jordan9.miz



    begin

    reserve i,j,k,n for Nat,

D for non empty set,

f,g for FinSequence of D;

    

     Lm1: for n st 1 <= n holds ((n -' 1) + 2) = (n + 1)

    proof

      let n;

      assume 1 <= n;

      

      hence ((n -' 1) + 2) = ((n + 2) -' 1) by NAT_D: 38

      .= (((n + 1) + 1) - 1) by NAT_D: 37

      .= (n + 1);

    end;

    theorem :: JORDAN9:1

    

     Th1: for T be non empty TopSpace holds for B,C1,C2,D be Subset of T st B is connected & C1 is_a_component_of D & C2 is_a_component_of D & B meets C1 & B meets C2 & B c= D holds C1 = C2

    proof

      let T be non empty TopSpace;

      let B,C1,C2,D be Subset of T;

      assume that

       A1: B is connected and

       A2: C1 is_a_component_of D & C2 is_a_component_of D and

       A3: B meets C1 and

       A4: B meets C2 & B c= D;

      

       A5: B <> {} by A3, XBOOLE_1: 65;

      B c= C1 & B c= C2 by A1, A2, A3, A4, GOBOARD9: 4;

      hence thesis by A2, A5, GOBOARD9: 1, XBOOLE_1: 68;

    end;

    theorem :: JORDAN9:2

    

     Th2: (for n holds (f | n) = (g | n)) implies f = g

    proof

      assume

       A1: for n holds (f | n) = (g | n);

       A2:

      now

        assume

         A3: ( len f) <> ( len g);

        per cases by A3, XXREAL_0: 1;

          suppose

           A4: ( len f) < ( len g);

          

           A5: (g | ( len g)) = g by FINSEQ_1: 58;

          (f | ( len g)) = f by A4, FINSEQ_1: 58;

          hence contradiction by A1, A4, A5;

        end;

          suppose

           A6: ( len g) < ( len f);

          then (f | ( len f)) = f & (g | ( len f)) = g by FINSEQ_1: 58;

          hence contradiction by A1, A6;

        end;

      end;

      (f | ( len f)) = f & (g | ( len g)) = g by FINSEQ_1: 58;

      hence thesis by A1, A2;

    end;

    theorem :: JORDAN9:3

    

     Th3: n in ( dom f) implies ex k st k in ( dom ( Rev f)) & (n + k) = (( len f) + 1) & (f /. n) = (( Rev f) /. k)

    proof

      assume

       A1: n in ( dom f);

      take k = ((( len f) + 1) -' n);

      1 <= n by A1, FINSEQ_3: 25;

      then (k + 1) <= (k + n) by XREAL_1: 6;

      then

       A2: ((k + 1) -' 1) <= ((k + n) -' 1) by NAT_D: 42;

      

       A3: n <= ( len f) by A1, FINSEQ_3: 25;

      then (n + 1) <= (( len f) + 1) by XREAL_1: 6;

      then

       A4: 1 <= k by NAT_D: 55;

      n <= (( len f) + 1) by A3, XREAL_1: 145;

      then

       A5: (k + n) = (( len f) + 1) by XREAL_1: 235;

      then ((k + n) -' 1) = ( len f) by NAT_D: 34;

      then k <= ( len f) by A2, NAT_D: 34;

      then k in ( dom f) by A4, FINSEQ_3: 25;

      hence thesis by A1, A5, FINSEQ_5: 57, FINSEQ_5: 66;

    end;

    theorem :: JORDAN9:4

    

     Th4: n in ( dom ( Rev f)) implies ex k st k in ( dom f) & (n + k) = (( len f) + 1) & (( Rev f) /. n) = (f /. k)

    proof

      assume n in ( dom ( Rev f));

      then n in ( dom f) by FINSEQ_5: 57;

      then

      consider k such that

       A1: k in ( dom ( Rev f)) & (n + k) = (( len f) + 1) and (f /. n) = (( Rev f) /. k) by Th3;

      

       A2: ( dom f) = ( dom ( Rev f)) by FINSEQ_5: 57;

      then (( Rev f) /. n) = (f /. k) by A1, FINSEQ_5: 66;

      hence thesis by A1, A2;

    end;

    begin

    reserve G for Go-board,

f,g for FinSequence of ( TOP-REAL 2),

p for Point of ( TOP-REAL 2),

r,s for Real,

x for set;

    theorem :: JORDAN9:5

    

     Th5: for D be non empty set holds for G be Matrix of D holds for f be FinSequence of D holds f is_sequence_on G iff ( Rev f) is_sequence_on G

    proof

      let D be non empty set;

      let G be Matrix of D;

      let f be FinSequence of D;

      hereby

        assume

         A1: f is_sequence_on G;

        

         A2: for n be Nat st n in ( dom ( Rev f)) & (n + 1) in ( dom ( Rev f)) holds for m,k,i,j be Nat st [m, k] in ( Indices G) & [i, j] in ( Indices G) & (( Rev f) /. n) = (G * (m,k)) & (( Rev f) /. (n + 1)) = (G * (i,j)) holds ( |.(m - i).| + |.(k - j).|) = 1

        proof

          let n be Nat such that

           A3: n in ( dom ( Rev f)) and

           A4: (n + 1) in ( dom ( Rev f));

          consider l be Nat such that

           A5: l in ( dom f) and

           A6: (n + l) = (( len f) + 1) and

           A7: (( Rev f) /. n) = (f /. l) by A3, Th4;

          let m,k,i,j be Nat such that

           A8: [m, k] in ( Indices G) & [i, j] in ( Indices G) & (( Rev f) /. n) = (G * (m,k)) & (( Rev f) /. (n + 1)) = (G * (i,j));

          

           A9: |.(i - m).| = |.(m - i).| & |.(j - k).| = |.(k - j).| by UNIFORM1: 11;

          consider l9 be Nat such that

           A10: l9 in ( dom f) and

           A11: ((n + 1) + l9) = (( len f) + 1) and

           A12: (( Rev f) /. (n + 1)) = (f /. l9) by A4, Th4;

          (n + (1 + l9)) = (n + l) by A6, A11;

          hence thesis by A1, A8, A5, A7, A10, A12, A9, GOBOARD1:def 9;

        end;

        for n be Nat st n in ( dom ( Rev f)) holds ex i,j be Nat st [i, j] in ( Indices G) & (( Rev f) /. n) = (G * (i,j))

        proof

          let n be Nat;

          assume n in ( dom ( Rev f));

          then

          consider k such that

           A13: k in ( dom f) and (n + k) = (( len f) + 1) and

           A14: (( Rev f) /. n) = (f /. k) by Th4;

          consider i,j be Nat such that

           A15: [i, j] in ( Indices G) & (f /. k) = (G * (i,j)) by A1, A13, GOBOARD1:def 9;

          take i, j;

          thus thesis by A14, A15;

        end;

        hence ( Rev f) is_sequence_on G by A2, GOBOARD1:def 9;

      end;

      assume

       A16: ( Rev f) is_sequence_on G;

      

       A17: for n be Nat st n in ( dom f) & (n + 1) in ( dom f) holds for m,k,i,j be Nat st [m, k] in ( Indices G) & [i, j] in ( Indices G) & (f /. n) = (G * (m,k)) & (f /. (n + 1)) = (G * (i,j)) holds ( |.(m - i).| + |.(k - j).|) = 1

      proof

        let n be Nat such that

         A18: n in ( dom f) and

         A19: (n + 1) in ( dom f);

        consider l be Nat such that

         A20: l in ( dom ( Rev f)) and

         A21: (n + l) = (( len f) + 1) and

         A22: (f /. n) = (( Rev f) /. l) by A18, Th3;

        let m,k,i,j be Nat such that

         A23: [m, k] in ( Indices G) & [i, j] in ( Indices G) & (f /. n) = (G * (m,k)) & (f /. (n + 1)) = (G * (i,j));

        

         A24: |.(i - m).| = |.(m - i).| & |.(j - k).| = |.(k - j).| by UNIFORM1: 11;

        consider l9 be Nat such that

         A25: l9 in ( dom ( Rev f)) and

         A26: ((n + 1) + l9) = (( len f) + 1) and

         A27: (f /. (n + 1)) = (( Rev f) /. l9) by A19, Th3;

        (n + (1 + l9)) = (n + l) by A21, A26;

        hence thesis by A16, A23, A20, A22, A25, A27, A24, GOBOARD1:def 9;

      end;

      for n be Nat st n in ( dom f) holds ex i,j be Nat st [i, j] in ( Indices G) & (f /. n) = (G * (i,j))

      proof

        let n be Nat;

        assume n in ( dom f);

        then

        consider k such that

         A28: k in ( dom ( Rev f)) and (n + k) = (( len f) + 1) and

         A29: (f /. n) = (( Rev f) /. k) by Th3;

        consider i,j be Nat such that

         A30: [i, j] in ( Indices G) & (( Rev f) /. k) = (G * (i,j)) by A16, A28, GOBOARD1:def 9;

        take i, j;

        thus thesis by A29, A30;

      end;

      hence thesis by A17, GOBOARD1:def 9;

    end;

    theorem :: JORDAN9:6

    

     Th6: for G be Matrix of ( TOP-REAL 2) holds for f be FinSequence of ( TOP-REAL 2) st f is_sequence_on G holds for k be Nat st 1 <= k & k <= ( len f) holds (f /. k) in ( Values G)

    proof

      let G be Matrix of ( TOP-REAL 2);

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

      assume that

       A1: f is_sequence_on G;

      let k be Nat such that

       A2: 1 <= k & k <= ( len f);

      

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

      then (f /. k) = (f . k) by PARTFUN1:def 6;

      then

       A4: (f /. k) in ( rng f) by A3, FUNCT_1:def 3;

      ( rng f) c= ( Values G) by A1, GOBRD13: 8;

      hence thesis by A4;

    end;

    

     Lm2: f is_sequence_on G & 1 <= k & k <= ( len f) implies ex i,j be Nat st [i, j] in ( Indices G) & (f /. k) = (G * (i,j))

    proof

      assume that

       A1: f is_sequence_on G and

       A2: 1 <= k & k <= ( len f);

      k in ( dom f) by A2, FINSEQ_3: 25;

      then

      consider i,j be Nat such that

       A3: [i, j] in ( Indices G) & (f /. k) = (G * (i,j)) by A1, GOBOARD1:def 9;

      take i, j;

      thus thesis by A3;

    end;

    theorem :: JORDAN9:7

    

     Th7: n <= ( len f) & x in ( L~ (f /^ n)) implies ex i be Nat st (n + 1) <= i & (i + 1) <= ( len f) & x in ( LSeg (f,i))

    proof

      assume that

       A1: n <= ( len f) and

       A2: x in ( L~ (f /^ n));

      consider j be Nat such that

       A3: 1 <= j and

       A4: (j + 1) <= ( len (f /^ n)) and

       A5: x in ( LSeg ((f /^ n),j)) by A2, SPPOL_2: 13;

      (j + 1) <= (( len f) - n) by A1, A4, RFINSEQ:def 1;

      then

       A6: (n + (j + 1)) <= ( len f) by XREAL_1: 19;

      take (n + j);

      (j + 1) <= (( len f) - n) by A1, A4, RFINSEQ:def 1;

      hence thesis by A3, A5, A6, SPPOL_2: 5, XREAL_1: 6;

    end;

    theorem :: JORDAN9:8

    

     Th8: f is_sequence_on G implies for k be Nat st 1 <= k & (k + 1) <= ( len f) holds (f /. k) in ( left_cell (f,k,G)) & (f /. k) in ( right_cell (f,k,G))

    proof

      assume that

       A1: f is_sequence_on G;

      let k be Nat such that

       A2: 1 <= k & (k + 1) <= ( len f);

      set p = (f /. k);

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

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

      then p in (( left_cell (f,k,G)) /\ ( right_cell (f,k,G))) by A1, A2, GOBRD13: 29;

      hence thesis by XBOOLE_0:def 4;

    end;

    theorem :: JORDAN9:9

    

     Th9: f is_sequence_on G implies for k be Nat st 1 <= k & (k + 1) <= ( len f) holds ( Int ( left_cell (f,k,G))) <> {} & ( Int ( right_cell (f,k,G))) <> {}

    proof

      assume

       A1: f is_sequence_on G;

      let k be Nat;

      assume

       A2: 1 <= k & (k + 1) <= ( len f);

      then

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

       A3: [i1, j1] in ( Indices G) and

       A4: (f /. k) = (G * (i1,j1)) and

       A5: [i2, j2] in ( Indices G) and

       A6: (f /. (k + 1)) = (G * (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, JORDAN8: 3;

      

       A8: i2 <= ( len G) by A5, MATRIX_0: 32;

      

       A9: i1 <= ( len G) by A3, MATRIX_0: 32;

      then

       A10: (i1 -' 1) <= ( len G) by NAT_D: 44;

      

       A11: j2 <= ( width G) by A5, MATRIX_0: 32;

      then

       A12: (j2 -' 1) <= ( width G) by NAT_D: 44;

      

       A13: (j1 + 1) > j1 & (j2 + 1) > j2 by NAT_1: 13;

      

       A14: j1 <= ( width G) by A3, MATRIX_0: 32;

      then

       A15: (j1 -' 1) <= ( width G) by NAT_D: 44;

      

       A16: (i1 + 1) > i1 & (i2 + 1) > i2 by NAT_1: 13;

      per cases by A7;

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

        then ( right_cell (f,k,G)) = ( cell (G,i1,j1)) & ( left_cell (f,k,G)) = ( cell (G,(i1 -' 1),j1)) by A1, A3, A4, A5, A6, A13, GOBRD13:def 2, GOBRD13:def 3, A2;

        hence thesis by A9, A14, A10, GOBOARD9: 14;

      end;

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

        then ( right_cell (f,k,G)) = ( cell (G,i1,(j1 -' 1))) & ( left_cell (f,k,G)) = ( cell (G,i1,j1)) by A1, A3, A4, A5, A6, A16, GOBRD13:def 2, GOBRD13:def 3, A2;

        hence thesis by A9, A14, A15, GOBOARD9: 14;

      end;

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

        then ( right_cell (f,k,G)) = ( cell (G,i2,j2)) & ( left_cell (f,k,G)) = ( cell (G,i2,(j2 -' 1))) by A1, A3, A4, A5, A6, A16, GOBRD13:def 2, GOBRD13:def 3, A2;

        hence thesis by A8, A11, A12, GOBOARD9: 14;

      end;

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

        then ( right_cell (f,k,G)) = ( cell (G,(i1 -' 1),j2)) & ( left_cell (f,k,G)) = ( cell (G,i1,j2)) by A1, A3, A4, A5, A6, A13, GOBRD13:def 2, GOBRD13:def 3, A2;

        hence thesis by A9, A11, A10, GOBOARD9: 14;

      end;

    end;

    theorem :: JORDAN9:10

    

     Th10: f is_sequence_on G implies for k be Nat st 1 <= k & (k + 1) <= ( len f) holds ( Int ( left_cell (f,k,G))) is convex & ( Int ( right_cell (f,k,G))) is convex

    proof

      assume

       A1: f is_sequence_on G;

      let k be Nat;

      assume

       A2: 1 <= k & (k + 1) <= ( len f);

      then

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

       A3: [i1, j1] in ( Indices G) and

       A4: (f /. k) = (G * (i1,j1)) and

       A5: [i2, j2] in ( Indices G) and

       A6: (f /. (k + 1)) = (G * (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, JORDAN8: 3;

      

       A8: i2 <= ( len G) by A5, MATRIX_0: 32;

      

       A9: i1 <= ( len G) by A3, MATRIX_0: 32;

      then

       A10: (i1 -' 1) <= ( len G) by NAT_D: 44;

      

       A11: j2 <= ( width G) by A5, MATRIX_0: 32;

      then

       A12: (j2 -' 1) <= ( width G) by NAT_D: 44;

      

       A13: (j1 + 1) > j1 & (j2 + 1) > j2 by NAT_1: 13;

      

       A14: j1 <= ( width G) by A3, MATRIX_0: 32;

      then

       A15: (j1 -' 1) <= ( width G) by NAT_D: 44;

      

       A16: (i1 + 1) > i1 & (i2 + 1) > i2 by NAT_1: 13;

      per cases by A7;

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

        then ( right_cell (f,k,G)) = ( cell (G,i1,j1)) & ( left_cell (f,k,G)) = ( cell (G,(i1 -' 1),j1)) by A1, A3, A4, A5, A6, A13, GOBRD13:def 2, GOBRD13:def 3, A2;

        hence thesis by A9, A14, A10, GOBOARD9: 17;

      end;

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

        then ( right_cell (f,k,G)) = ( cell (G,i1,(j1 -' 1))) & ( left_cell (f,k,G)) = ( cell (G,i1,j1)) by A1, A3, A4, A5, A6, A16, GOBRD13:def 2, GOBRD13:def 3, A2;

        hence thesis by A9, A14, A15, GOBOARD9: 17;

      end;

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

        then ( right_cell (f,k,G)) = ( cell (G,i2,j2)) & ( left_cell (f,k,G)) = ( cell (G,i2,(j2 -' 1))) by A1, A3, A4, A5, A6, A16, GOBRD13:def 2, GOBRD13:def 3, A2;

        hence thesis by A8, A11, A12, GOBOARD9: 17;

      end;

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

        then ( right_cell (f,k,G)) = ( cell (G,(i1 -' 1),j2)) & ( left_cell (f,k,G)) = ( cell (G,i1,j2)) by A1, A3, A4, A5, A6, A13, GOBRD13:def 2, GOBRD13:def 3, A2;

        hence thesis by A9, A11, A10, GOBOARD9: 17;

      end;

    end;

    theorem :: JORDAN9:11

    

     Th11: f is_sequence_on G & 1 <= k & (k + 1) <= ( len f) implies ( Cl ( Int ( left_cell (f,k,G)))) = ( left_cell (f,k,G)) & ( Cl ( Int ( right_cell (f,k,G)))) = ( right_cell (f,k,G))

    proof

      assume

       A1: f is_sequence_on G & 1 <= k & (k + 1) <= ( len f);

      then

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

       A2: [i1, j1] in ( Indices G) and

       A3: (f /. k) = (G * (i1,j1)) and

       A4: [i2, j2] in ( Indices G) and

       A5: (f /. (k + 1)) = (G * (i2,j2)) and

       A6: i1 = i2 & (j1 + 1) = j2 or (i1 + 1) = i2 & j1 = j2 or i1 = (i2 + 1) & j1 = j2 or i1 = i2 & j1 = (j2 + 1) by JORDAN8: 3;

      

       A7: i2 <= ( len G) by A4, MATRIX_0: 32;

      

       A8: i1 <= ( len G) by A2, MATRIX_0: 32;

      then

       A9: (i1 -' 1) <= ( len G) by NAT_D: 44;

      

       A10: j2 <= ( width G) by A4, MATRIX_0: 32;

      then

       A11: (j2 -' 1) <= ( width G) by NAT_D: 44;

      

       A12: (j1 + 1) > j1 & (j2 + 1) > j2 by NAT_1: 13;

      

       A13: j1 <= ( width G) by A2, MATRIX_0: 32;

      then

       A14: (j1 -' 1) <= ( width G) by NAT_D: 44;

      

       A15: (i1 + 1) > i1 & (i2 + 1) > i2 by NAT_1: 13;

      per cases by A6;

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

        then ( right_cell (f,k,G)) = ( cell (G,i1,j1)) & ( left_cell (f,k,G)) = ( cell (G,(i1 -' 1),j1)) by A1, A2, A3, A4, A5, A12, GOBRD13:def 2, GOBRD13:def 3;

        hence thesis by A8, A13, A9, GOBRD11: 35;

      end;

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

        then ( right_cell (f,k,G)) = ( cell (G,i1,(j1 -' 1))) & ( left_cell (f,k,G)) = ( cell (G,i1,j1)) by A1, A2, A3, A4, A5, A15, GOBRD13:def 2, GOBRD13:def 3;

        hence thesis by A8, A13, A14, GOBRD11: 35;

      end;

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

        then ( right_cell (f,k,G)) = ( cell (G,i2,j2)) & ( left_cell (f,k,G)) = ( cell (G,i2,(j2 -' 1))) by A1, A2, A3, A4, A5, A15, GOBRD13:def 2, GOBRD13:def 3;

        hence thesis by A7, A10, A11, GOBRD11: 35;

      end;

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

        then ( right_cell (f,k,G)) = ( cell (G,(i1 -' 1),j2)) & ( left_cell (f,k,G)) = ( cell (G,i1,j2)) by A1, A2, A3, A4, A5, A12, GOBRD13:def 2, GOBRD13:def 3;

        hence thesis by A8, A10, A9, GOBRD11: 35;

      end;

    end;

    theorem :: JORDAN9:12

    

     Th12: f is_sequence_on G & ( LSeg (f,k)) is horizontal implies ex j st 1 <= j & j <= ( width G) & for p st p in ( LSeg (f,k)) holds (p `2 ) = ((G * (1,j)) `2 )

    proof

      assume that

       A1: f is_sequence_on G and

       A2: ( LSeg (f,k)) is horizontal;

      per cases ;

        suppose

         A3: 1 <= k & (k + 1) <= ( len f);

        k <= (k + 1) by NAT_1: 11;

        then k <= ( len f) by A3, XXREAL_0: 2;

        then

        consider i,j be Nat such that

         A4: [i, j] in ( Indices G) and

         A5: (f /. k) = (G * (i,j)) by A1, A3, Lm2;

        reconsider j as Nat;

        take j;

        thus

         A6: 1 <= j & j <= ( width G) by A4, MATRIX_0: 32;

        

         A7: (f /. k) in ( LSeg (f,k)) by A3, TOPREAL1: 21;

        let p;

        

         A8: 1 <= i & i <= ( len G) by A4, MATRIX_0: 32;

        assume p in ( LSeg (f,k));

        

        hence (p `2 ) = ((f /. k) `2 ) by A2, A7, SPPOL_1:def 2

        .= ((G * (1,j)) `2 ) by A5, A6, A8, GOBOARD5: 1;

      end;

        suppose

         A9: not (1 <= k & (k + 1) <= ( len f));

        take 1;

        ( width G) <> 0 by MATRIX_0:def 10;

        hence 1 <= 1 & 1 <= ( width G) by NAT_1: 14;

        thus thesis by A9, TOPREAL1:def 3;

      end;

    end;

    theorem :: JORDAN9:13

    

     Th13: f is_sequence_on G & ( LSeg (f,k)) is vertical implies ex i st 1 <= i & i <= ( len G) & for p st p in ( LSeg (f,k)) holds (p `1 ) = ((G * (i,1)) `1 )

    proof

      assume that

       A1: f is_sequence_on G and

       A2: ( LSeg (f,k)) is vertical;

      per cases ;

        suppose

         A3: 1 <= k & (k + 1) <= ( len f);

        k <= (k + 1) by NAT_1: 11;

        then k <= ( len f) by A3, XXREAL_0: 2;

        then

        consider i,j be Nat such that

         A4: [i, j] in ( Indices G) and

         A5: (f /. k) = (G * (i,j)) by A1, A3, Lm2;

        reconsider i as Nat;

        take i;

        thus

         A6: 1 <= i & i <= ( len G) by A4, MATRIX_0: 32;

        

         A7: (f /. k) in ( LSeg (f,k)) by A3, TOPREAL1: 21;

        let p;

        

         A8: 1 <= j & j <= ( width G) by A4, MATRIX_0: 32;

        assume p in ( LSeg (f,k));

        

        hence (p `1 ) = ((f /. k) `1 ) by A2, A7, SPPOL_1:def 3

        .= ((G * (i,1)) `1 ) by A5, A6, A8, GOBOARD5: 2;

      end;

        suppose

         A9: not (1 <= k & (k + 1) <= ( len f));

        take 1;

         0 <> ( len G) by MATRIX_0:def 10;

        hence 1 <= 1 & 1 <= ( len G) by NAT_1: 14;

        thus thesis by A9, TOPREAL1:def 3;

      end;

    end;

    theorem :: JORDAN9:14

    

     Th14: f is_sequence_on G & f is special implies for i,j be Nat st i <= ( len G) & j <= ( width G) holds ( Int ( cell (G,i,j))) misses ( L~ f)

    proof

      assume that

       A1: f is_sequence_on G and

       A2: f is special;

      let i,j be Nat such that

       A3: i <= ( len G) and

       A4: j <= ( width G);

      

       A5: ( Int ( cell (G,i,j))) = (( Int ( v_strip (G,i))) /\ ( Int ( h_strip (G,j)))) by TOPS_1: 17;

      assume ( Int ( cell (G,i,j))) meets ( L~ f);

      then

      consider x be object such that

       A6: x in ( Int ( cell (G,i,j))) and

       A7: x in ( L~ f) by XBOOLE_0: 3;

      ( L~ f) = ( union { ( LSeg (f,k)) where k be Nat : 1 <= k & (k + 1) <= ( len f) }) by TOPREAL1:def 4;

      then

      consider X be set such that

       A8: x in X and

       A9: X in { ( LSeg (f,k)) where k be Nat : 1 <= k & (k + 1) <= ( len f) } by A7, TARSKI:def 4;

      consider k be Nat such that

       A10: X = ( LSeg (f,k)) and 1 <= k and (k + 1) <= ( len f) by A9;

      reconsider p = x as Point of ( TOP-REAL 2) by A8, A10;

      per cases by A2, SPPOL_1: 19;

        suppose ( LSeg (f,k)) is horizontal;

        then

        consider j0 be Nat such that

         A11: 1 <= j0 and

         A12: j0 <= ( width G) and

         A13: for p be Point of ( TOP-REAL 2) st p in ( LSeg (f,k)) holds (p `2 ) = ((G * (1,j0)) `2 ) by A1, Th12;

        now

          

           A14: j0 > j implies j0 >= (j + 1) by NAT_1: 13;

          assume

           A15: p in ( Int ( h_strip (G,j)));

          per cases by A14, XXREAL_0: 1;

            suppose

             A16: j0 < j;

             0 <> ( len G) by MATRIX_0:def 10;

            then 1 <= ( len G) by NAT_1: 14;

            then

             A17: ((G * (1,j)) `2 ) > ((G * (1,j0)) `2 ) by A4, A11, A16, GOBOARD5: 4;

            j >= 1 by A11, A16, XXREAL_0: 2;

            then (p `2 ) > ((G * (1,j)) `2 ) by A4, A15, GOBOARD6: 27;

            hence contradiction by A8, A10, A13, A17;

          end;

            suppose j0 = j;

            then (p `2 ) > ((G * (1,j0)) `2 ) by A11, A12, A15, GOBOARD6: 27;

            hence contradiction by A8, A10, A13;

          end;

            suppose

             A18: j0 > (j + 1);

            then (j + 1) <= ( width G) by A12, XXREAL_0: 2;

            then j < ( width G) by NAT_1: 13;

            then

             A19: (p `2 ) < ((G * (1,(j + 1))) `2 ) by A15, GOBOARD6: 28;

             0 <> ( len G) by MATRIX_0:def 10;

            then

             A20: 1 <= ( len G) by NAT_1: 14;

            (j + 1) >= 1 by NAT_1: 14;

            then ((G * (1,(j + 1))) `2 ) < ((G * (1,j0)) `2 ) by A12, A18, A20, GOBOARD5: 4;

            hence contradiction by A8, A10, A13, A19;

          end;

            suppose

             A21: j0 = (j + 1);

            then j < ( width G) by A12, NAT_1: 13;

            then (p `2 ) < ((G * (1,j0)) `2 ) by A15, A21, GOBOARD6: 28;

            hence contradiction by A8, A10, A13;

          end;

        end;

        hence contradiction by A6, A5, XBOOLE_0:def 4;

      end;

        suppose ( LSeg (f,k)) is vertical;

        then

        consider i0 be Nat such that

         A22: 1 <= i0 and

         A23: i0 <= ( len G) and

         A24: for p be Point of ( TOP-REAL 2) st p in ( LSeg (f,k)) holds (p `1 ) = ((G * (i0,1)) `1 ) by A1, Th13;

        now

          

           A25: i0 > i implies i0 >= (i + 1) by NAT_1: 13;

          assume

           A26: p in ( Int ( v_strip (G,i)));

          per cases by A25, XXREAL_0: 1;

            suppose

             A27: i0 < i;

             0 <> ( width G) by MATRIX_0:def 10;

            then 1 <= ( width G) by NAT_1: 14;

            then

             A28: ((G * (i,1)) `1 ) > ((G * (i0,1)) `1 ) by A3, A22, A27, GOBOARD5: 3;

            i >= 1 by A22, A27, XXREAL_0: 2;

            then (p `1 ) > ((G * (i,1)) `1 ) by A3, A26, GOBOARD6: 29;

            hence contradiction by A8, A10, A24, A28;

          end;

            suppose i0 = i;

            then (p `1 ) > ((G * (i0,1)) `1 ) by A22, A23, A26, GOBOARD6: 29;

            hence contradiction by A8, A10, A24;

          end;

            suppose

             A29: i0 > (i + 1);

            then (i + 1) <= ( len G) by A23, XXREAL_0: 2;

            then i < ( len G) by NAT_1: 13;

            then

             A30: (p `1 ) < ((G * ((i + 1),1)) `1 ) by A26, GOBOARD6: 30;

             0 <> ( width G) by MATRIX_0:def 10;

            then

             A31: 1 <= ( width G) by NAT_1: 14;

            (i + 1) >= 1 by NAT_1: 14;

            then ((G * ((i + 1),1)) `1 ) < ((G * (i0,1)) `1 ) by A23, A29, A31, GOBOARD5: 3;

            hence contradiction by A8, A10, A24, A30;

          end;

            suppose

             A32: i0 = (i + 1);

            then i < ( len G) by A23, NAT_1: 13;

            then (p `1 ) < ((G * (i0,1)) `1 ) by A26, A32, GOBOARD6: 30;

            hence contradiction by A8, A10, A24;

          end;

        end;

        hence contradiction by A6, A5, XBOOLE_0:def 4;

      end;

    end;

    theorem :: JORDAN9:15

    

     Th15: f is_sequence_on G & f is special implies for k be Nat st 1 <= k & (k + 1) <= ( len f) holds ( Int ( left_cell (f,k,G))) misses ( L~ f) & ( Int ( right_cell (f,k,G))) misses ( L~ f)

    proof

      assume that

       A1: f is_sequence_on G and

       A2: f is special;

      let k be Nat such that

       A3: 1 <= k & (k + 1) <= ( len f);

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

       A4: [i1, j1] in ( Indices G) and

       A5: (f /. k) = (G * (i1,j1)) and

       A6: [i2, j2] in ( Indices G) and

       A7: (f /. (k + 1)) = (G * (i2,j2)) and

       A8: 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, A3, JORDAN8: 3;

      

       A9: i2 <= ( len G) by A6, MATRIX_0: 32;

      

       A10: i1 <= ( len G) by A4, MATRIX_0: 32;

      then

       A11: (i1 -' 1) <= ( len G) by NAT_D: 44;

      

       A12: j2 <= ( width G) by A6, MATRIX_0: 32;

      then

       A13: (j2 -' 1) <= ( width G) by NAT_D: 44;

      

       A14: (j1 + 1) > j1 & (j2 + 1) > j2 by NAT_1: 13;

      

       A15: j1 <= ( width G) by A4, MATRIX_0: 32;

      then

       A16: (j1 -' 1) <= ( width G) by NAT_D: 44;

      

       A17: (i1 + 1) > i1 & (i2 + 1) > i2 by NAT_1: 13;

      per cases by A8;

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

        then ( right_cell (f,k,G)) = ( cell (G,i1,j1)) & ( left_cell (f,k,G)) = ( cell (G,(i1 -' 1),j1)) by A1, A3, A4, A5, A6, A7, A14, GOBRD13:def 2, GOBRD13:def 3;

        hence thesis by A1, A2, A10, A15, A11, Th14;

      end;

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

        then ( right_cell (f,k,G)) = ( cell (G,i1,(j1 -' 1))) & ( left_cell (f,k,G)) = ( cell (G,i1,j1)) by A1, A3, A4, A5, A6, A7, A17, GOBRD13:def 2, GOBRD13:def 3;

        hence thesis by A1, A2, A10, A15, A16, Th14;

      end;

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

        then ( right_cell (f,k,G)) = ( cell (G,i2,j2)) & ( left_cell (f,k,G)) = ( cell (G,i2,(j2 -' 1))) by A1, A3, A4, A5, A6, A7, A17, GOBRD13:def 2, GOBRD13:def 3;

        hence thesis by A1, A2, A9, A12, A13, Th14;

      end;

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

        then ( right_cell (f,k,G)) = ( cell (G,(i1 -' 1),j2)) & ( left_cell (f,k,G)) = ( cell (G,i1,j2)) by A1, A3, A4, A5, A6, A7, A14, GOBRD13:def 2, GOBRD13:def 3;

        hence thesis by A1, A2, A10, A12, A11, Th14;

      end;

    end;

    theorem :: JORDAN9:16

    

     Th16: 1 <= i & (i + 1) <= ( len G) & 1 <= j & (j + 1) <= ( width G) implies ((G * (i,j)) `1 ) = ((G * (i,(j + 1))) `1 ) & ((G * (i,j)) `2 ) = ((G * ((i + 1),j)) `2 ) & ((G * ((i + 1),(j + 1))) `1 ) = ((G * ((i + 1),j)) `1 ) & ((G * ((i + 1),(j + 1))) `2 ) = ((G * (i,(j + 1))) `2 )

    proof

      assume that

       A1: 1 <= i and

       A2: (i + 1) <= ( len G) and

       A3: 1 <= j and

       A4: (j + 1) <= ( width G);

      

       A5: j < ( width G) by A4, NAT_1: 13;

      

       A6: 1 <= (j + 1) by NAT_1: 11;

      

       A7: i < ( len G) by A2, NAT_1: 13;

      

      hence ((G * (i,j)) `1 ) = ((G * (i,1)) `1 ) by A1, A3, A5, GOBOARD5: 2

      .= ((G * (i,(j + 1))) `1 ) by A1, A4, A7, A6, GOBOARD5: 2;

      

       A8: 1 <= (i + 1) by NAT_1: 11;

      

      thus ((G * (i,j)) `2 ) = ((G * (1,j)) `2 ) by A1, A3, A7, A5, GOBOARD5: 1

      .= ((G * ((i + 1),j)) `2 ) by A2, A3, A5, A8, GOBOARD5: 1;

      

      thus ((G * ((i + 1),(j + 1))) `1 ) = ((G * ((i + 1),1)) `1 ) by A2, A4, A8, A6, GOBOARD5: 2

      .= ((G * ((i + 1),j)) `1 ) by A2, A3, A5, A8, GOBOARD5: 2;

      

      thus ((G * ((i + 1),(j + 1))) `2 ) = ((G * (1,(j + 1))) `2 ) by A2, A4, A8, A6, GOBOARD5: 1

      .= ((G * (i,(j + 1))) `2 ) by A1, A4, A7, A6, GOBOARD5: 1;

    end;

    theorem :: JORDAN9:17

    

     Th17: for i,j be Nat st 1 <= i & (i + 1) <= ( len G) & 1 <= j & (j + 1) <= ( width G) holds p in ( cell (G,i,j)) iff ((G * (i,j)) `1 ) <= (p `1 ) & (p `1 ) <= ((G * ((i + 1),j)) `1 ) & ((G * (i,j)) `2 ) <= (p `2 ) & (p `2 ) <= ((G * (i,(j + 1))) `2 )

    proof

      let i,j be Nat such that

       A1: 1 <= i and

       A2: (i + 1) <= ( len G) and

       A3: 1 <= j and

       A4: (j + 1) <= ( width G);

      

       A5: i < ( len G) & j < ( width G) by A2, A4, NAT_1: 13;

      then

       A6: ( h_strip (G,j)) = { |[r, s]| : ((G * (i,j)) `2 ) <= s & s <= ((G * (i,(j + 1))) `2 ) } by A1, A3, GOBOARD5: 5;

      

       A7: ( v_strip (G,i)) = { |[r, s]| : ((G * (i,j)) `1 ) <= r & r <= ((G * ((i + 1),j)) `1 ) } by A1, A3, A5, GOBOARD5: 8;

      hereby

        assume

         A8: p in ( cell (G,i,j));

        then p in ( v_strip (G,i)) by XBOOLE_0:def 4;

        then ex r, s st |[r, s]| = p & ((G * (i,j)) `1 ) <= r & r <= ((G * ((i + 1),j)) `1 ) by A7;

        hence ((G * (i,j)) `1 ) <= (p `1 ) & (p `1 ) <= ((G * ((i + 1),j)) `1 ) by EUCLID: 52;

        p in ( h_strip (G,j)) by A8, XBOOLE_0:def 4;

        then ex r, s st |[r, s]| = p & ((G * (i,j)) `2 ) <= s & s <= ((G * (i,(j + 1))) `2 ) by A6;

        hence ((G * (i,j)) `2 ) <= (p `2 ) & (p `2 ) <= ((G * (i,(j + 1))) `2 ) by EUCLID: 52;

      end;

      assume that

       A9: ((G * (i,j)) `1 ) <= (p `1 ) & (p `1 ) <= ((G * ((i + 1),j)) `1 ) and

       A10: ((G * (i,j)) `2 ) <= (p `2 ) & (p `2 ) <= ((G * (i,(j + 1))) `2 );

      

       A11: p = |[(p `1 ), (p `2 )]| by EUCLID: 53;

      then

       A12: p in ( h_strip (G,j)) by A6, A10;

      p in ( v_strip (G,i)) by A7, A9, A11;

      hence thesis by A12, XBOOLE_0:def 4;

    end;

    theorem :: JORDAN9:18

    1 <= i & (i + 1) <= ( len G) & 1 <= j & (j + 1) <= ( width G) implies ( cell (G,i,j)) = { |[r, s]| : ((G * (i,j)) `1 ) <= r & r <= ((G * ((i + 1),j)) `1 ) & ((G * (i,j)) `2 ) <= s & s <= ((G * (i,(j + 1))) `2 ) }

    proof

      set A = { |[r, s]| : ((G * (i,j)) `1 ) <= r & r <= ((G * ((i + 1),j)) `1 ) & ((G * (i,j)) `2 ) <= s & s <= ((G * (i,(j + 1))) `2 ) };

      assume

       A1: 1 <= i & (i + 1) <= ( len G) & 1 <= j & (j + 1) <= ( width G);

      now

        let p be object;

        assume

         A2: p in ( cell (G,i,j));

        then

        reconsider q = p as Point of ( TOP-REAL 2);

        

         A3: ((G * (i,j)) `2 ) <= (q `2 ) & (q `2 ) <= ((G * (i,(j + 1))) `2 ) by A1, A2, Th17;

        

         A4: p = |[(q `1 ), (q `2 )]| by EUCLID: 53;

        ((G * (i,j)) `1 ) <= (q `1 ) & (q `1 ) <= ((G * ((i + 1),j)) `1 ) by A1, A2, Th17;

        hence p in A by A4, A3;

      end;

      hence ( cell (G,i,j)) c= A;

      now

        let p be object;

        assume p in A;

        then

        consider r, s such that

         A5: |[r, s]| = p and

         A6: ((G * (i,j)) `1 ) <= r & r <= ((G * ((i + 1),j)) `1 ) & ((G * (i,j)) `2 ) <= s & s <= ((G * (i,(j + 1))) `2 );

        reconsider q = p as Point of ( TOP-REAL 2) by A5;

        r = (q `1 ) & s = (q `2 ) by A5, EUCLID: 52;

        hence p in ( cell (G,i,j)) by A1, A6, Th17;

      end;

      hence thesis;

    end;

    theorem :: JORDAN9:19

    

     Th19: 1 <= i & (i + 1) <= ( len G) & 1 <= j & (j + 1) <= ( width G) & p in ( Values G) & p in ( cell (G,i,j)) implies p = (G * (i,j)) or p = (G * (i,(j + 1))) or p = (G * ((i + 1),(j + 1))) or p = (G * ((i + 1),j))

    proof

      assume that

       A1: 1 <= i and

       A2: (i + 1) <= ( len G) and

       A3: 1 <= j and

       A4: (j + 1) <= ( width G) and

       A5: p in ( Values G) and

       A6: p in ( cell (G,i,j));

      

       A7: ( Values G) = { (G * (k,l)) where k,l be Nat : [k, l] in ( Indices G) } by MATRIX_0: 39;

      

       A8: i < ( len G) by A2, NAT_1: 13;

      

       A9: j < ( width G) by A4, NAT_1: 13;

      consider k,l be Nat such that

       A10: p = (G * (k,l)) and

       A11: [k, l] in ( Indices G) by A5, A7;

      

       A12: 1 <= k by A11, MATRIX_0: 32;

      

       A13: l <= ( width G) by A11, MATRIX_0: 32;

      

       A14: 1 <= l by A11, MATRIX_0: 32;

      

       A15: k <= ( len G) by A11, MATRIX_0: 32;

      

       A16: 1 <= (j + 1) by NAT_1: 11;

       A17:

      now

        assume

         A18: l <> j & l <> (j + 1);

        per cases by A18, NAT_1: 9;

          suppose l < j;

          then ((G * (k,l)) `2 ) < ((G * (k,j)) `2 ) by A9, A12, A15, A14, GOBOARD5: 4;

          then ((G * (k,l)) `2 ) < ((G * (1,j)) `2 ) by A3, A9, A12, A15, GOBOARD5: 1;

          then ((G * (k,l)) `2 ) < ((G * (i,j)) `2 ) by A1, A3, A8, A9, GOBOARD5: 1;

          hence contradiction by A1, A2, A3, A4, A6, A10, Th17;

        end;

          suppose (j + 1) < l;

          then ((G * (k,(j + 1))) `2 ) < ((G * (k,l)) `2 ) by A16, A12, A15, A13, GOBOARD5: 4;

          then ((G * (1,(j + 1))) `2 ) < ((G * (k,l)) `2 ) by A4, A16, A12, A15, GOBOARD5: 1;

          then ((G * (i,(j + 1))) `2 ) < ((G * (k,l)) `2 ) by A1, A4, A8, A16, GOBOARD5: 1;

          hence contradiction by A1, A2, A3, A4, A6, A10, Th17;

        end;

      end;

      

       A19: 1 <= (i + 1) by NAT_1: 11;

      now

        assume

         A20: k <> i & k <> (i + 1);

        per cases by A20, NAT_1: 9;

          suppose k < i;

          then ((G * (k,l)) `1 ) < ((G * (i,l)) `1 ) by A8, A12, A14, A13, GOBOARD5: 3;

          then ((G * (k,l)) `1 ) < ((G * (i,1)) `1 ) by A1, A8, A14, A13, GOBOARD5: 2;

          then ((G * (k,l)) `1 ) < ((G * (i,j)) `1 ) by A1, A3, A8, A9, GOBOARD5: 2;

          hence contradiction by A1, A2, A3, A4, A6, A10, Th17;

        end;

          suppose (i + 1) < k;

          then ((G * ((i + 1),l)) `1 ) < ((G * (k,l)) `1 ) by A19, A15, A14, A13, GOBOARD5: 3;

          then ((G * ((i + 1),1)) `1 ) < ((G * (k,l)) `1 ) by A2, A19, A14, A13, GOBOARD5: 2;

          then ((G * ((i + 1),j)) `1 ) < ((G * (k,l)) `1 ) by A2, A3, A9, A19, GOBOARD5: 2;

          hence contradiction by A1, A2, A3, A4, A6, A10, Th17;

        end;

      end;

      hence thesis by A10, A17;

    end;

    theorem :: JORDAN9:20

    

     Th20: 1 <= i & (i + 1) <= ( len G) & 1 <= j & (j + 1) <= ( width G) implies (G * (i,j)) in ( cell (G,i,j)) & (G * (i,(j + 1))) in ( cell (G,i,j)) & (G * ((i + 1),(j + 1))) in ( cell (G,i,j)) & (G * ((i + 1),j)) in ( cell (G,i,j))

    proof

      assume that

       A1: 1 <= i and

       A2: (i + 1) <= ( len G) and

       A3: 1 <= j and

       A4: (j + 1) <= ( width G);

      

       A5: i < (i + 1) & j < ( width G) by A4, NAT_1: 13;

      then

       A6: ((G * (i,j)) `1 ) <= ((G * ((i + 1),j)) `1 ) by A1, A2, A3, GOBOARD5: 3;

      

       A7: ((G * (i,j)) `1 ) <= ((G * ((i + 1),j)) `1 ) by A1, A2, A3, A5, GOBOARD5: 3;

      

       A8: j < (j + 1) & i < ( len G) by A2, NAT_1: 13;

      then

       A9: ((G * (i,j)) `2 ) <= ((G * (i,(j + 1))) `2 ) by A1, A3, A4, GOBOARD5: 4;

      

       A10: ((G * ((i + 1),(j + 1))) `1 ) = ((G * ((i + 1),j)) `1 ) by A1, A2, A3, A4, Th16;

      then

       A11: ((G * (i,j)) `1 ) <= ((G * ((i + 1),(j + 1))) `1 ) by A1, A2, A3, A5, GOBOARD5: 3;

      ((G * (i,j)) `2 ) <= ((G * (i,(j + 1))) `2 ) by A1, A3, A4, A8, GOBOARD5: 4;

      hence (G * (i,j)) in ( cell (G,i,j)) by A1, A2, A3, A4, A6, Th17;

      

       A12: ((G * (i,j)) `1 ) = ((G * (i,(j + 1))) `1 ) by A1, A2, A3, A4, Th16;

      then ((G * (i,(j + 1))) `1 ) <= ((G * ((i + 1),j)) `1 ) by A1, A2, A3, A5, GOBOARD5: 3;

      hence (G * (i,(j + 1))) in ( cell (G,i,j)) by A1, A2, A3, A4, A12, A9, Th17;

      

       A13: ((G * ((i + 1),(j + 1))) `2 ) = ((G * (i,(j + 1))) `2 ) by A1, A2, A3, A4, Th16;

      then ((G * (i,j)) `2 ) <= ((G * ((i + 1),(j + 1))) `2 ) by A1, A3, A4, A8, GOBOARD5: 4;

      hence (G * ((i + 1),(j + 1))) in ( cell (G,i,j)) by A1, A2, A3, A4, A10, A11, A13, Th17;

      

       A14: ((G * (i,j)) `2 ) = ((G * ((i + 1),j)) `2 ) by A1, A2, A3, A4, Th16;

      then ((G * ((i + 1),j)) `2 ) <= ((G * (i,(j + 1))) `2 ) by A1, A3, A4, A8, GOBOARD5: 4;

      hence thesis by A1, A2, A3, A4, A7, A14, Th17;

    end;

    theorem :: JORDAN9:21

    

     Th21: 1 <= i & (i + 1) <= ( len G) & 1 <= j & (j + 1) <= ( width G) & p in ( Values G) & p in ( cell (G,i,j)) implies p is_extremal_in ( cell (G,i,j))

    proof

      assume that

       A1: 1 <= i & (i + 1) <= ( len G) & 1 <= j & (j + 1) <= ( width G) and

       A2: p in ( Values G) and

       A3: p in ( cell (G,i,j));

      for a,b be Point of ( TOP-REAL 2) st p in ( LSeg (a,b)) & ( LSeg (a,b)) c= ( cell (G,i,j)) holds p = a or p = b

      proof

        let a,b be Point of ( TOP-REAL 2) such that

         A4: p in ( LSeg (a,b)) and

         A5: ( LSeg (a,b)) c= ( cell (G,i,j));

        

         A6: a in ( LSeg (a,b)) by RLTOPSP1: 68;

        

         A7: b in ( LSeg (a,b)) by RLTOPSP1: 68;

        assume that

         A8: a <> p and

         A9: b <> p;

        per cases by A1, A2, A3, Th19;

          suppose

           A10: p = (G * (i,j));

          then

           A11: (p `2 ) <= (b `2 ) by A1, A5, A7, Th17;

          

           A12: (p `1 ) <= (a `1 ) by A1, A5, A6, A10, Th17;

          

           A13: (p `1 ) <= (b `1 ) by A1, A5, A7, A10, Th17;

          

           A14: (p `2 ) <= (a `2 ) by A1, A5, A6, A10, Th17;

          now

            per cases ;

              suppose

               A15: (a `1 ) <= (b `1 ) & (a `2 ) <= (b `2 );

              then (a `2 ) <= (p `2 ) by A4, TOPREAL1: 4;

              then

               A16: (a `2 ) = (p `2 ) by A14, XXREAL_0: 1;

              (a `1 ) <= (p `1 ) by A4, A15, TOPREAL1: 3;

              then (a `1 ) = (p `1 ) by A12, XXREAL_0: 1;

              hence contradiction by A8, A16, TOPREAL3: 6;

            end;

              suppose

               A17: (a `1 ) <= (b `1 ) & (b `2 ) < (a `2 );

              then (b `2 ) <= (p `2 ) by A4, TOPREAL1: 4;

              then

               A18: (b `2 ) = (p `2 ) by A11, XXREAL_0: 1;

              

               A19: (a `1 ) <= (p `1 ) by A4, A17, TOPREAL1: 3;

              then

               A20: (a `1 ) = (p `1 ) by A12, XXREAL_0: 1;

              then (a `2 ) <> (p `2 ) by A8, TOPREAL3: 6;

              then ( LSeg (a,b)) is vertical by A4, A6, A12, A19, SPPOL_1: 18, XXREAL_0: 1;

              then (a `1 ) = (b `1 ) by SPPOL_1: 16;

              hence contradiction by A9, A20, A18, TOPREAL3: 6;

            end;

              suppose

               A21: (b `1 ) < (a `1 ) & (a `2 ) <= (b `2 );

              then (a `2 ) <= (p `2 ) by A4, TOPREAL1: 4;

              then

               A22: (a `2 ) = (p `2 ) by A14, XXREAL_0: 1;

              

               A23: (b `1 ) <= (p `1 ) by A4, A21, TOPREAL1: 3;

              then

               A24: (b `1 ) = (p `1 ) by A13, XXREAL_0: 1;

              then (b `2 ) <> (p `2 ) by A9, TOPREAL3: 6;

              then ( LSeg (a,b)) is vertical by A4, A7, A13, A23, SPPOL_1: 18, XXREAL_0: 1;

              then (a `1 ) = (b `1 ) by SPPOL_1: 16;

              hence contradiction by A8, A24, A22, TOPREAL3: 6;

            end;

              suppose

               A25: (b `1 ) < (a `1 ) & (b `2 ) < (a `2 );

              then (b `2 ) <= (p `2 ) by A4, TOPREAL1: 4;

              then

               A26: (b `2 ) = (p `2 ) by A11, XXREAL_0: 1;

              (b `1 ) <= (p `1 ) by A4, A25, TOPREAL1: 3;

              then (b `1 ) = (p `1 ) by A13, XXREAL_0: 1;

              hence contradiction by A9, A26, TOPREAL3: 6;

            end;

          end;

          hence contradiction;

        end;

          suppose

           A27: p = (G * (i,(j + 1)));

          then

           A28: (b `2 ) <= (p `2 ) by A1, A5, A7, Th17;

          

           A29: (p `1 ) = ((G * (i,j)) `1 ) by A1, A27, Th16;

          then

           A30: (p `1 ) <= (a `1 ) by A1, A5, A6, Th17;

          

           A31: (p `1 ) <= (b `1 ) by A1, A5, A7, A29, Th17;

          

           A32: (a `2 ) <= (p `2 ) by A1, A5, A6, A27, Th17;

          now

            per cases ;

              suppose

               A33: (a `1 ) <= (b `1 ) & (a `2 ) <= (b `2 );

              then (p `2 ) <= (b `2 ) by A4, TOPREAL1: 4;

              then

               A34: (b `2 ) = (p `2 ) by A28, XXREAL_0: 1;

              

               A35: (a `1 ) <= (p `1 ) by A4, A33, TOPREAL1: 3;

              then

               A36: (a `1 ) = (p `1 ) by A30, XXREAL_0: 1;

              then (a `2 ) <> (p `2 ) by A8, TOPREAL3: 6;

              then ( LSeg (a,b)) is vertical by A4, A6, A30, A35, SPPOL_1: 18, XXREAL_0: 1;

              then (a `1 ) = (b `1 ) by SPPOL_1: 16;

              hence contradiction by A9, A36, A34, TOPREAL3: 6;

            end;

              suppose

               A37: (a `1 ) <= (b `1 ) & (b `2 ) < (a `2 );

              then (p `2 ) <= (a `2 ) by A4, TOPREAL1: 4;

              then

               A38: (a `2 ) = (p `2 ) by A32, XXREAL_0: 1;

              (a `1 ) <= (p `1 ) by A4, A37, TOPREAL1: 3;

              then (a `1 ) = (p `1 ) by A30, XXREAL_0: 1;

              hence contradiction by A8, A38, TOPREAL3: 6;

            end;

              suppose

               A39: (b `1 ) < (a `1 ) & (a `2 ) <= (b `2 );

              then (p `2 ) <= (b `2 ) by A4, TOPREAL1: 4;

              then

               A40: (b `2 ) = (p `2 ) by A28, XXREAL_0: 1;

              (b `1 ) <= (p `1 ) by A4, A39, TOPREAL1: 3;

              then (b `1 ) = (p `1 ) by A31, XXREAL_0: 1;

              hence contradiction by A9, A40, TOPREAL3: 6;

            end;

              suppose

               A41: (b `1 ) < (a `1 ) & (b `2 ) < (a `2 );

              then (p `2 ) <= (a `2 ) by A4, TOPREAL1: 4;

              then

               A42: (a `2 ) = (p `2 ) by A32, XXREAL_0: 1;

              

               A43: (b `1 ) <= (p `1 ) by A4, A41, TOPREAL1: 3;

              then

               A44: (b `1 ) = (p `1 ) by A31, XXREAL_0: 1;

              then (b `2 ) <> (p `2 ) by A9, TOPREAL3: 6;

              then ( LSeg (a,b)) is vertical by A4, A7, A31, A43, SPPOL_1: 18, XXREAL_0: 1;

              then (a `1 ) = (b `1 ) by SPPOL_1: 16;

              hence contradiction by A8, A44, A42, TOPREAL3: 6;

            end;

          end;

          hence contradiction;

        end;

          suppose

           A45: p = (G * ((i + 1),(j + 1)));

          then

           A46: (p `1 ) = ((G * ((i + 1),j)) `1 ) by A1, Th16;

          then

           A47: (a `1 ) <= (p `1 ) by A1, A5, A6, Th17;

          

           A48: (p `2 ) = ((G * (i,(j + 1))) `2 ) by A1, A45, Th16;

          then

           A49: (b `2 ) <= (p `2 ) by A1, A5, A7, Th17;

          

           A50: (b `1 ) <= (p `1 ) by A1, A5, A7, A46, Th17;

          

           A51: (a `2 ) <= (p `2 ) by A1, A5, A6, A48, Th17;

          now

            per cases ;

              suppose

               A52: (a `1 ) <= (b `1 ) & (a `2 ) <= (b `2 );

              then (p `2 ) <= (b `2 ) by A4, TOPREAL1: 4;

              then

               A53: (b `2 ) = (p `2 ) by A49, XXREAL_0: 1;

              (p `1 ) <= (b `1 ) by A4, A52, TOPREAL1: 3;

              then (b `1 ) = (p `1 ) by A50, XXREAL_0: 1;

              hence contradiction by A9, A53, TOPREAL3: 6;

            end;

              suppose

               A54: (a `1 ) <= (b `1 ) & (b `2 ) < (a `2 );

              then (p `2 ) <= (a `2 ) by A4, TOPREAL1: 4;

              then

               A55: (a `2 ) = (p `2 ) by A51, XXREAL_0: 1;

              

               A56: (p `1 ) <= (b `1 ) by A4, A54, TOPREAL1: 3;

              then

               A57: (b `1 ) = (p `1 ) by A50, XXREAL_0: 1;

              then (b `2 ) <> (p `2 ) by A9, TOPREAL3: 6;

              then ( LSeg (a,b)) is vertical by A4, A7, A50, A56, SPPOL_1: 18, XXREAL_0: 1;

              then (a `1 ) = (b `1 ) by SPPOL_1: 16;

              hence contradiction by A8, A57, A55, TOPREAL3: 6;

            end;

              suppose

               A58: (b `1 ) < (a `1 ) & (a `2 ) <= (b `2 );

              then (p `2 ) <= (b `2 ) by A4, TOPREAL1: 4;

              then

               A59: (b `2 ) = (p `2 ) by A49, XXREAL_0: 1;

              

               A60: (p `1 ) <= (a `1 ) by A4, A58, TOPREAL1: 3;

              then

               A61: (a `1 ) = (p `1 ) by A47, XXREAL_0: 1;

              then (a `2 ) <> (p `2 ) by A8, TOPREAL3: 6;

              then ( LSeg (a,b)) is vertical by A4, A6, A47, A60, SPPOL_1: 18, XXREAL_0: 1;

              then (a `1 ) = (b `1 ) by SPPOL_1: 16;

              hence contradiction by A9, A61, A59, TOPREAL3: 6;

            end;

              suppose

               A62: (b `1 ) < (a `1 ) & (b `2 ) < (a `2 );

              then (p `2 ) <= (a `2 ) by A4, TOPREAL1: 4;

              then

               A63: (a `2 ) = (p `2 ) by A51, XXREAL_0: 1;

              (p `1 ) <= (a `1 ) by A4, A62, TOPREAL1: 3;

              then (a `1 ) = (p `1 ) by A47, XXREAL_0: 1;

              hence contradiction by A8, A63, TOPREAL3: 6;

            end;

          end;

          hence contradiction;

        end;

          suppose

           A64: p = (G * ((i + 1),j));

          then

           A65: (p `2 ) = ((G * (i,j)) `2 ) by A1, Th16;

          then

           A66: (p `2 ) <= (b `2 ) by A1, A5, A7, Th17;

          

           A67: (a `1 ) <= (p `1 ) by A1, A5, A6, A64, Th17;

          

           A68: (b `1 ) <= (p `1 ) by A1, A5, A7, A64, Th17;

          

           A69: (p `2 ) <= (a `2 ) by A1, A5, A6, A65, Th17;

          now

            per cases ;

              suppose

               A70: (a `1 ) <= (b `1 ) & (a `2 ) <= (b `2 );

              then (a `2 ) <= (p `2 ) by A4, TOPREAL1: 4;

              then

               A71: (a `2 ) = (p `2 ) by A69, XXREAL_0: 1;

              

               A72: (p `1 ) <= (b `1 ) by A4, A70, TOPREAL1: 3;

              then

               A73: (b `1 ) = (p `1 ) by A68, XXREAL_0: 1;

              then (b `2 ) <> (p `2 ) by A9, TOPREAL3: 6;

              then ( LSeg (a,b)) is vertical by A4, A7, A68, A72, SPPOL_1: 18, XXREAL_0: 1;

              then (a `1 ) = (b `1 ) by SPPOL_1: 16;

              hence contradiction by A8, A73, A71, TOPREAL3: 6;

            end;

              suppose

               A74: (a `1 ) <= (b `1 ) & (b `2 ) < (a `2 );

              then (b `2 ) <= (p `2 ) by A4, TOPREAL1: 4;

              then

               A75: (b `2 ) = (p `2 ) by A66, XXREAL_0: 1;

              (p `1 ) <= (b `1 ) by A4, A74, TOPREAL1: 3;

              then (b `1 ) = (p `1 ) by A68, XXREAL_0: 1;

              hence contradiction by A9, A75, TOPREAL3: 6;

            end;

              suppose

               A76: (b `1 ) < (a `1 ) & (a `2 ) <= (b `2 );

              then (a `2 ) <= (p `2 ) by A4, TOPREAL1: 4;

              then

               A77: (a `2 ) = (p `2 ) by A69, XXREAL_0: 1;

              (p `1 ) <= (a `1 ) by A4, A76, TOPREAL1: 3;

              then (a `1 ) = (p `1 ) by A67, XXREAL_0: 1;

              hence contradiction by A8, A77, TOPREAL3: 6;

            end;

              suppose

               A78: (b `1 ) < (a `1 ) & (b `2 ) < (a `2 );

              then (b `2 ) <= (p `2 ) by A4, TOPREAL1: 4;

              then

               A79: (b `2 ) = (p `2 ) by A66, XXREAL_0: 1;

              

               A80: (p `1 ) <= (a `1 ) by A4, A78, TOPREAL1: 3;

              then

               A81: (a `1 ) = (p `1 ) by A67, XXREAL_0: 1;

              then (a `2 ) <> (p `2 ) by A8, TOPREAL3: 6;

              then ( LSeg (a,b)) is vertical by A4, A6, A67, A80, SPPOL_1: 18, XXREAL_0: 1;

              then (a `1 ) = (b `1 ) by SPPOL_1: 16;

              hence contradiction by A9, A81, A79, TOPREAL3: 6;

            end;

          end;

          hence contradiction;

        end;

      end;

      hence thesis by A3, SPPOL_1:def 1;

    end;

    theorem :: JORDAN9:22

    

     Th22: 2 <= ( len G) & 2 <= ( width G) & f is_sequence_on G & 1 <= k & (k + 1) <= ( len f) implies ex i,j be Nat st 1 <= i & (i + 1) <= ( len G) & 1 <= j & (j + 1) <= ( width G) & ( LSeg (f,k)) c= ( cell (G,i,j))

    proof

      assume that

       A1: 2 <= ( len G) and

       A2: 2 <= ( width G) and

       A3: f is_sequence_on G and

       A4: 1 <= k & (k + 1) <= ( len f);

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

       A5: [i1, j1] in ( Indices G) and

       A6: (f /. k) = (G * (i1,j1)) and

       A7: [i2, j2] in ( Indices G) and

       A8: (f /. (k + 1)) = (G * (i2,j2)) and

       A9: i1 = i2 & (j1 + 1) = j2 or (i1 + 1) = i2 & j1 = j2 or i1 = (i2 + 1) & j1 = j2 or i1 = i2 & j1 = (j2 + 1) by A3, A4, JORDAN8: 3;

      

       A10: ( LSeg (f,k)) = ( LSeg ((f /. k),(f /. (k + 1)))) by A4, TOPREAL1:def 3;

      

       A11: 1 <= i2 by A7, MATRIX_0: 32;

      

       A12: 1 <= i1 by A5, MATRIX_0: 32;

      

       A13: 1 <= j2 by A7, MATRIX_0: 32;

      

       A14: 1 <= j1 by A5, MATRIX_0: 32;

      

       A15: i2 <= ( len G) by A7, MATRIX_0: 32;

      

       A16: i1 <= ( len G) by A5, MATRIX_0: 32;

      

       A17: j2 <= ( width G) by A7, MATRIX_0: 32;

      

       A18: j1 <= ( width G) by A5, MATRIX_0: 32;

      per cases by A9;

        suppose

         A19: i1 = i2 & (j1 + 1) = j2;

        then

         A20: j1 < ( width G) by A17, XREAL_1: 145;

        now

          per cases by A16, XXREAL_0: 1;

            suppose

             A21: i1 < ( len G);

            take i1, j1;

            

             A22: (i1 + 1) <= ( len G) by A21, NAT_1: 13;

            ( LSeg (f,k)) c= ( cell (G,i1,j1)) by A10, A6, A8, A12, A16, A14, A17, A19, GOBOARD5: 19, XREAL_1: 145;

            hence thesis by A12, A14, A17, A19, A22;

          end;

            suppose

             A23: i1 = ( len G);

            reconsider i19 = (i1 -' 1), j1 as Nat;

            take i19, j1;

            (2 - 1) <= (2 -' 1) & (2 -' 1) <= i19 by A1, A23, NAT_D: 42, XREAL_0:def 2;

            then

             A24: 1 <= i19 by XXREAL_0: 2;

            

             A25: (i19 + 1) = i1 by A12, XREAL_1: 235;

            then i19 < ( len G) by A16, NAT_1: 13;

            then ( LSeg (f,k)) c= ( cell (G,i19,j1)) by A10, A6, A8, A14, A19, A20, A25, GOBOARD5: 18;

            hence thesis by A16, A14, A17, A19, A24, A25;

          end;

        end;

        hence thesis;

      end;

        suppose

         A26: (i1 + 1) = i2 & j1 = j2;

        then

         A27: i1 < ( len G) by A15, XREAL_1: 145;

        now

          per cases by A18, XXREAL_0: 1;

            suppose

             A28: j1 < ( width G);

            take i1, j1;

            

             A29: (j1 + 1) <= ( width G) by A28, NAT_1: 13;

            ( LSeg (f,k)) c= ( cell (G,i1,j1)) by A10, A6, A8, A12, A14, A18, A15, A26, GOBOARD5: 22, XREAL_1: 145;

            hence thesis by A12, A14, A15, A26, A29;

          end;

            suppose

             A30: j1 = ( width G);

            reconsider i1, j19 = (j1 -' 1) as Nat;

            take i1, j19;

            (2 - 1) <= (2 -' 1) & (2 -' 1) <= j19 by A2, A30, NAT_D: 42, XREAL_0:def 2;

            then

             A31: 1 <= j19 by XXREAL_0: 2;

            

             A32: (j19 + 1) = j1 by A14, XREAL_1: 235;

            then j19 < ( width G) by A30, NAT_1: 13;

            then ( LSeg (f,k)) c= ( cell (G,i1,j19)) by A10, A6, A8, A12, A26, A27, A32, GOBOARD5: 21;

            hence thesis by A12, A18, A15, A26, A31, A32;

          end;

        end;

        hence thesis;

      end;

        suppose

         A33: i1 = (i2 + 1) & j1 = j2;

        then

         A34: i2 < ( len G) by A16, XREAL_1: 145;

        now

          per cases by A18, XXREAL_0: 1;

            suppose

             A35: j1 < ( width G);

            take i2, j1;

            

             A36: (j1 + 1) <= ( width G) by A35, NAT_1: 13;

            ( LSeg (f,k)) c= ( cell (G,i2,j1)) by A10, A6, A8, A16, A11, A13, A17, A33, GOBOARD5: 22, XREAL_1: 145;

            hence thesis by A16, A14, A11, A33, A36;

          end;

            suppose

             A37: j1 = ( width G);

            reconsider i2, j19 = (j1 -' 1) as Nat;

            take i2, j19;

            (2 - 1) <= (2 -' 1) & (2 -' 1) <= j19 by A2, A37, NAT_D: 42, XREAL_0:def 2;

            then

             A38: 1 <= j19 by XXREAL_0: 2;

            

             A39: (j19 + 1) = j1 by A14, XREAL_1: 235;

            then j19 < ( width G) by A37, NAT_1: 13;

            then ( LSeg (f,k)) c= ( cell (G,i2,j19)) by A10, A6, A8, A11, A33, A34, A39, GOBOARD5: 21;

            hence thesis by A16, A18, A11, A33, A38, A39;

          end;

        end;

        hence thesis;

      end;

        suppose

         A40: i1 = i2 & j1 = (j2 + 1);

        then

         A41: j2 < ( width G) by A18, XREAL_1: 145;

        now

          per cases by A16, XXREAL_0: 1;

            suppose

             A42: i1 < ( len G);

            take i1, j2;

            

             A43: (i1 + 1) <= ( len G) by A42, NAT_1: 13;

            ( LSeg (f,k)) c= ( cell (G,i1,j2)) by A10, A6, A8, A18, A11, A15, A13, A40, GOBOARD5: 19, XREAL_1: 145;

            hence thesis by A12, A18, A13, A40, A43;

          end;

            suppose

             A44: i1 = ( len G);

            reconsider i19 = (i1 -' 1), j2 as Nat;

            take i19, j2;

            (2 - 1) <= (2 -' 1) & (2 -' 1) <= i19 by A1, A44, NAT_D: 42, XREAL_0:def 2;

            then

             A45: 1 <= i19 by XXREAL_0: 2;

            

             A46: (i19 + 1) = i1 by A12, XREAL_1: 235;

            then i19 < ( len G) by A16, NAT_1: 13;

            then ( LSeg (f,k)) c= ( cell (G,i19,j2)) by A10, A6, A8, A13, A40, A41, A46, GOBOARD5: 18;

            hence thesis by A16, A18, A13, A40, A45, A46;

          end;

        end;

        hence thesis;

      end;

    end;

    theorem :: JORDAN9:23

    

     Th23: 2 <= ( len G) & 2 <= ( width G) & f is_sequence_on G & 1 <= k & (k + 1) <= ( len f) & p in ( Values G) & p in ( LSeg (f,k)) implies p = (f /. k) or p = (f /. (k + 1))

    proof

      assume that

       A1: 2 <= ( len G) & 2 <= ( width G) & f is_sequence_on G and

       A2: 1 <= k & (k + 1) <= ( len f) and

       A3: p in ( Values G) and

       A4: p in ( LSeg (f,k));

      

       A5: ( LSeg (f,k)) = ( LSeg ((f /. k),(f /. (k + 1)))) by A2, TOPREAL1:def 3;

      consider i,j be Nat such that

       A6: 1 <= i & (i + 1) <= ( len G) & 1 <= j & (j + 1) <= ( width G) and

       A7: ( LSeg (f,k)) c= ( cell (G,i,j)) by A1, A2, Th22;

      p is_extremal_in ( cell (G,i,j)) by A3, A4, A6, A7, Th21;

      hence thesis by A4, A7, A5, SPPOL_1:def 1;

    end;

    theorem :: JORDAN9:24

     [i, j] in ( Indices G) & 1 <= k & k <= ( width G) implies ((G * (i,j)) `1 ) <= ((G * (( len G),k)) `1 )

    proof

      assume that

       A1: [i, j] in ( Indices G) and

       A2: 1 <= k & k <= ( width G);

      

       A3: 1 <= i by A1, MATRIX_0: 32;

      

       A4: i <= ( len G) by A1, MATRIX_0: 32;

      then

       A5: i < ( len G) or i = ( len G) by XXREAL_0: 1;

      1 <= j & j <= ( width G) by A1, MATRIX_0: 32;

      

      then ((G * (i,j)) `1 ) = ((G * (i,1)) `1 ) by A3, A4, GOBOARD5: 2

      .= ((G * (i,k)) `1 ) by A2, A3, A4, GOBOARD5: 2;

      hence thesis by A2, A3, A5, GOBOARD5: 3;

    end;

    theorem :: JORDAN9:25

     [i, j] in ( Indices G) & 1 <= k & k <= ( len G) implies ((G * (i,j)) `2 ) <= ((G * (k,( width G))) `2 )

    proof

      assume that

       A1: [i, j] in ( Indices G) and

       A2: 1 <= k & k <= ( len G);

      

       A3: 1 <= j by A1, MATRIX_0: 32;

      

       A4: j <= ( width G) by A1, MATRIX_0: 32;

      then

       A5: j < ( width G) or j = ( width G) by XXREAL_0: 1;

      1 <= i & i <= ( len G) by A1, MATRIX_0: 32;

      

      then ((G * (i,j)) `2 ) = ((G * (1,j)) `2 ) by A3, A4, GOBOARD5: 1

      .= ((G * (k,j)) `2 ) by A2, A3, A4, GOBOARD5: 1;

      hence thesis by A2, A3, A5, GOBOARD5: 4;

    end;

    theorem :: JORDAN9:26

    

     Th26: f is_sequence_on G & f is special & ( L~ g) c= ( L~ f) & 1 <= k & (k + 1) <= ( len f) implies for A be Subset of ( TOP-REAL 2) st A = (( right_cell (f,k,G)) \ ( L~ g)) or A = (( left_cell (f,k,G)) \ ( L~ g)) holds A is connected

    proof

      assume that

       A1: f is_sequence_on G and

       A2: f is special and

       A3: ( L~ g) c= ( L~ f) and

       A4: 1 <= k & (k + 1) <= ( len f);

      let A be Subset of ( TOP-REAL 2) such that

       A5: A = (( right_cell (f,k,G)) \ ( L~ g)) or A = (( left_cell (f,k,G)) \ ( L~ g));

      per cases by A5;

        suppose

         A6: A = (( right_cell (f,k,G)) \ ( L~ g));

        ( Int ( right_cell (f,k,G))) misses ( L~ f) by A1, A2, A4, Th15;

        then ( Int ( right_cell (f,k,G))) misses ( L~ g) by A3, XBOOLE_1: 63;

        then

         A7: ( Int ( right_cell (f,k,G))) c= (( L~ g) ` ) by SUBSET_1: 23;

        A c= ( right_cell (f,k,G)) by A6, XBOOLE_1: 36;

        then

         A8: A c= ( Cl ( Int ( right_cell (f,k,G)))) by A1, A4, Th11;

        

         A9: A = (( right_cell (f,k,G)) /\ (( L~ g) ` )) by A6, SUBSET_1: 13;

        ( Int ( right_cell (f,k,G))) is convex & ( Int ( right_cell (f,k,G))) c= ( right_cell (f,k,G)) by A1, A4, Th10, TOPS_1: 16;

        hence thesis by A9, A7, A8, CONNSP_1: 18, XBOOLE_1: 19;

      end;

        suppose

         A10: A = (( left_cell (f,k,G)) \ ( L~ g));

        ( Int ( left_cell (f,k,G))) misses ( L~ f) by A1, A2, A4, Th15;

        then ( Int ( left_cell (f,k,G))) misses ( L~ g) by A3, XBOOLE_1: 63;

        then

         A11: ( Int ( left_cell (f,k,G))) c= (( L~ g) ` ) by SUBSET_1: 23;

        A c= ( left_cell (f,k,G)) by A10, XBOOLE_1: 36;

        then

         A12: A c= ( Cl ( Int ( left_cell (f,k,G)))) by A1, A4, Th11;

        

         A13: A = (( left_cell (f,k,G)) /\ (( L~ g) ` )) by A10, SUBSET_1: 13;

        ( Int ( left_cell (f,k,G))) is convex & ( Int ( left_cell (f,k,G))) c= ( left_cell (f,k,G)) by A1, A4, Th10, TOPS_1: 16;

        hence thesis by A13, A11, A12, CONNSP_1: 18, XBOOLE_1: 19;

      end;

    end;

    theorem :: JORDAN9:27

    

     Th27: for f be non constant standard special_circular_sequence st f is_sequence_on G holds for k st 1 <= k & (k + 1) <= ( len f) holds (( right_cell (f,k,G)) \ ( L~ f)) c= ( RightComp f) & (( left_cell (f,k,G)) \ ( L~ f)) c= ( LeftComp f)

    proof

      let f be non constant standard special_circular_sequence such that

       A1: f is_sequence_on G;

      let k such that

       A2: 1 <= k & (k + 1) <= ( len f);

      

       A3: ( Int ( right_cell (f,k,G))) <> {} by A1, A2, Th9;

      set rc = (( right_cell (f,k,G)) \ ( L~ f));

      (rc \/ ( L~ f)) = (( right_cell (f,k,G)) \/ ( L~ f)) by XBOOLE_1: 39;

      then ( Int ( right_cell (f,k,G))) c= ( right_cell (f,k,G)) & ( right_cell (f,k,G)) c= (rc \/ ( L~ f)) by TOPS_1: 16, XBOOLE_1: 7;

      then

       A4: ( Int ( right_cell (f,k,G))) c= (rc \/ ( L~ f));

      set lc = (( left_cell (f,k,G)) \ ( L~ f));

      rc = (( right_cell (f,k,G)) /\ (( L~ f) ` )) by SUBSET_1: 13;

      then

       A5: ( RightComp f) is_a_component_of (( L~ f) ` ) & rc c= (( L~ f) ` ) by GOBOARD9:def 2, XBOOLE_1: 17;

      rc c= ( right_cell (f,k,G)) & ( right_cell (f,k,G)) c= ( right_cell (f,k)) by A1, A2, GOBRD13: 33, XBOOLE_1: 36;

      then rc c= ( right_cell (f,k));

      then

       A6: ( Int rc) c= ( Int ( right_cell (f,k))) by TOPS_1: 19;

      ( Int ( right_cell (f,k))) c= ( RightComp f) by A2, GOBOARD9: 25;

      then

       A7: ( Int rc) c= ( RightComp f) by A6;

      ( Int ( right_cell (f,k,G))) misses ( L~ f) by A1, A2, Th15;

      then

       A8: ( Int ( Int ( right_cell (f,k,G)))) c= ( Int rc) by A4, TOPS_1: 19, XBOOLE_1: 73;

      ( Int ( right_cell (f,k,G))) c= rc by A1, A2, A4, Th15, XBOOLE_1: 73;

      then

       A9: rc meets ( Int rc) by A3, A8, XBOOLE_1: 68;

      rc is connected by A1, A2, Th26;

      hence (( right_cell (f,k,G)) \ ( L~ f)) c= ( RightComp f) by A7, A9, A5, GOBOARD9: 4;

      lc = (( left_cell (f,k,G)) /\ (( L~ f) ` )) by SUBSET_1: 13;

      then

       A10: ( LeftComp f) is_a_component_of (( L~ f) ` ) & lc c= (( L~ f) ` ) by GOBOARD9:def 1, XBOOLE_1: 17;

      (lc \/ ( L~ f)) = (( left_cell (f,k,G)) \/ ( L~ f)) by XBOOLE_1: 39;

      then ( Int ( left_cell (f,k,G))) c= ( left_cell (f,k,G)) & ( left_cell (f,k,G)) c= (lc \/ ( L~ f)) by TOPS_1: 16, XBOOLE_1: 7;

      then

       A11: ( Int ( left_cell (f,k,G))) c= (lc \/ ( L~ f));

      lc c= ( left_cell (f,k,G)) & ( left_cell (f,k,G)) c= ( left_cell (f,k)) by A1, A2, GOBRD13: 33, XBOOLE_1: 36;

      then lc c= ( left_cell (f,k));

      then

       A12: ( Int lc) c= ( Int ( left_cell (f,k))) by TOPS_1: 19;

      ( Int ( left_cell (f,k))) c= ( LeftComp f) by A2, GOBOARD9: 21;

      then

       A13: ( Int lc) c= ( LeftComp f) by A12;

      

       A14: ( Int ( left_cell (f,k,G))) <> {} by A1, A2, Th9;

      ( Int ( left_cell (f,k,G))) misses ( L~ f) by A1, A2, Th15;

      then

       A15: ( Int ( Int ( left_cell (f,k,G)))) c= ( Int lc) by A11, TOPS_1: 19, XBOOLE_1: 73;

      ( Int ( left_cell (f,k,G))) c= lc by A1, A2, A11, Th15, XBOOLE_1: 73;

      then

       A16: lc meets ( Int lc) by A14, A15, XBOOLE_1: 68;

      lc is connected by A1, A2, Th26;

      hence thesis by A13, A16, A10, GOBOARD9: 4;

    end;

    begin

    reserve C for compact non vertical non horizontal non empty Subset of ( TOP-REAL 2),

l,m,i1,i2,j1,j2 for Nat;

    theorem :: JORDAN9:28

    

     Th28: for n be Nat holds ex i st 1 <= i & (i + 1) <= ( len ( Gauge (C,n))) & ( N-min C) in ( cell (( Gauge (C,n)),i,(( width ( Gauge (C,n))) -' 1))) & ( N-min C) <> (( Gauge (C,n)) * (i,(( width ( Gauge (C,n))) -' 1)))

    proof

      let n be Nat;

      set G = ( Gauge (C,n));

      defpred P[ Nat] means 1 <= $1 & $1 < ( len G) & ((G * ($1,(( width G) -' 1))) `1 ) < (( N-min C) `1 );

      

       A1: for k be Nat st P[k] holds k <= ( len G);

      

       A2: ( len G) = ( width G) by JORDAN8:def 1;

      (( NW-corner C) `1 ) <= (( N-min C) `1 ) by PSCOMP_1: 38;

      then

       A3: ( W-bound C) <= (( N-min C) `1 ) by EUCLID: 52;

      

       A4: ( len G) >= 4 by JORDAN8: 10;

      then

       A5: (( len G) -' 1) <= ( len G) & 2 <= ( len G) by NAT_D: 35, XXREAL_0: 2;

      

       A6: 1 < ( len G) by A4, XXREAL_0: 2;

      then

       A7: 1 <= (( len G) -' 1) by NAT_D: 49;

      

       A8: n in NAT & ( len G) = ( width G) by JORDAN8:def 1, ORDINAL1:def 12;

      then ((G * (2,(( width G) -' 1))) `1 ) = ( W-bound C) by A7, JORDAN8: 11, NAT_D: 35;

      then ((G * (1,(( width G) -' 1))) `1 ) < ( W-bound C) by A2, A7, A5, GOBOARD5: 3;

      then ((G * (1,(( width G) -' 1))) `1 ) < (( N-min C) `1 ) by A3, XXREAL_0: 2;

      then

       A9: ex k be Nat st P[k] by A6;

      ex i be Nat st P[i] & for n be Nat st P[n] holds n <= i from NAT_1:sch 6( A1, A9);

      then

      consider i be Nat such that

       A10: 1 <= i and

       A11: i < ( len G) and

       A12: ((G * (i,(( width G) -' 1))) `1 ) < (( N-min C) `1 ) and

       A13: for n be Nat st P[n] holds n <= i;

      reconsider i as Nat;

      

       A14: 1 <= (i + 1) & i < (i + 1) by NAT_1: 12, NAT_1: 13;

      

       A15: (( N-min C) `2 ) = ( N-bound C) by EUCLID: 52;

      

       A16: (i + 1) <= ( len G) by A11, NAT_1: 13;

      then

       A17: (( N-min C) `2 ) = ((G * ((i + 1),(( width G) -' 1))) `2 ) by A8, A15, JORDAN8: 14, NAT_1: 12;

      now

        assume (i + 1) = ( len G);

        then (( len G) -' 1) = i by NAT_D: 34;

        then

         A18: ((G * (i,(( width G) -' 1))) `1 ) = ( E-bound C) by A8, A7, JORDAN8: 12, NAT_D: 35;

        (( NE-corner C) `1 ) >= (( N-min C) `1 ) by PSCOMP_1: 38;

        hence contradiction by A12, A18, EUCLID: 52;

      end;

      then (i + 1) < ( len G) by A16, XXREAL_0: 1;

      then

       A19: (( N-min C) `1 ) <= ((G * ((i + 1),(( width G) -' 1))) `1 ) by A13, A14;

      ((G * (i,(( width G) -' 1))) `2 ) = (( N-min C) `2 ) by A8, A10, A11, A15, JORDAN8: 14;

      then

       A20: ( N-min C) in ( LSeg ((G * (i,(( width G) -' 1))),(G * ((i + 1),(( width G) -' 1))))) by A12, A17, A19, GOBOARD7: 8;

      take i;

      thus 1 <= i & (i + 1) <= ( len G) by A10, A11, NAT_1: 13;

      ( LSeg ((G * (i,(( width G) -' 1))),(G * ((i + 1),(( width G) -' 1))))) c= ( cell (G,i,(( width G) -' 1))) by A2, A7, A10, A11, GOBOARD5: 22, NAT_D: 35;

      hence ( N-min C) in ( cell (G,i,(( width G) -' 1))) by A20;

      thus thesis by A12;

    end;

    theorem :: JORDAN9:29

    

     Th29: for n,i1,i2 be Nat holds 1 <= i1 & (i1 + 1) <= ( len ( Gauge (C,n))) & ( N-min C) in ( cell (( Gauge (C,n)),i1,(( width ( Gauge (C,n))) -' 1))) & ( N-min C) <> (( Gauge (C,n)) * (i1,(( width ( Gauge (C,n))) -' 1))) & 1 <= i2 & (i2 + 1) <= ( len ( Gauge (C,n))) & ( N-min C) in ( cell (( Gauge (C,n)),i2,(( width ( Gauge (C,n))) -' 1))) & ( N-min C) <> (( Gauge (C,n)) * (i2,(( width ( Gauge (C,n))) -' 1))) implies i1 = i2

    proof

      let n,i1,i2 be Nat;

      set G = ( Gauge (C,n)), j = (( width G) -' 1);

      

       A1: (2 |^ n) >= (n + 1) by NEWTON: 85;

      

       A2: (1 + (n + 3)) > (1 + 0 ) by XREAL_1: 6;

      

       A3: ( len G) = ( width G) by JORDAN8:def 1;

      

       A4: ( len G) = ((2 |^ n) + 3) by JORDAN8:def 1;

      then

       A5: ( len G) >= ((n + 1) + 3) by A1, XREAL_1: 6;

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

      then

       A6: ( len G) >= (1 + 1) by NAT_1: 13;

      then

       A7: 1 <= j by A3, JORDAN5B: 2;

      

       A8: (j + 1) = ( len G) by A3, A5, A2, XREAL_1: 235, XXREAL_0: 2;

      then

       A9: j < ( len G) by NAT_1: 13;

      assume that

       A10: 1 <= i1 and

       A11: (i1 + 1) <= ( len G) and

       A12: ( N-min C) in ( cell (G,i1,j)) and

       A13: ( N-min C) <> (G * (i1,j)) and

       A14: 1 <= i2 and

       A15: (i2 + 1) <= ( len G) and

       A16: ( N-min C) in ( cell (G,i2,j)) and

       A17: ( N-min C) <> (G * (i2,j)) and

       A18: i1 <> i2;

      

       A19: ( cell (G,i1,j)) meets ( cell (G,i2,j)) by A12, A16, XBOOLE_0: 3;

      

       A20: i1 < ( len G) by A11, NAT_1: 13;

      

       A21: i2 < ( len G) by A15, NAT_1: 13;

      per cases by A18, XXREAL_0: 1;

        suppose

         A22: i1 < i2;

        then

         A23: ((i2 -' i1) + i1) = i2 by XREAL_1: 235;

        then (i2 -' i1) <= 1 by A21, A3, A19, A7, A9, JORDAN8: 7;

        then (i2 -' i1) < 1 or (i2 -' i1) = 1 by XXREAL_0: 1;

        then (i2 -' i1) = 0 or (i2 -' i1) = 1 by NAT_1: 14;

        then (( cell (G,i1,j)) /\ ( cell (G,i2,j))) = ( LSeg ((G * (i2,j)),(G * (i2,(j + 1))))) by A20, A3, A6, A9, A22, A23, GOBOARD5: 25, JORDAN5B: 2;

        then

         A24: ( N-min C) in ( LSeg ((G * (i2,j)),(G * (i2,(j + 1))))) by A12, A16, XBOOLE_0:def 4;

        1 <= (j + 1) by NAT_1: 12;

        then

         A25: [i2, (j + 1)] in ( Indices G) by A14, A21, A3, A8, MATRIX_0: 30;

        set y2 = (( S-bound C) + (((( N-bound C) - ( S-bound C)) / (2 |^ n)) * (j - 1)));

        set y1 = (( S-bound C) + (((( N-bound C) - ( S-bound C)) / (2 |^ n)) * (j - 2)));

        set x = (( W-bound C) + (((( E-bound C) - ( W-bound C)) / (2 |^ n)) * (i2 - 2)));

        j = ((((2 |^ n) + 2) + 1) -' 1) by A4, JORDAN8:def 1

        .= ((2 |^ n) + 2) by NAT_D: 34;

        then

         A26: (((( N-bound C) - ( S-bound C)) / (2 |^ n)) * (j - 2)) = (( N-bound C) - ( S-bound C)) by A1, XCMPLX_1: 87;

         [i2, j] in ( Indices G) by A14, A21, A3, A7, A9, MATRIX_0: 30;

        then

         A27: (G * (i2,j)) = |[x, y1]| by JORDAN8:def 1;

        then

         A28: ((G * (i2,j)) `1 ) = x by EUCLID: 52;

        ((j + 1) - (1 + 1)) = (j - 1);

        then (G * (i2,(j + 1))) = |[x, y2]| by A25, JORDAN8:def 1;

        then ((G * (i2,(j + 1))) `1 ) = x by EUCLID: 52;

        then ( LSeg ((G * (i2,j)),(G * (i2,(j + 1))))) is vertical by A28, SPPOL_1: 16;

        then (( N-min C) `1 ) = ((G * (i2,j)) `1 ) by A24, SPPOL_1: 41;

        hence contradiction by A17, A27, A28, A26, EUCLID: 52;

      end;

        suppose

         A29: i2 < i1;

        then

         A30: ((i1 -' i2) + i2) = i1 by XREAL_1: 235;

        then (i1 -' i2) <= 1 by A20, A3, A19, A7, A9, JORDAN8: 7;

        then (i1 -' i2) < 1 or (i1 -' i2) = 1 by XXREAL_0: 1;

        then (i1 -' i2) = 0 or (i1 -' i2) = 1 by NAT_1: 14;

        then (( cell (G,i2,j)) /\ ( cell (G,i1,j))) = ( LSeg ((G * (i1,j)),(G * (i1,(j + 1))))) by A21, A3, A6, A9, A29, A30, GOBOARD5: 25, JORDAN5B: 2;

        then

         A31: ( N-min C) in ( LSeg ((G * (i1,j)),(G * (i1,(j + 1))))) by A12, A16, XBOOLE_0:def 4;

        1 <= (j + 1) by NAT_1: 12;

        then

         A32: [i1, (j + 1)] in ( Indices G) by A10, A20, A3, A8, MATRIX_0: 30;

        set y2 = (( S-bound C) + (((( N-bound C) - ( S-bound C)) / (2 |^ n)) * (j - 1)));

        set y1 = (( S-bound C) + (((( N-bound C) - ( S-bound C)) / (2 |^ n)) * (j - 2)));

        set x = (( W-bound C) + (((( E-bound C) - ( W-bound C)) / (2 |^ n)) * (i1 - 2)));

        j = ((((2 |^ n) + 2) + 1) -' 1) by A4, JORDAN8:def 1

        .= ((2 |^ n) + 2) by NAT_D: 34;

        then

         A33: (((( N-bound C) - ( S-bound C)) / (2 |^ n)) * (j - 2)) = (( N-bound C) - ( S-bound C)) by A1, XCMPLX_1: 87;

         [i1, j] in ( Indices G) by A10, A20, A3, A7, A9, MATRIX_0: 30;

        then

         A34: (G * (i1,j)) = |[x, y1]| by JORDAN8:def 1;

        then

         A35: ((G * (i1,j)) `1 ) = x by EUCLID: 52;

        ((j + 1) - (1 + 1)) = (j - 1);

        then (G * (i1,(j + 1))) = |[x, y2]| by A32, JORDAN8:def 1;

        then ((G * (i1,(j + 1))) `1 ) = x by EUCLID: 52;

        then ( LSeg ((G * (i1,j)),(G * (i1,(j + 1))))) is vertical by A35, SPPOL_1: 16;

        then (( N-min C) `1 ) = ((G * (i1,j)) `1 ) by A31, SPPOL_1: 41;

        hence contradiction by A13, A34, A35, A33, EUCLID: 52;

      end;

    end;

    theorem :: JORDAN9:30

    

     Th30: for n be Nat holds for f be standard non constant special_circular_sequence st f is_sequence_on ( Gauge (C,n)) & (for k be Nat st 1 <= k & (k + 1) <= ( len f) holds ( left_cell (f,k,( Gauge (C,n)))) misses C & ( right_cell (f,k,( Gauge (C,n)))) meets C) & (ex i be Nat st 1 <= i & (i + 1) <= ( len ( Gauge (C,n))) & (f /. 1) = (( Gauge (C,n)) * (i,( width ( Gauge (C,n))))) & (f /. 2) = (( Gauge (C,n)) * ((i + 1),( width ( Gauge (C,n))))) & ( N-min C) in ( cell (( Gauge (C,n)),i,(( width ( Gauge (C,n))) -' 1))) & ( N-min C) <> (( Gauge (C,n)) * (i,(( width ( Gauge (C,n))) -' 1)))) holds ( N-min ( L~ f)) = (f /. 1)

    proof

      let n be Nat;

      set G = ( Gauge (C,n));

      let f be standard non constant special_circular_sequence such that

       A1: f is_sequence_on G and

       A2: for k be Nat st 1 <= k & (k + 1) <= ( len f) holds ( left_cell (f,k,G)) misses C & ( right_cell (f,k,G)) meets C;

      ( N-min ( L~ f)) in ( rng f) by SPRECT_2: 39;

      then

      consider m be Nat such that

       A3: m in ( dom f) and

       A4: (f . m) = ( N-min ( L~ f)) by FINSEQ_2: 10;

      reconsider m as Nat;

      consider i,j be Nat such that

       A5: [i, j] in ( Indices G) and

       A6: (f /. m) = (G * (i,j)) by A1, A3, GOBOARD1:def 9;

      

       A7: (f /. m) = (f . m) by A3, PARTFUN1:def 6;

      

       A8: (( N-min ( L~ f)) `2 ) = ( N-bound ( L~ f)) by EUCLID: 52;

      set W = ( W-bound C), S = ( S-bound C), E = ( E-bound C), N = ( N-bound C);

      given i9 be Nat such that

       A9: 1 <= i9 and

       A10: (i9 + 1) <= ( len G) and

       A11: (f /. 1) = (G * (i9,( width G))) and

       A12: (f /. 2) = (G * ((i9 + 1),( width G))) and

       A13: ( N-min C) in ( cell (G,i9,(( width G) -' 1))) and

       A14: ( N-min C) <> (G * (i9,(( width G) -' 1)));

      

       A15: (G * (i9,(( len G) -' 1))) = |[((G * (i9,(( len G) -' 1))) `1 ), ((G * (i9,(( len G) -' 1))) `2 )]| & (( N-min C) `2 ) = N by EUCLID: 52, EUCLID: 53;

      (G * (i,j)) = |[(W + (((E - W) / (2 |^ n)) * (i - 2))), (S + (((N - S) / (2 |^ n)) * (j - 2)))]| by A5, JORDAN8:def 1;

      then

       A16: (S + (((N - S) / (2 |^ n)) * (j - 2))) = ( N-bound ( L~ f)) by A4, A7, A8, A6, EUCLID: 52;

      N > S by JORDAN8: 9;

      then (2 |^ n) > 0 & (N - S) > 0 by NEWTON: 83, XREAL_1: 50;

      then

       A17: ((N - S) / (2 |^ n)) > 0 by XREAL_1: 139;

      

       A18: (( NW-corner ( L~ f)) `1 ) = ( W-bound ( L~ f)) & (( NE-corner ( L~ f)) `1 ) = ( E-bound ( L~ f)) by EUCLID: 52;

      

       A19: 1 <= i by A5, MATRIX_0: 32;

      

       A20: (( NW-corner ( L~ f)) `2 ) = ( N-bound ( L~ f)) & (( NE-corner ( L~ f)) `2 ) = ( N-bound ( L~ f)) by EUCLID: 52;

      

       A21: m <= ( len f) by A3, FINSEQ_3: 25;

      

       A22: 1 <= j by A5, MATRIX_0: 32;

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

      then

       A23: ( len G) >= 3 by NAT_1: 12;

      then

       A24: 1 < ( len G) by XXREAL_0: 2;

      then

       A25: 1 <= (( len G) -' 1) by NAT_D: 49;

      then

       A26: (( len G) -' 1) < ( len G) by NAT_D: 51;

      

       A27: i <= ( len G) by A5, MATRIX_0: 32;

      

       A28: j <= ( width G) by A5, MATRIX_0: 32;

      then

       A29: ((G * (i,j)) `2 ) = ((G * (1,j)) `2 ) by A19, A27, A22, GOBOARD5: 1;

      

       A30: ( len f) > 4 by GOBOARD7: 34;

      1 in ( dom f) by FINSEQ_5: 6;

      then

       A31: (f /. 1) in ( L~ f) by A30, GOBOARD1: 1, XXREAL_0: 2;

      then

       A32: ( N-bound ( L~ f)) >= ((f /. 1) `2 ) by PSCOMP_1: 24;

      

       A33: ( len G) = ( width G) by JORDAN8:def 1;

      

       A34: i9 < ( len G) by A10, NAT_1: 13;

      then ((G * (i9,j)) `2 ) = ((G * (1,j)) `2 ) by A9, A22, A28, GOBOARD5: 1;

      then ((G * (i,j)) `2 ) <= ((G * (i9,( len G))) `2 ) by A9, A34, A33, A22, A28, A29, SPRECT_3: 12;

      then

       A35: ( N-bound ( L~ f)) = ((f /. 1) `2 ) by A11, A33, A4, A7, A8, A6, A32, XXREAL_0: 1;

       [i9, ( len G)] in ( Indices G) by A9, A34, A33, A24, MATRIX_0: 30;

      then (G * (i9,( len G))) = |[(W + (((E - W) / (2 |^ n)) * (i9 - 2))), (S + (((N - S) / (2 |^ n)) * (( len G) - 2)))]| by JORDAN8:def 1;

      then (S + (((N - S) / (2 |^ n)) * (( len G) - 2))) = ( N-bound ( L~ f)) by A11, A33, A35, EUCLID: 52;

      then

       A36: (( len G) - 2) = (j - 2) by A17, A16, XCMPLX_1: 5;

      then

       A37: ((G * (i9,( len G))) `1 ) = ((G * (i9,1)) `1 ) by A9, A34, A33, A22, GOBOARD5: 2;

      ( W-bound ( L~ f)) <= ((f /. 1) `1 ) & ((f /. 1) `1 ) <= ( E-bound ( L~ f)) by A31, PSCOMP_1: 24;

      then (f /. 1) in ( LSeg (( NW-corner ( L~ f)),( NE-corner ( L~ f)))) by A35, A18, A20, GOBOARD7: 8;

      then (f /. 1) in (( LSeg (( NW-corner ( L~ f)),( NE-corner ( L~ f)))) /\ ( L~ f)) by A31, XBOOLE_0:def 4;

      then

       A38: (( N-min ( L~ f)) `1 ) <= ((f /. 1) `1 ) by PSCOMP_1: 39;

      then

       A39: i <= i9 by A9, A11, A33, A4, A7, A6, A27, A22, A36, GOBOARD5: 3;

      then

       A40: i < ( len G) by A34, XXREAL_0: 2;

      then

       A41: (i + 1) <= ( len G) by NAT_1: 13;

      

       A42: ((( len G) -' 1) + 1) = ( len G) by A23, XREAL_1: 235, XXREAL_0: 2;

      then ( N-min C) in { |[r9, s9]| where r9,s9 be Real : ((G * (i9,1)) `1 ) <= r9 & r9 <= ((G * ((i9 + 1),1)) `1 ) & ((G * (1,(( len G) -' 1))) `2 ) <= s9 & s9 <= ((G * (1,( len G))) `2 ) } by A9, A13, A34, A33, A25, A26, GOBRD11: 32;

      then ex r9,s9 be Real st ( N-min C) = |[r9, s9]| & ((G * (i9,1)) `1 ) <= r9 & r9 <= ((G * ((i9 + 1),1)) `1 ) & ((G * (1,(( len G) -' 1))) `2 ) <= s9 & s9 <= ((G * (1,( len G))) `2 );

      then

       A43: ((f /. 1) `1 ) <= (( N-min C) `1 ) by A11, A33, A37, EUCLID: 52;

      then

       A44: (( N-min ( L~ f)) `1 ) <= (( N-min C) `1 ) by A38, XXREAL_0: 2;

      

       A45: 1 <= m by A3, FINSEQ_3: 25;

      

       A46: ((G * (i9,(( len G) -' 1))) `2 ) = N by A9, A34, JORDAN8: 14;

      

       A47: ( N-min C) = |[(( N-min C) `1 ), (( N-min C) `2 )]| by EUCLID: 53;

      

       A48: (( NW-corner C) `2 ) = N & (( NE-corner C) `2 ) = N by EUCLID: 52;

      

       A49: (( NW-corner C) `1 ) = W & (( NE-corner C) `1 ) = E by EUCLID: 52;

      

       A50: ( len G) = ( width G) by JORDAN8:def 1;

      ((G * (i9,(( len G) -' 1))) `1 ) = ((G * (i9,1)) `1 ) by A9, A34, A33, A25, A26, GOBOARD5: 2;

      then

       A51: ((G * (i9,(( len G) -' 1))) `1 ) < (( N-min C) `1 ) by A11, A14, A33, A37, A43, A47, A15, A46, XXREAL_0: 1;

      

       A52: ((G * (i,( len G))) `1 ) = ((G * (i,1)) `1 ) by A19, A27, A22, A28, A36, GOBOARD5: 2;

      per cases by A21, XXREAL_0: 1;

        suppose m = ( len f);

        hence thesis by A4, A7, FINSEQ_6:def 1;

      end;

        suppose m < ( len f);

        then

         A53: (m + 1) <= ( len f) by NAT_1: 13;

        then

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

         A54: [i1, j1] in ( Indices G) & (f /. m) = (G * (i1,j1)) and

         A55: [i2, j2] in ( Indices G) and

         A56: (f /. (m + 1)) = (G * (i2,j2)) and

         A57: 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, A45, JORDAN8: 3;

        

         A58: ( right_cell (f,m,G)) meets C by A2, A45, A53;

        then

        consider p be object such that

         A59: p in ( right_cell (f,m,G)) and

         A60: p in C by XBOOLE_0: 3;

        reconsider p as Point of ( TOP-REAL 2) by A59;

        

         A61: W <= (p `1 ) & (p `1 ) <= E by A60, PSCOMP_1: 24;

        

         A62: (( N-min C) `2 ) = N by EUCLID: 52;

        then

         A63: (p `2 ) <= (( N-min C) `2 ) by A60, PSCOMP_1: 24;

        

         A64: ((G * (1,(( len G) -' 1))) `2 ) < ((G * (1,( len G))) `2 ) by A50, A24, A25, A26, GOBOARD5: 4;

        

         A65: ((G * (1,(( len G) -' 1))) `2 ) = N by A24, JORDAN8: 14;

        

         A66: j2 <= ( len G) by A50, A55, MATRIX_0: 32;

        now

          per cases by A5, A6, A36, A54, A57, GOBOARD1: 5;

            suppose i = i2 & (( len G) + 1) = j2;

            hence thesis by A66, NAT_1: 13;

          end;

            suppose

             A67: (i + 1) = i2 & ( len G) = j2;

            

             A68: ( cell (G,i,(( len G) -' 1))) = { |[r, s]| : ((G * (i,1)) `1 ) <= r & r <= ((G * ((i + 1),1)) `1 ) & ((G * (1,(( len G) -' 1))) `2 ) <= s & s <= ((G * (1,((( len G) -' 1) + 1))) `2 ) } by A50, A19, A25, A26, A40, GOBRD11: 32;

            ( right_cell (f,m,G)) = ( cell (G,i,(( len G) -' 1))) by A1, A45, A5, A6, A36, A53, A55, A56, A67, GOBRD13: 24;

            then

            consider r, s such that

             A69: p = |[r, s]| and ((G * (i,1)) `1 ) <= r and

             A70: r <= ((G * ((i + 1),1)) `1 ) and

             A71: ((G * (1,(( len G) -' 1))) `2 ) <= s and s <= ((G * (1,((( len G) -' 1) + 1))) `2 ) by A59, A68;

            (p `2 ) = s by A69, EUCLID: 52;

            then (p `2 ) = N by A62, A63, A65, A71, XXREAL_0: 1;

            then p in ( LSeg (( NW-corner C),( NE-corner C))) by A49, A48, A61, GOBOARD7: 8;

            then p in (( LSeg (( NW-corner C),( NE-corner C))) /\ C) by A60, XBOOLE_0:def 4;

            then

             A72: (( N-min C) `1 ) <= (p `1 ) by PSCOMP_1: 39;

            (p `1 ) = r by A69, EUCLID: 52;

            then (( N-min C) `1 ) <= ((G * ((i + 1),1)) `1 ) by A70, A72, XXREAL_0: 2;

            then

             A73: ( N-min C) in ( cell (G,i,(( width G) -' 1))) by A33, A4, A7, A6, A36, A52, A42, A44, A47, A62, A65, A64, A68;

            ( N-min C) <> (G * (i,(( len G) -' 1))) by A34, A33, A19, A25, A26, A51, A39, SPRECT_3: 13;

            hence thesis by A9, A10, A11, A13, A14, A33, A4, A7, A6, A19, A36, A41, A73, Th29;

          end;

            suppose i = (i2 + 1) & ( len G) = j2;

            then ( right_cell (f,m,G)) = ( cell (G,i2,( len G))) & i2 < ( len G) by A1, A45, A5, A6, A27, A36, A53, A55, A56, GOBRD13: 26, NAT_1: 13;

            hence thesis by A2, A45, A53, JORDAN8: 15;

          end;

            suppose

             A74: i = i2 & ( len G) = (j2 + 1);

            then

             A75: j2 = (( len G) -' 1) by NAT_D: 34;

            then

             A76: ( right_cell (f,m,G)) = ( cell (G,(i -' 1),(( len G) -' 1))) by A1, A45, A5, A6, A36, A53, A55, A56, A74, GOBRD13: 28;

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

            then

             A77: (m -' 1) <= ( len f) by A21, XXREAL_0: 2;

            now

              1 <= (i9 + 1) by A9, NAT_1: 13;

              then

               A78: ((G * ((i9 + 1),( len G))) `2 ) = ((G * (1,( len G))) `2 ) by A10, A33, A24, GOBOARD5: 1;

              assume

               A79: m = 1;

              ((G * (i9,( len G))) `2 ) = ((G * (1,( len G))) `2 ) by A9, A34, A33, A24, GOBOARD5: 1;

              hence contradiction by A11, A12, A33, A6, A19, A27, A36, A25, A26, A56, A74, A75, A79, A78, GOBOARD5: 4;

            end;

            then m > 1 by A45, XXREAL_0: 1;

            then

             A80: (m -' 1) >= 1 by NAT_D: 49;

            

             A81: ((m -' 1) + 1) = m by A45, XREAL_1: 235;

            then

            consider i19,j19,i29,j29 be Nat such that

             A82: [i19, j19] in ( Indices G) and

             A83: (f /. (m -' 1)) = (G * (i19,j19)) and

             A84: [i29, j29] in ( Indices G) & (f /. m) = (G * (i29,j29)) & (i19 = i29 & (j19 + 1) = j29 or (i19 + 1) = i29 & j19 = j29 or i19 = (i29 + 1) & j19 = j29 or i19 = i29 & j19 = (j29 + 1)) by A1, A21, A80, JORDAN8: 3;

            

             A85: 1 <= i19 by A82, MATRIX_0: 32;

            

             A86: i19 <= ( len G) by A82, MATRIX_0: 32;

            now

              per cases by A5, A6, A36, A84, GOBOARD1: 5;

                suppose

                 A87: i19 = i & (j19 + 1) = ( len G);

                then j19 = (( len G) -' 1) by NAT_D: 34;

                then ( left_cell (f,(m -' 1),G)) = ( cell (G,(i -' 1),(( len G) -' 1))) by A1, A21, A5, A6, A36, A80, A81, A82, A83, A87, GOBRD13: 21;

                hence contradiction by A2, A21, A58, A76, A80, A81;

              end;

                suppose

                 A88: (i19 + 1) = i & j19 = ( len G);

                

                 A89: ((G * (i19,j)) `2 ) = ((G * (1,j)) `2 ) & ((G * (i,j)) `2 ) = ((G * (1,j)) `2 ) by A19, A27, A22, A28, A85, A86, GOBOARD5: 1;

                (m -' 1) in ( dom f) by A80, A77, FINSEQ_3: 25;

                then

                 A90: (f /. (m -' 1)) in ( L~ f) by A30, GOBOARD1: 1, XXREAL_0: 2;

                then ( W-bound ( L~ f)) <= ((f /. (m -' 1)) `1 ) & ((f /. (m -' 1)) `1 ) <= ( E-bound ( L~ f)) by PSCOMP_1: 24;

                then (f /. (m -' 1)) in ( LSeg (( NW-corner ( L~ f)),( NE-corner ( L~ f)))) by A4, A7, A8, A6, A36, A18, A20, A83, A88, A89, GOBOARD7: 8;

                then

                 A91: (f /. (m -' 1)) in (( LSeg (( NW-corner ( L~ f)),( NE-corner ( L~ f)))) /\ ( L~ f)) by A90, XBOOLE_0:def 4;

                i19 < i by A88, NAT_1: 13;

                then ((f /. (m -' 1)) `1 ) < ((f /. m) `1 ) by A6, A27, A22, A28, A36, A83, A85, A88, GOBOARD5: 3;

                hence contradiction by A4, A7, A91, PSCOMP_1: 39;

              end;

                suppose i19 = (i + 1) & j19 = ( len G);

                then ( right_cell (f,(m -' 1),G)) = ( cell (G,i,( len G))) by A1, A21, A5, A6, A36, A80, A81, A82, A83, GOBRD13: 26;

                hence contradiction by A2, A21, A27, A80, A81, JORDAN8: 15;

              end;

                suppose i19 = i & j19 = (( len G) + 1);

                then (( len G) + 1) <= (( len G) + 0 ) by A50, A82, MATRIX_0: 32;

                hence contradiction by XREAL_1: 6;

              end;

            end;

            hence thesis;

          end;

        end;

        hence thesis;

      end;

    end;

    definition

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

      let n be Nat;

      assume

       A1: C is connected;

      :: JORDAN9:def1

      func Cage (C,n) -> clockwise_oriented standard non constant special_circular_sequence means

      : Def1: it is_sequence_on ( Gauge (C,n)) & (ex i be Nat st 1 <= i & (i + 1) <= ( len ( Gauge (C,n))) & (it /. 1) = (( Gauge (C,n)) * (i,( width ( Gauge (C,n))))) & (it /. 2) = (( Gauge (C,n)) * ((i + 1),( width ( Gauge (C,n))))) & ( N-min C) in ( cell (( Gauge (C,n)),i,(( width ( Gauge (C,n))) -' 1))) & ( N-min C) <> (( Gauge (C,n)) * (i,(( width ( Gauge (C,n))) -' 1)))) & for k be Nat st 1 <= k & (k + 2) <= ( len it ) holds (( front_left_cell (it ,k,( Gauge (C,n)))) misses C & ( front_right_cell (it ,k,( Gauge (C,n)))) misses C implies it turns_right (k,( Gauge (C,n)))) & (( front_left_cell (it ,k,( Gauge (C,n)))) misses C & ( front_right_cell (it ,k,( Gauge (C,n)))) meets C implies it goes_straight (k,( Gauge (C,n)))) & (( front_left_cell (it ,k,( Gauge (C,n)))) meets C implies it turns_left (k,( Gauge (C,n))));

      existence

      proof

        set W = ( W-bound C), E = ( E-bound C), S = ( S-bound C), N = ( N-bound C);

        set G = ( Gauge (C,n));

        defpred P[ Nat, set, set] means ($1 = 0 implies ex i st 1 <= i & (i + 1) <= ( len G) & ( N-min C) in ( cell (G,i,(( width G) -' 1))) & ( N-min C) <> (G * (i,(( width G) -' 1))) & $3 = <*(G * (i,( width G)))*>) & ($1 = 1 implies ex i st 1 <= i & (i + 1) <= ( len G) & ( N-min C) in ( cell (G,i,(( width G) -' 1))) & ( N-min C) <> (G * (i,(( width G) -' 1))) & $3 = <*(G * (i,( width G))), (G * ((i + 1),( width G)))*>) & ($1 > 1 & $2 is FinSequence of ( TOP-REAL 2) implies for f be FinSequence of ( TOP-REAL 2) st $2 = f holds (( len f) = $1 implies (f is_sequence_on G & ( right_cell (f,(( len f) -' 1),G)) meets C implies (( front_left_cell (f,(( len f) -' 1),G)) misses C & ( front_right_cell (f,(( len f) -' 1),G)) misses C implies ex i, j st (f ^ <*(G * (i,j))*>) turns_right ((( len f) -' 1),G) & $3 = (f ^ <*(G * (i,j))*>)) & (( front_left_cell (f,(( len f) -' 1),G)) misses C & ( front_right_cell (f,(( len f) -' 1),G)) meets C implies ex i, j st (f ^ <*(G * (i,j))*>) goes_straight ((( len f) -' 1),G) & $3 = (f ^ <*(G * (i,j))*>)) & (( front_left_cell (f,(( len f) -' 1),G)) meets C implies ex i, j st (f ^ <*(G * (i,j))*>) turns_left ((( len f) -' 1),G) & $3 = (f ^ <*(G * (i,j))*>))) & ( not f is_sequence_on G or ( right_cell (f,(( len f) -' 1),G)) misses C implies $3 = (f ^ <*(G * (1,1))*>))) & (( len f) <> $1 implies $3 = {} )) & ($1 > 1 & not $2 is FinSequence of ( TOP-REAL 2) implies $3 = {} );

        

         A2: ( len G) = ( width G) by JORDAN8:def 1;

        

         A3: for k be Nat, x be set holds ex y be set st P[k, x, y]

        proof

          let k be Nat, x be set;

          consider m be Nat such that

           A4: 1 <= m & (m + 1) <= ( len G) & ( N-min C) in ( cell (G,m,(( width G) -' 1))) & ( N-min C) <> (G * (m,(( width G) -' 1))) by Th28;

          per cases by NAT_1: 25;

            suppose

             A5: k = 0 ;

            take <*(G * (m,( width G)))*>;

            thus thesis by A4, A5;

          end;

            suppose

             A6: k = 1;

            take <*(G * (m,( width G))), (G * ((m + 1),( width G)))*>;

            thus thesis by A4, A6;

          end;

            suppose that

             A7: k > 1 and

             A8: x is FinSequence of ( TOP-REAL 2);

            reconsider f = x as FinSequence of ( TOP-REAL 2) by A8;

            per cases ;

              suppose

               A9: ( len f) = k;

              per cases ;

                suppose

                 A10: f is_sequence_on G & ( right_cell (f,(( len f) -' 1),G)) meets C;

                

                 A11: ((( len f) -' 1) + 1) = ( len f) by A7, A9, XREAL_1: 235;

                then

                 A12: ((( len f) -' 1) + (1 + 1)) = (( len f) + 1);

                

                 A13: ((( len f) -' 1) + 1) in ( dom f) by A7, A9, A11, FINSEQ_3: 25;

                

                 A14: 1 <= (( len f) -' 1) by A7, A9, NAT_D: 49;

                then

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

                 A15: [i1, j1] in ( Indices G) and

                 A16: (f /. (( len f) -' 1)) = (G * (i1,j1)) and

                 A17: [i2, j2] in ( Indices G) and

                 A18: (f /. ((( len f) -' 1) + 1)) = (G * (i2,j2)) and

                 A19: i1 = i2 & (j1 + 1) = j2 or (i1 + 1) = i2 & j1 = j2 or i1 = (i2 + 1) & j1 = j2 or i1 = i2 & j1 = (j2 + 1) by A10, A11, JORDAN8: 3;

                

                 A20: i1 <= ( len G) by A15, MATRIX_0: 32;

                

                 A21: 1 <= (j2 + 1) by NAT_1: 12;

                

                 A22: 1 <= i2 by A17, MATRIX_0: 32;

                

                 A23: j1 <= ( width G) by A15, MATRIX_0: 32;

                

                 A24: 1 <= (i2 + 1) by NAT_1: 12;

                

                 A25: 1 <= j2 by A17, MATRIX_0: 32;

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

                then

                 A26: (( len f) -' 1) in ( dom f) by A14, FINSEQ_3: 25;

                

                 A27: j2 <= ( width G) by A17, MATRIX_0: 32;

                then

                 A28: (j2 -' 1) <= ( width G) by NAT_D: 44;

                

                 A29: i2 <= ( len G) by A17, MATRIX_0: 32;

                then

                 A30: (i2 -' 1) <= ( len G) by NAT_D: 44;

                per cases ;

                  suppose

                   A31: ( front_left_cell (f,(( len f) -' 1),G)) misses C & ( front_right_cell (f,(( len f) -' 1),G)) misses C;

                  per cases by A19;

                    suppose

                     A32: i1 = i2 & (j1 + 1) = j2;

                    take f1 = (f ^ <*(G * ((i2 + 1),j2))*>);

                    now

                      take i = (i2 + 1), j = j2;

                      thus f1 turns_right ((( len f) -' 1),G)

                      proof

                        let i19,j19,i29,j29 be Nat;

                        assume that

                         A33: [i19, j19] in ( Indices G) and

                         A34: [i29, j29] in ( Indices G) and

                         A35: (f1 /. (( len f) -' 1)) = (G * (i19,j19)) and

                         A36: (f1 /. ((( len f) -' 1) + 1)) = (G * (i29,j29));

                        

                         A37: (f /. (( len f) -' 1)) = (G * (i19,j19)) by A26, A35, FINSEQ_4: 68;

                        then

                         A38: i1 = i19 by A15, A16, A33, GOBOARD1: 5;

                        

                         A39: j1 = j19 by A15, A16, A33, A37, GOBOARD1: 5;

                        

                         A40: (f /. ((( len f) -' 1) + 1)) = (G * (i29,j29)) by A13, A36, FINSEQ_4: 68;

                        then

                         A41: i2 = i29 by A17, A18, A34, GOBOARD1: 5;

                        

                         A42: j2 = j29 by A17, A18, A34, A40, GOBOARD1: 5;

                        per cases by A15, A16, A19, A33, A37, A41, A42, GOBOARD1: 5;

                          case i19 = i29 & (j19 + 1) = j29;

                          now

                            assume (i2 + 1) > ( len G);

                            then

                             A43: (( len G) + 1) <= (i2 + 1) by NAT_1: 13;

                            (i2 + 1) <= (( len G) + 1) by A29, XREAL_1: 6;

                            then (i2 + 1) = (( len G) + 1) by A43, XXREAL_0: 1;

                            then ( cell (G,( len G),j1)) meets C by A10, A14, A11, A15, A16, A17, A18, A32, GOBRD13: 22;

                            hence contradiction by A2, A23, JORDAN8: 16;

                          end;

                          hence [(i29 + 1), j29] in ( Indices G) by A25, A27, A24, A41, A42, MATRIX_0: 30;

                          thus thesis by A12, A41, A42, FINSEQ_4: 67;

                        end;

                          case (i19 + 1) = i29 & j19 = j29;

                          hence thesis by A32, A38, A41;

                        end;

                          case i19 = (i29 + 1) & j19 = j29;

                          hence thesis by A32, A38, A41;

                        end;

                          case i19 = i29 & j19 = (j29 + 1);

                          hence thesis by A32, A39, A42;

                        end;

                      end;

                    end;

                    hence thesis by A7, A9, A10, A31;

                  end;

                    suppose

                     A44: (i1 + 1) = i2 & j1 = j2;

                    take f1 = (f ^ <*(G * (i2,(j2 -' 1)))*>);

                    now

                      take i = i2, j = (j2 -' 1);

                      thus f1 turns_right ((( len f) -' 1),G)

                      proof

                        let i19,j19,i29,j29 be Nat;

                        assume that

                         A45: [i19, j19] in ( Indices G) and

                         A46: [i29, j29] in ( Indices G) and

                         A47: (f1 /. (( len f) -' 1)) = (G * (i19,j19)) and

                         A48: (f1 /. ((( len f) -' 1) + 1)) = (G * (i29,j29));

                        

                         A49: (f /. ((( len f) -' 1) + 1)) = (G * (i29,j29)) by A13, A48, FINSEQ_4: 68;

                        then

                         A50: i2 = i29 by A17, A18, A46, GOBOARD1: 5;

                        

                         A51: (f /. (( len f) -' 1)) = (G * (i19,j19)) by A26, A47, FINSEQ_4: 68;

                        then

                         A52: i1 = i19 by A15, A16, A45, GOBOARD1: 5;

                        

                         A53: j2 = j29 by A17, A18, A46, A49, GOBOARD1: 5;

                        per cases by A15, A16, A19, A45, A51, A50, A53, GOBOARD1: 5;

                          case i19 = i29 & (j19 + 1) = j29;

                          hence thesis by A44, A52, A50;

                        end;

                          case (i19 + 1) = i29 & j19 = j29;

                          now

                            assume (j2 -' 1) < 1;

                            then j2 <= 1 by NAT_1: 14, NAT_D: 36;

                            then j2 = 1 by A25, XXREAL_0: 1;

                            then ( cell (G,i1,(1 -' 1))) meets C by A10, A14, A11, A15, A16, A17, A18, A44, GOBRD13: 24;

                            then ( cell (G,i1, 0 )) meets C by XREAL_1: 232;

                            hence contradiction by A20, JORDAN8: 17;

                          end;

                          hence [i29, (j29 -' 1)] in ( Indices G) by A22, A29, A28, A50, A53, MATRIX_0: 30;

                          thus thesis by A12, A50, A53, FINSEQ_4: 67;

                        end;

                          case i19 = (i29 + 1) & j19 = j29;

                          hence thesis by A44, A52, A50;

                        end;

                          case i19 = i29 & j19 = (j29 + 1);

                          hence thesis by A44, A52, A50;

                        end;

                      end;

                    end;

                    hence thesis by A7, A9, A10, A31;

                  end;

                    suppose

                     A54: i1 = (i2 + 1) & j1 = j2;

                    take f1 = (f ^ <*(G * (i2,(j2 + 1)))*>);

                    now

                      take i = i2, j = (j2 + 1);

                      thus f1 turns_right ((( len f) -' 1),G)

                      proof

                        let i19,j19,i29,j29 be Nat;

                        assume that

                         A55: [i19, j19] in ( Indices G) and

                         A56: [i29, j29] in ( Indices G) and

                         A57: (f1 /. (( len f) -' 1)) = (G * (i19,j19)) and

                         A58: (f1 /. ((( len f) -' 1) + 1)) = (G * (i29,j29));

                        

                         A59: (f /. ((( len f) -' 1) + 1)) = (G * (i29,j29)) by A13, A58, FINSEQ_4: 68;

                        then

                         A60: i2 = i29 by A17, A18, A56, GOBOARD1: 5;

                        

                         A61: (f /. (( len f) -' 1)) = (G * (i19,j19)) by A26, A57, FINSEQ_4: 68;

                        then

                         A62: i1 = i19 by A15, A16, A55, GOBOARD1: 5;

                        

                         A63: j2 = j29 by A17, A18, A56, A59, GOBOARD1: 5;

                        per cases by A15, A16, A19, A55, A61, A60, A63, GOBOARD1: 5;

                          case i19 = i29 & (j19 + 1) = j29;

                          hence thesis by A54, A62, A60;

                        end;

                          case (i19 + 1) = i29 & j19 = j29;

                          hence thesis by A54, A62, A60;

                        end;

                          case i19 = (i29 + 1) & j19 = j29;

                          now

                            assume (j2 + 1) > ( len G);

                            then

                             A64: (( len G) + 1) <= (j2 + 1) by NAT_1: 13;

                            (j2 + 1) <= (( len G) + 1) by A2, A27, XREAL_1: 6;

                            then (j2 + 1) = (( len G) + 1) by A64, XXREAL_0: 1;

                            then ( cell (G,i2,( len G))) meets C by A10, A14, A11, A15, A16, A17, A18, A54, GOBRD13: 26;

                            hence contradiction by A29, JORDAN8: 15;

                          end;

                          hence [i29, (j29 + 1)] in ( Indices G) by A2, A22, A29, A21, A60, A63, MATRIX_0: 30;

                          thus thesis by A12, A60, A63, FINSEQ_4: 67;

                        end;

                          case i19 = i29 & j19 = (j29 + 1);

                          hence thesis by A54, A62, A60;

                        end;

                      end;

                    end;

                    hence thesis by A7, A9, A10, A31;

                  end;

                    suppose

                     A65: i1 = i2 & j1 = (j2 + 1);

                    take f1 = (f ^ <*(G * ((i2 -' 1),j2))*>);

                    now

                      reconsider i = (i2 -' 1), j = j2 as Nat;

                      take i, j;

                      thus f1 turns_right ((( len f) -' 1),G)

                      proof

                        let i19,j19,i29,j29 be Nat;

                        assume that

                         A66: [i19, j19] in ( Indices G) and

                         A67: [i29, j29] in ( Indices G) and

                         A68: (f1 /. (( len f) -' 1)) = (G * (i19,j19)) and

                         A69: (f1 /. ((( len f) -' 1) + 1)) = (G * (i29,j29));

                        

                         A70: (f /. (( len f) -' 1)) = (G * (i19,j19)) by A26, A68, FINSEQ_4: 68;

                        then

                         A71: i1 = i19 by A15, A16, A66, GOBOARD1: 5;

                        

                         A72: j1 = j19 by A15, A16, A66, A70, GOBOARD1: 5;

                        

                         A73: (f /. ((( len f) -' 1) + 1)) = (G * (i29,j29)) by A13, A69, FINSEQ_4: 68;

                        then

                         A74: i2 = i29 by A17, A18, A67, GOBOARD1: 5;

                        

                         A75: j2 = j29 by A17, A18, A67, A73, GOBOARD1: 5;

                        per cases by A15, A16, A19, A66, A70, A74, A75, GOBOARD1: 5;

                          case i19 = i29 & (j19 + 1) = j29;

                          hence thesis by A65, A72, A75;

                        end;

                          case (i19 + 1) = i29 & j19 = j29;

                          hence thesis by A65, A71, A74;

                        end;

                          case i19 = (i29 + 1) & j19 = j29;

                          hence thesis by A65, A71, A74;

                        end;

                          case i19 = i29 & j19 = (j29 + 1);

                          now

                            assume (i2 -' 1) < 1;

                            then i2 <= 1 by NAT_1: 14, NAT_D: 36;

                            then i2 = 1 by A22, XXREAL_0: 1;

                            then ( cell (G,(1 -' 1),j2)) meets C by A10, A14, A11, A15, A16, A17, A18, A65, GOBRD13: 28;

                            then ( cell (G, 0 ,j2)) meets C by XREAL_1: 232;

                            hence contradiction by A2, A27, JORDAN8: 18;

                          end;

                          hence [(i29 -' 1), j29] in ( Indices G) by A25, A27, A30, A74, A75, MATRIX_0: 30;

                          thus thesis by A12, A74, A75, FINSEQ_4: 67;

                        end;

                      end;

                    end;

                    hence thesis by A7, A9, A10, A31;

                  end;

                end;

                  suppose

                   A76: ( front_left_cell (f,(( len f) -' 1),G)) misses C & ( front_right_cell (f,(( len f) -' 1),G)) meets C;

                  per cases by A19;

                    suppose

                     A77: i1 = i2 & (j1 + 1) = j2;

                    take f1 = (f ^ <*(G * (i2,(j2 + 1)))*>);

                    now

                      take i = i2, j = (j2 + 1);

                      thus f1 goes_straight ((( len f) -' 1),G)

                      proof

                        let i19,j19,i29,j29 be Nat;

                        assume that

                         A78: [i19, j19] in ( Indices G) and

                         A79: [i29, j29] in ( Indices G) and

                         A80: (f1 /. (( len f) -' 1)) = (G * (i19,j19)) and

                         A81: (f1 /. ((( len f) -' 1) + 1)) = (G * (i29,j29));

                        

                         A82: (f /. (( len f) -' 1)) = (G * (i19,j19)) by A26, A80, FINSEQ_4: 68;

                        then

                         A83: i1 = i19 by A15, A16, A78, GOBOARD1: 5;

                        

                         A84: j1 = j19 by A15, A16, A78, A82, GOBOARD1: 5;

                        

                         A85: (f /. ((( len f) -' 1) + 1)) = (G * (i29,j29)) by A13, A81, FINSEQ_4: 68;

                        then

                         A86: i2 = i29 by A17, A18, A79, GOBOARD1: 5;

                        

                         A87: j2 = j29 by A17, A18, A79, A85, GOBOARD1: 5;

                        per cases by A15, A16, A19, A78, A82, A86, A87, GOBOARD1: 5;

                          case i19 = i29 & (j19 + 1) = j29;

                          now

                            assume (j2 + 1) > ( len G);

                            then

                             A88: (( len G) + 1) <= (j2 + 1) by NAT_1: 13;

                            (j2 + 1) <= (( len G) + 1) by A2, A27, XREAL_1: 6;

                            then (j2 + 1) = (( len G) + 1) by A88, XXREAL_0: 1;

                            then ( cell (G,i1,( len G))) meets C by A10, A14, A11, A15, A16, A17, A18, A76, A77, GOBRD13: 35;

                            hence contradiction by A20, JORDAN8: 15;

                          end;

                          hence [i29, (j29 + 1)] in ( Indices G) by A2, A22, A29, A21, A86, A87, MATRIX_0: 30;

                          thus thesis by A12, A86, A87, FINSEQ_4: 67;

                        end;

                          case (i19 + 1) = i29 & j19 = j29;

                          hence thesis by A77, A83, A86;

                        end;

                          case i19 = (i29 + 1) & j19 = j29;

                          hence thesis by A77, A83, A86;

                        end;

                          case i19 = i29 & j19 = (j29 + 1);

                          hence thesis by A77, A84, A87;

                        end;

                      end;

                    end;

                    hence thesis by A7, A9, A10, A76;

                  end;

                    suppose

                     A89: (i1 + 1) = i2 & j1 = j2;

                    take f1 = (f ^ <*(G * ((i2 + 1),j2))*>);

                    now

                      take i = (i2 + 1), j = j2;

                      thus f1 goes_straight ((( len f) -' 1),G)

                      proof

                        let i19,j19,i29,j29 be Nat;

                        assume that

                         A90: [i19, j19] in ( Indices G) and

                         A91: [i29, j29] in ( Indices G) and

                         A92: (f1 /. (( len f) -' 1)) = (G * (i19,j19)) and

                         A93: (f1 /. ((( len f) -' 1) + 1)) = (G * (i29,j29));

                        

                         A94: (f /. ((( len f) -' 1) + 1)) = (G * (i29,j29)) by A13, A93, FINSEQ_4: 68;

                        then

                         A95: i2 = i29 by A17, A18, A91, GOBOARD1: 5;

                        

                         A96: (f /. (( len f) -' 1)) = (G * (i19,j19)) by A26, A92, FINSEQ_4: 68;

                        then

                         A97: i1 = i19 by A15, A16, A90, GOBOARD1: 5;

                        

                         A98: j2 = j29 by A17, A18, A91, A94, GOBOARD1: 5;

                        per cases by A15, A16, A19, A90, A96, A95, A98, GOBOARD1: 5;

                          case i19 = i29 & (j19 + 1) = j29;

                          hence thesis by A89, A97, A95;

                        end;

                          case (i19 + 1) = i29 & j19 = j29;

                          now

                            assume (i2 + 1) > ( len G);

                            then

                             A99: (( len G) + 1) <= (i2 + 1) by NAT_1: 13;

                            (i2 + 1) <= (( len G) + 1) by A29, XREAL_1: 6;

                            then (i2 + 1) = (( len G) + 1) by A99, XXREAL_0: 1;

                            then ( cell (G,( len G),(j1 -' 1))) meets C by A10, A14, A11, A15, A16, A17, A18, A76, A89, GOBRD13: 37;

                            hence contradiction by A2, A23, JORDAN8: 16, NAT_D: 44;

                          end;

                          hence [(i29 + 1), j29] in ( Indices G) by A25, A27, A24, A95, A98, MATRIX_0: 30;

                          thus thesis by A12, A95, A98, FINSEQ_4: 67;

                        end;

                          case i19 = (i29 + 1) & j19 = j29;

                          hence thesis by A89, A97, A95;

                        end;

                          case i19 = i29 & j19 = (j29 + 1);

                          hence thesis by A89, A97, A95;

                        end;

                      end;

                    end;

                    hence thesis by A7, A9, A10, A76;

                  end;

                    suppose

                     A100: i1 = (i2 + 1) & j1 = j2;

                    take f1 = (f ^ <*(G * ((i2 -' 1),j2))*>);

                    now

                      take i = (i2 -' 1), j = j2;

                      thus f1 goes_straight ((( len f) -' 1),G)

                      proof

                        let i19,j19,i29,j29 be Nat;

                        assume that

                         A101: [i19, j19] in ( Indices G) and

                         A102: [i29, j29] in ( Indices G) and

                         A103: (f1 /. (( len f) -' 1)) = (G * (i19,j19)) and

                         A104: (f1 /. ((( len f) -' 1) + 1)) = (G * (i29,j29));

                        

                         A105: (f /. ((( len f) -' 1) + 1)) = (G * (i29,j29)) by A13, A104, FINSEQ_4: 68;

                        then

                         A106: i2 = i29 by A17, A18, A102, GOBOARD1: 5;

                        

                         A107: (f /. (( len f) -' 1)) = (G * (i19,j19)) by A26, A103, FINSEQ_4: 68;

                        then

                         A108: i1 = i19 by A15, A16, A101, GOBOARD1: 5;

                        

                         A109: j2 = j29 by A17, A18, A102, A105, GOBOARD1: 5;

                        per cases by A15, A16, A19, A101, A107, A106, A109, GOBOARD1: 5;

                          case i19 = i29 & (j19 + 1) = j29;

                          hence thesis by A100, A108, A106;

                        end;

                          case (i19 + 1) = i29 & j19 = j29;

                          hence thesis by A100, A108, A106;

                        end;

                          case i19 = (i29 + 1) & j19 = j29;

                          now

                            assume (i2 -' 1) < 1;

                            then i2 <= 1 by NAT_1: 14, NAT_D: 36;

                            then i2 = 1 by A22, XXREAL_0: 1;

                            then ( cell (G,(1 -' 1),j1)) meets C by A10, A14, A11, A15, A16, A17, A18, A76, A100, GOBRD13: 39;

                            then ( cell (G, 0 ,j1)) meets C by XREAL_1: 232;

                            hence contradiction by A2, A23, JORDAN8: 18;

                          end;

                          hence [(i29 -' 1), j29] in ( Indices G) by A25, A27, A30, A106, A109, MATRIX_0: 30;

                          thus thesis by A12, A106, A109, FINSEQ_4: 67;

                        end;

                          case i19 = i29 & j19 = (j29 + 1);

                          hence thesis by A100, A108, A106;

                        end;

                      end;

                    end;

                    hence thesis by A7, A9, A10, A76;

                  end;

                    suppose

                     A110: i1 = i2 & j1 = (j2 + 1);

                    take f1 = (f ^ <*(G * (i2,(j2 -' 1)))*>);

                    now

                      take i = i2, j = (j2 -' 1);

                      thus f1 goes_straight ((( len f) -' 1),G)

                      proof

                        let i19,j19,i29,j29 be Nat;

                        assume that

                         A111: [i19, j19] in ( Indices G) and

                         A112: [i29, j29] in ( Indices G) and

                         A113: (f1 /. (( len f) -' 1)) = (G * (i19,j19)) and

                         A114: (f1 /. ((( len f) -' 1) + 1)) = (G * (i29,j29));

                        

                         A115: (f /. (( len f) -' 1)) = (G * (i19,j19)) by A26, A113, FINSEQ_4: 68;

                        then

                         A116: i1 = i19 by A15, A16, A111, GOBOARD1: 5;

                        

                         A117: j1 = j19 by A15, A16, A111, A115, GOBOARD1: 5;

                        

                         A118: (f /. ((( len f) -' 1) + 1)) = (G * (i29,j29)) by A13, A114, FINSEQ_4: 68;

                        then

                         A119: i2 = i29 by A17, A18, A112, GOBOARD1: 5;

                        

                         A120: j2 = j29 by A17, A18, A112, A118, GOBOARD1: 5;

                        per cases by A15, A16, A19, A111, A115, A119, A120, GOBOARD1: 5;

                          case i19 = i29 & (j19 + 1) = j29;

                          hence thesis by A110, A117, A120;

                        end;

                          case (i19 + 1) = i29 & j19 = j29;

                          hence thesis by A110, A116, A119;

                        end;

                          case i19 = (i29 + 1) & j19 = j29;

                          hence thesis by A110, A116, A119;

                        end;

                          case i19 = i29 & j19 = (j29 + 1);

                          now

                            assume (j2 -' 1) < 1;

                            then j2 <= 1 by NAT_1: 14, NAT_D: 36;

                            then j2 = 1 by A25, XXREAL_0: 1;

                            then ( cell (G,(i1 -' 1),(1 -' 1))) meets C by A10, A14, A11, A15, A16, A17, A18, A76, A110, GOBRD13: 41;

                            then ( cell (G,(i1 -' 1), 0 )) meets C by XREAL_1: 232;

                            hence contradiction by A20, JORDAN8: 17, NAT_D: 44;

                          end;

                          hence [i29, (j29 -' 1)] in ( Indices G) by A22, A29, A28, A119, A120, MATRIX_0: 30;

                          thus thesis by A12, A119, A120, FINSEQ_4: 67;

                        end;

                      end;

                    end;

                    hence thesis by A7, A9, A10, A76;

                  end;

                end;

                  suppose

                   A121: ( front_left_cell (f,(( len f) -' 1),G)) meets C;

                  per cases by A19;

                    suppose

                     A122: i1 = i2 & (j1 + 1) = j2;

                    take f1 = (f ^ <*(G * ((i2 -' 1),j2))*>);

                    now

                      take i = (i2 -' 1), j = j2;

                      thus f1 turns_left ((( len f) -' 1),G)

                      proof

                        let i19,j19,i29,j29 be Nat;

                        assume that

                         A123: [i19, j19] in ( Indices G) and

                         A124: [i29, j29] in ( Indices G) and

                         A125: (f1 /. (( len f) -' 1)) = (G * (i19,j19)) and

                         A126: (f1 /. ((( len f) -' 1) + 1)) = (G * (i29,j29));

                        

                         A127: (f /. (( len f) -' 1)) = (G * (i19,j19)) by A26, A125, FINSEQ_4: 68;

                        then

                         A128: i1 = i19 by A15, A16, A123, GOBOARD1: 5;

                        

                         A129: j1 = j19 by A15, A16, A123, A127, GOBOARD1: 5;

                        

                         A130: (f /. ((( len f) -' 1) + 1)) = (G * (i29,j29)) by A13, A126, FINSEQ_4: 68;

                        then

                         A131: i2 = i29 by A17, A18, A124, GOBOARD1: 5;

                        

                         A132: j2 = j29 by A17, A18, A124, A130, GOBOARD1: 5;

                        per cases by A15, A16, A19, A123, A127, A131, A132, GOBOARD1: 5;

                          case i19 = i29 & (j19 + 1) = j29;

                          now

                            assume (i2 -' 1) < 1;

                            then i2 <= 1 by NAT_1: 14, NAT_D: 36;

                            then i2 = 1 by A22, XXREAL_0: 1;

                            then ( cell (G,(1 -' 1),j2)) meets C by A10, A14, A11, A15, A16, A17, A18, A121, A122, GOBRD13: 34;

                            then ( cell (G, 0 ,j2)) meets C by XREAL_1: 232;

                            hence contradiction by A2, A27, JORDAN8: 18;

                          end;

                          hence [(i29 -' 1), j29] in ( Indices G) by A25, A27, A30, A131, A132, MATRIX_0: 30;

                          thus thesis by A12, A131, A132, FINSEQ_4: 67;

                        end;

                          case (i19 + 1) = i29 & j19 = j29;

                          hence thesis by A122, A128, A131;

                        end;

                          case i19 = (i29 + 1) & j19 = j29;

                          hence thesis by A122, A128, A131;

                        end;

                          case i19 = i29 & j19 = (j29 + 1);

                          hence thesis by A122, A129, A132;

                        end;

                      end;

                    end;

                    hence thesis by A7, A9, A10, A121;

                  end;

                    suppose

                     A133: (i1 + 1) = i2 & j1 = j2;

                    take f1 = (f ^ <*(G * (i2,(j2 + 1)))*>);

                    now

                      take i = i2, j = (j2 + 1);

                      thus f1 turns_left ((( len f) -' 1),G)

                      proof

                        let i19,j19,i29,j29 be Nat;

                        assume that

                         A134: [i19, j19] in ( Indices G) and

                         A135: [i29, j29] in ( Indices G) and

                         A136: (f1 /. (( len f) -' 1)) = (G * (i19,j19)) and

                         A137: (f1 /. ((( len f) -' 1) + 1)) = (G * (i29,j29));

                        

                         A138: (f /. ((( len f) -' 1) + 1)) = (G * (i29,j29)) by A13, A137, FINSEQ_4: 68;

                        then

                         A139: i2 = i29 by A17, A18, A135, GOBOARD1: 5;

                        

                         A140: (f /. (( len f) -' 1)) = (G * (i19,j19)) by A26, A136, FINSEQ_4: 68;

                        then

                         A141: i1 = i19 by A15, A16, A134, GOBOARD1: 5;

                        

                         A142: j2 = j29 by A17, A18, A135, A138, GOBOARD1: 5;

                        per cases by A15, A16, A19, A134, A140, A139, A142, GOBOARD1: 5;

                          case i19 = i29 & (j19 + 1) = j29;

                          hence thesis by A133, A141, A139;

                        end;

                          case (i19 + 1) = i29 & j19 = j29;

                          now

                            assume (j2 + 1) > ( len G);

                            then

                             A143: (( len G) + 1) <= (j2 + 1) by NAT_1: 13;

                            (j2 + 1) <= (( len G) + 1) by A2, A27, XREAL_1: 6;

                            then (j2 + 1) = (( len G) + 1) by A143, XXREAL_0: 1;

                            then ( cell (G,i2,( len G))) meets C by A10, A14, A11, A15, A16, A17, A18, A121, A133, GOBRD13: 36;

                            hence contradiction by A29, JORDAN8: 15;

                          end;

                          hence [i29, (j29 + 1)] in ( Indices G) by A2, A22, A29, A21, A139, A142, MATRIX_0: 30;

                          thus thesis by A12, A139, A142, FINSEQ_4: 67;

                        end;

                          case i19 = (i29 + 1) & j19 = j29;

                          hence thesis by A133, A141, A139;

                        end;

                          case i19 = i29 & j19 = (j29 + 1);

                          hence thesis by A133, A141, A139;

                        end;

                      end;

                    end;

                    hence thesis by A7, A9, A10, A121;

                  end;

                    suppose

                     A144: i1 = (i2 + 1) & j1 = j2;

                    take f1 = (f ^ <*(G * (i2,(j2 -' 1)))*>);

                    now

                      take i = i2, j = (j2 -' 1);

                      thus f1 turns_left ((( len f) -' 1),G)

                      proof

                        let i19,j19,i29,j29 be Nat;

                        assume that

                         A145: [i19, j19] in ( Indices G) and

                         A146: [i29, j29] in ( Indices G) and

                         A147: (f1 /. (( len f) -' 1)) = (G * (i19,j19)) and

                         A148: (f1 /. ((( len f) -' 1) + 1)) = (G * (i29,j29));

                        

                         A149: (f /. ((( len f) -' 1) + 1)) = (G * (i29,j29)) by A13, A148, FINSEQ_4: 68;

                        then

                         A150: i2 = i29 by A17, A18, A146, GOBOARD1: 5;

                        

                         A151: (f /. (( len f) -' 1)) = (G * (i19,j19)) by A26, A147, FINSEQ_4: 68;

                        then

                         A152: i1 = i19 by A15, A16, A145, GOBOARD1: 5;

                        

                         A153: j2 = j29 by A17, A18, A146, A149, GOBOARD1: 5;

                        per cases by A15, A16, A19, A145, A151, A150, A153, GOBOARD1: 5;

                          case i19 = i29 & (j19 + 1) = j29;

                          hence thesis by A144, A152, A150;

                        end;

                          case (i19 + 1) = i29 & j19 = j29;

                          hence thesis by A144, A152, A150;

                        end;

                          case i19 = (i29 + 1) & j19 = j29;

                          now

                            assume (j2 -' 1) < 1;

                            then j2 <= 1 by NAT_1: 14, NAT_D: 36;

                            then j2 = 1 by A25, XXREAL_0: 1;

                            then ( cell (G,(i2 -' 1),(1 -' 1))) meets C by A10, A14, A11, A15, A16, A17, A18, A121, A144, GOBRD13: 38;

                            then ( cell (G,(i2 -' 1), 0 )) meets C by XREAL_1: 232;

                            hence contradiction by A29, JORDAN8: 17, NAT_D: 44;

                          end;

                          hence [i29, (j29 -' 1)] in ( Indices G) by A22, A29, A28, A150, A153, MATRIX_0: 30;

                          thus thesis by A12, A150, A153, FINSEQ_4: 67;

                        end;

                          case i19 = i29 & j19 = (j29 + 1);

                          hence thesis by A144, A152, A150;

                        end;

                      end;

                    end;

                    hence thesis by A7, A9, A10, A121;

                  end;

                    suppose

                     A154: i1 = i2 & j1 = (j2 + 1);

                    take f1 = (f ^ <*(G * ((i2 + 1),j2))*>);

                    now

                      take i = (i2 + 1), j = j2;

                      thus f1 turns_left ((( len f) -' 1),G)

                      proof

                        let i19,j19,i29,j29 be Nat;

                        assume that

                         A155: [i19, j19] in ( Indices G) and

                         A156: [i29, j29] in ( Indices G) and

                         A157: (f1 /. (( len f) -' 1)) = (G * (i19,j19)) and

                         A158: (f1 /. ((( len f) -' 1) + 1)) = (G * (i29,j29));

                        

                         A159: (f /. (( len f) -' 1)) = (G * (i19,j19)) by A26, A157, FINSEQ_4: 68;

                        then

                         A160: i1 = i19 by A15, A16, A155, GOBOARD1: 5;

                        

                         A161: j1 = j19 by A15, A16, A155, A159, GOBOARD1: 5;

                        

                         A162: (f /. ((( len f) -' 1) + 1)) = (G * (i29,j29)) by A13, A158, FINSEQ_4: 68;

                        then

                         A163: i2 = i29 by A17, A18, A156, GOBOARD1: 5;

                        

                         A164: j2 = j29 by A17, A18, A156, A162, GOBOARD1: 5;

                        per cases by A15, A16, A19, A155, A159, A163, A164, GOBOARD1: 5;

                          case i19 = i29 & (j19 + 1) = j29;

                          hence thesis by A154, A161, A164;

                        end;

                          case (i19 + 1) = i29 & j19 = j29;

                          hence thesis by A154, A160, A163;

                        end;

                          case i19 = (i29 + 1) & j19 = j29;

                          hence thesis by A154, A160, A163;

                        end;

                          case i19 = i29 & j19 = (j29 + 1);

                          now

                            assume (i2 + 1) > ( len G);

                            then

                             A165: (( len G) + 1) <= (i2 + 1) by NAT_1: 13;

                            (i2 + 1) <= (( len G) + 1) by A29, XREAL_1: 6;

                            then (i2 + 1) = (( len G) + 1) by A165, XXREAL_0: 1;

                            then ( cell (G,( len G),(j2 -' 1))) meets C by A10, A14, A11, A15, A16, A17, A18, A121, A154, GOBRD13: 40;

                            hence contradiction by A2, A27, JORDAN8: 16, NAT_D: 44;

                          end;

                          hence [(i29 + 1), j29] in ( Indices G) by A25, A27, A24, A163, A164, MATRIX_0: 30;

                          thus thesis by A12, A163, A164, FINSEQ_4: 67;

                        end;

                      end;

                    end;

                    hence thesis by A7, A9, A10, A121;

                  end;

                end;

              end;

                suppose

                 A166: not f is_sequence_on G or ( right_cell (f,(( len f) -' 1),G)) misses C;

                take (f ^ <*(G * (1,1))*>);

                thus thesis by A7, A9, A166;

              end;

            end;

              suppose

               A167: ( len f) <> k;

              take {} ;

              thus thesis by A7, A167;

            end;

          end;

            suppose

             A168: k > 1 & not x is FinSequence of ( TOP-REAL 2);

            take {} ;

            thus thesis by A168;

          end;

        end;

        consider F be Function such that

         A169: ( dom F) = NAT and

         A170: (F . 0 ) = {} and

         A171: for k be Nat holds P[k, (F . k), (F . (k + 1))] from RECDEF_1:sch 1( A3);

        defpred P[ Nat] means (F . $1) is FinSequence of ( TOP-REAL 2);

        

         A172: {} = ( <*> the carrier of ( TOP-REAL 2));

        

         A173: for k st P[k] holds P[(k + 1)]

        proof

          let k such that

           A174: (F . k) is FinSequence of ( TOP-REAL 2);

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

          

           A175: P[k, (F . k), (F . (k + 1))] by A171;

          per cases by NAT_1: 25;

            suppose k = 0 ;

            hence thesis by A175;

          end;

            suppose k = 1;

            hence thesis by A175;

          end;

            suppose

             A176: k > 1;

            reconsider f = (F . k) as FinSequence of ( TOP-REAL 2) by A174;

            per cases ;

              suppose

               A177: ( len f) = k;

              per cases ;

                suppose

                 A178: f is_sequence_on G & ( right_cell (f,(( len f) -' 1),G)) meets C;

                then

                 A179: ( front_left_cell (f,(( len f) -' 1),G)) meets C implies ex i, j st (f ^ <*(G * (i,j))*>) turns_left ((( len f) -' 1),G) & (F . (k + 1)) = (f ^ <*(G * (i,j))*>) by A171, A176, A177;

                

                 A180: ( front_left_cell (f,(( len f) -' 1),G)) misses C & ( front_right_cell (f,(( len f) -' 1),G)) meets C implies ex i, j st (f ^ <*(G * (i,j))*>) goes_straight ((( len f) -' 1),G) & (F . (k + 1)) = (f ^ <*(G * (i,j))*>) by A171, A176, A177, A178;

                ( front_left_cell (f,(( len f) -' 1),G)) misses C & ( front_right_cell (f,(( len f) -' 1),G)) misses C implies ex i, j st (f ^ <*(G * (i,j))*>) turns_right ((( len f) -' 1),G) & (F . (k + 1)) = (f ^ <*(G * (i,j))*>) by A171, A176, A177, A178;

                hence thesis by A180, A179;

              end;

                suppose

                 A181: not f is_sequence_on G or ( right_cell (f,(( len f) -' 1),G)) misses C;

                (f ^ <*(G * (1,1))*>) is FinSequence of ( TOP-REAL 2);

                hence thesis by A171, A176, A177, A181;

              end;

            end;

              suppose ( len f) <> k;

              hence thesis by A171, A172, A176;

            end;

          end;

        end;

        

         A182: P[ 0 ] by A170, A172;

        

         A183: for k holds P[k] from NAT_1:sch 2( A182, A173);

        ( rng F) c= (the carrier of ( TOP-REAL 2) * )

        proof

          let y be object;

          assume y in ( rng F);

          then ex x be object st x in ( dom F) & (F . x) = y by FUNCT_1:def 3;

          then y is FinSequence of ( TOP-REAL 2) by A169, A183;

          hence thesis by FINSEQ_1:def 11;

        end;

        then

        reconsider F as sequence of (the carrier of ( TOP-REAL 2) * ) by A169, FUNCT_2:def 1, RELSET_1: 4;

        defpred P[ Nat] means ( len (F . $1)) = $1;

        

         A184: for k st P[k] holds P[(k + 1)]

        proof

          let k such that

           A185: ( len (F . k)) = k;

          

           A186: P[k, (F . k), (F . (k + 1))] by A171;

          per cases by NAT_1: 25;

            suppose k = 0 ;

            hence thesis by A186, FINSEQ_1: 39;

          end;

            suppose k = 1;

            hence thesis by A186, FINSEQ_1: 44;

          end;

            suppose

             A187: k > 1;

            thus thesis

            proof

              per cases ;

                suppose

                 A188: (F . k) is_sequence_on G & ( right_cell ((F . k),(( len (F . k)) -' 1),G)) meets C;

                then

                 A189: ( front_left_cell ((F . k),(( len (F . k)) -' 1),G)) meets C implies ex i, j st ((F . k) ^ <*(G * (i,j))*>) turns_left ((( len (F . k)) -' 1),G) & (F . (k + 1)) = ((F . k) ^ <*(G * (i,j))*>) by A171, A185, A187;

                

                 A190: ( front_left_cell ((F . k),(( len (F . k)) -' 1),G)) misses C & ( front_right_cell ((F . k),(( len (F . k)) -' 1),G)) meets C implies ex i, j st ((F . k) ^ <*(G * (i,j))*>) goes_straight ((( len (F . k)) -' 1),G) & (F . (k + 1)) = ((F . k) ^ <*(G * (i,j))*>) by A171, A185, A187, A188;

                ( front_left_cell ((F . k),(( len (F . k)) -' 1),G)) misses C & ( front_right_cell ((F . k),(( len (F . k)) -' 1),G)) misses C implies ex i, j st ((F . k) ^ <*(G * (i,j))*>) turns_right ((( len (F . k)) -' 1),G) & (F . (k + 1)) = ((F . k) ^ <*(G * (i,j))*>) by A171, A185, A187, A188;

                hence thesis by A185, A190, A189, FINSEQ_2: 16;

              end;

                suppose not (F . k) is_sequence_on G or ( right_cell ((F . k),(( len (F . k)) -' 1),G)) misses C;

                then (F . (k + 1)) = ((F . k) ^ <*(G * (1,1))*>) by A171, A185, A187;

                hence thesis by A185, FINSEQ_2: 16;

              end;

            end;

          end;

        end;

        

         A191: P[ 0 ] by A170, CARD_1: 27;

        

         A192: for k holds P[k] from NAT_1:sch 2( A191, A184);

         A193:

        now

          let k such that

           A194: (F . k) is_sequence_on G and

           A195: for m st 1 <= m & (m + 1) <= ( len (F . k)) holds ( left_cell ((F . k),m,G)) misses C & ( right_cell ((F . k),m,G)) meets C and

           A196: k > 1;

          ( len (F . k)) = k by A192;

          then

           A197: 1 <= (( len (F . k)) -' 1) & ((( len (F . k)) -' 1) + 1) = ( len (F . k)) by A196, NAT_D: 49, XREAL_1: 235;

          then

           A198: ( right_cell ((F . k),(( len (F . k)) -' 1),G)) meets C by A195;

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

           A199: [i1, j1] in ( Indices G) and

           A200: ((F . k) /. (( len (F . k)) -' 1)) = (G * (i1,j1)) and

           A201: [i2, j2] in ( Indices G) and

           A202: ((F . k) /. ( len (F . k))) = (G * (i2,j2));

          

           A203: i2 <= ( len G) by A201, MATRIX_0: 32;

          

           A204: 1 <= (i2 + 1) by NAT_1: 12;

          

           A205: 1 <= j2 by A201, MATRIX_0: 32;

          

           A206: j2 <= ( width G) by A201, MATRIX_0: 32;

          

           A207: j1 <= ( width G) by A199, MATRIX_0: 32;

          hereby

            assume

             A208: i1 = i2 & (j1 + 1) = j2;

            now

              assume (i2 + 1) > ( len G);

              then

               A209: (( len G) + 1) <= (i2 + 1) by NAT_1: 13;

              (i2 + 1) <= (( len G) + 1) by A203, XREAL_1: 6;

              then (i2 + 1) = (( len G) + 1) by A209, XXREAL_0: 1;

              then ( cell (G,( len G),j1)) meets C by A194, A199, A200, A201, A202, A197, A198, A208, GOBRD13: 22;

              hence contradiction by A2, A207, JORDAN8: 16;

            end;

            hence [(i2 + 1), j2] in ( Indices G) by A205, A206, A204, MATRIX_0: 30;

          end;

          

           A210: i1 <= ( len G) by A199, MATRIX_0: 32;

          

           A211: 1 <= i2 by A201, MATRIX_0: 32;

          

           A212: (j2 -' 1) <= ( width G) by A206, NAT_D: 44;

          hereby

            assume

             A213: (i1 + 1) = i2 & j1 = j2;

            now

              assume (j2 -' 1) < 1;

              then j2 <= 1 by NAT_1: 14, NAT_D: 36;

              then j2 = 1 by A205, XXREAL_0: 1;

              then ( cell (G,i1,(1 -' 1))) meets C by A194, A199, A200, A201, A202, A197, A198, A213, GOBRD13: 24;

              then ( cell (G,i1, 0 )) meets C by XREAL_1: 232;

              hence contradiction by A210, JORDAN8: 17;

            end;

            hence [i2, (j2 -' 1)] in ( Indices G) by A211, A203, A212, MATRIX_0: 30;

          end;

          

           A214: 1 <= (j2 + 1) by NAT_1: 12;

          hereby

            assume

             A215: i1 = (i2 + 1) & j1 = j2;

            now

              assume (j2 + 1) > ( len G);

              then

               A216: (( len G) + 1) <= (j2 + 1) by NAT_1: 13;

              (j2 + 1) <= (( len G) + 1) by A2, A206, XREAL_1: 6;

              then (j2 + 1) = (( len G) + 1) by A216, XXREAL_0: 1;

              then ( cell (G,i2,( len G))) meets C by A194, A199, A200, A201, A202, A197, A198, A215, GOBRD13: 26;

              hence contradiction by A203, JORDAN8: 15;

            end;

            hence [i2, (j2 + 1)] in ( Indices G) by A2, A211, A203, A214, MATRIX_0: 30;

          end;

          

           A217: (i2 -' 1) <= ( len G) by A203, NAT_D: 44;

          hereby

            assume

             A218: i1 = i2 & j1 = (j2 + 1);

            now

              assume (i2 -' 1) < 1;

              then i2 <= 1 by NAT_1: 14, NAT_D: 36;

              then i2 = 1 by A211, XXREAL_0: 1;

              then ( cell (G,(1 -' 1),j2)) meets C by A194, A199, A200, A201, A202, A197, A198, A218, GOBRD13: 28;

              then ( cell (G, 0 ,j2)) meets C by XREAL_1: 232;

              hence contradiction by A2, A206, JORDAN8: 18;

            end;

            hence [(i2 -' 1), j2] in ( Indices G) by A205, A206, A217, MATRIX_0: 30;

          end;

          hereby

            assume

             A219: ( front_right_cell ((F . k),(( len (F . k)) -' 1),G)) meets C & i1 = i2 & (j1 + 1) = j2;

            now

              assume (j2 + 1) > ( len G);

              then

               A220: (( len G) + 1) <= (j2 + 1) by NAT_1: 13;

              (j2 + 1) <= (( len G) + 1) by A2, A206, XREAL_1: 6;

              then (j2 + 1) = (( len G) + 1) by A220, XXREAL_0: 1;

              then ( cell (G,i1,( len G))) meets C by A194, A199, A200, A201, A202, A197, A219, GOBRD13: 35;

              hence contradiction by A210, JORDAN8: 15;

            end;

            hence [i2, (j2 + 1)] in ( Indices G) by A2, A211, A203, A214, MATRIX_0: 30;

          end;

          hereby

            assume

             A221: ( front_right_cell ((F . k),(( len (F . k)) -' 1),G)) meets C & (i1 + 1) = i2 & j1 = j2;

            now

              assume (i2 + 1) > ( len G);

              then

               A222: (( len G) + 1) <= (i2 + 1) by NAT_1: 13;

              (i2 + 1) <= (( len G) + 1) by A203, XREAL_1: 6;

              then (i2 + 1) = (( len G) + 1) by A222, XXREAL_0: 1;

              then ( cell (G,( len G),(j1 -' 1))) meets C by A194, A199, A200, A201, A202, A197, A221, GOBRD13: 37;

              hence contradiction by A2, A207, JORDAN8: 16, NAT_D: 44;

            end;

            hence [(i2 + 1), j2] in ( Indices G) by A205, A206, A204, MATRIX_0: 30;

          end;

          hereby

            assume

             A223: ( front_right_cell ((F . k),(( len (F . k)) -' 1),G)) meets C & i1 = (i2 + 1) & j1 = j2;

            now

              assume (i2 -' 1) < 1;

              then i2 <= 1 by NAT_1: 14, NAT_D: 36;

              then i2 = 1 by A211, XXREAL_0: 1;

              then ( cell (G,(1 -' 1),j1)) meets C by A194, A199, A200, A201, A202, A197, A223, GOBRD13: 39;

              then ( cell (G, 0 ,j1)) meets C by XREAL_1: 232;

              hence contradiction by A2, A207, JORDAN8: 18;

            end;

            hence [(i2 -' 1), j2] in ( Indices G) by A205, A206, A217, MATRIX_0: 30;

          end;

          hereby

            assume

             A224: ( front_right_cell ((F . k),(( len (F . k)) -' 1),G)) meets C & i1 = i2 & j1 = (j2 + 1);

            now

              assume (j2 -' 1) < 1;

              then j2 <= 1 by NAT_1: 14, NAT_D: 36;

              then j2 = 1 by A205, XXREAL_0: 1;

              then ( cell (G,(i1 -' 1),(1 -' 1))) meets C by A194, A199, A200, A201, A202, A197, A224, GOBRD13: 41;

              then ( cell (G,(i1 -' 1), 0 )) meets C by XREAL_1: 232;

              hence contradiction by A210, JORDAN8: 17, NAT_D: 44;

            end;

            hence [i2, (j2 -' 1)] in ( Indices G) by A211, A203, A212, MATRIX_0: 30;

          end;

          hereby

            assume

             A225: ( front_left_cell ((F . k),(( len (F . k)) -' 1),G)) meets C & i1 = i2 & (j1 + 1) = j2;

            now

              assume (i2 -' 1) < 1;

              then i2 <= 1 by NAT_1: 14, NAT_D: 36;

              then i2 = 1 by A211, XXREAL_0: 1;

              then ( cell (G,(1 -' 1),j2)) meets C by A194, A199, A200, A201, A202, A197, A225, GOBRD13: 34;

              then ( cell (G, 0 ,j2)) meets C by XREAL_1: 232;

              hence contradiction by A2, A206, JORDAN8: 18;

            end;

            hence [(i2 -' 1), j2] in ( Indices G) by A205, A206, A217, MATRIX_0: 30;

          end;

          hereby

            assume

             A226: ( front_left_cell ((F . k),(( len (F . k)) -' 1),G)) meets C & (i1 + 1) = i2 & j1 = j2;

            now

              assume (j2 + 1) > ( len G);

              then

               A227: (( len G) + 1) <= (j2 + 1) by NAT_1: 13;

              (j2 + 1) <= (( len G) + 1) by A2, A206, XREAL_1: 6;

              then (j2 + 1) = (( len G) + 1) by A227, XXREAL_0: 1;

              then ( cell (G,i2,( len G))) meets C by A194, A199, A200, A201, A202, A197, A226, GOBRD13: 36;

              hence contradiction by A203, JORDAN8: 15;

            end;

            hence [i2, (j2 + 1)] in ( Indices G) by A2, A211, A203, A214, MATRIX_0: 30;

          end;

          hereby

            assume

             A228: ( front_left_cell ((F . k),(( len (F . k)) -' 1),G)) meets C & i1 = (i2 + 1) & j1 = j2;

            now

              assume (j2 -' 1) < 1;

              then j2 <= 1 by NAT_1: 14, NAT_D: 36;

              then j2 = 1 by A205, XXREAL_0: 1;

              then ( cell (G,(i2 -' 1),(1 -' 1))) meets C by A194, A199, A200, A201, A202, A197, A228, GOBRD13: 38;

              then ( cell (G,(i2 -' 1), 0 )) meets C by XREAL_1: 232;

              hence contradiction by A203, JORDAN8: 17, NAT_D: 44;

            end;

            hence [i2, (j2 -' 1)] in ( Indices G) by A211, A203, A212, MATRIX_0: 30;

          end;

          hereby

            assume

             A229: ( front_left_cell ((F . k),(( len (F . k)) -' 1),G)) meets C & i1 = i2 & j1 = (j2 + 1);

            now

              assume (i2 + 1) > ( len G);

              then

               A230: (( len G) + 1) <= (i2 + 1) by NAT_1: 13;

              (i2 + 1) <= (( len G) + 1) by A203, XREAL_1: 6;

              then (i2 + 1) = (( len G) + 1) by A230, XXREAL_0: 1;

              then ( cell (G,( len G),(j2 -' 1))) meets C by A194, A199, A200, A201, A202, A197, A229, GOBRD13: 40;

              hence contradiction by A2, A206, JORDAN8: 16, NAT_D: 44;

            end;

            hence [(i2 + 1), j2] in ( Indices G) by A205, A206, A204, MATRIX_0: 30;

          end;

        end;

        defpred P[ Nat] means (F . $1) is_sequence_on G & for m st 1 <= m & (m + 1) <= ( len (F . $1)) holds ( left_cell ((F . $1),m,G)) misses C & ( right_cell ((F . $1),m,G)) meets C;

        

         A231: ( len G) = ((2 |^ n) + 3) by JORDAN8:def 1;

        

         A232: for k st P[k] holds P[(k + 1)]

        proof

          

           A233: (2 |^ n) > 0 by NEWTON: 83;

          

           A234: 1 <= ( len G) by A231, NAT_1: 12;

          let k such that

           A235: (F . k) is_sequence_on G and

           A236: for m st 1 <= m & (m + 1) <= ( len (F . k)) holds ( left_cell ((F . k),m,G)) misses C & ( right_cell ((F . k),m,G)) meets C;

          

           A237: ( len (F . k)) = k by A192;

          

           A238: ( len (F . (k + 1))) = (k + 1) by A192;

          per cases by NAT_1: 25;

            suppose

             A239: k = 0 ;

            then

            consider i such that

             A240: 1 <= i and

             A241: (i + 1) <= ( len G) and ( N-min C) in ( cell (G,i,(( width G) -' 1))) and ( N-min C) <> (G * (i,(( width G) -' 1))) and

             A242: (F . (k + 1)) = <*(G * (i,( width G)))*> by A171;

            i < ( len G) by A241, NAT_1: 13;

            then

             A243: [i, ( len G)] in ( Indices G) by A2, A234, A240, MATRIX_0: 30;

             A244:

            now

              let l;

              assume l in ( dom (F . (k + 1)));

              then 1 <= l & l <= 1 by A238, A239, FINSEQ_3: 25;

              then l = 1 by XXREAL_0: 1;

              hence ex i, j st [i, j] in ( Indices G) & ((F . (k + 1)) /. l) = (G * (i,j)) by A2, A242, A243, FINSEQ_4: 16;

            end;

            now

              let l;

              assume that

               A245: l in ( dom (F . (k + 1))) and

               A246: (l + 1) in ( dom (F . (k + 1)));

              1 <= l & l <= 1 by A238, A239, A245, FINSEQ_3: 25;

              then l = 1 by XXREAL_0: 1;

              hence for i1, j1, i2, j2 st [i1, j1] in ( Indices G) & [i2, j2] in ( Indices G) & ((F . (k + 1)) /. l) = (G * (i1,j1)) & ((F . (k + 1)) /. (l + 1)) = (G * (i2,j2)) holds ( |.(i1 - i2).| + |.(j1 - j2).|) = 1 by A238, A239, A246, FINSEQ_3: 25;

            end;

            hence (F . (k + 1)) is_sequence_on G by A244, GOBOARD1:def 9;

            let m;

            assume that

             A247: 1 <= m and

             A248: (m + 1) <= ( len (F . (k + 1)));

            1 <= (m + 1) by NAT_1: 12;

            then (m + 1) = ( 0 + 1) by A238, A239, A248, XXREAL_0: 1;

            hence thesis by A247;

          end;

            suppose

             A249: k = 1;

            then

            consider i such that

             A250: 1 <= i and

             A251: (i + 1) <= ( len G) and

             A252: ( N-min C) in ( cell (G,i,(( width G) -' 1))) and ( N-min C) <> (G * (i,(( width G) -' 1))) and

             A253: (F . (k + 1)) = <*(G * (i,( width G))), (G * ((i + 1),( width G)))*> by A171;

            

             A254: i < ( len G) by A251, NAT_1: 13;

            then

             A255: [i, ( len G)] in ( Indices G) by A2, A234, A250, MATRIX_0: 30;

            1 <= (i + 1) by A250, NAT_1: 13;

            then

             A256: [(i + 1), ( len G)] in ( Indices G) by A2, A234, A251, MATRIX_0: 30;

            

             A257: ((F . (k + 1)) /. 1) = (G * (i,( width G))) & ((F . (k + 1)) /. 2) = (G * ((i + 1),( width G))) by A253, FINSEQ_4: 17;

             A258:

            now

              let l;

              assume that

               A259: l in ( dom (F . (k + 1))) and

               A260: (l + 1) in ( dom (F . (k + 1)));

              l <= 2 by A238, A249, A259, FINSEQ_3: 25;

              then

               A261: l = 0 or ... or l = 2;

              let i1, j1, i2, j2 such that

               A262: [i1, j1] in ( Indices G) & [i2, j2] in ( Indices G) & ((F . (k + 1)) /. l) = (G * (i1,j1)) & ((F . (k + 1)) /. (l + 1)) = (G * (i2,j2));

              j1 = ( len G) & j2 = ( len G) by A2, A238, A249, A257, A255, A256, A259, A260, A261, A262, FINSEQ_3: 25, GOBOARD1: 5;

              then

               A263: |.(j1 - j2).| = 0 by ABSVALUE:def 1;

              i1 = i & i2 = (i + 1) by A2, A238, A249, A257, A255, A256, A259, A260, A261, A262, FINSEQ_3: 25, GOBOARD1: 5;

              then |.(i2 - i1).| = 1 by ABSVALUE:def 1;

              hence ( |.(i1 - i2).| + |.(j1 - j2).|) = 1 by A263, UNIFORM1: 11;

            end;

            now

              let l;

              assume

               A264: l in ( dom (F . (k + 1)));

              then l <= 2 by A238, A249, FINSEQ_3: 25;

              then l = 0 or ... or l = 2;

              hence ex i, j st [i, j] in ( Indices G) & ((F . (k + 1)) /. l) = (G * (i,j)) by A2, A257, A255, A256, A264, FINSEQ_3: 25;

            end;

            hence

             A265: (F . (k + 1)) is_sequence_on G by A258, GOBOARD1:def 9;

            

             A266: i < (i + 1) & (i + 1) < ((i + 1) + 1) by NAT_1: 13;

            let m;

            assume that

             A267: 1 <= m and

             A268: (m + 1) <= ( len (F . (k + 1)));

            (1 + 1) <= (m + 1) by A267, XREAL_1: 6;

            then

             A269: (m + 1) = (1 + 1) by A238, A249, A268, XXREAL_0: 1;

            then

             A270: ( left_cell ((F . (k + 1)),m,G)) = ( cell (G,i,( len G))) by A2, A257, A255, A256, A265, A268, A266, GOBRD13:def 3;

            now

              N > S by JORDAN8: 9;

              then (N - S) > (S - S) by XREAL_1: 9;

              then ((N - S) / (2 |^ n)) > 0 by A233, XREAL_1: 139;

              then

               A271: (N + 0 ) < (N + ((N - S) / (2 |^ n))) by XREAL_1: 6;

               [1, ( len G)] in ( Indices G) by A2, A234, MATRIX_0: 30;

              then (G * (1,( len G))) = |[(W + (((E - W) / (2 |^ n)) * (1 - 2))), (S + (((N - S) / (2 |^ n)) * (( len G) - 2)))]| by JORDAN8:def 1;

              then

               A272: ((G * (1,( len G))) `2 ) = (S + (((N - S) / (2 |^ n)) * (( len G) - 2))) by EUCLID: 52;

              

               A273: ( cell (G,i,( len G))) = { |[r, s]| : ((G * (i,1)) `1 ) <= r & r <= ((G * ((i + 1),1)) `1 ) & ((G * (1,( len G))) `2 ) <= s } by A2, A250, A254, GOBRD11: 31;

              assume ( left_cell ((F . (k + 1)),m,G)) meets C;

              then

              consider p be object such that

               A274: p in ( cell (G,i,( len G))) and

               A275: p in C by A270, XBOOLE_0: 3;

              reconsider p as Point of ( TOP-REAL 2) by A274;

              reconsider p as Element of ( TOP-REAL 2);

              

               A276: (p `2 ) <= N by A275, PSCOMP_1: 24;

              consider r, s such that

               A277: p = |[r, s]| and ((G * (i,1)) `1 ) <= r and r <= ((G * ((i + 1),1)) `1 ) and

               A278: ((G * (1,( len G))) `2 ) <= s by A274, A273;

              (((N - S) / (2 |^ n)) * (( len G) - 2)) = ((((N - S) / (2 |^ n)) * (2 |^ n)) + (((N - S) / (2 |^ n)) * 1)) by A231

              .= ((N - S) + ((N - S) / (2 |^ n))) by A233, XCMPLX_1: 87;

              then N < s by A278, A272, A271, XXREAL_0: 2;

              hence contradiction by A277, A276, EUCLID: 52;

            end;

            hence ( left_cell ((F . (k + 1)),m,G)) misses C;

            ( N-min C) in C & ( N-min C) in ( right_cell ((F . (k + 1)),m,G)) by A2, A252, A257, A255, A256, A265, A268, A269, A266, GOBRD13:def 2, SPRECT_1: 11;

            hence thesis by XBOOLE_0: 3;

          end;

            suppose

             A279: k > 1;

            then

             A280: ( len (F . k)) in ( dom (F . k)) by A237, FINSEQ_3: 25;

            

             A281: ((( len (F . k)) -' 1) + 1) = ( len (F . k)) by A237, A279, XREAL_1: 235;

            

             A282: 1 <= (( len (F . k)) -' 1) by A237, A279, NAT_D: 49;

            then

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

             A283: [i1, j1] in ( Indices G) and

             A284: ((F . k) /. (( len (F . k)) -' 1)) = (G * (i1,j1)) and

             A285: [i2, j2] in ( Indices G) and

             A286: ((F . k) /. ( len (F . k))) = (G * (i2,j2)) and i1 = i2 & (j1 + 1) = j2 or (i1 + 1) = i2 & j1 = j2 or i1 = (i2 + 1) & j1 = j2 or i1 = i2 & j1 = (j2 + 1) by A235, A281, JORDAN8: 3;

            

             A287: i1 = i2 & (j1 + 1) = j2 implies [(i2 + 1), j2] in ( Indices G) by A193, A235, A236, A279, A283, A284, A285, A286;

            

             A288: ( front_left_cell ((F . k),(( len (F . k)) -' 1),G)) meets C & i1 = (i2 + 1) & j1 = j2 implies [i2, (j2 -' 1)] in ( Indices G) by A193, A235, A236, A279, A283, A284, A285, A286;

            

             A289: ( front_left_cell ((F . k),(( len (F . k)) -' 1),G)) meets C & (i1 + 1) = i2 & j1 = j2 implies [i2, (j2 + 1)] in ( Indices G) by A193, A235, A236, A279, A283, A284, A285, A286;

            

             A290: ( front_left_cell ((F . k),(( len (F . k)) -' 1),G)) meets C & i1 = i2 & (j1 + 1) = j2 implies [(i2 -' 1), j2] in ( Indices G) by A193, A235, A236, A279, A283, A284, A285, A286;

            

             A291: ( front_right_cell ((F . k),(( len (F . k)) -' 1),G)) meets C & i1 = i2 & j1 = (j2 + 1) implies [i2, (j2 -' 1)] in ( Indices G) by A193, A235, A236, A279, A283, A284, A285, A286;

            

             A292: i1 = (i2 + 1) & j1 = j2 implies [i2, (j2 + 1)] in ( Indices G) by A193, A235, A236, A279, A283, A284, A285, A286;

            

             A293: (i1 + 1) = i2 & j1 = j2 implies [i2, (j2 -' 1)] in ( Indices G) by A193, A235, A236, A279, A283, A284, A285, A286;

            

             A294: 1 <= j2 by A285, MATRIX_0: 32;

            

             A295: ( right_cell ((F . k),(( len (F . k)) -' 1),G)) meets C by A236, A282, A281;

            

             A296: ( front_right_cell ((F . k),(( len (F . k)) -' 1),G)) meets C & i1 = (i2 + 1) & j1 = j2 implies [(i2 -' 1), j2] in ( Indices G) by A193, A235, A236, A279, A283, A284, A285, A286;

            

             A297: ( front_right_cell ((F . k),(( len (F . k)) -' 1),G)) meets C & (i1 + 1) = i2 & j1 = j2 implies [(i2 + 1), j2] in ( Indices G) by A193, A235, A236, A279, A283, A284, A285, A286;

            

             A298: ( front_right_cell ((F . k),(( len (F . k)) -' 1),G)) meets C & i1 = i2 & (j1 + 1) = j2 implies [i2, (j2 + 1)] in ( Indices G) by A193, A235, A236, A279, A283, A284, A285, A286;

            

             A299: i1 = i2 & j1 = (j2 + 1) implies [(i2 -' 1), j2] in ( Indices G) by A193, A235, A236, A279, A283, A284, A285, A286;

            

             A300: ( front_left_cell ((F . k),(( len (F . k)) -' 1),G)) meets C & i1 = i2 & j1 = (j2 + 1) implies [(i2 + 1), j2] in ( Indices G) by A193, A235, A236, A279, A283, A284, A285, A286;

            (( len (F . k)) -' 1) <= ( len (F . k)) by NAT_D: 35;

            then

             A301: (( len (F . k)) -' 1) in ( dom (F . k)) by A282, FINSEQ_3: 25;

            

             A302: 1 <= i2 by A285, MATRIX_0: 32;

            thus

             A303: (F . (k + 1)) is_sequence_on G

            proof

              per cases ;

                suppose ( front_left_cell ((F . k),(( len (F . k)) -' 1),G)) misses C & ( front_right_cell ((F . k),(( len (F . k)) -' 1),G)) misses C;

                then

                consider i, j such that

                 A304: ((F . k) ^ <*(G * (i,j))*>) turns_right ((( len (F . k)) -' 1),G) and

                 A305: (F . (k + 1)) = ((F . k) ^ <*(G * (i,j))*>) by A171, A235, A237, A279, A295;

                thus thesis

                proof

                  set f = ((F . k) ^ <*(G * (i,j))*>);

                  

                   A306: (f /. (( len (F . k)) + 1)) = (G * (i,j)) by FINSEQ_4: 67;

                  

                   A307: (f /. (( len (F . k)) -' 1)) = (G * (i1,j1)) & (f /. ( len (F . k))) = (G * (i2,j2)) by A284, A286, A301, A280, FINSEQ_4: 68;

                  per cases by A281, A283, A285, A304, A307;

                    suppose that

                     A308: i1 = i2 & (j1 + 1) = j2 and

                     A309: (f /. (( len (F . k)) + 1)) = (G * ((i2 + 1),j2));

                    now

                      let i19,j19,i29,j29 be Nat;

                      assume

                       A310: [i19, j19] in ( Indices G) & [i29, j29] in ( Indices G) & ((F . k) /. ( len (F . k))) = (G * (i19,j19)) & (G * ((i2 + 1),j2)) = (G * (i29,j29));

                      then j2 = j19 & j2 = j29 by A285, A286, A287, A308, GOBOARD1: 5;

                      then

                       A311: |.(j29 - j19).| = 0 by ABSVALUE:def 1;

                      i2 = i19 & (i2 + 1) = i29 by A285, A286, A287, A308, A310, GOBOARD1: 5;

                      hence ( |.(i29 - i19).| + |.(j29 - j19).|) = 1 by A311, ABSVALUE:def 1;

                    end;

                    hence thesis by A235, A237, A279, A287, A305, A306, A308, A309, CARD_1: 27, JORDAN8: 6;

                  end;

                    suppose that

                     A312: (i1 + 1) = i2 & j1 = j2 and

                     A313: (f /. (( len (F . k)) + 1)) = (G * (i2,(j2 -' 1)));

                    now

                      let i19,j19,i29,j29 be Nat;

                      assume

                       A314: [i19, j19] in ( Indices G) & [i29, j29] in ( Indices G) & ((F . k) /. ( len (F . k))) = (G * (i19,j19)) & (G * (i2,(j2 -' 1))) = (G * (i29,j29));

                      then j2 = j19 & (j2 -' 1) = j29 by A285, A286, A293, A312, GOBOARD1: 5;

                      then (j19 - j29) = (j2 - (j2 - 1)) by A294, XREAL_1: 233;

                      then

                       A315: |.(j19 - j29).| = 1 by ABSVALUE:def 1;

                      i2 = i19 & i2 = i29 by A285, A286, A293, A312, A314, GOBOARD1: 5;

                      then |.(i29 - i19).| = 0 by ABSVALUE:def 1;

                      hence ( |.(i29 - i19).| + |.(j29 - j19).|) = 1 by A315, UNIFORM1: 11;

                    end;

                    hence thesis by A235, A237, A279, A293, A305, A306, A312, A313, CARD_1: 27, JORDAN8: 6;

                  end;

                    suppose that

                     A316: i1 = (i2 + 1) & j1 = j2 and

                     A317: (f /. (( len (F . k)) + 1)) = (G * (i2,(j2 + 1)));

                    now

                      let i19,j19,i29,j29 be Nat;

                      assume

                       A318: [i19, j19] in ( Indices G) & [i29, j29] in ( Indices G) & ((F . k) /. ( len (F . k))) = (G * (i19,j19)) & (G * (i2,(j2 + 1))) = (G * (i29,j29));

                      then i2 = i19 & i2 = i29 by A285, A286, A292, A316, GOBOARD1: 5;

                      then

                       A319: |.(i29 - i19).| = 0 by ABSVALUE:def 1;

                      j2 = j19 & (j2 + 1) = j29 by A285, A286, A292, A316, A318, GOBOARD1: 5;

                      hence ( |.(i29 - i19).| + |.(j29 - j19).|) = 1 by A319, ABSVALUE:def 1;

                    end;

                    hence thesis by A235, A237, A279, A292, A305, A306, A316, A317, CARD_1: 27, JORDAN8: 6;

                  end;

                    suppose that

                     A320: i1 = i2 & j1 = (j2 + 1) and

                     A321: (f /. (( len (F . k)) + 1)) = (G * ((i2 -' 1),j2));

                    now

                      let i19,j19,i29,j29 be Nat;

                      assume

                       A322: [i19, j19] in ( Indices G) & [i29, j29] in ( Indices G) & ((F . k) /. ( len (F . k))) = (G * (i19,j19)) & (G * ((i2 -' 1),j2)) = (G * (i29,j29));

                      then i2 = i19 & (i2 -' 1) = i29 by A285, A286, A299, A320, GOBOARD1: 5;

                      then (i19 - i29) = (i2 - (i2 - 1)) by A302, XREAL_1: 233;

                      then

                       A323: |.(i19 - i29).| = 1 by ABSVALUE:def 1;

                      j2 = j19 & j2 = j29 by A285, A286, A299, A320, A322, GOBOARD1: 5;

                      then |.(j29 - j19).| = 0 by ABSVALUE:def 1;

                      hence ( |.(i29 - i19).| + |.(j29 - j19).|) = 1 by A323, UNIFORM1: 11;

                    end;

                    hence thesis by A235, A237, A279, A299, A305, A306, A320, A321, CARD_1: 27, JORDAN8: 6;

                  end;

                end;

              end;

                suppose

                 A324: ( front_left_cell ((F . k),(( len (F . k)) -' 1),G)) misses C & ( front_right_cell ((F . k),(( len (F . k)) -' 1),G)) meets C;

                then

                consider i, j such that

                 A325: ((F . k) ^ <*(G * (i,j))*>) goes_straight ((( len (F . k)) -' 1),G) and

                 A326: (F . (k + 1)) = ((F . k) ^ <*(G * (i,j))*>) by A171, A235, A237, A279, A295;

                thus thesis

                proof

                  set f = ((F . k) ^ <*(G * (i,j))*>);

                  

                   A327: (f /. (( len (F . k)) + 1)) = (G * (i,j)) by FINSEQ_4: 67;

                  

                   A328: (f /. (( len (F . k)) -' 1)) = (G * (i1,j1)) & (f /. ( len (F . k))) = (G * (i2,j2)) by A284, A286, A301, A280, FINSEQ_4: 68;

                  per cases by A281, A283, A285, A325, A328;

                    suppose that

                     A329: i1 = i2 & (j1 + 1) = j2 and

                     A330: (f /. (( len (F . k)) + 1)) = (G * (i2,(j2 + 1)));

                    now

                      let i19,j19,i29,j29 be Nat;

                      assume

                       A331: [i19, j19] in ( Indices G) & [i29, j29] in ( Indices G) & ((F . k) /. ( len (F . k))) = (G * (i19,j19)) & (G * (i2,(j2 + 1))) = (G * (i29,j29));

                      then i2 = i19 & i2 = i29 by A285, A286, A298, A324, A329, GOBOARD1: 5;

                      then

                       A332: |.(i29 - i19).| = 0 by ABSVALUE:def 1;

                      j2 = j19 & (j2 + 1) = j29 by A285, A286, A298, A324, A329, A331, GOBOARD1: 5;

                      hence ( |.(i29 - i19).| + |.(j29 - j19).|) = 1 by A332, ABSVALUE:def 1;

                    end;

                    hence thesis by A235, A237, A279, A298, A324, A326, A327, A329, A330, CARD_1: 27, JORDAN8: 6;

                  end;

                    suppose that

                     A333: (i1 + 1) = i2 & j1 = j2 and

                     A334: (f /. (( len (F . k)) + 1)) = (G * ((i2 + 1),j2));

                    now

                      let i19,j19,i29,j29 be Nat;

                      assume

                       A335: [i19, j19] in ( Indices G) & [i29, j29] in ( Indices G) & ((F . k) /. ( len (F . k))) = (G * (i19,j19)) & (G * ((i2 + 1),j2)) = (G * (i29,j29));

                      then j2 = j19 & j2 = j29 by A285, A286, A297, A324, A333, GOBOARD1: 5;

                      then

                       A336: |.(j29 - j19).| = 0 by ABSVALUE:def 1;

                      i2 = i19 & (i2 + 1) = i29 by A285, A286, A297, A324, A333, A335, GOBOARD1: 5;

                      hence ( |.(i29 - i19).| + |.(j29 - j19).|) = 1 by A336, ABSVALUE:def 1;

                    end;

                    hence thesis by A235, A237, A279, A297, A324, A326, A327, A333, A334, CARD_1: 27, JORDAN8: 6;

                  end;

                    suppose that

                     A337: i1 = (i2 + 1) & j1 = j2 and

                     A338: (f /. (( len (F . k)) + 1)) = (G * ((i2 -' 1),j2));

                    now

                      let i19,j19,i29,j29 be Nat;

                      assume

                       A339: [i19, j19] in ( Indices G) & [i29, j29] in ( Indices G) & ((F . k) /. ( len (F . k))) = (G * (i19,j19)) & (G * ((i2 -' 1),j2)) = (G * (i29,j29));

                      then i2 = i19 & (i2 -' 1) = i29 by A285, A286, A296, A324, A337, GOBOARD1: 5;

                      then (i19 - i29) = (i2 - (i2 - 1)) by A302, XREAL_1: 233;

                      then

                       A340: |.(i19 - i29).| = 1 by ABSVALUE:def 1;

                      j2 = j19 & j2 = j29 by A285, A286, A296, A324, A337, A339, GOBOARD1: 5;

                      then |.(j29 - j19).| = 0 by ABSVALUE:def 1;

                      hence ( |.(i29 - i19).| + |.(j29 - j19).|) = 1 by A340, UNIFORM1: 11;

                    end;

                    hence thesis by A235, A237, A279, A296, A324, A326, A327, A337, A338, CARD_1: 27, JORDAN8: 6;

                  end;

                    suppose that

                     A341: i1 = i2 & j1 = (j2 + 1) and

                     A342: (f /. (( len (F . k)) + 1)) = (G * (i2,(j2 -' 1)));

                    now

                      let i19,j19,i29,j29 be Nat;

                      assume

                       A343: [i19, j19] in ( Indices G) & [i29, j29] in ( Indices G) & ((F . k) /. ( len (F . k))) = (G * (i19,j19)) & (G * (i2,(j2 -' 1))) = (G * (i29,j29));

                      then j2 = j19 & (j2 -' 1) = j29 by A285, A286, A291, A324, A341, GOBOARD1: 5;

                      then (j19 - j29) = (j2 - (j2 - 1)) by A294, XREAL_1: 233;

                      then

                       A344: |.(j19 - j29).| = 1 by ABSVALUE:def 1;

                      i2 = i19 & i2 = i29 by A285, A286, A291, A324, A341, A343, GOBOARD1: 5;

                      then |.(i29 - i19).| = 0 by ABSVALUE:def 1;

                      hence ( |.(i29 - i19).| + |.(j29 - j19).|) = 1 by A344, UNIFORM1: 11;

                    end;

                    hence thesis by A235, A237, A279, A291, A324, A326, A327, A341, A342, CARD_1: 27, JORDAN8: 6;

                  end;

                end;

              end;

                suppose

                 A345: ( front_left_cell ((F . k),(( len (F . k)) -' 1),G)) meets C;

                then

                consider i, j such that

                 A346: ((F . k) ^ <*(G * (i,j))*>) turns_left ((( len (F . k)) -' 1),G) and

                 A347: (F . (k + 1)) = ((F . k) ^ <*(G * (i,j))*>) by A171, A235, A237, A279, A295;

                thus thesis

                proof

                  set f = ((F . k) ^ <*(G * (i,j))*>);

                  

                   A348: (f /. (( len (F . k)) + 1)) = (G * (i,j)) by FINSEQ_4: 67;

                  

                   A349: (f /. (( len (F . k)) -' 1)) = (G * (i1,j1)) & (f /. ( len (F . k))) = (G * (i2,j2)) by A284, A286, A301, A280, FINSEQ_4: 68;

                  per cases by A281, A283, A285, A346, A349;

                    suppose that

                     A350: i1 = i2 & (j1 + 1) = j2 and

                     A351: (f /. (( len (F . k)) + 1)) = (G * ((i2 -' 1),j2));

                    now

                      let i19,j19,i29,j29 be Nat;

                      assume

                       A352: [i19, j19] in ( Indices G) & [i29, j29] in ( Indices G) & ((F . k) /. ( len (F . k))) = (G * (i19,j19)) & (G * ((i2 -' 1),j2)) = (G * (i29,j29));

                      then i2 = i19 & (i2 -' 1) = i29 by A285, A286, A290, A345, A350, GOBOARD1: 5;

                      then (i19 - i29) = (i2 - (i2 - 1)) by A302, XREAL_1: 233;

                      then

                       A353: |.(i19 - i29).| = 1 by ABSVALUE:def 1;

                      j2 = j19 & j2 = j29 by A285, A286, A290, A345, A350, A352, GOBOARD1: 5;

                      then |.(j29 - j19).| = 0 by ABSVALUE:def 1;

                      hence ( |.(i29 - i19).| + |.(j29 - j19).|) = 1 by A353, UNIFORM1: 11;

                    end;

                    hence thesis by A235, A237, A279, A290, A345, A347, A348, A350, A351, CARD_1: 27, JORDAN8: 6;

                  end;

                    suppose that

                     A354: (i1 + 1) = i2 & j1 = j2 and

                     A355: (f /. (( len (F . k)) + 1)) = (G * (i2,(j2 + 1)));

                    now

                      let i19,j19,i29,j29 be Nat;

                      assume

                       A356: [i19, j19] in ( Indices G) & [i29, j29] in ( Indices G) & ((F . k) /. ( len (F . k))) = (G * (i19,j19)) & (G * (i2,(j2 + 1))) = (G * (i29,j29));

                      then i2 = i19 & i2 = i29 by A285, A286, A289, A345, A354, GOBOARD1: 5;

                      then

                       A357: |.(i29 - i19).| = 0 by ABSVALUE:def 1;

                      j2 = j19 & (j2 + 1) = j29 by A285, A286, A289, A345, A354, A356, GOBOARD1: 5;

                      hence ( |.(i29 - i19).| + |.(j29 - j19).|) = 1 by A357, ABSVALUE:def 1;

                    end;

                    hence thesis by A235, A237, A279, A289, A345, A347, A348, A354, A355, CARD_1: 27, JORDAN8: 6;

                  end;

                    suppose that

                     A358: i1 = (i2 + 1) & j1 = j2 and

                     A359: (f /. (( len (F . k)) + 1)) = (G * (i2,(j2 -' 1)));

                    now

                      let i19,j19,i29,j29 be Nat;

                      assume

                       A360: [i19, j19] in ( Indices G) & [i29, j29] in ( Indices G) & ((F . k) /. ( len (F . k))) = (G * (i19,j19)) & (G * (i2,(j2 -' 1))) = (G * (i29,j29));

                      then j2 = j19 & (j2 -' 1) = j29 by A285, A286, A288, A345, A358, GOBOARD1: 5;

                      then (j19 - j29) = (j2 - (j2 - 1)) by A294, XREAL_1: 233;

                      then

                       A361: |.(j19 - j29).| = 1 by ABSVALUE:def 1;

                      i2 = i19 & i2 = i29 by A285, A286, A288, A345, A358, A360, GOBOARD1: 5;

                      then |.(i29 - i19).| = 0 by ABSVALUE:def 1;

                      hence ( |.(i29 - i19).| + |.(j29 - j19).|) = 1 by A361, UNIFORM1: 11;

                    end;

                    hence thesis by A235, A237, A279, A288, A345, A347, A348, A358, A359, CARD_1: 27, JORDAN8: 6;

                  end;

                    suppose that

                     A362: i1 = i2 & j1 = (j2 + 1) and

                     A363: (f /. (( len (F . k)) + 1)) = (G * ((i2 + 1),j2));

                    now

                      let i19,j19,i29,j29 be Nat;

                      assume

                       A364: [i19, j19] in ( Indices G) & [i29, j29] in ( Indices G) & ((F . k) /. ( len (F . k))) = (G * (i19,j19)) & (G * ((i2 + 1),j2)) = (G * (i29,j29));

                      then j2 = j19 & j2 = j29 by A285, A286, A300, A345, A362, GOBOARD1: 5;

                      then

                       A365: |.(j29 - j19).| = 0 by ABSVALUE:def 1;

                      i2 = i19 & (i2 + 1) = i29 by A285, A286, A300, A345, A362, A364, GOBOARD1: 5;

                      hence ( |.(i29 - i19).| + |.(j29 - j19).|) = 1 by A365, ABSVALUE:def 1;

                    end;

                    hence thesis by A235, A237, A279, A300, A345, A347, A348, A362, A363, CARD_1: 27, JORDAN8: 6;

                  end;

                end;

              end;

            end;

            let m such that

             A366: 1 <= m and

             A367: (m + 1) <= ( len (F . (k + 1)));

            

             A368: ( left_cell ((F . k),(( len (F . k)) -' 1),G)) misses C by A236, A282, A281;

            now

              per cases ;

                suppose

                 A369: (m + 1) = ( len (F . (k + 1)));

                

                 A370: ((j2 -' 1) + 1) = j2 by A294, XREAL_1: 235;

                

                 A371: ((i2 -' 1) + 1) = i2 by A302, XREAL_1: 235;

                thus thesis

                proof

                  per cases ;

                    suppose

                     A372: ( front_left_cell ((F . k),(( len (F . k)) -' 1),G)) misses C & ( front_right_cell ((F . k),(( len (F . k)) -' 1),G)) misses C;

                    then

                     A373: ex i, j st ((F . k) ^ <*(G * (i,j))*>) turns_right ((( len (F . k)) -' 1),G) & (F . (k + 1)) = ((F . k) ^ <*(G * (i,j))*>) by A171, A235, A237, A279, A295;

                    then

                     A374: ((F . (k + 1)) /. ( len (F . k))) = (G * (i2,j2)) by A286, A280, FINSEQ_4: 68;

                    

                     A375: ((F . (k + 1)) /. (( len (F . k)) -' 1)) = (G * (i1,j1)) by A284, A301, A373, FINSEQ_4: 68;

                    now

                      per cases by A281, A283, A285, A373, A375, A374;

                        suppose that

                         A376: i1 = i2 & (j1 + 1) = j2 and

                         A377: ((F . (k + 1)) /. (( len (F . k)) + 1)) = (G * ((i2 + 1),j2));

                        ( front_right_cell ((F . k),(( len (F . k)) -' 1),G)) = ( cell (G,i1,j2)) by A235, A282, A281, A283, A284, A285, A286, A376, GOBRD13: 35;

                        hence ( left_cell ((F . (k + 1)),m,G)) misses C by A237, A238, A285, A287, A303, A366, A369, A372, A374, A376, A377, GOBRD13: 23;

                        (j2 -' 1) = j1 & ( cell (G,i1,j1)) meets C by A235, A282, A281, A283, A284, A285, A286, A295, A376, GOBRD13: 22, NAT_D: 34;

                        hence ( right_cell ((F . (k + 1)),m,G)) meets C by A237, A238, A285, A287, A303, A366, A369, A374, A376, A377, GOBRD13: 24;

                      end;

                        suppose that

                         A378: (i1 + 1) = i2 & j1 = j2 and

                         A379: ((F . (k + 1)) /. (( len (F . k)) + 1)) = (G * (i2,(j2 -' 1)));

                        ( front_right_cell ((F . k),(( len (F . k)) -' 1),G)) = ( cell (G,i2,(j2 -' 1))) by A235, A282, A281, A283, A284, A285, A286, A378, GOBRD13: 37;

                        hence ( left_cell ((F . (k + 1)),m,G)) misses C by A237, A238, A285, A293, A303, A366, A369, A370, A372, A374, A378, A379, GOBRD13: 27;

                        (i2 -' 1) = i1 & ( cell (G,i1,(j1 -' 1))) meets C by A235, A282, A281, A283, A284, A285, A286, A295, A378, GOBRD13: 24, NAT_D: 34;

                        hence ( right_cell ((F . (k + 1)),m,G)) meets C by A237, A238, A285, A293, A303, A366, A369, A370, A374, A378, A379, GOBRD13: 28;

                      end;

                        suppose that

                         A380: i1 = (i2 + 1) & j1 = j2 and

                         A381: ((F . (k + 1)) /. (( len (F . k)) + 1)) = (G * (i2,(j2 + 1)));

                        ( front_right_cell ((F . k),(( len (F . k)) -' 1),G)) = ( cell (G,(i2 -' 1),j2)) by A235, A282, A281, A283, A284, A285, A286, A380, GOBRD13: 39;

                        hence ( left_cell ((F . (k + 1)),m,G)) misses C by A237, A238, A285, A292, A303, A366, A369, A372, A374, A380, A381, GOBRD13: 21;

                        ( cell (G,i2,j2)) meets C by A235, A282, A281, A283, A284, A285, A286, A295, A380, GOBRD13: 26;

                        hence ( right_cell ((F . (k + 1)),m,G)) meets C by A237, A238, A285, A292, A303, A366, A369, A374, A380, A381, GOBRD13: 22;

                      end;

                        suppose that

                         A382: i1 = i2 & j1 = (j2 + 1) and

                         A383: ((F . (k + 1)) /. (( len (F . k)) + 1)) = (G * ((i2 -' 1),j2));

                        ( front_right_cell ((F . k),(( len (F . k)) -' 1),G)) = ( cell (G,(i2 -' 1),(j2 -' 1))) by A235, A282, A281, A283, A284, A285, A286, A382, GOBRD13: 41;

                        hence ( left_cell ((F . (k + 1)),m,G)) misses C by A237, A238, A285, A299, A303, A366, A369, A371, A372, A374, A382, A383, GOBRD13: 25;

                        ( cell (G,(i2 -' 1),j2)) meets C by A235, A282, A281, A283, A284, A285, A286, A295, A382, GOBRD13: 28;

                        hence ( right_cell ((F . (k + 1)),m,G)) meets C by A237, A238, A285, A299, A303, A366, A369, A371, A374, A382, A383, GOBRD13: 26;

                      end;

                    end;

                    hence thesis;

                  end;

                    suppose

                     A384: ( front_left_cell ((F . k),(( len (F . k)) -' 1),G)) misses C & ( front_right_cell ((F . k),(( len (F . k)) -' 1),G)) meets C;

                    then

                     A385: ex i, j st ((F . k) ^ <*(G * (i,j))*>) goes_straight ((( len (F . k)) -' 1),G) & (F . (k + 1)) = ((F . k) ^ <*(G * (i,j))*>) by A171, A235, A237, A279, A295;

                    then

                     A386: ((F . (k + 1)) /. ( len (F . k))) = (G * (i2,j2)) by A286, A280, FINSEQ_4: 68;

                    

                     A387: ((F . (k + 1)) /. (( len (F . k)) -' 1)) = (G * (i1,j1)) by A284, A301, A385, FINSEQ_4: 68;

                    now

                      per cases by A281, A283, A285, A385, A387, A386;

                        suppose that

                         A388: i1 = i2 & (j1 + 1) = j2 and

                         A389: ((F . (k + 1)) /. (( len (F . k)) + 1)) = (G * (i2,(j2 + 1)));

                        ( front_left_cell ((F . k),(( len (F . k)) -' 1),G)) = ( cell (G,(i1 -' 1),j2)) by A235, A282, A281, A283, A284, A285, A286, A388, GOBRD13: 34;

                        hence ( left_cell ((F . (k + 1)),m,G)) misses C by A237, A238, A285, A298, A303, A366, A369, A384, A386, A388, A389, GOBRD13: 21;

                        ( front_right_cell ((F . k),(( len (F . k)) -' 1),G)) = ( cell (G,i1,j2)) by A235, A282, A281, A283, A284, A285, A286, A388, GOBRD13: 35;

                        hence ( right_cell ((F . (k + 1)),m,G)) meets C by A237, A238, A285, A298, A303, A366, A369, A384, A386, A388, A389, GOBRD13: 22;

                      end;

                        suppose that

                         A390: (i1 + 1) = i2 & j1 = j2 and

                         A391: ((F . (k + 1)) /. (( len (F . k)) + 1)) = (G * ((i2 + 1),j2));

                        ( front_left_cell ((F . k),(( len (F . k)) -' 1),G)) = ( cell (G,i2,j2)) by A235, A282, A281, A283, A284, A285, A286, A390, GOBRD13: 36;

                        hence ( left_cell ((F . (k + 1)),m,G)) misses C by A237, A238, A285, A297, A303, A366, A369, A384, A386, A390, A391, GOBRD13: 23;

                        ( front_right_cell ((F . k),(( len (F . k)) -' 1),G)) = ( cell (G,i2,(j2 -' 1))) by A235, A282, A281, A283, A284, A285, A286, A390, GOBRD13: 37;

                        hence ( right_cell ((F . (k + 1)),m,G)) meets C by A237, A238, A285, A297, A303, A366, A369, A384, A386, A390, A391, GOBRD13: 24;

                      end;

                        suppose that

                         A392: i1 = (i2 + 1) & j1 = j2 and

                         A393: ((F . (k + 1)) /. (( len (F . k)) + 1)) = (G * ((i2 -' 1),j2));

                        ( front_left_cell ((F . k),(( len (F . k)) -' 1),G)) = ( cell (G,(i2 -' 1),(j2 -' 1))) by A235, A282, A281, A283, A284, A285, A286, A392, GOBRD13: 38;

                        hence ( left_cell ((F . (k + 1)),m,G)) misses C by A237, A238, A285, A296, A303, A366, A369, A371, A384, A386, A392, A393, GOBRD13: 25;

                        ( front_right_cell ((F . k),(( len (F . k)) -' 1),G)) = ( cell (G,(i2 -' 1),j2)) by A235, A282, A281, A283, A284, A285, A286, A392, GOBRD13: 39;

                        hence ( right_cell ((F . (k + 1)),m,G)) meets C by A237, A238, A285, A296, A303, A366, A369, A371, A384, A386, A392, A393, GOBRD13: 26;

                      end;

                        suppose that

                         A394: i1 = i2 & j1 = (j2 + 1) and

                         A395: ((F . (k + 1)) /. (( len (F . k)) + 1)) = (G * (i2,(j2 -' 1)));

                        ( front_left_cell ((F . k),(( len (F . k)) -' 1),G)) = ( cell (G,i2,(j2 -' 1))) by A235, A282, A281, A283, A284, A285, A286, A394, GOBRD13: 40;

                        hence ( left_cell ((F . (k + 1)),m,G)) misses C by A237, A238, A285, A291, A303, A366, A369, A370, A384, A386, A394, A395, GOBRD13: 27;

                        ( front_right_cell ((F . k),(( len (F . k)) -' 1),G)) = ( cell (G,(i2 -' 1),(j2 -' 1))) by A235, A282, A281, A283, A284, A285, A286, A394, GOBRD13: 41;

                        hence ( right_cell ((F . (k + 1)),m,G)) meets C by A237, A238, A285, A291, A303, A366, A369, A370, A384, A386, A394, A395, GOBRD13: 28;

                      end;

                    end;

                    hence thesis;

                  end;

                    suppose

                     A396: ( front_left_cell ((F . k),(( len (F . k)) -' 1),G)) meets C;

                    then

                     A397: ex i, j st ((F . k) ^ <*(G * (i,j))*>) turns_left ((( len (F . k)) -' 1),G) & (F . (k + 1)) = ((F . k) ^ <*(G * (i,j))*>) by A171, A235, A237, A279, A295;

                    then

                     A398: ((F . (k + 1)) /. ( len (F . k))) = (G * (i2,j2)) by A286, A280, FINSEQ_4: 68;

                    

                     A399: ((F . (k + 1)) /. (( len (F . k)) -' 1)) = (G * (i1,j1)) by A284, A301, A397, FINSEQ_4: 68;

                    now

                      per cases by A281, A283, A285, A397, A399, A398;

                        suppose that

                         A400: i1 = i2 & (j1 + 1) = j2 and

                         A401: ((F . (k + 1)) /. (( len (F . k)) + 1)) = (G * ((i2 -' 1),j2));

                        (j2 -' 1) = j1 & ( cell (G,(i1 -' 1),j1)) misses C by A235, A282, A281, A283, A284, A285, A286, A368, A400, GOBRD13: 21, NAT_D: 34;

                        hence ( left_cell ((F . (k + 1)),m,G)) misses C by A237, A238, A285, A290, A303, A366, A369, A371, A396, A398, A400, A401, GOBRD13: 25;

                        ( front_left_cell ((F . k),(( len (F . k)) -' 1),G)) = ( cell (G,(i1 -' 1),j2)) by A235, A282, A281, A283, A284, A285, A286, A400, GOBRD13: 34;

                        hence ( right_cell ((F . (k + 1)),m,G)) meets C by A237, A238, A285, A290, A303, A366, A369, A371, A396, A398, A400, A401, GOBRD13: 26;

                      end;

                        suppose that

                         A402: (i1 + 1) = i2 & j1 = j2 and

                         A403: ((F . (k + 1)) /. (( len (F . k)) + 1)) = (G * (i2,(j2 + 1)));

                        (i2 -' 1) = i1 & ( cell (G,i1,j1)) misses C by A235, A282, A281, A283, A284, A285, A286, A368, A402, GOBRD13: 23, NAT_D: 34;

                        hence ( left_cell ((F . (k + 1)),m,G)) misses C by A237, A238, A285, A289, A303, A366, A369, A396, A398, A402, A403, GOBRD13: 21;

                        ( front_left_cell ((F . k),(( len (F . k)) -' 1),G)) = ( cell (G,i2,j2)) by A235, A282, A281, A283, A284, A285, A286, A402, GOBRD13: 36;

                        hence ( right_cell ((F . (k + 1)),m,G)) meets C by A237, A238, A285, A289, A303, A366, A369, A396, A398, A402, A403, GOBRD13: 22;

                      end;

                        suppose that

                         A404: i1 = (i2 + 1) & j1 = j2 and

                         A405: ((F . (k + 1)) /. (( len (F . k)) + 1)) = (G * (i2,(j2 -' 1)));

                        ( cell (G,i2,(j2 -' 1))) misses C by A235, A282, A281, A283, A284, A285, A286, A368, A404, GOBRD13: 25;

                        hence ( left_cell ((F . (k + 1)),m,G)) misses C by A237, A238, A285, A288, A303, A366, A369, A370, A396, A398, A404, A405, GOBRD13: 27;

                        ( front_left_cell ((F . k),(( len (F . k)) -' 1),G)) = ( cell (G,(i2 -' 1),(j2 -' 1))) by A235, A282, A281, A283, A284, A285, A286, A404, GOBRD13: 38;

                        hence ( right_cell ((F . (k + 1)),m,G)) meets C by A237, A238, A285, A288, A303, A366, A369, A370, A396, A398, A404, A405, GOBRD13: 28;

                      end;

                        suppose that

                         A406: i1 = i2 & j1 = (j2 + 1) and

                         A407: ((F . (k + 1)) /. (( len (F . k)) + 1)) = (G * ((i2 + 1),j2));

                        ( cell (G,i2,j2)) misses C by A235, A282, A281, A283, A284, A285, A286, A368, A406, GOBRD13: 27;

                        hence ( left_cell ((F . (k + 1)),m,G)) misses C by A237, A238, A285, A300, A303, A366, A369, A396, A398, A406, A407, GOBRD13: 23;

                        ( front_left_cell ((F . k),(( len (F . k)) -' 1),G)) = ( cell (G,i2,(j2 -' 1))) by A235, A282, A281, A283, A284, A285, A286, A406, GOBRD13: 40;

                        hence ( right_cell ((F . (k + 1)),m,G)) meets C by A237, A238, A285, A300, A303, A366, A369, A396, A398, A406, A407, GOBRD13: 24;

                      end;

                    end;

                    hence thesis;

                  end;

                end;

              end;

                suppose (m + 1) <> ( len (F . (k + 1)));

                then (m + 1) < ( len (F . (k + 1))) by A367, XXREAL_0: 1;

                then

                 A408: (m + 1) <= ( len (F . k)) by A237, A238, NAT_1: 13;

                then

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

                 A409: [i1, j1] in ( Indices G) and

                 A410: ((F . k) /. m) = (G * (i1,j1)) and

                 A411: [i2, j2] in ( Indices G) and

                 A412: ((F . k) /. (m + 1)) = (G * (i2,j2)) and

                 A413: i1 = i2 & (j1 + 1) = j2 or (i1 + 1) = i2 & j1 = j2 or i1 = (i2 + 1) & j1 = j2 or i1 = i2 & j1 = (j2 + 1) by A235, A366, JORDAN8: 3;

                

                 A414: ( left_cell ((F . k),m,G)) misses C & ( right_cell ((F . k),m,G)) meets C by A236, A366, A408;

                 A415:

                now

                  per cases ;

                    suppose ( front_left_cell ((F . k),(( len (F . k)) -' 1),G)) misses C & ( front_right_cell ((F . k),(( len (F . k)) -' 1),G)) misses C;

                    then

                    consider i, j such that ((F . k) ^ <*(G * (i,j))*>) turns_right ((( len (F . k)) -' 1),G) and

                     A416: (F . (k + 1)) = ((F . k) ^ <*(G * (i,j))*>) by A171, A235, A237, A279, A295;

                    take i, j;

                    thus (F . (k + 1)) = ((F . k) ^ <*(G * (i,j))*>) by A416;

                  end;

                    suppose ( front_left_cell ((F . k),(( len (F . k)) -' 1),G)) misses C & ( front_right_cell ((F . k),(( len (F . k)) -' 1),G)) meets C;

                    then

                    consider i, j such that ((F . k) ^ <*(G * (i,j))*>) goes_straight ((( len (F . k)) -' 1),G) and

                     A417: (F . (k + 1)) = ((F . k) ^ <*(G * (i,j))*>) by A171, A235, A237, A279, A295;

                    take i, j;

                    thus (F . (k + 1)) = ((F . k) ^ <*(G * (i,j))*>) by A417;

                  end;

                    suppose ( front_left_cell ((F . k),(( len (F . k)) -' 1),G)) meets C;

                    then

                    consider i, j such that ((F . k) ^ <*(G * (i,j))*>) turns_left ((( len (F . k)) -' 1),G) and

                     A418: (F . (k + 1)) = ((F . k) ^ <*(G * (i,j))*>) by A171, A235, A237, A279, A295;

                    take i, j;

                    thus (F . (k + 1)) = ((F . k) ^ <*(G * (i,j))*>) by A418;

                  end;

                end;

                1 <= (m + 1) by NAT_1: 12;

                then (m + 1) in ( dom (F . k)) by A408, FINSEQ_3: 25;

                then

                 A419: ((F . (k + 1)) /. (m + 1)) = (G * (i2,j2)) by A412, A415, FINSEQ_4: 68;

                m <= ( len (F . k)) by A408, NAT_1: 13;

                then m in ( dom (F . k)) by A366, FINSEQ_3: 25;

                then

                 A420: ((F . (k + 1)) /. m) = (G * (i1,j1)) by A410, A415, FINSEQ_4: 68;

                now

                  per cases by A413;

                    suppose

                     A421: i1 = i2 & (j1 + 1) = j2;

                    then ( left_cell ((F . k),m,G)) = ( cell (G,(i1 -' 1),j1)) & ( right_cell ((F . k),m,G)) = ( cell (G,i1,j1)) by A235, A366, A408, A409, A410, A411, A412, GOBRD13: 21, GOBRD13: 22;

                    hence thesis by A303, A366, A367, A409, A411, A414, A420, A419, A421, GOBRD13: 21, GOBRD13: 22;

                  end;

                    suppose

                     A422: (i1 + 1) = i2 & j1 = j2;

                    then ( left_cell ((F . k),m,G)) = ( cell (G,i1,j1)) & ( right_cell ((F . k),m,G)) = ( cell (G,i1,(j1 -' 1))) by A235, A366, A408, A409, A410, A411, A412, GOBRD13: 23, GOBRD13: 24;

                    hence thesis by A303, A366, A367, A409, A411, A414, A420, A419, A422, GOBRD13: 23, GOBRD13: 24;

                  end;

                    suppose

                     A423: i1 = (i2 + 1) & j1 = j2;

                    then ( left_cell ((F . k),m,G)) = ( cell (G,i2,(j2 -' 1))) & ( right_cell ((F . k),m,G)) = ( cell (G,i2,j2)) by A235, A366, A408, A409, A410, A411, A412, GOBRD13: 25, GOBRD13: 26;

                    hence thesis by A303, A366, A367, A409, A411, A414, A420, A419, A423, GOBRD13: 25, GOBRD13: 26;

                  end;

                    suppose

                     A424: i1 = i2 & j1 = (j2 + 1);

                    then ( left_cell ((F . k),m,G)) = ( cell (G,i2,j2)) & ( right_cell ((F . k),m,G)) = ( cell (G,(i1 -' 1),j2)) by A235, A366, A408, A409, A410, A411, A412, GOBRD13: 27, GOBRD13: 28;

                    hence thesis by A303, A366, A367, A409, A411, A414, A420, A419, A424, GOBRD13: 27, GOBRD13: 28;

                  end;

                end;

                hence thesis;

              end;

            end;

            hence thesis;

          end;

        end;

        defpred Q[ Nat] means ex w be Nat st w = $1 & $1 >= 1 & ex m st m in ( dom (F . w)) & m <> ( len (F . w)) & ((F . w) /. m) = ((F . w) /. ( len (F . w)));

        

         A425: P[ 0 ]

        proof

          (for n st n in ( dom (F . 0 )) holds ex i, j st [i, j] in ( Indices G) & ((F . 0 ) /. n) = (G * (i,j))) & for n st n in ( dom (F . 0 )) & (n + 1) in ( dom (F . 0 )) holds for m, k, i, j st [m, k] in ( Indices G) & [i, j] in ( Indices G) & ((F . 0 ) /. n) = (G * (m,k)) & ((F . 0 ) /. (n + 1)) = (G * (i,j)) holds ( |.(m - i).| + |.(k - j).|) = 1 by A170;

          hence (F . 0 ) is_sequence_on G by GOBOARD1:def 9;

          let m;

          assume that 1 <= m and

           A426: (m + 1) <= ( len (F . 0 ));

          thus thesis by A170, A426, CARD_1: 27;

        end;

        

         A427: for k holds P[k] from NAT_1:sch 2( A425, A232);

        

         A428: for k, i1, i2, j1, j2 st k > 1 & [i1, j1] in ( Indices G) & ((F . k) /. (( len (F . k)) -' 1)) = (G * (i1,j1)) & [i2, j2] in ( Indices G) & ((F . k) /. ( len (F . k))) = (G * (i2,j2)) holds (( front_left_cell ((F . k),(( len (F . k)) -' 1),G)) misses C & ( front_right_cell ((F . k),(( len (F . k)) -' 1),G)) misses C implies (F . (k + 1)) turns_right ((( len (F . k)) -' 1),G) & (i1 = i2 & (j1 + 1) = j2 implies [(i2 + 1), j2] in ( Indices G) & (F . (k + 1)) = ((F . k) ^ <*(G * ((i2 + 1),j2))*>)) & ((i1 + 1) = i2 & j1 = j2 implies [i2, (j2 -' 1)] in ( Indices G) & (F . (k + 1)) = ((F . k) ^ <*(G * (i2,(j2 -' 1)))*>)) & (i1 = (i2 + 1) & j1 = j2 implies [i2, (j2 + 1)] in ( Indices G) & (F . (k + 1)) = ((F . k) ^ <*(G * (i2,(j2 + 1)))*>)) & (i1 = i2 & j1 = (j2 + 1) implies [(i2 -' 1), j2] in ( Indices G) & (F . (k + 1)) = ((F . k) ^ <*(G * ((i2 -' 1),j2))*>))) & (( front_left_cell ((F . k),(( len (F . k)) -' 1),G)) misses C & ( front_right_cell ((F . k),(( len (F . k)) -' 1),G)) meets C implies (F . (k + 1)) goes_straight ((( len (F . k)) -' 1),G) & (i1 = i2 & (j1 + 1) = j2 implies [i2, (j2 + 1)] in ( Indices G) & (F . (k + 1)) = ((F . k) ^ <*(G * (i2,(j2 + 1)))*>)) & ((i1 + 1) = i2 & j1 = j2 implies [(i2 + 1), j2] in ( Indices G) & (F . (k + 1)) = ((F . k) ^ <*(G * ((i2 + 1),j2))*>)) & (i1 = (i2 + 1) & j1 = j2 implies [(i2 -' 1), j2] in ( Indices G) & (F . (k + 1)) = ((F . k) ^ <*(G * ((i2 -' 1),j2))*>)) & (i1 = i2 & j1 = (j2 + 1) implies [i2, (j2 -' 1)] in ( Indices G) & (F . (k + 1)) = ((F . k) ^ <*(G * (i2,(j2 -' 1)))*>))) & (( front_left_cell ((F . k),(( len (F . k)) -' 1),G)) meets C implies (F . (k + 1)) turns_left ((( len (F . k)) -' 1),G) & (i1 = i2 & (j1 + 1) = j2 implies [(i2 -' 1), j2] in ( Indices G) & (F . (k + 1)) = ((F . k) ^ <*(G * ((i2 -' 1),j2))*>)) & ((i1 + 1) = i2 & j1 = j2 implies [i2, (j2 + 1)] in ( Indices G) & (F . (k + 1)) = ((F . k) ^ <*(G * (i2,(j2 + 1)))*>)) & (i1 = (i2 + 1) & j1 = j2 implies [i2, (j2 -' 1)] in ( Indices G) & (F . (k + 1)) = ((F . k) ^ <*(G * (i2,(j2 -' 1)))*>)) & (i1 = i2 & j1 = (j2 + 1) implies [(i2 + 1), j2] in ( Indices G) & (F . (k + 1)) = ((F . k) ^ <*(G * ((i2 + 1),j2))*>)))

        proof

          let k, i1, i2, j1, j2 such that

           A429: k > 1 and

           A430: [i1, j1] in ( Indices G) and

           A431: ((F . k) /. (( len (F . k)) -' 1)) = (G * (i1,j1)) and

           A432: [i2, j2] in ( Indices G) and

           A433: ((F . k) /. ( len (F . k))) = (G * (i2,j2));

          

           A434: ( len (F . k)) = k by A192;

          then

           A435: ((( len (F . k)) -' 1) + 1) = ( len (F . k)) by A429, XREAL_1: 235;

          

           A436: 1 <= (( len (F . k)) -' 1) by A429, A434, NAT_D: 49;

          then

           A437: ( right_cell ((F . k),(( len (F . k)) -' 1),G)) meets C by A427, A435;

          (( len (F . k)) -' 1) <= ( len (F . k)) by NAT_D: 35;

          then

           A438: (( len (F . k)) -' 1) in ( dom (F . k)) by A436, FINSEQ_3: 25;

          

           A439: (j1 + 1) > j1 & (j2 + 1) > j2 by NAT_1: 13;

          

           A440: (F . k) is_sequence_on G by A427;

          

           A441: (i1 + 1) > i1 & (i2 + 1) > i2 by NAT_1: 13;

          

           A442: ( len (F . k)) in ( dom (F . k)) by A429, A434, FINSEQ_3: 25;

          hereby

            assume ( front_left_cell ((F . k),(( len (F . k)) -' 1),G)) misses C & ( front_right_cell ((F . k),(( len (F . k)) -' 1),G)) misses C;

            then

            consider i, j such that

             A443: ((F . k) ^ <*(G * (i,j))*>) turns_right ((( len (F . k)) -' 1),G) and

             A444: (F . (k + 1)) = ((F . k) ^ <*(G * (i,j))*>) by A171, A429, A440, A434, A437;

            thus (F . (k + 1)) turns_right ((( len (F . k)) -' 1),G) by A443, A444;

            

             A445: ((F . (k + 1)) /. (( len (F . k)) + 1)) = (G * (i,j)) by A444, FINSEQ_4: 67;

            

             A446: ((F . (k + 1)) /. (( len (F . k)) -' 1)) = (G * (i1,j1)) & ((F . (k + 1)) /. ( len (F . k))) = (G * (i2,j2)) by A431, A433, A438, A442, A444, FINSEQ_4: 68;

            hence i1 = i2 & (j1 + 1) = j2 implies [(i2 + 1), j2] in ( Indices G) & (F . (k + 1)) = ((F . k) ^ <*(G * ((i2 + 1),j2))*>) by A430, A432, A435, A439, A443, A444, A445;

            thus (i1 + 1) = i2 & j1 = j2 implies [i2, (j2 -' 1)] in ( Indices G) & (F . (k + 1)) = ((F . k) ^ <*(G * (i2,(j2 -' 1)))*>) by A430, A432, A435, A441, A443, A444, A446, A445;

            thus i1 = (i2 + 1) & j1 = j2 implies [i2, (j2 + 1)] in ( Indices G) & (F . (k + 1)) = ((F . k) ^ <*(G * (i2,(j2 + 1)))*>) by A430, A432, A435, A441, A443, A444, A446, A445;

            thus i1 = i2 & j1 = (j2 + 1) implies [(i2 -' 1), j2] in ( Indices G) & (F . (k + 1)) = ((F . k) ^ <*(G * ((i2 -' 1),j2))*>) by A430, A432, A435, A439, A443, A444, A446, A445;

          end;

          hereby

            assume ( front_left_cell ((F . k),(( len (F . k)) -' 1),G)) misses C & ( front_right_cell ((F . k),(( len (F . k)) -' 1),G)) meets C;

            then

            consider i, j such that

             A447: ((F . k) ^ <*(G * (i,j))*>) goes_straight ((( len (F . k)) -' 1),G) and

             A448: (F . (k + 1)) = ((F . k) ^ <*(G * (i,j))*>) by A171, A429, A440, A434, A437;

            thus (F . (k + 1)) goes_straight ((( len (F . k)) -' 1),G) by A447, A448;

            

             A449: ((F . (k + 1)) /. (( len (F . k)) + 1)) = (G * (i,j)) by A448, FINSEQ_4: 67;

            

             A450: ((F . (k + 1)) /. (( len (F . k)) -' 1)) = (G * (i1,j1)) & ((F . (k + 1)) /. ( len (F . k))) = (G * (i2,j2)) by A431, A433, A438, A442, A448, FINSEQ_4: 68;

            hence i1 = i2 & (j1 + 1) = j2 implies [i2, (j2 + 1)] in ( Indices G) & (F . (k + 1)) = ((F . k) ^ <*(G * (i2,(j2 + 1)))*>) by A430, A432, A435, A439, A447, A448, A449;

            thus (i1 + 1) = i2 & j1 = j2 implies [(i2 + 1), j2] in ( Indices G) & (F . (k + 1)) = ((F . k) ^ <*(G * ((i2 + 1),j2))*>) by A430, A432, A435, A441, A447, A448, A450, A449;

            thus i1 = (i2 + 1) & j1 = j2 implies [(i2 -' 1), j2] in ( Indices G) & (F . (k + 1)) = ((F . k) ^ <*(G * ((i2 -' 1),j2))*>) by A430, A432, A435, A441, A447, A448, A450, A449;

            thus i1 = i2 & j1 = (j2 + 1) implies [i2, (j2 -' 1)] in ( Indices G) & (F . (k + 1)) = ((F . k) ^ <*(G * (i2,(j2 -' 1)))*>) by A430, A432, A435, A439, A447, A448, A450, A449;

          end;

          assume ( front_left_cell ((F . k),(( len (F . k)) -' 1),G)) meets C;

          then

          consider i, j such that

           A451: ((F . k) ^ <*(G * (i,j))*>) turns_left ((( len (F . k)) -' 1),G) and

           A452: (F . (k + 1)) = ((F . k) ^ <*(G * (i,j))*>) by A171, A429, A440, A434, A437;

          

           A453: ((F . (k + 1)) /. ( len (F . k))) = (G * (i2,j2)) by A433, A442, A452, FINSEQ_4: 68;

          thus (F . (k + 1)) turns_left ((( len (F . k)) -' 1),G) by A451, A452;

          

           A454: ((F . (k + 1)) /. (( len (F . k)) + 1)) = (G * (i,j)) by A452, FINSEQ_4: 67;

          

           A455: ((F . (k + 1)) /. (( len (F . k)) -' 1)) = (G * (i1,j1)) by A431, A438, A452, FINSEQ_4: 68;

          hence i1 = i2 & (j1 + 1) = j2 implies [(i2 -' 1), j2] in ( Indices G) & (F . (k + 1)) = ((F . k) ^ <*(G * ((i2 -' 1),j2))*>) by A430, A432, A435, A439, A451, A452, A453, A454;

          thus (i1 + 1) = i2 & j1 = j2 implies [i2, (j2 + 1)] in ( Indices G) & (F . (k + 1)) = ((F . k) ^ <*(G * (i2,(j2 + 1)))*>) by A430, A432, A435, A441, A451, A452, A455, A453, A454;

          thus i1 = (i2 + 1) & j1 = j2 implies [i2, (j2 -' 1)] in ( Indices G) & (F . (k + 1)) = ((F . k) ^ <*(G * (i2,(j2 -' 1)))*>) by A430, A432, A435, A441, A451, A452, A455, A453, A454;

          thus thesis by A430, A432, A435, A439, A451, A452, A455, A453, A454;

        end;

        

         A456: for k st k > 1 holds (( front_left_cell ((F . k),(k -' 1),( Gauge (C,n)))) misses C & ( front_right_cell ((F . k),(k -' 1),( Gauge (C,n)))) misses C implies (F . (k + 1)) turns_right ((k -' 1),( Gauge (C,n)))) & (( front_left_cell ((F . k),(k -' 1),( Gauge (C,n)))) misses C & ( front_right_cell ((F . k),(k -' 1),( Gauge (C,n)))) meets C implies (F . (k + 1)) goes_straight ((k -' 1),( Gauge (C,n)))) & (( front_left_cell ((F . k),(k -' 1),( Gauge (C,n)))) meets C implies (F . (k + 1)) turns_left ((k -' 1),( Gauge (C,n))))

        proof

          let k such that

           A457: k > 1;

          

           A458: (F . k) is_sequence_on G by A427;

          

           A459: ( len (F . k)) = k by A192;

          then 1 <= (( len (F . k)) -' 1) & ((( len (F . k)) -' 1) + 1) = ( len (F . k)) by A457, NAT_D: 49, XREAL_1: 235;

          then ex i1,j1,i2,j2 be Nat st [i1, j1] in ( Indices G) & ((F . k) /. (( len (F . k)) -' 1)) = (G * (i1,j1)) & [i2, j2] in ( Indices G) & ((F . k) /. ( len (F . k))) = (G * (i2,j2)) & (i1 = i2 & (j1 + 1) = j2 or (i1 + 1) = i2 & j1 = j2 or i1 = (i2 + 1) & j1 = j2 or i1 = i2 & j1 = (j2 + 1)) by A458, JORDAN8: 3;

          hence thesis by A428, A457, A459;

        end;

        defpred P[ Nat] means for m st m <= $1 holds ((F . $1) | m) = (F . m);

        

         A460: P[ 0 ]

        proof

          let m;

          assume

           A461: m <= 0 ;

          then ((F . 0 ) | 0 ) = ((F . 0 ) | m);

          hence thesis by A170, A461;

        end;

        

         A462: for k holds ex i, j st [i, j] in ( Indices G) & (F . (k + 1)) = ((F . k) ^ <*(G * (i,j))*>)

        proof

          let k;

          

           A463: (F . k) is_sequence_on G by A427;

          

           A464: ( len (F . k)) = k by A192;

          ( len G) >= 4 by JORDAN8: 10;

          then

           A465: ( len G) = ( width G) & 1 < ( len G) by JORDAN8:def 1, XXREAL_0: 2;

          per cases by XXREAL_0: 1;

            suppose

             A466: k < 1;

            consider i such that

             A467: 1 <= i and

             A468: (i + 1) <= ( len G) and ( N-min C) in ( cell (G,i,(( width G) -' 1))) and ( N-min C) <> (G * (i,(( width G) -' 1))) and

             A469: (F . ( 0 + 1)) = <*(G * (i,( width G)))*> by A171;

            take i, j = ( width G);

            i < ( len G) by A468, NAT_1: 13;

            hence [i, j] in ( Indices G) by A465, A467, MATRIX_0: 30;

            k = 0 by A466, NAT_1: 14;

            hence thesis by A170, A469, FINSEQ_1: 34;

          end;

            suppose

             A470: k = 1;

            consider i such that

             A471: 1 <= i and

             A472: (i + 1) <= ( len G) and

             A473: ( N-min C) in ( cell (G,i,(( width G) -' 1))) & ( N-min C) <> (G * (i,(( width G) -' 1))) and

             A474: (F . ( 0 + 1)) = <*(G * (i,( width G)))*> by A171;

            take (i + 1), j = ( width G);

            1 <= (i + 1) by A471, NAT_1: 13;

            hence [(i + 1), j] in ( Indices G) by A465, A472, MATRIX_0: 30;

            consider i9 be Nat such that

             A475: 1 <= i9 & (i9 + 1) <= ( len G) & ( N-min C) in ( cell (G,i9,(( width G) -' 1))) & ( N-min C) <> (G * (i9,(( width G) -' 1))) and

             A476: (F . (1 + 1)) = <*(G * (i9,( width G))), (G * ((i9 + 1),( width G)))*> by A171;

            i = i9 by A471, A472, A473, A475, Th29;

            hence thesis by A470, A474, A476, FINSEQ_1:def 9;

          end;

            suppose

             A477: k > 1;

            then 1 <= (( len (F . k)) -' 1) & ((( len (F . k)) -' 1) + 1) = ( len (F . k)) by A464, NAT_D: 49, XREAL_1: 235;

            then

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

             A478: [i1, j1] in ( Indices G) & ((F . k) /. (( len (F . k)) -' 1)) = (G * (i1,j1)) & [i2, j2] in ( Indices G) & ((F . k) /. ( len (F . k))) = (G * (i2,j2)) and

             A479: i1 = i2 & (j1 + 1) = j2 or (i1 + 1) = i2 & j1 = j2 or i1 = (i2 + 1) & j1 = j2 or i1 = i2 & j1 = (j2 + 1) by A463, JORDAN8: 3;

            now

              per cases ;

                suppose

                 A480: ( front_left_cell ((F . k),(( len (F . k)) -' 1),G)) misses C & ( front_right_cell ((F . k),(( len (F . k)) -' 1),G)) misses C;

                now

                  per cases by A479;

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

                    then [(i2 + 1), j2] in ( Indices G) & (F . (k + 1)) = ((F . k) ^ <*(G * ((i2 + 1),j2))*>) by A428, A477, A478, A480;

                    hence thesis;

                  end;

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

                    then [i2, (j2 -' 1)] in ( Indices G) & (F . (k + 1)) = ((F . k) ^ <*(G * (i2,(j2 -' 1)))*>) by A428, A477, A478, A480;

                    hence thesis;

                  end;

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

                    then [i2, (j2 + 1)] in ( Indices G) & (F . (k + 1)) = ((F . k) ^ <*(G * (i2,(j2 + 1)))*>) by A428, A477, A478, A480;

                    hence thesis;

                  end;

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

                    then [(i2 -' 1), j2] in ( Indices G) & (F . (k + 1)) = ((F . k) ^ <*(G * ((i2 -' 1),j2))*>) by A428, A477, A478, A480;

                    hence thesis;

                  end;

                end;

                hence thesis;

              end;

                suppose

                 A481: ( front_left_cell ((F . k),(( len (F . k)) -' 1),G)) misses C & ( front_right_cell ((F . k),(( len (F . k)) -' 1),G)) meets C;

                now

                  per cases by A479;

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

                    then [i2, (j2 + 1)] in ( Indices G) & (F . (k + 1)) = ((F . k) ^ <*(G * (i2,(j2 + 1)))*>) by A428, A477, A478, A481;

                    hence thesis;

                  end;

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

                    then [(i2 + 1), j2] in ( Indices G) & (F . (k + 1)) = ((F . k) ^ <*(G * ((i2 + 1),j2))*>) by A428, A477, A478, A481;

                    hence thesis;

                  end;

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

                    then [(i2 -' 1), j2] in ( Indices G) & (F . (k + 1)) = ((F . k) ^ <*(G * ((i2 -' 1),j2))*>) by A428, A477, A478, A481;

                    hence thesis;

                  end;

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

                    then [i2, (j2 -' 1)] in ( Indices G) & (F . (k + 1)) = ((F . k) ^ <*(G * (i2,(j2 -' 1)))*>) by A428, A477, A478, A481;

                    hence thesis;

                  end;

                end;

                hence thesis;

              end;

                suppose

                 A482: ( front_left_cell ((F . k),(( len (F . k)) -' 1),G)) meets C;

                now

                  per cases by A479;

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

                    then [(i2 -' 1), j2] in ( Indices G) & (F . (k + 1)) = ((F . k) ^ <*(G * ((i2 -' 1),j2))*>) by A428, A477, A478, A482;

                    hence thesis;

                  end;

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

                    then [i2, (j2 + 1)] in ( Indices G) & (F . (k + 1)) = ((F . k) ^ <*(G * (i2,(j2 + 1)))*>) by A428, A477, A478, A482;

                    hence thesis;

                  end;

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

                    then [i2, (j2 -' 1)] in ( Indices G) & (F . (k + 1)) = ((F . k) ^ <*(G * (i2,(j2 -' 1)))*>) by A428, A477, A478, A482;

                    hence thesis;

                  end;

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

                    then [(i2 + 1), j2] in ( Indices G) & (F . (k + 1)) = ((F . k) ^ <*(G * ((i2 + 1),j2))*>) by A428, A477, A478, A482;

                    hence thesis;

                  end;

                end;

                hence thesis;

              end;

            end;

            hence thesis;

          end;

        end;

        

         A483: for k st P[k] holds P[(k + 1)]

        proof

          let k such that

           A484: for m st m <= k holds ((F . k) | m) = (F . m);

          let m such that

           A485: m <= (k + 1);

          per cases by A485, XXREAL_0: 1;

            suppose m < (k + 1);

            then

             A486: m <= k by NAT_1: 13;

            ( len (F . k)) = k & ex i, j st [i, j] in ( Indices G) & (F . (k + 1)) = ((F . k) ^ <*(G * (i,j))*>) by A192, A462;

            then ((F . (k + 1)) | m) = ((F . k) | m) by A486, FINSEQ_5: 22;

            hence thesis by A484, A486;

          end;

            suppose

             A487: m = (k + 1);

            ( len (F . (k + 1))) = (k + 1) by A192;

            hence thesis by A487, FINSEQ_1: 58;

          end;

        end;

        

         A488: for k holds P[k] from NAT_1:sch 2( A460, A483);

        defpred P[ Nat] means (F . $1) is unfolded;

        

         A489: for k st P[k] holds P[(k + 1)]

        proof

          let k such that

           A490: (F . k) is unfolded;

          

           A491: (F . k) is_sequence_on G by A427;

          per cases ;

            suppose k <= 1;

            then (k + 1) <= (1 + 1) by XREAL_1: 6;

            then ( len (F . (k + 1))) <= 2 by A192;

            hence thesis by SPPOL_2: 26;

          end;

            suppose

             A492: k > 1;

            set m = (k -' 1);

            

             A493: (m + 1) = k by A492, XREAL_1: 235;

            

             A494: ( len (F . k)) = k by A192;

            

             A495: 1 <= m by A492, NAT_D: 49;

            then

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

             A496: [i1, j1] in ( Indices G) and

             A497: ((F . k) /. m) = (G * (i1,j1)) and

             A498: [i2, j2] in ( Indices G) and

             A499: ((F . k) /. ( len (F . k))) = (G * (i2,j2)) and

             A500: i1 = i2 & (j1 + 1) = j2 or (i1 + 1) = i2 & j1 = j2 or i1 = (i2 + 1) & j1 = j2 or i1 = i2 & j1 = (j2 + 1) by A491, A493, A494, JORDAN8: 3;

            

             A501: ( LSeg ((F . k),m)) = ( LSeg ((G * (i1,j1)),(G * (i2,j2)))) by A495, A493, A494, A497, A499, TOPREAL1:def 3;

            

             A502: 1 <= j2 by A498, MATRIX_0: 32;

            then

             A503: ((j2 -' 1) + 1) = j2 by XREAL_1: 235;

            

             A504: 1 <= j1 by A496, MATRIX_0: 32;

            

             A505: 1 <= i2 by A498, MATRIX_0: 32;

            then

             A506: ((i2 -' 1) + 1) = i2 by XREAL_1: 235;

            

             A507: i1 <= ( len G) by A496, MATRIX_0: 32;

            

             A508: j2 <= ( width G) by A498, MATRIX_0: 32;

            

             A509: 1 <= i1 by A496, MATRIX_0: 32;

            

             A510: j1 <= ( width G) by A496, MATRIX_0: 32;

            

             A511: i2 <= ( len G) by A498, MATRIX_0: 32;

            now

              per cases ;

                suppose

                 A512: ( front_left_cell ((F . k),(( len (F . k)) -' 1),G)) misses C & ( front_right_cell ((F . k),(( len (F . k)) -' 1),G)) misses C;

                now

                  per cases by A500;

                    suppose

                     A513: i1 = i2 & (j1 + 1) = j2;

                    then [(i2 + 1), j2] in ( Indices G) by A428, A492, A494, A496, A497, A498, A499, A512;

                    then (i2 + 1) <= ( len G) by MATRIX_0: 32;

                    then

                     A514: (( LSeg ((F . k),m)) /\ ( LSeg (((F . k) /. ( len (F . k))),(G * ((i2 + 1),j2))))) = {((F . k) /. ( len (F . k)))} by A499, A509, A504, A508, A501, A513, GOBOARD7: 15;

                    (F . (k + 1)) = ((F . k) ^ <*(G * ((i2 + 1),j2))*>) by A428, A492, A494, A496, A497, A498, A499, A512, A513;

                    hence thesis by A490, A493, A494, A514, SPPOL_2: 30;

                  end;

                    suppose

                     A515: (i1 + 1) = i2 & j1 = j2;

                    then [i2, (j2 -' 1)] in ( Indices G) by A428, A492, A494, A496, A497, A498, A499, A512;

                    then 1 <= (j2 -' 1) by MATRIX_0: 32;

                    then

                     A516: (( LSeg ((F . k),m)) /\ ( LSeg (((F . k) /. ( len (F . k))),(G * (i2,(j2 -' 1)))))) = {((F . k) /. ( len (F . k)))} by A499, A509, A510, A511, A503, A501, A515, GOBOARD7: 16;

                    (F . (k + 1)) = ((F . k) ^ <*(G * (i2,(j2 -' 1)))*>) by A428, A492, A494, A496, A497, A498, A499, A512, A515;

                    hence thesis by A490, A493, A494, A516, SPPOL_2: 30;

                  end;

                    suppose

                     A517: i1 = (i2 + 1) & j1 = j2;

                    then [i2, (j2 + 1)] in ( Indices G) by A428, A492, A494, A496, A497, A498, A499, A512;

                    then (j2 + 1) <= ( width G) by MATRIX_0: 32;

                    then

                     A518: (( LSeg ((F . k),m)) /\ ( LSeg (((F . k) /. ( len (F . k))),(G * (i2,(j2 + 1)))))) = {((F . k) /. ( len (F . k)))} by A499, A507, A504, A505, A501, A517, GOBOARD7: 17;

                    (F . (k + 1)) = ((F . k) ^ <*(G * (i2,(j2 + 1)))*>) by A428, A492, A494, A496, A497, A498, A499, A512, A517;

                    hence thesis by A490, A493, A494, A518, SPPOL_2: 30;

                  end;

                    suppose

                     A519: i1 = i2 & j1 = (j2 + 1);

                    then [(i2 -' 1), j2] in ( Indices G) by A428, A492, A494, A496, A497, A498, A499, A512;

                    then 1 <= (i2 -' 1) by MATRIX_0: 32;

                    then

                     A520: (( LSeg ((F . k),m)) /\ ( LSeg (((F . k) /. ( len (F . k))),(G * ((i2 -' 1),j2))))) = {((F . k) /. ( len (F . k)))} by A499, A507, A510, A502, A506, A501, A519, GOBOARD7: 18;

                    (F . (k + 1)) = ((F . k) ^ <*(G * ((i2 -' 1),j2))*>) by A428, A492, A494, A496, A497, A498, A499, A512, A519;

                    hence thesis by A490, A493, A494, A520, SPPOL_2: 30;

                  end;

                end;

                hence thesis;

              end;

                suppose

                 A521: ( front_left_cell ((F . k),(( len (F . k)) -' 1),G)) misses C & ( front_right_cell ((F . k),(( len (F . k)) -' 1),G)) meets C;

                now

                  per cases by A500;

                    suppose

                     A522: i1 = i2 & (j1 + 1) = j2;

                    then [i2, (j2 + 1)] in ( Indices G) by A428, A492, A494, A496, A497, A498, A499, A521;

                    then

                     A523: (j2 + 1) <= ( width G) by MATRIX_0: 32;

                    (j2 + 1) = (j1 + (1 + 1)) by A522;

                    then

                     A524: (( LSeg ((F . k),m)) /\ ( LSeg (((F . k) /. ( len (F . k))),(G * (i2,(j2 + 1)))))) = {((F . k) /. ( len (F . k)))} by A499, A509, A507, A504, A501, A522, A523, GOBOARD7: 13;

                    (F . (k + 1)) = ((F . k) ^ <*(G * (i2,(j2 + 1)))*>) by A428, A492, A494, A496, A497, A498, A499, A521, A522;

                    hence thesis by A490, A493, A494, A524, SPPOL_2: 30;

                  end;

                    suppose

                     A525: (i1 + 1) = i2 & j1 = j2;

                    then [(i2 + 1), j2] in ( Indices G) by A428, A492, A494, A496, A497, A498, A499, A521;

                    then

                     A526: (i2 + 1) <= ( len G) by MATRIX_0: 32;

                    (i2 + 1) = (i1 + (1 + 1)) by A525;

                    then

                     A527: (( LSeg ((F . k),m)) /\ ( LSeg (((F . k) /. ( len (F . k))),(G * ((i2 + 1),j2))))) = {((F . k) /. ( len (F . k)))} by A499, A509, A504, A510, A501, A525, A526, GOBOARD7: 14;

                    (F . (k + 1)) = ((F . k) ^ <*(G * ((i2 + 1),j2))*>) by A428, A492, A494, A496, A497, A498, A499, A521, A525;

                    hence thesis by A490, A493, A494, A527, SPPOL_2: 30;

                  end;

                    suppose

                     A528: i1 = (i2 + 1) & j1 = j2;

                    then [(i2 -' 1), j2] in ( Indices G) by A428, A492, A494, A496, A497, A498, A499, A521;

                    then

                     A529: 1 <= (i2 -' 1) by MATRIX_0: 32;

                    (((i2 -' 1) + 1) + 1) = ((i2 -' 1) + (1 + 1));

                    then

                     A530: (( LSeg ((F . k),m)) /\ ( LSeg (((F . k) /. ( len (F . k))),(G * ((i2 -' 1),j2))))) = {((F . k) /. ( len (F . k)))} by A499, A507, A504, A510, A506, A501, A528, A529, GOBOARD7: 14;

                    (F . (k + 1)) = ((F . k) ^ <*(G * ((i2 -' 1),j2))*>) by A428, A492, A494, A496, A497, A498, A499, A521, A528;

                    hence thesis by A490, A493, A494, A530, SPPOL_2: 30;

                  end;

                    suppose

                     A531: i1 = i2 & j1 = (j2 + 1);

                    then [i2, (j2 -' 1)] in ( Indices G) by A428, A492, A494, A496, A497, A498, A499, A521;

                    then

                     A532: 1 <= (j2 -' 1) by MATRIX_0: 32;

                    (((j2 -' 1) + 1) + 1) = ((j2 -' 1) + (1 + 1));

                    then

                     A533: (( LSeg ((F . k),m)) /\ ( LSeg (((F . k) /. ( len (F . k))),(G * (i2,(j2 -' 1)))))) = {((F . k) /. ( len (F . k)))} by A499, A509, A507, A510, A503, A501, A531, A532, GOBOARD7: 13;

                    (F . (k + 1)) = ((F . k) ^ <*(G * (i2,(j2 -' 1)))*>) by A428, A492, A494, A496, A497, A498, A499, A521, A531;

                    hence thesis by A490, A493, A494, A533, SPPOL_2: 30;

                  end;

                end;

                hence thesis;

              end;

                suppose

                 A534: ( front_left_cell ((F . k),(( len (F . k)) -' 1),G)) meets C;

                now

                  per cases by A500;

                    suppose

                     A535: i1 = i2 & (j1 + 1) = j2;

                    then [(i2 -' 1), j2] in ( Indices G) by A428, A492, A494, A496, A497, A498, A499, A534;

                    then 1 <= (i2 -' 1) by MATRIX_0: 32;

                    then

                     A536: (( LSeg ((F . k),m)) /\ ( LSeg (((F . k) /. ( len (F . k))),(G * ((i2 -' 1),j2))))) = {((F . k) /. ( len (F . k)))} by A499, A507, A504, A508, A506, A501, A535, GOBOARD7: 16;

                    (F . (k + 1)) = ((F . k) ^ <*(G * ((i2 -' 1),j2))*>) by A428, A492, A494, A496, A497, A498, A499, A534, A535;

                    hence thesis by A490, A493, A494, A536, SPPOL_2: 30;

                  end;

                    suppose

                     A537: (i1 + 1) = i2 & j1 = j2;

                    then [i2, (j2 + 1)] in ( Indices G) by A428, A492, A494, A496, A497, A498, A499, A534;

                    then (j2 + 1) <= ( width G) by MATRIX_0: 32;

                    then

                     A538: (( LSeg ((F . k),m)) /\ ( LSeg (((F . k) /. ( len (F . k))),(G * (i2,(j2 + 1)))))) = {((F . k) /. ( len (F . k)))} by A499, A509, A504, A511, A501, A537, GOBOARD7: 18;

                    (F . (k + 1)) = ((F . k) ^ <*(G * (i2,(j2 + 1)))*>) by A428, A492, A494, A496, A497, A498, A499, A534, A537;

                    hence thesis by A490, A493, A494, A538, SPPOL_2: 30;

                  end;

                    suppose

                     A539: i1 = (i2 + 1) & j1 = j2;

                    then [i2, (j2 -' 1)] in ( Indices G) by A428, A492, A494, A496, A497, A498, A499, A534;

                    then 1 <= (j2 -' 1) by MATRIX_0: 32;

                    then

                     A540: (( LSeg ((F . k),m)) /\ ( LSeg (((F . k) /. ( len (F . k))),(G * (i2,(j2 -' 1)))))) = {((F . k) /. ( len (F . k)))} by A499, A507, A510, A505, A503, A501, A539, GOBOARD7: 15;

                    (F . (k + 1)) = ((F . k) ^ <*(G * (i2,(j2 -' 1)))*>) by A428, A492, A494, A496, A497, A498, A499, A534, A539;

                    hence thesis by A490, A493, A494, A540, SPPOL_2: 30;

                  end;

                    suppose

                     A541: i1 = i2 & j1 = (j2 + 1);

                    then [(i2 + 1), j2] in ( Indices G) by A428, A492, A494, A496, A497, A498, A499, A534;

                    then (i2 + 1) <= ( len G) by MATRIX_0: 32;

                    then

                     A542: (( LSeg ((F . k),m)) /\ ( LSeg (((F . k) /. ( len (F . k))),(G * ((i2 + 1),j2))))) = {((F . k) /. ( len (F . k)))} by A499, A509, A510, A502, A501, A541, GOBOARD7: 17;

                    (F . (k + 1)) = ((F . k) ^ <*(G * ((i2 + 1),j2))*>) by A428, A492, A494, A496, A497, A498, A499, A534, A541;

                    hence thesis by A490, A493, A494, A542, SPPOL_2: 30;

                  end;

                end;

                hence thesis;

              end;

            end;

            hence thesis;

          end;

        end;

        now

          defpred P[ Nat] means (F . $1) is one-to-one;

          assume

           A543: for k st k >= 1 holds for m st m in ( dom (F . k)) & m <> ( len (F . k)) holds ((F . k) /. m) <> ((F . k) /. ( len (F . k)));

          

           A544: for k st P[k] holds P[(k + 1)]

          proof

            let k;

            assume

             A545: (F . k) is one-to-one;

            now

              let n,m be Element of NAT such that

               A546: n in ( dom (F . (k + 1))) and

               A547: m in ( dom (F . (k + 1))) and

               A548: ((F . (k + 1)) /. n) = ((F . (k + 1)) /. m);

              

               A549: n <= ( len (F . (k + 1))) & m <= ( len (F . (k + 1))) by A546, A547, FINSEQ_3: 25;

              

               A550: 1 <= m by A547, FINSEQ_3: 25;

              

               A551: 1 <= n by A546, FINSEQ_3: 25;

              

               A552: ex i, j st [i, j] in ( Indices G) & (F . (k + 1)) = ((F . k) ^ <*(G * (i,j))*>) by A462;

              

               A553: ( len (F . k)) = k by A192;

              

               A554: ( len (F . (k + 1))) = (k + 1) by A192;

              per cases by A549, A554, NAT_1: 8;

                suppose

                 A555: n <= k & m <= k;

                then

                 A556: m in ( dom (F . k)) by A550, A553, FINSEQ_3: 25;

                then

                 A557: ((F . (k + 1)) /. m) = ((F . k) /. m) by A552, FINSEQ_4: 68;

                

                 A558: n in ( dom (F . k)) by A551, A553, A555, FINSEQ_3: 25;

                then ((F . (k + 1)) /. n) = ((F . k) /. n) by A552, FINSEQ_4: 68;

                hence n = m by A545, A548, A558, A556, A557, PARTFUN2: 10;

              end;

                suppose n = (k + 1) & m <= k;

                hence n = m by A543, A547, A548, A554, NAT_1: 12;

              end;

                suppose n <= k & m = (k + 1);

                hence n = m by A543, A546, A548, A554, NAT_1: 12;

              end;

                suppose n = (k + 1) & m = (k + 1);

                hence n = m;

              end;

            end;

            hence thesis by PARTFUN2: 9;

          end;

          

           A559: P[ 0 ] by A170;

          

           A560: for k holds P[k] from NAT_1:sch 2( A559, A544);

          

           A561: for k holds ( card ( rng (F . k))) = k

          proof

            let k;

            (F . k) is one-to-one by A560;

            

            hence ( card ( rng (F . k))) = ( len (F . k)) by FINSEQ_4: 62

            .= k by A192;

          end;

          set k = ((( len G) * ( width G)) + 1);

          (F . k) is_sequence_on G by A427;

          then ( card ( Values G)) <= (( len G) * ( width G)) & ( card ( rng (F . k))) <= ( card ( Values G)) by GOBRD13: 8, MATRIX_0: 40, NAT_1: 43;

          then k > (( len G) * ( width G)) & ( card ( rng (F . k))) <= (( len G) * ( width G)) by NAT_1: 13, XXREAL_0: 2;

          hence contradiction by A561;

        end;

        then

         A562: ex k be Nat st Q[k];

        consider k be Nat such that

         A563: Q[k] and

         A564: for l be Nat st Q[l] holds k <= l from NAT_1:sch 5( A562);

        reconsider k as Nat;

        consider m such that

         A565: m in ( dom (F . k)) and

         A566: m <> ( len (F . k)) and

         A567: ((F . k) /. m) = ((F . k) /. ( len (F . k))) by A563;

        

         A568: 1 <= m by A565, FINSEQ_3: 25;

        reconsider f = (F . k) as non empty FinSequence of ( TOP-REAL 2) by A563;

        

         A569: f is_sequence_on G by A427;

        

         A570: m <= ( len f) by A565, FINSEQ_3: 25;

        then

         A571: m < ( len f) by A566, XXREAL_0: 1;

        then 1 < ( len f) by A568, XXREAL_0: 2;

        then

         A572: ( len f) >= (1 + 1) by NAT_1: 13;

        

         A573: P[ 0 ] by A170, CARD_1: 27, SPPOL_2: 26;

        for k holds P[k] from NAT_1:sch 2( A573, A489);

        then

        reconsider f as non constant special unfolded non empty FinSequence of ( TOP-REAL 2) by A569, A572, JORDAN8: 4, JORDAN8: 5;

        

         A574: (m + 1) <= ( len f) by A571, NAT_1: 13;

        set g = (f /^ (m -' 1));

        (m -' 1) <= m by NAT_D: 44;

        then (m -' 1) < (m + 1) by NAT_1: 13;

        then

         A575: (m -' 1) < ( len f) by A574, XXREAL_0: 2;

        then

         A576: ( len g) = (( len f) - (m -' 1)) by RFINSEQ:def 1;

        then ((m -' 1) - (m -' 1)) < ( len g) by A575, XREAL_1: 9;

        then

        reconsider g as non empty FinSequence of ( TOP-REAL 2) by CARD_1: 27;

        ( len g) in ( dom g) by FINSEQ_5: 6;

        

        then

         A577: (g /. ( len g)) = (f /. ((m -' 1) + ( len g))) by FINSEQ_5: 27

        .= (f /. ( len f)) by A576;

        

         A578: ( len (F . k)) = k by A192;

        

         A579: for i st 1 <= i & i < ( len g) & 1 <= j & j < ( len g) & (g /. i) = (g /. j) holds i = j

        proof

          let i such that

           A580: 1 <= i and

           A581: i < ( len g) and

           A582: 1 <= j and

           A583: j < ( len g) and

           A584: (g /. i) = (g /. j) and

           A585: i <> j;

          

           A586: i in ( dom g) by A580, A581, FINSEQ_3: 25;

          then

           A587: (g /. i) = (f /. ((m -' 1) + i)) by FINSEQ_5: 27;

          

           A588: j in ( dom g) by A582, A583, FINSEQ_3: 25;

          then

           A589: (g /. j) = (f /. ((m -' 1) + j)) by FINSEQ_5: 27;

          per cases by A585, XXREAL_0: 1;

            suppose

             A590: i < j;

            set l = ((m -' 1) + j), m9 = ((m -' 1) + i);

            

             A591: m9 < l by A590, XREAL_1: 6;

            

             A592: ( len (F . l)) = l by A192;

            

             A593: l < k by A578, A576, A583, XREAL_1: 20;

            then

             A594: (f | l) = (F . l) by A488;

            ( 0 + j) <= l by XREAL_1: 6;

            then

             A595: 1 <= l by A582, XXREAL_0: 2;

            then l in ( dom (F . l)) by A592, FINSEQ_3: 25;

            then

             A596: ((F . l) /. l) = (f /. l) by A594, FINSEQ_4: 70;

            ( 0 + i) <= m9 by XREAL_1: 6;

            then 1 <= m9 by A580, XXREAL_0: 2;

            then

             A597: m9 in ( dom (F . l)) by A591, A592, FINSEQ_3: 25;

            then ((F . l) /. m9) = (f /. m9) by A594, FINSEQ_4: 70;

            hence contradiction by A564, A584, A587, A588, A591, A593, A595, A592, A597, A596, FINSEQ_5: 27;

          end;

            suppose

             A598: j < i;

            set l = ((m -' 1) + i), m9 = ((m -' 1) + j);

            

             A599: m9 < l by A598, XREAL_1: 6;

            

             A600: ( len (F . l)) = l by A192;

            

             A601: l < k by A578, A576, A581, XREAL_1: 20;

            then

             A602: (f | l) = (F . l) by A488;

            ( 0 + i) <= l by XREAL_1: 6;

            then

             A603: 1 <= l by A580, XXREAL_0: 2;

            then l in ( dom (F . l)) by A600, FINSEQ_3: 25;

            then

             A604: ((F . l) /. l) = (f /. l) by A602, FINSEQ_4: 70;

            ( 0 + j) <= m9 by XREAL_1: 6;

            then 1 <= m9 by A582, XXREAL_0: 2;

            then

             A605: m9 in ( dom (F . l)) by A599, A600, FINSEQ_3: 25;

            then ((F . l) /. m9) = (f /. m9) by A602, FINSEQ_4: 70;

            hence contradiction by A564, A584, A586, A589, A599, A601, A603, A600, A605, A604, FINSEQ_5: 27;

          end;

        end;

         A606:

        now

          consider i such that

           A607: 1 <= i & (i + 1) <= ( len G) and

           A608: ( N-min C) in ( cell (G,i,(( width G) -' 1))) & ( N-min C) <> (G * (i,(( width G) -' 1))) and

           A609: (F . (1 + 1)) = <*(G * (i,( width G))), (G * ((i + 1),( width G)))*> by A171;

          take i;

          thus 1 <= i & (i + 1) <= ( len G) by A607;

          

           A610: (f | 2) = (F . 2) by A488, A578, A572;

          

           A611: ( len (f | 2)) = 2 by A572, FINSEQ_1: 59;

          then 1 in ( dom (f | 2)) by FINSEQ_3: 25;

          

          hence (f /. 1) = ((f | 2) /. 1) by FINSEQ_4: 70

          .= (G * (i,( width G))) by A609, A610, FINSEQ_4: 17;

          2 in ( dom (f | 2)) by A611, FINSEQ_3: 25;

          

          hence (f /. 2) = ((f | 2) /. 2) by FINSEQ_4: 70

          .= (G * ((i + 1),( width G))) by A609, A610, FINSEQ_4: 17;

          thus ( N-min C) in ( cell (G,i,(( width G) -' 1))) & ( N-min C) <> (G * (i,(( width G) -' 1))) by A608;

        end;

        1 in ( dom g) by FINSEQ_5: 6;

        

        then

         A612: (g /. 1) = (f /. ((m -' 1) + 1)) by FINSEQ_5: 27

        .= (f /. m) by A568, XREAL_1: 235;

        

         A613: for i st 1 < i & i < j & j <= ( len g) holds (g /. i) <> (g /. j)

        proof

          let i such that

           A614: 1 < i and

           A615: i < j and

           A616: j <= ( len g) and

           A617: (g /. i) = (g /. j);

          

           A618: 1 < j by A614, A615, XXREAL_0: 2;

          

           A619: i < ( len g) by A615, A616, XXREAL_0: 2;

          then

           A620: 1 < ( len g) by A614, XXREAL_0: 2;

          per cases ;

            suppose j <> ( len g);

            then j < ( len g) by A616, XXREAL_0: 1;

            hence contradiction by A579, A614, A615, A617, A618, A619;

          end;

            suppose j = ( len g);

            hence contradiction by A567, A612, A577, A579, A614, A615, A617, A620;

          end;

        end;

        ((m + 1) - (m -' 1)) <= ( len g) by A574, A576, XREAL_1: 9;

        then

         A621: ((m + 1) - (m - 1)) <= ( len g) by A568, XREAL_1: 233;

        then

         A622: (((1 + m) - m) + 1) <= ( len g);

        

         A623: g is_sequence_on G by A427, JORDAN8: 2;

        then

         A624: g is standard by JORDAN8: 4;

        

         A625: g is non constant

        proof

          take 1, 2;

          thus

           A626: 1 in ( dom g) by FINSEQ_5: 6;

          thus

           A627: 2 in ( dom g) by A621, FINSEQ_3: 25;

          then (g /. 1) <> (g /. (1 + 1)) by A624, FINSEQ_5: 6, GOBOARD7: 29;

          then (g . 1) <> (g /. (1 + 1)) by A626, PARTFUN1:def 6;

          hence thesis by A627, PARTFUN1:def 6;

        end;

        

         A628: for i st 1 <= i & i < j & j < ( len g) holds (g /. i) <> (g /. j)

        proof

          let i such that

           A629: 1 <= i & i < j & j < ( len g) and

           A630: (g /. i) = (g /. j);

          1 < j & i < ( len g) by A629, XXREAL_0: 2;

          hence contradiction by A579, A629, A630;

        end;

        g is s.c.c.

        proof

          let i, j such that

           A631: (i + 1) < j and

           A632: i > 1 & j < ( len g) or (j + 1) < ( len g);

          

           A633: 1 < j by A631, NAT_1: 12;

          

           A634: 1 <= (i + 1) by NAT_1: 12;

          

           A635: j <= (j + 1) by NAT_1: 12;

          then

           A636: (i + 1) < (j + 1) by A631, XXREAL_0: 2;

          i < j by A631, NAT_1: 13;

          then

           A637: i < (j + 1) by A635, XXREAL_0: 2;

          per cases by A632, NAT_1: 14;

            suppose

             A638: i > 1 & j < ( len g);

            then

             A639: (j + 1) <= ( len g) by NAT_1: 13;

            then

             A640: ( LSeg (g,j)) = ( LSeg ((g /. j),(g /. (j + 1)))) by A633, TOPREAL1:def 3;

            consider i19,j19,i29,j29 be Nat such that

             A641: [i19, j19] in ( Indices G) and

             A642: (g /. j) = (G * (i19,j19)) and

             A643: [i29, j29] in ( Indices G) and

             A644: (g /. (j + 1)) = (G * (i29,j29)) and

             A645: i19 = i29 & (j19 + 1) = j29 or (i19 + 1) = i29 & j19 = j29 or i19 = (i29 + 1) & j19 = j29 or i19 = i29 & j19 = (j29 + 1) by A623, A633, A639, JORDAN8: 3;

            

             A646: 1 <= i19 by A641, MATRIX_0: 32;

            

             A647: j29 <= ( width G) by A643, MATRIX_0: 32;

            

             A648: 1 <= i29 by A643, MATRIX_0: 32;

            

             A649: i19 <= ( len G) by A641, MATRIX_0: 32;

            

             A650: 1 <= j29 by A643, MATRIX_0: 32;

            

             A651: j19 <= ( width G) by A641, MATRIX_0: 32;

            

             A652: i29 <= ( len G) by A643, MATRIX_0: 32;

            

             A653: 1 <= j19 by A641, MATRIX_0: 32;

            

             A654: (i + 1) < ( len g) by A631, A638, XXREAL_0: 2;

            then

             A655: ( LSeg (g,i)) = ( LSeg ((g /. i),(g /. (i + 1)))) by A638, TOPREAL1:def 3;

            

             A656: i < ( len g) by A654, NAT_1: 13;

            consider i1, j1, i2, j2 such that

             A657: [i1, j1] in ( Indices G) and

             A658: (g /. i) = (G * (i1,j1)) and

             A659: [i2, j2] in ( Indices G) and

             A660: (g /. (i + 1)) = (G * (i2,j2)) and

             A661: i1 = i2 & (j1 + 1) = j2 or (i1 + 1) = i2 & j1 = j2 or i1 = (i2 + 1) & j1 = j2 or i1 = i2 & j1 = (j2 + 1) by A623, A638, A654, JORDAN8: 3;

            

             A662: 1 <= i1 by A657, MATRIX_0: 32;

            

             A663: j2 <= ( width G) by A659, MATRIX_0: 32;

            

             A664: j1 <= ( width G) by A657, MATRIX_0: 32;

            

             A665: 1 <= j2 by A659, MATRIX_0: 32;

            

             A666: 1 <= j1 by A657, MATRIX_0: 32;

            

             A667: i2 <= ( len G) by A659, MATRIX_0: 32;

            

             A668: i1 <= ( len G) by A657, MATRIX_0: 32;

            

             A669: 1 < (i + 1) by A638, NAT_1: 13;

            assume

             A670: ( LSeg (g,i)) meets ( LSeg (g,j));

            

             A671: 1 <= i2 by A659, MATRIX_0: 32;

            now

              per cases by A661, A645;

                suppose

                 A672: i1 = i2 & (j1 + 1) = j2 & i19 = i29 & (j19 + 1) = j29;

                then

                 A673: j1 = j19 or j1 = (j19 + 1) or (j1 + 1) = j19 by A655, A658, A660, A662, A668, A666, A663, A640, A642, A644, A646, A649, A653, A647, A670, GOBOARD7: 22;

                i1 = i19 by A655, A658, A660, A662, A668, A666, A663, A640, A642, A644, A646, A649, A653, A647, A670, A672, GOBOARD7: 19;

                hence contradiction by A579, A613, A631, A635, A634, A633, A637, A638, A654, A656, A658, A660, A639, A642, A644, A672, A673;

              end;

                suppose

                 A674: i1 = i2 & (j1 + 1) = j2 & (i19 + 1) = i29 & j19 = j29;

                then i1 = i19 & j1 = j19 or i1 = i19 & (j1 + 1) = j19 or i1 = (i19 + 1) & j1 = j19 or i1 = (i19 + 1) & (j1 + 1) = j19 by A655, A658, A660, A662, A668, A666, A663, A640, A642, A644, A646, A653, A651, A652, A670, GOBOARD7: 21;

                hence contradiction by A579, A613, A631, A636, A633, A637, A638, A669, A656, A658, A660, A639, A642, A644, A674;

              end;

                suppose

                 A675: i1 = i2 & (j1 + 1) = j2 & i19 = (i29 + 1) & j19 = j29;

                then i1 = i29 & j19 = j1 or i1 = i29 & (j1 + 1) = j19 or i1 = (i29 + 1) & j19 = j1 or i1 = (i29 + 1) & (j1 + 1) = j19 by A655, A658, A660, A662, A668, A666, A663, A640, A642, A644, A649, A653, A651, A648, A670, GOBOARD7: 21;

                hence contradiction by A579, A613, A631, A636, A633, A637, A638, A669, A656, A658, A660, A639, A642, A644, A675;

              end;

                suppose

                 A676: i1 = i2 & (j1 + 1) = j2 & i19 = i29 & j19 = (j29 + 1);

                then

                 A677: j1 = j29 or j1 = (j29 + 1) or (j1 + 1) = j29 by A655, A658, A660, A662, A668, A666, A663, A640, A642, A644, A646, A649, A651, A650, A670, GOBOARD7: 22;

                i1 = i19 by A655, A658, A660, A662, A668, A666, A663, A640, A642, A644, A646, A649, A651, A650, A670, A676, GOBOARD7: 19;

                hence contradiction by A579, A613, A631, A636, A633, A638, A669, A656, A658, A660, A639, A642, A644, A676, A677;

              end;

                suppose

                 A678: (i1 + 1) = i2 & j1 = j2 & i19 = i29 & (j19 + 1) = j29;

                then i19 = i1 & j1 = j19 or i19 = i1 & (j19 + 1) = j1 or i19 = (i1 + 1) & j1 = j19 or i19 = (i1 + 1) & (j19 + 1) = j1 by A655, A658, A660, A662, A666, A664, A667, A640, A642, A644, A646, A649, A653, A647, A670, GOBOARD7: 21;

                hence contradiction by A579, A613, A631, A636, A633, A637, A638, A669, A656, A658, A660, A639, A642, A644, A678;

              end;

                suppose

                 A679: (i1 + 1) = i2 & j1 = j2 & (i19 + 1) = i29 & j19 = j29;

                then

                 A680: i1 = i19 or i1 = (i19 + 1) or (i1 + 1) = i19 by A655, A658, A660, A662, A666, A664, A667, A640, A642, A644, A646, A653, A651, A652, A670, GOBOARD7: 23;

                j1 = j19 by A655, A658, A660, A662, A666, A664, A667, A640, A642, A644, A646, A653, A651, A652, A670, A679, GOBOARD7: 20;

                hence contradiction by A579, A613, A631, A635, A634, A633, A637, A638, A654, A656, A658, A660, A639, A642, A644, A679, A680;

              end;

                suppose

                 A681: (i1 + 1) = i2 & j1 = j2 & i19 = (i29 + 1) & j19 = j29;

                then

                 A682: i1 = i29 or i1 = (i29 + 1) or (i1 + 1) = i29 by A655, A658, A660, A662, A666, A664, A667, A640, A642, A644, A649, A653, A651, A648, A670, GOBOARD7: 23;

                j1 = j19 by A655, A658, A660, A662, A666, A664, A667, A640, A642, A644, A649, A653, A651, A648, A670, A681, GOBOARD7: 20;

                hence contradiction by A579, A613, A636, A633, A637, A638, A669, A656, A658, A660, A639, A642, A644, A681, A682;

              end;

                suppose

                 A683: (i1 + 1) = i2 & j1 = j2 & i19 = i29 & j19 = (j29 + 1);

                then i19 = i1 & j1 = j29 or i19 = i1 & (j29 + 1) = j1 or i19 = (i1 + 1) & j1 = j29 or i19 = (i1 + 1) & (j29 + 1) = j1 by A655, A658, A660, A662, A666, A664, A667, A640, A642, A644, A646, A649, A651, A650, A670, GOBOARD7: 21;

                hence contradiction by A579, A613, A631, A636, A633, A637, A638, A669, A656, A658, A660, A639, A642, A644, A683;

              end;

                suppose

                 A684: i1 = (i2 + 1) & j1 = j2 & i19 = i29 & (j19 + 1) = j29;

                then i19 = i2 & j19 = j1 or i19 = i2 & (j19 + 1) = j1 or i19 = (i2 + 1) & j19 = j1 or i19 = (i2 + 1) & (j19 + 1) = j1 by A655, A658, A660, A668, A666, A664, A671, A640, A642, A644, A646, A649, A653, A647, A670, GOBOARD7: 21;

                hence contradiction by A579, A613, A631, A636, A633, A637, A638, A669, A656, A658, A660, A639, A642, A644, A684;

              end;

                suppose

                 A685: i1 = (i2 + 1) & j1 = j2 & (i19 + 1) = i29 & j19 = j29;

                then

                 A686: i2 = i19 or i2 = (i19 + 1) or (i2 + 1) = i19 by A655, A658, A660, A668, A666, A664, A671, A640, A642, A644, A646, A653, A651, A652, A670, GOBOARD7: 23;

                j1 = j19 by A655, A658, A660, A668, A666, A664, A671, A640, A642, A644, A646, A653, A651, A652, A670, A685, GOBOARD7: 20;

                hence contradiction by A579, A613, A631, A636, A633, A638, A669, A656, A658, A660, A639, A642, A644, A685, A686;

              end;

                suppose

                 A687: i1 = (i2 + 1) & j1 = j2 & i19 = (i29 + 1) & j19 = j29;

                then

                 A688: i2 = i29 or i2 = (i29 + 1) or (i2 + 1) = i29 by A655, A658, A660, A668, A666, A664, A671, A640, A642, A644, A649, A653, A651, A648, A670, GOBOARD7: 23;

                j1 = j19 by A655, A658, A660, A668, A666, A664, A671, A640, A642, A644, A649, A653, A651, A648, A670, A687, GOBOARD7: 20;

                hence contradiction by A613, A631, A636, A637, A638, A669, A658, A660, A639, A642, A644, A687, A688;

              end;

                suppose

                 A689: i1 = (i2 + 1) & j1 = j2 & i19 = i29 & j19 = (j29 + 1);

                then i19 = i2 & j29 = j1 or i19 = i2 & (j29 + 1) = j1 or i19 = (i2 + 1) & j29 = j1 or i19 = (i2 + 1) & (j29 + 1) = j1 by A655, A658, A660, A668, A666, A664, A671, A640, A642, A644, A646, A649, A651, A650, A670, GOBOARD7: 21;

                hence contradiction by A579, A613, A631, A636, A633, A637, A638, A669, A656, A658, A660, A639, A642, A644, A689;

              end;

                suppose

                 A690: i1 = i2 & j1 = (j2 + 1) & i19 = i29 & (j19 + 1) = j29;

                then

                 A691: j2 = j19 or j2 = (j19 + 1) or (j2 + 1) = j19 by A655, A658, A660, A662, A668, A664, A665, A640, A642, A644, A646, A649, A653, A647, A670, GOBOARD7: 22;

                i1 = i19 by A655, A658, A660, A662, A668, A664, A665, A640, A642, A644, A646, A649, A653, A647, A670, A690, GOBOARD7: 19;

                hence contradiction by A579, A613, A631, A636, A633, A638, A669, A656, A658, A660, A639, A642, A644, A690, A691;

              end;

                suppose

                 A692: i1 = i2 & j1 = (j2 + 1) & (i19 + 1) = i29 & j19 = j29;

                then i1 = i19 & j2 = j19 or i1 = i19 & (j2 + 1) = j19 or i1 = (i19 + 1) & j2 = j19 or i1 = (i19 + 1) & (j2 + 1) = j19 by A655, A658, A660, A662, A668, A664, A665, A640, A642, A644, A646, A653, A651, A652, A670, GOBOARD7: 21;

                hence contradiction by A579, A613, A631, A636, A633, A637, A638, A669, A656, A658, A660, A639, A642, A644, A692;

              end;

                suppose

                 A693: i1 = i2 & j1 = (j2 + 1) & i19 = (i29 + 1) & j19 = j29;

                then i1 = i29 & j2 = j19 or i1 = i29 & (j2 + 1) = j19 or i1 = (i29 + 1) & j2 = j19 or i1 = (i29 + 1) & (j2 + 1) = j19 by A655, A658, A660, A662, A668, A664, A665, A640, A642, A644, A649, A653, A651, A648, A670, GOBOARD7: 21;

                hence contradiction by A579, A613, A631, A636, A633, A637, A638, A669, A656, A658, A660, A639, A642, A644, A693;

              end;

                suppose

                 A694: i1 = i2 & j1 = (j2 + 1) & i19 = i29 & j19 = (j29 + 1);

                then

                 A695: j2 = j29 or j2 = (j29 + 1) or (j2 + 1) = j29 by A655, A658, A660, A662, A668, A664, A665, A640, A642, A644, A646, A649, A651, A650, A670, GOBOARD7: 22;

                i1 = i19 by A655, A658, A660, A662, A668, A664, A665, A640, A642, A644, A646, A649, A651, A650, A670, A694, GOBOARD7: 19;

                hence contradiction by A613, A631, A636, A637, A638, A669, A658, A660, A639, A642, A644, A694, A695;

              end;

            end;

            hence contradiction;

          end;

            suppose i = 0 & (j + 1) < ( len g);

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

            hence thesis by XBOOLE_1: 65;

          end;

            suppose

             A696: 1 <= i & (j + 1) < ( len g);

            then

             A697: (i + 1) < ( len g) by A636, XXREAL_0: 2;

            then

             A698: ( LSeg (g,i)) = ( LSeg ((g /. i),(g /. (i + 1)))) by A696, TOPREAL1:def 3;

            

             A699: i < ( len g) by A697, NAT_1: 13;

            consider i1, j1, i2, j2 such that

             A700: [i1, j1] in ( Indices G) and

             A701: (g /. i) = (G * (i1,j1)) and

             A702: [i2, j2] in ( Indices G) and

             A703: (g /. (i + 1)) = (G * (i2,j2)) and

             A704: i1 = i2 & (j1 + 1) = j2 or (i1 + 1) = i2 & j1 = j2 or i1 = (i2 + 1) & j1 = j2 or i1 = i2 & j1 = (j2 + 1) by A623, A696, A697, JORDAN8: 3;

            

             A705: 1 <= i1 by A700, MATRIX_0: 32;

            

             A706: j2 <= ( width G) by A702, MATRIX_0: 32;

            

             A707: j1 <= ( width G) by A700, MATRIX_0: 32;

            

             A708: 1 <= j2 by A702, MATRIX_0: 32;

            

             A709: 1 <= j1 by A700, MATRIX_0: 32;

            

             A710: i2 <= ( len G) by A702, MATRIX_0: 32;

            

             A711: i1 <= ( len G) by A700, MATRIX_0: 32;

            

             A712: 1 < (i + 1) by A696, NAT_1: 13;

            assume

             A713: ( LSeg (g,i)) meets ( LSeg (g,j));

            consider i19,j19,i29,j29 be Nat such that

             A714: [i19, j19] in ( Indices G) and

             A715: (g /. j) = (G * (i19,j19)) and

             A716: [i29, j29] in ( Indices G) and

             A717: (g /. (j + 1)) = (G * (i29,j29)) and

             A718: i19 = i29 & (j19 + 1) = j29 or (i19 + 1) = i29 & j19 = j29 or i19 = (i29 + 1) & j19 = j29 or i19 = i29 & j19 = (j29 + 1) by A623, A633, A696, JORDAN8: 3;

            

             A719: 1 <= i19 by A714, MATRIX_0: 32;

            

             A720: j29 <= ( width G) by A716, MATRIX_0: 32;

            

             A721: 1 <= i29 by A716, MATRIX_0: 32;

            

             A722: i19 <= ( len G) by A714, MATRIX_0: 32;

            

             A723: 1 <= j29 by A716, MATRIX_0: 32;

            

             A724: j19 <= ( width G) by A714, MATRIX_0: 32;

            

             A725: i29 <= ( len G) by A716, MATRIX_0: 32;

            

             A726: 1 <= j19 by A714, MATRIX_0: 32;

            

             A727: j < ( len g) by A696, NAT_1: 12;

            

             A728: ( LSeg (g,j)) = ( LSeg ((g /. j),(g /. (j + 1)))) by A633, A696, TOPREAL1:def 3;

            

             A729: 1 <= i2 by A702, MATRIX_0: 32;

            now

              per cases by A704, A718;

                suppose

                 A730: i1 = i2 & (j1 + 1) = j2 & i19 = i29 & (j19 + 1) = j29;

                then

                 A731: j1 = j19 or j1 = (j19 + 1) or (j1 + 1) = j19 by A698, A701, A703, A705, A711, A709, A706, A728, A715, A717, A719, A722, A726, A720, A713, GOBOARD7: 22;

                i1 = i19 by A698, A701, A703, A705, A711, A709, A706, A728, A715, A717, A719, A722, A726, A720, A713, A730, GOBOARD7: 19;

                hence contradiction by A628, A631, A636, A634, A637, A696, A727, A701, A703, A715, A717, A730, A731;

              end;

                suppose

                 A732: i1 = i2 & (j1 + 1) = j2 & (i19 + 1) = i29 & j19 = j29;

                then i1 = i19 & j1 = j19 or i1 = i19 & (j1 + 1) = j19 or i1 = (i19 + 1) & j1 = j19 or i1 = (i19 + 1) & (j1 + 1) = j19 by A698, A701, A703, A705, A711, A709, A706, A728, A715, A717, A719, A726, A724, A725, A713, GOBOARD7: 21;

                hence contradiction by A579, A628, A631, A636, A633, A637, A696, A712, A699, A727, A701, A703, A715, A717, A732;

              end;

                suppose

                 A733: i1 = i2 & (j1 + 1) = j2 & i19 = (i29 + 1) & j19 = j29;

                then i1 = i29 & j19 = j1 or i1 = i29 & (j1 + 1) = j19 or i1 = (i29 + 1) & j19 = j1 or i1 = (i29 + 1) & (j1 + 1) = j19 by A698, A701, A703, A705, A711, A709, A706, A728, A715, A717, A722, A726, A724, A721, A713, GOBOARD7: 21;

                hence contradiction by A579, A628, A631, A636, A633, A637, A696, A712, A699, A727, A701, A703, A715, A717, A733;

              end;

                suppose

                 A734: i1 = i2 & (j1 + 1) = j2 & i19 = i29 & j19 = (j29 + 1);

                then

                 A735: j1 = j29 or j1 = (j29 + 1) or (j1 + 1) = j29 by A698, A701, A703, A705, A711, A709, A706, A728, A715, A717, A719, A722, A724, A723, A713, GOBOARD7: 22;

                i1 = i19 by A698, A701, A703, A705, A711, A709, A706, A728, A715, A717, A719, A722, A724, A723, A713, A734, GOBOARD7: 19;

                hence contradiction by A579, A613, A631, A636, A633, A696, A712, A699, A727, A701, A703, A715, A717, A734, A735;

              end;

                suppose

                 A736: (i1 + 1) = i2 & j1 = j2 & i19 = i29 & (j19 + 1) = j29;

                then i19 = i1 & j1 = j19 or i19 = i1 & (j19 + 1) = j1 or i19 = (i1 + 1) & j1 = j19 or i19 = (i1 + 1) & (j19 + 1) = j1 by A698, A701, A703, A705, A709, A707, A710, A728, A715, A717, A719, A722, A726, A720, A713, GOBOARD7: 21;

                hence contradiction by A579, A628, A631, A636, A634, A633, A637, A696, A699, A727, A701, A703, A715, A717, A736;

              end;

                suppose

                 A737: (i1 + 1) = i2 & j1 = j2 & (i19 + 1) = i29 & j19 = j29;

                then

                 A738: i1 = i19 or i1 = (i19 + 1) or (i1 + 1) = i19 by A698, A701, A703, A705, A709, A707, A710, A728, A715, A717, A719, A726, A724, A725, A713, GOBOARD7: 23;

                j1 = j19 by A698, A701, A703, A705, A709, A707, A710, A728, A715, A717, A719, A726, A724, A725, A713, A737, GOBOARD7: 20;

                hence contradiction by A628, A631, A636, A634, A637, A696, A727, A701, A703, A715, A717, A737, A738;

              end;

                suppose

                 A739: (i1 + 1) = i2 & j1 = j2 & i19 = (i29 + 1) & j19 = j29;

                then

                 A740: i1 = i29 or i1 = (i29 + 1) or (i1 + 1) = i29 by A698, A701, A703, A705, A709, A707, A710, A728, A715, A717, A722, A726, A724, A721, A713, GOBOARD7: 23;

                j1 = j19 by A698, A701, A703, A705, A709, A707, A710, A728, A715, A717, A722, A726, A724, A721, A713, A739, GOBOARD7: 20;

                hence contradiction by A579, A628, A636, A633, A637, A696, A712, A699, A727, A701, A703, A715, A717, A739, A740;

              end;

                suppose

                 A741: (i1 + 1) = i2 & j1 = j2 & i19 = i29 & j19 = (j29 + 1);

                then i19 = i1 & j1 = j29 or i19 = i1 & (j29 + 1) = j1 or i19 = (i1 + 1) & j1 = j29 or i19 = (i1 + 1) & (j29 + 1) = j1 by A698, A701, A703, A705, A709, A707, A710, A728, A715, A717, A719, A722, A724, A723, A713, GOBOARD7: 21;

                hence contradiction by A579, A628, A631, A636, A633, A637, A696, A712, A699, A727, A701, A703, A715, A717, A741;

              end;

                suppose

                 A742: i1 = (i2 + 1) & j1 = j2 & i19 = i29 & (j19 + 1) = j29;

                then i19 = i2 & j19 = j1 or i19 = i2 & (j19 + 1) = j1 or i19 = (i2 + 1) & j19 = j1 or i19 = (i2 + 1) & (j19 + 1) = j1 by A698, A701, A703, A711, A709, A707, A729, A728, A715, A717, A719, A722, A726, A720, A713, GOBOARD7: 21;

                hence contradiction by A579, A628, A631, A636, A633, A637, A696, A712, A699, A727, A701, A703, A715, A717, A742;

              end;

                suppose

                 A743: i1 = (i2 + 1) & j1 = j2 & (i19 + 1) = i29 & j19 = j29;

                then

                 A744: i2 = i19 or i2 = (i19 + 1) or (i2 + 1) = i19 by A698, A701, A703, A711, A709, A707, A729, A728, A715, A717, A719, A726, A724, A725, A713, GOBOARD7: 23;

                j1 = j19 by A698, A701, A703, A711, A709, A707, A729, A728, A715, A717, A719, A726, A724, A725, A713, A743, GOBOARD7: 20;

                hence contradiction by A579, A613, A631, A636, A633, A696, A712, A699, A727, A701, A703, A715, A717, A743, A744;

              end;

                suppose

                 A745: i1 = (i2 + 1) & j1 = j2 & i19 = (i29 + 1) & j19 = j29;

                then

                 A746: i2 = i29 or i2 = (i29 + 1) or (i2 + 1) = i29 by A698, A701, A703, A711, A709, A707, A729, A728, A715, A717, A722, A726, A724, A721, A713, GOBOARD7: 23;

                j1 = j19 by A698, A701, A703, A711, A709, A707, A729, A728, A715, A717, A722, A726, A724, A721, A713, A745, GOBOARD7: 20;

                hence contradiction by A628, A631, A636, A637, A696, A712, A727, A701, A703, A715, A717, A745, A746;

              end;

                suppose

                 A747: i1 = (i2 + 1) & j1 = j2 & i19 = i29 & j19 = (j29 + 1);

                then i19 = i2 & j29 = j1 or i19 = i2 & (j29 + 1) = j1 or i19 = (i2 + 1) & j29 = j1 or i19 = (i2 + 1) & (j29 + 1) = j1 by A698, A701, A703, A711, A709, A707, A729, A728, A715, A717, A719, A722, A724, A723, A713, GOBOARD7: 21;

                hence contradiction by A579, A628, A631, A636, A633, A637, A696, A712, A699, A727, A701, A703, A715, A717, A747;

              end;

                suppose

                 A748: i1 = i2 & j1 = (j2 + 1) & i19 = i29 & (j19 + 1) = j29;

                then

                 A749: j2 = j19 or j2 = (j19 + 1) or (j2 + 1) = j19 by A698, A701, A703, A705, A711, A707, A708, A728, A715, A717, A719, A722, A726, A720, A713, GOBOARD7: 22;

                i1 = i19 by A698, A701, A703, A705, A711, A707, A708, A728, A715, A717, A719, A722, A726, A720, A713, A748, GOBOARD7: 19;

                hence contradiction by A579, A613, A631, A636, A633, A696, A712, A699, A727, A701, A703, A715, A717, A748, A749;

              end;

                suppose

                 A750: i1 = i2 & j1 = (j2 + 1) & (i19 + 1) = i29 & j19 = j29;

                then i1 = i19 & j2 = j19 or i1 = i19 & (j2 + 1) = j19 or i1 = (i19 + 1) & j2 = j19 or i1 = (i19 + 1) & (j2 + 1) = j19 by A698, A701, A703, A705, A711, A707, A708, A728, A715, A717, A719, A726, A724, A725, A713, GOBOARD7: 21;

                hence contradiction by A579, A628, A631, A636, A633, A637, A696, A712, A699, A727, A701, A703, A715, A717, A750;

              end;

                suppose

                 A751: i1 = i2 & j1 = (j2 + 1) & i19 = (i29 + 1) & j19 = j29;

                then i1 = i29 & j2 = j19 or i1 = i29 & (j2 + 1) = j19 or i1 = (i29 + 1) & j2 = j19 or i1 = (i29 + 1) & (j2 + 1) = j19 by A698, A701, A703, A705, A711, A707, A708, A728, A715, A717, A722, A726, A724, A721, A713, GOBOARD7: 21;

                hence contradiction by A579, A628, A631, A636, A633, A637, A696, A712, A699, A727, A701, A703, A715, A717, A751;

              end;

                suppose

                 A752: i1 = i2 & j1 = (j2 + 1) & i19 = i29 & j19 = (j29 + 1);

                then

                 A753: j2 = j29 or j2 = (j29 + 1) or (j2 + 1) = j29 by A698, A701, A703, A705, A711, A707, A708, A728, A715, A717, A719, A722, A724, A723, A713, GOBOARD7: 22;

                i1 = i19 by A698, A701, A703, A705, A711, A707, A708, A728, A715, A717, A719, A722, A724, A723, A713, A752, GOBOARD7: 19;

                hence contradiction by A628, A631, A636, A637, A696, A712, A727, A701, A703, A715, A717, A752, A753;

              end;

            end;

            hence contradiction;

          end;

        end;

        then

        reconsider g as standard non constant special_circular_sequence by A567, A612, A577, A623, A625, FINSEQ_6:def 1, JORDAN8: 4;

        

         A754: for i st 1 <= i & (i + 1) <= ( len f) holds ( right_cell (f,i,G)) = ( Cl ( Int ( right_cell (f,i,G))))

        proof

          let i such that

           A755: 1 <= i & (i + 1) <= ( len f);

          consider i1, j1, i2, j2 such that

           A756: [i1, j1] in ( Indices G) and

           A757: (f /. i) = (G * (i1,j1)) and

           A758: [i2, j2] in ( Indices G) and

           A759: (f /. (i + 1)) = (G * (i2,j2)) and

           A760: i1 = i2 & (j1 + 1) = j2 or (i1 + 1) = i2 & j1 = j2 or i1 = (i2 + 1) & j1 = j2 or i1 = i2 & j1 = (j2 + 1) by A569, A755, JORDAN8: 3;

          

           A761: i1 <= ( len G) by A756, MATRIX_0: 32;

          

           A762: j2 <= ( width G) by A758, MATRIX_0: 32;

          

           A763: j1 <= ( width G) by A756, MATRIX_0: 32;

          

           A764: (j1 + 1) > j1 & (j2 + 1) > j2 by NAT_1: 13;

          

           A765: i2 <= ( len G) by A758, MATRIX_0: 32;

          

           A766: (i1 + 1) > i1 & (i2 + 1) > i2 by NAT_1: 13;

          per cases by A760;

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

            then ( right_cell (f,i,G)) = ( cell (G,i1,j1)) by A569, A755, A756, A757, A758, A759, A764, GOBRD13:def 2;

            hence thesis by A761, A763, GOBRD11: 35;

          end;

            suppose

             A767: (i1 + 1) = i2 & j1 = j2;

            

             A768: (j1 -' 1) <= ( width G) by A763, NAT_D: 44;

            ( right_cell (f,i,G)) = ( cell (G,i1,(j1 -' 1))) by A569, A755, A756, A757, A758, A759, A766, A767, GOBRD13:def 2;

            hence thesis by A761, A768, GOBRD11: 35;

          end;

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

            then ( right_cell (f,i,G)) = ( cell (G,i2,j2)) by A569, A755, A756, A757, A758, A759, A766, GOBRD13:def 2;

            hence thesis by A765, A762, GOBRD11: 35;

          end;

            suppose

             A769: i1 = i2 & j1 = (j2 + 1);

            

             A770: (i1 -' 1) <= ( len G) by A761, NAT_D: 44;

            ( right_cell (f,i,G)) = ( cell (G,(i1 -' 1),j2)) by A569, A755, A756, A757, A758, A759, A764, A769, GOBRD13:def 2;

            hence thesis by A762, A770, GOBRD11: 35;

          end;

        end;

        now

          

           A771: for h be standard non constant special_circular_sequence st ( L~ h) c= ( L~ f) holds for Comp be Subset of ( TOP-REAL 2) st Comp is_a_component_of (( L~ h) ` ) holds for n st 1 <= n & (n + 1) <= ( len f) & (f /. n) in Comp & not (f /. n) in ( L~ h) holds C meets Comp

          proof

            let h be standard non constant special_circular_sequence such that

             A772: ( L~ h) c= ( L~ f);

            let Comp be Subset of ( TOP-REAL 2) such that

             A773: Comp is_a_component_of (( L~ h) ` );

            let n such that

             A774: 1 <= n & (n + 1) <= ( len f) and

             A775: (f /. n) in Comp and

             A776: not (f /. n) in ( L~ h);

            reconsider rc = (( right_cell (f,n,G)) \ ( L~ h)) as Subset of ( TOP-REAL 2);

            (f /. n) in ( right_cell (f,n,G)) by A569, A774, Th8;

            then (f /. n) in rc by A776, XBOOLE_0:def 5;

            then

             A777: rc meets Comp by A775, XBOOLE_0: 3;

            

             A778: rc meets C

            proof

              ( right_cell (f,n,G)) meets C by A427, A774;

              then

              consider p be object such that

               A779: p in ( right_cell (f,n,G)) and

               A780: p in C by XBOOLE_0: 3;

              reconsider p as Element of ( TOP-REAL 2) by A779;

              now

                take a = p;

                now

                  assume p in ( L~ h);

                  then

                  consider j such that

                   A781: 1 <= j & (j + 1) <= ( len f) and

                   A782: p in ( LSeg (f,j)) by A772, SPPOL_2: 13;

                  p in (( left_cell (f,j,G)) /\ ( right_cell (f,j,G))) by A427, A781, A782, GOBRD13: 29;

                  then

                   A783: p in ( left_cell (f,j,G)) by XBOOLE_0:def 4;

                  ( left_cell (f,j,G)) misses C by A427, A781;

                  hence contradiction by A780, A783, XBOOLE_0: 3;

                end;

                hence a in rc by A779, XBOOLE_0:def 5;

                thus a in C by A780;

              end;

              hence thesis by XBOOLE_0: 3;

            end;

            ( Int ( right_cell (f,n,G))) misses ( L~ f) by A569, A774, Th15;

            then ( Int ( right_cell (f,n,G))) misses ( L~ h) by A772, XBOOLE_1: 63;

            then

             A784: ( Int ( right_cell (f,n,G))) c= (( L~ h) ` ) by SUBSET_1: 23;

            

             A785: rc = (( right_cell (f,n,G)) /\ (( L~ h) ` )) by SUBSET_1: 13;

            then

             A786: rc c= (( L~ h) ` ) by XBOOLE_1: 17;

            rc c= ( right_cell (f,n,G)) by XBOOLE_1: 36;

            then

             A787: ( Int ( right_cell (f,n,G))) c= ( right_cell (f,n,G)) & rc c= ( Cl ( Int ( right_cell (f,n,G)))) by A569, A774, Th11, TOPS_1: 16;

            ( Int ( right_cell (f,n,G))) is convex by A569, A774, Th10;

            then rc is connected by A785, A784, A787, CONNSP_1: 18, XBOOLE_1: 19;

            then rc c= Comp by A773, A777, A786, GOBOARD9: 4;

            hence thesis by A778, XBOOLE_1: 63;

          end;

          (( L~ g) ` ) is open by TOPS_1: 3;

          then

           A788: (( L~ g) ` ) = ( Int (( L~ g) ` )) by TOPS_1: 23;

          

           A789: ( L~ g) c= ( L~ f) by JORDAN3: 40;

          

           A790: for j, k st 1 <= j & j <= k holds ((F . k) /. j) = ((F . j) /. j)

          proof

            let j, k;

            assume that

             A791: 1 <= j and

             A792: j <= k;

            j <= ( len (F . k)) by A192, A792;

            then ( len ((F . k) | j)) = j by FINSEQ_1: 59;

            then

             A793: j in ( dom ((F . k) | j)) by A791, FINSEQ_3: 25;

            ((F . k) | j) = (F . j) by A488, A792;

            hence thesis by A793, FINSEQ_4: 70;

          end;

          assume m <> 1;

          then

           A794: 1 < m by A568, XXREAL_0: 1;

          

           A795: for n st 1 <= n & n <= (m -' 1) holds not (f /. n) in ( L~ g)

          proof

            

             A796: 2 <= ( len G) by A231, NAT_1: 12;

            let n such that

             A797: 1 <= n and

             A798: n <= (m -' 1);

            set p = (f /. n);

            

             A799: n <= ( len f) by A575, A798, XXREAL_0: 2;

            then

             A800: p in ( Values G) by A427, A797, Th6;

            assume p in ( L~ g);

            then

            consider j such that

             A801: ((m -' 1) + 1) <= j and

             A802: (j + 1) <= ( len f) and

             A803: p in ( LSeg (f,j)) by A575, Th7;

            

             A804: j < k by A578, A802, NAT_1: 13;

            

             A805: n < ((m -' 1) + 1) by A798, NAT_1: 13;

            then

             A806: n < j by A801, XXREAL_0: 2;

            

             A807: ((m -' 1) + 1) = m by A568, XREAL_1: 235;

            then

             A808: 1 < j by A794, A801, XXREAL_0: 2;

            per cases by A2, A427, A802, A803, A808, A796, A800, Th23;

              suppose

               A809: p = (f /. j);

              

               A810: n <> ( len (F . j)) by A192, A801, A805;

              n <= ( len (F . j)) by A192, A806;

              then

               A811: n in ( dom (F . j)) by A797, FINSEQ_3: 25;

              ((F . j) /. n) = ((F . n) /. n) by A790, A797, A806

              .= p by A578, A790, A797, A799

              .= ((F . j) /. j) by A790, A808, A804, A809

              .= ((F . j) /. ( len (F . j))) by A192;

              hence contradiction by A564, A808, A804, A811, A810;

            end;

              suppose

               A812: p = (f /. (j + 1));

              now

                per cases by A578, A802, XXREAL_0: 1;

                  suppose

                   A813: (j + 1) = k;

                  

                   A814: n <> ( len (F . m)) by A192, A805, A807;

                  n <= ( len (F . m)) by A192, A805, A807;

                  then

                   A815: n in ( dom (F . m)) by A797, FINSEQ_3: 25;

                  ((F . m) /. n) = ((F . n) /. n) by A790, A797, A805, A807

                  .= ((F . k) /. k) by A578, A790, A797, A799, A812, A813

                  .= ((F . m) /. m) by A567, A578, A568, A570, A790

                  .= ((F . m) /. ( len (F . m))) by A192;

                  hence contradiction by A564, A578, A568, A571, A815, A814;

                end;

                  suppose

                   A816: (j + 1) < k;

                  set l = (j + 1);

                  

                   A817: 1 <= l by NAT_1: 11;

                  

                   A818: n < (n + 1) & (n + 1) < l by A806, XREAL_1: 6, XREAL_1: 29;

                  then

                   A819: n <> ( len (F . l)) by A192;

                  

                   A820: n < l by A818, XXREAL_0: 2;

                  then n <= ( len (F . l)) by A192;

                  then

                   A821: n in ( dom (F . l)) by A797, FINSEQ_3: 25;

                  ((F . l) /. n) = ((F . n) /. n) by A790, A797, A820

                  .= p by A578, A790, A797, A799

                  .= ((F . l) /. l) by A790, A812, A816, A817

                  .= ((F . l) /. ( len (F . l))) by A192;

                  hence contradiction by A564, A816, A821, A819, NAT_1: 11;

                end;

              end;

              hence contradiction;

            end;

          end;

          C meets ( RightComp ( Rev g))

          proof

            1 <= ( len g) by A622, XREAL_1: 145;

            then

             A822: ((( len g) -' 1) + 2) = (( len g) + 1) by Lm1;

            

             A823: (1 - 1) < (m - 1) by A794, XREAL_1: 9;

            

             A824: ((m -' 1) + 2) = (m + 1) by A568, Lm1;

            set l = ((m -' 1) + (( len g) -' 1));

            set a = (f /. (m -' 1));

            set rg = ( Rev g);

            set p = (rg /. 1), q = (rg /. 2);

            

             A825: ((1 + 1) - 1) <= (( len g) - 1) by A621, XREAL_1: 9;

            ((1 + 1) -' 1) <= (( len g) -' 1) by A621, NAT_D: 42;

            then

             A826: 1 <= (( len g) -' 1) by NAT_D: 34;

            then ((m -' 1) + 1) <= l by XREAL_1: 6;

            then (m -' 1) < l by NAT_1: 13;

            then

             A827: (m -' 1) <> ( len (F . l)) by A192;

            

             A828: (1 + 1) <= ( len rg) by A621, FINSEQ_5:def 3;

            then ((1 + 1) -' 1) <= (( len rg) -' 1) by NAT_D: 42;

            then

             A829: 1 <= (( len rg) -' 1) by NAT_D: 34;

            1 < ( len rg) by A828, NAT_1: 13;

            then

             A830: ((( len rg) -' 1) + 1) = ( len rg) by XREAL_1: 235;

            

             A831: rg is_sequence_on G by A623, Th5;

            then

            consider p1,p2,q1,q2 be Nat such that

             A832: [p1, p2] in ( Indices G) and

             A833: p = (G * (p1,p2)) and

             A834: [q1, q2] in ( Indices G) and

             A835: q = (G * (q1,q2)) and

             A836: p1 = q1 & (p2 + 1) = q2 or (p1 + 1) = q1 & p2 = q2 or p1 = (q1 + 1) & p2 = q2 or p1 = q1 & p2 = (q2 + 1) by A828, JORDAN8: 3;

            

             A837: 1 <= p1 by A832, MATRIX_0: 32;

            

             A838: p2 <= ( width G) by A832, MATRIX_0: 32;

            

             A839: p1 <= ( len G) by A832, MATRIX_0: 32;

            

             A840: 1 <= p2 by A832, MATRIX_0: 32;

            

             A841: p = (f /. m) by A567, A577, FINSEQ_5: 65;

            (( len g) -' 1) <= ( len g) by NAT_D: 44;

            then

             A842: (( len g) -' 1) in ( dom g) by A826, FINSEQ_3: 25;

            

            then

             A843: q = (g /. (( len g) -' 1)) by A822, FINSEQ_5: 66

            .= (f /. l) by A842, FINSEQ_5: 27;

            l = ((m + (( len g) -' 1)) -' 1) by A568, NAT_D: 38

            .= (((( len g) -' 1) + m) - 1) by A826, NAT_D: 37

            .= (((( len g) - 1) + m) - 1) by A825, XREAL_0:def 2

            .= ((((k - (m - 1)) - 1) + m) - 1) by A578, A576, A823, XREAL_0:def 2

            .= (k - 1);

            then

             A844: k = (l + 1);

            then

             A845: l < k by XREAL_1: 29;

            (( len g) -' 1) <= l by NAT_1: 11;

            then

             A846: 1 <= l by A826, XXREAL_0: 2;

            then

             A847: ( right_cell (f,l,G)) meets C by A427, A578, A844;

            

             A848: ((m -' 1) + 1) = m by A568, XREAL_1: 235;

            then

             A849: 1 <= (m -' 1) by A794, NAT_1: 13;

            then

             A850: ( right_cell (f,(m -' 1),G)) meets C by A427, A570, A848;

            (m -' 1) <= l by NAT_1: 11;

            then (m -' 1) <= ( len (F . l)) by A192;

            then

             A851: (m -' 1) in ( dom (F . l)) by A849, FINSEQ_3: 25;

             not a in ( L~ g) by A795, A849;

            then

             A852: not a in ( L~ rg) by SPPOL_2: 22;

            per cases by A836;

              suppose

               A853: p1 = q1 & (p2 + 1) = q2;

              consider a1,a2,p91,p92 be Nat such that

               A854: [a1, a2] in ( Indices G) and

               A855: a = (G * (a1,a2)) and

               A856: [p91, p92] in ( Indices G) & p = (G * (p91,p92)) and

               A857: a1 = p91 & (a2 + 1) = p92 or (a1 + 1) = p91 & a2 = p92 or a1 = (p91 + 1) & a2 = p92 or a1 = p91 & a2 = (p92 + 1) by A569, A570, A841, A848, A849, JORDAN8: 3;

              

               A858: 1 <= a2 by A854, MATRIX_0: 32;

              thus thesis

              proof

                per cases by A857;

                  suppose

                   A859: a1 = p91 & (a2 + 1) = p92;

                  

                   A860: ((m -' 1) + 1) <= ( len (F . m)) & (f /. ((m -' 1) + 1)) = ((F . m) /. m) by A192, A578, A568, A570, A790, A848;

                  

                   A861: ((F . k) | (m + 1)) = (F . (m + 1)) by A488, A578, A574;

                  

                   A862: a1 = p1 by A832, A833, A856, A859, GOBOARD1: 5;

                  

                   A863: (m -' 1) <= m by A848, NAT_1: 11;

                  

                   A864: (f /. (m -' 1)) = ((F . (m -' 1)) /. (m -' 1)) by A578, A575, A790, A849

                  .= ((F . m) /. (m -' 1)) by A790, A849, A863;

                  

                   A865: 2 in ( dom g) by A621, FINSEQ_3: 25;

                  

                   A866: (a2 + 1) = p2 by A832, A833, A856, A859, GOBOARD1: 5;

                  then

                   A867: (p2 -' 1) = a2 by NAT_D: 34;

                  ( right_cell (f,l,G)) = ( cell (G,(p1 -' 1),p2)) by A427, A567, A578, A841, A844, A846, A843, A832, A833, A834, A835, A853, GOBRD13: 28

                  .= ( front_left_cell ((F . m),(m -' 1),G)) by A427, A841, A848, A849, A832, A833, A854, A855, A862, A866, A864, A860, GOBRD13: 34;

                  then (F . (m + 1)) turns_left ((m -' 1),G) by A456, A794, A847;

                  then

                   A868: (a2 + 1) > a2 & f turns_left ((m -' 1),G) by A849, A824, A861, GOBRD13: 44, NAT_1: 13;

                  ((( len rg) -' 1) + 2) = (( len g) + 1) by A822, FINSEQ_5:def 3;

                  

                  then

                   A869: (rg /. (( len rg) -' 1)) = (g /. 2) by A865, FINSEQ_5: 66

                  .= (f /. (m + 1)) by A824, A865, FINSEQ_5: 27;

                  

                   A870: p = (g /. 1) by A567, A612, A577, FINSEQ_5: 65

                  .= (rg /. ( len g)) by FINSEQ_5: 65

                  .= (rg /. ( len rg)) by FINSEQ_5:def 3;

                  set rc = (( right_cell (rg,(( len rg) -' 1),G)) \ ( L~ rg));

                  

                   A871: ( RightComp rg) is_a_component_of (( L~ rg) ` ) by GOBOARD9:def 2;

                  

                   A872: ((p1 -' 1) + 1) = p1 by A837, XREAL_1: 235;

                  

                   A873: (p2 + 1) > (a2 + 1) by A866, NAT_1: 13;

                  then

                   A874: [(p1 -' 1), p2] in ( Indices G) by A841, A848, A832, A833, A854, A855, A868;

                  then

                   A875: 1 <= (p1 -' 1) by MATRIX_0: 32;

                  (f /. (m + 1)) = (G * ((p1 -' 1),p2)) by A841, A848, A832, A833, A854, A855, A873, A868;

                  then ( right_cell (rg,(( len rg) -' 1),G)) = ( cell (G,(p1 -' 1),a2)) by A831, A829, A830, A832, A833, A874, A867, A872, A869, A870, GOBRD13: 24;

                  then a in ( right_cell (rg,(( len rg) -' 1),G)) by A839, A838, A855, A858, A862, A866, A875, A872, Th20;

                  then

                   A876: a in rc by A852, XBOOLE_0:def 5;

                  rc c= ( RightComp rg) & ( L~ rg) c= ( L~ f) by A789, A831, A829, A830, Th27, SPPOL_2: 22;

                  hence thesis by A570, A771, A848, A849, A852, A876, A871;

                end;

                  suppose

                   A877: (a1 + 1) = p91 & a2 = p92;

                  then (a1 + 1) = p1 by A832, A833, A856, GOBOARD1: 5;

                  then

                   A878: (q1 -' 1) = a1 by A853, NAT_D: 34;

                  a2 = p2 by A832, A833, A856, A877, GOBOARD1: 5;

                  

                  then ( right_cell (f,l,G)) = ( cell (G,a1,a2)) by A427, A567, A578, A841, A844, A846, A843, A832, A833, A834, A835, A853, A878, GOBRD13: 28

                  .= ( left_cell (f,(m -' 1),G)) by A427, A570, A841, A848, A849, A854, A855, A856, A877, GOBRD13: 23;

                  hence thesis by A427, A570, A848, A849, A847;

                end;

                  suppose a1 = (p91 + 1) & a2 = p92;

                  then a1 = (p1 + 1) & a2 = p2 by A832, A833, A856, GOBOARD1: 5;

                  

                  then ( right_cell (f,(m -' 1),G)) = ( cell (G,p1,p2)) by A427, A570, A841, A848, A849, A832, A833, A854, A855, GOBRD13: 26

                  .= ( left_cell (f,l,G)) by A427, A567, A578, A841, A844, A846, A843, A832, A833, A834, A835, A853, GOBRD13: 27;

                  hence thesis by A427, A578, A844, A846, A850;

                end;

                  suppose a1 = p91 & a2 = (p92 + 1);

                  then

                   A879: a1 = q1 & a2 = q2 by A832, A833, A853, A856, GOBOARD1: 5;

                  ((F . l) /. (m -' 1)) = ((F . (m -' 1)) /. (m -' 1)) by A790, A849, NAT_1: 11

                  .= q by A578, A575, A790, A849, A835, A855, A879

                  .= ((F . l) /. l) by A790, A845, A846, A843

                  .= ((F . l) /. ( len (F . l))) by A192;

                  hence thesis by A564, A845, A846, A851, A827;

                end;

              end;

            end;

              suppose

               A880: (p1 + 1) = q1 & p2 = q2;

              consider a1,a2,p91,p92 be Nat such that

               A881: [a1, a2] in ( Indices G) and

               A882: a = (G * (a1,a2)) and

               A883: [p91, p92] in ( Indices G) & p = (G * (p91,p92)) and

               A884: a1 = p91 & (a2 + 1) = p92 or (a1 + 1) = p91 & a2 = p92 or a1 = (p91 + 1) & a2 = p92 or a1 = p91 & a2 = (p92 + 1) by A569, A570, A841, A848, A849, JORDAN8: 3;

              

               A885: 1 <= a1 by A881, MATRIX_0: 32;

              thus thesis

              proof

                per cases by A884;

                  suppose

                   A886: a1 = p91 & (a2 + 1) = p92;

                  then (a2 + 1) = p2 by A832, A833, A883, GOBOARD1: 5;

                  then

                   A887: (q2 -' 1) = a2 by A880, NAT_D: 34;

                  

                   A888: a1 = p1 by A832, A833, A883, A886, GOBOARD1: 5;

                  ( right_cell (f,(m -' 1),G)) = ( cell (G,a1,a2)) by A427, A570, A841, A848, A849, A881, A882, A883, A886, GOBRD13: 22

                  .= ( left_cell (f,l,G)) by A427, A567, A578, A841, A844, A846, A843, A832, A833, A834, A835, A880, A888, A887, GOBRD13: 25;

                  hence thesis by A427, A578, A844, A846, A850;

                end;

                  suppose

                   A889: (a1 + 1) = p91 & a2 = p92;

                  

                   A890: (m -' 1) <= m by A848, NAT_1: 11;

                  

                   A891: (f /. (m -' 1)) = ((F . (m -' 1)) /. (m -' 1)) by A578, A575, A790, A849

                  .= ((F . m) /. (m -' 1)) by A790, A849, A890;

                  

                   A892: 2 in ( dom g) by A621, FINSEQ_3: 25;

                  ((( len rg) -' 1) + 2) = (( len g) + 1) by A822, FINSEQ_5:def 3;

                  

                  then

                   A893: (rg /. (( len rg) -' 1)) = (g /. 2) by A892, FINSEQ_5: 66

                  .= (f /. (m + 1)) by A824, A892, FINSEQ_5: 27;

                  

                   A894: p = (g /. 1) by A567, A612, A577, FINSEQ_5: 65

                  .= (rg /. ( len g)) by FINSEQ_5: 65

                  .= (rg /. ( len rg)) by FINSEQ_5:def 3;

                  

                   A895: a2 = p2 by A832, A833, A883, A889, GOBOARD1: 5;

                  

                   A896: ((m -' 1) + 1) <= ( len (F . m)) & (f /. ((m -' 1) + 1)) = ((F . m) /. m) by A192, A578, A568, A570, A790, A848;

                  

                   A897: ((F . k) | (m + 1)) = (F . (m + 1)) by A488, A578, A574;

                  set rc = (( right_cell (rg,(( len rg) -' 1),G)) \ ( L~ rg));

                  

                   A898: a1 < (a1 + 1) & p1 < (p1 + 1) by XREAL_1: 29;

                  

                   A899: (a1 + 1) = p1 by A832, A833, A883, A889, GOBOARD1: 5;

                  then

                   A900: a1 = (p1 -' 1) by NAT_D: 34;

                  ( right_cell (f,l,G)) = ( cell (G,p1,p2)) by A427, A567, A578, A841, A844, A846, A843, A832, A833, A834, A835, A880, GOBRD13: 26

                  .= ( front_left_cell ((F . m),(m -' 1),G)) by A427, A841, A848, A849, A832, A833, A881, A882, A899, A895, A891, A896, GOBRD13: 36;

                  then (F . (m + 1)) turns_left ((m -' 1),G) by A456, A794, A847;

                  then

                   A901: f turns_left ((m -' 1),G) by A849, A824, A897, GOBRD13: 44;

                  then

                   A902: [p1, (p2 + 1)] in ( Indices G) by A841, A848, A832, A833, A881, A882, A899, A898;

                  then

                   A903: (p2 + 1) <= ( width G) by MATRIX_0: 32;

                  (f /. (m + 1)) = (G * (p1,(p2 + 1))) by A841, A848, A832, A833, A881, A882, A899, A898, A901;

                  then ( right_cell (rg,(( len rg) -' 1),G)) = ( cell (G,(p1 -' 1),a2)) by A831, A829, A830, A832, A833, A895, A902, A893, A894, GOBRD13: 28;

                  then a in ( right_cell (rg,(( len rg) -' 1),G)) by A839, A840, A882, A885, A899, A895, A903, A900, Th20;

                  then

                   A904: a in rc by A852, XBOOLE_0:def 5;

                  

                   A905: ( RightComp rg) is_a_component_of (( L~ rg) ` ) by GOBOARD9:def 2;

                  rc c= ( RightComp rg) & ( L~ rg) c= ( L~ f) by A789, A831, A829, A830, Th27, SPPOL_2: 22;

                  hence thesis by A570, A771, A848, A849, A852, A904, A905;

                end;

                  suppose a1 = (p91 + 1) & a2 = p92;

                  then

                   A906: a1 = q1 & a2 = q2 by A832, A833, A880, A883, GOBOARD1: 5;

                  ((F . l) /. (m -' 1)) = ((F . (m -' 1)) /. (m -' 1)) by A790, A849, NAT_1: 11

                  .= q by A578, A575, A790, A849, A835, A882, A906

                  .= ((F . l) /. l) by A790, A845, A846, A843

                  .= ((F . l) /. ( len (F . l))) by A192;

                  hence thesis by A564, A845, A846, A851, A827;

                end;

                  suppose a1 = p91 & a2 = (p92 + 1);

                  then

                   A907: a1 = p1 & a2 = (p2 + 1) by A832, A833, A883, GOBOARD1: 5;

                  ( right_cell (f,l,G)) = ( cell (G,p1,p2)) by A427, A567, A578, A841, A844, A846, A843, A832, A833, A834, A835, A880, GOBRD13: 26

                  .= ( left_cell (f,(m -' 1),G)) by A427, A570, A841, A848, A849, A832, A833, A881, A882, A907, GOBRD13: 27;

                  hence thesis by A427, A570, A848, A849, A847;

                end;

              end;

            end;

              suppose

               A908: p1 = (q1 + 1) & p2 = q2;

              consider a1,a2,p91,p92 be Nat such that

               A909: [a1, a2] in ( Indices G) and

               A910: a = (G * (a1,a2)) and

               A911: [p91, p92] in ( Indices G) & p = (G * (p91,p92)) and

               A912: a1 = p91 & (a2 + 1) = p92 or (a1 + 1) = p91 & a2 = p92 or a1 = (p91 + 1) & a2 = p92 or a1 = p91 & a2 = (p92 + 1) by A569, A570, A841, A848, A849, JORDAN8: 3;

              

               A913: a1 <= ( len G) by A909, MATRIX_0: 32;

              thus thesis

              proof

                per cases by A912;

                  suppose

                   A914: a1 = p91 & (a2 + 1) = p92;

                  then (a2 + 1) = p2 by A832, A833, A911, GOBOARD1: 5;

                  then

                   A915: (q2 -' 1) = a2 by A908, NAT_D: 34;

                  a1 = p1 by A832, A833, A911, A914, GOBOARD1: 5;

                  then

                   A916: q1 = (a1 -' 1) by A908, NAT_D: 34;

                  ( right_cell (f,l,G)) = ( cell (G,q1,(q2 -' 1))) by A427, A567, A578, A841, A844, A846, A843, A832, A833, A834, A835, A908, GOBRD13: 24

                  .= ( left_cell (f,(m -' 1),G)) by A427, A570, A841, A848, A849, A909, A910, A911, A914, A916, A915, GOBRD13: 21;

                  hence thesis by A427, A570, A848, A849, A847;

                end;

                  suppose (a1 + 1) = p91 & a2 = p92;

                  then

                   A917: (a1 + 1) = p1 & a2 = p2 by A832, A833, A911, GOBOARD1: 5;

                  ((F . l) /. (m -' 1)) = ((F . (m -' 1)) /. (m -' 1)) by A790, A849, NAT_1: 11

                  .= q by A578, A575, A790, A849, A835, A908, A910, A917

                  .= ((F . l) /. l) by A790, A845, A846, A843

                  .= ((F . l) /. ( len (F . l))) by A192;

                  hence thesis by A564, A845, A846, A851, A827;

                end;

                  suppose

                   A918: a1 = (p91 + 1) & a2 = p92;

                  

                   A919: p = (g /. 1) by A567, A612, A577, FINSEQ_5: 65

                  .= (rg /. ( len g)) by FINSEQ_5: 65

                  .= (rg /. ( len rg)) by FINSEQ_5:def 3;

                  

                   A920: a1 = (p1 + 1) by A832, A833, A911, A918, GOBOARD1: 5;

                  (p1 + 1) > p1 by XREAL_1: 29;

                  then

                   A921: (a1 + 1) > p1 by A920, NAT_1: 13;

                  

                   A922: (m -' 1) <= m by A848, NAT_1: 11;

                  

                   A923: (f /. (m -' 1)) = ((F . (m -' 1)) /. (m -' 1)) by A578, A575, A790, A849

                  .= ((F . m) /. (m -' 1)) by A790, A849, A922;

                  

                   A924: 2 in ( dom g) by A621, FINSEQ_3: 25;

                  ((( len rg) -' 1) + 2) = (( len g) + 1) by A822, FINSEQ_5:def 3;

                  

                  then

                   A925: (rg /. (( len rg) -' 1)) = (g /. 2) by A924, FINSEQ_5: 66

                  .= (f /. (m + 1)) by A824, A924, FINSEQ_5: 27;

                  

                   A926: ((F . k) | (m + 1)) = (F . (m + 1)) by A488, A578, A574;

                  set rc = (( right_cell (rg,(( len rg) -' 1),G)) \ ( L~ rg));

                  

                   A927: ( RightComp rg) is_a_component_of (( L~ rg) ` ) by GOBOARD9:def 2;

                  

                   A928: ((p2 -' 1) + 1) = p2 by A840, XREAL_1: 235;

                  

                   A929: (f /. ((m -' 1) + 1)) = ((F . m) /. m) by A578, A568, A570, A790, A848;

                  

                   A930: (p1 -' 1) = q1 & ((m -' 1) + 1) <= ( len (F . m)) by A192, A848, A908, NAT_D: 34;

                  

                   A931: a2 = p2 by A832, A833, A911, A918, GOBOARD1: 5;

                  ( right_cell (f,l,G)) = ( cell (G,q1,(q2 -' 1))) by A427, A567, A578, A841, A844, A846, A843, A832, A833, A834, A835, A908, GOBRD13: 24

                  .= ( front_left_cell ((F . m),(m -' 1),G)) by A427, A841, A848, A849, A832, A833, A908, A909, A910, A920, A931, A930, A923, A929, GOBRD13: 38;

                  then (F . (m + 1)) turns_left ((m -' 1),G) by A456, A794, A847;

                  then

                   A932: f turns_left ((m -' 1),G) by A849, A824, A926, GOBRD13: 44;

                  

                   A933: (a2 + 1) > p2 by A931, NAT_1: 13;

                  then

                   A934: [p1, (p2 -' 1)] in ( Indices G) by A841, A848, A832, A833, A909, A910, A931, A921, A932;

                  then

                   A935: 1 <= (p2 -' 1) by MATRIX_0: 32;

                  (f /. (m + 1)) = (G * (p1,(p2 -' 1))) by A841, A848, A832, A833, A909, A910, A931, A933, A921, A932;

                  then ( right_cell (rg,(( len rg) -' 1),G)) = ( cell (G,p1,(p2 -' 1))) by A831, A829, A830, A832, A833, A934, A928, A925, A919, GOBRD13: 22;

                  then a in ( right_cell (rg,(( len rg) -' 1),G)) by A837, A838, A910, A913, A920, A931, A935, A928, Th20;

                  then

                   A936: a in rc by A852, XBOOLE_0:def 5;

                  rc c= ( RightComp rg) & ( L~ rg) c= ( L~ f) by A789, A831, A829, A830, Th27, SPPOL_2: 22;

                  hence thesis by A570, A771, A848, A849, A852, A936, A927;

                end;

                  suppose

                   A937: a1 = p91 & a2 = (p92 + 1);

                  then a1 = p1 by A832, A833, A911, GOBOARD1: 5;

                  then

                   A938: q1 = (a1 -' 1) by A908, NAT_D: 34;

                  a2 = (p2 + 1) by A832, A833, A911, A937, GOBOARD1: 5;

                  

                  then ( right_cell (f,(m -' 1),G)) = ( cell (G,q1,q2)) by A427, A570, A841, A848, A849, A908, A909, A910, A911, A937, A938, GOBRD13: 28

                  .= ( left_cell (f,l,G)) by A427, A567, A578, A841, A844, A846, A843, A832, A833, A834, A835, A908, GOBRD13: 23;

                  hence thesis by A427, A578, A844, A846, A850;

                end;

              end;

            end;

              suppose

               A939: p1 = q1 & p2 = (q2 + 1);

              consider a1,a2,p91,p92 be Nat such that

               A940: [a1, a2] in ( Indices G) and

               A941: a = (G * (a1,a2)) and

               A942: [p91, p92] in ( Indices G) & p = (G * (p91,p92)) and

               A943: a1 = p91 & (a2 + 1) = p92 or (a1 + 1) = p91 & a2 = p92 or a1 = (p91 + 1) & a2 = p92 or a1 = p91 & a2 = (p92 + 1) by A569, A570, A841, A848, A849, JORDAN8: 3;

              

               A944: a2 <= ( width G) by A940, MATRIX_0: 32;

              thus thesis

              proof

                per cases by A943;

                  suppose a1 = p91 & (a2 + 1) = p92;

                  then

                   A945: a1 = p1 & (a2 + 1) = p2 by A832, A833, A942, GOBOARD1: 5;

                  ((F . l) /. (m -' 1)) = ((F . (m -' 1)) /. (m -' 1)) by A790, A849, NAT_1: 11

                  .= q by A578, A575, A790, A849, A835, A939, A941, A945

                  .= ((F . l) /. l) by A790, A845, A846, A843

                  .= ((F . l) /. ( len (F . l))) by A192;

                  hence thesis by A564, A845, A846, A851, A827;

                end;

                  suppose

                   A946: (a1 + 1) = p91 & a2 = p92;

                  then a2 = p2 by A832, A833, A942, GOBOARD1: 5;

                  then

                   A947: (a2 -' 1) = q2 by A939, NAT_D: 34;

                  (a1 + 1) = p1 by A832, A833, A942, A946, GOBOARD1: 5;

                  then

                   A948: a1 = (q1 -' 1) by A939, NAT_D: 34;

                  ( right_cell (f,(m -' 1),G)) = ( cell (G,a1,(a2 -' 1))) by A427, A570, A841, A848, A849, A940, A941, A942, A946, GOBRD13: 24

                  .= ( left_cell (f,l,G)) by A427, A567, A578, A841, A844, A846, A843, A832, A833, A834, A835, A939, A948, A947, GOBRD13: 21;

                  hence thesis by A427, A578, A844, A846, A850;

                end;

                  suppose

                   A949: a1 = (p91 + 1) & a2 = p92;

                  then a2 = p2 by A832, A833, A942, GOBOARD1: 5;

                  then

                   A950: (a2 -' 1) = q2 by A939, NAT_D: 34;

                  

                   A951: a1 = (p1 + 1) by A832, A833, A942, A949, GOBOARD1: 5;

                  ( right_cell (f,l,G)) = ( cell (G,q1,q2)) by A427, A567, A578, A841, A844, A846, A843, A832, A833, A834, A835, A939, GOBRD13: 22

                  .= ( left_cell (f,(m -' 1),G)) by A427, A570, A841, A848, A849, A939, A940, A941, A942, A949, A951, A950, GOBRD13: 25;

                  hence thesis by A427, A570, A848, A849, A847;

                end;

                  suppose

                   A952: a1 = p91 & a2 = (p92 + 1);

                  set rc = (( right_cell (rg,(( len rg) -' 1),G)) \ ( L~ rg));

                  

                   A953: ( RightComp rg) is_a_component_of (( L~ rg) ` ) by GOBOARD9:def 2;

                  

                   A954: 2 in ( dom g) by A621, FINSEQ_3: 25;

                  ((( len rg) -' 1) + 2) = (( len g) + 1) by A822, FINSEQ_5:def 3;

                  

                  then

                   A955: (rg /. (( len rg) -' 1)) = (g /. 2) by A954, FINSEQ_5: 66

                  .= (f /. (m + 1)) by A824, A954, FINSEQ_5: 27;

                  

                   A956: p = (g /. 1) by A567, A612, A577, FINSEQ_5: 65

                  .= (rg /. ( len g)) by FINSEQ_5: 65

                  .= (rg /. ( len rg)) by FINSEQ_5:def 3;

                  

                   A957: a1 = p1 by A832, A833, A942, A952, GOBOARD1: 5;

                  

                   A958: (m -' 1) <= m by A848, NAT_1: 11;

                  

                   A959: (f /. (m -' 1)) = ((F . (m -' 1)) /. (m -' 1)) by A578, A575, A790, A849

                  .= ((F . m) /. (m -' 1)) by A790, A849, A958;

                  

                   A960: (p2 -' 1) = q2 & ((m -' 1) + 1) <= ( len (F . m)) by A192, A848, A939, NAT_D: 34;

                  

                   A961: (f /. ((m -' 1) + 1)) = ((F . m) /. m) by A578, A568, A570, A790, A848;

                  

                   A962: ((F . k) | (m + 1)) = (F . (m + 1)) by A488, A578, A574;

                  

                   A963: a2 = (p2 + 1) by A832, A833, A942, A952, GOBOARD1: 5;

                  ( right_cell (f,l,G)) = ( cell (G,q1,q2)) by A427, A567, A578, A841, A844, A846, A843, A832, A833, A834, A835, A939, GOBRD13: 22

                  .= ( front_left_cell ((F . m),(m -' 1),G)) by A427, A841, A848, A849, A832, A833, A939, A940, A941, A957, A963, A960, A959, A961, GOBRD13: 40;

                  then (F . (m + 1)) turns_left ((m -' 1),G) by A456, A794, A847;

                  then

                   A964: (p2 + 1) > p2 & f turns_left ((m -' 1),G) by A849, A824, A962, GOBRD13: 44, NAT_1: 13;

                  

                   A965: (a2 + 1) > (p2 + 1) by A963, NAT_1: 13;

                  then

                   A966: [(p1 + 1), p2] in ( Indices G) by A841, A848, A832, A833, A940, A941, A964;

                  then

                   A967: (p1 + 1) <= ( len G) by MATRIX_0: 32;

                  (f /. (m + 1)) = (G * ((p1 + 1),p2)) by A841, A848, A832, A833, A940, A941, A965, A964;

                  then ( right_cell (rg,(( len rg) -' 1),G)) = ( cell (G,p1,p2)) by A831, A829, A830, A832, A833, A966, A955, A956, GOBRD13: 26;

                  then a in ( right_cell (rg,(( len rg) -' 1),G)) by A837, A840, A941, A944, A957, A963, A967, Th20;

                  then

                   A968: a in rc by A852, XBOOLE_0:def 5;

                  rc c= ( RightComp rg) & ( L~ rg) c= ( L~ f) by A789, A831, A829, A830, Th27, SPPOL_2: 22;

                  hence thesis by A570, A771, A848, A849, A852, A968, A953;

                end;

              end;

            end;

          end;

          then

           A969: ( LeftComp g) is_a_component_of (( L~ g) ` ) & C meets ( LeftComp g) by GOBOARD9: 24, GOBOARD9:def 1;

          reconsider Lg9 = (( L~ g) ` ) as Subset of ( TOP-REAL 2);

          

           A970: ( RightComp g) is_a_component_of (( L~ g) ` ) by GOBOARD9:def 2;

          

           A971: C c= Lg9

          proof

            let c be object;

            assume that

             A972: c in C and

             A973: not c in Lg9;

            reconsider c as Point of ( TOP-REAL 2) by A972;

            consider i such that

             A974: 1 <= i and

             A975: (i + 1) <= ( len g) and

             A976: c in ( LSeg ((g /. i),(g /. (i + 1)))) by A973, SPPOL_2: 14, SUBSET_1: 29;

            

             A977: 1 <= (i + (m -' 1)) by A974, NAT_1: 12;

            (i + 1) in ( dom g) by A974, A975, SEQ_4: 134;

            then

             A978: (g /. (i + 1)) = (f /. ((i + 1) + (m -' 1))) by FINSEQ_5: 27;

            ((i + 1) + (m -' 1)) = ((i + (m -' 1)) + 1);

            then

             A979: ((i + (m -' 1)) + 1) <= (( len g) + (m -' 1)) by A975, XREAL_1: 6;

            i in ( dom g) by A974, A975, SEQ_4: 134;

            then (g /. i) = (f /. (i + (m -' 1))) by FINSEQ_5: 27;

            then c in ( LSeg (f,(i + (m -' 1)))) by A576, A976, A978, A977, A979, TOPREAL1:def 3;

            then c in (( left_cell (f,(i + (m -' 1)),G)) /\ ( right_cell (f,(i + (m -' 1)),G))) by A427, A576, A977, A979, GOBRD13: 29;

            then c in ( left_cell (f,(i + (m -' 1)),G)) by XBOOLE_0:def 4;

            then ( left_cell (f,(i + (m -' 1)),G)) meets C by A972, XBOOLE_0: 3;

            hence contradiction by A427, A576, A977, A979;

          end;

          

           A980: the TopStruct of ( TOP-REAL 2) = ( TopSpaceMetr ( Euclid 2)) by EUCLID:def 8;

          C meets ( RightComp g)

          proof

            ( right_cell (f,m,G)) meets C by A427, A568, A574;

            then

            consider p be object such that

             A981: p in ( right_cell (f,m,G)) and

             A982: p in C by XBOOLE_0: 3;

            reconsider p as Element of ( TOP-REAL 2) by A981;

            now

              reconsider u = p as Element of ( Euclid 2) by TOPREAL3: 8;

              take a = p;

              thus a in C by A982;

              consider r be Real such that

               A983: r > 0 and

               A984: ( Ball (u,r)) c= (( L~ g) ` ) by A971, A788, A982, GOBOARD6: 5;

              reconsider r as Real;

              

               A985: p in ( Ball (u,r)) by A983, GOBOARD6: 1;

              reconsider B = ( Ball (u,r)) as non empty Subset of ( TOP-REAL 2) by A980, A983, GOBOARD6: 1, TOPMETR: 12;

              

               A986: p in B by A983, GOBOARD6: 1;

              ( right_cell (f,m,G)) = ( Cl ( Int ( right_cell (f,m,G)))) & B is open by A568, A574, A754, GOBOARD6: 3;

              then

               A987: ( Int ( right_cell (f,m,G))) meets B by A981, A985, TOPS_1: 12;

              ( Int ( right_cell (g,1,G))) c= ( Int ( right_cell (g,1))) & ( Int ( right_cell (g,1))) c= ( RightComp g) by A623, A622, GOBOARD9: 25, GOBRD13: 33, TOPS_1: 19;

              then ( Int ( right_cell (g,1,G))) c= ( RightComp g);

              then ( Int ( right_cell (f,((m -' 1) + 1),G))) c= ( RightComp g) by A569, A575, A622, GOBRD13: 32;

              then B is connected & ( Int ( right_cell (f,m,G))) c= ( RightComp g) by A568, SPRECT_3: 7, XREAL_1: 235;

              then B c= ( RightComp g) by A970, A984, A987, GOBOARD9: 4;

              hence a in ( RightComp g) by A986;

            end;

            hence thesis by XBOOLE_0: 3;

          end;

          hence contradiction by A1, A970, A971, A969, Th1, SPRECT_4: 6;

        end;

        

        then g = (f /^ 0 ) by XREAL_1: 232

        .= f by FINSEQ_5: 28;

        then

        reconsider f as standard non constant special_circular_sequence;

        f is clockwise_oriented

        proof

          (f /. 2) in ( LSeg ((f /. 1),(f /. (1 + 1)))) by RLTOPSP1: 68;

          then

           A988: (f /. 2) in ( L~ f) by A572, SPPOL_2: 15;

          (( NW-corner ( L~ f)) `1 ) = ( W-bound ( L~ f)) by EUCLID: 52;

          then

           A989: (( NW-corner ( L~ f)) `1 ) <= ((f /. 2) `1 ) by A988, PSCOMP_1: 24;

          ( len G) >= 3 by A231, NAT_1: 12;

          then

           A990: 1 < ( len G) by XXREAL_0: 2;

          (( NE-corner ( L~ f)) `1 ) = ( E-bound ( L~ f)) by EUCLID: 52;

          then

           A991: ((f /. 2) `1 ) <= (( NE-corner ( L~ f)) `1 ) by A988, PSCOMP_1: 24;

          for k st 1 <= k & (k + 1) <= ( len f) holds ( left_cell (f,k,G)) misses C & ( right_cell (f,k,G)) meets C by A427;

          then

           A992: ( N-min ( L~ f)) = (f /. 1) by A569, A606, Th30;

          consider i such that

           A993: 1 <= i and

           A994: (i + 1) <= ( len G) and

           A995: (f /. 1) = (G * (i,( width G))) & (f /. 2) = (G * ((i + 1),( width G))) and ( N-min C) in ( cell (G,i,(( width G) -' 1))) and ( N-min C) <> (G * (i,(( width G) -' 1))) by A606;

          i < ( len G) by A994, NAT_1: 13;

          then

           A996: (( N-min ( L~ f)) `2 ) = ( N-bound ( L~ f)) & ((G * (i,( width G))) `2 ) = ((G * (1,( width G))) `2 ) by A2, A990, A993, EUCLID: 52, GOBOARD5: 1;

          1 <= (i + 1) by NAT_1: 12;

          then

           A997: ((G * ((i + 1),( width G))) `2 ) = ((G * (1,( width G))) `2 ) by A2, A990, A994, GOBOARD5: 1;

          (( NW-corner ( L~ f)) `2 ) = (( NE-corner ( L~ f)) `2 ) & (( NE-corner ( L~ f)) `2 ) = ( N-bound ( L~ f)) by EUCLID: 52, PSCOMP_1: 27;

          then (f /. 2) in ( LSeg (( NW-corner ( L~ f)),( NE-corner ( L~ f)))) by A992, A995, A996, A997, A989, A991, GOBOARD7: 8;

          then (f /. 2) in (( LSeg (( NW-corner ( L~ f)),( NE-corner ( L~ f)))) /\ ( L~ f)) by A988, XBOOLE_0:def 4;

          hence thesis by A992, SPRECT_2: 30;

        end;

        then

        reconsider f as clockwise_oriented standard non constant special_circular_sequence;

        take f;

        thus f is_sequence_on G by A427;

        thus ex i st 1 <= i & (i + 1) <= ( len G) & (f /. 1) = (G * (i,( width G))) & (f /. 2) = (G * ((i + 1),( width G))) & ( N-min C) in ( cell (G,i,(( width G) -' 1))) & ( N-min C) <> (G * (i,(( width G) -' 1))) by A606;

        let m such that

         A998: 1 <= m and

         A999: (m + 2) <= ( len f);

        

         A1000: (F . ((m + 1) + 1)) = (f | ((m + 1) + 1)) by A488, A578, A999;

        

         A1001: (m + 1) < (m + 2) by XREAL_1: 6;

        then

         A1002: (f | (m + 1)) = (F . (m + 1)) by A488, A578, A999, XXREAL_0: 2;

        

         A1003: (m + 1) <= ( len f) by A999, A1001, XXREAL_0: 2;

        then

         A1004: ( front_left_cell ((F . (m + 1)),m,G)) = ( front_left_cell (f,m,G)) by A569, A998, A1002, GOBRD13: 42;

        

         A1005: m = ((m + 1) -' 1) & (m + 1) > 1 by A998, NAT_1: 13, NAT_D: 34;

        

         A1006: ( front_right_cell ((F . (m + 1)),m,G)) = ( front_right_cell (f,m,G)) by A569, A998, A1003, A1002, GOBRD13: 42;

        hereby

          assume ( front_left_cell (f,m,G)) misses C & ( front_right_cell (f,m,G)) misses C;

          then (F . ((m + 1) + 1)) turns_right (m,G) by A456, A1005, A1004, A1006;

          hence f turns_right (m,G) by A998, A1000, GOBRD13: 43;

        end;

        hereby

          assume ( front_left_cell (f,m,G)) misses C & ( front_right_cell (f,m,G)) meets C;

          then (F . ((m + 1) + 1)) goes_straight (m,G) by A456, A1005, A1004, A1006;

          hence f goes_straight (m,G) by A998, A1000, GOBRD13: 45;

        end;

        assume ( front_left_cell (f,m,G)) meets C;

        then (F . ((m + 1) + 1)) turns_left (m,G) by A456, A1005, A1004;

        hence thesis by A998, A1000, GOBRD13: 44;

      end;

      uniqueness

      proof

        let f1,f2 be clockwise_oriented standard non constant special_circular_sequence such that

         A1007: f1 is_sequence_on ( Gauge (C,n));

        defpred P[ Nat] means (f1 | $1) = (f2 | $1);

        given i1 such that

         A1008: 1 <= i1 & (i1 + 1) <= ( len ( Gauge (C,n))) and

         A1009: (f1 /. 1) = (( Gauge (C,n)) * (i1,( width ( Gauge (C,n))))) and

         A1010: (f1 /. 2) = (( Gauge (C,n)) * ((i1 + 1),( width ( Gauge (C,n))))) and

         A1011: ( N-min C) in ( cell (( Gauge (C,n)),i1,(( width ( Gauge (C,n))) -' 1))) & ( N-min C) <> (( Gauge (C,n)) * (i1,(( width ( Gauge (C,n))) -' 1)));

        assume that

         A1012: for k st 1 <= k & (k + 2) <= ( len f1) holds (( front_left_cell (f1,k,( Gauge (C,n)))) misses C & ( front_right_cell (f1,k,( Gauge (C,n)))) misses C implies f1 turns_right (k,( Gauge (C,n)))) & (( front_left_cell (f1,k,( Gauge (C,n)))) misses C & ( front_right_cell (f1,k,( Gauge (C,n)))) meets C implies f1 goes_straight (k,( Gauge (C,n)))) & (( front_left_cell (f1,k,( Gauge (C,n)))) meets C implies f1 turns_left (k,( Gauge (C,n)))) and

         A1013: f2 is_sequence_on ( Gauge (C,n));

        given i2 such that

         A1014: 1 <= i2 & (i2 + 1) <= ( len ( Gauge (C,n))) and

         A1015: (f2 /. 1) = (( Gauge (C,n)) * (i2,( width ( Gauge (C,n))))) and

         A1016: (f2 /. 2) = (( Gauge (C,n)) * ((i2 + 1),( width ( Gauge (C,n))))) and

         A1017: ( N-min C) in ( cell (( Gauge (C,n)),i2,(( width ( Gauge (C,n))) -' 1))) & ( N-min C) <> (( Gauge (C,n)) * (i2,(( width ( Gauge (C,n))) -' 1)));

        assume

         A1018: for k st 1 <= k & (k + 2) <= ( len f2) holds (( front_left_cell (f2,k,( Gauge (C,n)))) misses C & ( front_right_cell (f2,k,( Gauge (C,n)))) misses C implies f2 turns_right (k,( Gauge (C,n)))) & (( front_left_cell (f2,k,( Gauge (C,n)))) misses C & ( front_right_cell (f2,k,( Gauge (C,n)))) meets C implies f2 goes_straight (k,( Gauge (C,n)))) & (( front_left_cell (f2,k,( Gauge (C,n)))) meets C implies f2 turns_left (k,( Gauge (C,n))));

        

         A1019: for k st P[k] holds P[(k + 1)]

        proof

          

           A1020: ( len f1) > 4 by GOBOARD7: 34;

          

           A1021: (f1 | 1) = <*(f1 . 1)*> & (f2 | 1) = <*(f2 . 1)*> by FINSEQ_5: 20;

          

           A1022: i1 = i2 & ( len f2) > 4 by A1008, A1011, A1014, A1017, Th29, GOBOARD7: 34;

          1 <= ( len f1) & 1 <= ( len f2) by A1020, A1022, XXREAL_0: 2;

          then 1 in ( dom f1) & 1 in ( dom f2) by FINSEQ_3: 25;

          then

           S: (f1 . 1) = (f1 /. 1) & (f2 . 1) = (f2 /. 1) by PARTFUN1:def 6;

          let k such that

           A1023: (f1 | k) = (f2 | k);

          per cases by NAT_1: 25;

            suppose k = 0 ;

            hence thesis by S, A1008, A1009, A1011, A1014, A1015, A1017, A1021, Th29;

          end;

            suppose

             A1024: k = 1;

            (f1 | 2) = <*(f1 /. 1), (f1 /. 2)*> by A1020, FINSEQ_5: 81, XXREAL_0: 2;

            hence thesis by A1009, A1010, A1015, A1016, A1022, A1024, FINSEQ_5: 81, XXREAL_0: 2;

          end;

            suppose

             A1025: k > 1;

            

             A1026: (f1 /. 1) = (f1 /. ( len f1)) & (f2 /. 1) = (f2 /. ( len f2)) by FINSEQ_6:def 1;

            now

              per cases ;

                suppose

                 A1027: ( len f1) > k;

                set m = (k -' 1);

                

                 A1028: 1 <= m by A1025, NAT_D: 49;

                then

                 A1029: (m + 1) = k by NAT_D: 43;

                then

                 A1030: ( front_left_cell (f1,m,( Gauge (C,n)))) = ( front_left_cell ((f1 | k),m,( Gauge (C,n)))) by A1007, A1027, A1028, GOBRD13: 42;

                

                 A1031: (m + (1 + 1)) = (k + 1) by A1029;

                

                 A1032: (k + 1) <= ( len f1) by A1027, NAT_1: 13;

                 A1033:

                now

                  

                   A1034: 1 < ( len f2) by GOBOARD7: 34, XXREAL_0: 2;

                  assume

                   A1035: ( len f2) <= k;

                  then

                   A1036: f2 = (f2 | k) by FINSEQ_1: 58;

                  then ( len f2) in ( dom (f2 | k)) by FINSEQ_5: 6;

                  then

                   A1037: ((f1 | k) /. ( len f2)) = (f1 /. ( len f2)) by A1023, FINSEQ_4: 70;

                  1 in ( dom (f2 | k)) & ( len f2) <= ( len f1) by A1023, A1036, FINSEQ_5: 6, FINSEQ_5: 16;

                  hence contradiction by A1023, A1026, A1027, A1035, A1036, A1037, A1034, FINSEQ_4: 70, GOBOARD7: 38;

                end;

                then

                 A1038: (k + 1) <= ( len f2) by NAT_1: 13;

                

                 A1039: ( front_right_cell (f2,m,( Gauge (C,n)))) = ( front_right_cell ((f2 | k),m,( Gauge (C,n)))) by A1013, A1028, A1029, A1033, GOBRD13: 42;

                

                 A1040: ( front_left_cell (f2,m,( Gauge (C,n)))) = ( front_left_cell ((f2 | k),m,( Gauge (C,n)))) by A1013, A1028, A1029, A1033, GOBRD13: 42;

                

                 A1041: ( front_right_cell (f1,m,( Gauge (C,n)))) = ( front_right_cell ((f1 | k),m,( Gauge (C,n)))) by A1007, A1027, A1028, A1029, GOBRD13: 42;

                now

                  per cases ;

                    suppose ( front_left_cell (f1,m,( Gauge (C,n)))) misses C & ( front_right_cell (f1,m,( Gauge (C,n)))) misses C;

                    then f1 turns_right (m,( Gauge (C,n))) & f2 turns_right (m,( Gauge (C,n))) by A1012, A1018, A1023, A1028, A1038, A1032, A1030, A1041, A1040, A1039, A1031;

                    hence thesis by A1013, A1023, A1025, A1038, A1032, GOBRD13: 46;

                  end;

                    suppose ( front_left_cell (f1,m,( Gauge (C,n)))) misses C & ( front_right_cell (f1,m,( Gauge (C,n)))) meets C;

                    then f1 goes_straight (m,( Gauge (C,n))) & f2 goes_straight (m,( Gauge (C,n))) by A1012, A1018, A1023, A1028, A1038, A1032, A1030, A1041, A1040, A1039, A1031;

                    hence thesis by A1013, A1023, A1025, A1038, A1032, GOBRD13: 48;

                  end;

                    suppose ( front_left_cell (f1,m,( Gauge (C,n)))) meets C;

                    then f1 turns_left (m,( Gauge (C,n))) & f2 turns_left (m,( Gauge (C,n))) by A1012, A1018, A1023, A1028, A1038, A1032, A1030, A1040, A1031;

                    hence thesis by A1013, A1023, A1025, A1038, A1032, GOBRD13: 47;

                  end;

                end;

                hence thesis;

              end;

                suppose

                 A1042: k >= ( len f1);

                

                 A1043: 1 < ( len f1) by GOBOARD7: 34, XXREAL_0: 2;

                

                 A1044: f1 = (f1 | k) by A1042, FINSEQ_1: 58;

                then ( len f1) in ( dom (f1 | k)) by FINSEQ_5: 6;

                then

                 A1045: ((f2 | k) /. ( len f1)) = (f2 /. ( len f1)) by A1023, FINSEQ_4: 70;

                1 in ( dom (f1 | k)) & ( len f1) <= ( len f2) by A1023, A1044, FINSEQ_5: 6, FINSEQ_5: 16;

                then

                 A1046: ( len f2) = ( len f1) by A1023, A1026, A1044, A1045, A1043, FINSEQ_4: 70, GOBOARD7: 38;

                

                 A1047: (k + 1) > ( len f1) by A1042, NAT_1: 13;

                

                hence (f1 | (k + 1)) = f1 by FINSEQ_1: 58

                .= f2 by A1023, A1042, A1044, A1046, FINSEQ_1: 58

                .= (f2 | (k + 1)) by A1046, A1047, FINSEQ_1: 58;

              end;

            end;

            hence thesis;

          end;

        end;

        

         A1048: P[ 0 ];

        for k holds P[k] from NAT_1:sch 2( A1048, A1019);

        hence thesis by Th2;

      end;

    end

    theorem :: JORDAN9:31

    

     Th31: C is connected implies for n,k be Nat st 1 <= k & (k + 1) <= ( len ( Cage (C,n))) holds ( left_cell (( Cage (C,n)),k,( Gauge (C,n)))) misses C & ( right_cell (( Cage (C,n)),k,( Gauge (C,n)))) meets C

    proof

      assume

       A1: C is connected;

      let n be Nat;

      set G = ( Gauge (C,n)), f = ( Cage (C,n));

      set W = ( W-bound C), E = ( E-bound C), S = ( S-bound C), N = ( N-bound C);

      

       A2: f is_sequence_on G by A1, Def1;

      defpred P[ Nat] means for m be Nat st 1 <= m & (m + 1) <= ( len (f | $1)) holds ( left_cell ((f | $1),m,G)) misses C & ( right_cell ((f | $1),m,G)) meets C;

      

       A3: ( len G) = ( width G) by JORDAN8:def 1;

      

       A4: ( len G) = ((2 |^ n) + 3) by JORDAN8:def 1;

      

       A5: for k be Nat st P[k] holds P[(k + 1)]

      proof

        let k be Nat such that

         A6: for m be Nat st 1 <= m & (m + 1) <= ( len (f | k)) holds ( left_cell ((f | k),m,G)) misses C & ( right_cell ((f | k),m,G)) meets C;

        per cases ;

          suppose k >= ( len f);

          then (f | k) = f & (f | (k + 1)) = f by FINSEQ_1: 58, NAT_1: 12;

          hence thesis by A6;

        end;

          suppose

           A7: k < ( len f);

          then

           A8: ( len (f | k)) = k by FINSEQ_1: 59;

          

           A9: 1 <= ( len G) by A4, NAT_1: 12;

          

           A10: (f | k) is_sequence_on G by A2, GOBOARD1: 22;

          

           A11: (f | (k + 1)) is_sequence_on G by A2, GOBOARD1: 22;

          consider i be Nat such that

           A12: 1 <= i and

           A13: (i + 1) <= ( len G) and

           A14: (f /. 1) = (G * (i,( width G))) & (f /. 2) = (G * ((i + 1),( width G))) and

           A15: ( N-min C) in ( cell (G,i,(( width G) -' 1))) and ( N-min C) <> (G * (i,(( width G) -' 1))) by A1, Def1;

          let m be Nat such that

           A16: 1 <= m and

           A17: (m + 1) <= ( len (f | (k + 1)));

          

           A18: (k + 1) <= ( len f) by A7, NAT_1: 13;

          then

           A19: ( len (f | (k + 1))) = (k + 1) by FINSEQ_1: 59;

          

           A20: (2 |^ n) >= (n + 1) by NEWTON: 85;

          now

            per cases by NAT_1: 25;

              suppose

               A21: k = 0 ;

              1 <= (m + 1) by NAT_1: 12;

              then (m + 1) = ( 0 + 1) by A17, A21, XXREAL_0: 1;

              hence thesis by A16;

            end;

              suppose

               A22: k = 1;

              (1 + 1) <= (m + 1) by A16, XREAL_1: 6;

              then

               A23: (m + 1) = (1 + 1) by A19, A17, A22, XXREAL_0: 1;

              (f | (k + 1)) = <*(G * (i,( width G))), (G * ((i + 1),( width G)))*> by A18, A14, A22, FINSEQ_5: 81;

              then

               A24: ((f | (k + 1)) /. 1) = (G * (i,( width G))) & ((f | (k + 1)) /. 2) = (G * ((i + 1),( width G))) by FINSEQ_4: 17;

              1 <= (i + 1) by A12, NAT_1: 13;

              then

               A25: [(i + 1), ( len G)] in ( Indices G) by A3, A13, A9, MATRIX_0: 30;

              

               A26: i < ( len G) by A13, NAT_1: 13;

              then

               A27: [i, ( len G)] in ( Indices G) by A3, A12, A9, MATRIX_0: 30;

              

               A28: i < (i + 1) & (i + 1) < ((i + 1) + 1) by NAT_1: 13;

              then

               A29: ( left_cell ((f | (k + 1)),m,G)) = ( cell (G,i,( len G))) by A3, A11, A17, A24, A27, A25, A23, GOBRD13:def 3;

              now

                N > S by JORDAN8: 9;

                then (N - S) > (S - S) by XREAL_1: 9;

                then ((N - S) / (2 |^ n)) > 0 by A20, XREAL_1: 139;

                then

                 A30: (N + 0 ) < (N + ((N - S) / (2 |^ n))) by XREAL_1: 6;

                assume ( left_cell ((f | (k + 1)),m,G)) meets C;

                then

                consider p be object such that

                 A31: p in ( cell (G,i,( len G))) and

                 A32: p in C by A29, XBOOLE_0: 3;

                reconsider p as Element of ( TOP-REAL 2) by A31;

                

                 A33: (p `2 ) <= N by A32, PSCOMP_1: 24;

                 [1, ( len G)] in ( Indices G) by A3, A9, MATRIX_0: 30;

                then (G * (1,( len G))) = |[(W + (((E - W) / (2 |^ n)) * (1 - 2))), (S + (((N - S) / (2 |^ n)) * (( len G) - 2)))]| by JORDAN8:def 1;

                then

                 A34: ((G * (1,( len G))) `2 ) = (S + (((N - S) / (2 |^ n)) * (( len G) - 2))) by EUCLID: 52;

                ( cell (G,i,( len G))) = { |[r, s]| : ((G * (i,1)) `1 ) <= r & r <= ((G * ((i + 1),1)) `1 ) & ((G * (1,( len G))) `2 ) <= s } by A3, A12, A26, GOBRD11: 31;

                then

                consider r, s such that

                 A35: p = |[r, s]| and ((G * (i,1)) `1 ) <= r and r <= ((G * ((i + 1),1)) `1 ) and

                 A36: ((G * (1,( len G))) `2 ) <= s by A31;

                (((N - S) / (2 |^ n)) * (( len G) - 2)) = ((((N - S) / (2 |^ n)) * (2 |^ n)) + (((N - S) / (2 |^ n)) * 1)) by A4

                .= ((N - S) + ((N - S) / (2 |^ n))) by A20, XCMPLX_1: 87;

                then N < s by A36, A34, A30, XXREAL_0: 2;

                hence contradiction by A35, A33, EUCLID: 52;

              end;

              hence ( left_cell ((f | (k + 1)),m,G)) misses C;

              ( N-min C) in C & ( N-min C) in ( right_cell ((f | (k + 1)),m,G)) by A3, A11, A15, A17, A24, A27, A25, A23, A28, GOBRD13:def 2, SPRECT_1: 11;

              hence ( right_cell ((f | (k + 1)),m,G)) meets C by XBOOLE_0: 3;

            end;

              suppose

               A37: k > 1;

              then

               A38: ((( len (f | k)) -' 1) + 1) = ( len (f | k)) by A8, XREAL_1: 235;

              

               A39: 1 <= (( len (f | k)) -' 1) by A8, A37, NAT_D: 49;

              now

                per cases ;

                  suppose

                   A40: (m + 1) = ( len (f | (k + 1)));

                  

                   A41: ( len (f | k)) <= ( len f) by FINSEQ_5: 16;

                  now

                    ( left_cell ((f | k),(( len (f | k)) -' 1),G)) misses C by A6, A39, A38;

                    then

                     A42: ( left_cell (f,(( len (f | k)) -' 1),G)) misses C by A2, A8, A39, A38, A41, GOBRD13: 31;

                    

                     A43: ((( len (f | k)) -' 1) + (1 + 1)) = (( len (f | k)) + 1) by A38;

                    ( right_cell ((f | k),(( len (f | k)) -' 1),G)) meets C by A6, A39, A38;

                    then

                     A44: ( right_cell (f,(( len (f | k)) -' 1),G)) meets C by A2, A8, A39, A38, A41, GOBRD13: 31;

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

                     A45: [i1, j1] in ( Indices G) & (f /. (( len (f | k)) -' 1)) = (G * (i1,j1)) and

                     A46: [i2, j2] in ( Indices G) and

                     A47: (f /. ( len (f | k))) = (G * (i2,j2)) and i1 = i2 & (j1 + 1) = j2 or (i1 + 1) = i2 & j1 = j2 or i1 = (i2 + 1) & j1 = j2 or i1 = i2 & j1 = (j2 + 1) by A2, A7, A8, A39, A38, JORDAN8: 3;

                    1 <= i2 by A46, MATRIX_0: 32;

                    then

                     A48: ((i2 -' 1) + 1) = i2 by XREAL_1: 235;

                    1 <= j2 by A46, MATRIX_0: 32;

                    then

                     A49: ((j2 -' 1) + 1) = j2 by XREAL_1: 235;

                    per cases ;

                      suppose

                       A50: ( front_left_cell (f,(( len (f | k)) -' 1),G)) misses C & ( front_right_cell (f,(( len (f | k)) -' 1),G)) misses C;

                      then

                       A51: f turns_right ((( len (f | k)) -' 1),G) by A1, A18, A8, A39, A43, Def1;

                      now

                        per cases by A38, A45, A46, A47, A51;

                          suppose that

                           A52: i1 = i2 & (j1 + 1) = j2 and

                           A53: [(i2 + 1), j2] in ( Indices G) & (f /. (( len (f | k)) + 1)) = (G * ((i2 + 1),j2));

                          ( front_right_cell (f,(( len (f | k)) -' 1),G)) = ( cell (G,i1,j2)) by A2, A39, A38, A41, A45, A46, A47, A52, GOBRD13: 35;

                          then ( left_cell (f,m,G)) misses C by A2, A18, A8, A19, A16, A40, A46, A47, A50, A52, A53, GOBRD13: 23;

                          hence ( left_cell ((f | (k + 1)),m,G)) misses C by A2, A18, A19, A16, A40, GOBRD13: 31;

                          (j2 -' 1) = j1 & ( cell (G,i1,j1)) meets C by A2, A39, A38, A41, A45, A46, A47, A44, A52, GOBRD13: 22, NAT_D: 34;

                          then ( right_cell (f,m,G)) meets C by A2, A18, A8, A19, A16, A40, A46, A47, A52, A53, GOBRD13: 24;

                          hence ( right_cell ((f | (k + 1)),m,G)) meets C by A2, A18, A19, A16, A40, GOBRD13: 31;

                        end;

                          suppose that

                           A54: (i1 + 1) = i2 & j1 = j2 and

                           A55: [i2, (j2 -' 1)] in ( Indices G) & (f /. (( len (f | k)) + 1)) = (G * (i2,(j2 -' 1)));

                          ( front_right_cell (f,(( len (f | k)) -' 1),G)) = ( cell (G,i2,(j2 -' 1))) by A2, A39, A38, A41, A45, A46, A47, A54, GOBRD13: 37;

                          then ( left_cell (f,m,G)) misses C by A2, A18, A8, A19, A16, A40, A46, A47, A49, A50, A55, GOBRD13: 27;

                          hence ( left_cell ((f | (k + 1)),m,G)) misses C by A2, A18, A19, A16, A40, GOBRD13: 31;

                          (i2 -' 1) = i1 & ( cell (G,i1,(j1 -' 1))) meets C by A2, A39, A38, A41, A45, A46, A47, A44, A54, GOBRD13: 24, NAT_D: 34;

                          then ( right_cell (f,m,G)) meets C by A2, A18, A8, A19, A16, A40, A46, A47, A49, A54, A55, GOBRD13: 28;

                          hence ( right_cell ((f | (k + 1)),m,G)) meets C by A2, A18, A19, A16, A40, GOBRD13: 31;

                        end;

                          suppose that

                           A56: i1 = (i2 + 1) & j1 = j2 and

                           A57: [i2, (j2 + 1)] in ( Indices G) & (f /. (( len (f | k)) + 1)) = (G * (i2,(j2 + 1)));

                          ( front_right_cell (f,(( len (f | k)) -' 1),G)) = ( cell (G,(i2 -' 1),j2)) by A2, A39, A38, A41, A45, A46, A47, A56, GOBRD13: 39;

                          then ( left_cell (f,m,G)) misses C by A2, A18, A8, A19, A16, A40, A46, A47, A50, A57, GOBRD13: 21;

                          hence ( left_cell ((f | (k + 1)),m,G)) misses C by A2, A18, A19, A16, A40, GOBRD13: 31;

                          ( cell (G,i2,j2)) meets C by A2, A39, A38, A41, A45, A46, A47, A44, A56, GOBRD13: 26;

                          then ( right_cell (f,m,G)) meets C by A2, A18, A8, A19, A16, A40, A46, A47, A57, GOBRD13: 22;

                          hence ( right_cell ((f | (k + 1)),m,G)) meets C by A2, A18, A19, A16, A40, GOBRD13: 31;

                        end;

                          suppose that

                           A58: i1 = i2 & j1 = (j2 + 1) and

                           A59: [(i2 -' 1), j2] in ( Indices G) & (f /. (( len (f | k)) + 1)) = (G * ((i2 -' 1),j2));

                          ( front_right_cell (f,(( len (f | k)) -' 1),G)) = ( cell (G,(i2 -' 1),(j2 -' 1))) by A2, A39, A38, A41, A45, A46, A47, A58, GOBRD13: 41;

                          then ( left_cell (f,m,G)) misses C by A2, A18, A8, A19, A16, A40, A46, A47, A48, A50, A59, GOBRD13: 25;

                          hence ( left_cell ((f | (k + 1)),m,G)) misses C by A2, A18, A19, A16, A40, GOBRD13: 31;

                          ( cell (G,(i2 -' 1),j2)) meets C by A2, A39, A38, A41, A45, A46, A47, A44, A58, GOBRD13: 28;

                          then ( right_cell (f,m,G)) meets C by A2, A18, A8, A19, A16, A40, A46, A47, A48, A59, GOBRD13: 26;

                          hence ( right_cell ((f | (k + 1)),m,G)) meets C by A2, A18, A19, A16, A40, GOBRD13: 31;

                        end;

                      end;

                      hence thesis;

                    end;

                      suppose

                       A60: ( front_left_cell (f,(( len (f | k)) -' 1),G)) misses C & ( front_right_cell (f,(( len (f | k)) -' 1),G)) meets C;

                      then

                       A61: f goes_straight ((( len (f | k)) -' 1),G) by A1, A18, A8, A39, A43, Def1;

                      now

                        per cases by A38, A45, A46, A47, A61;

                          suppose that

                           A62: i1 = i2 & (j1 + 1) = j2 and

                           A63: [i2, (j2 + 1)] in ( Indices G) & (f /. (( len (f | k)) + 1)) = (G * (i2,(j2 + 1)));

                          ( front_left_cell (f,(( len (f | k)) -' 1),G)) = ( cell (G,(i1 -' 1),j2)) by A2, A39, A38, A41, A45, A46, A47, A62, GOBRD13: 34;

                          then ( left_cell (f,m,G)) misses C by A2, A18, A8, A19, A16, A40, A46, A47, A60, A62, A63, GOBRD13: 21;

                          hence ( left_cell ((f | (k + 1)),m,G)) misses C by A2, A18, A19, A16, A40, GOBRD13: 31;

                          ( front_right_cell (f,(( len (f | k)) -' 1),G)) = ( cell (G,i1,j2)) by A2, A39, A38, A41, A45, A46, A47, A62, GOBRD13: 35;

                          then ( right_cell (f,m,G)) meets C by A2, A18, A8, A19, A16, A40, A46, A47, A60, A62, A63, GOBRD13: 22;

                          hence ( right_cell ((f | (k + 1)),m,G)) meets C by A2, A18, A19, A16, A40, GOBRD13: 31;

                        end;

                          suppose that

                           A64: (i1 + 1) = i2 & j1 = j2 and

                           A65: [(i2 + 1), j2] in ( Indices G) & (f /. (( len (f | k)) + 1)) = (G * ((i2 + 1),j2));

                          ( front_left_cell (f,(( len (f | k)) -' 1),G)) = ( cell (G,i2,j2)) by A2, A39, A38, A41, A45, A46, A47, A64, GOBRD13: 36;

                          then ( left_cell (f,m,G)) misses C by A2, A18, A8, A19, A16, A40, A46, A47, A60, A65, GOBRD13: 23;

                          hence ( left_cell ((f | (k + 1)),m,G)) misses C by A2, A18, A19, A16, A40, GOBRD13: 31;

                          ( front_right_cell (f,(( len (f | k)) -' 1),G)) = ( cell (G,i2,(j2 -' 1))) by A2, A39, A38, A41, A45, A46, A47, A64, GOBRD13: 37;

                          then ( right_cell (f,m,G)) meets C by A2, A18, A8, A19, A16, A40, A46, A47, A60, A65, GOBRD13: 24;

                          hence ( right_cell ((f | (k + 1)),m,G)) meets C by A2, A18, A19, A16, A40, GOBRD13: 31;

                        end;

                          suppose that

                           A66: i1 = (i2 + 1) & j1 = j2 and

                           A67: [(i2 -' 1), j2] in ( Indices G) & (f /. (( len (f | k)) + 1)) = (G * ((i2 -' 1),j2));

                          ( front_left_cell (f,(( len (f | k)) -' 1),G)) = ( cell (G,(i2 -' 1),(j2 -' 1))) by A2, A39, A38, A41, A45, A46, A47, A66, GOBRD13: 38;

                          then ( left_cell (f,m,G)) misses C by A2, A18, A8, A19, A16, A40, A46, A47, A48, A60, A67, GOBRD13: 25;

                          hence ( left_cell ((f | (k + 1)),m,G)) misses C by A2, A18, A19, A16, A40, GOBRD13: 31;

                          ( front_right_cell (f,(( len (f | k)) -' 1),G)) = ( cell (G,(i2 -' 1),j2)) by A2, A39, A38, A41, A45, A46, A47, A66, GOBRD13: 39;

                          then ( right_cell (f,m,G)) meets C by A2, A18, A8, A19, A16, A40, A46, A47, A48, A60, A67, GOBRD13: 26;

                          hence ( right_cell ((f | (k + 1)),m,G)) meets C by A2, A18, A19, A16, A40, GOBRD13: 31;

                        end;

                          suppose that

                           A68: i1 = i2 & j1 = (j2 + 1) and

                           A69: [i2, (j2 -' 1)] in ( Indices G) & (f /. (( len (f | k)) + 1)) = (G * (i2,(j2 -' 1)));

                          ( front_left_cell (f,(( len (f | k)) -' 1),G)) = ( cell (G,i2,(j2 -' 1))) by A2, A39, A38, A41, A45, A46, A47, A68, GOBRD13: 40;

                          then ( left_cell (f,m,G)) misses C by A2, A18, A8, A19, A16, A40, A46, A47, A49, A60, A69, GOBRD13: 27;

                          hence ( left_cell ((f | (k + 1)),m,G)) misses C by A2, A18, A19, A16, A40, GOBRD13: 31;

                          ( front_right_cell (f,(( len (f | k)) -' 1),G)) = ( cell (G,(i2 -' 1),(j2 -' 1))) by A2, A39, A38, A41, A45, A46, A47, A68, GOBRD13: 41;

                          then ( right_cell (f,m,G)) meets C by A2, A18, A8, A19, A16, A40, A46, A47, A49, A60, A69, GOBRD13: 28;

                          hence ( right_cell ((f | (k + 1)),m,G)) meets C by A2, A18, A19, A16, A40, GOBRD13: 31;

                        end;

                      end;

                      hence thesis;

                    end;

                      suppose

                       A70: ( front_left_cell (f,(( len (f | k)) -' 1),G)) meets C;

                      then

                       A71: f turns_left ((( len (f | k)) -' 1),G) by A1, A18, A8, A39, A43, Def1;

                      now

                        per cases by A38, A45, A46, A47, A71;

                          suppose that

                           A72: i1 = i2 & (j1 + 1) = j2 and

                           A73: [(i2 -' 1), j2] in ( Indices G) & (f /. (( len (f | k)) + 1)) = (G * ((i2 -' 1),j2));

                          (j2 -' 1) = j1 & ( cell (G,(i1 -' 1),j1)) misses C by A2, A39, A38, A41, A45, A46, A47, A42, A72, GOBRD13: 21, NAT_D: 34;

                          then ( left_cell (f,m,G)) misses C by A2, A18, A8, A19, A16, A40, A46, A47, A48, A72, A73, GOBRD13: 25;

                          hence ( left_cell ((f | (k + 1)),m,G)) misses C by A2, A18, A19, A16, A40, GOBRD13: 31;

                          ( front_left_cell (f,(( len (f | k)) -' 1),G)) = ( cell (G,(i1 -' 1),j2)) by A2, A39, A38, A41, A45, A46, A47, A72, GOBRD13: 34;

                          then ( right_cell (f,m,G)) meets C by A2, A18, A8, A19, A16, A40, A46, A47, A48, A70, A72, A73, GOBRD13: 26;

                          hence ( right_cell ((f | (k + 1)),m,G)) meets C by A2, A18, A19, A16, A40, GOBRD13: 31;

                        end;

                          suppose that

                           A74: (i1 + 1) = i2 & j1 = j2 and

                           A75: [i2, (j2 + 1)] in ( Indices G) & (f /. (( len (f | k)) + 1)) = (G * (i2,(j2 + 1)));

                          (i2 -' 1) = i1 & ( cell (G,i1,j1)) misses C by A2, A39, A38, A41, A45, A46, A47, A42, A74, GOBRD13: 23, NAT_D: 34;

                          then ( left_cell (f,m,G)) misses C by A2, A18, A8, A19, A16, A40, A46, A47, A74, A75, GOBRD13: 21;

                          hence ( left_cell ((f | (k + 1)),m,G)) misses C by A2, A18, A19, A16, A40, GOBRD13: 31;

                          ( front_left_cell (f,(( len (f | k)) -' 1),G)) = ( cell (G,i2,j2)) by A2, A39, A38, A41, A45, A46, A47, A74, GOBRD13: 36;

                          then ( right_cell (f,m,G)) meets C by A2, A18, A8, A19, A16, A40, A46, A47, A70, A75, GOBRD13: 22;

                          hence ( right_cell ((f | (k + 1)),m,G)) meets C by A2, A18, A19, A16, A40, GOBRD13: 31;

                        end;

                          suppose that

                           A76: i1 = (i2 + 1) & j1 = j2 and

                           A77: [i2, (j2 -' 1)] in ( Indices G) & (f /. (( len (f | k)) + 1)) = (G * (i2,(j2 -' 1)));

                          ( cell (G,i2,(j2 -' 1))) misses C by A2, A39, A38, A41, A45, A46, A47, A42, A76, GOBRD13: 25;

                          then ( left_cell (f,m,G)) misses C by A2, A18, A8, A19, A16, A40, A46, A47, A49, A77, GOBRD13: 27;

                          hence ( left_cell ((f | (k + 1)),m,G)) misses C by A2, A18, A19, A16, A40, GOBRD13: 31;

                          ( front_left_cell (f,(( len (f | k)) -' 1),G)) = ( cell (G,(i2 -' 1),(j2 -' 1))) by A2, A39, A38, A41, A45, A46, A47, A76, GOBRD13: 38;

                          then ( right_cell (f,m,G)) meets C by A2, A18, A8, A19, A16, A40, A46, A47, A49, A70, A77, GOBRD13: 28;

                          hence ( right_cell ((f | (k + 1)),m,G)) meets C by A2, A18, A19, A16, A40, GOBRD13: 31;

                        end;

                          suppose that

                           A78: i1 = i2 & j1 = (j2 + 1) and

                           A79: [(i2 + 1), j2] in ( Indices G) & (f /. (( len (f | k)) + 1)) = (G * ((i2 + 1),j2));

                          ( cell (G,i2,j2)) misses C by A2, A39, A38, A41, A45, A46, A47, A42, A78, GOBRD13: 27;

                          then ( left_cell (f,m,G)) misses C by A2, A18, A8, A19, A16, A40, A46, A47, A79, GOBRD13: 23;

                          hence ( left_cell ((f | (k + 1)),m,G)) misses C by A2, A18, A19, A16, A40, GOBRD13: 31;

                          ( front_left_cell (f,(( len (f | k)) -' 1),G)) = ( cell (G,i2,(j2 -' 1))) by A2, A39, A38, A41, A45, A46, A47, A78, GOBRD13: 40;

                          then ( right_cell (f,m,G)) meets C by A2, A18, A8, A19, A16, A40, A46, A47, A70, A79, GOBRD13: 24;

                          hence ( right_cell ((f | (k + 1)),m,G)) meets C by A2, A18, A19, A16, A40, GOBRD13: 31;

                        end;

                      end;

                      hence thesis;

                    end;

                  end;

                  hence thesis;

                end;

                  suppose (m + 1) <> ( len (f | (k + 1)));

                  then (m + 1) < ( len (f | (k + 1))) by A17, XXREAL_0: 1;

                  then

                   A80: (m + 1) <= ( len (f | k)) by A8, A19, NAT_1: 13;

                  then

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

                   A81: [i1, j1] in ( Indices G) and

                   A82: ((f | k) /. m) = (G * (i1,j1)) and

                   A83: [i2, j2] in ( Indices G) and

                   A84: ((f | k) /. (m + 1)) = (G * (i2,j2)) and

                   A85: i1 = i2 & (j1 + 1) = j2 or (i1 + 1) = i2 & j1 = j2 or i1 = (i2 + 1) & j1 = j2 or i1 = i2 & j1 = (j2 + 1) by A10, A16, JORDAN8: 3;

                  

                   A86: ( left_cell ((f | k),m,G)) misses C & ( right_cell ((f | k),m,G)) meets C by A6, A16, A80;

                  

                   A87: (f | (k + 1)) = ((f | k) ^ <*(f /. (k + 1))*>) by A18, FINSEQ_5: 82;

                  1 <= (m + 1) by NAT_1: 12;

                  then (m + 1) in ( dom (f | k)) by A80, FINSEQ_3: 25;

                  then

                   A88: ((f | (k + 1)) /. (m + 1)) = (G * (i2,j2)) by A84, A87, FINSEQ_4: 68;

                  m <= ( len (f | k)) by A80, NAT_1: 13;

                  then m in ( dom (f | k)) by A16, FINSEQ_3: 25;

                  then

                   A89: ((f | (k + 1)) /. m) = (G * (i1,j1)) by A82, A87, FINSEQ_4: 68;

                  now

                    per cases by A85;

                      suppose

                       A90: i1 = i2 & (j1 + 1) = j2;

                      then ( left_cell ((f | k),m,G)) = ( cell (G,(i1 -' 1),j1)) & ( right_cell ((f | k),m,G)) = ( cell (G,i1,j1)) by A10, A16, A80, A81, A82, A83, A84, GOBRD13: 21, GOBRD13: 22;

                      hence thesis by A11, A16, A17, A81, A83, A86, A89, A88, A90, GOBRD13: 21, GOBRD13: 22;

                    end;

                      suppose

                       A91: (i1 + 1) = i2 & j1 = j2;

                      then ( left_cell ((f | k),m,G)) = ( cell (G,i1,j1)) & ( right_cell ((f | k),m,G)) = ( cell (G,i1,(j1 -' 1))) by A10, A16, A80, A81, A82, A83, A84, GOBRD13: 23, GOBRD13: 24;

                      hence thesis by A11, A16, A17, A81, A83, A86, A89, A88, A91, GOBRD13: 23, GOBRD13: 24;

                    end;

                      suppose

                       A92: i1 = (i2 + 1) & j1 = j2;

                      then ( left_cell ((f | k),m,G)) = ( cell (G,i2,(j2 -' 1))) & ( right_cell ((f | k),m,G)) = ( cell (G,i2,j2)) by A10, A16, A80, A81, A82, A83, A84, GOBRD13: 25, GOBRD13: 26;

                      hence thesis by A11, A16, A17, A81, A83, A86, A89, A88, A92, GOBRD13: 25, GOBRD13: 26;

                    end;

                      suppose

                       A93: i1 = i2 & j1 = (j2 + 1);

                      then ( left_cell ((f | k),m,G)) = ( cell (G,i2,j2)) & ( right_cell ((f | k),m,G)) = ( cell (G,(i1 -' 1),j2)) by A10, A16, A80, A81, A82, A83, A84, GOBRD13: 27, GOBRD13: 28;

                      hence thesis by A11, A16, A17, A81, A83, A86, A89, A88, A93, GOBRD13: 27, GOBRD13: 28;

                    end;

                  end;

                  hence thesis;

                end;

              end;

              hence thesis;

            end;

          end;

          hence thesis;

        end;

      end;

      

       A94: (f | ( len f)) = f by FINSEQ_1: 58;

      

       A95: P[ 0 ] by CARD_1: 27;

      for k be Nat holds P[k] from NAT_1:sch 2( A95, A5);

      hence thesis by A94;

    end;

    theorem :: JORDAN9:32

    C is connected implies for n be Nat holds ( N-min ( L~ ( Cage (C,n)))) = (( Cage (C,n)) /. 1)

    proof

      assume

       A1: C is connected;

      let n be Nat;

      set f = ( Cage (C,n));

      

       A2: for k be Nat st 1 <= k & (k + 1) <= ( len f) holds ( left_cell (f,k,( Gauge (C,n)))) misses C & ( right_cell (f,k,( Gauge (C,n)))) meets C by A1, Th31;

      f is_sequence_on ( Gauge (C,n)) & ex i be Nat st 1 <= i & (i + 1) <= ( len ( Gauge (C,n))) & (f /. 1) = (( Gauge (C,n)) * (i,( width ( Gauge (C,n))))) & (f /. 2) = (( Gauge (C,n)) * ((i + 1),( width ( Gauge (C,n))))) & ( N-min C) in ( cell (( Gauge (C,n)),i,(( width ( Gauge (C,n))) -' 1))) & ( N-min C) <> (( Gauge (C,n)) * (i,(( width ( Gauge (C,n))) -' 1))) by A1, Def1;

      hence thesis by A2, Th30;

    end;