeg (Gij,Gi1k)) \/ ( LSeg (Gi1k,Gik))) by XBOOLE_0:def 3;

        then Gik in ( L~ <*Gij, Gi1k, Gik*>) by SPRECT_1: 8;

        hence thesis by A59, A60, XBOOLE_0:def 4;

      end;

      then

       A61: (( L~ <*Gij, Gi1k, Gik*>) /\ ( LSeg (co,1))) = {Gik} by A57;

      

       A62: (go /. 1) = (US /. 1) by A14, SPRECT_3: 22

      .= Wmin by JORDAN1F: 5;

      

      then

       A63: (go /. 1) = (LS /. ( len LS)) by JORDAN1F: 8

      .= (co /. ( len co)) by A13, JORDAN1J: 35;

      

       A64: ( rng go) c= ( L~ go) by A36, SPPOL_2: 18;

      

       A65: ( rng co) c= ( L~ co) by A39, SPPOL_2: 18;

      

       A66: {(go /. 1)} c= (( L~ go) /\ ( L~ co))

      proof

        let x be object;

        assume x in {(go /. 1)};

        then

         A67: x = (go /. 1) by TARSKI:def 1;

        then

         A68: x in ( rng go) by FINSEQ_6: 42;

        x in ( rng co) by A63, A67, FINSEQ_6: 168;

        hence thesis by A64, A65, A68, XBOOLE_0:def 4;

      end;

      

       A69: (LS . 1) = (LS /. 1) by A30, PARTFUN1:def 6

      .= Emax by JORDAN1F: 6;

      

       A70: [( len G), j] in ( Indices G) by A4, A15, A24, MATRIX_0: 30;

      (( 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~ go) by XBOOLE_0:def 4;

        

         A73: x in ( L~ co) by A71, XBOOLE_0:def 4;

        then x in (( L~ US) /\ ( L~ LS)) by A49, A56, A72, XBOOLE_0:def 4;

        then x in {Wmin, Emax} by JORDAN1E: 16;

        then

         A74: x = Wmin or x = Emax by TARSKI:def 2;

        now

          assume x = Emax;

          then

           A75: Emax = Gik by A13, A69, A73, JORDAN1E: 7;

          ((G * (( len G),j)) `1 ) = Ebo by A4, A15, A20, JORDAN1A: 71;

          then (Emax `1 ) <> Ebo by A3, A18, A70, A75, JORDAN1G: 7;

          hence contradiction by EUCLID: 52;

        end;

        hence thesis by A62, A74, TARSKI:def 1;

      end;

      then

       A76: (( L~ go) /\ ( L~ co)) = {(go /. 1)} by A66;

      set W2 = (go /. 2);

      

       A77: 2 in ( dom go) by A36, FINSEQ_3: 25;

       A78:

      now

        assume (Gij `1 ) = Wbo;

        then ((G * (1,j)) `1 ) = ((G * (i1,j)) `1 ) by A4, A15, A20, JORDAN1A: 73;

        hence contradiction by A1, A17, A27, JORDAN1G: 7;

      end;

      go = ( mid (US,1,(Gij .. US))) by A37, JORDAN1G: 49

      .= (US | (Gij .. US)) by A37, FINSEQ_4: 21, FINSEQ_6: 116;

      then

       A79: W2 = (US /. 2) by A77, FINSEQ_4: 70;

      

       A80: Wmin in ( rng go) by A62, FINSEQ_6: 42;

      set pion = <*Gij, Gi1k, Gik*>;

       A81:

      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 A17, A18, A19, FINSEQ_4: 18;

      end;

      

       A82: (Gi1k `1 ) = ((G * (i1,1)) `1 ) by A1, A6, A10, A16, GOBOARD5: 2

      .= (Gij `1 ) by A1, A4, A10, A15, GOBOARD5: 2;

      (Gi1k `2 ) = ((G * (1,k)) `2 ) by A1, A6, A10, A16, GOBOARD5: 1

      .= (Gik `2 ) by A3, A6, A11, A16, GOBOARD5: 1;

      then

       A83: Gi1k = |[(Gij `1 ), (Gik `2 )]| by A82, EUCLID: 53;

      

       A84: Gi1k in pio by RLTOPSP1: 68;

      

       A85: Gi1k in poz by RLTOPSP1: 68;

      now

        per cases ;

          suppose (Gik `1 ) <> (Gij `1 ) & (Gik `2 ) <> (Gij `2 );

          then pion is being_S-Seq by A83, TOPREAL3: 34;

          then

          consider pion1 be FinSequence of ( TOP-REAL 2) such that

           A86: pion1 is_sequence_on G and

           A87: pion1 is being_S-Seq and

           A88: ( L~ pion) = ( L~ pion1) and

           A89: (pion /. 1) = (pion1 /. 1) and

           A90: (pion /. ( len pion)) = (pion1 /. ( len pion1)) and

           A91: ( len pion) <= ( len pion1) by A81, GOBOARD3: 2;

          reconsider pion1 as being_S-Seq FinSequence of ( TOP-REAL 2) by A87;

          set godo = ((go ^' pion1) ^' co);

          

           A92: (Gi1k `1 ) = ((G * (i1,1)) `1 ) by A1, A6, A10, A16, GOBOARD5: 2

          .= (Gij `1 ) by A1, A4, A10, A15, GOBOARD5: 2;

          

           A93: (Gi1k `1 ) <= (Gik `1 ) by A1, A2, A3, A6, A16, JORDAN1A: 18;

          then

           A94: ( W-bound poz) = (Gi1k `1 ) by SPRECT_1: 54;

          

           A95: ( W-bound pio) = (Gij `1 ) by A92, SPRECT_1: 54;

          ( W-bound (poz \/ pio)) = ( min (( W-bound poz),( W-bound pio))) by SPRECT_1: 47

          .= (Gij `1 ) by A92, A94, A95;

          then

           A96: ( W-bound ( L~ pion1)) = (Gij `1 ) by A88, TOPREAL3: 16;

          

           A97: (1 + 1) <= ( len ( Cage (C,n))) by GOBOARD7: 34, XXREAL_0: 2;

          

           A98: (1 + 1) <= ( len ( Rotate (( Cage (C,n)),Wmin))) by GOBOARD7: 34, XXREAL_0: 2;

          ( len (go ^' pion1)) >= ( len go) by TOPREAL8: 7;

          then

           A99: ( len (go ^' pion1)) >= (1 + 1) by A36, XXREAL_0: 2;

          then

           A100: ( len (go ^' pion1)) > (1 + 0 ) by NAT_1: 13;

          

           A101: ( len godo) >= ( len (go ^' pion1)) by TOPREAL8: 7;

          then

           A102: (1 + 1) <= ( len godo) by A99, XXREAL_0: 2;

          

           A103: US is_sequence_on G by JORDAN1G: 4;

          

           A104: (go /. ( len go)) = (pion1 /. 1) by A44, A89, FINSEQ_4: 18;

          then

           A105: (go ^' pion1) is_sequence_on G by A38, A86, TOPREAL8: 12;

          

           A106: ((go ^' pion1) /. ( len (go ^' pion1))) = (pion /. ( len pion)) by A90, FINSEQ_6: 156

          .= (pion /. 3) by FINSEQ_1: 45

          .= (co /. 1) by A45, FINSEQ_4: 18;

          then

           A107: godo is_sequence_on G by A41, A105, TOPREAL8: 12;

          ( LSeg (pion1,1)) c= ( L~ pion) by A88, TOPREAL3: 19;

          then

           A108: (( LSeg (go,(( len go) -' 1))) /\ ( LSeg (pion1,1))) c= {Gij} by A47, A54, XBOOLE_1: 27;

          ( len pion1) >= (2 + 1) by A91, FINSEQ_1: 45;

          then

           A109: ( len pion1) > (1 + 1) by NAT_1: 13;

           {Gij} c= (( LSeg (go,m)) /\ ( LSeg (pion1,1)))

          proof

            let x be object;

            assume x in {Gij};

            then

             A110: x = Gij by TARSKI:def 1;

            

             A111: Gij in ( LSeg (go,m)) by A51, RLTOPSP1: 68;

            Gij in ( LSeg (pion1,1)) by A44, A104, A109, TOPREAL1: 21;

            hence thesis by A110, A111, XBOOLE_0:def 4;

          end;

          then (( LSeg (go,(( len go) -' 1))) /\ ( LSeg (pion1,1))) = {(go /. ( len go))} by A44, A47, A108;

          then

           A112: (go ^' pion1) is unfolded by A104, TOPREAL8: 34;

          ( len pion1) >= (2 + 1) by A91, FINSEQ_1: 45;

          then

           A113: (( 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 A113, XREAL_0:def 2;

          then

           A114: (( len (go ^' pion1)) -' 1) = (( len go) + (( len pion1) -' 2)) by XREAL_0:def 2;

          

           A115: (( len pion1) - 1) >= 1 by A109, XREAL_1: 19;

          then

           A116: (( len pion1) -' 1) = (( len pion1) - 1) by XREAL_0:def 2;

          

           A117: ((( len pion1) -' 2) + 1) = ((( len pion1) - 2) + 1) by A113, XREAL_0:def 2

          .= (( len pion1) -' 1) by A115, XREAL_0:def 2;

          ((( len pion1) - 1) + 1) <= ( len pion1);

          then

           A118: (( len pion1) -' 1) < ( len pion1) by A116, NAT_1: 13;

          ( LSeg (pion1,(( len pion1) -' 1))) c= ( L~ pion) by A88, TOPREAL3: 19;

          then

           A119: (( LSeg (pion1,(( len pion1) -' 1))) /\ ( LSeg (co,1))) c= {Gik} by A61, XBOOLE_1: 27;

           {Gik} c= (( LSeg (pion1,(( len pion1) -' 1))) /\ ( LSeg (co,1)))

          proof

            let x be object;

            assume x in {Gik};

            then

             A120: x = Gik by TARSKI:def 1;

            

             A121: Gik in ( LSeg (co,1)) by A58, RLTOPSP1: 68;

            (pion1 /. ((( len pion1) -' 1) + 1)) = (pion /. 3) by A90, A116, FINSEQ_1: 45

            .= Gik by FINSEQ_4: 18;

            then Gik in ( LSeg (pion1,(( len pion1) -' 1))) by A115, A116, TOPREAL1: 21;

            hence thesis by A120, A121, XBOOLE_0:def 4;

          end;

          then (( LSeg (pion1,(( len pion1) -' 1))) /\ ( LSeg (co,1))) = {Gik} by A119;

          then

           A122: (( LSeg ((go ^' pion1),(( len go) + (( len pion1) -' 2)))) /\ ( LSeg (co,1))) = {((go ^' pion1) /. ( len (go ^' pion1)))} by A45, A104, A106, A117, A118, TOPREAL8: 31;

          

           A123: (go ^' pion1) is non trivial by A99, NAT_D: 60;

          

           A124: ( rng pion1) c= ( L~ pion1) by A109, SPPOL_2: 18;

          

           A125: {(pion1 /. 1)} c= (( L~ go) /\ ( L~ pion1))

          proof

            let x be object;

            assume x in {(pion1 /. 1)};

            then

             A126: x = (pion1 /. 1) by TARSKI:def 1;

            then

             A127: x in ( rng go) by A104, FINSEQ_6: 168;

            x in ( rng pion1) by A126, FINSEQ_6: 42;

            hence thesis by A64, A124, A127, XBOOLE_0:def 4;

          end;

          (( L~ go) /\ ( L~ pion1)) c= {(pion1 /. 1)}

          proof

            let x be object;

            assume

             A128: x in (( L~ go) /\ ( L~ pion1));

            then

             A129: x in ( L~ go) by XBOOLE_0:def 4;

            x in ( L~ pion1) by A128, XBOOLE_0:def 4;

            hence thesis by A7, A12, A44, A49, A88, A104, A129, XBOOLE_0:def 4;

          end;

          then

           A130: (( L~ go) /\ ( L~ pion1)) = {(pion1 /. 1)} by A125;

          then

           A131: (go ^' pion1) is s.n.c. by A104, JORDAN1J: 54;

          (( rng go) /\ ( rng pion1)) c= {(pion1 /. 1)} by A64, A124, A130, XBOOLE_1: 27;

          then

           A132: (go ^' pion1) is one-to-one by JORDAN1J: 55;

          

           A133: (pion /. ( len pion)) = (pion /. 3) by FINSEQ_1: 45

          .= (co /. 1) by A45, FINSEQ_4: 18;

          

           A134: {(pion1 /. ( len pion1))} c= (( L~ co) /\ ( L~ pion1))

          proof

            let x be object;

            assume x in {(pion1 /. ( len pion1))};

            then

             A135: x = (pion1 /. ( len pion1)) by TARSKI:def 1;

            then

             A136: x in ( rng co) by A90, A133, FINSEQ_6: 42;

            x in ( rng pion1) by A135, FINSEQ_6: 168;

            hence thesis by A65, A124, A136, XBOOLE_0:def 4;

          end;

          (( L~ co) /\ ( L~ pion1)) c= {(pion1 /. ( len pion1))}

          proof

            let x be object;

            assume

             A137: x in (( L~ co) /\ ( L~ pion1));

            then

             A138: x in ( L~ co) by XBOOLE_0:def 4;

            x in ( L~ pion1) by A137, XBOOLE_0:def 4;

            hence thesis by A8, A12, A45, A56, A88, A90, A133, A138, XBOOLE_0:def 4;

          end;

          then

           A139: (( L~ co) /\ ( L~ pion1)) = {(pion1 /. ( len pion1))} by A134;

          

           A140: (( L~ (go ^' pion1)) /\ ( L~ co)) = ((( L~ go) \/ ( L~ pion1)) /\ ( L~ co)) by A104, TOPREAL8: 35

          .= ( {(go /. 1)} \/ {(co /. 1)}) by A76, A90, A133, A139, 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 A63, FINSEQ_6: 155;

          then

          reconsider godo as non constant standard special_circular_sequence by A102, A106, A107, A112, A114, A122, A123, A131, A132, A140, JORDAN8: 4, JORDAN8: 5, TOPREAL8: 11, TOPREAL8: 33, TOPREAL8: 34;

          

           A141: UA is_an_arc_of (( E-max C),( W-min C)) by JORDAN6:def 9;

          then

           A142: UA is connected by JORDAN6: 10;

          

           A143: ( W-min C) in UA by A141, TOPREAL1: 1;

          

           A144: ( E-max C) in UA by A141, TOPREAL1: 1;

          set ff = ( Rotate (( Cage (C,n)),Wmin));

          Wmin in ( rng ( Cage (C,n))) by SPRECT_2: 43;

          then

           A145: (ff /. 1) = Wmin by FINSEQ_6: 92;

          

           A146: ( L~ ff) = ( L~ ( Cage (C,n))) by REVROT_1: 33;

          then (( W-max ( L~ ff)) .. ff) > 1 by A145, SPRECT_5: 22;

          then (( N-min ( L~ ff)) .. ff) > 1 by A145, A146, SPRECT_5: 23, XXREAL_0: 2;

          then (( N-max ( L~ ff)) .. ff) > 1 by A145, A146, SPRECT_5: 24, XXREAL_0: 2;

          then

           A147: (Emax .. ff) > 1 by A145, A146, SPRECT_5: 25, XXREAL_0: 2;

           A148:

          now

            assume

             A149: (Gij .. US) <= 1;

            (Gij .. US) >= 1 by A37, FINSEQ_4: 21;

            then (Gij .. US) = 1 by A149, XXREAL_0: 1;

            then Gij = (US /. 1) by A37, FINSEQ_5: 38;

            hence contradiction by A22, A26, JORDAN1F: 5;

          end;

          

           A150: ( Cage (C,n)) is_sequence_on G by JORDAN9:def 1;

          then

           A151: ff is_sequence_on G by REVROT_1: 34;

          

           A152: (( right_cell (godo,1,G)) \ ( L~ godo)) c= ( RightComp godo) by A102, A107, JORDAN9: 27;

          

           A153: ( L~ godo) = (( L~ (go ^' pion1)) \/ ( L~ co)) by A106, TOPREAL8: 35

          .= ((( L~ go) \/ ( L~ pion1)) \/ ( L~ co)) by A104, TOPREAL8: 35;

          

           A154: ( L~ ( Cage (C,n))) = (( L~ US) \/ ( L~ LS)) by JORDAN1E: 13;

          then

           A155: ( L~ US) c= ( L~ ( Cage (C,n))) by XBOOLE_1: 7;

          

           A156: ( L~ LS) c= ( L~ ( Cage (C,n))) by A154, XBOOLE_1: 7;

          

           A157: ( L~ go) c= ( L~ ( Cage (C,n))) by A49, A155;

          

           A158: ( L~ co) c= ( L~ ( Cage (C,n))) by A56, A156;

          

           A159: ( W-min C) in C by SPRECT_1: 13;

           A160:

          now

            assume ( W-min C) in ( L~ godo);

            then

             A161: ( W-min C) in (( L~ go) \/ ( L~ pion1)) or ( W-min C) in ( L~ co) by A153, XBOOLE_0:def 3;

            per cases by A161, XBOOLE_0:def 3;

              suppose ( W-min C) in ( L~ go);

              then C meets ( L~ ( Cage (C,n))) by A157, A159, XBOOLE_0: 3;

              hence contradiction by JORDAN10: 5;

            end;

              suppose ( W-min C) in ( L~ pion1);

              hence contradiction by A9, A12, A88, A143, XBOOLE_0: 3;

            end;

              suppose ( W-min C) in ( L~ co);

              then C meets ( L~ ( Cage (C,n))) by A158, A159, 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 A98, 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 A147, A151, JORDAN1J: 53

          .= ( right_cell (US,1,G)) by JORDAN1E:def 1

          .= ( right_cell (( R_Cut (US,Gij)),1,G)) by A37, A103, A148, JORDAN1J: 52

          .= ( right_cell ((go ^' pion1),1,G)) by A42, A105, JORDAN1J: 51

          .= ( right_cell (godo,1,G)) by A100, A107, JORDAN1J: 51;

          then ( W-min C) in ( right_cell (godo,1,G)) by JORDAN1I: 6;

          then

           A162: ( W-min C) in (( right_cell (godo,1,G)) \ ( L~ godo)) by A160, XBOOLE_0:def 5;

          

           A163: (godo /. 1) = ((go ^' pion1) /. 1) by FINSEQ_6: 155

          .= Wmin by A62, FINSEQ_6: 155;

          

           A164: ( len US) >= 2 by A21, XXREAL_0: 2;

          

           A165: (godo /. 2) = ((go ^' pion1) /. 2) by A99, FINSEQ_6: 159

          .= (US /. 2) by A36, A79, FINSEQ_6: 159

          .= ((US ^' LS) /. 2) by A164, FINSEQ_6: 159

          .= (( Rotate (( Cage (C,n)),Wmin)) /. 2) by JORDAN1E: 11;

          

           A166: (( L~ go) \/ ( L~ co)) is compact by COMPTS_1: 10;

          Wmin in (( L~ go) \/ ( L~ co)) by A64, A80, XBOOLE_0:def 3;

          then

           A167: ( W-min (( L~ go) \/ ( L~ co))) = Wmin by A157, A158, A166, JORDAN1J: 21, XBOOLE_1: 8;

          

           A168: (( W-min (( L~ go) \/ ( L~ co))) `1 ) = ( W-bound (( L~ go) \/ ( L~ co))) by EUCLID: 52;

          

           A169: (Wmin `1 ) = Wbo by EUCLID: 52;

          (Gij `1 ) >= Wbo by A14, A155, PSCOMP_1: 24;

          then (Gij `1 ) > Wbo by A78, XXREAL_0: 1;

          then ( W-min ((( L~ go) \/ ( L~ co)) \/ ( L~ pion1))) = ( W-min (( L~ go) \/ ( L~ co))) by A96, A166, A167, A168, A169, JORDAN1J: 33;

          then

           A170: ( W-min ( L~ godo)) = Wmin by A153, A167, XBOOLE_1: 4;

          

           A171: ( rng godo) c= ( L~ godo) by A99, A101, SPPOL_2: 18, XXREAL_0: 2;

          2 in ( dom godo) by A102, FINSEQ_3: 25;

          then

           A172: (godo /. 2) in ( rng godo) by PARTFUN2: 2;

          (godo /. 2) in ( W-most ( L~ ( Cage (C,n)))) by A165, JORDAN1I: 25;

          

          then ((godo /. 2) `1 ) = (( W-min ( L~ godo)) `1 ) by A170, PSCOMP_1: 31

          .= ( W-bound ( L~ godo)) by EUCLID: 52;

          then (godo /. 2) in ( W-most ( L~ godo)) by A171, A172, SPRECT_2: 12;

          then (( Rotate (godo,( W-min ( L~ godo)))) /. 2) in ( W-most ( L~ godo)) by A163, A170, 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

           A173: (US . ( len US)) = (US /. ( len US)) by PARTFUN1:def 6

          .= Emax by JORDAN1F: 7;

          

           A174: ( 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

             A175: p in ( east_halfline ( E-max C)) and

             A176: p in ( L~ go) by XBOOLE_0: 3;

            reconsider p as Point of ( TOP-REAL 2) by A175;

            p in ( L~ US) by A49, A176;

            then p in (( east_halfline ( E-max C)) /\ ( L~ ( Cage (C,n)))) by A155, A175, XBOOLE_0:def 4;

            then

             A177: (p `1 ) = Ebo by JORDAN1A: 83, PSCOMP_1: 50;

            then

             A178: p = Emax by A49, A176, JORDAN1J: 46;

            then Emax = Gij by A14, A173, A176, JORDAN1J: 43;

            then (Gij `1 ) = ((G * (( len G),k)) `1 ) by A6, A16, A20, A177, A178, JORDAN1A: 71;

            hence contradiction by A2, A3, A17, A33, JORDAN1G: 7;

          end;

          now

            assume ( east_halfline ( E-max C)) meets ( L~ godo);

            then

             A179: ( east_halfline ( E-max C)) meets (( L~ go) \/ ( L~ pion1)) or ( east_halfline ( E-max C)) meets ( L~ co) by A153, XBOOLE_1: 70;

            per cases by A179, XBOOLE_1: 70;

              suppose ( east_halfline ( E-max C)) meets ( L~ go);

              hence contradiction by A174;

            end;

              suppose ( east_halfline ( E-max C)) meets ( L~ pion1);

              then

              consider p be object such that

               A180: p in ( east_halfline ( E-max C)) and

               A181: p in ( L~ pion1) by XBOOLE_0: 3;

              reconsider p as Point of ( TOP-REAL 2) by A180;

               A182:

              now

                per cases by A12, A88, A181, XBOOLE_0:def 3;

                  suppose p in poz;

                  hence (p `1 ) <= (Gik `1 ) by A93, TOPREAL1: 3;

                end;

                  suppose p in pio;

                  hence (p `1 ) <= (Gik `1 ) by A92, A93, GOBOARD7: 5;

                end;

              end;

              (i2 + 1) <= ( len G) by A3, NAT_1: 13;

              then i2 <= (( len G) - 1) by XREAL_1: 19;

              then

               A183: 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, A11, A16, A20, A24, A183, JORDAN1A: 18;

              then (p `1 ) <= ((G * ((( len G) -' 1),1)) `1 ) by A182, XXREAL_0: 2;

              then (p `1 ) <= ( E-bound C) by A24, JORDAN8: 12;

              then

               A184: (p `1 ) <= (( E-max C) `1 ) by EUCLID: 52;

              (p `1 ) >= (( E-max C) `1 ) by A180, TOPREAL1:def 11;

              then

               A185: (p `1 ) = (( E-max C) `1 ) by A184, XXREAL_0: 1;

              (p `2 ) = (( E-max C) `2 ) by A180, TOPREAL1:def 11;

              then p = ( E-max C) by A185, TOPREAL3: 6;

              hence contradiction by A9, A12, A88, A144, A181, XBOOLE_0: 3;

            end;

              suppose ( east_halfline ( E-max C)) meets ( L~ co);

              then

              consider p be object such that

               A186: p in ( east_halfline ( E-max C)) and

               A187: p in ( L~ co) by XBOOLE_0: 3;

              reconsider p as Point of ( TOP-REAL 2) by A186;

              p in ( L~ LS) by A56, A187;

              then p in (( east_halfline ( E-max C)) /\ ( L~ ( Cage (C,n)))) by A156, A186, XBOOLE_0:def 4;

              then

               A188: (p `1 ) = Ebo by JORDAN1A: 83, PSCOMP_1: 50;

              

               A189: (( E-max C) `2 ) = (p `2 ) by A186, TOPREAL1:def 11;

              set RC = ( Rotate (( Cage (C,n)),Emax));

              

               A190: ( E-max C) in ( right_cell (RC,1)) by JORDAN1I: 7;

              

               A191: (1 + 1) <= ( len LS) by A28, XXREAL_0: 2;

              LS = (RC -: Wmin) by JORDAN1G: 18;

              then

               A192: ( LSeg (LS,1)) = ( LSeg (RC,1)) by A191, SPPOL_2: 9;

              

               A193: ( L~ RC) = ( L~ ( Cage (C,n))) by REVROT_1: 33;

              

               A194: ( len RC) = ( len ( Cage (C,n))) by FINSEQ_6: 179;

              

               A195: ( GoB RC) = ( GoB ( Cage (C,n))) by REVROT_1: 28

              .= G by JORDAN1H: 44;

              

               A196: Emax in ( rng ( Cage (C,n))) by SPRECT_2: 46;

              

               A197: RC is_sequence_on G by A150, REVROT_1: 34;

              

               A198: (RC /. 1) = ( E-max ( L~ RC)) by A193, A196, FINSEQ_6: 92;

              consider ii,jj be Nat such that

               A199: [ii, (jj + 1)] in ( Indices G) and

               A200: [ii, jj] in ( Indices G) and

               A201: (RC /. 1) = (G * (ii,(jj + 1))) and

               A202: (RC /. (1 + 1)) = (G * (ii,jj)) by A97, A193, A194, A196, A197, FINSEQ_6: 92, JORDAN1I: 23;

              consider jj2 be Nat such that

               A203: 1 <= jj2 and

               A204: jj2 <= ( width G) and

               A205: Emax = (G * (( len G),jj2)) by JORDAN1D: 25;

              

               A206: ( len G) >= 4 by JORDAN8: 10;

              then ( len G) >= 1 by XXREAL_0: 2;

              then [( len G), jj2] in ( Indices G) by A203, A204, MATRIX_0: 30;

              then

               A207: ii = ( len G) by A193, A198, A199, A201, A205, GOBOARD1: 5;

              

               A208: 1 <= ii by A199, MATRIX_0: 32;

              

               A209: ii <= ( len G) by A199, MATRIX_0: 32;

              

               A210: 1 <= (jj + 1) by A199, MATRIX_0: 32;

              

               A211: (jj + 1) <= ( width G) by A199, MATRIX_0: 32;

              

               A212: 1 <= ii by A200, MATRIX_0: 32;

              

               A213: ii <= ( len G) by A200, MATRIX_0: 32;

              

               A214: 1 <= jj by A200, MATRIX_0: 32;

              

               A215: jj <= ( width G) by A200, MATRIX_0: 32;

              

               A216: (ii + 1) <> ii;

              ((jj + 1) + 1) <> jj;

              then

               A217: ( right_cell (RC,1)) = ( cell (G,(ii -' 1),jj)) by A97, A194, A195, A199, A200, A201, A202, A216, GOBOARD5:def 6;

              

               A218: ((ii -' 1) + 1) = ii by A208, XREAL_1: 235;

              (ii - 1) >= (4 - 1) by A206, A207, XREAL_1: 9;

              then

               A219: (ii - 1) >= 1 by XXREAL_0: 2;

              then

               A220: 1 <= (ii -' 1) by XREAL_0:def 2;

              

               A221: ((G * ((ii -' 1),jj)) `2 ) <= (p `2 ) by A189, A190, A209, A211, A214, A217, A218, A219, JORDAN9: 17;

              

               A222: (p `2 ) <= ((G * ((ii -' 1),(jj + 1))) `2 ) by A189, A190, A209, A211, A214, A217, A218, A219, JORDAN9: 17;

              

               A223: (ii -' 1) < ( len G) by A209, A218, NAT_1: 13;

              

              then

               A224: ((G * ((ii -' 1),jj)) `2 ) = ((G * (1,jj)) `2 ) by A214, A215, A220, GOBOARD5: 1

              .= ((G * (ii,jj)) `2 ) by A212, A213, A214, A215, GOBOARD5: 1;

              

               A225: ((G * ((ii -' 1),(jj + 1))) `2 ) = ((G * (1,(jj + 1))) `2 ) by A210, A211, A220, A223, GOBOARD5: 1

              .= ((G * (ii,(jj + 1))) `2 ) by A208, A209, A210, A211, GOBOARD5: 1;

              

               A226: ((G * (( len G),jj)) `1 ) = Ebo by A20, A214, A215, JORDAN1A: 71;

              Ebo = ((G * (( len G),(jj + 1))) `1 ) by A20, A210, A211, JORDAN1A: 71;

              then p in ( LSeg ((RC /. 1),(RC /. (1 + 1)))) by A188, A201, A202, A207, A221, A222, A224, A225, A226, GOBOARD7: 7;

              then

               A227: p in ( LSeg (LS,1)) by A97, A192, A194, TOPREAL1:def 3;

              

               A228: p in ( LSeg (co,( Index (p,co)))) by A187, JORDAN3: 9;

              

               A229: co = ( mid (LS,(Gik .. LS),( len LS))) by A40, JORDAN1J: 37;

              

               A230: 1 <= (Gik .. LS) by A40, FINSEQ_4: 21;

              

               A231: (Gik .. LS) <= ( len LS) by A40, FINSEQ_4: 21;

              (Gik .. LS) <> ( len LS) by A32, A40, FINSEQ_4: 19;

              then

               A232: (Gik .. LS) < ( len LS) by A231, XXREAL_0: 1;

              

               A233: 1 <= ( Index (p,co)) by A187, JORDAN3: 8;

              

               A234: ( Index (p,co)) < ( len co) by A187, JORDAN3: 8;

              

               A235: (( Index (Gik,LS)) + 1) = (Gik .. LS) by A35, A40, JORDAN1J: 56;

              consider t be Nat such that

               A236: t in ( dom LS) and

               A237: (LS . t) = Gik by A40, FINSEQ_2: 10;

              

               A238: 1 <= t by A236, FINSEQ_3: 25;

              

               A239: t <= ( len LS) by A236, FINSEQ_3: 25;

              1 < t by A35, A237, A238, XXREAL_0: 1;

              then (( Index (Gik,LS)) + 1) = t by A237, A239, JORDAN3: 12;

              then

               A240: ( len ( L_Cut (LS,Gik))) = (( len LS) - ( Index (Gik,LS))) by A13, A237, JORDAN3: 26;

              set tt = ((( Index (p,co)) + (Gik .. LS)) -' 1);

              

               A241: 1 <= ( Index (Gik,LS)) by A13, JORDAN3: 8;

              ( 0 + ( Index (Gik,LS))) < ( len LS) by A13, JORDAN3: 8;

              then

               A242: (( len LS) - ( Index (Gik,LS))) > 0 by XREAL_1: 20;

              ( Index (p,co)) < (( len LS) -' ( Index (Gik,LS))) by A234, A240, XREAL_0:def 2;

              then (( Index (p,co)) + 1) <= (( len LS) -' ( Index (Gik,LS))) by NAT_1: 13;

              then ( Index (p,co)) <= ((( len LS) -' ( Index (Gik,LS))) - 1) by XREAL_1: 19;

              then ( Index (p,co)) <= ((( len LS) - ( Index (Gik,LS))) - 1) by A242, XREAL_0:def 2;

              then ( Index (p,co)) <= (( len LS) - (Gik .. LS)) by A235;

              then ( Index (p,co)) <= (( len LS) -' (Gik .. LS)) by XREAL_0:def 2;

              then ( Index (p,co)) < ((( len LS) -' (Gik .. LS)) + 1) by NAT_1: 13;

              then

               A243: ( LSeg (( mid (LS,(Gik .. LS),( len LS))),( Index (p,co)))) = ( LSeg (LS,((( Index (p,co)) + (Gik .. LS)) -' 1))) by A230, A232, A233, JORDAN4: 19;

              

               A244: (1 + 1) <= (Gik .. LS) by A235, A241, XREAL_1: 7;

              then (( Index (p,co)) + (Gik .. LS)) >= ((1 + 1) + 1) by A233, XREAL_1: 7;

              then ((( Index (p,co)) + (Gik .. LS)) - 1) >= (((1 + 1) + 1) - 1) by XREAL_1: 9;

              then

               A245: tt >= (1 + 1) by XREAL_0:def 2;

              

               A246: 2 in ( dom LS) by A191, FINSEQ_3: 25;

              now

                per cases by A245, XXREAL_0: 1;

                  suppose tt > (1 + 1);

                  then ( LSeg (LS,1)) misses ( LSeg (LS,tt)) by TOPREAL1:def 7;

                  hence contradiction by A227, A228, A229, A243, XBOOLE_0: 3;

                end;

                  suppose

                   A247: tt = (1 + 1);

                  then (( LSeg (LS,1)) /\ ( LSeg (LS,tt))) = {(LS /. 2)} by A28, TOPREAL1:def 6;

                  then p in {(LS /. 2)} by A227, A228, A229, A243, XBOOLE_0:def 4;

                  then

                   A248: p = (LS /. 2) by TARSKI:def 1;

                  then

                   A249: (p .. LS) = 2 by A246, FINSEQ_5: 41;

                  (1 + 1) = ((( Index (p,co)) + (Gik .. LS)) - 1) by A247, XREAL_0:def 2;

                  then ((1 + 1) + 1) = (( Index (p,co)) + (Gik .. LS));

                  then

                   A250: (Gik .. LS) = 2 by A233, A244, JORDAN1E: 6;

                  p in ( rng LS) by A246, A248, PARTFUN2: 2;

                  then p = Gik by A40, A249, A250, FINSEQ_5: 9;

                  then (Gik `1 ) = Ebo by A248, JORDAN1G: 32;

                  then (Gik `1 ) = ((G * (( len G),j)) `1 ) by A4, A15, A20, JORDAN1A: 71;

                  hence contradiction by A3, A18, A70, 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

           A251: W is_a_component_of (( L~ godo) ` ) and

           A252: ( east_halfline ( E-max C)) c= W by GOBOARD9: 3;

           not W is bounded by A252, JORDAN2C: 121, RLTOPSP1: 42;

          then W is_outside_component_of ( L~ godo) by A251, JORDAN2C:def 3;

          then W c= ( UBD ( L~ godo)) by JORDAN2C: 23;

          then

           A253: ( east_halfline ( E-max C)) c= ( UBD ( L~ godo)) by A252;

          ( E-max C) in ( east_halfline ( E-max C)) by TOPREAL1: 38;

          then ( E-max C) in ( UBD ( L~ godo)) by A253;

          then ( E-max C) in ( LeftComp godo) by GOBRD14: 36;

          then UA meets ( L~ godo) by A142, A143, A144, A152, A162, JORDAN1J: 36;

          then

           A254: UA meets (( L~ go) \/ ( L~ pion1)) or UA meets ( L~ co) by A153, XBOOLE_1: 70;

          

           A255: UA c= C by JORDAN6: 61;

          now

            per cases by A254, XBOOLE_1: 70;

              suppose UA meets ( L~ go);

              then UA meets ( L~ ( Cage (C,n))) by A49, A155, XBOOLE_1: 1, XBOOLE_1: 63;

              hence contradiction by A255, JORDAN10: 5, XBOOLE_1: 63;

            end;

              suppose UA meets ( L~ pion1);

              hence contradiction by A9, A12, A88;

            end;

              suppose UA meets ( L~ co);

              then UA meets ( L~ ( Cage (C,n))) by A56, A156, XBOOLE_1: 1, XBOOLE_1: 63;

              hence contradiction by A255, JORDAN10: 5, XBOOLE_1: 63;

            end;

          end;

          hence contradiction;

        end;

          suppose (Gik `1 ) = (Gij `1 );

          then

           A256: i1 = i2 by A17, A18, JORDAN1G: 7;

          then poz = {Gi1k} by RLTOPSP1: 70;

          then poz c= pio by A84, ZFMISC_1: 31;

          then (pio \/ poz) = pio by XBOOLE_1: 12;

          hence contradiction by A1, A3, A4, A5, A6, A7, A8, A9, A256, Th13;

        end;

          suppose (Gik `2 ) = (Gij `2 );

          then

           A257: j = k by A17, A18, JORDAN1G: 6;

          then pio = {Gi1k} by RLTOPSP1: 70;

          then pio c= poz by A85, ZFMISC_1: 31;

          then (pio \/ poz) = poz by XBOOLE_1: 12;

          hence contradiction by A1, A2, A3, A4, A6, A7, A8, A9, A257, JORDAN15: 36;

        end;

      end;

      hence contradiction;

    end;

    theorem :: JORDAN19:22

    

     Th22: 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)) * (i1,j))} & ((( 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)) * (i2,k))} 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 * (i1,j))} and

       A8: ((pio \/ poz) /\ ( L~ LS)) = {(G * (i2,k))} and

       A9: (pio \/ poz) misses ( Upper_Arc C);

      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 Gik = (G * (i2,k));

      set Gij = (G * (i1,j));

      set Gi1k = (G * (i1,k));

      

       A10: 1 < i1 by A1, A2, XXREAL_0: 2;

      

       A11: i2 < ( len G) by A2, A3, XXREAL_0: 2;

      

       A12: ( L~ <*Gij, Gi1k, Gik*>) = (poz \/ pio) by TOPREAL3: 16;

      Gik in {Gik} by TARSKI:def 1;

      then

       A13: Gik in ( L~ LS) by A8, XBOOLE_0:def 4;

      Gij in {Gij} by TARSKI:def 1;

      then

       A14: Gij in ( L~ US) by A7, XBOOLE_0:def 4;

      

       A15: j <= ( width G) by A5, A6, XXREAL_0: 2;

      

       A16: 1 <= k by A4, A5, XXREAL_0: 2;

      

       A17: [i1, j] in ( Indices G) by A3, A4, A10, A15, MATRIX_0: 30;

      

       A18: [i2, k] in ( Indices G) by A1, A6, A11, A16, MATRIX_0: 30;

      

       A19: [i1, k] in ( Indices G) by A3, A6, A10, A16, MATRIX_0: 30;

      set go = ( R_Cut (US,Gij));

      set co = ( L_Cut (LS,Gik));

      

       A20: ( len G) = ( width G) by JORDAN8:def 1;

      

       A21: ( len US) >= 3 by JORDAN1E: 15;

      then ( len US) >= 1 by XXREAL_0: 2;

      then 1 in ( dom US) by FINSEQ_3: 25;

      

      then

       A22: (US . 1) = (US /. 1) by PARTFUN1:def 6

      .= Wmin by JORDAN1F: 5;

      

       A23: (Wmin `1 ) = Wbo by EUCLID: 52

      .= ((G * (1,k)) `1 ) by A6, A16, A20, JORDAN1A: 73;

      ( len G) >= 4 by JORDAN8: 10;

      then

       A24: ( len G) >= 1 by XXREAL_0: 2;

      then

       A25: [1, k] in ( Indices G) by A6, A16, MATRIX_0: 30;

      then

       A26: Gij <> (US . 1) by A1, A2, A17, A22, A23, JORDAN1G: 7;

      then

      reconsider go as being_S-Seq FinSequence of ( TOP-REAL 2) by A14, JORDAN3: 35;

      

       A27: ( len LS) >= (1 + 2) by JORDAN1E: 15;

      then

       A28: ( len LS) >= 1 by XXREAL_0: 2;

      then

       A29: 1 in ( dom LS) by FINSEQ_3: 25;

      ( len LS) in ( dom LS) by A28, FINSEQ_3: 25;

      

      then

       A30: (LS . ( len LS)) = (LS /. ( len LS)) by PARTFUN1:def 6

      .= Wmin by JORDAN1F: 8;

      (Wmin `1 ) = Wbo by EUCLID: 52

      .= ((G * (1,k)) `1 ) by A6, A16, A20, JORDAN1A: 73;

      then

       A31: Gik <> (LS . ( len LS)) by A1, A18, A25, A30, JORDAN1G: 7;

      then

      reconsider co as being_S-Seq FinSequence of ( TOP-REAL 2) by A13, JORDAN3: 34;

      

       A32: [( len G), k] in ( Indices G) by A6, A16, A24, MATRIX_0: 30;

      

       A33: (LS . 1) = (LS /. 1) by A29, PARTFUN1:def 6

      .= Emax by JORDAN1F: 6;

      (Emax `1 ) = Ebo by EUCLID: 52

      .= ((G * (( len G),k)) `1 ) by A6, A16, A20, JORDAN1A: 71;

      then

       A34: Gik <> (LS . 1) by A2, A3, A18, A32, A33, JORDAN1G: 7;

      

       A35: ( len go) >= (1 + 1) by TOPREAL1:def 8;

      

       A36: Gij in ( rng US) by A3, A4, A10, A14, A15, JORDAN1G: 4, JORDAN1J: 40;

      then

       A37: go is_sequence_on G by JORDAN1G: 4, JORDAN1J: 38;

      

       A38: ( len co) >= (1 + 1) by TOPREAL1:def 8;

      

       A39: Gik in ( rng LS) by A1, A6, A11, A13, A16, JORDAN1G: 5, JORDAN1J: 40;

      then

       A40: co is_sequence_on G 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

      .= Gij by A14, 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

      .= Gik by A13, 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 A14, JORDAN3: 41;

      then ( LSeg (go,m)) c= ( L~ US) by A47;

      then

       A49: (( LSeg (go,m)) /\ ( L~ <*Gij, Gi1k, Gik*>)) c= {Gij} by A7, A12, XBOOLE_1: 26;

      m >= 1 by A35, XREAL_1: 19;

      then

       A50: ( LSeg (go,m)) = ( LSeg ((go /. m),Gij)) by A43, A45, TOPREAL1:def 3;

       {Gij} c= (( LSeg (go,m)) /\ ( L~ <*Gij, Gi1k, Gik*>))

      proof

        let x be object;

        assume x in {Gij};

        then

         A51: x = Gij by TARSKI:def 1;

        

         A52: Gij in ( LSeg (go,m)) by A50, RLTOPSP1: 68;

        Gij in ( LSeg (Gij,Gi1k)) by RLTOPSP1: 68;

        then Gij in (( LSeg (Gij,Gi1k)) \/ ( LSeg (Gi1k,Gik))) by XBOOLE_0:def 3;

        then Gij in ( L~ <*Gij, Gi1k, Gik*>) by SPRECT_1: 8;

        hence thesis by A51, A52, XBOOLE_0:def 4;

      end;

      then

       A53: (( LSeg (go,m)) /\ ( L~ <*Gij, Gi1k, Gik*>)) = {Gij} by A49;

      

       A54: ( LSeg (co,1)) c= ( L~ co) by TOPREAL3: 19;

      

       A55: ( L~ co) c= ( L~ LS) by A13, JORDAN3: 42;

      then ( LSeg (co,1)) c= ( L~ LS) by A54;

      then

       A56: (( LSeg (co,1)) /\ ( L~ <*Gij, Gi1k, Gik*>)) c= {Gik} by A8, A12, XBOOLE_1: 26;

      

       A57: ( LSeg (co,1)) = ( LSeg (Gik,(co /. (1 + 1)))) by A38, A44, TOPREAL1:def 3;

       {Gik} c= (( LSeg (co,1)) /\ ( L~ <*Gij, Gi1k, Gik*>))

      proof

        let x be object;

        assume x in {Gik};

        then

         A58: x = Gik by TARSKI:def 1;

        

         A59: Gik in ( LSeg (co,1)) by A57, RLTOPSP1: 68;

        Gik in ( LSeg (Gi1k,Gik)) by RLTOPSP1: 68;

        then Gik in (( LSeg (Gij,Gi1k)) \/ ( LSeg (Gi1k,Gik))) by XBOOLE_0:def 3;

        then Gik in ( L~ <*Gij, Gi1k, Gik*>) by SPRECT_1: 8;

        hence thesis by A58, A59, XBOOLE_0:def 4;

      end;

      then

       A60: (( L~ <*Gij, Gi1k, Gik*>) /\ ( LSeg (co,1))) = {Gik} by A56;

      

       A61: (go /. 1) = (US /. 1) by A14, SPRECT_3: 22

      .= Wmin by JORDAN1F: 5;

      

      then

       A62: (go /. 1) = (LS /. ( len LS)) by JORDAN1F: 8

      .= (co /. ( len co)) by A13, 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 A29, PARTFUN1:def 6

      .= Emax by JORDAN1F: 6;

      

       A69: [( len G), j] in ( Indices G) by A4, A15, A24, 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~ go) by XBOOLE_0:def 4;

        

         A72: x in ( L~ co) 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

         A73: x = Wmin or x = Emax by TARSKI:def 2;

        now

          assume x = Emax;

          then

           A74: Emax = Gik by A13, A68, A72, JORDAN1E: 7;

          ((G * (( len G),j)) `1 ) = Ebo by A4, A15, A20, JORDAN1A: 71;

          then (Emax `1 ) <> Ebo by A2, A3, A18, A69, A74, JORDAN1G: 7;

          hence contradiction by EUCLID: 52;

        end;

        hence thesis by A61, A73, TARSKI:def 1;

      end;

      then

       A75: (( L~ go) /\ ( L~ co)) = {(go /. 1)} by A65;

      set W2 = (go /. 2);

      

       A76: 2 in ( dom go) by A35, FINSEQ_3: 25;

       A77:

      now

        assume (Gik `1 ) = Wbo;

        then ((G * (1,k)) `1 ) = ((G * (i2,k)) `1 ) by A6, A16, A20, JORDAN1A: 73;

        hence contradiction by A1, A18, A25, JORDAN1G: 7;

      end;

      go = ( mid (US,1,(Gij .. US))) by A36, JORDAN1G: 49

      .= (US | (Gij .. US)) by A36, FINSEQ_4: 21, FINSEQ_6: 116;

      then

       A78: W2 = (US /. 2) by A76, FINSEQ_4: 70;

      

       A79: Wmin in ( rng go) by A61, FINSEQ_6: 42;

      set pion = <*Gij, Gi1k, Gik*>;

       A80:

      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 A17, A18, A19, FINSEQ_4: 18;

      end;

      

       A81: (Gi1k `1 ) = ((G * (i1,1)) `1 ) by A3, A6, A10, A16, GOBOARD5: 2

      .= (Gij `1 ) by A3, A4, A10, A15, GOBOARD5: 2;

      (Gi1k `2 ) = ((G * (1,k)) `2 ) by A3, A6, A10, A16, GOBOARD5: 1

      .= (Gik `2 ) by A1, A6, A11, A16, GOBOARD5: 1;

      then

       A82: Gi1k = |[(Gij `1 ), (Gik `2 )]| by A81, EUCLID: 53;

      

       A83: Gi1k in pio by RLTOPSP1: 68;

      

       A84: Gi1k in poz by RLTOPSP1: 68;

      now

        per cases ;

          suppose (Gik `1 ) <> (Gij `1 ) & (Gik `2 ) <> (Gij `2 );

          then pion is being_S-Seq by A82, TOPREAL3: 34;

          then

          consider pion1 be FinSequence of ( TOP-REAL 2) such that

           A85: pion1 is_sequence_on G and

           A86: pion1 is being_S-Seq and

           A87: ( L~ pion) = ( L~ pion1) and

           A88: (pion /. 1) = (pion1 /. 1) and

           A89: (pion /. ( len pion)) = (pion1 /. ( len pion1)) and

           A90: ( len pion) <= ( len pion1) by A80, GOBOARD3: 2;

          reconsider pion1 as being_S-Seq FinSequence of ( TOP-REAL 2) by A86;

          set godo = ((go ^' pion1) ^' co);

          

           A91: (Gi1k `1 ) = ((G * (i1,1)) `1 ) by A3, A6, A10, A16, GOBOARD5: 2

          .= (Gij `1 ) by A3, A4, A10, A15, GOBOARD5: 2;

          

           A92: (Gik `1 ) <= (Gi1k `1 ) by A1, A2, A3, A6, A16, JORDAN1A: 18;

          then

           A93: ( W-bound poz) = (Gik `1 ) by SPRECT_1: 54;

          

           A94: ( W-bound pio) = (Gij `1 ) by A91, SPRECT_1: 54;

          ( W-bound (poz \/ pio)) = ( min (( W-bound poz),( W-bound pio))) by SPRECT_1: 47

          .= (Gik `1 ) by A91, A92, A93, A94, XXREAL_0:def 9;

          then

           A95: ( W-bound ( L~ pion1)) = (Gik `1 ) by A87, TOPREAL3: 16;

          

           A96: (1 + 1) <= ( len ( Cage (C,n))) by GOBOARD7: 34, XXREAL_0: 2;

          

           A97: (1 + 1) <= ( len ( Rotate (( Cage (C,n)),Wmin))) by GOBOARD7: 34, XXREAL_0: 2;

          ( len (go ^' pion1)) >= ( len go) by TOPREAL8: 7;

          then

           A98: ( len (go ^' pion1)) >= (1 + 1) by A35, XXREAL_0: 2;

          then

           A99: ( len (go ^' pion1)) > (1 + 0 ) by NAT_1: 13;

          

           A100: ( len godo) >= ( len (go ^' pion1)) by TOPREAL8: 7;

          then

           A101: (1 + 1) <= ( len godo) by A98, XXREAL_0: 2;

          

           A102: US is_sequence_on G by JORDAN1G: 4;

          

           A103: (go /. ( len go)) = (pion1 /. 1) by A43, A88, FINSEQ_4: 18;

          then

           A104: (go ^' pion1) is_sequence_on G by A37, A85, TOPREAL8: 12;

          

           A105: ((go ^' pion1) /. ( len (go ^' pion1))) = (pion /. ( len pion)) by A89, FINSEQ_6: 156

          .= (pion /. 3) by FINSEQ_1: 45

          .= (co /. 1) by A44, FINSEQ_4: 18;

          then

           A106: godo is_sequence_on G by A40, A104, TOPREAL8: 12;

          ( LSeg (pion1,1)) c= ( L~ pion) by A87, TOPREAL3: 19;

          then

           A107: (( LSeg (go,(( len go) -' 1))) /\ ( LSeg (pion1,1))) c= {Gij} by A46, A53, XBOOLE_1: 27;

          ( len pion1) >= (2 + 1) by A90, FINSEQ_1: 45;

          then

           A108: ( len pion1) > (1 + 1) by NAT_1: 13;

           {Gij} c= (( LSeg (go,m)) /\ ( LSeg (pion1,1)))

          proof

            let x be object;

            assume x in {Gij};

            then

             A109: x = Gij by TARSKI:def 1;

            

             A110: Gij in ( LSeg (go,m)) by A50, RLTOPSP1: 68;

            Gij in ( LSeg (pion1,1)) by A43, A103, A108, TOPREAL1: 21;

            hence thesis by A109, A110, XBOOLE_0:def 4;

          end;

          then (( LSeg (go,(( len go) -' 1))) /\ ( LSeg (pion1,1))) = {(go /. ( len go))} by A43, A46, A107;

          then

           A111: (go ^' pion1) is unfolded by A103, TOPREAL8: 34;

          ( len pion1) >= (2 + 1) by A90, FINSEQ_1: 45;

          then

           A112: (( 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 A112, XREAL_0:def 2;

          then

           A113: (( len (go ^' pion1)) -' 1) = (( len go) + (( len pion1) -' 2)) by XREAL_0:def 2;

          

           A114: (( len pion1) - 1) >= 1 by A108, XREAL_1: 19;

          then

           A115: (( len pion1) -' 1) = (( len pion1) - 1) by XREAL_0:def 2;

          

           A116: ((( len pion1) -' 2) + 1) = ((( len pion1) - 2) + 1) by A112, XREAL_0:def 2

          .= (( len pion1) -' 1) by A114, XREAL_0:def 2;

          ((( len pion1) - 1) + 1) <= ( len pion1);

          then

           A117: (( len pion1) -' 1) < ( len pion1) by A115, NAT_1: 13;

          ( LSeg (pion1,(( len pion1) -' 1))) c= ( L~ pion) by A87, TOPREAL3: 19;

          then

           A118: (( LSeg (pion1,(( len pion1) -' 1))) /\ ( LSeg (co,1))) c= {Gik} by A60, XBOOLE_1: 27;

           {Gik} c= (( LSeg (pion1,(( len pion1) -' 1))) /\ ( LSeg (co,1)))

          proof

            let x be object;

            assume x in {Gik};

            then

             A119: x = Gik by TARSKI:def 1;

            

             A120: Gik in ( LSeg (co,1)) by A57, RLTOPSP1: 68;

            (pion1 /. ((( len pion1) -' 1) + 1)) = (pion /. 3) by A89, A115, FINSEQ_1: 45

            .= Gik by FINSEQ_4: 18;

            then Gik in ( LSeg (pion1,(( len pion1) -' 1))) by A114, A115, TOPREAL1: 21;

            hence thesis by A119, A120, XBOOLE_0:def 4;

          end;

          then (( LSeg (pion1,(( len pion1) -' 1))) /\ ( LSeg (co,1))) = {Gik} by A118;

          then

           A121: (( LSeg ((go ^' pion1),(( len go) + (( len pion1) -' 2)))) /\ ( LSeg (co,1))) = {((go ^' pion1) /. ( len (go ^' pion1)))} by A44, A103, A105, A116, A117, TOPREAL8: 31;

          

           A122: (go ^' pion1) is non trivial by A98, NAT_D: 60;

          

           A123: ( rng pion1) c= ( L~ pion1) by A108, SPPOL_2: 18;

          

           A124: {(pion1 /. 1)} c= (( L~ go) /\ ( L~ pion1))

          proof

            let x be object;

            assume x in {(pion1 /. 1)};

            then

             A125: x = (pion1 /. 1) by TARSKI:def 1;

            then

             A126: x in ( rng go) by A103, FINSEQ_6: 168;

            x in ( rng pion1) by A125, FINSEQ_6: 42;

            hence thesis by A63, A123, A126, XBOOLE_0:def 4;

          end;

          (( L~ go) /\ ( L~ pion1)) c= {(pion1 /. 1)}

          proof

            let x be object;

            assume

             A127: x in (( L~ go) /\ ( L~ pion1));

            then

             A128: x in ( L~ go) by XBOOLE_0:def 4;

            x in ( L~ pion1) by A127, XBOOLE_0:def 4;

            hence thesis by A7, A12, A43, A48, A87, A103, A128, XBOOLE_0:def 4;

          end;

          then

           A129: (( L~ go) /\ ( L~ pion1)) = {(pion1 /. 1)} by A124;

          then

           A130: (go ^' pion1) is s.n.c. by A103, JORDAN1J: 54;

          (( rng go) /\ ( rng pion1)) c= {(pion1 /. 1)} by A63, A123, A129, XBOOLE_1: 27;

          then

           A131: (go ^' pion1) is one-to-one by JORDAN1J: 55;

          

           A132: (pion /. ( len pion)) = (pion /. 3) by FINSEQ_1: 45

          .= (co /. 1) by A44, FINSEQ_4: 18;

          

           A133: {(pion1 /. ( len pion1))} c= (( L~ co) /\ ( L~ pion1))

          proof

            let x be object;

            assume x in {(pion1 /. ( len pion1))};

            then

             A134: x = (pion1 /. ( len pion1)) by TARSKI:def 1;

            then

             A135: x in ( rng co) by A89, A132, FINSEQ_6: 42;

            x in ( rng pion1) by A134, FINSEQ_6: 168;

            hence thesis by A64, A123, A135, XBOOLE_0:def 4;

          end;

          (( L~ co) /\ ( L~ pion1)) c= {(pion1 /. ( len pion1))}

          proof

            let x be object;

            assume

             A136: x in (( L~ co) /\ ( L~ pion1));

            then

             A137: x in ( L~ co) by XBOOLE_0:def 4;

            x in ( L~ pion1) by A136, XBOOLE_0:def 4;

            hence thesis by A8, A12, A44, A55, A87, A89, A132, A137, XBOOLE_0:def 4;

          end;

          then

           A138: (( L~ co) /\ ( L~ pion1)) = {(pion1 /. ( len pion1))} by A133;

          

           A139: (( L~ (go ^' pion1)) /\ ( L~ co)) = ((( L~ go) \/ ( L~ pion1)) /\ ( L~ co)) by A103, TOPREAL8: 35

          .= ( {(go /. 1)} \/ {(co /. 1)}) by A75, A89, A132, A138, 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 A101, A105, A106, A111, A113, A121, A122, A130, A131, A139, JORDAN8: 4, JORDAN8: 5, TOPREAL8: 11, TOPREAL8: 33, TOPREAL8: 34;

          

           A140: UA is_an_arc_of (( W-min C),( E-max C)) by JORDAN6:def 8;

          then

           A141: UA is connected by JORDAN6: 10;

          

           A142: ( W-min C) in UA by A140, TOPREAL1: 1;

          

           A143: ( E-max C) in UA by A140, TOPREAL1: 1;

          set ff = ( Rotate (( Cage (C,n)),Wmin));

          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:

          now

            assume

             A148: (Gij .. US) <= 1;

            (Gij .. US) >= 1 by A36, FINSEQ_4: 21;

            then (Gij .. US) = 1 by A148, XXREAL_0: 1;

            then Gij = (US /. 1) by A36, FINSEQ_5: 38;

            hence contradiction by A22, A26, JORDAN1F: 5;

          end;

          

           A149: ( Cage (C,n)) is_sequence_on G by JORDAN9:def 1;

          then

           A150: ff is_sequence_on G by REVROT_1: 34;

          

           A151: (( right_cell (godo,1,G)) \ ( L~ godo)) c= ( RightComp godo) by A101, A106, JORDAN9: 27;

          

           A152: ( L~ godo) = (( L~ (go ^' pion1)) \/ ( L~ co)) by A105, TOPREAL8: 35

          .= ((( L~ go) \/ ( L~ pion1)) \/ ( L~ co)) by A103, TOPREAL8: 35;

          

           A153: ( L~ ( Cage (C,n))) = (( L~ US) \/ ( L~ LS)) by JORDAN1E: 13;

          then

           A154: ( L~ US) c= ( L~ ( Cage (C,n))) by XBOOLE_1: 7;

          

           A155: ( L~ LS) c= ( L~ ( Cage (C,n))) by A153, XBOOLE_1: 7;

          

           A156: ( L~ go) c= ( L~ ( Cage (C,n))) by A48, A154;

          

           A157: ( L~ co) c= ( L~ ( Cage (C,n))) by A55, A155;

          

           A158: ( W-min C) in C by SPRECT_1: 13;

           A159:

          now

            assume ( W-min C) in ( L~ godo);

            then

             A160: ( W-min C) in (( L~ go) \/ ( L~ pion1)) or ( W-min C) in ( L~ co) by A152, XBOOLE_0:def 3;

            per cases by A160, XBOOLE_0:def 3;

              suppose ( W-min C) in ( L~ go);

              then C meets ( L~ ( Cage (C,n))) by A156, A158, XBOOLE_0: 3;

              hence contradiction by JORDAN10: 5;

            end;

              suppose ( W-min C) in ( L~ pion1);

              hence contradiction by A9, A12, A87, A142, XBOOLE_0: 3;

            end;

              suppose ( W-min C) in ( L~ co);

              then C meets ( L~ ( Cage (C,n))) by A157, A158, 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 A97, 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, A150, JORDAN1J: 53

          .= ( right_cell (US,1,G)) by JORDAN1E:def 1

          .= ( right_cell (( R_Cut (US,Gij)),1,G)) by A36, A102, A147, JORDAN1J: 52

          .= ( right_cell ((go ^' pion1),1,G)) by A41, A104, JORDAN1J: 51

          .= ( right_cell (godo,1,G)) by A99, A106, JORDAN1J: 51;

          then ( W-min C) in ( right_cell (godo,1,G)) by JORDAN1I: 6;

          then

           A161: ( W-min C) in (( right_cell (godo,1,G)) \ ( L~ godo)) by A159, XBOOLE_0:def 5;

          

           A162: (godo /. 1) = ((go ^' pion1) /. 1) by FINSEQ_6: 155

          .= Wmin by A61, FINSEQ_6: 155;

          

           A163: ( len US) >= 2 by A21, XXREAL_0: 2;

          

           A164: (godo /. 2) = ((go ^' pion1) /. 2) by A98, FINSEQ_6: 159

          .= (US /. 2) by A35, A78, FINSEQ_6: 159

          .= ((US ^' LS) /. 2) by A163, FINSEQ_6: 159

          .= (( Rotate (( Cage (C,n)),Wmin)) /. 2) by JORDAN1E: 11;

          

           A165: (( L~ go) \/ ( L~ co)) is compact by COMPTS_1: 10;

          Wmin in (( L~ go) \/ ( L~ co)) by A63, A79, XBOOLE_0:def 3;

          then

           A166: ( W-min (( L~ go) \/ ( L~ co))) = Wmin by A156, A157, A165, JORDAN1J: 21, XBOOLE_1: 8;

          

           A167: (( W-min (( L~ go) \/ ( L~ co))) `1 ) = ( W-bound (( L~ go) \/ ( L~ co))) by EUCLID: 52;

          

           A168: (Wmin `1 ) = Wbo by EUCLID: 52;

          (Gik `1 ) >= Wbo by A13, A155, PSCOMP_1: 24;

          then (Gik `1 ) > Wbo by A77, XXREAL_0: 1;

          then ( W-min ((( L~ go) \/ ( L~ co)) \/ ( L~ pion1))) = ( W-min (( L~ go) \/ ( L~ co))) by A95, A165, A166, A167, A168, JORDAN1J: 33;

          then

           A169: ( W-min ( L~ godo)) = Wmin by A152, A166, XBOOLE_1: 4;

          

           A170: ( rng godo) c= ( L~ godo) by A98, A100, SPPOL_2: 18, XXREAL_0: 2;

          2 in ( dom godo) by A101, FINSEQ_3: 25;

          then

           A171: (godo /. 2) in ( rng godo) by PARTFUN2: 2;

          (godo /. 2) in ( W-most ( L~ ( Cage (C,n)))) by A164, JORDAN1I: 25;

          

          then ((godo /. 2) `1 ) = (( W-min ( L~ godo)) `1 ) by A169, PSCOMP_1: 31

          .= ( W-bound ( L~ godo)) by EUCLID: 52;

          then (godo /. 2) in ( W-most ( L~ godo)) by A170, A171, SPRECT_2: 12;

          then (( Rotate (godo,( W-min ( L~ godo)))) /. 2) in ( W-most ( L~ godo)) by A162, A169, 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

           A172: (US . ( len US)) = (US /. ( len US)) by PARTFUN1:def 6

          .= Emax by JORDAN1F: 7;

          

           A173: ( 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

             A174: p in ( east_halfline ( E-max C)) and

             A175: p in ( L~ go) by XBOOLE_0: 3;

            reconsider p as Point of ( TOP-REAL 2) by A174;

            p in ( L~ US) by A48, A175;

            then p in (( east_halfline ( E-max C)) /\ ( L~ ( Cage (C,n)))) by A154, A174, XBOOLE_0:def 4;

            then

             A176: (p `1 ) = Ebo by JORDAN1A: 83, PSCOMP_1: 50;

            then

             A177: p = Emax by A48, A175, JORDAN1J: 46;

            then Emax = Gij by A14, A172, A175, JORDAN1J: 43;

            then (Gij `1 ) = ((G * (( len G),k)) `1 ) by A6, A16, A20, A176, A177, JORDAN1A: 71;

            hence contradiction by A3, A17, A32, JORDAN1G: 7;

          end;

          now

            assume ( east_halfline ( E-max C)) meets ( L~ godo);

            then

             A178: ( east_halfline ( E-max C)) meets (( L~ go) \/ ( L~ pion1)) or ( east_halfline ( E-max C)) meets ( L~ co) by A152, XBOOLE_1: 70;

            per cases by A178, XBOOLE_1: 70;

              suppose ( east_halfline ( E-max C)) meets ( L~ go);

              hence contradiction by A173;

            end;

              suppose ( east_halfline ( E-max C)) meets ( L~ pion1);

              then

              consider p be object such that

               A179: p in ( east_halfline ( E-max C)) and

               A180: p in ( L~ pion1) by XBOOLE_0: 3;

              reconsider p as Point of ( TOP-REAL 2) by A179;

               A181:

              now

                per cases by A12, A87, A180, XBOOLE_0:def 3;

                  suppose p in poz;

                  hence (p `1 ) <= (Gij `1 ) by A91, A92, TOPREAL1: 3;

                end;

                  suppose p in pio;

                  hence (p `1 ) <= (Gij `1 ) by A91, GOBOARD7: 5;

                end;

              end;

              (i1 + 1) <= ( len G) by A3, NAT_1: 13;

              then i1 <= (( len G) - 1) by XREAL_1: 19;

              then

               A182: i1 <= (( len G) -' 1) by XREAL_0:def 2;

              (( len G) -' 1) <= ( len G) by NAT_D: 35;

              then (Gij `1 ) <= ((G * ((( len G) -' 1),1)) `1 ) by A4, A10, A15, A20, A24, A182, JORDAN1A: 18;

              then (p `1 ) <= ((G * ((( len G) -' 1),1)) `1 ) by A181, XXREAL_0: 2;

              then (p `1 ) <= ( E-bound C) by A24, JORDAN8: 12;

              then

               A183: (p `1 ) <= (( E-max C) `1 ) by EUCLID: 52;

              (p `1 ) >= (( E-max C) `1 ) by A179, TOPREAL1:def 11;

              then

               A184: (p `1 ) = (( E-max C) `1 ) by A183, XXREAL_0: 1;

              (p `2 ) = (( E-max C) `2 ) by A179, TOPREAL1:def 11;

              then p = ( E-max C) by A184, TOPREAL3: 6;

              hence contradiction by A9, A12, A87, A143, A180, XBOOLE_0: 3;

            end;

              suppose ( east_halfline ( E-max C)) meets ( L~ co);

              then

              consider p be object such that

               A185: p in ( east_halfline ( E-max C)) and

               A186: p in ( L~ co) by XBOOLE_0: 3;

              reconsider p as Point of ( TOP-REAL 2) by A185;

              p in ( L~ LS) by A55, A186;

              then p in (( east_halfline ( E-max C)) /\ ( L~ ( Cage (C,n)))) by A155, A185, XBOOLE_0:def 4;

              then

               A187: (p `1 ) = Ebo by JORDAN1A: 83, PSCOMP_1: 50;

              

               A188: (( E-max C) `2 ) = (p `2 ) by A185, TOPREAL1:def 11;

              set RC = ( Rotate (( Cage (C,n)),Emax));

              

               A189: ( E-max C) in ( right_cell (RC,1)) by JORDAN1I: 7;

              

               A190: (1 + 1) <= ( len LS) by A27, XXREAL_0: 2;

              LS = (RC -: Wmin) by JORDAN1G: 18;

              then

               A191: ( LSeg (LS,1)) = ( LSeg (RC,1)) by A190, SPPOL_2: 9;

              

               A192: ( L~ RC) = ( L~ ( Cage (C,n))) by REVROT_1: 33;

              

               A193: ( len RC) = ( len ( Cage (C,n))) by FINSEQ_6: 179;

              

               A194: ( GoB RC) = ( GoB ( Cage (C,n))) by REVROT_1: 28

              .= G by JORDAN1H: 44;

              

               A195: Emax in ( rng ( Cage (C,n))) by SPRECT_2: 46;

              

               A196: RC is_sequence_on G by A149, REVROT_1: 34;

              

               A197: (RC /. 1) = ( E-max ( L~ RC)) by A192, A195, FINSEQ_6: 92;

              consider ii,jj be Nat such that

               A198: [ii, (jj + 1)] in ( Indices G) and

               A199: [ii, jj] in ( Indices G) and

               A200: (RC /. 1) = (G * (ii,(jj + 1))) and

               A201: (RC /. (1 + 1)) = (G * (ii,jj)) by A96, A192, A193, A195, A196, FINSEQ_6: 92, JORDAN1I: 23;

              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 [( len G), jj2] in ( Indices G) by A202, A203, MATRIX_0: 30;

              then

               A206: ii = ( len G) by A192, A197, A198, A200, A204, GOBOARD1: 5;

              

               A207: 1 <= ii by A198, MATRIX_0: 32;

              

               A208: ii <= ( len G) by A198, MATRIX_0: 32;

              

               A209: 1 <= (jj + 1) by A198, MATRIX_0: 32;

              

               A210: (jj + 1) <= ( width G) by A198, MATRIX_0: 32;

              

               A211: 1 <= ii by A199, MATRIX_0: 32;

              

               A212: ii <= ( len G) by A199, MATRIX_0: 32;

              

               A213: 1 <= jj by A199, MATRIX_0: 32;

              

               A214: jj <= ( width G) by A199, MATRIX_0: 32;

              

               A215: (ii + 1) <> ii;

              ((jj + 1) + 1) <> jj;

              then

               A216: ( right_cell (RC,1)) = ( cell (G,(ii -' 1),jj)) by A96, A193, A194, A198, A199, A200, A201, A215, GOBOARD5:def 6;

              

               A217: ((ii -' 1) + 1) = ii by A207, XREAL_1: 235;

              (ii - 1) >= (4 - 1) by A205, A206, XREAL_1: 9;

              then

               A218: (ii - 1) >= 1 by XXREAL_0: 2;

              then

               A219: 1 <= (ii -' 1) by XREAL_0:def 2;

              

               A220: ((G * ((ii -' 1),jj)) `2 ) <= (p `2 ) by A188, A189, A208, A210, A213, A216, A217, A218, JORDAN9: 17;

              

               A221: (p `2 ) <= ((G * ((ii -' 1),(jj + 1))) `2 ) by A188, A189, A208, A210, A213, A216, A217, A218, JORDAN9: 17;

              

               A222: (ii -' 1) < ( len G) by A208, A217, NAT_1: 13;

              

              then

               A223: ((G * ((ii -' 1),jj)) `2 ) = ((G * (1,jj)) `2 ) by A213, A214, A219, GOBOARD5: 1

              .= ((G * (ii,jj)) `2 ) by A211, A212, A213, A214, GOBOARD5: 1;

              

               A224: ((G * ((ii -' 1),(jj + 1))) `2 ) = ((G * (1,(jj + 1))) `2 ) by A209, A210, A219, A222, GOBOARD5: 1

              .= ((G * (ii,(jj + 1))) `2 ) by A207, A208, A209, A210, GOBOARD5: 1;

              

               A225: ((G * (( len G),jj)) `1 ) = Ebo by A20, A213, A214, JORDAN1A: 71;

              Ebo = ((G * (( len G),(jj + 1))) `1 ) by A20, A209, A210, JORDAN1A: 71;

              then p in ( LSeg ((RC /. 1),(RC /. (1 + 1)))) by A187, A200, A201, A206, A220, A221, A223, A224, A225, GOBOARD7: 7;

              then

               A226: p in ( LSeg (LS,1)) by A96, A191, A193, TOPREAL1:def 3;

              

               A227: p in ( LSeg (co,( Index (p,co)))) by A186, JORDAN3: 9;

              

               A228: co = ( mid (LS,(Gik .. LS),( len LS))) by A39, JORDAN1J: 37;

              

               A229: 1 <= (Gik .. LS) by A39, FINSEQ_4: 21;

              

               A230: (Gik .. LS) <= ( len LS) by A39, FINSEQ_4: 21;

              (Gik .. LS) <> ( len LS) by A31, A39, FINSEQ_4: 19;

              then

               A231: (Gik .. LS) < ( len LS) by A230, XXREAL_0: 1;

              

               A232: 1 <= ( Index (p,co)) by A186, JORDAN3: 8;

              

               A233: ( Index (p,co)) < ( len co) by A186, JORDAN3: 8;

              

               A234: (( Index (Gik,LS)) + 1) = (Gik .. LS) by A34, A39, JORDAN1J: 56;

              consider t be Nat such that

               A235: t in ( dom LS) and

               A236: (LS . t) = Gik by A39, FINSEQ_2: 10;

              

               A237: 1 <= t by A235, FINSEQ_3: 25;

              

               A238: t <= ( len LS) by A235, FINSEQ_3: 25;

              1 < t by A34, A236, A237, XXREAL_0: 1;

              then (( Index (Gik,LS)) + 1) = t by A236, A238, JORDAN3: 12;

              then

               A239: ( len ( L_Cut (LS,Gik))) = (( len LS) - ( Index (Gik,LS))) by A13, A236, JORDAN3: 26;

              set tt = ((( Index (p,co)) + (Gik .. LS)) -' 1);

              

               A240: 1 <= ( Index (Gik,LS)) by A13, JORDAN3: 8;

              ( 0 + ( Index (Gik,LS))) < ( len LS) by A13, JORDAN3: 8;

              then

               A241: (( len LS) - ( Index (Gik,LS))) > 0 by XREAL_1: 20;

              ( Index (p,co)) < (( len LS) -' ( Index (Gik,LS))) by A233, A239, XREAL_0:def 2;

              then (( Index (p,co)) + 1) <= (( len LS) -' ( Index (Gik,LS))) by NAT_1: 13;

              then ( Index (p,co)) <= ((( len LS) -' ( Index (Gik,LS))) - 1) by XREAL_1: 19;

              then ( Index (p,co)) <= ((( len LS) - ( Index (Gik,LS))) - 1) by A241, XREAL_0:def 2;

              then ( Index (p,co)) <= (( len LS) - (Gik .. LS)) by A234;

              then ( Index (p,co)) <= (( len LS) -' (Gik .. LS)) by XREAL_0:def 2;

              then ( Index (p,co)) < ((( len LS) -' (Gik .. LS)) + 1) by NAT_1: 13;

              then

               A242: ( LSeg (( mid (LS,(Gik .. LS),( len LS))),( Index (p,co)))) = ( LSeg (LS,((( Index (p,co)) + (Gik .. LS)) -' 1))) by A229, A231, A232, JORDAN4: 19;

              

               A243: (1 + 1) <= (Gik .. LS) by A234, A240, XREAL_1: 7;

              then (( Index (p,co)) + (Gik .. LS)) >= ((1 + 1) + 1) by A232, XREAL_1: 7;

              then ((( Index (p,co)) + (Gik .. LS)) - 1) >= (((1 + 1) + 1) - 1) by XREAL_1: 9;

              then

               A244: tt >= (1 + 1) by XREAL_0:def 2;

              

               A245: 2 in ( dom LS) by A190, FINSEQ_3: 25;

              now

                per cases by A244, XXREAL_0: 1;

                  suppose tt > (1 + 1);

                  then ( LSeg (LS,1)) misses ( LSeg (LS,tt)) by TOPREAL1:def 7;

                  hence contradiction by A226, A227, A228, A242, XBOOLE_0: 3;

                end;

                  suppose

                   A246: tt = (1 + 1);

                  then (( LSeg (LS,1)) /\ ( LSeg (LS,tt))) = {(LS /. 2)} by A27, TOPREAL1:def 6;

                  then p in {(LS /. 2)} by A226, A227, A228, A242, XBOOLE_0:def 4;

                  then

                   A247: p = (LS /. 2) by TARSKI:def 1;

                  then

                   A248: (p .. LS) = 2 by A245, FINSEQ_5: 41;

                  (1 + 1) = ((( Index (p,co)) + (Gik .. LS)) - 1) by A246, XREAL_0:def 2;

                  then ((1 + 1) + 1) = (( Index (p,co)) + (Gik .. LS));

                  then

                   A249: (Gik .. LS) = 2 by A232, A243, JORDAN1E: 6;

                  p in ( rng LS) by A245, A247, PARTFUN2: 2;

                  then p = Gik by A39, A248, A249, FINSEQ_5: 9;

                  then (Gik `1 ) = Ebo by A247, JORDAN1G: 32;

                  then (Gik `1 ) = ((G * (( len G),j)) `1 ) by A4, A15, A20, JORDAN1A: 71;

                  hence contradiction by A2, A3, A18, 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

           A250: W is_a_component_of (( L~ godo) ` ) and

           A251: ( east_halfline ( E-max C)) c= W by GOBOARD9: 3;

           not W is bounded by A251, JORDAN2C: 121, RLTOPSP1: 42;

          then W is_outside_component_of ( L~ godo) by A250, JORDAN2C:def 3;

          then W c= ( UBD ( L~ godo)) by JORDAN2C: 23;

          then

           A252: ( east_halfline ( E-max C)) c= ( UBD ( L~ godo)) by A251;

          ( E-max C) in ( east_halfline ( E-max C)) by TOPREAL1: 38;

          then ( E-max C) in ( UBD ( L~ godo)) by A252;

          then ( E-max C) in ( LeftComp godo) by GOBRD14: 36;

          then UA meets ( L~ godo) by A141, A142, A143, A151, A161, JORDAN1J: 36;

          then

           A253: UA meets (( L~ go) \/ ( L~ pion1)) or UA meets ( L~ co) by A152, XBOOLE_1: 70;

          

           A254: UA c= C by JORDAN6: 61;

          now

            per cases by A253, XBOOLE_1: 70;

              suppose UA meets ( L~ go);

              then UA meets ( L~ ( Cage (C,n))) by A48, A154, XBOOLE_1: 1, XBOOLE_1: 63;

              hence contradiction by A254, JORDAN10: 5, XBOOLE_1: 63;

            end;

              suppose UA meets ( L~ pion1);

              hence contradiction by A9, A12, A87;

            end;

              suppose UA meets ( L~ co);

              then UA meets ( L~ ( Cage (C,n))) by A55, A155, XBOOLE_1: 1, XBOOLE_1: 63;

              hence contradiction by A254, JORDAN10: 5, XBOOLE_1: 63;

            end;

          end;

          hence contradiction;

        end;

          suppose (Gik `1 ) = (Gij `1 );

          then

           A255: i1 = i2 by A17, A18, JORDAN1G: 7;

          then poz = {Gi1k} by RLTOPSP1: 70;

          then poz c= pio by A83, ZFMISC_1: 31;

          then (pio \/ poz) = pio by XBOOLE_1: 12;

          hence contradiction by A1, A3, A4, A5, A6, A7, A8, A9, A255, Th12;

        end;

          suppose (Gik `2 ) = (Gij `2 );

          then

           A256: j = k by A17, A18, JORDAN1G: 6;

          then pio = {Gi1k} by RLTOPSP1: 70;

          then pio c= poz by A84, ZFMISC_1: 31;

          then (pio \/ poz) = poz by XBOOLE_1: 12;

          hence contradiction by A1, A2, A3, A4, A6, A7, A8, A9, A256, JORDAN15: 29;

        end;

      end;

      hence contradiction;

    end;

    theorem :: JORDAN19:23

    

     Th23: 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)) * (i1,j))} & ((( 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)) * (i2,k))} 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 * (i1,j))} and

       A8: ((pio \/ poz) /\ ( L~ LS)) = {(G * (i2,k))} and

       A9: (pio \/ poz) misses ( Lower_Arc C);

      set UA = ( 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 Gik = (G * (i2,k));

      set Gij = (G * (i1,j));

      set Gi1k = (G * (i1,k));

      

       A10: 1 < i1 by A1, A2, XXREAL_0: 2;

      

       A11: i2 < ( len G) by A2, A3, XXREAL_0: 2;

      

       A12: ( L~ <*Gij, Gi1k, Gik*>) = (poz \/ pio) by TOPREAL3: 16;

      Gik in {Gik} by TARSKI:def 1;

      then

       A13: Gik in ( L~ LS) by A8, XBOOLE_0:def 4;

      Gij in {Gij} by TARSKI:def 1;

      then

       A14: Gij in ( L~ US) by A7, XBOOLE_0:def 4;

      

       A15: j <= ( width G) by A5, A6, XXREAL_0: 2;

      

       A16: 1 <= k by A4, A5, XXREAL_0: 2;

      

       A17: [i1, j] in ( Indices G) by A3, A4, A10, A15, MATRIX_0: 30;

      

       A18: [i2, k] in ( Indices G) by A1, A6, A11, A16, MATRIX_0: 30;

      

       A19: [i1, k] in ( Indices G) by A3, A6, A10, A16, MATRIX_0: 30;

      set go = ( R_Cut (US,Gij));

      set co = ( L_Cut (LS,Gik));

      

       A20: ( len G) = ( width G) by JORDAN8:def 1;

      

       A21: ( len US) >= 3 by JORDAN1E: 15;

      then ( len US) >= 1 by XXREAL_0: 2;

      then 1 in ( dom US) by FINSEQ_3: 25;

      

      then

       A22: (US . 1) = (US /. 1) by PARTFUN1:def 6

      .= Wmin by JORDAN1F: 5;

      

       A23: (Wmin `1 ) = Wbo by EUCLID: 52

      .= ((G * (1,k)) `1 ) by A6, A16, A20, JORDAN1A: 73;

      ( len G) >= 4 by JORDAN8: 10;

      then

       A24: ( len G) >= 1 by XXREAL_0: 2;

      then

       A25: [1, k] in ( Indices G) by A6, A16, MATRIX_0: 30;

      then

       A26: Gij <> (US . 1) by A1, A2, A17, A22, A23, JORDAN1G: 7;

      then

      reconsider go as being_S-Seq FinSequence of ( TOP-REAL 2) by A14, JORDAN3: 35;

      

       A27: ( len LS) >= (1 + 2) by JORDAN1E: 15;

      then

       A28: ( len LS) >= 1 by XXREAL_0: 2;

      then

       A29: 1 in ( dom LS) by FINSEQ_3: 25;

      ( len LS) in ( dom LS) by A28, FINSEQ_3: 25;

      

      then

       A30: (LS . ( len LS)) = (LS /. ( len LS)) by PARTFUN1:def 6

      .= Wmin by JORDAN1F: 8;

      (Wmin `1 ) = Wbo by EUCLID: 52

      .= ((G * (1,k)) `1 ) by A6, A16, A20, JORDAN1A: 73;

      then

       A31: Gik <> (LS . ( len LS)) by A1, A18, A25, A30, JORDAN1G: 7;

      then

      reconsider co as being_S-Seq FinSequence of ( TOP-REAL 2) by A13, JORDAN3: 34;

      

       A32: [( len G), k] in ( Indices G) by A6, A16, A24, MATRIX_0: 30;

      

       A33: (LS . 1) = (LS /. 1) by A29, PARTFUN1:def 6

      .= Emax by JORDAN1F: 6;

      (Emax `1 ) = Ebo by EUCLID: 52

      .= ((G * (( len G),k)) `1 ) by A6, A16, A20, JORDAN1A: 71;

      then

       A34: Gik <> (LS . 1) by A2, A3, A18, A32, A33, JORDAN1G: 7;

      

       A35: ( len go) >= (1 + 1) by TOPREAL1:def 8;

      

       A36: Gij in ( rng US) by A3, A4, A10, A14, A15, JORDAN1G: 4, JORDAN1J: 40;

      then

       A37: go is_sequence_on G by JORDAN1G: 4, JORDAN1J: 38;

      

       A38: ( len co) >= (1 + 1) by TOPREAL1:def 8;

      

       A39: Gik in ( rng LS) by A1, A6, A11, A13, A16, JORDAN1G: 5, JORDAN1J: 40;

      then

       A40: co is_sequence_on G 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

      .= Gij by A14, 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

      .= Gik by A13, 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 A14, JORDAN3: 41;

      then ( LSeg (go,m)) c= ( L~ US) by A47;

      then

       A49: (( LSeg (go,m)) /\ ( L~ <*Gij, Gi1k, Gik*>)) c= {Gij} by A7, A12, XBOOLE_1: 26;

      m >= 1 by A35, XREAL_1: 19;

      then

       A50: ( LSeg (go,m)) = ( LSeg ((go /. m),Gij)) by A43, A45, TOPREAL1:def 3;

       {Gij} c= (( LSeg (go,m)) /\ ( L~ <*Gij, Gi1k, Gik*>))

      proof

        let x be object;

        assume x in {Gij};

        then

         A51: x = Gij by TARSKI:def 1;

        

         A52: Gij in ( LSeg (go,m)) by A50, RLTOPSP1: 68;

        Gij in ( LSeg (Gij,Gi1k)) by RLTOPSP1: 68;

        then Gij in (( LSeg (Gij,Gi1k)) \/ ( LSeg (Gi1k,Gik))) by XBOOLE_0:def 3;

        then Gij in ( L~ <*Gij, Gi1k, Gik*>) by SPRECT_1: 8;

        hence thesis by A51, A52, XBOOLE_0:def 4;

      end;

      then

       A53: (( LSeg (go,m)) /\ ( L~ <*Gij, Gi1k, Gik*>)) = {Gij} by A49;

      

       A54: ( LSeg (co,1)) c= ( L~ co) by TOPREAL3: 19;

      

       A55: ( L~ co) c= ( L~ LS) by A13, JORDAN3: 42;

      then ( LSeg (co,1)) c= ( L~ LS) by A54;

      then

       A56: (( LSeg (co,1)) /\ ( L~ <*Gij, Gi1k, Gik*>)) c= {Gik} by A8, A12, XBOOLE_1: 26;

      

       A57: ( LSeg (co,1)) = ( LSeg (Gik,(co /. (1 + 1)))) by A38, A44, TOPREAL1:def 3;

       {Gik} c= (( LSeg (co,1)) /\ ( L~ <*Gij, Gi1k, Gik*>))

      proof

        let x be object;

        assume x in {Gik};

        then

         A58: x = Gik by TARSKI:def 1;

        

         A59: Gik in ( LSeg (co,1)) by A57, RLTOPSP1: 68;

        Gik in ( LSeg (Gi1k,Gik)) by RLTOPSP1: 68;

        then Gik in (( LSeg (Gij,Gi1k)) \/ ( LSeg (Gi1k,Gik))) by XBOOLE_0:def 3;

        then Gik in ( L~ <*Gij, Gi1k, Gik*>) by SPRECT_1: 8;

        hence thesis by A58, A59, XBOOLE_0:def 4;

      end;

      then

       A60: (( L~ <*Gij, Gi1k, Gik*>) /\ ( LSeg (co,1))) = {Gik} by A56;

      

       A61: (go /. 1) = (US /. 1) by A14, SPRECT_3: 22

      .= Wmin by JORDAN1F: 5;

      

      then

       A62: (go /. 1) = (LS /. ( len LS)) by JORDAN1F: 8

      .= (co /. ( len co)) by A13, 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 A29, PARTFUN1:def 6

      .= Emax by JORDAN1F: 6;

      

       A69: [( len G), j] in ( Indices G) by A4, A15, A24, 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~ go) by XBOOLE_0:def 4;

        

         A72: x in ( L~ co) 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

         A73: x = Wmin or x = Emax by TARSKI:def 2;

        now

          assume x = Emax;

          then

           A74: Emax = Gik by A13, A68, A72, JORDAN1E: 7;

          ((G * (( len G),j)) `1 ) = Ebo by A4, A15, A20, JORDAN1A: 71;

          then (Emax `1 ) <> Ebo by A2, A3, A18, A69, A74, JORDAN1G: 7;

          hence contradiction by EUCLID: 52;

        end;

        hence thesis by A61, A73, TARSKI:def 1;

      end;

      then

       A75: (( L~ go) /\ ( L~ co)) = {(go /. 1)} by A65;

      set W2 = (go /. 2);

      

       A76: 2 in ( dom go) by A35, FINSEQ_3: 25;

       A77:

      now

        assume (Gik `1 ) = Wbo;

        then ((G * (1,k)) `1 ) = ((G * (i2,k)) `1 ) by A6, A16, A20, JORDAN1A: 73;

        hence contradiction by A1, A18, A25, JORDAN1G: 7;

      end;

      go = ( mid (US,1,(Gij .. US))) by A36, JORDAN1G: 49

      .= (US | (Gij .. US)) by A36, FINSEQ_4: 21, FINSEQ_6: 116;

      then

       A78: W2 = (US /. 2) by A76, FINSEQ_4: 70;

      

       A79: Wmin in ( rng go) by A61, FINSEQ_6: 42;

      set pion = <*Gij, Gi1k, Gik*>;

       A80:

      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 A17, A18, A19, FINSEQ_4: 18;

      end;

      

       A81: (Gi1k `1 ) = ((G * (i1,1)) `1 ) by A3, A6, A10, A16, GOBOARD5: 2

      .= (Gij `1 ) by A3, A4, A10, A15, GOBOARD5: 2;

      (Gi1k `2 ) = ((G * (1,k)) `2 ) by A3, A6, A10, A16, GOBOARD5: 1

      .= (Gik `2 ) by A1, A6, A11, A16, GOBOARD5: 1;

      then

       A82: Gi1k = |[(Gij `1 ), (Gik `2 )]| by A81, EUCLID: 53;

      

       A83: Gi1k in pio by RLTOPSP1: 68;

      

       A84: Gi1k in poz by RLTOPSP1: 68;

      now

        per cases ;

          suppose (Gik `1 ) <> (Gij `1 ) & (Gik `2 ) <> (Gij `2 );

          then pion is being_S-Seq by A82, TOPREAL3: 34;

          then

          consider pion1 be FinSequence of ( TOP-REAL 2) such that

           A85: pion1 is_sequence_on G and

           A86: pion1 is being_S-Seq and

           A87: ( L~ pion) = ( L~ pion1) and

           A88: (pion /. 1) = (pion1 /. 1) and

           A89: (pion /. ( len pion)) = (pion1 /. ( len pion1)) and

           A90: ( len pion) <= ( len pion1) by A80, GOBOARD3: 2;

          reconsider pion1 as being_S-Seq FinSequence of ( TOP-REAL 2) by A86;

          set godo = ((go ^' pion1) ^' co);

          

           A91: (Gi1k `1 ) = ((G * (i1,1)) `1 ) by A3, A6, A10, A16, GOBOARD5: 2

          .= (Gij `1 ) by A3, A4, A10, A15, GOBOARD5: 2;

          

           A92: (Gik `1 ) <= (Gi1k `1 ) by A1, A2, A3, A6, A16, JORDAN1A: 18;

          then

           A93: ( W-bound poz) = (Gik `1 ) by SPRECT_1: 54;

          

           A94: ( W-bound pio) = (Gij `1 ) by A91, SPRECT_1: 54;

          ( W-bound (poz \/ pio)) = ( min (( W-bound poz),( W-bound pio))) by SPRECT_1: 47

          .= (Gik `1 ) by A91, A92, A93, A94, XXREAL_0:def 9;

          then

           A95: ( W-bound ( L~ pion1)) = (Gik `1 ) by A87, TOPREAL3: 16;

          

           A96: (1 + 1) <= ( len ( Cage (C,n))) by GOBOARD7: 34, XXREAL_0: 2;

          

           A97: (1 + 1) <= ( len ( Rotate (( Cage (C,n)),Wmin))) by GOBOARD7: 34, XXREAL_0: 2;

          ( len (go ^' pion1)) >= ( len go) by TOPREAL8: 7;

          then

           A98: ( len (go ^' pion1)) >= (1 + 1) by A35, XXREAL_0: 2;

          then

           A99: ( len (go ^' pion1)) > (1 + 0 ) by NAT_1: 13;

          

           A100: ( len godo) >= ( len (go ^' pion1)) by TOPREAL8: 7;

          then

           A101: (1 + 1) <= ( len godo) by A98, XXREAL_0: 2;

          

           A102: US is_sequence_on G by JORDAN1G: 4;

          

           A103: (go /. ( len go)) = (pion1 /. 1) by A43, A88, FINSEQ_4: 18;

          then

           A104: (go ^' pion1) is_sequence_on G by A37, A85, TOPREAL8: 12;

          

           A105: ((go ^' pion1) /. ( len (go ^' pion1))) = (pion /. ( len pion)) by A89, FINSEQ_6: 156

          .= (pion /. 3) by FINSEQ_1: 45

          .= (co /. 1) by A44, FINSEQ_4: 18;

          then

           A106: godo is_sequence_on G by A40, A104, TOPREAL8: 12;

          ( LSeg (pion1,1)) c= ( L~ pion) by A87, TOPREAL3: 19;

          then

           A107: (( LSeg (go,(( len go) -' 1))) /\ ( LSeg (pion1,1))) c= {Gij} by A46, A53, XBOOLE_1: 27;

          ( len pion1) >= (2 + 1) by A90, FINSEQ_1: 45;

          then

           A108: ( len pion1) > (1 + 1) by NAT_1: 13;

           {Gij} c= (( LSeg (go,m)) /\ ( LSeg (pion1,1)))

          proof

            let x be object;

            assume x in {Gij};

            then

             A109: x = Gij by TARSKI:def 1;

            

             A110: Gij in ( LSeg (go,m)) by A50, RLTOPSP1: 68;

            Gij in ( LSeg (pion1,1)) by A43, A103, A108, TOPREAL1: 21;

            hence thesis by A109, A110, XBOOLE_0:def 4;

          end;

          then (( LSeg (go,(( len go) -' 1))) /\ ( LSeg (pion1,1))) = {(go /. ( len go))} by A43, A46, A107;

          then

           A111: (go ^' pion1) is unfolded by A103, TOPREAL8: 34;

          ( len pion1) >= (2 + 1) by A90, FINSEQ_1: 45;

          then

           A112: (( 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 A112, XREAL_0:def 2;

          then

           A113: (( len (go ^' pion1)) -' 1) = (( len go) + (( len pion1) -' 2)) by XREAL_0:def 2;

          

           A114: (( len pion1) - 1) >= 1 by A108, XREAL_1: 19;

          then

           A115: (( len pion1) -' 1) = (( len pion1) - 1) by XREAL_0:def 2;

          

           A116: ((( len pion1) -' 2) + 1) = ((( len pion1) - 2) + 1) by A112, XREAL_0:def 2

          .= (( len pion1) -' 1) by A114, XREAL_0:def 2;

          ((( len pion1) - 1) + 1) <= ( len pion1);

          then

           A117: (( len pion1) -' 1) < ( len pion1) by A115, NAT_1: 13;

          ( LSeg (pion1,(( len pion1) -' 1))) c= ( L~ pion) by A87, TOPREAL3: 19;

          then

           A118: (( LSeg (pion1,(( len pion1) -' 1))) /\ ( LSeg (co,1))) c= {Gik} by A60, XBOOLE_1: 27;

           {Gik} c= (( LSeg (pion1,(( len pion1) -' 1))) /\ ( LSeg (co,1)))

          proof

            let x be object;

            assume x in {Gik};

            then

             A119: x = Gik by TARSKI:def 1;

            

             A120: Gik in ( LSeg (co,1)) by A57, RLTOPSP1: 68;

            (pion1 /. ((( len pion1) -' 1) + 1)) = (pion /. 3) by A89, A115, FINSEQ_1: 45

            .= Gik by FINSEQ_4: 18;

            then Gik in ( LSeg (pion1,(( len pion1) -' 1))) by A114, A115, TOPREAL1: 21;

            hence thesis by A119, A120, XBOOLE_0:def 4;

          end;

          then (( LSeg (pion1,(( len pion1) -' 1))) /\ ( LSeg (co,1))) = {Gik} by A118;

          then

           A121: (( LSeg ((go ^' pion1),(( len go) + (( len pion1) -' 2)))) /\ ( LSeg (co,1))) = {((go ^' pion1) /. ( len (go ^' pion1)))} by A44, A103, A105, A116, A117, TOPREAL8: 31;

          

           A122: (go ^' pion1) is non trivial by A98, NAT_D: 60;

          

           A123: ( rng pion1) c= ( L~ pion1) by A108, SPPOL_2: 18;

          

           A124: {(pion1 /. 1)} c= (( L~ go) /\ ( L~ pion1))

          proof

            let x be object;

            assume x in {(pion1 /. 1)};

            then

             A125: x = (pion1 /. 1) by TARSKI:def 1;

            then

             A126: x in ( rng go) by A103, FINSEQ_6: 168;

            x in ( rng pion1) by A125, FINSEQ_6: 42;

            hence thesis by A63, A123, A126, XBOOLE_0:def 4;

          end;

          (( L~ go) /\ ( L~ pion1)) c= {(pion1 /. 1)}

          proof

            let x be object;

            assume

             A127: x in (( L~ go) /\ ( L~ pion1));

            then

             A128: x in ( L~ go) by XBOOLE_0:def 4;

            x in ( L~ pion1) by A127, XBOOLE_0:def 4;

            hence thesis by A7, A12, A43, A48, A87, A103, A128, XBOOLE_0:def 4;

          end;

          then

           A129: (( L~ go) /\ ( L~ pion1)) = {(pion1 /. 1)} by A124;

          then

           A130: (go ^' pion1) is s.n.c. by A103, JORDAN1J: 54;

          (( rng go) /\ ( rng pion1)) c= {(pion1 /. 1)} by A63, A123, A129, XBOOLE_1: 27;

          then

           A131: (go ^' pion1) is one-to-one by JORDAN1J: 55;

          

           A132: (pion /. ( len pion)) = (pion /. 3) by FINSEQ_1: 45

          .= (co /. 1) by A44, FINSEQ_4: 18;

          

           A133: {(pion1 /. ( len pion1))} c= (( L~ co) /\ ( L~ pion1))

          proof

            let x be object;

            assume x in {(pion1 /. ( len pion1))};

            then

             A134: x = (pion1 /. ( len pion1)) by TARSKI:def 1;

            then

             A135: x in ( rng co) by A89, A132, FINSEQ_6: 42;

            x in ( rng pion1) by A134, FINSEQ_6: 168;

            hence thesis by A64, A123, A135, XBOOLE_0:def 4;

          end;

          (( L~ co) /\ ( L~ pion1)) c= {(pion1 /. ( len pion1))}

          proof

            let x be object;

            assume

             A136: x in (( L~ co) /\ ( L~ pion1));

            then

             A137: x in ( L~ co) by XBOOLE_0:def 4;

            x in ( L~ pion1) by A136, XBOOLE_0:def 4;

            hence thesis by A8, A12, A44, A55, A87, A89, A132, A137, XBOOLE_0:def 4;

          end;

          then

           A138: (( L~ co) /\ ( L~ pion1)) = {(pion1 /. ( len pion1))} by A133;

          

           A139: (( L~ (go ^' pion1)) /\ ( L~ co)) = ((( L~ go) \/ ( L~ pion1)) /\ ( L~ co)) by A103, TOPREAL8: 35

          .= ( {(go /. 1)} \/ {(co /. 1)}) by A75, A89, A132, A138, 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 A101, A105, A106, A111, A113, A121, A122, A130, A131, A139, JORDAN8: 4, JORDAN8: 5, TOPREAL8: 11, TOPREAL8: 33, TOPREAL8: 34;

          

           A140: UA is_an_arc_of (( E-max C),( W-min C)) by JORDAN6:def 9;

          then

           A141: UA is connected by JORDAN6: 10;

          

           A142: ( W-min C) in UA by A140, TOPREAL1: 1;

          

           A143: ( E-max C) in UA by A140, TOPREAL1: 1;

          set ff = ( Rotate (( Cage (C,n)),Wmin));

          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:

          now

            assume

             A148: (Gij .. US) <= 1;

            (Gij .. US) >= 1 by A36, FINSEQ_4: 21;

            then (Gij .. US) = 1 by A148, XXREAL_0: 1;

            then Gij = (US /. 1) by A36, FINSEQ_5: 38;

            hence contradiction by A22, A26, JORDAN1F: 5;

          end;

          

           A149: ( Cage (C,n)) is_sequence_on G by JORDAN9:def 1;

          then

           A150: ff is_sequence_on G by REVROT_1: 34;

          

           A151: (( right_cell (godo,1,G)) \ ( L~ godo)) c= ( RightComp godo) by A101, A106, JORDAN9: 27;

          

           A152: ( L~ godo) = (( L~ (go ^' pion1)) \/ ( L~ co)) by A105, TOPREAL8: 35

          .= ((( L~ go) \/ ( L~ pion1)) \/ ( L~ co)) by A103, TOPREAL8: 35;

          

           A153: ( L~ ( Cage (C,n))) = (( L~ US) \/ ( L~ LS)) by JORDAN1E: 13;

          then

           A154: ( L~ US) c= ( L~ ( Cage (C,n))) by XBOOLE_1: 7;

          

           A155: ( L~ LS) c= ( L~ ( Cage (C,n))) by A153, XBOOLE_1: 7;

          

           A156: ( L~ go) c= ( L~ ( Cage (C,n))) by A48, A154;

          

           A157: ( L~ co) c= ( L~ ( Cage (C,n))) by A55, A155;

          

           A158: ( W-min C) in C by SPRECT_1: 13;

           A159:

          now

            assume ( W-min C) in ( L~ godo);

            then

             A160: ( W-min C) in (( L~ go) \/ ( L~ pion1)) or ( W-min C) in ( L~ co) by A152, XBOOLE_0:def 3;

            per cases by A160, XBOOLE_0:def 3;

              suppose ( W-min C) in ( L~ go);

              then C meets ( L~ ( Cage (C,n))) by A156, A158, XBOOLE_0: 3;

              hence contradiction by JORDAN10: 5;

            end;

              suppose ( W-min C) in ( L~ pion1);

              hence contradiction by A9, A12, A87, A142, XBOOLE_0: 3;

            end;

              suppose ( W-min C) in ( L~ co);

              then C meets ( L~ ( Cage (C,n))) by A157, A158, 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 A97, 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, A150, JORDAN1J: 53

          .= ( right_cell (US,1,G)) by JORDAN1E:def 1

          .= ( right_cell (( R_Cut (US,Gij)),1,G)) by A36, A102, A147, JORDAN1J: 52

          .= ( right_cell ((go ^' pion1),1,G)) by A41, A104, JORDAN1J: 51

          .= ( right_cell (godo,1,G)) by A99, A106, JORDAN1J: 51;

          then ( W-min C) in ( right_cell (godo,1,G)) by JORDAN1I: 6;

          then

           A161: ( W-min C) in (( right_cell (godo,1,G)) \ ( L~ godo)) by A159, XBOOLE_0:def 5;

          

           A162: (godo /. 1) = ((go ^' pion1) /. 1) by FINSEQ_6: 155

          .= Wmin by A61, FINSEQ_6: 155;

          

           A163: ( len US) >= 2 by A21, XXREAL_0: 2;

          

           A164: (godo /. 2) = ((go ^' pion1) /. 2) by A98, FINSEQ_6: 159

          .= (US /. 2) by A35, A78, FINSEQ_6: 159

          .= ((US ^' LS) /. 2) by A163, FINSEQ_6: 159

          .= (( Rotate (( Cage (C,n)),Wmin)) /. 2) by JORDAN1E: 11;

          

           A165: (( L~ go) \/ ( L~ co)) is compact by COMPTS_1: 10;

          Wmin in (( L~ go) \/ ( L~ co)) by A63, A79, XBOOLE_0:def 3;

          then

           A166: ( W-min (( L~ go) \/ ( L~ co))) = Wmin by A156, A157, A165, JORDAN1J: 21, XBOOLE_1: 8;

          

           A167: (( W-min (( L~ go) \/ ( L~ co))) `1 ) = ( W-bound (( L~ go) \/ ( L~ co))) by EUCLID: 52;

          

           A168: (Wmin `1 ) = Wbo by EUCLID: 52;

          (Gik `1 ) >= Wbo by A13, A155, PSCOMP_1: 24;

          then (Gik `1 ) > Wbo by A77, XXREAL_0: 1;

          then ( W-min ((( L~ go) \/ ( L~ co)) \/ ( L~ pion1))) = ( W-min (( L~ go) \/ ( L~ co))) by A95, A165, A166, A167, A168, JORDAN1J: 33;

          then

           A169: ( W-min ( L~ godo)) = Wmin by A152, A166, XBOOLE_1: 4;

          

           A170: ( rng godo) c= ( L~ godo) by A98, A100, SPPOL_2: 18, XXREAL_0: 2;

          2 in ( dom godo) by A101, FINSEQ_3: 25;

          then

           A171: (godo /. 2) in ( rng godo) by PARTFUN2: 2;

          (godo /. 2) in ( W-most ( L~ ( Cage (C,n)))) by A164, JORDAN1I: 25;

          

          then ((godo /. 2) `1 ) = (( W-min ( L~ godo)) `1 ) by A169, PSCOMP_1: 31

          .= ( W-bound ( L~ godo)) by EUCLID: 52;

          then (godo /. 2) in ( W-most ( L~ godo)) by A170, A171, SPRECT_2: 12;

          then (( Rotate (godo,( W-min ( L~ godo)))) /. 2) in ( W-most ( L~ godo)) by A162, A169, 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

           A172: (US . ( len US)) = (US /. ( len US)) by PARTFUN1:def 6

          .= Emax by JORDAN1F: 7;

          

           A173: ( 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

             A174: p in ( east_halfline ( E-max C)) and

             A175: p in ( L~ go) by XBOOLE_0: 3;

            reconsider p as Point of ( TOP-REAL 2) by A174;

            p in ( L~ US) by A48, A175;

            then p in (( east_halfline ( E-max C)) /\ ( L~ ( Cage (C,n)))) by A154, A174, XBOOLE_0:def 4;

            then

             A176: (p `1 ) = Ebo by JORDAN1A: 83, PSCOMP_1: 50;

            then

             A177: p = Emax by A48, A175, JORDAN1J: 46;

            then Emax = Gij by A14, A172, A175, JORDAN1J: 43;

            then (Gij `1 ) = ((G * (( len G),k)) `1 ) by A6, A16, A20, A176, A177, JORDAN1A: 71;

            hence contradiction by A3, A17, A32, JORDAN1G: 7;

          end;

          now

            assume ( east_halfline ( E-max C)) meets ( L~ godo);

            then

             A178: ( east_halfline ( E-max C)) meets (( L~ go) \/ ( L~ pion1)) or ( east_halfline ( E-max C)) meets ( L~ co) by A152, XBOOLE_1: 70;

            per cases by A178, XBOOLE_1: 70;

              suppose ( east_halfline ( E-max C)) meets ( L~ go);

              hence contradiction by A173;

            end;

              suppose ( east_halfline ( E-max C)) meets ( L~ pion1);

              then

              consider p be object such that

               A179: p in ( east_halfline ( E-max C)) and

               A180: p in ( L~ pion1) by XBOOLE_0: 3;

              reconsider p as Point of ( TOP-REAL 2) by A179;

               A181:

              now

                per cases by A12, A87, A180, XBOOLE_0:def 3;

                  suppose p in poz;

                  hence (p `1 ) <= (Gij `1 ) by A91, A92, TOPREAL1: 3;

                end;

                  suppose p in pio;

                  hence (p `1 ) <= (Gij `1 ) by A91, GOBOARD7: 5;

                end;

              end;

              (i1 + 1) <= ( len G) by A3, NAT_1: 13;

              then i1 <= (( len G) - 1) by XREAL_1: 19;

              then

               A182: i1 <= (( len G) -' 1) by XREAL_0:def 2;

              (( len G) -' 1) <= ( len G) by NAT_D: 35;

              then (Gij `1 ) <= ((G * ((( len G) -' 1),1)) `1 ) by A4, A10, A15, A20, A24, A182, JORDAN1A: 18;

              then (p `1 ) <= ((G * ((( len G) -' 1),1)) `1 ) by A181, XXREAL_0: 2;

              then (p `1 ) <= ( E-bound C) by A24, JORDAN8: 12;

              then

               A183: (p `1 ) <= (( E-max C) `1 ) by EUCLID: 52;

              (p `1 ) >= (( E-max C) `1 ) by A179, TOPREAL1:def 11;

              then

               A184: (p `1 ) = (( E-max C) `1 ) by A183, XXREAL_0: 1;

              (p `2 ) = (( E-max C) `2 ) by A179, TOPREAL1:def 11;

              then p = ( E-max C) by A184, TOPREAL3: 6;

              hence contradiction by A9, A12, A87, A143, A180, XBOOLE_0: 3;

            end;

              suppose ( east_halfline ( E-max C)) meets ( L~ co);

              then

              consider p be object such that

               A185: p in ( east_halfline ( E-max C)) and

               A186: p in ( L~ co) by XBOOLE_0: 3;

              reconsider p as Point of ( TOP-REAL 2) by A185;

              p in ( L~ LS) by A55, A186;

              then p in (( east_halfline ( E-max C)) /\ ( L~ ( Cage (C,n)))) by A155, A185, XBOOLE_0:def 4;

              then

               A187: (p `1 ) = Ebo by JORDAN1A: 83, PSCOMP_1: 50;

              

               A188: (( E-max C) `2 ) = (p `2 ) by A185, TOPREAL1:def 11;

              set RC = ( Rotate (( Cage (C,n)),Emax));

              

               A189: ( E-max C) in ( right_cell (RC,1)) by JORDAN1I: 7;

              

               A190: (1 + 1) <= ( len LS) by A27, XXREAL_0: 2;

              LS = (RC -: Wmin) by JORDAN1G: 18;

              then

               A191: ( LSeg (LS,1)) = ( LSeg (RC,1)) by A190, SPPOL_2: 9;

              

               A192: ( L~ RC) = ( L~ ( Cage (C,n))) by REVROT_1: 33;

              

               A193: ( len RC) = ( len ( Cage (C,n))) by FINSEQ_6: 179;

              

               A194: ( GoB RC) = ( GoB ( Cage (C,n))) by REVROT_1: 28

              .= G by JORDAN1H: 44;

              

               A195: Emax in ( rng ( Cage (C,n))) by SPRECT_2: 46;

              

               A196: RC is_sequence_on G by A149, REVROT_1: 34;

              

               A197: (RC /. 1) = ( E-max ( L~ RC)) by A192, A195, FINSEQ_6: 92;

              consider ii,jj be Nat such that

               A198: [ii, (jj + 1)] in ( Indices G) and

               A199: [ii, jj] in ( Indices G) and

               A200: (RC /. 1) = (G * (ii,(jj + 1))) and

               A201: (RC /. (1 + 1)) = (G * (ii,jj)) by A96, A192, A193, A195, A196, FINSEQ_6: 92, JORDAN1I: 23;

              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 [( len G), jj2] in ( Indices G) by A202, A203, MATRIX_0: 30;

              then

               A206: ii = ( len G) by A192, A197, A198, A200, A204, GOBOARD1: 5;

              

               A207: 1 <= ii by A198, MATRIX_0: 32;

              

               A208: ii <= ( len G) by A198, MATRIX_0: 32;

              

               A209: 1 <= (jj + 1) by A198, MATRIX_0: 32;

              

               A210: (jj + 1) <= ( width G) by A198, MATRIX_0: 32;

              

               A211: 1 <= ii by A199, MATRIX_0: 32;

              

               A212: ii <= ( len G) by A199, MATRIX_0: 32;

              

               A213: 1 <= jj by A199, MATRIX_0: 32;

              

               A214: jj <= ( width G) by A199, MATRIX_0: 32;

              

               A215: (ii + 1) <> ii;

              ((jj + 1) + 1) <> jj;

              then

               A216: ( right_cell (RC,1)) = ( cell (G,(ii -' 1),jj)) by A96, A193, A194, A198, A199, A200, A201, A215, GOBOARD5:def 6;

              

               A217: ((ii -' 1) + 1) = ii by A207, XREAL_1: 235;

              (ii - 1) >= (4 - 1) by A205, A206, XREAL_1: 9;

              then

               A218: (ii - 1) >= 1 by XXREAL_0: 2;

              then

               A219: 1 <= (ii -' 1) by XREAL_0:def 2;

              

               A220: ((G * ((ii -' 1),jj)) `2 ) <= (p `2 ) by A188, A189, A208, A210, A213, A216, A217, A218, JORDAN9: 17;

              

               A221: (p `2 ) <= ((G * ((ii -' 1),(jj + 1))) `2 ) by A188, A189, A208, A210, A213, A216, A217, A218, JORDAN9: 17;

              

               A222: (ii -' 1) < ( len G) by A208, A217, NAT_1: 13;

              

              then

               A223: ((G * ((ii -' 1),jj)) `2 ) = ((G * (1,jj)) `2 ) by A213, A214, A219, GOBOARD5: 1

              .= ((G * (ii,jj)) `2 ) by A211, A212, A213, A214, GOBOARD5: 1;

              

               A224: ((G * ((ii -' 1),(jj + 1))) `2 ) = ((G * (1,(jj + 1))) `2 ) by A209, A210, A219, A222, GOBOARD5: 1

              .= ((G * (ii,(jj + 1))) `2 ) by A207, A208, A209, A210, GOBOARD5: 1;

              

               A225: ((G * (( len G),jj)) `1 ) = Ebo by A20, A213, A214, JORDAN1A: 71;

              Ebo = ((G * (( len G),(jj + 1))) `1 ) by A20, A209, A210, JORDAN1A: 71;

              then p in ( LSeg ((RC /. 1),(RC /. (1 + 1)))) by A187, A200, A201, A206, A220, A221, A223, A224, A225, GOBOARD7: 7;

              then

               A226: p in ( LSeg (LS,1)) by A96, A191, A193, TOPREAL1:def 3;

              

               A227: p in ( LSeg (co,( Index (p,co)))) by A186, JORDAN3: 9;

              

               A228: co = ( mid (LS,(Gik .. LS),( len LS))) by A39, JORDAN1J: 37;

              

               A229: 1 <= (Gik .. LS) by A39, FINSEQ_4: 21;

              

               A230: (Gik .. LS) <= ( len LS) by A39, FINSEQ_4: 21;

              (Gik .. LS) <> ( len LS) by A31, A39, FINSEQ_4: 19;

              then

               A231: (Gik .. LS) < ( len LS) by A230, XXREAL_0: 1;

              

               A232: 1 <= ( Index (p,co)) by A186, JORDAN3: 8;

              

               A233: ( Index (p,co)) < ( len co) by A186, JORDAN3: 8;

              

               A234: (( Index (Gik,LS)) + 1) = (Gik .. LS) by A34, A39, JORDAN1J: 56;

              consider t be Nat such that

               A235: t in ( dom LS) and

               A236: (LS . t) = Gik by A39, FINSEQ_2: 10;

              

               A237: 1 <= t by A235, FINSEQ_3: 25;

              

               A238: t <= ( len LS) by A235, FINSEQ_3: 25;

              1 < t by A34, A236, A237, XXREAL_0: 1;

              then (( Index (Gik,LS)) + 1) = t by A236, A238, JORDAN3: 12;

              then

               A239: ( len ( L_Cut (LS,Gik))) = (( len LS) - ( Index (Gik,LS))) by A13, A236, JORDAN3: 26;

              set tt = ((( Index (p,co)) + (Gik .. LS)) -' 1);

              

               A240: 1 <= ( Index (Gik,LS)) by A13, JORDAN3: 8;

              ( 0 + ( Index (Gik,LS))) < ( len LS) by A13, JORDAN3: 8;

              then

               A241: (( len LS) - ( Index (Gik,LS))) > 0 by XREAL_1: 20;

              ( Index (p,co)) < (( len LS) -' ( Index (Gik,LS))) by A233, A239, XREAL_0:def 2;

              then (( Index (p,co)) + 1) <= (( len LS) -' ( Index (Gik,LS))) by NAT_1: 13;

              then ( Index (p,co)) <= ((( len LS) -' ( Index (Gik,LS))) - 1) by XREAL_1: 19;

              then ( Index (p,co)) <= ((( len LS) - ( Index (Gik,LS))) - 1) by A241, XREAL_0:def 2;

              then ( Index (p,co)) <= (( len LS) - (Gik .. LS)) by A234;

              then ( Index (p,co)) <= (( len LS) -' (Gik .. LS)) by XREAL_0:def 2;

              then ( Index (p,co)) < ((( len LS) -' (Gik .. LS)) + 1) by NAT_1: 13;

              then

               A242: ( LSeg (( mid (LS,(Gik .. LS),( len LS))),( Index (p,co)))) = ( LSeg (LS,((( Index (p,co)) + (Gik .. LS)) -' 1))) by A229, A231, A232, JORDAN4: 19;

              

               A243: (1 + 1) <= (Gik .. LS) by A234, A240, XREAL_1: 7;

              then (( Index (p,co)) + (Gik .. LS)) >= ((1 + 1) + 1) by A232, XREAL_1: 7;

              then ((( Index (p,co)) + (Gik .. LS)) - 1) >= (((1 + 1) + 1) - 1) by XREAL_1: 9;

              then

               A244: tt >= (1 + 1) by XREAL_0:def 2;

              

               A245: 2 in ( dom LS) by A190, FINSEQ_3: 25;

              now

                per cases by A244, XXREAL_0: 1;

                  suppose tt > (1 + 1);

                  then ( LSeg (LS,1)) misses ( LSeg (LS,tt)) by TOPREAL1:def 7;

                  hence contradiction by A226, A227, A228, A242, XBOOLE_0: 3;

                end;

                  suppose

                   A246: tt = (1 + 1);

                  then (( LSeg (LS,1)) /\ ( LSeg (LS,tt))) = {(LS /. 2)} by A27, TOPREAL1:def 6;

                  then p in {(LS /. 2)} by A226, A227, A228, A242, XBOOLE_0:def 4;

                  then

                   A247: p = (LS /. 2) by TARSKI:def 1;

                  then

                   A248: (p .. LS) = 2 by A245, FINSEQ_5: 41;

                  (1 + 1) = ((( Index (p,co)) + (Gik .. LS)) - 1) by A246, XREAL_0:def 2;

                  then ((1 + 1) + 1) = (( Index (p,co)) + (Gik .. LS));

                  then

                   A249: (Gik .. LS) = 2 by A232, A243, JORDAN1E: 6;

                  p in ( rng LS) by A245, A247, PARTFUN2: 2;

                  then p = Gik by A39, A248, A249, FINSEQ_5: 9;

                  then (Gik `1 ) = Ebo by A247, JORDAN1G: 32;

                  then (Gik `1 ) = ((G * (( len G),j)) `1 ) by A4, A15, A20, JORDAN1A: 71;

                  hence contradiction by A2, A3, A18, 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

           A250: W is_a_component_of (( L~ godo) ` ) and

           A251: ( east_halfline ( E-max C)) c= W by GOBOARD9: 3;

           not W is bounded by A251, JORDAN2C: 121, RLTOPSP1: 42;

          then W is_outside_component_of ( L~ godo) by A250, JORDAN2C:def 3;

          then W c= ( UBD ( L~ godo)) by JORDAN2C: 23;

          then

           A252: ( east_halfline ( E-max C)) c= ( UBD ( L~ godo)) by A251;

          ( E-max C) in ( east_halfline ( E-max C)) by TOPREAL1: 38;

          then ( E-max C) in ( UBD ( L~ godo)) by A252;

          then ( E-max C) in ( LeftComp godo) by GOBRD14: 36;

          then UA meets ( L~ godo) by A141, A142, A143, A151, A161, JORDAN1J: 36;

          then

           A253: UA meets (( L~ go) \/ ( L~ pion1)) or UA meets ( L~ co) by A152, XBOOLE_1: 70;

          

           A254: UA c= C by JORDAN6: 61;

          now

            per cases by A253, XBOOLE_1: 70;

              suppose UA meets ( L~ go);

              then UA meets ( L~ ( Cage (C,n))) by A48, A154, XBOOLE_1: 1, XBOOLE_1: 63;

              hence contradiction by A254, JORDAN10: 5, XBOOLE_1: 63;

            end;

              suppose UA meets ( L~ pion1);

              hence contradiction by A9, A12, A87;

            end;

              suppose UA meets ( L~ co);

              then UA meets ( L~ ( Cage (C,n))) by A55, A155, XBOOLE_1: 1, XBOOLE_1: 63;

              hence contradiction by A254, JORDAN10: 5, XBOOLE_1: 63;

            end;

          end;

          hence contradiction;

        end;

          suppose (Gik `1 ) = (Gij `1 );

          then

           A255: i1 = i2 by A17, A18, JORDAN1G: 7;

          then poz = {Gi1k} by RLTOPSP1: 70;

          then poz c= pio by A83, ZFMISC_1: 31;

          then (pio \/ poz) = pio by XBOOLE_1: 12;

          hence contradiction by A1, A3, A4, A5, A6, A7, A8, A9, A255, Th13;

        end;

          suppose (Gik `2 ) = (Gij `2 );

          then

           A256: j = k by A17, A18, JORDAN1G: 6;

          then pio = {Gi1k} by RLTOPSP1: 70;

          then pio c= poz by A84, ZFMISC_1: 31;

          then (pio \/ poz) = poz by XBOOLE_1: 12;

          hence contradiction by A1, A2, A3, A4, A6, A7, A8, A9, A256, JORDAN15: 28;

        end;

      end;

      hence contradiction;

    end;

    theorem :: JORDAN19:24

    

     Th24: 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 ( Lower_Arc ( L~ ( Cage (C,(n + 1))))) & (( Gauge (C,(n + 1))) * (i2,j)) in ( Upper_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 ( Lower_Arc ( L~ ( Cage (C,(n + 1))))) and

       A9: (G * (i2,j)) in ( Upper_Arc ( L~ ( Cage (C,(n + 1)))));

      

       A10: ( Lower_Arc ( L~ ( Cage (C,(n + 1))))) = ( L~ ( Lower_Seq (C,(n + 1)))) by JORDAN1G: 56;

      

       A11: ( Upper_Arc ( L~ ( Cage (C,(n + 1))))) = ( L~ ( Upper_Seq (C,(n + 1)))) by JORDAN1G: 55;

      

       A12: j <= ( width G) by A6, A7, XXREAL_0: 2;

      then

       A13: [i2, j] in ( Indices G) by A3, A4, A5, MATRIX_0: 30;

      

       A14: 1 <= k by A5, A6, XXREAL_0: 2;

      then

       A15: [i2, k] in ( Indices G) by A3, A4, A7, MATRIX_0: 30;

      ((G * (i2,j)) `1 ) = ((G * (i2,1)) `1 ) by A3, A4, A5, A12, GOBOARD5: 2

      .= ((G * (i2,k)) `1 ) by A3, A4, A7, A14, GOBOARD5: 2;

      then

       A16: ( LSeg ((G * (i2,j)),(G * (i2,k)))) is vertical by SPPOL_1: 16;

      ((G * (i2,k)) `2 ) = ((G * (1,k)) `2 ) by A3, A4, A7, A14, GOBOARD5: 1

      .= ((G * (i1,k)) `2 ) by A1, A2, A7, A14, GOBOARD5: 1;

      then

       A17: ( LSeg ((G * (i2,k)),(G * (i1,k)))) is horizontal by SPPOL_1: 15;

      

       A18: [i2, k] in ( Indices G) by A3, A4, A7, A14, MATRIX_0: 30;

      

       A19: [i1, k] in ( Indices G) by A1, A2, A7, A14, MATRIX_0: 30;

      now

        per cases ;

          suppose

           A20: ( LSeg ((G * (i2,j)),(G * (i2,k)))) meets ( Lower_Arc ( L~ ( Cage (C,(n + 1)))));

          then

          consider m be Nat such that

           A21: j <= m and

           A22: m <= k and

           A23: ((G * (i2,m)) `2 ) = ( lower_bound ( proj2 .: (( LSeg ((G * (i2,j)),(G * (i2,k)))) /\ ( L~ ( Lower_Seq (C,(n + 1))))))) by A6, A10, A13, A15, JORDAN1F: 1, JORDAN1G: 5;

          

           A24: 1 <= m by A5, A21, XXREAL_0: 2;

          

           A25: m <= ( width G) by A7, A22, XXREAL_0: 2;

          set X = (( LSeg ((G * (i2,j)),(G * (i2,k)))) /\ ( L~ ( Lower_Seq (C,(n + 1)))));

          

           A26: ((G * (i2,m)) `1 ) = ((G * (i2,1)) `1 ) by A3, A4, A24, A25, GOBOARD5: 2;

          then

           A27: |[((G * (i2,1)) `1 ), ( lower_bound ( proj2 .: X))]| = (G * (i2,m)) by A23, EUCLID: 53;

          then

           A28: ((G * (i2,j)) `1 ) = ( |[((G * (i2,1)) `1 ), ( lower_bound ( proj2 .: X))]| `1 ) by A3, A4, A5, A12, A26, GOBOARD5: 2;

          ex x be object st x in ( LSeg ((G * (i2,j)),(G * (i2,k)))) & x in ( L~ ( Lower_Seq (C,(n + 1)))) by A10, 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

           A29: pp in ( S-most X1) by XBOOLE_0:def 1;

          reconsider pp as Point of ( TOP-REAL 2) by A29;

          

           A30: pp in X by A29, XBOOLE_0:def 4;

          then

           A31: pp in ( L~ ( Lower_Seq (C,(n + 1)))) by XBOOLE_0:def 4;

          pp in ( LSeg ((G * (i2,j)),(G * (i2,k)))) by A30, XBOOLE_0:def 4;

          then

           A32: (pp `1 ) = ( |[((G * (i2,1)) `1 ), ( lower_bound ( proj2 .: X))]| `1 ) by A16, A28, SPPOL_1: 41;

          ( |[((G * (i2,1)) `1 ), ( lower_bound ( proj2 .: X))]| `2 ) = ( S-bound X) by A23, A27, SPRECT_1: 44

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

          .= (pp `2 ) by A29, PSCOMP_1: 55;

          then (G * (i2,m)) in ( Lower_Arc ( L~ ( Cage (C,(n + 1))))) by A10, A27, A31, A32, TOPREAL3: 6;

          then ( LSeg ((G * (i2,j)),(G * (i2,m)))) meets ( Lower_Arc C) by A3, A4, A5, A9, A21, A25, Th19;

          then ( LSeg ((G * (i2,j)),(G * (i2,k)))) meets ( Lower_Arc C) by A3, A4, A5, A7, A21, A22, JORDAN15: 5, XBOOLE_1: 63;

          hence thesis by XBOOLE_1: 70;

        end;

          suppose

           A33: ( LSeg ((G * (i2,k)),(G * (i1,k)))) meets ( Upper_Arc ( L~ ( Cage (C,(n + 1))))) & i2 <= i1;

          then

          consider m be Nat such that

           A34: i2 <= m and

           A35: m <= i1 and

           A36: ((G * (m,k)) `1 ) = ( upper_bound ( proj1 .: (( LSeg ((G * (i2,k)),(G * (i1,k)))) /\ ( L~ ( Upper_Seq (C,(n + 1))))))) by A11, A18, A19, JORDAN1F: 4, JORDAN1G: 4;

          

           A37: 1 < m by A3, A34, XXREAL_0: 2;

          

           A38: m < ( len G) by A2, A35, XXREAL_0: 2;

          set X = (( LSeg ((G * (i2,k)),(G * (i1,k)))) /\ ( L~ ( Upper_Seq (C,(n + 1)))));

          

           A39: ((G * (m,k)) `2 ) = ((G * (1,k)) `2 ) by A7, A14, A37, A38, GOBOARD5: 1;

          then

           A40: |[( upper_bound ( proj1 .: X)), ((G * (1,k)) `2 )]| = (G * (m,k)) by A36, EUCLID: 53;

          then

           A41: ((G * (i2,k)) `2 ) = ( |[( upper_bound ( proj1 .: X)), ((G * (1,k)) `2 )]| `2 ) by A3, A4, A7, A14, A39, GOBOARD5: 1;

          ex x be object st x in ( LSeg ((G * (i2,k)),(G * (i1,k)))) & x in ( L~ ( Upper_Seq (C,(n + 1)))) by A11, A33, 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

           A42: pp in ( E-most X1) by XBOOLE_0:def 1;

          reconsider pp as Point of ( TOP-REAL 2) by A42;

          

           A43: pp in X by A42, XBOOLE_0:def 4;

          then

           A44: pp in ( L~ ( Upper_Seq (C,(n + 1)))) by XBOOLE_0:def 4;

          pp in ( LSeg ((G * (i2,k)),(G * (i1,k)))) by A43, XBOOLE_0:def 4;

          then

           A45: (pp `2 ) = ( |[( upper_bound ( proj1 .: X)), ((G * (1,k)) `2 )]| `2 ) by A17, A41, SPPOL_1: 40;

          ( |[( upper_bound ( proj1 .: X)), ((G * (1,k)) `2 )]| `1 ) = ( E-bound X) by A36, A40, SPRECT_1: 46

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

          .= (pp `1 ) by A42, PSCOMP_1: 47;

          then (G * (m,k)) in ( Upper_Arc ( L~ ( Cage (C,(n + 1))))) by A11, A40, A44, A45, TOPREAL3: 6;

          then ( LSeg ((G * (m,k)),(G * (i1,k)))) meets ( Lower_Arc C) by A2, A7, A8, A14, A35, A37, JORDAN15: 40;

          then ( LSeg ((G * (i2,k)),(G * (i1,k)))) meets ( Lower_Arc C) by A2, A3, A7, A14, A34, A35, JORDAN15: 6, XBOOLE_1: 63;

          hence thesis by XBOOLE_1: 70;

        end;

          suppose

           A46: ( LSeg ((G * (i2,k)),(G * (i1,k)))) meets ( Upper_Arc ( L~ ( Cage (C,(n + 1))))) & i1 < i2;

          then

          consider m be Nat such that

           A47: i1 <= m and

           A48: m <= i2 and

           A49: ((G * (m,k)) `1 ) = ( lower_bound ( proj1 .: (( LSeg ((G * (i1,k)),(G * (i2,k)))) /\ ( L~ ( Upper_Seq (C,(n + 1))))))) by A11, A18, A19, JORDAN1F: 3, JORDAN1G: 4;

          

           A50: 1 < m by A1, A47, XXREAL_0: 2;

          

           A51: m < ( len G) by A4, A48, XXREAL_0: 2;

          set X = (( LSeg ((G * (i1,k)),(G * (i2,k)))) /\ ( L~ ( Upper_Seq (C,(n + 1)))));

          

           A52: ((G * (m,k)) `2 ) = ((G * (1,k)) `2 ) by A7, A14, A50, A51, GOBOARD5: 1;

          then

           A53: |[( lower_bound ( proj1 .: X)), ((G * (1,k)) `2 )]| = (G * (m,k)) by A49, EUCLID: 53;

          then

           A54: ((G * (i1,k)) `2 ) = ( |[( lower_bound ( proj1 .: X)), ((G * (1,k)) `2 )]| `2 ) by A1, A2, A7, A14, A52, GOBOARD5: 1;

          ex x be object st x in ( LSeg ((G * (i1,k)),(G * (i2,k)))) & x in ( L~ ( Upper_Seq (C,(n + 1)))) by A11, A46, 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

           A55: pp in ( W-most X1) by XBOOLE_0:def 1;

          reconsider pp as Point of ( TOP-REAL 2) by A55;

          

           A56: pp in X by A55, XBOOLE_0:def 4;

          then

           A57: pp in ( L~ ( Upper_Seq (C,(n + 1)))) by XBOOLE_0:def 4;

          pp in ( LSeg ((G * (i1,k)),(G * (i2,k)))) by A56, XBOOLE_0:def 4;

          then

           A58: (pp `2 ) = ( |[( lower_bound ( proj1 .: X)), ((G * (1,k)) `2 )]| `2 ) by A17, A54, SPPOL_1: 40;

          ( |[( lower_bound ( proj1 .: X)), ((G * (1,k)) `2 )]| `1 ) = ( W-bound X) by A49, A53, SPRECT_1: 43

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

          .= (pp `1 ) by A55, PSCOMP_1: 31;

          then (G * (m,k)) in ( Upper_Arc ( L~ ( Cage (C,(n + 1))))) by A11, A53, A57, A58, TOPREAL3: 6;

          then ( LSeg ((G * (i1,k)),(G * (m,k)))) meets ( Lower_Arc C) by A1, A7, A8, A14, A47, A51, JORDAN15: 32;

          then ( LSeg ((G * (i1,k)),(G * (i2,k)))) meets ( Lower_Arc C) by A1, A4, A7, A14, A47, A48, JORDAN15: 6, XBOOLE_1: 63;

          hence thesis by XBOOLE_1: 70;

        end;

          suppose

           A59: ( LSeg ((G * (i2,j)),(G * (i2,k)))) misses ( Lower_Arc ( L~ ( Cage (C,(n + 1))))) & ( LSeg ((( Gauge (C,(n + 1))) * (i2,k)),(( Gauge (C,(n + 1))) * (i1,k)))) misses ( Upper_Arc ( L~ ( Cage (C,(n + 1)))));

          consider j1 be Nat such that

           A60: j <= j1 and

           A61: j1 <= k and

           A62: (( LSeg ((G * (i2,j1)),(G * (i2,k)))) /\ ( L~ ( Upper_Seq (C,(n + 1))))) = {(G * (i2,j1))} by A3, A4, A5, A6, A7, A9, A11, JORDAN15: 15;

          (G * (i2,j1)) in (( LSeg ((G * (i2,j1)),(G * (i2,k)))) /\ ( L~ ( Upper_Seq (C,(n + 1))))) by A62, TARSKI:def 1;

          then

           A63: (G * (i2,j1)) in ( L~ ( Upper_Seq (C,(n + 1)))) by XBOOLE_0:def 4;

          

           A64: 1 <= j1 by A5, A60, XXREAL_0: 2;

          now

            per cases ;

              suppose i2 <= i1;

              then

              consider i3 be Nat such that

               A65: i2 <= i3 and

               A66: i3 <= i1 and

               A67: (( LSeg ((G * (i2,k)),(G * (i3,k)))) /\ ( L~ ( Lower_Seq (C,(n + 1))))) = {(G * (i3,k))} by A2, A3, A7, A8, A10, A14, JORDAN15: 19;

              

               A68: i3 < ( len G) by A2, A66, XXREAL_0: 2;

              (G * (i3,k)) in (( LSeg ((G * (i2,k)),(G * (i3,k)))) /\ ( L~ ( Lower_Seq (C,(n + 1))))) by A67, TARSKI:def 1;

              then

               A69: (G * (i3,k)) in ( L~ ( Lower_Seq (C,(n + 1)))) by XBOOLE_0:def 4;

              

               A70: ( LSeg ((G * (i2,j1)),(G * (i2,k)))) c= ( LSeg ((G * (i2,j)),(G * (i2,k)))) by A3, A4, A5, A7, A60, A61, JORDAN15: 5;

              

               A71: ( LSeg ((G * (i2,k)),(G * (i3,k)))) c= ( LSeg ((G * (i2,k)),(G * (i1,k)))) by A2, A3, A7, A14, A65, A66, JORDAN15: 6;

              then

               A72: (( 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 A70, XBOOLE_1: 13;

              

               A73: ((( LSeg ((G * (i2,j1)),(G * (i2,k)))) \/ ( LSeg ((G * (i2,k)),(G * (i3,k))))) /\ ( L~ ( Lower_Seq (C,(n + 1))))) = {(G * (i3,k))}

              proof

                thus ((( LSeg ((G * (i2,j1)),(G * (i2,k)))) \/ ( LSeg ((G * (i2,k)),(G * (i3,k))))) /\ ( L~ ( Lower_Seq (C,(n + 1))))) c= {(G * (i3,k))}

                proof

                  let x be object;

                  assume

                   A74: 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

                   A75: 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 A74, XBOOLE_0:def 4;

                  hence thesis by A10, A59, A67, A70, A75, XBOOLE_0:def 4;

                end;

                let x be object;

                assume x in {(G * (i3,k))};

                then

                 A76: x = (G * (i3,k)) by TARSKI:def 1;

                (G * (i3,k)) in ( LSeg ((G * (i2,k)),(G * (i3,k)))) by RLTOPSP1: 68;

                then (G * (i3,k)) in (( LSeg ((G * (i2,j1)),(G * (i2,k)))) \/ ( LSeg ((G * (i2,k)),(G * (i3,k))))) by XBOOLE_0:def 3;

                hence thesis by A69, A76, XBOOLE_0:def 4;

              end;

              ((( LSeg ((G * (i2,j1)),(G * (i2,k)))) \/ ( LSeg ((G * (i2,k)),(G * (i3,k))))) /\ ( L~ ( Upper_Seq (C,(n + 1))))) = {(G * (i2,j1))}

              proof

                thus ((( LSeg ((G * (i2,j1)),(G * (i2,k)))) \/ ( LSeg ((G * (i2,k)),(G * (i3,k))))) /\ ( L~ ( Upper_Seq (C,(n + 1))))) c= {(G * (i2,j1))}

                proof

                  let x be object;

                  assume

                   A77: 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

                   A78: 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 A77, XBOOLE_0:def 4;

                  hence thesis by A11, A59, A62, A71, A78, XBOOLE_0:def 4;

                end;

                let x be object;

                assume x in {(G * (i2,j1))};

                then

                 A79: x = (G * (i2,j1)) by TARSKI:def 1;

                (G * (i2,j1)) in ( LSeg ((G * (i2,j1)),(G * (i2,k)))) by RLTOPSP1: 68;

                then (G * (i2,j1)) in (( LSeg ((G * (i2,j1)),(G * (i2,k)))) \/ ( LSeg ((G * (i2,k)),(G * (i3,k))))) by XBOOLE_0:def 3;

                hence thesis by A63, A79, XBOOLE_0:def 4;

              end;

              hence thesis by A3, A7, A61, A64, A65, A68, A72, A73, Th21, XBOOLE_1: 63;

            end;

              suppose i1 < i2;

              then

              consider i3 be Nat such that

               A80: i1 <= i3 and

               A81: i3 <= i2 and

               A82: (( LSeg ((G * (i3,k)),(G * (i2,k)))) /\ ( L~ ( Lower_Seq (C,(n + 1))))) = {(G * (i3,k))} by A1, A4, A7, A8, A10, A14, JORDAN15: 12;

              

               A83: 1 < i3 by A1, A80, XXREAL_0: 2;

              (G * (i3,k)) in (( LSeg ((G * (i2,k)),(G * (i3,k)))) /\ ( L~ ( Lower_Seq (C,(n + 1))))) by A82, TARSKI:def 1;

              then

               A84: (G * (i3,k)) in ( L~ ( Lower_Seq (C,(n + 1)))) by XBOOLE_0:def 4;

              

               A85: ( LSeg ((G * (i2,j1)),(G * (i2,k)))) c= ( LSeg ((G * (i2,j)),(G * (i2,k)))) by A3, A4, A5, A7, A60, A61, JORDAN15: 5;

              

               A86: ( LSeg ((G * (i2,k)),(G * (i3,k)))) c= ( LSeg ((G * (i2,k)),(G * (i1,k)))) by A1, A4, A7, A14, A80, A81, JORDAN15: 6;

              then

               A87: (( 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 A85, XBOOLE_1: 13;

              

               A88: ((( LSeg ((G * (i2,j1)),(G * (i2,k)))) \/ ( LSeg ((G * (i2,k)),(G * (i3,k))))) /\ ( L~ ( Lower_Seq (C,(n + 1))))) = {(G * (i3,k))}

              proof

                thus ((( LSeg ((G * (i2,j1)),(G * (i2,k)))) \/ ( LSeg ((G * (i2,k)),(G * (i3,k))))) /\ ( L~ ( Lower_Seq (C,(n + 1))))) c= {(G * (i3,k))}

                proof

                  let x be object;

                  assume

                   A89: 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

                   A90: 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 A89, XBOOLE_0:def 4;

                  hence thesis by A10, A59, A82, A85, A90, XBOOLE_0:def 4;

                end;

                let x be object;

                assume x in {(G * (i3,k))};

                then

                 A91: x = (G * (i3,k)) by TARSKI:def 1;

                (G * (i3,k)) in ( LSeg ((G * (i2,k)),(G * (i3,k)))) by RLTOPSP1: 68;

                then (G * (i3,k)) in (( LSeg ((G * (i2,j1)),(G * (i2,k)))) \/ ( LSeg ((G * (i2,k)),(G * (i3,k))))) by XBOOLE_0:def 3;

                hence thesis by A84, A91, XBOOLE_0:def 4;

              end;

              ((( LSeg ((G * (i2,j1)),(G * (i2,k)))) \/ ( LSeg ((G * (i2,k)),(G * (i3,k))))) /\ ( L~ ( Upper_Seq (C,(n + 1))))) = {(G * (i2,j1))}

              proof

                thus ((( LSeg ((G * (i2,j1)),(G * (i2,k)))) \/ ( LSeg ((G * (i2,k)),(G * (i3,k))))) /\ ( L~ ( Upper_Seq (C,(n + 1))))) c= {(G * (i2,j1))}

                proof

                  let x be object;

                  assume

                   A92: 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

                   A93: 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 A92, XBOOLE_0:def 4;

                  hence thesis by A11, A59, A62, A86, A93, XBOOLE_0:def 4;

                end;

                let x be object;

                assume x in {(G * (i2,j1))};

                then

                 A94: x = (G * (i2,j1)) by TARSKI:def 1;

                (G * (i2,j1)) in ( LSeg ((G * (i2,j1)),(G * (i2,k)))) by RLTOPSP1: 68;

                then (G * (i2,j1)) in (( LSeg ((G * (i2,j1)),(G * (i2,k)))) \/ ( LSeg ((G * (i2,k)),(G * (i3,k))))) by XBOOLE_0:def 3;

                hence thesis by A63, A94, XBOOLE_0:def 4;

              end;

              hence thesis by A4, A7, A61, A64, A81, A83, A87, A88, Th23, XBOOLE_1: 63;

            end;

          end;

          hence thesis;

        end;

      end;

      hence thesis;

    end;

    theorem :: JORDAN19:25

    

     Th25: 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 ( Lower_Arc ( L~ ( Cage (C,(n + 1))))) & (( Gauge (C,(n + 1))) * (i2,j)) in ( Upper_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 ( Lower_Arc ( L~ ( Cage (C,(n + 1))))) and

       A9: (G * (i2,j)) in ( Upper_Arc ( L~ ( Cage (C,(n + 1)))));

      

       A10: ( Lower_Arc ( L~ ( Cage (C,(n + 1))))) = ( L~ ( Lower_Seq (C,(n + 1)))) by JORDAN1G: 56;

      

       A11: ( Upper_Arc ( L~ ( Cage (C,(n + 1))))) = ( L~ ( Upper_Seq (C,(n + 1)))) by JORDAN1G: 55;

      

       A12: j <= ( width G) by A6, A7, XXREAL_0: 2;

      then

       A13: [i2, j] in ( Indices G) by A3, A4, A5, MATRIX_0: 30;

      

       A14: 1 <= k by A5, A6, XXREAL_0: 2;

      then

       A15: [i2, k] in ( Indices G) by A3, A4, A7, MATRIX_0: 30;

      ((G * (i2,j)) `1 ) = ((G * (i2,1)) `1 ) by A3, A4, A5, A12, GOBOARD5: 2

      .= ((G * (i2,k)) `1 ) by A3, A4, A7, A14, GOBOARD5: 2;

      then

       A16: ( LSeg ((G * (i2,j)),(G * (i2,k)))) is vertical by SPPOL_1: 16;

      ((G * (i2,k)) `2 ) = ((G * (1,k)) `2 ) by A3, A4, A7, A14, GOBOARD5: 1

      .= ((G * (i1,k)) `2 ) by A1, A2, A7, A14, GOBOARD5: 1;

      then

       A17: ( LSeg ((G * (i2,k)),(G * (i1,k)))) is horizontal by SPPOL_1: 15;

      

       A18: [i2, k] in ( Indices G) by A3, A4, A7, A14, MATRIX_0: 30;

      

       A19: [i1, k] in ( Indices G) by A1, A2, A7, A14, MATRIX_0: 30;

      now

        per cases ;

          suppose

           A20: ( LSeg ((G * (i2,j)),(G * (i2,k)))) meets ( Lower_Arc ( L~ ( Cage (C,(n + 1)))));

          then

          consider m be Nat such that

           A21: j <= m and

           A22: m <= k and

           A23: ((G * (i2,m)) `2 ) = ( lower_bound ( proj2 .: (( LSeg ((G * (i2,j)),(G * (i2,k)))) /\ ( L~ ( Lower_Seq (C,(n + 1))))))) by A6, A10, A13, A15, JORDAN1F: 1, JORDAN1G: 5;

          

           A24: 1 <= m by A5, A21, XXREAL_0: 2;

          

           A25: m <= ( width G) by A7, A22, XXREAL_0: 2;

          set X = (( LSeg ((G * (i2,j)),(G * (i2,k)))) /\ ( L~ ( Lower_Seq (C,(n + 1)))));

          

           A26: ((G * (i2,m)) `1 ) = ((G * (i2,1)) `1 ) by A3, A4, A24, A25, GOBOARD5: 2;

          then

           A27: |[((G * (i2,1)) `1 ), ( lower_bound ( proj2 .: X))]| = (G * (i2,m)) by A23, EUCLID: 53;

          then

           A28: ((G * (i2,j)) `1 ) = ( |[((G * (i2,1)) `1 ), ( lower_bound ( proj2 .: X))]| `1 ) by A3, A4, A5, A12, A26, GOBOARD5: 2;

          ex x be object st x in ( LSeg ((G * (i2,j)),(G * (i2,k)))) & x in ( L~ ( Lower_Seq (C,(n + 1)))) by A10, 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

           A29: pp in ( S-most X1) by XBOOLE_0:def 1;

          reconsider pp as Point of ( TOP-REAL 2) by A29;

          

           A30: pp in X by A29, XBOOLE_0:def 4;

          then

           A31: pp in ( L~ ( Lower_Seq (C,(n + 1)))) by XBOOLE_0:def 4;

          pp in ( LSeg ((G * (i2,j)),(G * (i2,k)))) by A30, XBOOLE_0:def 4;

          then

           A32: (pp `1 ) = ( |[((G * (i2,1)) `1 ), ( lower_bound ( proj2 .: X))]| `1 ) by A16, A28, SPPOL_1: 41;

          ( |[((G * (i2,1)) `1 ), ( lower_bound ( proj2 .: X))]| `2 ) = ( S-bound X) by A23, A27, SPRECT_1: 44

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

          .= (pp `2 ) by A29, PSCOMP_1: 55;

          then (G * (i2,m)) in ( Lower_Arc ( L~ ( Cage (C,(n + 1))))) by A10, A27, A31, A32, TOPREAL3: 6;

          then ( LSeg ((G * (i2,j)),(G * (i2,m)))) meets ( Upper_Arc C) by A3, A4, A5, A9, A21, A25, Th18;

          then ( LSeg ((G * (i2,j)),(G * (i2,k)))) meets ( Upper_Arc C) by A3, A4, A5, A7, A21, A22, JORDAN15: 5, XBOOLE_1: 63;

          hence thesis by XBOOLE_1: 70;

        end;

          suppose

           A33: ( LSeg ((G * (i2,k)),(G * (i1,k)))) meets ( Upper_Arc ( L~ ( Cage (C,(n + 1))))) & i2 <= i1;

          then

          consider m be Nat such that

           A34: i2 <= m and

           A35: m <= i1 and

           A36: ((G * (m,k)) `1 ) = ( upper_bound ( proj1 .: (( LSeg ((G * (i2,k)),(G * (i1,k)))) /\ ( L~ ( Upper_Seq (C,(n + 1))))))) by A11, A18, A19, JORDAN1F: 4, JORDAN1G: 4;

          

           A37: 1 < m by A3, A34, XXREAL_0: 2;

          

           A38: m < ( len G) by A2, A35, XXREAL_0: 2;

          set X = (( LSeg ((G * (i2,k)),(G * (i1,k)))) /\ ( L~ ( Upper_Seq (C,(n + 1)))));

          

           A39: ((G * (m,k)) `2 ) = ((G * (1,k)) `2 ) by A7, A14, A37, A38, GOBOARD5: 1;

          then

           A40: |[( upper_bound ( proj1 .: X)), ((G * (1,k)) `2 )]| = (G * (m,k)) by A36, EUCLID: 53;

          then

           A41: ((G * (i2,k)) `2 ) = ( |[( upper_bound ( proj1 .: X)), ((G * (1,k)) `2 )]| `2 ) by A3, A4, A7, A14, A39, GOBOARD5: 1;

          ex x be object st x in ( LSeg ((G * (i2,k)),(G * (i1,k)))) & x in ( L~ ( Upper_Seq (C,(n + 1)))) by A11, A33, 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

           A42: pp in ( E-most X1) by XBOOLE_0:def 1;

          reconsider pp as Point of ( TOP-REAL 2) by A42;

          

           A43: pp in X by A42, XBOOLE_0:def 4;

          then

           A44: pp in ( L~ ( Upper_Seq (C,(n + 1)))) by XBOOLE_0:def 4;

          pp in ( LSeg ((G * (i2,k)),(G * (i1,k)))) by A43, XBOOLE_0:def 4;

          then

           A45: (pp `2 ) = ( |[( upper_bound ( proj1 .: X)), ((G * (1,k)) `2 )]| `2 ) by A17, A41, SPPOL_1: 40;

          ( |[( upper_bound ( proj1 .: X)), ((G * (1,k)) `2 )]| `1 ) = ( E-bound X) by A36, A40, SPRECT_1: 46

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

          .= (pp `1 ) by A42, PSCOMP_1: 47;

          then (G * (m,k)) in ( Upper_Arc ( L~ ( Cage (C,(n + 1))))) by A11, A40, A44, A45, TOPREAL3: 6;

          then ( LSeg ((G * (m,k)),(G * (i1,k)))) meets ( Upper_Arc C) by A2, A7, A8, A14, A35, A37, JORDAN15: 41;

          then ( LSeg ((G * (i2,k)),(G * (i1,k)))) meets ( Upper_Arc C) by A2, A3, A7, A14, A34, A35, JORDAN15: 6, XBOOLE_1: 63;

          hence thesis by XBOOLE_1: 70;

        end;

          suppose

           A46: ( LSeg ((G * (i2,k)),(G * (i1,k)))) meets ( Upper_Arc ( L~ ( Cage (C,(n + 1))))) & i1 < i2;

          then

          consider m be Nat such that

           A47: i1 <= m and

           A48: m <= i2 and

           A49: ((G * (m,k)) `1 ) = ( lower_bound ( proj1 .: (( LSeg ((G * (i1,k)),(G * (i2,k)))) /\ ( L~ ( Upper_Seq (C,(n + 1))))))) by A11, A18, A19, JORDAN1F: 3, JORDAN1G: 4;

          

           A50: 1 < m by A1, A47, XXREAL_0: 2;

          

           A51: m < ( len G) by A4, A48, XXREAL_0: 2;

          set X = (( LSeg ((G * (i1,k)),(G * (i2,k)))) /\ ( L~ ( Upper_Seq (C,(n + 1)))));

          

           A52: ((G * (m,k)) `2 ) = ((G * (1,k)) `2 ) by A7, A14, A50, A51, GOBOARD5: 1;

          then

           A53: |[( lower_bound ( proj1 .: X)), ((G * (1,k)) `2 )]| = (G * (m,k)) by A49, EUCLID: 53;

          then

           A54: ((G * (i1,k)) `2 ) = ( |[( lower_bound ( proj1 .: X)), ((G * (1,k)) `2 )]| `2 ) by A1, A2, A7, A14, A52, GOBOARD5: 1;

          ex x be object st x in ( LSeg ((G * (i1,k)),(G * (i2,k)))) & x in ( L~ ( Upper_Seq (C,(n + 1)))) by A11, A46, 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

           A55: pp in ( W-most X1) by XBOOLE_0:def 1;

          reconsider pp as Point of ( TOP-REAL 2) by A55;

          

           A56: pp in X by A55, XBOOLE_0:def 4;

          then

           A57: pp in ( L~ ( Upper_Seq (C,(n + 1)))) by XBOOLE_0:def 4;

          pp in ( LSeg ((G * (i1,k)),(G * (i2,k)))) by A56, XBOOLE_0:def 4;

          then

           A58: (pp `2 ) = ( |[( lower_bound ( proj1 .: X)), ((G * (1,k)) `2 )]| `2 ) by A17, A54, SPPOL_1: 40;

          ( |[( lower_bound ( proj1 .: X)), ((G * (1,k)) `2 )]| `1 ) = ( W-bound X) by A49, A53, SPRECT_1: 43

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

          .= (pp `1 ) by A55, PSCOMP_1: 31;

          then (G * (m,k)) in ( Upper_Arc ( L~ ( Cage (C,(n + 1))))) by A11, A53, A57, A58, TOPREAL3: 6;

          then ( LSeg ((G * (i1,k)),(G * (m,k)))) meets ( Upper_Arc C) by A1, A7, A8, A14, A47, A51, JORDAN15: 33;

          then ( LSeg ((G * (i1,k)),(G * (i2,k)))) meets ( Upper_Arc C) by A1, A4, A7, A14, A47, A48, JORDAN15: 6, XBOOLE_1: 63;

          hence thesis by XBOOLE_1: 70;

        end;

          suppose

           A59: ( LSeg ((G * (i2,j)),(G * (i2,k)))) misses ( Lower_Arc ( L~ ( Cage (C,(n + 1))))) & ( LSeg ((( Gauge (C,(n + 1))) * (i2,k)),(( Gauge (C,(n + 1))) * (i1,k)))) misses ( Upper_Arc ( L~ ( Cage (C,(n + 1)))));

          consider j1 be Nat such that

           A60: j <= j1 and

           A61: j1 <= k and

           A62: (( LSeg ((G * (i2,j1)),(G * (i2,k)))) /\ ( L~ ( Upper_Seq (C,(n + 1))))) = {(G * (i2,j1))} by A3, A4, A5, A6, A7, A9, A11, JORDAN15: 15;

          (G * (i2,j1)) in (( LSeg ((G * (i2,j1)),(G * (i2,k)))) /\ ( L~ ( Upper_Seq (C,(n + 1))))) by A62, TARSKI:def 1;

          then

           A63: (G * (i2,j1)) in ( L~ ( Upper_Seq (C,(n + 1)))) by XBOOLE_0:def 4;

          

           A64: 1 <= j1 by A5, A60, XXREAL_0: 2;

          now

            per cases ;

              suppose i2 <= i1;

              then

              consider i3 be Nat such that

               A65: i2 <= i3 and

               A66: i3 <= i1 and

               A67: (( LSeg ((G * (i2,k)),(G * (i3,k)))) /\ ( L~ ( Lower_Seq (C,(n + 1))))) = {(G * (i3,k))} by A2, A3, A7, A8, A10, A14, JORDAN15: 19;

              

               A68: i3 < ( len G) by A2, A66, XXREAL_0: 2;

              (G * (i3,k)) in (( LSeg ((G * (i2,k)),(G * (i3,k)))) /\ ( L~ ( Lower_Seq (C,(n + 1))))) by A67, TARSKI:def 1;

              then

               A69: (G * (i3,k)) in ( L~ ( Lower_Seq (C,(n + 1)))) by XBOOLE_0:def 4;

              

               A70: ( LSeg ((G * (i2,j1)),(G * (i2,k)))) c= ( LSeg ((G * (i2,j)),(G * (i2,k)))) by A3, A4, A5, A7, A60, A61, JORDAN15: 5;

              

               A71: ( LSeg ((G * (i2,k)),(G * (i3,k)))) c= ( LSeg ((G * (i2,k)),(G * (i1,k)))) by A2, A3, A7, A14, A65, A66, JORDAN15: 6;

              then

               A72: (( 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 A70, XBOOLE_1: 13;

              

               A73: ((( LSeg ((G * (i2,j1)),(G * (i2,k)))) \/ ( LSeg ((G * (i2,k)),(G * (i3,k))))) /\ ( L~ ( Lower_Seq (C,(n + 1))))) = {(G * (i3,k))}

              proof

                thus ((( LSeg ((G * (i2,j1)),(G * (i2,k)))) \/ ( LSeg ((G * (i2,k)),(G * (i3,k))))) /\ ( L~ ( Lower_Seq (C,(n + 1))))) c= {(G * (i3,k))}

                proof

                  let x be object;

                  assume

                   A74: 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

                   A75: 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 A74, XBOOLE_0:def 4;

                  hence thesis by A10, A59, A67, A70, A75, XBOOLE_0:def 4;

                end;

                let x be object;

                assume x in {(G * (i3,k))};

                then

                 A76: x = (G * (i3,k)) by TARSKI:def 1;

                (G * (i3,k)) in ( LSeg ((G * (i2,k)),(G * (i3,k)))) by RLTOPSP1: 68;

                then (G * (i3,k)) in (( LSeg ((G * (i2,j1)),(G * (i2,k)))) \/ ( LSeg ((G * (i2,k)),(G * (i3,k))))) by XBOOLE_0:def 3;

                hence thesis by A69, A76, XBOOLE_0:def 4;

              end;

              ((( LSeg ((G * (i2,j1)),(G * (i2,k)))) \/ ( LSeg ((G * (i2,k)),(G * (i3,k))))) /\ ( L~ ( Upper_Seq (C,(n + 1))))) = {(G * (i2,j1))}

              proof

                thus ((( LSeg ((G * (i2,j1)),(G * (i2,k)))) \/ ( LSeg ((G * (i2,k)),(G * (i3,k))))) /\ ( L~ ( Upper_Seq (C,(n + 1))))) c= {(G * (i2,j1))}

                proof

                  let x be object;

                  assume

                   A77: 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

                   A78: 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 A77, XBOOLE_0:def 4;

                  hence thesis by A11, A59, A62, A71, A78, XBOOLE_0:def 4;

                end;

                let x be object;

                assume x in {(G * (i2,j1))};

                then

                 A79: x = (G * (i2,j1)) by TARSKI:def 1;

                (G * (i2,j1)) in ( LSeg ((G * (i2,j1)),(G * (i2,k)))) by RLTOPSP1: 68;

                then (G * (i2,j1)) in (( LSeg ((G * (i2,j1)),(G * (i2,k)))) \/ ( LSeg ((G * (i2,k)),(G * (i3,k))))) by XBOOLE_0:def 3;

                hence thesis by A63, A79, XBOOLE_0:def 4;

              end;

              hence thesis by A3, A7, A61, A64, A65, A68, A72, A73, Th20, XBOOLE_1: 63;

            end;

              suppose i1 < i2;

              then

              consider i3 be Nat such that

               A80: i1 <= i3 and

               A81: i3 <= i2 and

               A82: (( LSeg ((G * (i3,k)),(G * (i2,k)))) /\ ( L~ ( Lower_Seq (C,(n + 1))))) = {(G * (i3,k))} by A1, A4, A7, A8, A10, A14, JORDAN15: 12;

              

               A83: 1 < i3 by A1, A80, XXREAL_0: 2;

              (G * (i3,k)) in (( LSeg ((G * (i2,k)),(G * (i3,k)))) /\ ( L~ ( Lower_Seq (C,(n + 1))))) by A82, TARSKI:def 1;

              then

               A84: (G * (i3,k)) in ( L~ ( Lower_Seq (C,(n + 1)))) by XBOOLE_0:def 4;

              

               A85: ( LSeg ((G * (i2,j1)),(G * (i2,k)))) c= ( LSeg ((G * (i2,j)),(G * (i2,k)))) by A3, A4, A5, A7, A60, A61, JORDAN15: 5;

              

               A86: ( LSeg ((G * (i2,k)),(G * (i3,k)))) c= ( LSeg ((G * (i2,k)),(G * (i1,k)))) by A1, A4, A7, A14, A80, A81, JORDAN15: 6;

              then

               A87: (( 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 A85, XBOOLE_1: 13;

              

               A88: ((( LSeg ((G * (i2,j1)),(G * (i2,k)))) \/ ( LSeg ((G * (i2,k)),(G * (i3,k))))) /\ ( L~ ( Lower_Seq (C,(n + 1))))) = {(G * (i3,k))}

              proof

                thus ((( LSeg ((G * (i2,j1)),(G * (i2,k)))) \/ ( LSeg ((G * (i2,k)),(G * (i3,k))))) /\ ( L~ ( Lower_Seq (C,(n + 1))))) c= {(G * (i3,k))}

                proof

                  let x be object;

                  assume

                   A89: 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

                   A90: 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 A89, XBOOLE_0:def 4;

                  hence thesis by A10, A59, A82, A85, A90, XBOOLE_0:def 4;

                end;

                let x be object;

                assume x in {(G * (i3,k))};

                then

                 A91: x = (G * (i3,k)) by TARSKI:def 1;

                (G * (i3,k)) in ( LSeg ((G * (i2,k)),(G * (i3,k)))) by RLTOPSP1: 68;

                then (G * (i3,k)) in (( LSeg ((G * (i2,j1)),(G * (i2,k)))) \/ ( LSeg ((G * (i2,k)),(G * (i3,k))))) by XBOOLE_0:def 3;

                hence thesis by A84, A91, XBOOLE_0:def 4;

              end;

              ((( LSeg ((G * (i2,j1)),(G * (i2,k)))) \/ ( LSeg ((G * (i2,k)),(G * (i3,k))))) /\ ( L~ ( Upper_Seq (C,(n + 1))))) = {(G * (i2,j1))}

              proof

                thus ((( LSeg ((G * (i2,j1)),(G * (i2,k)))) \/ ( LSeg ((G * (i2,k)),(G * (i3,k))))) /\ ( L~ ( Upper_Seq (C,(n + 1))))) c= {(G * (i2,j1))}

                proof

                  let x be object;

                  assume

                   A92: 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

                   A93: 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 A92, XBOOLE_0:def 4;

                  hence thesis by A11, A59, A62, A86, A93, XBOOLE_0:def 4;

                end;

                let x be object;

                assume x in {(G * (i2,j1))};

                then

                 A94: x = (G * (i2,j1)) by TARSKI:def 1;

                (G * (i2,j1)) in ( LSeg ((G * (i2,j1)),(G * (i2,k)))) by RLTOPSP1: 68;

                then (G * (i2,j1)) in (( LSeg ((G * (i2,j1)),(G * (i2,k)))) \/ ( LSeg ((G * (i2,k)),(G * (i3,k))))) by XBOOLE_0:def 3;

                hence thesis by A63, A94, XBOOLE_0:def 4;

              end;

              hence thesis by A4, A7, A61, A64, A81, A83, A87, A88, Th22, XBOOLE_1: 63;

            end;

          end;

          hence thesis;

        end;

      end;

      hence thesis;

    end;

    theorem :: JORDAN19:26

    

     Th26: for C be Simple_closed_curve holds for p be Point of ( TOP-REAL 2) st ( W-bound C) < (p `1 ) & (p `1 ) < ( E-bound C) holds not (p in ( North_Arc C) & p in ( South_Arc C))

    proof

      let C be Simple_closed_curve;

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

      reconsider p9 = p as Point of ( Euclid 2) by EUCLID: 22;

      assume that

       A1: ( W-bound C) < (p `1 ) and

       A2: (p `1 ) < ( E-bound C) and

       A3: p in ( North_Arc C) and

       A4: p in ( South_Arc C);

      set s = ( min (((p `1 ) - ( W-bound C)),(( E-bound C) - (p `1 ))));

      

       A5: ( W-bound C) = (( W-bound C) + 0 );

      

       A6: (p `1 ) = ((p `1 ) + 0 );

      

       A7: ((p `1 ) - ( W-bound C)) > 0 by A1, A5, XREAL_1: 20;

      (( E-bound C) - (p `1 )) > 0 by A2, A6, XREAL_1: 20;

      then

       A8: s > 0 by A7, XXREAL_0: 15;

      now

        let r be Real;

        reconsider rr = r as Real;

        assume that

         A9: 0 < r and

         A10: r < s;

        

         A11: (r / 8) > 0 by A9, XREAL_1: 139;

        reconsider G = ( Ball (p9,(r / 8))) as a_neighborhood of p by A9, GOBOARD6: 2, XREAL_1: 139;

        consider k1 be Nat such that

         A12: for m be Nat st m > k1 holds (( Upper_Appr C) . m) meets G by A3, KURATO_2:def 1;

        consider k2 be Nat such that

         A13: for m be Nat st m > k2 holds (( Lower_Appr C) . m) meets G by A4, KURATO_2:def 1;

        reconsider k = ( max (k1,k2)) as Nat by TARSKI: 1;

        

         A14: k >= k1 by XXREAL_0: 25;

        set z9 = ( max ((( N-bound C) - ( S-bound C)),(( E-bound C) - ( W-bound C))));

        set z = ( max (z9,(r / 8)));

        (z / (r / 8)) >= 1 by A11, XREAL_1: 181, XXREAL_0: 25;

        then ( log (2,(z / (r / 8)))) >= ( log (2,1)) by PRE_FF: 10;

        then ( log (2,(z / (r / 8)))) >= 0 by POWER: 51;

        then

        reconsider m9 = [\( log (2,(z / (r / 8))))/] as Nat by INT_1: 53;

        

         A15: (2 to_power (m9 + 1)) > 0 by POWER: 34;

        set N = (2 to_power (m9 + 1));

        ( log (2,(z / (r / 8)))) < ((m9 + 1) * 1) by INT_1: 29;

        then ( log (2,(z / (r / 8)))) < ((m9 + 1) * ( log (2,2))) by POWER: 52;

        then ( log (2,(z / (r / 8)))) < ( log (2,(2 to_power (m9 + 1)))) by POWER: 55;

        then (z / (r / 8)) < N by A15, PRE_FF: 10;

        then ((z / (r / 8)) * (r / 8)) < (N * (r / 8)) by A11, XREAL_1: 68;

        then z < (N * (r / 8)) by A11, XCMPLX_1: 87;

        then (z / N) < ((N * (r / 8)) / N) by A15, XREAL_1: 74;

        then (z / N) < (((r / 8) / N) * N);

        then

         A16: (z / N) < (r / 8) by A15, XCMPLX_1: 87;

        (z / N) >= (z9 / N) by A15, XREAL_1: 72, XXREAL_0: 25;

        then

         A17: (z9 / N) < (r / 8) by A16, XXREAL_0: 2;

        reconsider W = ( max (k,m9)) as Nat by TARSKI: 1;

        set m = (W + 1);

        

         A18: ( len ( Gauge (C,m))) = ( width ( Gauge (C,m))) by JORDAN8:def 1;

        ( max (k,m9)) >= k by XXREAL_0: 25;

        then ( max (k,m9)) >= k1 by A14, XXREAL_0: 2;

        then m > k1 by NAT_1: 13;

        then (( Upper_Appr C) . m) meets G by A12;

        then ( Upper_Arc ( L~ ( Cage (C,m)))) meets G by Def1;

        then

        consider p1 be object such that

         A19: p1 in ( Upper_Arc ( L~ ( Cage (C,m)))) and

         A20: p1 in G by XBOOLE_0: 3;

        reconsider p1 as Point of ( TOP-REAL 2) by A19;

        reconsider p19 = p1 as Point of ( Euclid 2) by EUCLID: 22;

        set f = ( Upper_Seq (C,m));

        

         A21: ( Upper_Arc ( L~ ( Cage (C,m)))) = ( L~ ( Upper_Seq (C,m))) by JORDAN1G: 55;

        then

        consider i1 be Nat such that

         A22: 1 <= i1 and

         A23: (i1 + 1) <= ( len f) and

         A24: p1 in ( LSeg ((f /. i1),(f /. (i1 + 1)))) by A19, SPPOL_2: 14;

        reconsider c1 = (f /. i1) as Point of ( Euclid 2) by EUCLID: 22;

        reconsider c2 = (f /. (i1 + 1)) as Point of ( Euclid 2) by EUCLID: 22;

        

         A25: f is_sequence_on ( Gauge (C,m)) by JORDAN1G: 4;

        i1 < ( len f) by A23, NAT_1: 13;

        then i1 in ( Seg ( len f)) by A22, FINSEQ_1: 1;

        then

         A26: i1 in ( dom f) by FINSEQ_1:def 3;

        then

        consider ii1,jj1 be Nat such that

         A27: [ii1, jj1] in ( Indices ( Gauge (C,m))) and

         A28: (f /. i1) = (( Gauge (C,m)) * (ii1,jj1)) by A25, GOBOARD1:def 9;

        

         A29: ( N-bound C) > (( S-bound C) + 0 ) by TOPREAL5: 16;

        

         A30: ( E-bound C) > (( W-bound C) + 0 ) by TOPREAL5: 17;

        

         A31: (( N-bound C) - ( S-bound C)) > 0 by A29, XREAL_1: 20;

        

         A32: (( E-bound C) - ( W-bound C)) > 0 by A30, XREAL_1: 20;

        

         A33: (2 |^ (m9 + 1)) > 0 by A15, POWER: 41;

        ( max (k,m9)) >= m9 by XXREAL_0: 25;

        then m > m9 by NAT_1: 13;

        then m >= (m9 + 1) by NAT_1: 13;

        then

         A34: (2 |^ m) >= (2 |^ (m9 + 1)) by PREPOWER: 93;

        then

         A35: ((( N-bound C) - ( S-bound C)) / (2 |^ m)) <= ((( N-bound C) - ( S-bound C)) / (2 |^ (m9 + 1))) by A31, A33, XREAL_1: 118;

        

         A36: ((( E-bound C) - ( W-bound C)) / (2 |^ m)) <= ((( E-bound C) - ( W-bound C)) / (2 |^ (m9 + 1))) by A32, A33, A34, XREAL_1: 118;

        

         A37: ((( N-bound C) - ( S-bound C)) / N) <= (z9 / N) by A15, XREAL_1: 72, XXREAL_0: 25;

        

         A38: ((( E-bound C) - ( W-bound C)) / N) <= (z9 / N) by A15, XREAL_1: 72, XXREAL_0: 25;

        

         A39: ((( N-bound C) - ( S-bound C)) / (2 |^ (m9 + 1))) <= (z9 / N) by A37, POWER: 41;

        

         A40: ((( E-bound C) - ( W-bound C)) / (2 |^ (m9 + 1))) <= (z9 / N) by A38, POWER: 41;

        

         A41: ((( N-bound C) - ( S-bound C)) / (2 |^ m)) <= (z9 / N) by A35, A39, XXREAL_0: 2;

        

         A42: ((( E-bound C) - ( W-bound C)) / (2 |^ m)) <= (z9 / N) by A36, A40, XXREAL_0: 2;

        then ( dist ((f /. i1),(f /. (i1 + 1)))) <= (z9 / N) by A22, A23, A25, A41, Th6;

        then ( dist ((f /. i1),(f /. (i1 + 1)))) < (r / 8) by A17, XXREAL_0: 2;

        then ( dist (c1,c2)) < (r / 8) by TOPREAL6:def 1;

        then

         A43: |.((f /. i1) - (f /. (i1 + 1))).| < (r / 8) by SPPOL_1: 39;

         |.(p1 - (f /. i1)).| <= |.((f /. i1) - (f /. (i1 + 1))).| by A24, JGRAPH_1: 36;

        then

         A44: |.(p1 - (f /. i1)).| < (r / 8) by A43, XXREAL_0: 2;

        ( dist (p19,p9)) < (r / 8) by A20, METRIC_1: 11;

        then |.(p - p1).| < (r / 8) by SPPOL_1: 39;

        then

         A45: ( |.(p - p1).| + |.(p1 - (f /. i1)).|) < ((r / (2 * 4)) + (r / (2 * 4))) by A44, XREAL_1: 8;

         |.(p - (f /. i1)).| <= ( |.(p - p1).| + |.(p1 - (f /. i1)).|) by TOPRNS_1: 34;

        then

         A46: |.(p - (f /. i1)).| < (r / 4) by A45, XXREAL_0: 2;

        then

         A47: ( dist (p9,c1)) < (r / 4) by SPPOL_1: 39;

        then

         A48: (f /. i1) in ( Ball (p9,(r / 4))) by METRIC_1: 11;

        

         A49: (f /. i1) in ( Upper_Arc ( L~ ( Cage (C,m)))) by A21, A26, SPPOL_2: 44;

        

         A50: k >= k2 by XXREAL_0: 25;

        ( max (k,m9)) >= k by XXREAL_0: 25;

        then ( max (k,m9)) >= k2 by A50, XXREAL_0: 2;

        then m > k2 by NAT_1: 13;

        then (( Lower_Appr C) . m) meets G by A13;

        then ( Lower_Arc ( L~ ( Cage (C,m)))) meets G by Def2;

        then

        consider p2 be object such that

         A51: p2 in ( Lower_Arc ( L~ ( Cage (C,m)))) and

         A52: p2 in G by XBOOLE_0: 3;

        reconsider p2 as Point of ( TOP-REAL 2) by A51;

        reconsider p29 = p2 as Point of ( Euclid 2) by EUCLID: 22;

        set g = ( Lower_Seq (C,m));

        

         A53: ( Lower_Arc ( L~ ( Cage (C,m)))) = ( L~ ( Lower_Seq (C,m))) by JORDAN1G: 56;

        then

        consider i2 be Nat such that

         A54: 1 <= i2 and

         A55: (i2 + 1) <= ( len g) and

         A56: p2 in ( LSeg ((g /. i2),(g /. (i2 + 1)))) by A51, SPPOL_2: 14;

        reconsider d1 = (g /. i2) as Point of ( Euclid 2) by EUCLID: 22;

        reconsider d2 = (g /. (i2 + 1)) as Point of ( Euclid 2) by EUCLID: 22;

        

         A57: g is_sequence_on ( Gauge (C,m)) by JORDAN1G: 5;

        i2 < ( len g) by A55, NAT_1: 13;

        then i2 in ( Seg ( len g)) by A54, FINSEQ_1: 1;

        then

         A58: i2 in ( dom g) by FINSEQ_1:def 3;

        then

        consider ii2,jj2 be Nat such that

         A59: [ii2, jj2] in ( Indices ( Gauge (C,m))) and

         A60: (g /. i2) = (( Gauge (C,m)) * (ii2,jj2)) by A57, GOBOARD1:def 9;

        ( dist ((g /. i2),(g /. (i2 + 1)))) <= (z9 / N) by A41, A42, A54, A55, A57, Th6;

        then ( dist ((g /. i2),(g /. (i2 + 1)))) < (r / 8) by A17, XXREAL_0: 2;

        then ( dist (d1,d2)) < (r / 8) by TOPREAL6:def 1;

        then

         A61: |.((g /. i2) - (g /. (i2 + 1))).| < (r / 8) by SPPOL_1: 39;

         |.(p2 - (g /. i2)).| <= |.((g /. i2) - (g /. (i2 + 1))).| by A56, JGRAPH_1: 36;

        then

         A62: |.(p2 - (g /. i2)).| < (r / 8) by A61, XXREAL_0: 2;

        ( dist (p29,p9)) < (r / 8) by A52, METRIC_1: 11;

        then |.(p - p2).| < (r / 8) by SPPOL_1: 39;

        then

         A63: ( |.(p - p2).| + |.(p2 - (g /. i2)).|) < ((r / (2 * 4)) + (r / (2 * 4))) by A62, XREAL_1: 8;

         |.(p - (g /. i2)).| <= ( |.(p - p2).| + |.(p2 - (g /. i2)).|) by TOPRNS_1: 34;

        then

         A64: |.(p - (g /. i2)).| < (r / 4) by A63, XXREAL_0: 2;

        then

         A65: ( dist (p9,d1)) < (r / 4) by SPPOL_1: 39;

        then

         A66: (g /. i2) in ( Ball (p9,(r / 4))) by METRIC_1: 11;

        

         A67: (g /. i2) in ( Lower_Arc ( L~ ( Cage (C,m)))) by A53, A58, SPPOL_2: 44;

        set Gij = (( Gauge (C,m)) * (ii2,jj1));

        set Gji = (( Gauge (C,m)) * (ii1,jj2));

        reconsider Gij9 = Gij, Gji9 = Gji as Point of ( Euclid 2) by EUCLID: 22;

        

         A68: 1 <= ii1 by A27, MATRIX_0: 32;

        

         A69: ii1 <= ( len ( Gauge (C,m))) by A27, MATRIX_0: 32;

        

         A70: 1 <= jj1 by A27, MATRIX_0: 32;

        

         A71: jj1 <= ( width ( Gauge (C,m))) by A27, MATRIX_0: 32;

        

         A72: 1 <= ii2 by A59, MATRIX_0: 32;

        

         A73: ii2 <= ( len ( Gauge (C,m))) by A59, MATRIX_0: 32;

        

         A74: 1 <= jj2 by A59, MATRIX_0: 32;

        

         A75: jj2 <= ( width ( Gauge (C,m))) by A59, MATRIX_0: 32;

        

         A76: ( len f) >= 3 by JORDAN1E: 15;

        

         A77: ( len g) >= 3 by JORDAN1E: 15;

        

         A78: ( len f) >= 1 by A76, XXREAL_0: 2;

        

         A79: ( len g) >= 1 by A77, XXREAL_0: 2;

        

         A80: ( len f) in ( Seg ( len f)) by A78, FINSEQ_1: 1;

        

         A81: ( len g) in ( Seg ( len g)) by A79, FINSEQ_1: 1;

        

         A82: ( len f) in ( dom f) by A80, FINSEQ_1:def 3;

        

         A83: ( len g) in ( dom g) by A81, FINSEQ_1:def 3;

        

         A84: (r / 4) < r by A9, XREAL_1: 223;

        

         A85: (r / 2) < r by A9, XREAL_1: 216;

        

         A86: s <= ((p `1 ) - ( W-bound C)) by XXREAL_0: 17;

        

         A87: s <= (( E-bound C) - (p `1 )) by XXREAL_0: 17;

         A88:

        now

          assume 1 >= ii1;

          then

           A89: ii1 = 1 by A68, XXREAL_0: 1;

          ( dist (p9,c1)) < r by A47, A84, XXREAL_0: 2;

          then ( dist (p9,c1)) < s by A10, XXREAL_0: 2;

          then

           A90: ( dist (p9,c1)) < ((p `1 ) - ( W-bound C)) by A86, XXREAL_0: 2;

          

           A91: ((p `1 ) - ((f /. i1) `1 )) <= |.((p `1 ) - ((f /. i1) `1 )).| by ABSVALUE: 4;

           |.((p `1 ) - ((f /. i1) `1 )).| <= |.(p - (f /. i1)).| by JGRAPH_1: 34;

          then ((p `1 ) - ((f /. i1) `1 )) <= |.(p - (f /. i1)).| by A91, XXREAL_0: 2;

          then ((p `1 ) - ( W-bound ( L~ ( Cage (C,m))))) <= |.(p - (f /. i1)).| by A18, A28, A70, A71, A89, JORDAN1A: 73;

          then ((p `1 ) - ( W-bound ( L~ ( Cage (C,m))))) <= ( dist (p9,c1)) by SPPOL_1: 39;

          then ((p `1 ) - ( W-bound ( L~ ( Cage (C,m))))) < ((p `1 ) - ( W-bound C)) by A90, XXREAL_0: 2;

          then ( W-bound ( L~ ( Cage (C,m)))) > ( W-bound C) by XREAL_1: 13;

          hence contradiction by Th11;

        end;

         A92:

        now

          assume ii1 >= ( len ( Gauge (C,m)));

          then

           A93: ii1 = ( len ( Gauge (C,m))) by A69, XXREAL_0: 1;

          ((( Gauge (C,m)) * (( len ( Gauge (C,m))),jj1)) `1 ) = ( E-bound ( L~ ( Cage (C,m)))) by A18, A70, A71, JORDAN1A: 71;

          

          then (f /. i1) = ( E-max ( L~ ( Cage (C,m)))) by A21, A26, A28, A93, JORDAN1J: 46, SPPOL_2: 44

          .= (f /. ( len f)) by JORDAN1F: 7;

          then i1 = ( len f) by A26, A82, PARTFUN2: 10;

          hence contradiction by A23, NAT_1: 13;

        end;

         A94:

        now

          assume ii2 <= 1;

          then

           A95: ii2 = 1 by A72, XXREAL_0: 1;

          ((( Gauge (C,m)) * (1,jj2)) `1 ) = ( W-bound ( L~ ( Cage (C,m)))) by A18, A74, A75, JORDAN1A: 73;

          

          then (g /. i2) = ( W-min ( L~ ( Cage (C,m)))) by A53, A58, A60, A95, JORDAN1J: 47, SPPOL_2: 44

          .= (g /. ( len g)) by JORDAN1F: 8;

          then i2 = ( len g) by A58, A83, PARTFUN2: 10;

          hence contradiction by A55, NAT_1: 13;

        end;

         A96:

        now

          assume ii2 >= ( len ( Gauge (C,m)));

          then

           A97: ii2 = ( len ( Gauge (C,m))) by A73, XXREAL_0: 1;

          ( dist (p9,d1)) < r by A65, A84, XXREAL_0: 2;

          then ( dist (p9,d1)) < s by A10, XXREAL_0: 2;

          then

           A98: ( dist (p9,d1)) < (( E-bound C) - (p `1 )) by A87, XXREAL_0: 2;

          

           A99: (((g /. i2) `1 ) - (p `1 )) <= |.(((g /. i2) `1 ) - (p `1 )).| by ABSVALUE: 4;

           |.(((g /. i2) `1 ) - (p `1 )).| <= |.((g /. i2) - p).| by JGRAPH_1: 34;

          then |.(((g /. i2) `1 ) - (p `1 )).| <= |.(p - (g /. i2)).| by TOPRNS_1: 27;

          then (((g /. i2) `1 ) - (p `1 )) <= |.(p - (g /. i2)).| by A99, XXREAL_0: 2;

          then (( E-bound ( L~ ( Cage (C,m)))) - (p `1 )) <= |.(p - (g /. i2)).| by A18, A60, A74, A75, A97, JORDAN1A: 71;

          then (( E-bound ( L~ ( Cage (C,m)))) - (p `1 )) <= ( dist (p9,d1)) by SPPOL_1: 39;

          then (( E-bound ( L~ ( Cage (C,m)))) - (p `1 )) < (( E-bound C) - (p `1 )) by A98, XXREAL_0: 2;

          then ( E-bound ( L~ ( Cage (C,m)))) < ( E-bound C) by XREAL_1: 13;

          hence contradiction by Th9;

        end;

        

         A100: ( Ball (p9,(rr / 4))) c= ( Ball (p9,rr)) by A84, PCOMPS_1: 1;

        

         A101: (Gij `1 ) = ((( Gauge (C,m)) * (ii2,1)) `1 ) by A70, A71, A72, A73, GOBOARD5: 2

        .= ((g /. i2) `1 ) by A60, A72, A73, A74, A75, GOBOARD5: 2;

        

         A102: (Gij `2 ) = ((( Gauge (C,m)) * (1,jj1)) `2 ) by A70, A71, A72, A73, GOBOARD5: 1

        .= ((f /. i1) `2 ) by A28, A68, A69, A70, A71, GOBOARD5: 1;

        

         A103: (Gji `1 ) = ((( Gauge (C,m)) * (ii1,1)) `1 ) by A68, A69, A74, A75, GOBOARD5: 2

        .= ((f /. i1) `1 ) by A28, A68, A69, A70, A71, GOBOARD5: 2;

        

         A104: (Gji `2 ) = ((( Gauge (C,m)) * (1,jj2)) `2 ) by A68, A69, A74, A75, GOBOARD5: 1

        .= ((g /. i2) `2 ) by A60, A72, A73, A74, A75, GOBOARD5: 1;

        

         A105: |.(((g /. i2) `1 ) - (p `1 )).| <= |.((g /. i2) - p).| by JGRAPH_1: 34;

        

         A106: |.(((f /. i1) `2 ) - (p `2 )).| <= |.((f /. i1) - p).| by JGRAPH_1: 34;

        

         A107: |.(((g /. i2) `1 ) - (p `1 )).| <= |.(p - (g /. i2)).| by A105, TOPRNS_1: 27;

        

         A108: |.(((f /. i1) `2 ) - (p `2 )).| <= |.(p - (f /. i1)).| by A106, TOPRNS_1: 27;

        

         A109: |.(((g /. i2) `1 ) - (p `1 )).| <= (r / 4) by A64, A107, XXREAL_0: 2;

         |.(((f /. i1) `2 ) - (p `2 )).| <= (r / 4) by A46, A108, XXREAL_0: 2;

        then ( |.(((g /. i2) `1 ) - (p `1 )).| + |.(((f /. i1) `2 ) - (p `2 )).|) <= ((r / (2 * 2)) + (r / (2 * 2))) by A109, XREAL_1: 7;

        then

         A110: ( |.(((g /. i2) `1 ) - (p `1 )).| + |.(((f /. i1) `2 ) - (p `2 )).|) < r by A85, XXREAL_0: 2;

        

         A111: |.(((f /. i1) `1 ) - (p `1 )).| <= |.((f /. i1) - p).| by JGRAPH_1: 34;

        

         A112: |.(((g /. i2) `2 ) - (p `2 )).| <= |.((g /. i2) - p).| by JGRAPH_1: 34;

        

         A113: |.(((f /. i1) `1 ) - (p `1 )).| <= |.(p - (f /. i1)).| by A111, TOPRNS_1: 27;

        

         A114: |.(((g /. i2) `2 ) - (p `2 )).| <= |.(p - (g /. i2)).| by A112, TOPRNS_1: 27;

        

         A115: |.(((f /. i1) `1 ) - (p `1 )).| <= (r / 4) by A46, A113, XXREAL_0: 2;

         |.(((g /. i2) `2 ) - (p `2 )).| <= (r / 4) by A64, A114, XXREAL_0: 2;

        then ( |.(((f /. i1) `1 ) - (p `1 )).| + |.(((g /. i2) `2 ) - (p `2 )).|) <= ((r / (2 * 2)) + (r / (2 * 2))) by A115, XREAL_1: 7;

        then

         A116: ( |.(((f /. i1) `1 ) - (p `1 )).| + |.(((g /. i2) `2 ) - (p `2 )).|) < r by A85, XXREAL_0: 2;

         |.(Gij - p).| <= ( |.(((g /. i2) `1 ) - (p `1 )).| + |.(((f /. i1) `2 ) - (p `2 )).|) by A101, A102, JGRAPH_1: 32;

        then |.(Gij - p).| < r by A110, XXREAL_0: 2;

        then ( dist (Gij9,p9)) < r by SPPOL_1: 39;

        then

         A117: Gij in ( Ball (p9,r)) by METRIC_1: 11;

         |.(Gji - p).| <= ( |.(((f /. i1) `1 ) - (p `1 )).| + |.(((g /. i2) `2 ) - (p `2 )).|) by A103, A104, JGRAPH_1: 32;

        then |.(Gji - p).| < r by A116, XXREAL_0: 2;

        then ( dist (Gji9,p9)) < r by SPPOL_1: 39;

        then

         A118: Gji in ( Ball (p9,r)) by METRIC_1: 11;

        

         A119: ( LSeg ((g /. i2),Gij)) c= ( Ball (p9,rr)) by A66, A100, A117, TOPREAL3: 21;

        

         A120: ( LSeg (Gij,(f /. i1))) c= ( Ball (p9,rr)) by A48, A100, A117, TOPREAL3: 21;

        

         A121: ( LSeg ((g /. i2),Gji)) c= ( Ball (p9,rr)) by A66, A100, A118, TOPREAL3: 21;

        

         A122: ( LSeg (Gji,(f /. i1))) c= ( Ball (p9,rr)) by A48, A100, A118, TOPREAL3: 21;

        now

          per cases ;

            suppose

             A123: jj2 <= jj1;

            (( LSeg ((g /. i2),Gij)) \/ ( LSeg (Gij,(f /. i1)))) c= ( Ball (p9,r))

            proof

              let x be object;

              assume

               A124: x in (( LSeg ((g /. i2),Gij)) \/ ( LSeg (Gij,(f /. i1))));

              then

              reconsider x9 = x as Point of ( TOP-REAL 2);

              now

                per cases by A124, XBOOLE_0:def 3;

                  suppose x9 in ( LSeg ((g /. i2),Gij));

                  hence x9 in ( Ball (p9,r)) by A119;

                end;

                  suppose x9 in ( LSeg (Gij,(f /. i1)));

                  hence x9 in ( Ball (p9,r)) by A120;

                end;

              end;

              hence thesis;

            end;

            hence ( Ball (p9,r)) meets ( Upper_Arc C) by A28, A49, A60, A67, A71, A74, A88, A92, A94, A96, A123, JORDAN15: 48, XBOOLE_1: 63;

          end;

            suppose

             A125: jj1 < jj2;

            (( LSeg ((f /. i1),Gji)) \/ ( LSeg (Gji,(g /. i2)))) c= ( Ball (p9,r))

            proof

              let x be object;

              assume

               A126: x in (( LSeg ((f /. i1),Gji)) \/ ( LSeg (Gji,(g /. i2))));

              then

              reconsider x9 = x as Point of ( TOP-REAL 2);

              now

                per cases by A126, XBOOLE_0:def 3;

                  suppose x9 in ( LSeg ((f /. i1),Gji));

                  hence x9 in ( Ball (p9,r)) by A122;

                end;

                  suppose x9 in ( LSeg (Gji,(g /. i2)));

                  hence x9 in ( Ball (p9,r)) by A121;

                end;

              end;

              hence thesis;

            end;

            hence ( Ball (p9,r)) meets ( Upper_Arc C) by A28, A49, A60, A67, A70, A75, A88, A92, A94, A96, A125, Th25, XBOOLE_1: 63;

          end;

        end;

        hence ( Ball (p9,r)) meets ( Upper_Arc C);

      end;

      then p in ( Cl ( Upper_Arc C)) by A8, GOBOARD6: 93;

      then

       A127: p in ( Upper_Arc C) by PRE_TOPC: 22;

      now

        let r be Real;

        reconsider rr = r as Real;

        assume that

         A128: 0 < r and

         A129: r < s;

        

         A130: (r / 8) > 0 by A128, XREAL_1: 139;

        reconsider G = ( Ball (p9,(r / 8))) as a_neighborhood of p by A128, GOBOARD6: 2, XREAL_1: 139;

        consider k1 be Nat such that

         A131: for m be Nat st m > k1 holds (( Upper_Appr C) . m) meets G by A3, KURATO_2:def 1;

        consider k2 be Nat such that

         A132: for m be Nat st m > k2 holds (( Lower_Appr C) . m) meets G by A4, KURATO_2:def 1;

        set k = ( max (k1,k2));

        

         A133: k >= k1 by XXREAL_0: 25;

        set z9 = ( max ((( N-bound C) - ( S-bound C)),(( E-bound C) - ( W-bound C))));

        set z = ( max (z9,(r / 8)));

        (z / (r / 8)) >= 1 by A130, XREAL_1: 181, XXREAL_0: 25;

        then ( log (2,(z / (r / 8)))) >= ( log (2,1)) by PRE_FF: 10;

        then ( log (2,(z / (r / 8)))) >= 0 by POWER: 51;

        then

        reconsider m9 = [\( log (2,(z / (r / 8))))/] as Nat by INT_1: 53;

        

         A134: (2 to_power (m9 + 1)) > 0 by POWER: 34;

        set N = (2 to_power (m9 + 1));

        ( log (2,(z / (r / 8)))) < ((m9 + 1) * 1) by INT_1: 29;

        then ( log (2,(z / (r / 8)))) < ((m9 + 1) * ( log (2,2))) by POWER: 52;

        then ( log (2,(z / (r / 8)))) < ( log (2,(2 to_power (m9 + 1)))) by POWER: 55;

        then (z / (r / 8)) < N by A134, PRE_FF: 10;

        then ((z / (r / 8)) * (r / 8)) < (N * (r / 8)) by A130, XREAL_1: 68;

        then z < (N * (r / 8)) by A130, XCMPLX_1: 87;

        then (z / N) < ((N * (r / 8)) / N) by A134, XREAL_1: 74;

        then (z / N) < (((r / 8) / N) * N);

        then

         A135: (z / N) < (r / 8) by A134, XCMPLX_1: 87;

        (z / N) >= (z9 / N) by A134, XREAL_1: 72, XXREAL_0: 25;

        then

         A136: (z9 / N) < (r / 8) by A135, XXREAL_0: 2;

        reconsider W = ( max (k,m9)) as Nat by TARSKI: 1;

        set m = (W + 1);

        reconsider m as Nat;

        

         A137: ( len ( Gauge (C,m))) = ( width ( Gauge (C,m))) by JORDAN8:def 1;

        ( max (k,m9)) >= k by XXREAL_0: 25;

        then ( max (k,m9)) >= k1 by A133, XXREAL_0: 2;

        then m > k1 by NAT_1: 13;

        then (( Upper_Appr C) . m) meets G by A131;

        then ( Upper_Arc ( L~ ( Cage (C,m)))) meets G by Def1;

        then

        consider p1 be object such that

         A138: p1 in ( Upper_Arc ( L~ ( Cage (C,m)))) and

         A139: p1 in G by XBOOLE_0: 3;

        reconsider p1 as Point of ( TOP-REAL 2) by A138;

        reconsider p19 = p1 as Point of ( Euclid 2) by EUCLID: 22;

        set f = ( Upper_Seq (C,m));

        

         A140: ( Upper_Arc ( L~ ( Cage (C,m)))) = ( L~ ( Upper_Seq (C,m))) by JORDAN1G: 55;

        then

        consider i1 be Nat such that

         A141: 1 <= i1 and

         A142: (i1 + 1) <= ( len f) and

         A143: p1 in ( LSeg ((f /. i1),(f /. (i1 + 1)))) by A138, SPPOL_2: 14;

        reconsider c1 = (f /. i1) as Point of ( Euclid 2) by EUCLID: 22;

        reconsider c2 = (f /. (i1 + 1)) as Point of ( Euclid 2) by EUCLID: 22;

        

         A144: f is_sequence_on ( Gauge (C,m)) by JORDAN1G: 4;

        i1 < ( len f) by A142, NAT_1: 13;

        then i1 in ( Seg ( len f)) by A141, FINSEQ_1: 1;

        then

         A145: i1 in ( dom f) by FINSEQ_1:def 3;

        then

        consider ii1,jj1 be Nat such that

         A146: [ii1, jj1] in ( Indices ( Gauge (C,m))) and

         A147: (f /. i1) = (( Gauge (C,m)) * (ii1,jj1)) by A144, GOBOARD1:def 9;

        

         A148: ( N-bound C) > (( S-bound C) + 0 ) by TOPREAL5: 16;

        

         A149: ( E-bound C) > (( W-bound C) + 0 ) by TOPREAL5: 17;

        

         A150: (( N-bound C) - ( S-bound C)) > 0 by A148, XREAL_1: 20;

        

         A151: (( E-bound C) - ( W-bound C)) > 0 by A149, XREAL_1: 20;

        

         A152: (2 |^ (m9 + 1)) > 0 by A134, POWER: 41;

        ( max (k,m9)) >= m9 by XXREAL_0: 25;

        then m > m9 by NAT_1: 13;

        then m >= (m9 + 1) by NAT_1: 13;

        then

         A153: (2 |^ m) >= (2 |^ (m9 + 1)) by PREPOWER: 93;

        then

         A154: ((( N-bound C) - ( S-bound C)) / (2 |^ m)) <= ((( N-bound C) - ( S-bound C)) / (2 |^ (m9 + 1))) by A150, A152, XREAL_1: 118;

        

         A155: ((( E-bound C) - ( W-bound C)) / (2 |^ m)) <= ((( E-bound C) - ( W-bound C)) / (2 |^ (m9 + 1))) by A151, A152, A153, XREAL_1: 118;

        

         A156: ((( N-bound C) - ( S-bound C)) / N) <= (z9 / N) by A134, XREAL_1: 72, XXREAL_0: 25;

        

         A157: ((( E-bound C) - ( W-bound C)) / N) <= (z9 / N) by A134, XREAL_1: 72, XXREAL_0: 25;

        

         A158: ((( N-bound C) - ( S-bound C)) / (2 |^ (m9 + 1))) <= (z9 / N) by A156, POWER: 41;

        

         A159: ((( E-bound C) - ( W-bound C)) / (2 |^ (m9 + 1))) <= (z9 / N) by A157, POWER: 41;

        

         A160: ((( N-bound C) - ( S-bound C)) / (2 |^ m)) <= (z9 / N) by A154, A158, XXREAL_0: 2;

        

         A161: ((( E-bound C) - ( W-bound C)) / (2 |^ m)) <= (z9 / N) by A155, A159, XXREAL_0: 2;

        then ( dist ((f /. i1),(f /. (i1 + 1)))) <= (z9 / N) by A141, A142, A144, A160, Th6;

        then ( dist ((f /. i1),(f /. (i1 + 1)))) < (r / 8) by A136, XXREAL_0: 2;

        then ( dist (c1,c2)) < (r / 8) by TOPREAL6:def 1;

        then

         A162: |.((f /. i1) - (f /. (i1 + 1))).| < (r / 8) by SPPOL_1: 39;

         |.(p1 - (f /. i1)).| <= |.((f /. i1) - (f /. (i1 + 1))).| by A143, JGRAPH_1: 36;

        then

         A163: |.(p1 - (f /. i1)).| < (r / 8) by A162, XXREAL_0: 2;

        ( dist (p19,p9)) < (r / 8) by A139, METRIC_1: 11;

        then |.(p - p1).| < (r / 8) by SPPOL_1: 39;

        then

         A164: ( |.(p - p1).| + |.(p1 - (f /. i1)).|) < ((r / (2 * 4)) + (r / (2 * 4))) by A163, XREAL_1: 8;

         |.(p - (f /. i1)).| <= ( |.(p - p1).| + |.(p1 - (f /. i1)).|) by TOPRNS_1: 34;

        then

         A165: |.(p - (f /. i1)).| < (r / 4) by A164, XXREAL_0: 2;

        then

         A166: ( dist (p9,c1)) < (r / 4) by SPPOL_1: 39;

        then

         A167: (f /. i1) in ( Ball (p9,(r / 4))) by METRIC_1: 11;

        

         A168: (f /. i1) in ( Upper_Arc ( L~ ( Cage (C,m)))) by A140, A145, SPPOL_2: 44;

        

         A169: k >= k2 by XXREAL_0: 25;

        ( max (k,m9)) >= k by XXREAL_0: 25;

        then ( max (k,m9)) >= k2 by A169, XXREAL_0: 2;

        then m > k2 by NAT_1: 13;

        then (( Lower_Appr C) . m) meets G by A132;

        then ( Lower_Arc ( L~ ( Cage (C,m)))) meets G by Def2;

        then

        consider p2 be object such that

         A170: p2 in ( Lower_Arc ( L~ ( Cage (C,m)))) and

         A171: p2 in G by XBOOLE_0: 3;

        reconsider p2 as Point of ( TOP-REAL 2) by A170;

        reconsider p29 = p2 as Point of ( Euclid 2) by EUCLID: 22;

        set g = ( Lower_Seq (C,m));

        

         A172: ( Lower_Arc ( L~ ( Cage (C,m)))) = ( L~ ( Lower_Seq (C,m))) by JORDAN1G: 56;

        then

        consider i2 be Nat such that

         A173: 1 <= i2 and

         A174: (i2 + 1) <= ( len g) and

         A175: p2 in ( LSeg ((g /. i2),(g /. (i2 + 1)))) by A170, SPPOL_2: 14;

        reconsider d1 = (g /. i2) as Point of ( Euclid 2) by EUCLID: 22;

        reconsider d2 = (g /. (i2 + 1)) as Point of ( Euclid 2) by EUCLID: 22;

        

         A176: g is_sequence_on ( Gauge (C,m)) by JORDAN1G: 5;

        i2 < ( len g) by A174, NAT_1: 13;

        then i2 in ( Seg ( len g)) by A173, FINSEQ_1: 1;

        then

         A177: i2 in ( dom g) by FINSEQ_1:def 3;

        then

        consider ii2,jj2 be Nat such that

         A178: [ii2, jj2] in ( Indices ( Gauge (C,m))) and

         A179: (g /. i2) = (( Gauge (C,m)) * (ii2,jj2)) by A176, GOBOARD1:def 9;

        ( dist ((g /. i2),(g /. (i2 + 1)))) <= (z9 / N) by A160, A161, A173, A174, A176, Th6;

        then ( dist ((g /. i2),(g /. (i2 + 1)))) < (r / 8) by A136, XXREAL_0: 2;

        then ( dist (d1,d2)) < (r / 8) by TOPREAL6:def 1;

        then

         A180: |.((g /. i2) - (g /. (i2 + 1))).| < (r / 8) by SPPOL_1: 39;

         |.(p2 - (g /. i2)).| <= |.((g /. i2) - (g /. (i2 + 1))).| by A175, JGRAPH_1: 36;

        then

         A181: |.(p2 - (g /. i2)).| < (r / 8) by A180, XXREAL_0: 2;

        ( dist (p29,p9)) < (r / 8) by A171, METRIC_1: 11;

        then |.(p - p2).| < (r / 8) by SPPOL_1: 39;

        then

         A182: ( |.(p - p2).| + |.(p2 - (g /. i2)).|) < ((r / (2 * 4)) + (r / (2 * 4))) by A181, XREAL_1: 8;

         |.(p - (g /. i2)).| <= ( |.(p - p2).| + |.(p2 - (g /. i2)).|) by TOPRNS_1: 34;

        then

         A183: |.(p - (g /. i2)).| < (r / 4) by A182, XXREAL_0: 2;

        then

         A184: ( dist (p9,d1)) < (r / 4) by SPPOL_1: 39;

        then

         A185: (g /. i2) in ( Ball (p9,(r / 4))) by METRIC_1: 11;

        

         A186: (g /. i2) in ( Lower_Arc ( L~ ( Cage (C,m)))) by A172, A177, SPPOL_2: 44;

        set Gij = (( Gauge (C,m)) * (ii2,jj1));

        set Gji = (( Gauge (C,m)) * (ii1,jj2));

        reconsider Gij9 = Gij, Gji9 = Gji as Point of ( Euclid 2) by EUCLID: 22;

        

         A187: 1 <= ii1 by A146, MATRIX_0: 32;

        

         A188: ii1 <= ( len ( Gauge (C,m))) by A146, MATRIX_0: 32;

        

         A189: 1 <= jj1 by A146, MATRIX_0: 32;

        

         A190: jj1 <= ( width ( Gauge (C,m))) by A146, MATRIX_0: 32;

        

         A191: 1 <= ii2 by A178, MATRIX_0: 32;

        

         A192: ii2 <= ( len ( Gauge (C,m))) by A178, MATRIX_0: 32;

        

         A193: 1 <= jj2 by A178, MATRIX_0: 32;

        

         A194: jj2 <= ( width ( Gauge (C,m))) by A178, MATRIX_0: 32;

        

         A195: ( len f) >= 3 by JORDAN1E: 15;

        

         A196: ( len g) >= 3 by JORDAN1E: 15;

        

         A197: ( len f) >= 1 by A195, XXREAL_0: 2;

        

         A198: ( len g) >= 1 by A196, XXREAL_0: 2;

        

         A199: ( len f) in ( Seg ( len f)) by A197, FINSEQ_1: 1;

        

         A200: ( len g) in ( Seg ( len g)) by A198, FINSEQ_1: 1;

        

         A201: ( len f) in ( dom f) by A199, FINSEQ_1:def 3;

        

         A202: ( len g) in ( dom g) by A200, FINSEQ_1:def 3;

        

         A203: (r / 4) < r by A128, XREAL_1: 223;

        

         A204: (r / 2) < r by A128, XREAL_1: 216;

        

         A205: s <= ((p `1 ) - ( W-bound C)) by XXREAL_0: 17;

        

         A206: s <= (( E-bound C) - (p `1 )) by XXREAL_0: 17;

         A207:

        now

          assume 1 >= ii1;

          then

           A208: ii1 = 1 by A187, XXREAL_0: 1;

          ( dist (p9,c1)) < r by A166, A203, XXREAL_0: 2;

          then ( dist (p9,c1)) < s by A129, XXREAL_0: 2;

          then

           A209: ( dist (p9,c1)) < ((p `1 ) - ( W-bound C)) by A205, XXREAL_0: 2;

          

           A210: ((p `1 ) - ((f /. i1) `1 )) <= |.((p `1 ) - ((f /. i1) `1 )).| by ABSVALUE: 4;

           |.((p `1 ) - ((f /. i1) `1 )).| <= |.(p - (f /. i1)).| by JGRAPH_1: 34;

          then ((p `1 ) - ((f /. i1) `1 )) <= |.(p - (f /. i1)).| by A210, XXREAL_0: 2;

          then ((p `1 ) - ( W-bound ( L~ ( Cage (C,m))))) <= |.(p - (f /. i1)).| by A137, A147, A189, A190, A208, JORDAN1A: 73;

          then ((p `1 ) - ( W-bound ( L~ ( Cage (C,m))))) <= ( dist (p9,c1)) by SPPOL_1: 39;

          then ((p `1 ) - ( W-bound ( L~ ( Cage (C,m))))) < ((p `1 ) - ( W-bound C)) by A209, XXREAL_0: 2;

          then ( W-bound ( L~ ( Cage (C,m)))) > ( W-bound C) by XREAL_1: 13;

          hence contradiction by Th11;

        end;

         A211:

        now

          assume ii1 >= ( len ( Gauge (C,m)));

          then

           A212: ii1 = ( len ( Gauge (C,m))) by A188, XXREAL_0: 1;

          ((( Gauge (C,m)) * (( len ( Gauge (C,m))),jj1)) `1 ) = ( E-bound ( L~ ( Cage (C,m)))) by A137, A189, A190, JORDAN1A: 71;

          

          then (f /. i1) = ( E-max ( L~ ( Cage (C,m)))) by A140, A145, A147, A212, JORDAN1J: 46, SPPOL_2: 44

          .= (f /. ( len f)) by JORDAN1F: 7;

          then i1 = ( len f) by A145, A201, PARTFUN2: 10;

          hence contradiction by A142, NAT_1: 13;

        end;

         A213:

        now

          assume ii2 <= 1;

          then

           A214: ii2 = 1 by A191, XXREAL_0: 1;

          ((( Gauge (C,m)) * (1,jj2)) `1 ) = ( W-bound ( L~ ( Cage (C,m)))) by A137, A193, A194, JORDAN1A: 73;

          

          then (g /. i2) = ( W-min ( L~ ( Cage (C,m)))) by A172, A177, A179, A214, JORDAN1J: 47, SPPOL_2: 44

          .= (g /. ( len g)) by JORDAN1F: 8;

          then i2 = ( len g) by A177, A202, PARTFUN2: 10;

          hence contradiction by A174, NAT_1: 13;

        end;

         A215:

        now

          assume ii2 >= ( len ( Gauge (C,m)));

          then

           A216: ii2 = ( len ( Gauge (C,m))) by A192, XXREAL_0: 1;

          ( dist (p9,d1)) < r by A184, A203, XXREAL_0: 2;

          then ( dist (p9,d1)) < s by A129, XXREAL_0: 2;

          then

           A217: ( dist (p9,d1)) < (( E-bound C) - (p `1 )) by A206, XXREAL_0: 2;

          

           A218: (((g /. i2) `1 ) - (p `1 )) <= |.(((g /. i2) `1 ) - (p `1 )).| by ABSVALUE: 4;

           |.(((g /. i2) `1 ) - (p `1 )).| <= |.((g /. i2) - p).| by JGRAPH_1: 34;

          then |.(((g /. i2) `1 ) - (p `1 )).| <= |.(p - (g /. i2)).| by TOPRNS_1: 27;

          then (((g /. i2) `1 ) - (p `1 )) <= |.(p - (g /. i2)).| by A218, XXREAL_0: 2;

          then (( E-bound ( L~ ( Cage (C,m)))) - (p `1 )) <= |.(p - (g /. i2)).| by A137, A179, A193, A194, A216, JORDAN1A: 71;

          then (( E-bound ( L~ ( Cage (C,m)))) - (p `1 )) <= ( dist (p9,d1)) by SPPOL_1: 39;

          then (( E-bound ( L~ ( Cage (C,m)))) - (p `1 )) < (( E-bound C) - (p `1 )) by A217, XXREAL_0: 2;

          then ( E-bound ( L~ ( Cage (C,m)))) < ( E-bound C) by XREAL_1: 13;

          hence contradiction by Th9;

        end;

        

         A219: ( Ball (p9,(rr / 4))) c= ( Ball (p9,rr)) by A203, PCOMPS_1: 1;

        

         A220: (Gij `1 ) = ((( Gauge (C,m)) * (ii2,1)) `1 ) by A189, A190, A191, A192, GOBOARD5: 2

        .= ((g /. i2) `1 ) by A179, A191, A192, A193, A194, GOBOARD5: 2;

        

         A221: (Gij `2 ) = ((( Gauge (C,m)) * (1,jj1)) `2 ) by A189, A190, A191, A192, GOBOARD5: 1

        .= ((f /. i1) `2 ) by A147, A187, A188, A189, A190, GOBOARD5: 1;

        

         A222: (Gji `1 ) = ((( Gauge (C,m)) * (ii1,1)) `1 ) by A187, A188, A193, A194, GOBOARD5: 2

        .= ((f /. i1) `1 ) by A147, A187, A188, A189, A190, GOBOARD5: 2;

        

         A223: (Gji `2 ) = ((( Gauge (C,m)) * (1,jj2)) `2 ) by A187, A188, A193, A194, GOBOARD5: 1

        .= ((g /. i2) `2 ) by A179, A191, A192, A193, A194, GOBOARD5: 1;

        

         A224: |.(((g /. i2) `1 ) - (p `1 )).| <= |.((g /. i2) - p).| by JGRAPH_1: 34;

        

         A225: |.(((f /. i1) `2 ) - (p `2 )).| <= |.((f /. i1) - p).| by JGRAPH_1: 34;

        

         A226: |.(((g /. i2) `1 ) - (p `1 )).| <= |.(p - (g /. i2)).| by A224, TOPRNS_1: 27;

        

         A227: |.(((f /. i1) `2 ) - (p `2 )).| <= |.(p - (f /. i1)).| by A225, TOPRNS_1: 27;

        

         A228: |.(((g /. i2) `1 ) - (p `1 )).| <= (r / 4) by A183, A226, XXREAL_0: 2;

         |.(((f /. i1) `2 ) - (p `2 )).| <= (r / 4) by A165, A227, XXREAL_0: 2;

        then ( |.(((g /. i2) `1 ) - (p `1 )).| + |.(((f /. i1) `2 ) - (p `2 )).|) <= ((r / (2 * 2)) + (r / (2 * 2))) by A228, XREAL_1: 7;

        then

         A229: ( |.(((g /. i2) `1 ) - (p `1 )).| + |.(((f /. i1) `2 ) - (p `2 )).|) < r by A204, XXREAL_0: 2;

        

         A230: |.(((f /. i1) `1 ) - (p `1 )).| <= |.((f /. i1) - p).| by JGRAPH_1: 34;

        

         A231: |.(((g /. i2) `2 ) - (p `2 )).| <= |.((g /. i2) - p).| by JGRAPH_1: 34;

        

         A232: |.(((f /. i1) `1 ) - (p `1 )).| <= |.(p - (f /. i1)).| by A230, TOPRNS_1: 27;

        

         A233: |.(((g /. i2) `2 ) - (p `2 )).| <= |.(p - (g /. i2)).| by A231, TOPRNS_1: 27;

        

         A234: |.(((f /. i1) `1 ) - (p `1 )).| <= (r / 4) by A165, A232, XXREAL_0: 2;

         |.(((g /. i2) `2 ) - (p `2 )).| <= (r / 4) by A183, A233, XXREAL_0: 2;

        then ( |.(((f /. i1) `1 ) - (p `1 )).| + |.(((g /. i2) `2 ) - (p `2 )).|) <= ((r / (2 * 2)) + (r / (2 * 2))) by A234, XREAL_1: 7;

        then

         A235: ( |.(((f /. i1) `1 ) - (p `1 )).| + |.(((g /. i2) `2 ) - (p `2 )).|) < r by A204, XXREAL_0: 2;

         |.(Gij - p).| <= ( |.(((g /. i2) `1 ) - (p `1 )).| + |.(((f /. i1) `2 ) - (p `2 )).|) by A220, A221, JGRAPH_1: 32;

        then |.(Gij - p).| < r by A229, XXREAL_0: 2;

        then ( dist (Gij9,p9)) < r by SPPOL_1: 39;

        then

         A236: Gij in ( Ball (p9,r)) by METRIC_1: 11;

         |.(Gji - p).| <= ( |.(((f /. i1) `1 ) - (p `1 )).| + |.(((g /. i2) `2 ) - (p `2 )).|) by A222, A223, JGRAPH_1: 32;

        then |.(Gji - p).| < r by A235, XXREAL_0: 2;

        then ( dist (Gji9,p9)) < r by SPPOL_1: 39;

        then

         A237: Gji in ( Ball (p9,r)) by METRIC_1: 11;

        

         A238: ( LSeg ((g /. i2),Gij)) c= ( Ball (p9,rr)) by A185, A219, A236, TOPREAL3: 21;

        

         A239: ( LSeg (Gij,(f /. i1))) c= ( Ball (p9,rr)) by A167, A219, A236, TOPREAL3: 21;

        

         A240: ( LSeg ((g /. i2),Gji)) c= ( Ball (p9,rr)) by A185, A219, A237, TOPREAL3: 21;

        

         A241: ( LSeg (Gji,(f /. i1))) c= ( Ball (p9,rr)) by A167, A219, A237, TOPREAL3: 21;

        now

          per cases ;

            suppose

             A242: jj2 <= jj1;

            (( LSeg ((g /. i2),Gij)) \/ ( LSeg (Gij,(f /. i1)))) c= ( Ball (p9,r))

            proof

              let x be object;

              assume

               A243: x in (( LSeg ((g /. i2),Gij)) \/ ( LSeg (Gij,(f /. i1))));

              then

              reconsider x9 = x as Point of ( TOP-REAL 2);

              now

                per cases by A243, XBOOLE_0:def 3;

                  suppose x9 in ( LSeg ((g /. i2),Gij));

                  hence x9 in ( Ball (p9,r)) by A238;

                end;

                  suppose x9 in ( LSeg (Gij,(f /. i1)));

                  hence x9 in ( Ball (p9,r)) by A239;

                end;

              end;

              hence thesis;

            end;

            hence ( Ball (p9,r)) meets ( Lower_Arc C) by A147, A168, A179, A186, A190, A193, A207, A211, A213, A215, A242, JORDAN15: 49, XBOOLE_1: 63;

          end;

            suppose

             A244: jj1 < jj2;

            (( LSeg ((f /. i1),Gji)) \/ ( LSeg (Gji,(g /. i2)))) c= ( Ball (p9,r))

            proof

              let x be object;

              assume

               A245: x in (( LSeg ((f /. i1),Gji)) \/ ( LSeg (Gji,(g /. i2))));

              then

              reconsider x9 = x as Point of ( TOP-REAL 2);

              now

                per cases by A245, XBOOLE_0:def 3;

                  suppose x9 in ( LSeg ((f /. i1),Gji));

                  hence x9 in ( Ball (p9,r)) by A241;

                end;

                  suppose x9 in ( LSeg (Gji,(g /. i2)));

                  hence x9 in ( Ball (p9,r)) by A240;

                end;

              end;

              hence thesis;

            end;

            hence ( Ball (p9,r)) meets ( Lower_Arc C) by A147, A168, A179, A186, A189, A194, A207, A211, A213, A215, A244, Th24, XBOOLE_1: 63;

          end;

        end;

        hence ( Ball (p9,r)) meets ( Lower_Arc C);

      end;

      then p in ( Cl ( Lower_Arc C)) by A8, GOBOARD6: 93;

      then p in ( Lower_Arc C) by PRE_TOPC: 22;

      then p in (( Upper_Arc C) /\ ( Lower_Arc C)) by A127, XBOOLE_0:def 4;

      then p in {( W-min C), ( E-max C)} by JORDAN6: 50;

      then p = ( W-min C) or p = ( E-max C) by TARSKI:def 2;

      hence contradiction by A1, A2, EUCLID: 52;

    end;

    theorem :: JORDAN19:27

    for C be Simple_closed_curve holds for p be Point of ( TOP-REAL 2) st (p `1 ) = ((( W-bound C) + ( E-bound C)) / 2) holds not (p in ( North_Arc C) & p in ( South_Arc C))

    proof

      let C be Simple_closed_curve;

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

      

       A1: ( W-bound C) < ( E-bound C) by SPRECT_1: 31;

      assume

       A2: (p `1 ) = ((( W-bound C) + ( E-bound C)) / 2);

      then

       A3: ( W-bound C) < (p `1 ) by A1, XREAL_1: 226;

      (p `1 ) < ( E-bound C) by A1, A2, XREAL_1: 226;

      hence thesis by A3, Th26;

    end;