goboard7.miz



    begin

    reserve f for non empty FinSequence of ( TOP-REAL 2),

i,j,k,k1,k2,n,i1,i2,j1,j2 for Nat,

r,s,r1,r2 for Real,

p,q,p1,q1 for Point of ( TOP-REAL 2),

G for Go-board;

    theorem :: GOBOARD7:1

    

     Th1: |.(r1 - r2).| > s implies (r1 + s) < r2 or (r2 + s) < r1

    proof

      assume

       A1: |.(r1 - r2).| > s;

      now

        per cases ;

          case r1 < r2;

          then (r1 - r2) < 0 by XREAL_1: 49;

          

          then |.(r1 - r2).| = ( - (r1 - r2)) by ABSVALUE:def 1

          .= (r2 - r1);

          hence (r1 + s) < r2 by A1, XREAL_1: 20;

        end;

          case r2 <= r1;

          then (r1 - r2) >= 0 by XREAL_1: 48;

          then |.(r1 - r2).| = (r1 - r2) by ABSVALUE:def 1;

          hence (r2 + s) < r1 by A1, XREAL_1: 20;

        end;

      end;

      hence thesis;

    end;

    theorem :: GOBOARD7:2

    

     Th2: |.(r - s).| = 0 iff r = s

    proof

      hereby

        assume |.(r - s).| = 0 ;

        then (r - s) = 0 by ABSVALUE: 2;

        hence r = s;

      end;

      assume r = s;

      hence thesis by ABSVALUE: 2;

    end;

    theorem :: GOBOARD7:3

    

     Th3: for p,p1,q be Point of ( TOP-REAL n) st (p + p1) = (q + p1) holds p = q

    proof

      let p,p1,q be Point of ( TOP-REAL n) such that

       A1: (p + p1) = (q + p1);

      

      thus p = (p + ( 0. ( TOP-REAL n))) by RLVECT_1: 4

      .= (p + (p1 - p1)) by RLVECT_1: 5

      .= ((p + p1) - p1) by RLVECT_1:def 3

      .= (q + (p1 - p1)) by A1, RLVECT_1:def 3

      .= (q + ( 0. ( TOP-REAL n))) by RLVECT_1: 5

      .= q by RLVECT_1: 4;

    end;

    theorem :: GOBOARD7:4

    for p,p1,q be Point of ( TOP-REAL n) st (p1 + p) = (p1 + q) holds p = q by Th3;

    theorem :: GOBOARD7:5

    

     Th5: p1 in ( LSeg (p,q)) & (p `1 ) = (q `1 ) implies (p1 `1 ) = (q `1 )

    proof

      assume p1 in ( LSeg (p,q));

      then

      consider r such that

       A1: p1 = (((1 - r) * p) + (r * q)) and 0 <= r and r <= 1;

      assume

       A2: (p `1 ) = (q `1 );

      (p1 `1 ) = ((((1 - r) * p) `1 ) + ((r * q) `1 )) by A1, TOPREAL3: 2

      .= ((((1 - r) * p) `1 ) + (r * (q `1 ))) by TOPREAL3: 4

      .= (((1 - r) * (p `1 )) + (r * (q `1 ))) by TOPREAL3: 4;

      hence thesis by A2;

    end;

    theorem :: GOBOARD7:6

    

     Th6: p1 in ( LSeg (p,q)) & (p `2 ) = (q `2 ) implies (p1 `2 ) = (q `2 )

    proof

      assume p1 in ( LSeg (p,q));

      then

      consider r such that

       A1: p1 = (((1 - r) * p) + (r * q)) and 0 <= r and r <= 1;

      assume

       A2: (p `2 ) = (q `2 );

      (p1 `2 ) = ((((1 - r) * p) `2 ) + ((r * q) `2 )) by A1, TOPREAL3: 2

      .= ((((1 - r) * p) `2 ) + (r * (q `2 ))) by TOPREAL3: 4

      .= (((1 - r) * (p `2 )) + (r * (q `2 ))) by TOPREAL3: 4;

      hence thesis by A2;

    end;

    theorem :: GOBOARD7:7

    

     Th7: (p `1 ) = (q `1 ) & (q `1 ) = (p1 `1 ) & (p `2 ) <= (q `2 ) & (q `2 ) <= (p1 `2 ) implies q in ( LSeg (p,p1))

    proof

      assume that

       A1: (p `1 ) = (q `1 ) and

       A2: (q `1 ) = (p1 `1 ) and

       A3: (p `2 ) <= (q `2 ) & (q `2 ) <= (p1 `2 );

      

       A4: (p `2 ) <= (p1 `2 ) by A3, XXREAL_0: 2;

      per cases by A4, XXREAL_0: 1;

        suppose

         A5: (p `2 ) = (p1 `2 );

        then (p `2 ) = (q `2 ) by A3, XXREAL_0: 1;

        

        then

         A6: q = |[(p `1 ), (p `2 )]| by A1, EUCLID: 53

        .= p by EUCLID: 53;

        p = |[(p1 `1 ), (p1 `2 )]| by A1, A2, A5, EUCLID: 53

        .= p1 by EUCLID: 53;

        then ( LSeg (p,p1)) = {p} by RLTOPSP1: 70;

        hence thesis by A6, TARSKI:def 1;

      end;

        suppose

         A7: (p `2 ) < (p1 `2 );

        

         A8: q in { q1 : (q1 `1 ) = (q `1 ) & (p `2 ) <= (q1 `2 ) & (q1 `2 ) <= (p1 `2 ) } by A3;

        p = |[(q `1 ), (p `2 )]| & p1 = |[(q `1 ), (p1 `2 )]| by A1, A2, EUCLID: 53;

        hence thesis by A7, A8, TOPREAL3: 9;

      end;

    end;

    theorem :: GOBOARD7:8

    

     Th8: (p `1 ) <= (q `1 ) & (q `1 ) <= (p1 `1 ) & (p `2 ) = (q `2 ) & (q `2 ) = (p1 `2 ) implies q in ( LSeg (p,p1))

    proof

      assume that

       A1: (p `1 ) <= (q `1 ) & (q `1 ) <= (p1 `1 ) and

       A2: (p `2 ) = (q `2 ) and

       A3: (q `2 ) = (p1 `2 );

      

       A4: (p `1 ) <= (p1 `1 ) by A1, XXREAL_0: 2;

      per cases by A4, XXREAL_0: 1;

        suppose

         A5: (p `1 ) = (p1 `1 );

        then (p `1 ) = (q `1 ) by A1, XXREAL_0: 1;

        

        then

         A6: q = |[(p `1 ), (p `2 )]| by A2, EUCLID: 53

        .= p by EUCLID: 53;

        p = |[(p1 `1 ), (p1 `2 )]| by A2, A3, A5, EUCLID: 53

        .= p1 by EUCLID: 53;

        then ( LSeg (p,p1)) = {p} by RLTOPSP1: 70;

        hence thesis by A6, TARSKI:def 1;

      end;

        suppose

         A7: (p `1 ) < (p1 `1 );

        

         A8: q in { q1 : (q1 `2 ) = (q `2 ) & (p `1 ) <= (q1 `1 ) & (q1 `1 ) <= (p1 `1 ) } by A1;

        p = |[(p `1 ), (q `2 )]| & p1 = |[(p1 `1 ), (q `2 )]| by A2, A3, EUCLID: 53;

        hence thesis by A7, A8, TOPREAL3: 10;

      end;

    end;

    theorem :: GOBOARD7:9

    1 <= i & (i + 1) <= ( len G) & 1 <= j & (j + 1) <= ( width G) implies ((1 / 2) * ((G * (i,j)) + (G * ((i + 1),(j + 1))))) = ((1 / 2) * ((G * (i,(j + 1))) + (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);

      

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

      

       A6: 1 <= (j + 1) by NAT_1: 11;

      

       A7: 1 <= (i + 1) by NAT_1: 11;

      

      then

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

      .= ((G * ((i + 1),(j + 1))) `1 ) by A2, A4, A7, A6, GOBOARD5: 2;

      

       A9: i < ( len G) by A2, NAT_1: 13;

      

      then

       A10: ((G * (i,j)) `1 ) = ((G * (i,1)) `1 ) by A1, A3, A5, GOBOARD5: 2

      .= ((G * (i,(j + 1))) `1 ) by A1, A4, A9, A6, GOBOARD5: 2;

      

       A11: ((G * ((i + 1),(j + 1))) `2 ) = ((G * (1,(j + 1))) `2 ) by A2, A4, A7, A6, GOBOARD5: 1

      .= ((G * (i,(j + 1))) `2 ) by A1, A4, A9, A6, GOBOARD5: 1;

      

       A12: ((G * (i,j)) `2 ) = ((G * (1,j)) `2 ) by A1, A3, A9, A5, GOBOARD5: 1

      .= ((G * ((i + 1),j)) `2 ) by A2, A3, A7, A5, GOBOARD5: 1;

      

       A13: (((1 / 2) * ((G * (i,j)) + (G * ((i + 1),(j + 1))))) `2 ) = ((1 / 2) * (((G * (i,j)) + (G * ((i + 1),(j + 1)))) `2 )) by TOPREAL3: 4

      .= ((1 / 2) * (((G * (i,j)) `2 ) + ((G * ((i + 1),(j + 1))) `2 ))) by TOPREAL3: 2

      .= ((1 / 2) * (((G * (i,(j + 1))) + (G * ((i + 1),j))) `2 )) by A12, A11, TOPREAL3: 2

      .= (((1 / 2) * ((G * (i,(j + 1))) + (G * ((i + 1),j)))) `2 ) by TOPREAL3: 4;

      (((1 / 2) * ((G * (i,j)) + (G * ((i + 1),(j + 1))))) `1 ) = ((1 / 2) * (((G * (i,j)) + (G * ((i + 1),(j + 1)))) `1 )) by TOPREAL3: 4

      .= ((1 / 2) * (((G * (i,j)) `1 ) + ((G * ((i + 1),(j + 1))) `1 ))) by TOPREAL3: 2

      .= ((1 / 2) * (((G * (i,(j + 1))) + (G * ((i + 1),j))) `1 )) by A10, A8, TOPREAL3: 2

      .= (((1 / 2) * ((G * (i,(j + 1))) + (G * ((i + 1),j)))) `1 ) by TOPREAL3: 4;

      

      hence ((1 / 2) * ((G * (i,j)) + (G * ((i + 1),(j + 1))))) = |[(((1 / 2) * ((G * (i,(j + 1))) + (G * ((i + 1),j)))) `1 ), (((1 / 2) * ((G * (i,(j + 1))) + (G * ((i + 1),j)))) `2 )]| by A13, EUCLID: 53

      .= ((1 / 2) * ((G * (i,(j + 1))) + (G * ((i + 1),j)))) by EUCLID: 53;

    end;

    theorem :: GOBOARD7:10

    

     Th10: ( LSeg (f,k)) is horizontal implies ex j st 1 <= j & j <= ( width ( GoB f)) & for p st p in ( LSeg (f,k)) holds (p `2 ) = ((( GoB f) * (1,j)) `2 )

    proof

      assume

       A1: ( LSeg (f,k)) is horizontal;

      per cases ;

        suppose

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

        k <= (k + 1) by NAT_1: 11;

        then k <= ( len f) by A2, XXREAL_0: 2;

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

        then

        consider i, j such that

         A3: [i, j] in ( Indices ( GoB f)) and

         A4: (f /. k) = (( GoB f) * (i,j)) by GOBOARD2: 14;

        take j;

        thus

         A5: 1 <= j & j <= ( width ( GoB f)) by A3, MATRIX_0: 32;

        

         A6: (f /. k) in ( LSeg (f,k)) by A2, TOPREAL1: 21;

        let p;

        

         A7: 1 <= i & i <= ( len ( GoB f)) by A3, MATRIX_0: 32;

        assume p in ( LSeg (f,k));

        

        hence (p `2 ) = ((f /. k) `2 ) by A1, A6, SPPOL_1:def 2

        .= ((( GoB f) * (1,j)) `2 ) by A4, A5, A7, GOBOARD5: 1;

      end;

        suppose

         A8: not (1 <= k & (k + 1) <= ( len f));

        take 1;

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

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

        thus thesis by A8, TOPREAL1:def 3;

      end;

    end;

    theorem :: GOBOARD7:11

    

     Th11: ( LSeg (f,k)) is vertical implies ex i st 1 <= i & i <= ( len ( GoB f)) & for p st p in ( LSeg (f,k)) holds (p `1 ) = ((( GoB f) * (i,1)) `1 )

    proof

      assume

       A1: ( LSeg (f,k)) is vertical;

      per cases ;

        suppose

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

        k <= (k + 1) by NAT_1: 11;

        then k <= ( len f) by A2, XXREAL_0: 2;

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

        then

        consider i, j such that

         A3: [i, j] in ( Indices ( GoB f)) and

         A4: (f /. k) = (( GoB f) * (i,j)) by GOBOARD2: 14;

        take i;

        thus

         A5: 1 <= i & i <= ( len ( GoB f)) by A3, MATRIX_0: 32;

        

         A6: (f /. k) in ( LSeg (f,k)) by A2, TOPREAL1: 21;

        let p;

        

         A7: 1 <= j & j <= ( width ( GoB f)) by A3, MATRIX_0: 32;

        assume p in ( LSeg (f,k));

        

        hence (p `1 ) = ((f /. k) `1 ) by A1, A6, SPPOL_1:def 3

        .= ((( GoB f) * (i,1)) `1 ) by A4, A5, A7, GOBOARD5: 2;

      end;

        suppose

         A8: not (1 <= k & (k + 1) <= ( len f));

        take 1;

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

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

        thus thesis by A8, TOPREAL1:def 3;

      end;

    end;

    theorem :: GOBOARD7:12

    f is special & i <= ( len ( GoB f)) & j <= ( width ( GoB f)) implies ( Int ( cell (( GoB f),i,j))) misses ( L~ f)

    proof

      assume that

       A1: f is special and

       A2: i <= ( len ( GoB f)) and

       A3: j <= ( width ( GoB f));

      

       A4: ( Int ( cell (( GoB f),i,j))) = ( Int (( v_strip (( GoB f),i)) /\ ( h_strip (( GoB f),j)))) by GOBOARD5:def 3

      .= (( Int ( v_strip (( GoB f),i))) /\ ( Int ( h_strip (( GoB f),j)))) by TOPS_1: 17;

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

      then

      consider x be object such that

       A5: x in ( Int ( cell (( GoB f),i,j))) and

       A6: x in ( L~ f) by XBOOLE_0: 3;

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

      then

      consider X be set such that

       A7: x in X and

       A8: X in { ( LSeg (f,k)) : 1 <= k & (k + 1) <= ( len f) } by A6, TARSKI:def 4;

      consider k such that

       A9: X = ( LSeg (f,k)) and 1 <= k and (k + 1) <= ( len f) by A8;

      reconsider p = x as Point of ( TOP-REAL 2) by A7, A9;

      per cases by A1, SPPOL_1: 19;

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

        then

        consider j0 be Nat such that

         A10: 1 <= j0 and

         A11: j0 <= ( width ( GoB f)) and

         A12: for p st p in ( LSeg (f,k)) holds (p `2 ) = ((( GoB f) * (1,j0)) `2 ) by Th10;

        now

          

           A13: j0 > j implies j0 >= (j + 1) by NAT_1: 13;

          assume

           A14: p in ( Int ( h_strip (( GoB f),j)));

          per cases by A13, XXREAL_0: 1;

            suppose

             A15: j0 < j;

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

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

            then

             A16: ((( GoB f) * (1,j)) `2 ) > ((( GoB f) * (1,j0)) `2 ) by A3, A10, A15, GOBOARD5: 4;

            j >= 1 by A10, A15, XXREAL_0: 2;

            then (p `2 ) > ((( GoB f) * (1,j)) `2 ) by A3, A14, GOBOARD6: 27;

            hence contradiction by A7, A9, A12, A16;

          end;

            suppose j0 = j;

            then (p `2 ) > ((( GoB f) * (1,j0)) `2 ) by A10, A11, A14, GOBOARD6: 27;

            hence contradiction by A7, A9, A12;

          end;

            suppose

             A17: j0 > (j + 1);

            then (j + 1) <= ( width ( GoB f)) by A11, XXREAL_0: 2;

            then j < ( width ( GoB f)) by NAT_1: 13;

            then

             A18: (p `2 ) < ((( GoB f) * (1,(j + 1))) `2 ) by A14, GOBOARD6: 28;

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

            then

             A19: 1 <= ( len ( GoB f)) by NAT_1: 14;

            (j + 1) >= 1 by NAT_1: 11;

            then ((( GoB f) * (1,(j + 1))) `2 ) < ((( GoB f) * (1,j0)) `2 ) by A11, A17, A19, GOBOARD5: 4;

            hence contradiction by A7, A9, A12, A18;

          end;

            suppose

             A20: j0 = (j + 1);

            then j < ( width ( GoB f)) by A11, NAT_1: 13;

            then (p `2 ) < ((( GoB f) * (1,j0)) `2 ) by A14, A20, GOBOARD6: 28;

            hence contradiction by A7, A9, A12;

          end;

        end;

        hence contradiction by A5, A4, XBOOLE_0:def 4;

      end;

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

        then

        consider i0 be Nat such that

         A21: 1 <= i0 and

         A22: i0 <= ( len ( GoB f)) and

         A23: for p st p in ( LSeg (f,k)) holds (p `1 ) = ((( GoB f) * (i0,1)) `1 ) by Th11;

        now

          

           A24: i0 > i implies i0 >= (i + 1) by NAT_1: 13;

          assume

           A25: p in ( Int ( v_strip (( GoB f),i)));

          per cases by A24, XXREAL_0: 1;

            suppose

             A26: i0 < i;

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

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

            then

             A27: ((( GoB f) * (i,1)) `1 ) > ((( GoB f) * (i0,1)) `1 ) by A2, A21, A26, GOBOARD5: 3;

            i >= 1 by A21, A26, XXREAL_0: 2;

            then (p `1 ) > ((( GoB f) * (i,1)) `1 ) by A2, A25, GOBOARD6: 29;

            hence contradiction by A7, A9, A23, A27;

          end;

            suppose i0 = i;

            then (p `1 ) > ((( GoB f) * (i0,1)) `1 ) by A21, A22, A25, GOBOARD6: 29;

            hence contradiction by A7, A9, A23;

          end;

            suppose

             A28: i0 > (i + 1);

            then (i + 1) <= ( len ( GoB f)) by A22, XXREAL_0: 2;

            then i < ( len ( GoB f)) by NAT_1: 13;

            then

             A29: (p `1 ) < ((( GoB f) * ((i + 1),1)) `1 ) by A25, GOBOARD6: 30;

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

            then

             A30: 1 <= ( width ( GoB f)) by NAT_1: 14;

            (i + 1) >= 1 by NAT_1: 11;

            then ((( GoB f) * ((i + 1),1)) `1 ) < ((( GoB f) * (i0,1)) `1 ) by A22, A28, A30, GOBOARD5: 3;

            hence contradiction by A7, A9, A23, A29;

          end;

            suppose

             A31: i0 = (i + 1);

            then i < ( len ( GoB f)) by A22, NAT_1: 13;

            then (p `1 ) < ((( GoB f) * (i0,1)) `1 ) by A25, A31, GOBOARD6: 30;

            hence contradiction by A7, A9, A23;

          end;

        end;

        hence contradiction by A5, A4, XBOOLE_0:def 4;

      end;

    end;

    begin

    theorem :: GOBOARD7:13

    

     Th13: 1 <= i & i <= ( len G) & 1 <= j & (j + 2) <= ( width G) implies (( LSeg ((G * (i,j)),(G * (i,(j + 1))))) /\ ( LSeg ((G * (i,(j + 1))),(G * (i,(j + 2)))))) = {(G * (i,(j + 1)))}

    proof

      assume that

       A1: 1 <= i & i <= ( len G) and

       A2: 1 <= j and

       A3: (j + 2) <= ( width G);

      now

        let x be object;

        hereby

          assume

           A4: x in (( LSeg ((G * (i,j)),(G * (i,(j + 1))))) /\ ( LSeg ((G * (i,(j + 1))),(G * (i,(j + 2))))));

          then

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

          

           A5: x in ( LSeg ((G * (i,j)),(G * (i,(j + 1))))) by A4, XBOOLE_0:def 4;

          

           A6: p in ( LSeg ((G * (i,(j + 1))),(G * (i,(j + 2))))) by A4, XBOOLE_0:def 4;

          j <= (j + 2) by NAT_1: 11;

          then

           A7: j <= ( width G) by A3, XXREAL_0: 2;

          

           A8: (j + 1) < (j + 2) by XREAL_1: 6;

          then

           A9: (j + 1) <= ( width G) by A3, XXREAL_0: 2;

          

           A10: 1 <= (j + 1) by NAT_1: 11;

          

          then ((G * (i,(j + 1))) `1 ) = ((G * (i,1)) `1 ) by A1, A9, GOBOARD5: 2

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

          then

           A11: (p `1 ) = ((G * (i,(j + 1))) `1 ) by A5, Th5;

          j < (j + 1) by XREAL_1: 29;

          then ((G * (i,j)) `2 ) < ((G * (i,(j + 1))) `2 ) by A1, A2, A9, GOBOARD5: 4;

          then

           A12: (p `2 ) <= ((G * (i,(j + 1))) `2 ) by A5, TOPREAL1: 4;

          ((G * (i,(j + 1))) `2 ) < ((G * (i,(j + 2))) `2 ) by A1, A3, A8, A10, GOBOARD5: 4;

          then (p `2 ) >= ((G * (i,(j + 1))) `2 ) by A6, TOPREAL1: 4;

          then (p `2 ) = ((G * (i,(j + 1))) `2 ) by A12, XXREAL_0: 1;

          hence x = (G * (i,(j + 1))) by A11, TOPREAL3: 6;

        end;

        assume x = (G * (i,(j + 1)));

        then x in ( LSeg ((G * (i,j)),(G * (i,(j + 1))))) & x in ( LSeg ((G * (i,(j + 1))),(G * (i,(j + 2))))) by RLTOPSP1: 68;

        hence x in (( LSeg ((G * (i,j)),(G * (i,(j + 1))))) /\ ( LSeg ((G * (i,(j + 1))),(G * (i,(j + 2)))))) by XBOOLE_0:def 4;

      end;

      hence thesis by TARSKI:def 1;

    end;

    theorem :: GOBOARD7:14

    

     Th14: 1 <= i & (i + 2) <= ( len G) & 1 <= j & j <= ( width G) implies (( LSeg ((G * (i,j)),(G * ((i + 1),j)))) /\ ( LSeg ((G * ((i + 1),j)),(G * ((i + 2),j))))) = {(G * ((i + 1),j))}

    proof

      assume that

       A1: 1 <= i and

       A2: (i + 2) <= ( len G) and

       A3: 1 <= j & j <= ( width G);

      now

        let x be object;

        hereby

          assume

           A4: x in (( LSeg ((G * (i,j)),(G * ((i + 1),j)))) /\ ( LSeg ((G * ((i + 1),j)),(G * ((i + 2),j)))));

          then

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

          

           A5: x in ( LSeg ((G * (i,j)),(G * ((i + 1),j)))) by A4, XBOOLE_0:def 4;

          

           A6: p in ( LSeg ((G * ((i + 1),j)),(G * ((i + 2),j)))) by A4, XBOOLE_0:def 4;

          i <= (i + 2) by NAT_1: 11;

          then

           A7: i <= ( len G) by A2, XXREAL_0: 2;

          

           A8: (i + 1) < (i + 2) by XREAL_1: 6;

          then

           A9: (i + 1) <= ( len G) by A2, XXREAL_0: 2;

          

           A10: 1 <= (i + 1) by NAT_1: 11;

          

          then ((G * ((i + 1),j)) `2 ) = ((G * (1,j)) `2 ) by A3, A9, GOBOARD5: 1

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

          then

           A11: (p `2 ) = ((G * ((i + 1),j)) `2 ) by A5, Th6;

          i < (i + 1) by XREAL_1: 29;

          then ((G * (i,j)) `1 ) < ((G * ((i + 1),j)) `1 ) by A1, A3, A9, GOBOARD5: 3;

          then

           A12: (p `1 ) <= ((G * ((i + 1),j)) `1 ) by A5, TOPREAL1: 3;

          ((G * ((i + 1),j)) `1 ) < ((G * ((i + 2),j)) `1 ) by A2, A3, A8, A10, GOBOARD5: 3;

          then (p `1 ) >= ((G * ((i + 1),j)) `1 ) by A6, TOPREAL1: 3;

          then (p `1 ) = ((G * ((i + 1),j)) `1 ) by A12, XXREAL_0: 1;

          hence x = (G * ((i + 1),j)) by A11, TOPREAL3: 6;

        end;

        assume x = (G * ((i + 1),j));

        then x in ( LSeg ((G * (i,j)),(G * ((i + 1),j)))) & x in ( LSeg ((G * ((i + 1),j)),(G * ((i + 2),j)))) by RLTOPSP1: 68;

        hence x in (( LSeg ((G * (i,j)),(G * ((i + 1),j)))) /\ ( LSeg ((G * ((i + 1),j)),(G * ((i + 2),j))))) by XBOOLE_0:def 4;

      end;

      hence thesis by TARSKI:def 1;

    end;

    theorem :: GOBOARD7:15

    

     Th15: 1 <= i & (i + 1) <= ( len G) & 1 <= j & (j + 1) <= ( width G) implies (( LSeg ((G * (i,j)),(G * (i,(j + 1))))) /\ ( LSeg ((G * (i,(j + 1))),(G * ((i + 1),(j + 1)))))) = {(G * (i,(j + 1)))}

    proof

      assume that

       A1: 1 <= i and

       A2: (i + 1) <= ( len G) and

       A3: 1 <= j and

       A4: (j + 1) <= ( width G);

      now

        let x be object;

        hereby

          assume

           A5: x in (( LSeg ((G * (i,j)),(G * (i,(j + 1))))) /\ ( LSeg ((G * (i,(j + 1))),(G * ((i + 1),(j + 1))))));

          then

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

          

           A6: x in ( LSeg ((G * (i,j)),(G * (i,(j + 1))))) by A5, XBOOLE_0:def 4;

          

           A7: p in ( LSeg ((G * (i,(j + 1))),(G * ((i + 1),(j + 1))))) by A5, XBOOLE_0:def 4;

          

           A8: 1 <= (i + 1) by NAT_1: 11;

          i <= (i + 1) by NAT_1: 11;

          then

           A9: i <= ( len G) by A2, XXREAL_0: 2;

          

           A10: 1 <= (j + 1) by NAT_1: 11;

          

          then ((G * (i,(j + 1))) `2 ) = ((G * (1,(j + 1))) `2 ) by A1, A4, A9, GOBOARD5: 1

          .= ((G * ((i + 1),(j + 1))) `2 ) by A2, A4, A10, A8, GOBOARD5: 1;

          then

           A11: (p `2 ) = ((G * (i,(j + 1))) `2 ) by A7, Th6;

          j < (j + 1) by XREAL_1: 29;

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

          

          then ((G * (i,j)) `1 ) = ((G * (i,1)) `1 ) by A1, A3, A9, GOBOARD5: 2

          .= ((G * (i,(j + 1))) `1 ) by A1, A4, A9, A10, GOBOARD5: 2;

          then (p `1 ) = ((G * (i,(j + 1))) `1 ) by A6, Th5;

          hence x = (G * (i,(j + 1))) by A11, TOPREAL3: 6;

        end;

        assume x = (G * (i,(j + 1)));

        then x in ( LSeg ((G * (i,j)),(G * (i,(j + 1))))) & x in ( LSeg ((G * (i,(j + 1))),(G * ((i + 1),(j + 1))))) by RLTOPSP1: 68;

        hence x in (( LSeg ((G * (i,j)),(G * (i,(j + 1))))) /\ ( LSeg ((G * (i,(j + 1))),(G * ((i + 1),(j + 1)))))) by XBOOLE_0:def 4;

      end;

      hence thesis by TARSKI:def 1;

    end;

    theorem :: GOBOARD7:16

    

     Th16: 1 <= i & (i + 1) <= ( len G) & 1 <= j & (j + 1) <= ( width G) implies (( LSeg ((G * (i,(j + 1))),(G * ((i + 1),(j + 1))))) /\ ( LSeg ((G * ((i + 1),j)),(G * ((i + 1),(j + 1)))))) = {(G * ((i + 1),(j + 1)))}

    proof

      assume that

       A1: 1 <= i and

       A2: (i + 1) <= ( len G) and

       A3: 1 <= j and

       A4: (j + 1) <= ( width G);

      now

        let x be object;

        hereby

          assume

           A5: x in (( LSeg ((G * (i,(j + 1))),(G * ((i + 1),(j + 1))))) /\ ( LSeg ((G * ((i + 1),j)),(G * ((i + 1),(j + 1))))));

          then

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

          

           A6: x in ( LSeg ((G * (i,(j + 1))),(G * ((i + 1),(j + 1))))) by A5, XBOOLE_0:def 4;

          

           A7: p in ( LSeg ((G * ((i + 1),j)),(G * ((i + 1),(j + 1))))) by A5, XBOOLE_0:def 4;

          

           A8: 1 <= (i + 1) by NAT_1: 11;

          

           A9: 1 <= (i + 1) by NAT_1: 11;

          

           A10: 1 <= (j + 1) by NAT_1: 11;

          j < (j + 1) by XREAL_1: 29;

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

          

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

          .= ((G * ((i + 1),(j + 1))) `1 ) by A2, A4, A10, A9, GOBOARD5: 2;

          then

           A11: (p `1 ) = ((G * ((i + 1),(j + 1))) `1 ) by A7, Th5;

          i <= (i + 1) by NAT_1: 11;

          then i <= ( len G) by A2, XXREAL_0: 2;

          

          then ((G * (i,(j + 1))) `2 ) = ((G * (1,(j + 1))) `2 ) by A1, A4, A10, GOBOARD5: 1

          .= ((G * ((i + 1),(j + 1))) `2 ) by A2, A4, A10, A8, GOBOARD5: 1;

          then (p `2 ) = ((G * ((i + 1),(j + 1))) `2 ) by A6, Th6;

          hence x = (G * ((i + 1),(j + 1))) by A11, TOPREAL3: 6;

        end;

        assume x = (G * ((i + 1),(j + 1)));

        then x in ( LSeg ((G * (i,(j + 1))),(G * ((i + 1),(j + 1))))) & x in ( LSeg ((G * ((i + 1),j)),(G * ((i + 1),(j + 1))))) by RLTOPSP1: 68;

        hence x in (( LSeg ((G * (i,(j + 1))),(G * ((i + 1),(j + 1))))) /\ ( LSeg ((G * ((i + 1),j)),(G * ((i + 1),(j + 1)))))) by XBOOLE_0:def 4;

      end;

      hence thesis by TARSKI:def 1;

    end;

    theorem :: GOBOARD7:17

    

     Th17: 1 <= i & (i + 1) <= ( len G) & 1 <= j & (j + 1) <= ( width G) implies (( LSeg ((G * (i,j)),(G * ((i + 1),j)))) /\ ( LSeg ((G * (i,j)),(G * (i,(j + 1)))))) = {(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);

      now

        let x be object;

        hereby

          assume

           A5: x in (( LSeg ((G * (i,j)),(G * ((i + 1),j)))) /\ ( LSeg ((G * (i,j)),(G * (i,(j + 1))))));

          then

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

          

           A6: x in ( LSeg ((G * (i,j)),(G * ((i + 1),j)))) by A5, XBOOLE_0:def 4;

          

           A7: p in ( LSeg ((G * (i,j)),(G * (i,(j + 1))))) by A5, XBOOLE_0:def 4;

          

           A8: 1 <= (i + 1) by NAT_1: 11;

          

           A9: 1 <= (j + 1) by NAT_1: 11;

          j < (j + 1) by XREAL_1: 29;

          then

           A10: j <= ( width G) by A4, XXREAL_0: 2;

          i <= (i + 1) by NAT_1: 11;

          then

           A11: i <= ( len G) by A2, XXREAL_0: 2;

          

          then ((G * (i,j)) `1 ) = ((G * (i,1)) `1 ) by A1, A3, A10, GOBOARD5: 2

          .= ((G * (i,(j + 1))) `1 ) by A1, A4, A11, A9, GOBOARD5: 2;

          then

           A12: (p `1 ) = ((G * (i,j)) `1 ) by A7, Th5;

          ((G * (i,j)) `2 ) = ((G * (1,j)) `2 ) by A1, A3, A11, A10, GOBOARD5: 1

          .= ((G * ((i + 1),j)) `2 ) by A2, A3, A8, A10, GOBOARD5: 1;

          then (p `2 ) = ((G * (i,j)) `2 ) by A6, Th6;

          hence x = (G * (i,j)) by A12, TOPREAL3: 6;

        end;

        assume x = (G * (i,j));

        then x in ( LSeg ((G * (i,j)),(G * ((i + 1),j)))) & x in ( LSeg ((G * (i,j)),(G * (i,(j + 1))))) by RLTOPSP1: 68;

        hence x in (( LSeg ((G * (i,j)),(G * ((i + 1),j)))) /\ ( LSeg ((G * (i,j)),(G * (i,(j + 1)))))) by XBOOLE_0:def 4;

      end;

      hence thesis by TARSKI:def 1;

    end;

    theorem :: GOBOARD7:18

    

     Th18: 1 <= i & (i + 1) <= ( len G) & 1 <= j & (j + 1) <= ( width G) implies (( LSeg ((G * (i,j)),(G * ((i + 1),j)))) /\ ( LSeg ((G * ((i + 1),j)),(G * ((i + 1),(j + 1)))))) = {(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);

      now

        let x be object;

        hereby

          assume

           A5: x in (( LSeg ((G * (i,j)),(G * ((i + 1),j)))) /\ ( LSeg ((G * ((i + 1),j)),(G * ((i + 1),(j + 1))))));

          then

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

          

           A6: x in ( LSeg ((G * (i,j)),(G * ((i + 1),j)))) by A5, XBOOLE_0:def 4;

          

           A7: p in ( LSeg ((G * ((i + 1),j)),(G * ((i + 1),(j + 1))))) by A5, XBOOLE_0:def 4;

          

           A8: 1 <= (j + 1) & 1 <= (i + 1) by NAT_1: 11;

          j < (j + 1) by XREAL_1: 29;

          then

           A9: j <= ( width G) by A4, XXREAL_0: 2;

          

           A10: 1 <= (i + 1) by NAT_1: 11;

          

          then ((G * ((i + 1),j)) `1 ) = ((G * ((i + 1),1)) `1 ) by A2, A3, A9, GOBOARD5: 2

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

          then

           A11: (p `1 ) = ((G * ((i + 1),j)) `1 ) by A7, Th5;

          i <= (i + 1) by NAT_1: 11;

          then i <= ( len G) by A2, XXREAL_0: 2;

          

          then ((G * (i,j)) `2 ) = ((G * (1,j)) `2 ) by A1, A3, A9, GOBOARD5: 1

          .= ((G * ((i + 1),j)) `2 ) by A2, A3, A10, A9, GOBOARD5: 1;

          then (p `2 ) = ((G * ((i + 1),j)) `2 ) by A6, Th6;

          hence x = (G * ((i + 1),j)) by A11, TOPREAL3: 6;

        end;

        assume x = (G * ((i + 1),j));

        then x in ( LSeg ((G * (i,j)),(G * ((i + 1),j)))) & x in ( LSeg ((G * ((i + 1),j)),(G * ((i + 1),(j + 1))))) by RLTOPSP1: 68;

        hence x in (( LSeg ((G * (i,j)),(G * ((i + 1),j)))) /\ ( LSeg ((G * ((i + 1),j)),(G * ((i + 1),(j + 1)))))) by XBOOLE_0:def 4;

      end;

      hence thesis by TARSKI:def 1;

    end;

    theorem :: GOBOARD7:19

    

     Th19: for i1,j1,i2,j2 be Nat st 1 <= i1 & i1 <= ( len G) & 1 <= j1 & (j1 + 1) <= ( width G) & 1 <= i2 & i2 <= ( len G) & 1 <= j2 & (j2 + 1) <= ( width G) & ( LSeg ((G * (i1,j1)),(G * (i1,(j1 + 1))))) meets ( LSeg ((G * (i2,j2)),(G * (i2,(j2 + 1))))) holds i1 = i2 & |.(j1 - j2).| <= 1

    proof

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

       A1: 1 <= i1 & i1 <= ( len G) and

       A2: 1 <= j1 and

       A3: (j1 + 1) <= ( width G) and

       A4: 1 <= i2 & i2 <= ( len G) and

       A5: 1 <= j2 and

       A6: (j2 + 1) <= ( width G);

      

       A7: 1 <= (j1 + 1) by A2, NAT_1: 13;

      

       A8: j1 < ( width G) by A3, NAT_1: 13;

      assume ( LSeg ((G * (i1,j1)),(G * (i1,(j1 + 1))))) meets ( LSeg ((G * (i2,j2)),(G * (i2,(j2 + 1)))));

      then

      consider x be object such that

       A9: x in ( LSeg ((G * (i1,j1)),(G * (i1,(j1 + 1))))) and

       A10: x in ( LSeg ((G * (i2,j2)),(G * (i2,(j2 + 1))))) by XBOOLE_0: 3;

      reconsider p = x as Point of ( TOP-REAL 2) by A9;

      consider r1 such that

       A11: p = (((1 - r1) * (G * (i1,j1))) + (r1 * (G * (i1,(j1 + 1))))) and

       A12: r1 >= 0 and

       A13: r1 <= 1 by A9;

      consider r2 such that

       A14: p = (((1 - r2) * (G * (i2,j2))) + (r2 * (G * (i2,(j2 + 1))))) and

       A15: r2 >= 0 and

       A16: r2 <= 1 by A10;

      

       A17: 1 <= (j2 + 1) by A5, NAT_1: 13;

      

       A18: j2 < ( width G) by A6, NAT_1: 13;

      assume

       A19: not thesis;

      per cases by A19;

        suppose i1 <> i2;

        then

         A20: i1 < i2 or i2 < i1 by XXREAL_0: 1;

        

         A21: ((G * (i2,j2)) `1 ) = ((G * (i2,1)) `1 ) by A4, A5, A18, GOBOARD5: 2

        .= ((G * (i2,(j2 + 1))) `1 ) by A4, A6, A17, GOBOARD5: 2;

        ((G * (i1,j1)) `1 ) = ((G * (i1,1)) `1 ) by A1, A2, A8, GOBOARD5: 2

        .= ((G * (i1,(j1 + 1))) `1 ) by A1, A3, A7, GOBOARD5: 2;

        

        then (1 * ((G * (i1,j1)) `1 )) = (((1 - r1) * ((G * (i1,j1)) `1 )) + (r1 * ((G * (i1,(j1 + 1))) `1 )))

        .= ((((1 - r1) * (G * (i1,j1))) `1 ) + (r1 * ((G * (i1,(j1 + 1))) `1 ))) by TOPREAL3: 4

        .= ((((1 - r1) * (G * (i1,j1))) `1 ) + ((r1 * (G * (i1,(j1 + 1)))) `1 )) by TOPREAL3: 4

        .= (p `1 ) by A11, TOPREAL3: 2

        .= ((((1 - r2) * (G * (i2,j2))) `1 ) + ((r2 * (G * (i2,(j2 + 1)))) `1 )) by A14, TOPREAL3: 2

        .= (((1 - r2) * ((G * (i2,j2)) `1 )) + ((r2 * (G * (i2,(j2 + 1)))) `1 )) by TOPREAL3: 4

        .= (((1 - r2) * ((G * (i2,j2)) `1 )) + (r2 * ((G * (i2,(j2 + 1))) `1 ))) by TOPREAL3: 4

        .= ((G * (i2,1)) `1 ) by A4, A6, A17, A21, GOBOARD5: 2

        .= ((G * (i2,j1)) `1 ) by A2, A4, A8, GOBOARD5: 2;

        hence contradiction by A1, A2, A4, A8, A20, GOBOARD5: 3;

      end;

        suppose

         A22: |.(j1 - j2).| > 1;

        

         A23: ((G * (i2,(j2 + 1))) `2 ) = ((G * (1,(j2 + 1))) `2 ) by A4, A6, A17, GOBOARD5: 1

        .= ((G * (i1,(j2 + 1))) `2 ) by A1, A6, A17, GOBOARD5: 1;

        

         A24: ((G * (i2,j2)) `2 ) = ((G * (1,j2)) `2 ) by A4, A5, A18, GOBOARD5: 1

        .= ((G * (i1,j2)) `2 ) by A1, A5, A18, GOBOARD5: 1;

        

         A25: (((1 - r1) * ((G * (i1,j1)) `2 )) + (r1 * ((G * (i1,(j1 + 1))) `2 ))) = ((((1 - r1) * (G * (i1,j1))) `2 ) + (r1 * ((G * (i1,(j1 + 1))) `2 ))) by TOPREAL3: 4

        .= ((((1 - r1) * (G * (i1,j1))) `2 ) + ((r1 * (G * (i1,(j1 + 1)))) `2 )) by TOPREAL3: 4

        .= (p `2 ) by A11, TOPREAL3: 2

        .= ((((1 - r2) * (G * (i2,j2))) `2 ) + ((r2 * (G * (i2,(j2 + 1)))) `2 )) by A14, TOPREAL3: 2

        .= (((1 - r2) * ((G * (i2,j2)) `2 )) + ((r2 * (G * (i2,(j2 + 1)))) `2 )) by TOPREAL3: 4

        .= (((1 - r2) * ((G * (i1,j2)) `2 )) + (r2 * ((G * (i1,(j2 + 1))) `2 ))) by A23, A24, TOPREAL3: 4;

        now

          per cases by A22, Th1;

            suppose

             A26: (j1 + 1) < j2;

            j2 < (j2 + 1) by XREAL_1: 29;

            then ((G * (i1,j2)) `2 ) < ((G * (i1,(j2 + 1))) `2 ) by A1, A5, A6, GOBOARD5: 4;

            then (((1 - r2) * ((G * (i1,j2)) `2 )) + (r2 * ((G * (i1,j2)) `2 ))) = (1 * ((G * (i1,j2)) `2 )) & (r2 * ((G * (i1,j2)) `2 )) <= (r2 * ((G * (i1,(j2 + 1))) `2 )) by A15, XREAL_1: 64;

            then

             A27: ((G * (i1,j2)) `2 ) <= (((1 - r2) * ((G * (i1,j2)) `2 )) + (r2 * ((G * (i1,(j2 + 1))) `2 ))) by XREAL_1: 6;

            j1 < (j1 + 1) by XREAL_1: 29;

            then

             A28: ((G * (i1,j1)) `2 ) <= ((G * (i1,(j1 + 1))) `2 ) by A1, A2, A3, GOBOARD5: 4;

            (1 - r1) >= 0 by A13, XREAL_1: 48;

            then (((1 - r1) * ((G * (i1,(j1 + 1))) `2 )) + (r1 * ((G * (i1,(j1 + 1))) `2 ))) = (1 * ((G * (i1,(j1 + 1))) `2 )) & ((1 - r1) * ((G * (i1,j1)) `2 )) <= ((1 - r1) * ((G * (i1,(j1 + 1))) `2 )) by A28, XREAL_1: 64;

            then

             A29: (((1 - r1) * ((G * (i1,j1)) `2 )) + (r1 * ((G * (i1,(j1 + 1))) `2 ))) <= ((G * (i1,(j1 + 1))) `2 ) by XREAL_1: 6;

            ((G * (i1,(j1 + 1))) `2 ) < ((G * (i1,j2)) `2 ) by A1, A7, A18, A26, GOBOARD5: 4;

            hence contradiction by A25, A29, A27, XXREAL_0: 2;

          end;

            suppose

             A30: (j2 + 1) < j1;

            j1 < (j1 + 1) by XREAL_1: 29;

            then ((G * (i1,j1)) `2 ) < ((G * (i1,(j1 + 1))) `2 ) by A1, A2, A3, GOBOARD5: 4;

            then (((1 - r1) * ((G * (i1,j1)) `2 )) + (r1 * ((G * (i1,j1)) `2 ))) = (1 * ((G * (i1,j1)) `2 )) & (r1 * ((G * (i1,j1)) `2 )) <= (r1 * ((G * (i1,(j1 + 1))) `2 )) by A12, XREAL_1: 64;

            then

             A31: ((G * (i1,j1)) `2 ) <= (((1 - r1) * ((G * (i1,j1)) `2 )) + (r1 * ((G * (i1,(j1 + 1))) `2 ))) by XREAL_1: 6;

            j2 < (j2 + 1) by XREAL_1: 29;

            then

             A32: ((G * (i1,j2)) `2 ) <= ((G * (i1,(j2 + 1))) `2 ) by A1, A5, A6, GOBOARD5: 4;

            (1 - r2) >= 0 by A16, XREAL_1: 48;

            then (((1 - r2) * ((G * (i1,(j2 + 1))) `2 )) + (r2 * ((G * (i1,(j2 + 1))) `2 ))) = (1 * ((G * (i1,(j2 + 1))) `2 )) & ((1 - r2) * ((G * (i1,j2)) `2 )) <= ((1 - r2) * ((G * (i1,(j2 + 1))) `2 )) by A32, XREAL_1: 64;

            then

             A33: (((1 - r2) * ((G * (i1,j2)) `2 )) + (r2 * ((G * (i1,(j2 + 1))) `2 ))) <= ((G * (i1,(j2 + 1))) `2 ) by XREAL_1: 6;

            ((G * (i1,(j2 + 1))) `2 ) < ((G * (i1,j1)) `2 ) by A1, A8, A17, A30, GOBOARD5: 4;

            hence contradiction by A25, A33, A31, XXREAL_0: 2;

          end;

        end;

        hence contradiction;

      end;

    end;

    theorem :: GOBOARD7:20

    

     Th20: for i1,j1,i2,j2 be Nat st 1 <= i1 & (i1 + 1) <= ( len G) & 1 <= j1 & j1 <= ( width G) & 1 <= i2 & (i2 + 1) <= ( len G) & 1 <= j2 & j2 <= ( width G) & ( LSeg ((G * (i1,j1)),(G * ((i1 + 1),j1)))) meets ( LSeg ((G * (i2,j2)),(G * ((i2 + 1),j2)))) holds j1 = j2 & |.(i1 - i2).| <= 1

    proof

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

       A1: 1 <= i1 and

       A2: (i1 + 1) <= ( len G) and

       A3: 1 <= j1 & j1 <= ( width G) and

       A4: 1 <= i2 and

       A5: (i2 + 1) <= ( len G) and

       A6: 1 <= j2 & j2 <= ( width G);

      

       A7: 1 <= (i1 + 1) by A1, NAT_1: 13;

      

       A8: i1 < ( len G) by A2, NAT_1: 13;

      assume ( LSeg ((G * (i1,j1)),(G * ((i1 + 1),j1)))) meets ( LSeg ((G * (i2,j2)),(G * ((i2 + 1),j2))));

      then

      consider x be object such that

       A9: x in ( LSeg ((G * (i1,j1)),(G * ((i1 + 1),j1)))) and

       A10: x in ( LSeg ((G * (i2,j2)),(G * ((i2 + 1),j2)))) by XBOOLE_0: 3;

      reconsider p = x as Point of ( TOP-REAL 2) by A9;

      consider r1 such that

       A11: p = (((1 - r1) * (G * (i1,j1))) + (r1 * (G * ((i1 + 1),j1)))) and

       A12: r1 >= 0 and

       A13: r1 <= 1 by A9;

      consider r2 such that

       A14: p = (((1 - r2) * (G * (i2,j2))) + (r2 * (G * ((i2 + 1),j2)))) and

       A15: r2 >= 0 and

       A16: r2 <= 1 by A10;

      

       A17: 1 <= (i2 + 1) by A4, NAT_1: 13;

      

       A18: i2 < ( len G) by A5, NAT_1: 13;

      assume

       A19: not thesis;

      per cases by A19;

        suppose j1 <> j2;

        then

         A20: j1 < j2 or j2 < j1 by XXREAL_0: 1;

        

         A21: ((G * (i2,j2)) `2 ) = ((G * (1,j2)) `2 ) by A4, A6, A18, GOBOARD5: 1

        .= ((G * ((i2 + 1),j2)) `2 ) by A5, A6, A17, GOBOARD5: 1;

        ((G * (i1,j1)) `2 ) = ((G * (1,j1)) `2 ) by A1, A3, A8, GOBOARD5: 1

        .= ((G * ((i1 + 1),j1)) `2 ) by A2, A3, A7, GOBOARD5: 1;

        

        then (1 * ((G * (i1,j1)) `2 )) = (((1 - r1) * ((G * (i1,j1)) `2 )) + (r1 * ((G * ((i1 + 1),j1)) `2 )))

        .= ((((1 - r1) * (G * (i1,j1))) `2 ) + (r1 * ((G * ((i1 + 1),j1)) `2 ))) by TOPREAL3: 4

        .= ((((1 - r1) * (G * (i1,j1))) `2 ) + ((r1 * (G * ((i1 + 1),j1))) `2 )) by TOPREAL3: 4

        .= (p `2 ) by A11, TOPREAL3: 2

        .= ((((1 - r2) * (G * (i2,j2))) `2 ) + ((r2 * (G * ((i2 + 1),j2))) `2 )) by A14, TOPREAL3: 2

        .= (((1 - r2) * ((G * (i2,j2)) `2 )) + ((r2 * (G * ((i2 + 1),j2))) `2 )) by TOPREAL3: 4

        .= (((1 - r2) * ((G * (i2,j2)) `2 )) + (r2 * ((G * ((i2 + 1),j2)) `2 ))) by TOPREAL3: 4

        .= ((G * (1,j2)) `2 ) by A5, A6, A17, A21, GOBOARD5: 1

        .= ((G * (i1,j2)) `2 ) by A1, A6, A8, GOBOARD5: 1;

        hence contradiction by A1, A3, A6, A8, A20, GOBOARD5: 4;

      end;

        suppose

         A22: |.(i1 - i2).| > 1;

        

         A23: ((G * ((i2 + 1),j2)) `1 ) = ((G * ((i2 + 1),1)) `1 ) by A5, A6, A17, GOBOARD5: 2

        .= ((G * ((i2 + 1),j1)) `1 ) by A3, A5, A17, GOBOARD5: 2;

        

         A24: ((G * (i2,j2)) `1 ) = ((G * (i2,1)) `1 ) by A4, A6, A18, GOBOARD5: 2

        .= ((G * (i2,j1)) `1 ) by A3, A4, A18, GOBOARD5: 2;

        

         A25: (((1 - r1) * ((G * (i1,j1)) `1 )) + (r1 * ((G * ((i1 + 1),j1)) `1 ))) = ((((1 - r1) * (G * (i1,j1))) `1 ) + (r1 * ((G * ((i1 + 1),j1)) `1 ))) by TOPREAL3: 4

        .= ((((1 - r1) * (G * (i1,j1))) `1 ) + ((r1 * (G * ((i1 + 1),j1))) `1 )) by TOPREAL3: 4

        .= (p `1 ) by A11, TOPREAL3: 2

        .= ((((1 - r2) * (G * (i2,j2))) `1 ) + ((r2 * (G * ((i2 + 1),j2))) `1 )) by A14, TOPREAL3: 2

        .= (((1 - r2) * ((G * (i2,j2)) `1 )) + ((r2 * (G * ((i2 + 1),j2))) `1 )) by TOPREAL3: 4

        .= (((1 - r2) * ((G * (i2,j1)) `1 )) + (r2 * ((G * ((i2 + 1),j1)) `1 ))) by A23, A24, TOPREAL3: 4;

        now

          per cases by A22, Th1;

            suppose

             A26: (i1 + 1) < i2;

            i2 < (i2 + 1) by XREAL_1: 29;

            then ((G * (i2,j1)) `1 ) < ((G * ((i2 + 1),j1)) `1 ) by A3, A4, A5, GOBOARD5: 3;

            then (((1 - r2) * ((G * (i2,j1)) `1 )) + (r2 * ((G * (i2,j1)) `1 ))) = (1 * ((G * (i2,j1)) `1 )) & (r2 * ((G * (i2,j1)) `1 )) <= (r2 * ((G * ((i2 + 1),j1)) `1 )) by A15, XREAL_1: 64;

            then

             A27: ((G * (i2,j1)) `1 ) <= (((1 - r2) * ((G * (i2,j1)) `1 )) + (r2 * ((G * ((i2 + 1),j1)) `1 ))) by XREAL_1: 6;

            i1 < (i1 + 1) by XREAL_1: 29;

            then

             A28: ((G * (i1,j1)) `1 ) <= ((G * ((i1 + 1),j1)) `1 ) by A1, A2, A3, GOBOARD5: 3;

            (1 - r1) >= 0 by A13, XREAL_1: 48;

            then (((1 - r1) * ((G * ((i1 + 1),j1)) `1 )) + (r1 * ((G * ((i1 + 1),j1)) `1 ))) = (1 * ((G * ((i1 + 1),j1)) `1 )) & ((1 - r1) * ((G * (i1,j1)) `1 )) <= ((1 - r1) * ((G * ((i1 + 1),j1)) `1 )) by A28, XREAL_1: 64;

            then

             A29: (((1 - r1) * ((G * (i1,j1)) `1 )) + (r1 * ((G * ((i1 + 1),j1)) `1 ))) <= ((G * ((i1 + 1),j1)) `1 ) by XREAL_1: 6;

            ((G * ((i1 + 1),j1)) `1 ) < ((G * (i2,j1)) `1 ) by A3, A7, A18, A26, GOBOARD5: 3;

            hence contradiction by A25, A29, A27, XXREAL_0: 2;

          end;

            suppose

             A30: (i2 + 1) < i1;

            i1 < (i1 + 1) by XREAL_1: 29;

            then ((G * (i1,j1)) `1 ) < ((G * ((i1 + 1),j1)) `1 ) by A1, A2, A3, GOBOARD5: 3;

            then (((1 - r1) * ((G * (i1,j1)) `1 )) + (r1 * ((G * (i1,j1)) `1 ))) = (1 * ((G * (i1,j1)) `1 )) & (r1 * ((G * (i1,j1)) `1 )) <= (r1 * ((G * ((i1 + 1),j1)) `1 )) by A12, XREAL_1: 64;

            then

             A31: ((G * (i1,j1)) `1 ) <= (((1 - r1) * ((G * (i1,j1)) `1 )) + (r1 * ((G * ((i1 + 1),j1)) `1 ))) by XREAL_1: 6;

            i2 < (i2 + 1) by XREAL_1: 29;

            then

             A32: ((G * (i2,j1)) `1 ) <= ((G * ((i2 + 1),j1)) `1 ) by A3, A4, A5, GOBOARD5: 3;

            (1 - r2) >= 0 by A16, XREAL_1: 48;

            then (((1 - r2) * ((G * ((i2 + 1),j1)) `1 )) + (r2 * ((G * ((i2 + 1),j1)) `1 ))) = (1 * ((G * ((i2 + 1),j1)) `1 )) & ((1 - r2) * ((G * (i2,j1)) `1 )) <= ((1 - r2) * ((G * ((i2 + 1),j1)) `1 )) by A32, XREAL_1: 64;

            then

             A33: (((1 - r2) * ((G * (i2,j1)) `1 )) + (r2 * ((G * ((i2 + 1),j1)) `1 ))) <= ((G * ((i2 + 1),j1)) `1 ) by XREAL_1: 6;

            ((G * ((i2 + 1),j1)) `1 ) < ((G * (i1,j1)) `1 ) by A3, A8, A17, A30, GOBOARD5: 3;

            hence contradiction by A25, A33, A31, XXREAL_0: 2;

          end;

        end;

        hence contradiction;

      end;

    end;

    theorem :: GOBOARD7:21

    

     Th21: for i1,j1,i2,j2 be Nat st 1 <= i1 & i1 <= ( len G) & 1 <= j1 & (j1 + 1) <= ( width G) & 1 <= i2 & (i2 + 1) <= ( len G) & 1 <= j2 & j2 <= ( width G) & ( LSeg ((G * (i1,j1)),(G * (i1,(j1 + 1))))) meets ( LSeg ((G * (i2,j2)),(G * ((i2 + 1),j2)))) holds (i1 = i2 or i1 = (i2 + 1)) & (j1 = j2 or (j1 + 1) = j2)

    proof

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

       A1: 1 <= i1 and

       A2: i1 <= ( len G) and

       A3: 1 <= j1 and

       A4: (j1 + 1) <= ( width G) and

       A5: 1 <= i2 and

       A6: (i2 + 1) <= ( len G) and

       A7: 1 <= j2 and

       A8: j2 <= ( width G);

      assume ( LSeg ((G * (i1,j1)),(G * (i1,(j1 + 1))))) meets ( LSeg ((G * (i2,j2)),(G * ((i2 + 1),j2))));

      then

      consider x be object such that

       A9: x in ( LSeg ((G * (i1,j1)),(G * (i1,(j1 + 1))))) and

       A10: x in ( LSeg ((G * (i2,j2)),(G * ((i2 + 1),j2)))) by XBOOLE_0: 3;

      reconsider p = x as Point of ( TOP-REAL 2) by A9;

      consider r1 such that

       A11: p = (((1 - r1) * (G * (i1,j1))) + (r1 * (G * (i1,(j1 + 1))))) and

       A12: r1 >= 0 and

       A13: r1 <= 1 by A9;

      consider r2 such that

       A14: p = (((1 - r2) * (G * (i2,j2))) + (r2 * (G * ((i2 + 1),j2)))) and

       A15: r2 >= 0 and

       A16: r2 <= 1 by A10;

      

       A17: i2 < ( len G) by A6, NAT_1: 13;

      

       A18: 1 <= (j1 + 1) by A3, NAT_1: 13;

      

      then

       A19: ((G * (i1,(j1 + 1))) `2 ) = ((G * (1,(j1 + 1))) `2 ) by A1, A2, A4, GOBOARD5: 1

      .= ((G * (i2,(j1 + 1))) `2 ) by A4, A5, A18, A17, GOBOARD5: 1;

      

       A20: j1 < ( width G) by A4, NAT_1: 13;

      

      then

       A21: ((G * (i1,j1)) `2 ) = ((G * (1,j1)) `2 ) by A1, A2, A3, GOBOARD5: 1

      .= ((G * (i2,j1)) `2 ) by A3, A5, A20, A17, GOBOARD5: 1;

      

       A22: (((1 - r2) * ((G * (i2,j2)) `2 )) + (r2 * ((G * ((i2 + 1),j2)) `2 ))) = ((((1 - r2) * (G * (i2,j2))) `2 ) + (r2 * ((G * ((i2 + 1),j2)) `2 ))) by TOPREAL3: 4

      .= ((((1 - r2) * (G * (i2,j2))) `2 ) + ((r2 * (G * ((i2 + 1),j2))) `2 )) by TOPREAL3: 4

      .= (p `2 ) by A14, TOPREAL3: 2

      .= ((((1 - r1) * (G * (i1,j1))) `2 ) + ((r1 * (G * (i1,(j1 + 1)))) `2 )) by A11, TOPREAL3: 2

      .= (((1 - r1) * ((G * (i1,j1)) `2 )) + ((r1 * (G * (i1,(j1 + 1)))) `2 )) by TOPREAL3: 4

      .= (((1 - r1) * ((G * (i2,j1)) `2 )) + (r1 * ((G * (i2,(j1 + 1))) `2 ))) by A19, A21, TOPREAL3: 4;

      

       A23: 1 <= (i2 + 1) by A5, NAT_1: 13;

      thus i1 = i2 or i1 = (i2 + 1)

      proof

        

         A24: ((G * (i2,j2)) `1 ) = ((G * (i2,1)) `1 ) by A5, A7, A8, A17, GOBOARD5: 2

        .= ((G * (i2,j1)) `1 ) by A3, A5, A20, A17, GOBOARD5: 2;

        

         A25: ((G * ((i2 + 1),j2)) `1 ) = ((G * ((i2 + 1),1)) `1 ) by A6, A7, A8, A23, GOBOARD5: 2

        .= ((G * ((i2 + 1),j1)) `1 ) by A3, A6, A20, A23, GOBOARD5: 2;

        

         A26: (((1 - r1) * ((G * (i1,j1)) `1 )) + (r1 * ((G * (i1,(j1 + 1))) `1 ))) = ((((1 - r1) * (G * (i1,j1))) `1 ) + (r1 * ((G * (i1,(j1 + 1))) `1 ))) by TOPREAL3: 4

        .= ((((1 - r1) * (G * (i1,j1))) `1 ) + ((r1 * (G * (i1,(j1 + 1)))) `1 )) by TOPREAL3: 4

        .= (p `1 ) by A11, TOPREAL3: 2

        .= ((((1 - r2) * (G * (i2,j2))) `1 ) + ((r2 * (G * ((i2 + 1),j2))) `1 )) by A14, TOPREAL3: 2

        .= (((1 - r2) * ((G * (i2,j2)) `1 )) + ((r2 * (G * ((i2 + 1),j2))) `1 )) by TOPREAL3: 4

        .= (((1 - r2) * ((G * (i2,j1)) `1 )) + (r2 * ((G * ((i2 + 1),j1)) `1 ))) by A25, A24, TOPREAL3: 4;

        

         A27: ((G * (i1,j1)) `1 ) = ((G * (i1,1)) `1 ) by A1, A2, A3, A20, GOBOARD5: 2

        .= ((G * (i1,(j1 + 1))) `1 ) by A1, A2, A4, A18, GOBOARD5: 2;

        assume

         A28: not thesis;

        per cases by A28, XXREAL_0: 1;

          suppose

           A29: i1 < i2 & i1 < (i2 + 1);

          i2 < (i2 + 1) by XREAL_1: 29;

          then ((G * (i2,j1)) `1 ) < ((G * ((i2 + 1),j1)) `1 ) by A3, A5, A6, A20, GOBOARD5: 3;

          then

           A30: (((1 - r2) * ((G * (i2,j1)) `1 )) + (r2 * ((G * (i2,j1)) `1 ))) = (1 * ((G * (i2,j1)) `1 )) & (r2 * ((G * (i2,j1)) `1 )) <= (r2 * ((G * ((i2 + 1),j1)) `1 )) by A15, XREAL_1: 64;

          ((G * (i1,j1)) `1 ) < ((G * (i2,j1)) `1 ) by A1, A3, A20, A17, A29, GOBOARD5: 3;

          hence contradiction by A26, A27, A30, XREAL_1: 6;

        end;

          suppose i1 < i2 & (i2 + 1) < i1;

          hence thesis by NAT_1: 13;

        end;

          suppose i2 < i1 & i1 < (i2 + 1);

          hence thesis by NAT_1: 13;

        end;

          suppose

           A31: (i2 + 1) < i1;

          i2 < (i2 + 1) by XREAL_1: 29;

          then

           A32: ((G * (i2,j1)) `1 ) <= ((G * ((i2 + 1),j1)) `1 ) by A3, A5, A6, A20, GOBOARD5: 3;

          (1 - r2) >= 0 by A16, XREAL_1: 48;

          then (((1 - r2) * ((G * ((i2 + 1),j1)) `1 )) + (r2 * ((G * ((i2 + 1),j1)) `1 ))) = (1 * ((G * ((i2 + 1),j1)) `1 )) & ((1 - r2) * ((G * (i2,j1)) `1 )) <= ((1 - r2) * ((G * ((i2 + 1),j1)) `1 )) by A32, XREAL_1: 64;

          then (((1 - r2) * ((G * (i2,j1)) `1 )) + (r2 * ((G * ((i2 + 1),j1)) `1 ))) <= ((G * ((i2 + 1),j1)) `1 ) by XREAL_1: 6;

          hence contradiction by A2, A3, A20, A23, A26, A27, A31, GOBOARD5: 3;

        end;

      end;

      

       A33: ((G * (i2,j2)) `2 ) = ((G * (1,j2)) `2 ) by A5, A7, A8, A17, GOBOARD5: 1

      .= ((G * ((i2 + 1),j2)) `2 ) by A6, A7, A8, A23, GOBOARD5: 1;

      assume

       A34: not thesis;

      per cases by A34, XXREAL_0: 1;

        suppose

         A35: j2 < j1 & j2 < (j1 + 1);

        j1 < (j1 + 1) by XREAL_1: 29;

        then ((G * (i2,j1)) `2 ) < ((G * (i2,(j1 + 1))) `2 ) by A3, A4, A5, A17, GOBOARD5: 4;

        then

         A36: (((1 - r1) * ((G * (i2,j1)) `2 )) + (r1 * ((G * (i2,j1)) `2 ))) = (1 * ((G * (i2,j1)) `2 )) & (r1 * ((G * (i2,j1)) `2 )) <= (r1 * ((G * (i2,(j1 + 1))) `2 )) by A12, XREAL_1: 64;

        ((G * (i2,j2)) `2 ) < ((G * (i2,j1)) `2 ) by A5, A7, A20, A17, A35, GOBOARD5: 4;

        hence contradiction by A22, A33, A36, XREAL_1: 6;

      end;

        suppose j2 < j1 & (j1 + 1) < j2;

        hence thesis by NAT_1: 13;

      end;

        suppose j1 < j2 & j2 < (j1 + 1);

        hence thesis by NAT_1: 13;

      end;

        suppose

         A37: (j1 + 1) < j2;

        j1 < (j1 + 1) by XREAL_1: 29;

        then

         A38: ((G * (i2,j1)) `2 ) <= ((G * (i2,(j1 + 1))) `2 ) by A3, A4, A5, A17, GOBOARD5: 4;

        (1 - r1) >= 0 by A13, XREAL_1: 48;

        then (((1 - r1) * ((G * (i2,(j1 + 1))) `2 )) + (r1 * ((G * (i2,(j1 + 1))) `2 ))) = (1 * ((G * (i2,(j1 + 1))) `2 )) & ((1 - r1) * ((G * (i2,j1)) `2 )) <= ((1 - r1) * ((G * (i2,(j1 + 1))) `2 )) by A38, XREAL_1: 64;

        then (((1 - r1) * ((G * (i2,j1)) `2 )) + (r1 * ((G * (i2,(j1 + 1))) `2 ))) <= ((G * (i2,(j1 + 1))) `2 ) by XREAL_1: 6;

        hence contradiction by A5, A8, A18, A17, A22, A33, A37, GOBOARD5: 4;

      end;

    end;

    theorem :: GOBOARD7:22

    for i1,j1,i2,j2 be Nat st 1 <= i1 & i1 <= ( len G) & 1 <= j1 & (j1 + 1) <= ( width G) & 1 <= i2 & i2 <= ( len G) & 1 <= j2 & (j2 + 1) <= ( width G) & ( LSeg ((G * (i1,j1)),(G * (i1,(j1 + 1))))) meets ( LSeg ((G * (i2,j2)),(G * (i2,(j2 + 1))))) holds j1 = j2 & ( LSeg ((G * (i1,j1)),(G * (i1,(j1 + 1))))) = ( LSeg ((G * (i2,j2)),(G * (i2,(j2 + 1))))) or j1 = (j2 + 1) & (( LSeg ((G * (i1,j1)),(G * (i1,(j1 + 1))))) /\ ( LSeg ((G * (i2,j2)),(G * (i2,(j2 + 1)))))) = {(G * (i1,j1))} or (j1 + 1) = j2 & (( LSeg ((G * (i1,j1)),(G * (i1,(j1 + 1))))) /\ ( LSeg ((G * (i2,j2)),(G * (i2,(j2 + 1)))))) = {(G * (i2,j2))}

    proof

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

       A1: 1 <= i1 & i1 <= ( len G) and

       A2: 1 <= j1 and

       A3: (j1 + 1) <= ( width G) and

       A4: 1 <= i2 & i2 <= ( len G) and

       A5: 1 <= j2 and

       A6: (j2 + 1) <= ( width G) and

       A7: ( LSeg ((G * (i1,j1)),(G * (i1,(j1 + 1))))) meets ( LSeg ((G * (i2,j2)),(G * (i2,(j2 + 1)))));

      

       A8: i1 = i2 by A1, A2, A3, A4, A5, A6, A7, Th19;

      

       A9: ((j1 + 1) + 1) = (j1 + (1 + 1));

      

       A10: ((j2 + 1) + 1) = (j2 + (1 + 1));

      

       A11: |.(j1 - j2).| = 0 or |.(j1 - j2).| = 1 by A1, A2, A3, A4, A5, A6, A7, Th19, NAT_1: 25;

      per cases by A11, Th2, SEQM_3: 41;

        case j1 = j2;

        hence thesis by A8;

      end;

        case j1 = (j2 + 1);

        hence thesis by A1, A3, A5, A8, A10, Th13;

      end;

        case (j1 + 1) = j2;

        hence thesis by A1, A2, A6, A8, A9, Th13;

      end;

    end;

    theorem :: GOBOARD7:23

    for i1,j1,i2,j2 be Nat st 1 <= i1 & (i1 + 1) <= ( len G) & 1 <= j1 & j1 <= ( width G) & 1 <= i2 & (i2 + 1) <= ( len G) & 1 <= j2 & j2 <= ( width G) & ( LSeg ((G * (i1,j1)),(G * ((i1 + 1),j1)))) meets ( LSeg ((G * (i2,j2)),(G * ((i2 + 1),j2)))) holds i1 = i2 & ( LSeg ((G * (i1,j1)),(G * ((i1 + 1),j1)))) = ( LSeg ((G * (i2,j2)),(G * ((i2 + 1),j2)))) or i1 = (i2 + 1) & (( LSeg ((G * (i1,j1)),(G * ((i1 + 1),j1)))) /\ ( LSeg ((G * (i2,j2)),(G * ((i2 + 1),j2))))) = {(G * (i1,j1))} or (i1 + 1) = i2 & (( LSeg ((G * (i1,j1)),(G * ((i1 + 1),j1)))) /\ ( LSeg ((G * (i2,j2)),(G * ((i2 + 1),j2))))) = {(G * (i2,j2))}

    proof

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

       A1: 1 <= i1 and

       A2: (i1 + 1) <= ( len G) and

       A3: 1 <= j1 & j1 <= ( width G) and

       A4: 1 <= i2 and

       A5: (i2 + 1) <= ( len G) and

       A6: 1 <= j2 & j2 <= ( width G) & ( LSeg ((G * (i1,j1)),(G * ((i1 + 1),j1)))) meets ( LSeg ((G * (i2,j2)),(G * ((i2 + 1),j2))));

      

       A7: j1 = j2 by A1, A2, A3, A4, A5, A6, Th20;

      

       A8: ((i1 + 1) + 1) = (i1 + (1 + 1));

      

       A9: ((i2 + 1) + 1) = (i2 + (1 + 1));

      

       A10: |.(i1 - i2).| = 0 or |.(i1 - i2).| = 1 by A1, A2, A3, A4, A5, A6, Th20, NAT_1: 25;

      per cases by A10, Th2, SEQM_3: 41;

        case i1 = i2;

        hence thesis by A7;

      end;

        case i1 = (i2 + 1);

        hence thesis by A2, A3, A4, A7, A9, Th14;

      end;

        case (i1 + 1) = i2;

        hence thesis by A1, A3, A5, A7, A8, Th14;

      end;

    end;

    theorem :: GOBOARD7:24

    for i1,j1,i2,j2 be Nat st 1 <= i1 & i1 <= ( len G) & 1 <= j1 & (j1 + 1) <= ( width G) & 1 <= i2 & (i2 + 1) <= ( len G) & 1 <= j2 & j2 <= ( width G) & ( LSeg ((G * (i1,j1)),(G * (i1,(j1 + 1))))) meets ( LSeg ((G * (i2,j2)),(G * ((i2 + 1),j2)))) holds j1 = j2 & (( LSeg ((G * (i1,j1)),(G * (i1,(j1 + 1))))) /\ ( LSeg ((G * (i2,j2)),(G * ((i2 + 1),j2))))) = {(G * (i1,j1))} or (j1 + 1) = j2 & (( LSeg ((G * (i1,j1)),(G * (i1,(j1 + 1))))) /\ ( LSeg ((G * (i2,j2)),(G * ((i2 + 1),j2))))) = {(G * (i1,(j1 + 1)))}

    proof

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

       A1: 1 <= i1 & i1 <= ( len G) and

       A2: 1 <= j1 & (j1 + 1) <= ( width G) & 1 <= i2 & (i2 + 1) <= ( len G) and

       A3: 1 <= j2 & j2 <= ( width G) & ( LSeg ((G * (i1,j1)),(G * (i1,(j1 + 1))))) meets ( LSeg ((G * (i2,j2)),(G * ((i2 + 1),j2))));

      per cases by A1, A2, A3, Th21;

        case

         A4: j1 = j2;

        now

          per cases by A1, A2, A3, Th21;

            suppose i1 = i2;

            hence thesis by A2, A4, Th17;

          end;

            suppose i1 = (i2 + 1);

            hence thesis by A2, A4, Th18;

          end;

        end;

        hence thesis;

      end;

        case

         A5: (j1 + 1) = j2;

        now

          per cases by A1, A2, A3, Th21;

            suppose i1 = i2;

            hence thesis by A2, A5, Th15;

          end;

            suppose i1 = (i2 + 1);

            hence thesis by A2, A5, Th16;

          end;

        end;

        hence thesis;

      end;

    end;

    

     Lm1: (1 - (1 / 2)) = (1 / 2);

    theorem :: GOBOARD7:25

    

     Th25: 1 <= i1 & i1 <= ( len G) & 1 <= j1 & (j1 + 1) <= ( width G) & 1 <= i2 & i2 <= ( len G) & 1 <= j2 & (j2 + 1) <= ( width G) & ((1 / 2) * ((G * (i1,j1)) + (G * (i1,(j1 + 1))))) in ( LSeg ((G * (i2,j2)),(G * (i2,(j2 + 1))))) implies i1 = i2 & j1 = j2

    proof

      assume that

       A1: 1 <= i1 & i1 <= ( len G) and

       A2: 1 <= j1 and

       A3: (j1 + 1) <= ( width G) and

       A4: 1 <= i2 & i2 <= ( len G) and

       A5: 1 <= j2 and

       A6: (j2 + 1) <= ( width G);

      set mi = ((1 / 2) * ((G * (i1,j1)) + (G * (i1,(j1 + 1)))));

      

       A7: (((1 / 2) * (G * (i1,j1))) + ((1 / 2) * (G * (i1,(j1 + 1))))) = ((1 / 2) * ((G * (i1,j1)) + (G * (i1,(j1 + 1))))) by RLVECT_1:def 5;

      then

       A8: mi in ( LSeg ((G * (i1,j1)),(G * (i1,(j1 + 1))))) by Lm1;

      assume

       A9: mi in ( LSeg ((G * (i2,j2)),(G * (i2,(j2 + 1)))));

      then

       A10: ( LSeg ((G * (i1,j1)),(G * (i1,(j1 + 1))))) meets ( LSeg ((G * (i2,j2)),(G * (i2,(j2 + 1))))) by A8, XBOOLE_0: 3;

      hence

       A11: i1 = i2 by A1, A2, A3, A4, A5, A6, Th19;

      now

        j1 < (j1 + 1) by XREAL_1: 29;

        then

         A12: ((G * (i1,(j1 + 1))) `2 ) > ((G * (i1,j1)) `2 ) by A1, A2, A3, GOBOARD5: 4;

        assume

         A13: |.(j1 - j2).| = 1;

        per cases by A13, SEQM_3: 41;

          suppose

           A14: j1 = (j2 + 1);

          then (( LSeg ((G * (i2,j2)),(G * (i2,(j2 + 1))))) /\ ( LSeg ((G * (i2,(j2 + 1))),(G * (i2,(j2 + 2)))))) = {(G * (i2,(j2 + 1)))} by A3, A4, A5, Th13;

          then mi in {(G * (i1,j1))} by A9, A8, A11, A14, XBOOLE_0:def 4;

          

          then (((1 / 2) * (G * (i1,j1))) + ((1 / 2) * (G * (i1,(j1 + 1))))) = (G * (i1,j1)) by A7, TARSKI:def 1

          .= (((1 / 2) + (1 / 2)) * (G * (i1,j1))) by RLVECT_1:def 8

          .= (((1 / 2) * (G * (i1,j1))) + ((1 / 2) * (G * (i1,j1)))) by RLVECT_1:def 6;

          then ((1 / 2) * (G * (i1,j1))) = ((1 / 2) * (G * (i1,(j1 + 1)))) by Th3;

          hence contradiction by A12, RLVECT_1: 36;

        end;

          suppose

           A15: (j1 + 1) = j2;

          then (( LSeg ((G * (i2,j1)),(G * (i2,(j1 + 1))))) /\ ( LSeg ((G * (i2,(j1 + 1))),(G * (i2,(j1 + 2)))))) = {(G * (i2,(j1 + 1)))} by A2, A4, A6, Th13;

          then mi in {(G * (i1,j2))} by A9, A8, A11, A15, XBOOLE_0:def 4;

          

          then (((1 / 2) * (G * (i1,j1))) + ((1 / 2) * (G * (i1,(j1 + 1))))) = (G * (i1,j2)) by A7, TARSKI:def 1

          .= (((1 / 2) + (1 / 2)) * (G * (i1,j2))) by RLVECT_1:def 8

          .= (((1 / 2) * (G * (i1,j2))) + ((1 / 2) * (G * (i1,j2)))) by RLVECT_1:def 6;

          then ((1 / 2) * (G * (i1,j1))) = ((1 / 2) * (G * (i1,(j1 + 1)))) by A15, Th3;

          hence contradiction by A12, RLVECT_1: 36;

        end;

      end;

      then |.(j1 - j2).| = 0 by A1, A2, A3, A4, A5, A6, A10, Th19, NAT_1: 25;

      hence thesis by Th2;

    end;

    theorem :: GOBOARD7:26

    

     Th26: 1 <= i1 & (i1 + 1) <= ( len G) & 1 <= j1 & j1 <= ( width G) & 1 <= i2 & (i2 + 1) <= ( len G) & 1 <= j2 & j2 <= ( width G) & ((1 / 2) * ((G * (i1,j1)) + (G * ((i1 + 1),j1)))) in ( LSeg ((G * (i2,j2)),(G * ((i2 + 1),j2)))) implies i1 = i2 & j1 = j2

    proof

      assume that

       A1: 1 <= i1 and

       A2: (i1 + 1) <= ( len G) and

       A3: 1 <= j1 & j1 <= ( width G) and

       A4: 1 <= i2 and

       A5: (i2 + 1) <= ( len G) and

       A6: 1 <= j2 & j2 <= ( width G);

      set mi = ((1 / 2) * ((G * (i1,j1)) + (G * ((i1 + 1),j1))));

      

       A7: (((1 / 2) * (G * (i1,j1))) + ((1 / 2) * (G * ((i1 + 1),j1)))) = ((1 / 2) * ((G * (i1,j1)) + (G * ((i1 + 1),j1)))) by RLVECT_1:def 5;

      then

       A8: mi in ( LSeg ((G * (i1,j1)),(G * ((i1 + 1),j1)))) by Lm1;

      assume

       A9: mi in ( LSeg ((G * (i2,j2)),(G * ((i2 + 1),j2))));

      then

       A10: ( LSeg ((G * (i1,j1)),(G * ((i1 + 1),j1)))) meets ( LSeg ((G * (i2,j2)),(G * ((i2 + 1),j2)))) by A8, XBOOLE_0: 3;

      then

       A11: j1 = j2 by A1, A2, A3, A4, A5, A6, Th20;

      now

        i1 < (i1 + 1) by XREAL_1: 29;

        then

         A12: ((G * ((i1 + 1),j1)) `1 ) > ((G * (i1,j1)) `1 ) by A1, A2, A3, GOBOARD5: 3;

        assume

         A13: |.(i1 - i2).| = 1;

        per cases by A13, SEQM_3: 41;

          suppose

           A14: i1 = (i2 + 1);

          then (( LSeg ((G * (i2,j2)),(G * ((i2 + 1),j2)))) /\ ( LSeg ((G * ((i2 + 1),j2)),(G * ((i2 + 2),j2))))) = {(G * ((i2 + 1),j2))} by A2, A4, A6, Th14;

          then mi in {(G * (i1,j1))} by A9, A8, A11, A14, XBOOLE_0:def 4;

          

          then (((1 / 2) * (G * (i1,j1))) + ((1 / 2) * (G * ((i1 + 1),j1)))) = (G * (i1,j1)) by A7, TARSKI:def 1

          .= (((1 / 2) + (1 / 2)) * (G * (i1,j1))) by RLVECT_1:def 8

          .= (((1 / 2) * (G * (i1,j1))) + ((1 / 2) * (G * (i1,j1)))) by RLVECT_1:def 6;

          then ((1 / 2) * (G * (i1,j1))) = ((1 / 2) * (G * ((i1 + 1),j1))) by Th3;

          hence contradiction by A12, RLVECT_1: 36;

        end;

          suppose

           A15: (i1 + 1) = i2;

          then (( LSeg ((G * (i1,j2)),(G * ((i1 + 1),j2)))) /\ ( LSeg ((G * ((i1 + 1),j2)),(G * ((i1 + 2),j2))))) = {(G * ((i1 + 1),j2))} by A1, A5, A6, Th14;

          then mi in {(G * (i2,j1))} by A9, A8, A11, A15, XBOOLE_0:def 4;

          

          then (((1 / 2) * (G * (i1,j1))) + ((1 / 2) * (G * ((i1 + 1),j1)))) = (G * (i2,j1)) by A7, TARSKI:def 1

          .= (((1 / 2) + (1 / 2)) * (G * (i2,j1))) by RLVECT_1:def 8

          .= (((1 / 2) * (G * (i2,j1))) + ((1 / 2) * (G * (i2,j1)))) by RLVECT_1:def 6;

          then ((1 / 2) * (G * (i1,j1))) = ((1 / 2) * (G * ((i1 + 1),j1))) by A15, Th3;

          hence contradiction by A12, RLVECT_1: 36;

        end;

      end;

      then |.(i1 - i2).| = 0 by A1, A2, A3, A4, A5, A6, A10, Th20, NAT_1: 25;

      hence i1 = i2 by Th2;

      thus thesis by A1, A2, A3, A4, A5, A6, A10, Th20;

    end;

    theorem :: GOBOARD7:27

    

     Th27: 1 <= i1 & (i1 + 1) <= ( len G) & 1 <= j1 & j1 <= ( width G) implies not ex i2, j2 st 1 <= i2 & i2 <= ( len G) & 1 <= j2 & (j2 + 1) <= ( width G) & ((1 / 2) * ((G * (i1,j1)) + (G * ((i1 + 1),j1)))) in ( LSeg ((G * (i2,j2)),(G * (i2,(j2 + 1)))))

    proof

      assume that

       A1: 1 <= i1 & (i1 + 1) <= ( len G) and

       A2: 1 <= j1 & j1 <= ( width G);

      

       A3: i1 < (i1 + 1) by XREAL_1: 29;

      set mi = ((1 / 2) * ((G * (i1,j1)) + (G * ((i1 + 1),j1))));

      given i2, j2 such that

       A4: 1 <= i2 & i2 <= ( len G) and

       A5: 1 <= j2 & (j2 + 1) <= ( width G) and

       A6: mi in ( LSeg ((G * (i2,j2)),(G * (i2,(j2 + 1)))));

      

       A7: (((1 / 2) * (G * (i1,j1))) + ((1 / 2) * (G * ((i1 + 1),j1)))) = ((1 / 2) * ((G * (i1,j1)) + (G * ((i1 + 1),j1)))) by RLVECT_1:def 5;

      then

       A8: mi in ( LSeg ((G * (i1,j1)),(G * ((i1 + 1),j1)))) by Lm1;

      then

       A9: ( LSeg ((G * (i1,j1)),(G * ((i1 + 1),j1)))) meets ( LSeg ((G * (i2,j2)),(G * (i2,(j2 + 1))))) by A6, XBOOLE_0: 3;

      per cases by A1, A2, A4, A5, A9, Th21;

        suppose

         A10: j1 = j2 & i1 = i2;

        then (( LSeg ((G * (i1,j1)),(G * ((i1 + 1),j1)))) /\ ( LSeg ((G * (i1,(j1 + 1))),(G * (i1,j1))))) = {(G * (i1,j1))} by A1, A5, Th17;

        then mi in {(G * (i1,j1))} by A6, A8, A10, XBOOLE_0:def 4;

        

        then (((1 / 2) * (G * (i1,j1))) + ((1 / 2) * (G * ((i1 + 1),j1)))) = (G * (i1,j1)) by A7, TARSKI:def 1

        .= (((1 / 2) + (1 / 2)) * (G * (i1,j1))) by RLVECT_1:def 8

        .= (((1 / 2) * (G * (i1,j1))) + ((1 / 2) * (G * (i1,j1)))) by RLVECT_1:def 6;

        then

         A11: ((1 / 2) * (G * ((i1 + 1),j1))) = ((1 / 2) * (G * (i1,j1))) by Th3;

        ((G * ((i1 + 1),j1)) `1 ) > ((G * (i1,j1)) `1 ) by A1, A2, A3, GOBOARD5: 3;

        hence contradiction by A11, RLVECT_1: 36;

      end;

        suppose

         A12: j1 = j2 & (i1 + 1) = i2;

        then (( LSeg ((G * (i1,j1)),(G * ((i1 + 1),j1)))) /\ ( LSeg ((G * ((i1 + 1),(j1 + 1))),(G * ((i1 + 1),j1))))) = {(G * ((i1 + 1),j1))} by A1, A5, Th18;

        then mi in {(G * ((i1 + 1),j1))} by A6, A8, A12, XBOOLE_0:def 4;

        

        then (((1 / 2) * (G * (i1,j1))) + ((1 / 2) * (G * ((i1 + 1),j1)))) = (G * ((i1 + 1),j1)) by A7, TARSKI:def 1

        .= (((1 / 2) + (1 / 2)) * (G * ((i1 + 1),j1))) by RLVECT_1:def 8

        .= (((1 / 2) * (G * ((i1 + 1),j1))) + ((1 / 2) * (G * ((i1 + 1),j1)))) by RLVECT_1:def 6;

        then

         A13: ((1 / 2) * (G * ((i1 + 1),j1))) = ((1 / 2) * (G * (i1,j1))) by Th3;

        ((G * ((i1 + 1),j1)) `1 ) > ((G * (i1,j1)) `1 ) by A1, A2, A3, GOBOARD5: 3;

        hence contradiction by A13, RLVECT_1: 36;

      end;

        suppose

         A14: j1 = (j2 + 1) & i1 = i2;

        then (( LSeg ((G * (i1,j1)),(G * ((i1 + 1),j1)))) /\ ( LSeg ((G * (i1,j1)),(G * (i1,j2))))) = {(G * (i1,j1))} by A1, A5, Th15;

        then mi in {(G * (i1,j1))} by A6, A8, A14, XBOOLE_0:def 4;

        

        then (((1 / 2) * (G * (i1,j1))) + ((1 / 2) * (G * ((i1 + 1),j1)))) = (G * (i1,j1)) by A7, TARSKI:def 1

        .= (((1 / 2) + (1 / 2)) * (G * (i1,j1))) by RLVECT_1:def 8

        .= (((1 / 2) * (G * (i1,j1))) + ((1 / 2) * (G * (i1,j1)))) by RLVECT_1:def 6;

        then

         A15: ((1 / 2) * (G * ((i1 + 1),j1))) = ((1 / 2) * (G * (i1,j1))) by Th3;

        ((G * ((i1 + 1),j1)) `1 ) > ((G * (i1,j1)) `1 ) by A1, A2, A3, GOBOARD5: 3;

        hence contradiction by A15, RLVECT_1: 36;

      end;

        suppose

         A16: j1 = (j2 + 1) & (i1 + 1) = i2;

        then (( LSeg ((G * (i1,j1)),(G * ((i1 + 1),j1)))) /\ ( LSeg ((G * ((i1 + 1),j1)),(G * ((i1 + 1),j2))))) = {(G * ((i1 + 1),j1))} by A1, A5, Th16;

        then mi in {(G * ((i1 + 1),j1))} by A6, A8, A16, XBOOLE_0:def 4;

        

        then (((1 / 2) * (G * (i1,j1))) + ((1 / 2) * (G * ((i1 + 1),j1)))) = (G * ((i1 + 1),j1)) by A7, TARSKI:def 1

        .= (((1 / 2) + (1 / 2)) * (G * ((i1 + 1),j1))) by RLVECT_1:def 8

        .= (((1 / 2) * (G * ((i1 + 1),j1))) + ((1 / 2) * (G * ((i1 + 1),j1)))) by RLVECT_1:def 6;

        then

         A17: ((1 / 2) * (G * ((i1 + 1),j1))) = ((1 / 2) * (G * (i1,j1))) by Th3;

        ((G * ((i1 + 1),j1)) `1 ) > ((G * (i1,j1)) `1 ) by A1, A2, A3, GOBOARD5: 3;

        hence contradiction by A17, RLVECT_1: 36;

      end;

    end;

    theorem :: GOBOARD7:28

    

     Th28: 1 <= i1 & i1 <= ( len G) & 1 <= j1 & (j1 + 1) <= ( width G) implies not ex i2, j2 st 1 <= i2 & (i2 + 1) <= ( len G) & 1 <= j2 & j2 <= ( width G) & ((1 / 2) * ((G * (i1,j1)) + (G * (i1,(j1 + 1))))) in ( LSeg ((G * (i2,j2)),(G * ((i2 + 1),j2))))

    proof

      assume that

       A1: 1 <= i1 & i1 <= ( len G) and

       A2: 1 <= j1 & (j1 + 1) <= ( width G);

      

       A3: j1 < (j1 + 1) by XREAL_1: 29;

      set mi = ((1 / 2) * ((G * (i1,j1)) + (G * (i1,(j1 + 1)))));

      given i2, j2 such that

       A4: 1 <= i2 & (i2 + 1) <= ( len G) and

       A5: 1 <= j2 & j2 <= ( width G) and

       A6: mi in ( LSeg ((G * (i2,j2)),(G * ((i2 + 1),j2))));

      

       A7: (((1 / 2) * (G * (i1,j1))) + ((1 / 2) * (G * (i1,(j1 + 1))))) = ((1 / 2) * ((G * (i1,j1)) + (G * (i1,(j1 + 1))))) by RLVECT_1:def 5;

      then

       A8: mi in ( LSeg ((G * (i1,j1)),(G * (i1,(j1 + 1))))) by Lm1;

      then

       A9: ( LSeg ((G * (i1,j1)),(G * (i1,(j1 + 1))))) meets ( LSeg ((G * (i2,j2)),(G * ((i2 + 1),j2)))) by A6, XBOOLE_0: 3;

      per cases by A1, A2, A4, A5, A9, Th21;

        suppose

         A10: i1 = i2 & j1 = j2;

        then (( LSeg ((G * (i1,j1)),(G * (i1,(j1 + 1))))) /\ ( LSeg ((G * ((i1 + 1),j1)),(G * (i1,j1))))) = {(G * (i1,j1))} by A2, A4, Th17;

        then mi in {(G * (i1,j1))} by A6, A8, A10, XBOOLE_0:def 4;

        

        then (((1 / 2) * (G * (i1,j1))) + ((1 / 2) * (G * (i1,(j1 + 1))))) = (G * (i1,j1)) by A7, TARSKI:def 1

        .= (((1 / 2) + (1 / 2)) * (G * (i1,j1))) by RLVECT_1:def 8

        .= (((1 / 2) * (G * (i1,j1))) + ((1 / 2) * (G * (i1,j1)))) by RLVECT_1:def 6;

        then

         A11: ((1 / 2) * (G * (i1,(j1 + 1)))) = ((1 / 2) * (G * (i1,j1))) by Th3;

        ((G * (i1,(j1 + 1))) `2 ) > ((G * (i1,j1)) `2 ) by A1, A2, A3, GOBOARD5: 4;

        hence contradiction by A11, RLVECT_1: 36;

      end;

        suppose

         A12: i1 = i2 & (j1 + 1) = j2;

        then (( LSeg ((G * (i1,j1)),(G * (i1,(j1 + 1))))) /\ ( LSeg ((G * ((i1 + 1),(j1 + 1))),(G * (i1,(j1 + 1)))))) = {(G * (i1,(j1 + 1)))} by A2, A4, Th15;

        then mi in {(G * (i1,(j1 + 1)))} by A6, A8, A12, XBOOLE_0:def 4;

        

        then (((1 / 2) * (G * (i1,j1))) + ((1 / 2) * (G * (i1,(j1 + 1))))) = (G * (i1,(j1 + 1))) by A7, TARSKI:def 1

        .= (((1 / 2) + (1 / 2)) * (G * (i1,(j1 + 1)))) by RLVECT_1:def 8

        .= (((1 / 2) * (G * (i1,(j1 + 1)))) + ((1 / 2) * (G * (i1,(j1 + 1))))) by RLVECT_1:def 6;

        then

         A13: ((1 / 2) * (G * (i1,(j1 + 1)))) = ((1 / 2) * (G * (i1,j1))) by Th3;

        ((G * (i1,(j1 + 1))) `2 ) > ((G * (i1,j1)) `2 ) by A1, A2, A3, GOBOARD5: 4;

        hence contradiction by A13, RLVECT_1: 36;

      end;

        suppose

         A14: i1 = (i2 + 1) & j1 = j2;

        then (( LSeg ((G * (i1,j1)),(G * (i1,(j1 + 1))))) /\ ( LSeg ((G * (i1,j1)),(G * (i2,j1))))) = {(G * (i1,j1))} by A2, A4, Th18;

        then mi in {(G * (i1,j1))} by A6, A8, A14, XBOOLE_0:def 4;

        

        then (((1 / 2) * (G * (i1,j1))) + ((1 / 2) * (G * (i1,(j1 + 1))))) = (G * (i1,j1)) by A7, TARSKI:def 1

        .= (((1 / 2) + (1 / 2)) * (G * (i1,j1))) by RLVECT_1:def 8

        .= (((1 / 2) * (G * (i1,j1))) + ((1 / 2) * (G * (i1,j1)))) by RLVECT_1:def 6;

        then

         A15: ((1 / 2) * (G * (i1,(j1 + 1)))) = ((1 / 2) * (G * (i1,j1))) by Th3;

        ((G * (i1,(j1 + 1))) `2 ) > ((G * (i1,j1)) `2 ) by A1, A2, A3, GOBOARD5: 4;

        hence contradiction by A15, RLVECT_1: 36;

      end;

        suppose

         A16: i1 = (i2 + 1) & (j1 + 1) = j2;

        then (( LSeg ((G * (i1,j1)),(G * (i1,(j1 + 1))))) /\ ( LSeg ((G * (i1,(j1 + 1))),(G * (i2,(j1 + 1)))))) = {(G * (i1,(j1 + 1)))} by A2, A4, Th16;

        then mi in {(G * (i1,(j1 + 1)))} by A6, A8, A16, XBOOLE_0:def 4;

        

        then (((1 / 2) * (G * (i1,j1))) + ((1 / 2) * (G * (i1,(j1 + 1))))) = (G * (i1,(j1 + 1))) by A7, TARSKI:def 1

        .= (((1 / 2) + (1 / 2)) * (G * (i1,(j1 + 1)))) by RLVECT_1:def 8

        .= (((1 / 2) * (G * (i1,(j1 + 1)))) + ((1 / 2) * (G * (i1,(j1 + 1))))) by RLVECT_1:def 6;

        then

         A17: ((1 / 2) * (G * (i1,(j1 + 1)))) = ((1 / 2) * (G * (i1,j1))) by Th3;

        ((G * (i1,(j1 + 1))) `2 ) > ((G * (i1,j1)) `2 ) by A1, A2, A3, GOBOARD5: 4;

        hence contradiction by A17, RLVECT_1: 36;

      end;

    end;

    begin

    reserve f for non constant standard special_circular_sequence;

    

     Lm2: ( len f) > 1

    proof

      consider n1,n2 be object such that

       A1: n1 in ( dom f) and

       A2: n2 in ( dom f) & (f . n1) <> (f . n2) by FUNCT_1:def 10;

      reconsider df = ( dom f) as finite set;

       A3:

      now

        assume

         A4: ( card df) <= 1;

        per cases by A4, NAT_1: 25;

          suppose ( card df) = 0 ;

          hence contradiction by A1;

        end;

          suppose ( card df) = 1;

          then

          consider x be object such that

           A5: ( dom f) = {x} by CARD_2: 42;

          n1 = x by A1, A5, TARSKI:def 1;

          hence contradiction by A2, A5, TARSKI:def 1;

        end;

      end;

      ( dom f) = ( Seg ( len f)) by FINSEQ_1:def 3;

      hence thesis by A3, FINSEQ_1: 57;

    end;

    theorem :: GOBOARD7:29

    

     Th29: for f be standard non empty FinSequence of ( TOP-REAL 2) st i in ( dom f) & (i + 1) in ( dom f) holds (f /. i) <> (f /. (i + 1))

    proof

      

       A1: |. 0 .| = 0 by ABSVALUE: 2;

      let f be standard non empty FinSequence of ( TOP-REAL 2) such that

       A2: i in ( dom f) and

       A3: (i + 1) in ( dom f);

      

       A4: f is_sequence_on ( GoB f) by GOBOARD5:def 5;

      then

      consider i1, j1 such that

       A5: [i1, j1] in ( Indices ( GoB f)) & (f /. i) = (( GoB f) * (i1,j1)) by A2, GOBOARD1:def 9;

      consider i2, j2 such that

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

       A7: (f /. (i + 1)) = (( GoB f) * (i2,j2)) by A3, A4, GOBOARD1:def 9;

      assume

       A8: (f /. i) = (f /. (i + 1));

      then j1 = j2 by A5, A6, A7, GOBOARD1: 5;

      then

       A9: (j1 - j2) = 0 ;

      i1 = i2 by A5, A6, A7, A8, GOBOARD1: 5;

      then (i1 - i2) = 0 ;

      then ( |. 0 .| + |. 0 .|) = 1 by A2, A3, A4, A5, A7, A9, GOBOARD1:def 9;

      hence contradiction by A1;

    end;

    theorem :: GOBOARD7:30

    

     Th30: ex i st i in ( dom f) & ((f /. i) `1 ) <> ((f /. 1) `1 )

    proof

      assume

       A1: for i st i in ( dom f) holds ((f /. i) `1 ) = ((f /. 1) `1 );

      

       A2: ( len f) > 1 by Lm2;

      then

       A3: ( len f) >= (1 + 1) by NAT_1: 13;

      then

       A4: (1 + 1) in ( dom f) by FINSEQ_3: 25;

       A5:

      now

        assume

         A6: ((f /. 2) `2 ) = ((f /. 1) `2 );

        ((f /. 2) `1 ) = ((f /. 1) `1 ) by A1, A4;

        

        then (f /. 2) = |[((f /. 1) `1 ), ((f /. 1) `2 )]| by A6, EUCLID: 53

        .= (f /. 1) by EUCLID: 53;

        hence contradiction by A4, Th29, FINSEQ_5: 6;

      end;

      ( len f) = 2 implies (f /. 2) = (f /. 1) by FINSEQ_6:def 1;

      then

       A7: 2 < ( len f) by A3, A5, XXREAL_0: 1;

      per cases by A5, XXREAL_0: 1;

        suppose

         A8: ((f /. 2) `2 ) < ((f /. 1) `2 );

        defpred P[ Nat] means 2 <= $1 & $1 < ( len f) implies ((f /. $1) `2 ) <= ((f /. 2) `2 ) & ((f /. ($1 + 1)) `2 ) < ((f /. $1) `2 );

        

         A9: for j st P[j] holds P[(j + 1)]

        proof

          let j such that

           A10: 2 <= j & j < ( len f) implies ((f /. j) `2 ) <= ((f /. 2) `2 ) & ((f /. (j + 1)) `2 ) < ((f /. j) `2 ) and

           A11: 2 <= (j + 1) and

           A12: (j + 1) < ( len f);

          (1 + 1) <= (j + 1) by A11;

          then

           A13: 1 <= j by XREAL_1: 6;

          thus ((f /. (j + 1)) `2 ) <= ((f /. 2) `2 )

          proof

            per cases by A11, XXREAL_0: 1;

              suppose 2 = (j + 1);

              hence thesis;

            end;

              suppose 2 < (j + 1);

              hence thesis by A10, A12, NAT_1: 13, XXREAL_0: 2;

            end;

          end;

          

           A14: ((j + 1) + 1) <= ( len f) by A12, NAT_1: 13;

           A15:

          now

            per cases by A11, XXREAL_0: 1;

              suppose (1 + 1) = (j + 1);

              hence ((f /. (j + 1)) `2 ) < ((f /. j) `2 ) by A8;

            end;

              suppose 2 < (j + 1);

              hence ((f /. (j + 1)) `2 ) < ((f /. j) `2 ) by A10, A12, NAT_1: 13;

            end;

          end;

          

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

          then

           A17: (j + 1) in ( dom f) by A12, FINSEQ_3: 25;

          then

           A18: ((f /. (j + 1)) `1 ) = ((f /. 1) `1 ) by A1;

          j < ( len f) by A12, NAT_1: 13;

          then

           A19: j in ( dom f) by A13, FINSEQ_3: 25;

          then

           A20: ((f /. j) `1 ) = ((f /. 1) `1 ) by A1;

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

          then

           A21: ((j + 1) + 1) in ( dom f) by A14, FINSEQ_3: 25;

          then

           A22: ((f /. ((j + 1) + 1)) `1 ) = ((f /. 1) `1 ) by A1;

          assume

           A23: ((f /. ((j + 1) + 1)) `2 ) >= ((f /. (j + 1)) `2 );

          per cases by A23, XXREAL_0: 1;

            suppose

             A24: ((f /. ((j + 1) + 1)) `2 ) > ((f /. (j + 1)) `2 );

            now

              per cases ;

                suppose ((f /. j) `2 ) <= ((f /. ((j + 1) + 1)) `2 );

                then (f /. j) in ( LSeg ((f /. (j + 1)),(f /. ((j + 1) + 1)))) by A15, A20, A18, A22, Th7;

                then

                 A25: (f /. j) in ( LSeg (f,(j + 1))) by A14, A16, TOPREAL1:def 3;

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

                then

                 A26: (( LSeg (f,j)) /\ ( LSeg (f,(j + 1)))) = {(f /. (j + 1))} by A14, A13, TOPREAL1:def 6;

                (f /. j) in ( LSeg (f,j)) by A12, A13, TOPREAL1: 21;

                then (f /. j) in (( LSeg (f,j)) /\ ( LSeg (f,(j + 1)))) by A25, XBOOLE_0:def 4;

                then (f /. j) = (f /. (j + 1)) by A26, TARSKI:def 1;

                hence contradiction by A19, A17, Th29;

              end;

                suppose ((f /. j) `2 ) >= ((f /. ((j + 1) + 1)) `2 );

                then (f /. ((j + 1) + 1)) in ( LSeg ((f /. j),(f /. (j + 1)))) by A20, A18, A22, A24, Th7;

                then

                 A27: (f /. ((j + 1) + 1)) in ( LSeg (f,j)) by A12, A13, TOPREAL1:def 3;

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

                then

                 A28: (( LSeg (f,j)) /\ ( LSeg (f,(j + 1)))) = {(f /. (j + 1))} by A14, A13, TOPREAL1:def 6;

                (f /. ((j + 1) + 1)) in ( LSeg (f,(j + 1))) by A14, A16, TOPREAL1: 21;

                then (f /. ((j + 1) + 1)) in (( LSeg (f,j)) /\ ( LSeg (f,(j + 1)))) by A27, XBOOLE_0:def 4;

                then (f /. ((j + 1) + 1)) = (f /. (j + 1)) by A28, TARSKI:def 1;

                hence contradiction by A17, A21, Th29;

              end;

            end;

            hence contradiction;

          end;

            suppose

             A29: ((f /. ((j + 1) + 1)) `2 ) = ((f /. (j + 1)) `2 );

            ((f /. ((j + 1) + 1)) `1 ) = ((f /. 1) `1 ) by A1, A21

            .= ((f /. (j + 1)) `1 ) by A1, A17;

            

            then (f /. ((j + 1) + 1)) = |[((f /. (j + 1)) `1 ), ((f /. (j + 1)) `2 )]| by A29, EUCLID: 53

            .= (f /. (j + 1)) by EUCLID: 53;

            hence contradiction by A17, A21, Th29;

          end;

        end;

        

         A30: ((( len f) -' 1) + 1) = ( len f) by A2, XREAL_1: 235;

        then

         A31: 2 <= (( len f) -' 1) & (( len f) -' 1) < ( len f) by A7, NAT_1: 13;

        

         A32: P[ 0 ];

        

         A33: for j holds P[j] from NAT_1:sch 2( A32, A9);

        then

         A34: ((f /. (( len f) -' 1)) `2 ) <= ((f /. 2) `2 ) by A31;

        ((f /. ( len f)) `2 ) < ((f /. (( len f) -' 1)) `2 ) by A33, A30, A31;

        then ((f /. ( len f)) `2 ) < ((f /. 2) `2 ) by A34, XXREAL_0: 2;

        hence contradiction by A8, FINSEQ_6:def 1;

      end;

        suppose

         A35: ((f /. 2) `2 ) > ((f /. 1) `2 );

        defpred P[ Nat] means 2 <= $1 & $1 < ( len f) implies ((f /. $1) `2 ) >= ((f /. 2) `2 ) & ((f /. ($1 + 1)) `2 ) > ((f /. $1) `2 );

        

         A36: for j st P[j] holds P[(j + 1)]

        proof

          let j such that

           A37: 2 <= j & j < ( len f) implies ((f /. j) `2 ) >= ((f /. 2) `2 ) & ((f /. (j + 1)) `2 ) > ((f /. j) `2 ) and

           A38: 2 <= (j + 1) and

           A39: (j + 1) < ( len f);

          (1 + 1) <= (j + 1) by A38;

          then

           A40: 1 <= j by XREAL_1: 6;

          thus ((f /. (j + 1)) `2 ) >= ((f /. 2) `2 )

          proof

            per cases by A38, XXREAL_0: 1;

              suppose 2 = (j + 1);

              hence thesis;

            end;

              suppose 2 < (j + 1);

              hence thesis by A37, A39, NAT_1: 13, XXREAL_0: 2;

            end;

          end;

          

           A41: ((j + 1) + 1) <= ( len f) by A39, NAT_1: 13;

           A42:

          now

            per cases by A38, XXREAL_0: 1;

              suppose (1 + 1) = (j + 1);

              hence ((f /. (j + 1)) `2 ) > ((f /. j) `2 ) by A35;

            end;

              suppose 2 < (j + 1);

              hence ((f /. (j + 1)) `2 ) > ((f /. j) `2 ) by A37, A39, NAT_1: 13;

            end;

          end;

          

           A43: 1 <= (j + 1) by NAT_1: 11;

          then

           A44: (j + 1) in ( dom f) by A39, FINSEQ_3: 25;

          then

           A45: ((f /. (j + 1)) `1 ) = ((f /. 1) `1 ) by A1;

          j < ( len f) by A39, NAT_1: 13;

          then

           A46: j in ( dom f) by A40, FINSEQ_3: 25;

          then

           A47: ((f /. j) `1 ) = ((f /. 1) `1 ) by A1;

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

          then

           A48: ((j + 1) + 1) in ( dom f) by A41, FINSEQ_3: 25;

          then

           A49: ((f /. ((j + 1) + 1)) `1 ) = ((f /. 1) `1 ) by A1;

          assume

           A50: ((f /. ((j + 1) + 1)) `2 ) <= ((f /. (j + 1)) `2 );

          per cases by A50, XXREAL_0: 1;

            suppose

             A51: ((f /. ((j + 1) + 1)) `2 ) < ((f /. (j + 1)) `2 );

            now

              per cases ;

                suppose ((f /. j) `2 ) >= ((f /. ((j + 1) + 1)) `2 );

                then (f /. j) in ( LSeg ((f /. (j + 1)),(f /. ((j + 1) + 1)))) by A42, A47, A45, A49, Th7;

                then

                 A52: (f /. j) in ( LSeg (f,(j + 1))) by A41, A43, TOPREAL1:def 3;

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

                then

                 A53: (( LSeg (f,j)) /\ ( LSeg (f,(j + 1)))) = {(f /. (j + 1))} by A41, A40, TOPREAL1:def 6;

                (f /. j) in ( LSeg (f,j)) by A39, A40, TOPREAL1: 21;

                then (f /. j) in (( LSeg (f,j)) /\ ( LSeg (f,(j + 1)))) by A52, XBOOLE_0:def 4;

                then (f /. j) = (f /. (j + 1)) by A53, TARSKI:def 1;

                hence contradiction by A46, A44, Th29;

              end;

                suppose ((f /. j) `2 ) <= ((f /. ((j + 1) + 1)) `2 );

                then (f /. ((j + 1) + 1)) in ( LSeg ((f /. j),(f /. (j + 1)))) by A47, A45, A49, A51, Th7;

                then

                 A54: (f /. ((j + 1) + 1)) in ( LSeg (f,j)) by A39, A40, TOPREAL1:def 3;

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

                then

                 A55: (( LSeg (f,j)) /\ ( LSeg (f,(j + 1)))) = {(f /. (j + 1))} by A41, A40, TOPREAL1:def 6;

                (f /. ((j + 1) + 1)) in ( LSeg (f,(j + 1))) by A41, A43, TOPREAL1: 21;

                then (f /. ((j + 1) + 1)) in (( LSeg (f,j)) /\ ( LSeg (f,(j + 1)))) by A54, XBOOLE_0:def 4;

                then (f /. ((j + 1) + 1)) = (f /. (j + 1)) by A55, TARSKI:def 1;

                hence contradiction by A44, A48, Th29;

              end;

            end;

            hence contradiction;

          end;

            suppose

             A56: ((f /. ((j + 1) + 1)) `2 ) = ((f /. (j + 1)) `2 );

            ((f /. ((j + 1) + 1)) `1 ) = ((f /. 1) `1 ) by A1, A48

            .= ((f /. (j + 1)) `1 ) by A1, A44;

            

            then (f /. ((j + 1) + 1)) = |[((f /. (j + 1)) `1 ), ((f /. (j + 1)) `2 )]| by A56, EUCLID: 53

            .= (f /. (j + 1)) by EUCLID: 53;

            hence contradiction by A44, A48, Th29;

          end;

        end;

        

         A57: ((( len f) -' 1) + 1) = ( len f) by A2, XREAL_1: 235;

        then

         A58: 2 <= (( len f) -' 1) & (( len f) -' 1) < ( len f) by A7, NAT_1: 13;

        

         A59: P[ 0 ];

        

         A60: for j holds P[j] from NAT_1:sch 2( A59, A36);

        then

         A61: ((f /. (( len f) -' 1)) `2 ) >= ((f /. 2) `2 ) by A58;

        ((f /. ( len f)) `2 ) > ((f /. (( len f) -' 1)) `2 ) by A60, A57, A58;

        then ((f /. ( len f)) `2 ) > ((f /. 2) `2 ) by A61, XXREAL_0: 2;

        hence contradiction by A35, FINSEQ_6:def 1;

      end;

    end;

    theorem :: GOBOARD7:31

    

     Th31: ex i st i in ( dom f) & ((f /. i) `2 ) <> ((f /. 1) `2 )

    proof

      assume

       A1: for i st i in ( dom f) holds ((f /. i) `2 ) = ((f /. 1) `2 );

      

       A2: ( len f) > 1 by Lm2;

      then

       A3: ( len f) >= (1 + 1) by NAT_1: 13;

      then

       A4: (1 + 1) in ( dom f) by FINSEQ_3: 25;

       A5:

      now

        assume

         A6: ((f /. 2) `1 ) = ((f /. 1) `1 );

        ((f /. 2) `2 ) = ((f /. 1) `2 ) by A1, A4;

        

        then (f /. 2) = |[((f /. 1) `1 ), ((f /. 1) `2 )]| by A6, EUCLID: 53

        .= (f /. 1) by EUCLID: 53;

        hence contradiction by A4, Th29, FINSEQ_5: 6;

      end;

      ( len f) = 2 implies (f /. 2) = (f /. 1) by FINSEQ_6:def 1;

      then

       A7: 2 < ( len f) by A3, A5, XXREAL_0: 1;

      per cases by A5, XXREAL_0: 1;

        suppose

         A8: ((f /. 2) `1 ) < ((f /. 1) `1 );

        defpred P[ Nat] means 2 <= $1 & $1 < ( len f) implies ((f /. $1) `1 ) <= ((f /. 2) `1 ) & ((f /. ($1 + 1)) `1 ) < ((f /. $1) `1 );

        

         A9: for j st P[j] holds P[(j + 1)]

        proof

          let j such that

           A10: 2 <= j & j < ( len f) implies ((f /. j) `1 ) <= ((f /. 2) `1 ) & ((f /. (j + 1)) `1 ) < ((f /. j) `1 ) and

           A11: 2 <= (j + 1) and

           A12: (j + 1) < ( len f);

          (1 + 1) <= (j + 1) by A11;

          then

           A13: 1 <= j by XREAL_1: 6;

          thus ((f /. (j + 1)) `1 ) <= ((f /. 2) `1 )

          proof

            per cases by A11, XXREAL_0: 1;

              suppose 2 = (j + 1);

              hence thesis;

            end;

              suppose 2 < (j + 1);

              hence thesis by A10, A12, NAT_1: 13, XXREAL_0: 2;

            end;

          end;

          

           A14: ((j + 1) + 1) <= ( len f) by A12, NAT_1: 13;

           A15:

          now

            per cases by A11, XXREAL_0: 1;

              suppose (1 + 1) = (j + 1);

              hence ((f /. (j + 1)) `1 ) < ((f /. j) `1 ) by A8;

            end;

              suppose 2 < (j + 1);

              hence ((f /. (j + 1)) `1 ) < ((f /. j) `1 ) by A10, A12, NAT_1: 13;

            end;

          end;

          

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

          then

           A17: (j + 1) in ( dom f) by A12, FINSEQ_3: 25;

          then

           A18: ((f /. (j + 1)) `2 ) = ((f /. 1) `2 ) by A1;

          j < ( len f) by A12, NAT_1: 13;

          then

           A19: j in ( dom f) by A13, FINSEQ_3: 25;

          then

           A20: ((f /. j) `2 ) = ((f /. 1) `2 ) by A1;

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

          then

           A21: ((j + 1) + 1) in ( dom f) by A14, FINSEQ_3: 25;

          then

           A22: ((f /. ((j + 1) + 1)) `2 ) = ((f /. 1) `2 ) by A1;

          assume

           A23: ((f /. ((j + 1) + 1)) `1 ) >= ((f /. (j + 1)) `1 );

          per cases by A23, XXREAL_0: 1;

            suppose

             A24: ((f /. ((j + 1) + 1)) `1 ) > ((f /. (j + 1)) `1 );

            now

              per cases ;

                suppose ((f /. j) `1 ) <= ((f /. ((j + 1) + 1)) `1 );

                then (f /. j) in ( LSeg ((f /. (j + 1)),(f /. ((j + 1) + 1)))) by A15, A20, A18, A22, Th8;

                then

                 A25: (f /. j) in ( LSeg (f,(j + 1))) by A14, A16, TOPREAL1:def 3;

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

                then

                 A26: (( LSeg (f,j)) /\ ( LSeg (f,(j + 1)))) = {(f /. (j + 1))} by A14, A13, TOPREAL1:def 6;

                (f /. j) in ( LSeg (f,j)) by A12, A13, TOPREAL1: 21;

                then (f /. j) in (( LSeg (f,j)) /\ ( LSeg (f,(j + 1)))) by A25, XBOOLE_0:def 4;

                then (f /. j) = (f /. (j + 1)) by A26, TARSKI:def 1;

                hence contradiction by A19, A17, Th29;

              end;

                suppose ((f /. j) `1 ) >= ((f /. ((j + 1) + 1)) `1 );

                then (f /. ((j + 1) + 1)) in ( LSeg ((f /. j),(f /. (j + 1)))) by A20, A18, A22, A24, Th8;

                then

                 A27: (f /. ((j + 1) + 1)) in ( LSeg (f,j)) by A12, A13, TOPREAL1:def 3;

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

                then

                 A28: (( LSeg (f,j)) /\ ( LSeg (f,(j + 1)))) = {(f /. (j + 1))} by A14, A13, TOPREAL1:def 6;

                (f /. ((j + 1) + 1)) in ( LSeg (f,(j + 1))) by A14, A16, TOPREAL1: 21;

                then (f /. ((j + 1) + 1)) in (( LSeg (f,j)) /\ ( LSeg (f,(j + 1)))) by A27, XBOOLE_0:def 4;

                then (f /. ((j + 1) + 1)) = (f /. (j + 1)) by A28, TARSKI:def 1;

                hence contradiction by A17, A21, Th29;

              end;

            end;

            hence contradiction;

          end;

            suppose

             A29: ((f /. ((j + 1) + 1)) `1 ) = ((f /. (j + 1)) `1 );

            ((f /. ((j + 1) + 1)) `2 ) = ((f /. 1) `2 ) by A1, A21

            .= ((f /. (j + 1)) `2 ) by A1, A17;

            

            then (f /. ((j + 1) + 1)) = |[((f /. (j + 1)) `1 ), ((f /. (j + 1)) `2 )]| by A29, EUCLID: 53

            .= (f /. (j + 1)) by EUCLID: 53;

            hence contradiction by A17, A21, Th29;

          end;

        end;

        

         A30: ((( len f) -' 1) + 1) = ( len f) by A2, XREAL_1: 235;

        then

         A31: 2 <= (( len f) -' 1) & (( len f) -' 1) < ( len f) by A7, NAT_1: 13;

        

         A32: P[ 0 ];

        

         A33: for j holds P[j] from NAT_1:sch 2( A32, A9);

        then

         A34: ((f /. (( len f) -' 1)) `1 ) <= ((f /. 2) `1 ) by A31;

        ((f /. ( len f)) `1 ) < ((f /. (( len f) -' 1)) `1 ) by A33, A30, A31;

        then ((f /. ( len f)) `1 ) < ((f /. 2) `1 ) by A34, XXREAL_0: 2;

        hence contradiction by A8, FINSEQ_6:def 1;

      end;

        suppose

         A35: ((f /. 2) `1 ) > ((f /. 1) `1 );

        defpred P[ Nat] means 2 <= $1 & $1 < ( len f) implies ((f /. $1) `1 ) >= ((f /. 2) `1 ) & ((f /. ($1 + 1)) `1 ) > ((f /. $1) `1 );

        

         A36: for j st P[j] holds P[(j + 1)]

        proof

          let j such that

           A37: 2 <= j & j < ( len f) implies ((f /. j) `1 ) >= ((f /. 2) `1 ) & ((f /. (j + 1)) `1 ) > ((f /. j) `1 ) and

           A38: 2 <= (j + 1) and

           A39: (j + 1) < ( len f);

          (1 + 1) <= (j + 1) by A38;

          then

           A40: 1 <= j by XREAL_1: 6;

          thus ((f /. (j + 1)) `1 ) >= ((f /. 2) `1 )

          proof

            per cases by A38, XXREAL_0: 1;

              suppose 2 = (j + 1);

              hence thesis;

            end;

              suppose 2 < (j + 1);

              hence thesis by A37, A39, NAT_1: 13, XXREAL_0: 2;

            end;

          end;

          

           A41: ((j + 1) + 1) <= ( len f) by A39, NAT_1: 13;

           A42:

          now

            per cases by A38, XXREAL_0: 1;

              suppose (1 + 1) = (j + 1);

              hence ((f /. (j + 1)) `1 ) > ((f /. j) `1 ) by A35;

            end;

              suppose 2 < (j + 1);

              hence ((f /. (j + 1)) `1 ) > ((f /. j) `1 ) by A37, A39, NAT_1: 13;

            end;

          end;

          

           A43: 1 <= (j + 1) by NAT_1: 11;

          then

           A44: (j + 1) in ( dom f) by A39, FINSEQ_3: 25;

          then

           A45: ((f /. (j + 1)) `2 ) = ((f /. 1) `2 ) by A1;

          j < ( len f) by A39, NAT_1: 13;

          then

           A46: j in ( dom f) by A40, FINSEQ_3: 25;

          then

           A47: ((f /. j) `2 ) = ((f /. 1) `2 ) by A1;

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

          then

           A48: ((j + 1) + 1) in ( dom f) by A41, FINSEQ_3: 25;

          then

           A49: ((f /. ((j + 1) + 1)) `2 ) = ((f /. 1) `2 ) by A1;

          assume

           A50: ((f /. ((j + 1) + 1)) `1 ) <= ((f /. (j + 1)) `1 );

          per cases by A50, XXREAL_0: 1;

            suppose

             A51: ((f /. ((j + 1) + 1)) `1 ) < ((f /. (j + 1)) `1 );

            now

              per cases ;

                suppose ((f /. j) `1 ) >= ((f /. ((j + 1) + 1)) `1 );

                then (f /. j) in ( LSeg ((f /. (j + 1)),(f /. ((j + 1) + 1)))) by A42, A47, A45, A49, Th8;

                then

                 A52: (f /. j) in ( LSeg (f,(j + 1))) by A41, A43, TOPREAL1:def 3;

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

                then

                 A53: (( LSeg (f,j)) /\ ( LSeg (f,(j + 1)))) = {(f /. (j + 1))} by A41, A40, TOPREAL1:def 6;

                (f /. j) in ( LSeg (f,j)) by A39, A40, TOPREAL1: 21;

                then (f /. j) in (( LSeg (f,j)) /\ ( LSeg (f,(j + 1)))) by A52, XBOOLE_0:def 4;

                then (f /. j) = (f /. (j + 1)) by A53, TARSKI:def 1;

                hence contradiction by A46, A44, Th29;

              end;

                suppose ((f /. j) `1 ) <= ((f /. ((j + 1) + 1)) `1 );

                then (f /. ((j + 1) + 1)) in ( LSeg ((f /. j),(f /. (j + 1)))) by A47, A45, A49, A51, Th8;

                then

                 A54: (f /. ((j + 1) + 1)) in ( LSeg (f,j)) by A39, A40, TOPREAL1:def 3;

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

                then

                 A55: (( LSeg (f,j)) /\ ( LSeg (f,(j + 1)))) = {(f /. (j + 1))} by A41, A40, TOPREAL1:def 6;

                (f /. ((j + 1) + 1)) in ( LSeg (f,(j + 1))) by A41, A43, TOPREAL1: 21;

                then (f /. ((j + 1) + 1)) in (( LSeg (f,j)) /\ ( LSeg (f,(j + 1)))) by A54, XBOOLE_0:def 4;

                then (f /. ((j + 1) + 1)) = (f /. (j + 1)) by A55, TARSKI:def 1;

                hence contradiction by A44, A48, Th29;

              end;

            end;

            hence contradiction;

          end;

            suppose

             A56: ((f /. ((j + 1) + 1)) `1 ) = ((f /. (j + 1)) `1 );

            ((f /. ((j + 1) + 1)) `2 ) = ((f /. 1) `2 ) by A1, A48

            .= ((f /. (j + 1)) `2 ) by A1, A44;

            

            then (f /. ((j + 1) + 1)) = |[((f /. (j + 1)) `1 ), ((f /. (j + 1)) `2 )]| by A56, EUCLID: 53

            .= (f /. (j + 1)) by EUCLID: 53;

            hence contradiction by A44, A48, Th29;

          end;

        end;

        

         A57: ((( len f) -' 1) + 1) = ( len f) by A2, XREAL_1: 235;

        then

         A58: 2 <= (( len f) -' 1) & (( len f) -' 1) < ( len f) by A7, NAT_1: 13;

        

         A59: P[ 0 ];

        

         A60: for j holds P[j] from NAT_1:sch 2( A59, A36);

        then

         A61: ((f /. (( len f) -' 1)) `1 ) >= ((f /. 2) `1 ) by A58;

        ((f /. ( len f)) `1 ) > ((f /. (( len f) -' 1)) `1 ) by A60, A57, A58;

        then ((f /. ( len f)) `1 ) > ((f /. 2) `1 ) by A61, XXREAL_0: 2;

        hence contradiction by A35, FINSEQ_6:def 1;

      end;

    end;

    theorem :: GOBOARD7:32

    ( len ( GoB f)) > 1

    proof

      

       A1: ( len ( GoB f)) <> 0 by MATRIX_0:def 10;

      1 in ( dom f) by FINSEQ_5: 6;

      then

      consider i2, j2 such that

       A2: [i2, j2] in ( Indices ( GoB f)) and

       A3: (f /. 1) = (( GoB f) * (i2,j2)) by GOBOARD2: 14;

      

       A4: 1 <= i2 by A2, MATRIX_0: 32;

      assume ( len ( GoB f)) <= 1;

      then

       A5: ( len ( GoB f)) = 1 by A1, NAT_1: 25;

      then i2 <= 1 by A2, MATRIX_0: 32;

      then

       A6: i2 = 1 by A4, XXREAL_0: 1;

      consider i such that

       A7: i in ( dom f) and

       A8: ((f /. i) `1 ) <> ((f /. 1) `1 ) by Th30;

      consider i1, j1 such that

       A9: [i1, j1] in ( Indices ( GoB f)) and

       A10: (f /. i) = (( GoB f) * (i1,j1)) by A7, GOBOARD2: 14;

      

       A11: 1 <= j1 & j1 <= ( width ( GoB f)) by A9, MATRIX_0: 32;

      

       A12: 1 <= i1 by A9, MATRIX_0: 32;

      i1 <= 1 by A5, A9, MATRIX_0: 32;

      then i1 = 1 by A12, XXREAL_0: 1;

      then

       A13: ((( GoB f) * (i1,j1)) `1 ) = ((( GoB f) * (1,1)) `1 ) by A5, A11, GOBOARD5: 2;

      1 <= j2 & j2 <= ( width ( GoB f)) by A2, MATRIX_0: 32;

      hence contradiction by A5, A8, A10, A3, A13, A6, GOBOARD5: 2;

    end;

    theorem :: GOBOARD7:33

    ( width ( GoB f)) > 1

    proof

      

       A1: ( width ( GoB f)) <> 0 by MATRIX_0:def 10;

      1 in ( dom f) by FINSEQ_5: 6;

      then

      consider i2, j2 such that

       A2: [i2, j2] in ( Indices ( GoB f)) and

       A3: (f /. 1) = (( GoB f) * (i2,j2)) by GOBOARD2: 14;

      

       A4: 1 <= j2 by A2, MATRIX_0: 32;

      assume ( width ( GoB f)) <= 1;

      then

       A5: ( width ( GoB f)) = 1 by A1, NAT_1: 25;

      then j2 <= 1 by A2, MATRIX_0: 32;

      then

       A6: j2 = 1 by A4, XXREAL_0: 1;

      consider i such that

       A7: i in ( dom f) and

       A8: ((f /. i) `2 ) <> ((f /. 1) `2 ) by Th31;

      consider i1, j1 such that

       A9: [i1, j1] in ( Indices ( GoB f)) and

       A10: (f /. i) = (( GoB f) * (i1,j1)) by A7, GOBOARD2: 14;

      

       A11: 1 <= i1 & i1 <= ( len ( GoB f)) by A9, MATRIX_0: 32;

      

       A12: 1 <= j1 by A9, MATRIX_0: 32;

      j1 <= 1 by A5, A9, MATRIX_0: 32;

      then j1 = 1 by A12, XXREAL_0: 1;

      then

       A13: ((( GoB f) * (i1,j1)) `2 ) = ((( GoB f) * (1,1)) `2 ) by A5, A11, GOBOARD5: 1;

      1 <= i2 & i2 <= ( len ( GoB f)) by A2, MATRIX_0: 32;

      hence contradiction by A5, A8, A10, A3, A13, A6, GOBOARD5: 1;

    end;

    theorem :: GOBOARD7:34

    

     Th34: ( len f) > 4

    proof

      assume

       A1: ( len f) <= 4;

      

       A2: ( len f) > 1 by Lm2;

      then

       A3: 1 in ( dom f) by FINSEQ_3: 25;

      

       A4: ( len f) >= (1 + 1) by A2, NAT_1: 13;

      then

       A5: 2 in ( dom f) by FINSEQ_3: 25;

      consider i2 such that

       A6: i2 in ( dom f) and

       A7: ((f /. i2) `2 ) <> ((f /. 1) `2 ) by Th31;

      consider i1 such that

       A8: i1 in ( dom f) and

       A9: ((f /. i1) `1 ) <> ((f /. 1) `1 ) by Th30;

      per cases by A4, TOPREAL1:def 5;

        suppose

         A10: ((f /. (1 + 1)) `1 ) = ((f /. 1) `1 );

        

         A11: i1 <= ( len f) by A8, FINSEQ_3: 25;

        

         A12: (f /. ( len f)) = (f /. 1) by FINSEQ_6:def 1;

        

         A13: i1 <> 0 by A8, FINSEQ_3: 25;

        now

          i1 <= 4 by A1, A11, XXREAL_0: 2;

          then i1 = 0 or ... or i1 = 4;

          per cases by A9, A10, A13;

            suppose

             A14: i1 = 3;

             A15:

            now

              assume ((f /. (1 + 1)) `2 ) = ((f /. 1) `2 );

              

              then (f /. (1 + 1)) = |[((f /. 1) `1 ), ((f /. 1) `2 )]| by A10, EUCLID: 53

              .= (f /. 1) by EUCLID: 53;

              hence contradiction by A3, A5, Th29;

            end;

            

             A16: ( len f) >= 3 by A8, A14, FINSEQ_3: 25;

            then ( len f) > 3 by A9, A12, A14, XXREAL_0: 1;

            then

             A17: ( len f) >= (3 + 1) by NAT_1: 13;

            then

             A18: ((f /. 3) `1 ) = ((f /. (3 + 1)) `1 ) or ((f /. 3) `2 ) = ((f /. (3 + 1)) `2 ) by TOPREAL1:def 5;

            

             A19: ( len f) = 4 by A1, A17, XXREAL_0: 1;

            ((f /. 2) `2 ) = ((f /. (2 + 1)) `2 ) by A9, A10, A14, A16, TOPREAL1:def 5;

            hence contradiction by A9, A14, A19, A15, A18, FINSEQ_6:def 1;

          end;

            suppose i1 = 4;

            hence contradiction by A1, A9, A11, A12, XXREAL_0: 1;

          end;

        end;

        hence contradiction;

      end;

        suppose

         A20: ((f /. (1 + 1)) `2 ) = ((f /. 1) `2 );

        

         A21: i2 <= ( len f) by A6, FINSEQ_3: 25;

        

         A22: (f /. ( len f)) = (f /. 1) by FINSEQ_6:def 1;

        

         A23: i2 <> 0 by A6, FINSEQ_3: 25;

        now

          i2 <= 4 by A1, A21, XXREAL_0: 2;

          then i2 = 0 or ... or i2 = 4;

          per cases by A7, A20, A23;

            suppose

             A24: i2 = 3;

             A25:

            now

              assume ((f /. (1 + 1)) `1 ) = ((f /. 1) `1 );

              

              then (f /. (1 + 1)) = |[((f /. 1) `1 ), ((f /. 1) `2 )]| by A20, EUCLID: 53

              .= (f /. 1) by EUCLID: 53;

              hence contradiction by A3, A5, Th29;

            end;

            

             A26: ( len f) >= 3 by A6, A24, FINSEQ_3: 25;

            then ( len f) > 3 by A7, A22, A24, XXREAL_0: 1;

            then

             A27: ( len f) >= (3 + 1) by NAT_1: 13;

            then

             A28: ((f /. 3) `2 ) = ((f /. (3 + 1)) `2 ) or ((f /. 3) `1 ) = ((f /. (3 + 1)) `1 ) by TOPREAL1:def 5;

            

             A29: ( len f) = 4 by A1, A27, XXREAL_0: 1;

            ((f /. 2) `1 ) = ((f /. (2 + 1)) `1 ) by A7, A20, A24, A26, TOPREAL1:def 5;

            hence contradiction by A7, A24, A29, A25, A28, FINSEQ_6:def 1;

          end;

            suppose i2 = 4;

            hence contradiction by A1, A7, A21, A22, XXREAL_0: 1;

          end;

        end;

        hence contradiction;

      end;

    end;

    theorem :: GOBOARD7:35

    

     Th35: for f be circular s.c.c. FinSequence of ( TOP-REAL 2) st ( len f) > 4 holds for i,j be Nat st 1 <= i & i < j & j < ( len f) holds (f /. i) <> (f /. j)

    proof

      let f be circular s.c.c. FinSequence of ( TOP-REAL 2) such that

       A1: ( len f) > 4;

      let i,j be Nat such that

       A2: 1 <= i and

       A3: i < j and

       A4: j < ( len f) and

       A5: (f /. i) = (f /. j);

      

       A6: (j + 1) <= ( len f) by A4, NAT_1: 13;

      

       A7: (i + 1) <= j & i <> 0 by A2, A3, NAT_1: 13;

      1 <= j by A2, A3, XXREAL_0: 2;

      then

       A8: (f /. j) in ( LSeg (f,j)) by A6, TOPREAL1: 21;

      

       A9: i < ( len f) by A3, A4, XXREAL_0: 2;

      then (i + 1) <= ( len f) by NAT_1: 13;

      then

       A10: (f /. i) in ( LSeg (f,i)) by A2, TOPREAL1: 21;

      i <= 2 implies i = 0 or ... or i = 2;

      per cases by A7, XXREAL_0: 1;

        suppose that

         A11: (i + 1) = j and

         A12: i = 1;

        

         A13: ((( len f) -' 1) + 1) = ( len f) by A1, XREAL_1: 235, XXREAL_0: 2;

        ((j + 1) + 1) < ( len f) by A1, A11, A12;

        then

         A14: (j + 1) < (( len f) -' 1) by A13, XREAL_1: 6;

        (( len f) -' 1) < ( len f) by A13, XREAL_1: 29;

        then ( LSeg (f,j)) misses ( LSeg (f,(( len f) -' 1))) by A11, A12, A14, GOBOARD5:def 4;

        then

         A15: (( LSeg (f,j)) /\ ( LSeg (f,(( len f) -' 1)))) = {} by XBOOLE_0:def 7;

        

         A16: (f /. i) = (f /. ( len f)) by A12, FINSEQ_6:def 1;

        (1 + 1) <= ( len f) by A1, XXREAL_0: 2;

        then 1 <= (( len f) -' 1) by A13, XREAL_1: 6;

        then (f /. i) in ( LSeg (f,(( len f) -' 1))) by A13, A16, TOPREAL1: 21;

        hence contradiction by A5, A8, A15, XBOOLE_0:def 4;

      end;

        suppose that

         A17: (i + 1) = j and

         A18: i = (1 + 1);

        

         A19: ((i -' 1) + 1) = i by A2, XREAL_1: 235;

        (j + 1) < ( len f) by A1, A17, A18;

        then ( LSeg (f,(i -' 1))) misses ( LSeg (f,j)) by A3, A19, GOBOARD5:def 4;

        then

         A20: (( LSeg (f,(i -' 1))) /\ ( LSeg (f,j))) = {} by XBOOLE_0:def 7;

        (f /. i) in ( LSeg (f,(i -' 1))) by A9, A18, A19, TOPREAL1: 21;

        hence contradiction by A5, A8, A20, XBOOLE_0:def 4;

      end;

        suppose that

         A21: i > (1 + 1);

        

         A22: ((i -' 1) + 1) = i by A2, XREAL_1: 235;

        then

         A23: 1 < (i -' 1) by A21, XREAL_1: 6;

        then ( LSeg (f,(i -' 1))) misses ( LSeg (f,j)) by A3, A4, A22, GOBOARD5:def 4;

        then

         A24: (( LSeg (f,(i -' 1))) /\ ( LSeg (f,j))) = {} by XBOOLE_0:def 7;

        (f /. i) in ( LSeg (f,(i -' 1))) by A9, A22, A23, TOPREAL1: 21;

        hence contradiction by A5, A8, A24, XBOOLE_0:def 4;

      end;

        suppose that

         A25: (i + 1) < j and

         A26: i <> 1;

        1 < i by A2, A26, XXREAL_0: 1;

        then ( LSeg (f,i)) misses ( LSeg (f,j)) by A4, A25, GOBOARD5:def 4;

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

        hence contradiction by A5, A8, A10, XBOOLE_0:def 4;

      end;

        suppose that

         A27: (i + 1) < j and

         A28: (j + 1) <> ( len f);

        (j + 1) < ( len f) by A6, A28, XXREAL_0: 1;

        then ( LSeg (f,i)) misses ( LSeg (f,j)) by A27, GOBOARD5:def 4;

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

        hence contradiction by A5, A8, A10, XBOOLE_0:def 4;

      end;

        suppose that

         A29: (i + 1) < j and

         A30: i = 1 and

         A31: (j + 1) = ( len f);

        

         A32: j < ( len f) by A31, NAT_1: 13;

        

         A33: ((j -' 1) + 1) = j by A2, A3, XREAL_1: 235, XXREAL_0: 2;

        then

         A34: (i + 1) <= (j -' 1) by A29, NAT_1: 13;

        (i + 1) <> (j -' 1) by A1, A30, A31, A33;

        then (i + 1) < (j -' 1) by A34, XXREAL_0: 1;

        then ( LSeg (f,1)) misses ( LSeg (f,(j -' 1))) by A30, A33, A32, GOBOARD5:def 4;

        then

         A35: (( LSeg (f,1)) /\ ( LSeg (f,(j -' 1)))) = {} by XBOOLE_0:def 7;

        1 <= (j -' 1) by A30, A34, XXREAL_0: 2;

        then (f /. j) in ( LSeg (f,(j -' 1))) by A4, A33, TOPREAL1: 21;

        hence contradiction by A5, A10, A30, A35, XBOOLE_0:def 4;

      end;

    end;

    theorem :: GOBOARD7:36

    

     Th36: for i,j be Nat st 1 <= i & i < j & j < ( len f) holds (f /. i) <> (f /. j)

    proof

      ( len f) > 4 by Th34;

      hence thesis by Th35;

    end;

    theorem :: GOBOARD7:37

    

     Th37: for i,j be Nat st 1 < i & i < j & j <= ( len f) holds (f /. i) <> (f /. j)

    proof

      let i,j be Nat such that

       A1: 1 < i and

       A2: i < j and

       A3: j <= ( len f);

      per cases by A3, XXREAL_0: 1;

        suppose j < ( len f);

        hence thesis by A1, A2, Th36;

      end;

        suppose j = ( len f);

        then

         A4: (f /. j) = (f /. 1) by FINSEQ_6:def 1;

        i < ( len f) by A2, A3, XXREAL_0: 2;

        hence thesis by A1, A4, Th36;

      end;

    end;

    theorem :: GOBOARD7:38

    

     Th38: for i be Nat st 1 < i & i <= ( len f) & (f /. i) = (f /. 1) holds i = ( len f)

    proof

      let i be Nat such that

       A1: 1 < i and

       A2: i <= ( len f) and

       A3: (f /. i) = (f /. 1);

      assume i <> ( len f);

      then i < ( len f) by A2, XXREAL_0: 1;

      hence contradiction by A1, A3, Th36;

    end;

    theorem :: GOBOARD7:39

    

     Th39: 1 <= i & i <= ( len ( GoB f)) & 1 <= j & (j + 1) <= ( width ( GoB f)) & ((1 / 2) * ((( GoB f) * (i,j)) + (( GoB f) * (i,(j + 1))))) in ( L~ f) implies ex k st 1 <= k & (k + 1) <= ( len f) & ( LSeg ((( GoB f) * (i,j)),(( GoB f) * (i,(j + 1))))) = ( LSeg (f,k))

    proof

      set mi = ((1 / 2) * ((( GoB f) * (i,j)) + (( GoB f) * (i,(j + 1)))));

      assume that

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

       A2: ((1 / 2) * ((( GoB f) * (i,j)) + (( GoB f) * (i,(j + 1))))) in ( L~ f);

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

      then

      consider x be set such that

       A3: ((1 / 2) * ((( GoB f) * (i,j)) + (( GoB f) * (i,(j + 1))))) in x and

       A4: x in { ( LSeg (f,k)) : 1 <= k & (k + 1) <= ( len f) } by A2, TARSKI:def 4;

      consider k such that

       A5: x = ( LSeg (f,k)) and

       A6: 1 <= k and

       A7: (k + 1) <= ( len f) by A4;

      

       A8: f is_sequence_on ( GoB f) by GOBOARD5:def 5;

      

       A9: mi in ( LSeg ((f /. k),(f /. (k + 1)))) by A3, A5, A6, A7, TOPREAL1:def 3;

      k <= (k + 1) by NAT_1: 11;

      then k <= ( len f) by A7, XXREAL_0: 2;

      then

       A10: k in ( dom f) by A6, FINSEQ_3: 25;

      then

      consider i1,j1 be Nat such that

       A11: [i1, j1] in ( Indices ( GoB f)) and

       A12: (f /. k) = (( GoB f) * (i1,j1)) by A8, GOBOARD1:def 9;

      

       A13: 1 <= i1 by A11, MATRIX_0: 32;

      take k;

      thus 1 <= k & (k + 1) <= ( len f) by A6, A7;

      1 <= (k + 1) by NAT_1: 11;

      then

       A14: (k + 1) in ( dom f) by A7, FINSEQ_3: 25;

      then

      consider i2,j2 be Nat such that

       A15: [i2, j2] in ( Indices ( GoB f)) and

       A16: (f /. (k + 1)) = (( GoB f) * (i2,j2)) by A8, GOBOARD1:def 9;

      

       A17: 1 <= i2 by A15, MATRIX_0: 32;

      

       A18: j2 <= ( width ( GoB f)) by A15, MATRIX_0: 32;

      ( |.(i1 - i2).| + |.(j1 - j2).|) = 1 by A8, A10, A11, A12, A14, A15, A16, GOBOARD1:def 9;

      then

       A19: |.(i1 - i2).| = 1 & j1 = j2 or |.(j1 - j2).| = 1 & i1 = i2 by SEQM_3: 42;

      

       A20: i1 <= ( len ( GoB f)) by A11, MATRIX_0: 32;

      

       A21: j1 <= ( width ( GoB f)) by A11, MATRIX_0: 32;

      

       A22: 1 <= j1 by A11, MATRIX_0: 32;

      

       A23: i2 <= ( len ( GoB f)) by A15, MATRIX_0: 32;

      

       A24: 1 <= j2 by A15, MATRIX_0: 32;

      per cases by A19, SEQM_3: 41;

        suppose

         A25: j1 = j2 & i1 = (i2 + 1);

        then mi in ( LSeg ((( GoB f) * (i2,j2)),(( GoB f) * ((i2 + 1),j2)))) by A3, A5, A6, A7, A12, A16, TOPREAL1:def 3;

        hence thesis by A1, A20, A17, A24, A18, A25, Th28;

      end;

        suppose

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

        then mi in ( LSeg ((( GoB f) * (i1,j1)),(( GoB f) * ((i1 + 1),j1)))) by A3, A5, A6, A7, A12, A16, TOPREAL1:def 3;

        hence thesis by A1, A13, A22, A21, A23, A26, Th28;

      end;

        suppose

         A27: j1 = (j2 + 1) & i1 = i2;

        then i = i2 & j = j2 by A1, A12, A16, A13, A20, A21, A24, A9, Th25;

        hence thesis by A6, A7, A12, A16, A27, TOPREAL1:def 3;

      end;

        suppose

         A28: (j1 + 1) = j2 & i1 = i2;

        then i = i1 & j = j1 by A1, A12, A16, A13, A20, A22, A18, A9, Th25;

        hence thesis by A6, A7, A12, A16, A28, TOPREAL1:def 3;

      end;

    end;

    theorem :: GOBOARD7:40

    

     Th40: 1 <= i & (i + 1) <= ( len ( GoB f)) & 1 <= j & j <= ( width ( GoB f)) & ((1 / 2) * ((( GoB f) * (i,j)) + (( GoB f) * ((i + 1),j)))) in ( L~ f) implies ex k st 1 <= k & (k + 1) <= ( len f) & ( LSeg ((( GoB f) * (i,j)),(( GoB f) * ((i + 1),j)))) = ( LSeg (f,k))

    proof

      set mi = ((1 / 2) * ((( GoB f) * (i,j)) + (( GoB f) * ((i + 1),j))));

      assume that

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

       A2: ((1 / 2) * ((( GoB f) * (i,j)) + (( GoB f) * ((i + 1),j)))) in ( L~ f);

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

      then

      consider x be set such that

       A3: ((1 / 2) * ((( GoB f) * (i,j)) + (( GoB f) * ((i + 1),j)))) in x and

       A4: x in { ( LSeg (f,k)) : 1 <= k & (k + 1) <= ( len f) } by A2, TARSKI:def 4;

      consider k such that

       A5: x = ( LSeg (f,k)) and

       A6: 1 <= k and

       A7: (k + 1) <= ( len f) by A4;

      

       A8: f is_sequence_on ( GoB f) by GOBOARD5:def 5;

      

       A9: mi in ( LSeg ((f /. k),(f /. (k + 1)))) by A3, A5, A6, A7, TOPREAL1:def 3;

      k <= (k + 1) by NAT_1: 11;

      then k <= ( len f) by A7, XXREAL_0: 2;

      then

       A10: k in ( dom f) by A6, FINSEQ_3: 25;

      then

      consider i1,j1 be Nat such that

       A11: [i1, j1] in ( Indices ( GoB f)) and

       A12: (f /. k) = (( GoB f) * (i1,j1)) by A8, GOBOARD1:def 9;

      

       A13: 1 <= j1 by A11, MATRIX_0: 32;

      take k;

      thus 1 <= k & (k + 1) <= ( len f) by A6, A7;

      1 <= (k + 1) by NAT_1: 11;

      then

       A14: (k + 1) in ( dom f) by A7, FINSEQ_3: 25;

      then

      consider i2,j2 be Nat such that

       A15: [i2, j2] in ( Indices ( GoB f)) and

       A16: (f /. (k + 1)) = (( GoB f) * (i2,j2)) by A8, GOBOARD1:def 9;

      

       A17: 1 <= j2 by A15, MATRIX_0: 32;

      

       A18: i2 <= ( len ( GoB f)) by A15, MATRIX_0: 32;

      ( |.(j1 - j2).| + |.(i1 - i2).|) = 1 by A8, A10, A11, A12, A14, A15, A16, GOBOARD1:def 9;

      then

       A19: |.(j1 - j2).| = 1 & i1 = i2 or |.(i1 - i2).| = 1 & j1 = j2 by SEQM_3: 42;

      

       A20: j1 <= ( width ( GoB f)) by A11, MATRIX_0: 32;

      

       A21: i1 <= ( len ( GoB f)) by A11, MATRIX_0: 32;

      

       A22: 1 <= i1 by A11, MATRIX_0: 32;

      

       A23: j2 <= ( width ( GoB f)) by A15, MATRIX_0: 32;

      

       A24: 1 <= i2 by A15, MATRIX_0: 32;

      per cases by A19, SEQM_3: 41;

        suppose

         A25: i1 = i2 & j1 = (j2 + 1);

        then mi in ( LSeg ((( GoB f) * (i2,j2)),(( GoB f) * (i2,(j2 + 1))))) by A3, A5, A6, A7, A12, A16, TOPREAL1:def 3;

        hence thesis by A1, A20, A17, A24, A18, A25, Th27;

      end;

        suppose

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

        then mi in ( LSeg ((( GoB f) * (i1,j1)),(( GoB f) * (i1,(j1 + 1))))) by A3, A5, A6, A7, A12, A16, TOPREAL1:def 3;

        hence thesis by A1, A13, A22, A21, A23, A26, Th27;

      end;

        suppose

         A27: i1 = (i2 + 1) & j1 = j2;

        then j = j2 & i = i2 by A1, A12, A16, A13, A20, A21, A24, A9, Th26;

        hence thesis by A6, A7, A12, A16, A27, TOPREAL1:def 3;

      end;

        suppose

         A28: (i1 + 1) = i2 & j1 = j2;

        then j = j1 & i = i1 by A1, A12, A16, A13, A20, A22, A18, A9, Th26;

        hence thesis by A6, A7, A12, A16, A28, TOPREAL1:def 3;

      end;

    end;

    theorem :: GOBOARD7:41

    1 <= i & (i + 1) <= ( len ( GoB f)) & 1 <= j & (j + 1) <= ( width ( GoB f)) & 1 <= k & (k + 1) < ( len f) & ( LSeg ((( GoB f) * (i,(j + 1))),(( GoB f) * ((i + 1),(j + 1))))) = ( LSeg (f,k)) & ( LSeg ((( GoB f) * ((i + 1),j)),(( GoB f) * ((i + 1),(j + 1))))) = ( LSeg (f,(k + 1))) implies (f /. k) = (( GoB f) * (i,(j + 1))) & (f /. (k + 1)) = (( GoB f) * ((i + 1),(j + 1))) & (f /. (k + 2)) = (( GoB f) * ((i + 1),j))

    proof

      assume that

       A1: 1 <= i and

       A2: (i + 1) <= ( len ( GoB f)) and

       A3: 1 <= j and

       A4: (j + 1) <= ( width ( GoB f)) and

       A5: 1 <= k and

       A6: (k + 1) < ( len f) and

       A7: ( LSeg ((( GoB f) * (i,(j + 1))),(( GoB f) * ((i + 1),(j + 1))))) = ( LSeg (f,k)) and

       A8: ( LSeg ((( GoB f) * ((i + 1),j)),(( GoB f) * ((i + 1),(j + 1))))) = ( LSeg (f,(k + 1)));

      

       A9: 1 <= (j + 1) by NAT_1: 11;

      

       A10: i < (i + 1) by NAT_1: 13;

      

       A11: 1 <= (i + 1) by NAT_1: 11;

      j < ( width ( GoB f)) by A4, NAT_1: 13;

      

      then ((( GoB f) * ((i + 1),j)) `1 ) = ((( GoB f) * ((i + 1),1)) `1 ) by A2, A3, A11, GOBOARD5: 2

      .= ((( GoB f) * ((i + 1),(j + 1))) `1 ) by A2, A4, A9, A11, GOBOARD5: 2;

      then

       A12: (( GoB f) * (i,(j + 1))) <> (( GoB f) * ((i + 1),j)) by A1, A2, A4, A9, A10, GOBOARD5: 3;

      

       A13: 1 <= (k + 1) by NAT_1: 11;

      

       A14: (k + (1 + 1)) = ((k + 1) + 1);

      then (k + 2) <= ( len f) by A6, NAT_1: 13;

      then

       A15: ( LSeg ((( GoB f) * ((i + 1),j)),(( GoB f) * ((i + 1),(j + 1))))) = ( LSeg ((f /. (k + 1)),(f /. (k + 2)))) by A8, A14, A13, TOPREAL1:def 3;

      then

       A16: (( GoB f) * ((i + 1),j)) = (f /. (k + 2)) & (( GoB f) * ((i + 1),(j + 1))) = (f /. (k + 1)) or (( GoB f) * ((i + 1),j)) = (f /. (k + 1)) & (( GoB f) * ((i + 1),(j + 1))) = (f /. (k + 2)) by SPPOL_1: 8;

      

       A17: ( LSeg ((( GoB f) * (i,(j + 1))),(( GoB f) * ((i + 1),(j + 1))))) = ( LSeg ((f /. k),(f /. (k + 1)))) by A5, A6, A7, TOPREAL1:def 3;

      then (( GoB f) * ((i + 1),(j + 1))) = (f /. (k + 1)) & (( GoB f) * (i,(j + 1))) = (f /. k) or (( GoB f) * ((i + 1),(j + 1))) = (f /. k) & (( GoB f) * (i,(j + 1))) = (f /. (k + 1)) by SPPOL_1: 8;

      hence (f /. k) = (( GoB f) * (i,(j + 1))) by A15, A12, SPPOL_1: 8;

      thus (f /. (k + 1)) = (( GoB f) * ((i + 1),(j + 1))) by A17, A16, A12, SPPOL_1: 8;

      thus thesis by A17, A16, A12, SPPOL_1: 8;

    end;

    theorem :: GOBOARD7:42

    1 <= i & i <= ( len ( GoB f)) & 1 <= j & (j + 1) < ( width ( GoB f)) & 1 <= k & (k + 1) < ( len f) & ( LSeg ((( GoB f) * (i,(j + 1))),(( GoB f) * (i,(j + 2))))) = ( LSeg (f,k)) & ( LSeg ((( GoB f) * (i,j)),(( GoB f) * (i,(j + 1))))) = ( LSeg (f,(k + 1))) implies (f /. k) = (( GoB f) * (i,(j + 2))) & (f /. (k + 1)) = (( GoB f) * (i,(j + 1))) & (f /. (k + 2)) = (( GoB f) * (i,j))

    proof

      assume that

       A1: 1 <= i & i <= ( len ( GoB f)) & 1 <= j and

       A2: (j + 1) < ( width ( GoB f)) and

       A3: 1 <= k and

       A4: (k + 1) < ( len f) and

       A5: ( LSeg ((( GoB f) * (i,(j + 1))),(( GoB f) * (i,(j + 2))))) = ( LSeg (f,k)) and

       A6: ( LSeg ((( GoB f) * (i,j)),(( GoB f) * (i,(j + 1))))) = ( LSeg (f,(k + 1)));

      

       A7: 1 <= (k + 1) by NAT_1: 11;

      

       A8: (k + (1 + 1)) = ((k + 1) + 1);

      then (k + 2) <= ( len f) by A4, NAT_1: 13;

      then

       A9: ( LSeg ((( GoB f) * (i,j)),(( GoB f) * (i,(j + 1))))) = ( LSeg ((f /. (k + 1)),(f /. (k + 2)))) by A6, A8, A7, TOPREAL1:def 3;

      then

       A10: (( GoB f) * (i,j)) = (f /. (k + 2)) & (( GoB f) * (i,(j + 1))) = (f /. (k + 1)) or (( GoB f) * (i,j)) = (f /. (k + 1)) & (( GoB f) * (i,(j + 1))) = (f /. (k + 2)) by SPPOL_1: 8;

      

       A11: j < (j + 2) by XREAL_1: 29;

      (j + (1 + 1)) = ((j + 1) + 1);

      then (j + 2) <= ( width ( GoB f)) by A2, NAT_1: 13;

      then

       A12: ((( GoB f) * (i,j)) `2 ) < ((( GoB f) * (i,(j + 2))) `2 ) by A1, A11, GOBOARD5: 4;

      

       A13: ( LSeg ((( GoB f) * (i,(j + 1))),(( GoB f) * (i,(j + 2))))) = ( LSeg ((f /. k),(f /. (k + 1)))) by A3, A4, A5, TOPREAL1:def 3;

      then (( GoB f) * (i,(j + 1))) = (f /. (k + 1)) & (( GoB f) * (i,(j + 2))) = (f /. k) or (( GoB f) * (i,(j + 1))) = (f /. k) & (( GoB f) * (i,(j + 2))) = (f /. (k + 1)) by SPPOL_1: 8;

      hence (f /. k) = (( GoB f) * (i,(j + 2))) by A9, A12, SPPOL_1: 8;

      thus (f /. (k + 1)) = (( GoB f) * (i,(j + 1))) by A13, A10, A12, SPPOL_1: 8;

      thus thesis by A13, A10, A12, SPPOL_1: 8;

    end;

    theorem :: GOBOARD7:43

    1 <= i & (i + 1) <= ( len ( GoB f)) & 1 <= j & (j + 1) <= ( width ( GoB f)) & 1 <= k & (k + 1) < ( len f) & ( LSeg ((( GoB f) * (i,(j + 1))),(( GoB f) * ((i + 1),(j + 1))))) = ( LSeg (f,k)) & ( LSeg ((( GoB f) * (i,j)),(( GoB f) * (i,(j + 1))))) = ( LSeg (f,(k + 1))) implies (f /. k) = (( GoB f) * ((i + 1),(j + 1))) & (f /. (k + 1)) = (( GoB f) * (i,(j + 1))) & (f /. (k + 2)) = (( GoB f) * (i,j))

    proof

      assume that

       A1: 1 <= i and

       A2: (i + 1) <= ( len ( GoB f)) and

       A3: 1 <= j and

       A4: (j + 1) <= ( width ( GoB f)) and

       A5: 1 <= k and

       A6: (k + 1) < ( len f) and

       A7: ( LSeg ((( GoB f) * (i,(j + 1))),(( GoB f) * ((i + 1),(j + 1))))) = ( LSeg (f,k)) and

       A8: ( LSeg ((( GoB f) * (i,j)),(( GoB f) * (i,(j + 1))))) = ( LSeg (f,(k + 1)));

      

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

      

       A10: j < (j + 1) by NAT_1: 13;

      

       A11: 1 <= (i + 1) by NAT_1: 11;

      i < ( len ( GoB f)) by A2, NAT_1: 13;

      

      then ((( GoB f) * (i,j)) `2 ) = ((( GoB f) * (1,j)) `2 ) by A1, A3, A9, GOBOARD5: 1

      .= ((( GoB f) * ((i + 1),j)) `2 ) by A2, A3, A11, A9, GOBOARD5: 1;

      then

       A12: (( GoB f) * (i,j)) <> (( GoB f) * ((i + 1),(j + 1))) by A2, A3, A4, A11, A10, GOBOARD5: 4;

      

       A13: 1 <= (k + 1) by NAT_1: 11;

      

       A14: (k + (1 + 1)) = ((k + 1) + 1);

      then (k + 2) <= ( len f) by A6, NAT_1: 13;

      then

       A15: ( LSeg ((( GoB f) * (i,j)),(( GoB f) * (i,(j + 1))))) = ( LSeg ((f /. (k + 1)),(f /. (k + 2)))) by A8, A14, A13, TOPREAL1:def 3;

      then

       A16: (( GoB f) * (i,j)) = (f /. (k + 2)) & (( GoB f) * (i,(j + 1))) = (f /. (k + 1)) or (( GoB f) * (i,j)) = (f /. (k + 1)) & (( GoB f) * (i,(j + 1))) = (f /. (k + 2)) by SPPOL_1: 8;

      

       A17: ( LSeg ((( GoB f) * (i,(j + 1))),(( GoB f) * ((i + 1),(j + 1))))) = ( LSeg ((f /. k),(f /. (k + 1)))) by A5, A6, A7, TOPREAL1:def 3;

      then (( GoB f) * ((i + 1),(j + 1))) = (f /. (k + 1)) & (( GoB f) * (i,(j + 1))) = (f /. k) or (( GoB f) * ((i + 1),(j + 1))) = (f /. k) & (( GoB f) * (i,(j + 1))) = (f /. (k + 1)) by SPPOL_1: 8;

      hence (f /. k) = (( GoB f) * ((i + 1),(j + 1))) by A15, A12, SPPOL_1: 8;

      thus (f /. (k + 1)) = (( GoB f) * (i,(j + 1))) by A17, A16, A12, SPPOL_1: 8;

      thus thesis by A17, A16, A12, SPPOL_1: 8;

    end;

    theorem :: GOBOARD7:44

    1 <= i & (i + 1) <= ( len ( GoB f)) & 1 <= j & (j + 1) <= ( width ( GoB f)) & 1 <= k & (k + 1) < ( len f) & ( LSeg ((( GoB f) * ((i + 1),j)),(( GoB f) * ((i + 1),(j + 1))))) = ( LSeg (f,k)) & ( LSeg ((( GoB f) * (i,(j + 1))),(( GoB f) * ((i + 1),(j + 1))))) = ( LSeg (f,(k + 1))) implies (f /. k) = (( GoB f) * ((i + 1),j)) & (f /. (k + 1)) = (( GoB f) * ((i + 1),(j + 1))) & (f /. (k + 2)) = (( GoB f) * (i,(j + 1)))

    proof

      assume that

       A1: 1 <= i and

       A2: (i + 1) <= ( len ( GoB f)) and

       A3: 1 <= j and

       A4: (j + 1) <= ( width ( GoB f)) and

       A5: 1 <= k and

       A6: (k + 1) < ( len f) and

       A7: ( LSeg ((( GoB f) * ((i + 1),j)),(( GoB f) * ((i + 1),(j + 1))))) = ( LSeg (f,k)) and

       A8: ( LSeg ((( GoB f) * (i,(j + 1))),(( GoB f) * ((i + 1),(j + 1))))) = ( LSeg (f,(k + 1)));

      

       A9: 1 <= (i + 1) by NAT_1: 11;

      

       A10: j < (j + 1) by NAT_1: 13;

      

       A11: 1 <= (j + 1) by NAT_1: 11;

      i < ( len ( GoB f)) by A2, NAT_1: 13;

      

      then ((( GoB f) * (i,(j + 1))) `2 ) = ((( GoB f) * (1,(j + 1))) `2 ) by A1, A4, A11, GOBOARD5: 1

      .= ((( GoB f) * ((i + 1),(j + 1))) `2 ) by A2, A4, A9, A11, GOBOARD5: 1;

      then

       A12: (( GoB f) * ((i + 1),j)) <> (( GoB f) * (i,(j + 1))) by A2, A3, A4, A9, A10, GOBOARD5: 4;

      

       A13: 1 <= (k + 1) by NAT_1: 11;

      

       A14: (k + (1 + 1)) = ((k + 1) + 1);

      then (k + 2) <= ( len f) by A6, NAT_1: 13;

      then

       A15: ( LSeg ((( GoB f) * (i,(j + 1))),(( GoB f) * ((i + 1),(j + 1))))) = ( LSeg ((f /. (k + 1)),(f /. (k + 2)))) by A8, A14, A13, TOPREAL1:def 3;

      then

       A16: (( GoB f) * (i,(j + 1))) = (f /. (k + 2)) & (( GoB f) * ((i + 1),(j + 1))) = (f /. (k + 1)) or (( GoB f) * (i,(j + 1))) = (f /. (k + 1)) & (( GoB f) * ((i + 1),(j + 1))) = (f /. (k + 2)) by SPPOL_1: 8;

      

       A17: ( LSeg ((( GoB f) * ((i + 1),j)),(( GoB f) * ((i + 1),(j + 1))))) = ( LSeg ((f /. k),(f /. (k + 1)))) by A5, A6, A7, TOPREAL1:def 3;

      then (( GoB f) * ((i + 1),(j + 1))) = (f /. (k + 1)) & (( GoB f) * ((i + 1),j)) = (f /. k) or (( GoB f) * ((i + 1),(j + 1))) = (f /. k) & (( GoB f) * ((i + 1),j)) = (f /. (k + 1)) by SPPOL_1: 8;

      hence (f /. k) = (( GoB f) * ((i + 1),j)) by A15, A12, SPPOL_1: 8;

      thus (f /. (k + 1)) = (( GoB f) * ((i + 1),(j + 1))) by A17, A16, A12, SPPOL_1: 8;

      thus thesis by A17, A16, A12, SPPOL_1: 8;

    end;

    theorem :: GOBOARD7:45

    1 <= i & (i + 1) < ( len ( GoB f)) & 1 <= j & j <= ( width ( GoB f)) & 1 <= k & (k + 1) < ( len f) & ( LSeg ((( GoB f) * ((i + 1),j)),(( GoB f) * ((i + 2),j)))) = ( LSeg (f,k)) & ( LSeg ((( GoB f) * (i,j)),(( GoB f) * ((i + 1),j)))) = ( LSeg (f,(k + 1))) implies (f /. k) = (( GoB f) * ((i + 2),j)) & (f /. (k + 1)) = (( GoB f) * ((i + 1),j)) & (f /. (k + 2)) = (( GoB f) * (i,j))

    proof

      assume that

       A1: 1 <= i and

       A2: (i + 1) < ( len ( GoB f)) and

       A3: 1 <= j & j <= ( width ( GoB f)) and

       A4: 1 <= k and

       A5: (k + 1) < ( len f) and

       A6: ( LSeg ((( GoB f) * ((i + 1),j)),(( GoB f) * ((i + 2),j)))) = ( LSeg (f,k)) and

       A7: ( LSeg ((( GoB f) * (i,j)),(( GoB f) * ((i + 1),j)))) = ( LSeg (f,(k + 1)));

      

       A8: i < (i + 2) by XREAL_1: 29;

      (i + (1 + 1)) = ((i + 1) + 1);

      then (i + 2) <= ( len ( GoB f)) by A2, NAT_1: 13;

      then

       A9: ((( GoB f) * (i,j)) `1 ) < ((( GoB f) * ((i + 2),j)) `1 ) by A1, A3, A8, GOBOARD5: 3;

      

       A10: 1 <= (k + 1) by NAT_1: 11;

      

       A11: (k + (1 + 1)) = ((k + 1) + 1);

      then (k + 2) <= ( len f) by A5, NAT_1: 13;

      then

       A12: ( LSeg ((( GoB f) * (i,j)),(( GoB f) * ((i + 1),j)))) = ( LSeg ((f /. (k + 1)),(f /. (k + 2)))) by A7, A11, A10, TOPREAL1:def 3;

      then

       A13: (( GoB f) * (i,j)) = (f /. (k + 2)) & (( GoB f) * ((i + 1),j)) = (f /. (k + 1)) or (( GoB f) * (i,j)) = (f /. (k + 1)) & (( GoB f) * ((i + 1),j)) = (f /. (k + 2)) by SPPOL_1: 8;

      

       A14: ( LSeg ((( GoB f) * ((i + 1),j)),(( GoB f) * ((i + 2),j)))) = ( LSeg ((f /. k),(f /. (k + 1)))) by A4, A5, A6, TOPREAL1:def 3;

      then (( GoB f) * ((i + 1),j)) = (f /. (k + 1)) & (( GoB f) * ((i + 2),j)) = (f /. k) or (( GoB f) * ((i + 1),j)) = (f /. k) & (( GoB f) * ((i + 2),j)) = (f /. (k + 1)) by SPPOL_1: 8;

      hence (f /. k) = (( GoB f) * ((i + 2),j)) by A12, A9, SPPOL_1: 8;

      thus (f /. (k + 1)) = (( GoB f) * ((i + 1),j)) by A14, A13, A9, SPPOL_1: 8;

      thus thesis by A14, A13, A9, SPPOL_1: 8;

    end;

    theorem :: GOBOARD7:46

    1 <= i & (i + 1) <= ( len ( GoB f)) & 1 <= j & (j + 1) <= ( width ( GoB f)) & 1 <= k & (k + 1) < ( len f) & ( LSeg ((( GoB f) * ((i + 1),j)),(( GoB f) * ((i + 1),(j + 1))))) = ( LSeg (f,k)) & ( LSeg ((( GoB f) * (i,j)),(( GoB f) * ((i + 1),j)))) = ( LSeg (f,(k + 1))) implies (f /. k) = (( GoB f) * ((i + 1),(j + 1))) & (f /. (k + 1)) = (( GoB f) * ((i + 1),j)) & (f /. (k + 2)) = (( GoB f) * (i,j))

    proof

      assume that

       A1: 1 <= i and

       A2: (i + 1) <= ( len ( GoB f)) and

       A3: 1 <= j and

       A4: (j + 1) <= ( width ( GoB f)) and

       A5: 1 <= k and

       A6: (k + 1) < ( len f) and

       A7: ( LSeg ((( GoB f) * ((i + 1),j)),(( GoB f) * ((i + 1),(j + 1))))) = ( LSeg (f,k)) and

       A8: ( LSeg ((( GoB f) * (i,j)),(( GoB f) * ((i + 1),j)))) = ( LSeg (f,(k + 1)));

      

       A9: i < ( len ( GoB f)) by A2, NAT_1: 13;

      

       A10: i < (i + 1) by NAT_1: 13;

      

       A11: 1 <= (j + 1) by NAT_1: 11;

      j < ( width ( GoB f)) by A4, NAT_1: 13;

      

      then ((( GoB f) * (i,j)) `1 ) = ((( GoB f) * (i,1)) `1 ) by A1, A3, A9, GOBOARD5: 2

      .= ((( GoB f) * (i,(j + 1))) `1 ) by A1, A4, A11, A9, GOBOARD5: 2;

      then

       A12: (( GoB f) * (i,j)) <> (( GoB f) * ((i + 1),(j + 1))) by A1, A2, A4, A11, A10, GOBOARD5: 3;

      

       A13: 1 <= (k + 1) by NAT_1: 11;

      

       A14: (k + (1 + 1)) = ((k + 1) + 1);

      then (k + 2) <= ( len f) by A6, NAT_1: 13;

      then

       A15: ( LSeg ((( GoB f) * (i,j)),(( GoB f) * ((i + 1),j)))) = ( LSeg ((f /. (k + 1)),(f /. (k + 2)))) by A8, A14, A13, TOPREAL1:def 3;

      then

       A16: (( GoB f) * (i,j)) = (f /. (k + 2)) & (( GoB f) * ((i + 1),j)) = (f /. (k + 1)) or (( GoB f) * (i,j)) = (f /. (k + 1)) & (( GoB f) * ((i + 1),j)) = (f /. (k + 2)) by SPPOL_1: 8;

      

       A17: ( LSeg ((( GoB f) * ((i + 1),j)),(( GoB f) * ((i + 1),(j + 1))))) = ( LSeg ((f /. k),(f /. (k + 1)))) by A5, A6, A7, TOPREAL1:def 3;

      then (( GoB f) * ((i + 1),(j + 1))) = (f /. (k + 1)) & (( GoB f) * ((i + 1),j)) = (f /. k) or (( GoB f) * ((i + 1),(j + 1))) = (f /. k) & (( GoB f) * ((i + 1),j)) = (f /. (k + 1)) by SPPOL_1: 8;

      hence (f /. k) = (( GoB f) * ((i + 1),(j + 1))) by A15, A12, SPPOL_1: 8;

      thus (f /. (k + 1)) = (( GoB f) * ((i + 1),j)) by A17, A16, A12, SPPOL_1: 8;

      thus thesis by A17, A16, A12, SPPOL_1: 8;

    end;

    theorem :: GOBOARD7:47

    1 <= i & (i + 1) <= ( len ( GoB f)) & 1 <= j & (j + 1) <= ( width ( GoB f)) & 1 <= k & (k + 1) < ( len f) & ( LSeg ((( GoB f) * ((i + 1),j)),(( GoB f) * ((i + 1),(j + 1))))) = ( LSeg (f,k)) & ( LSeg ((( GoB f) * (i,(j + 1))),(( GoB f) * ((i + 1),(j + 1))))) = ( LSeg (f,(k + 1))) implies (f /. k) = (( GoB f) * ((i + 1),j)) & (f /. (k + 1)) = (( GoB f) * ((i + 1),(j + 1))) & (f /. (k + 2)) = (( GoB f) * (i,(j + 1)))

    proof

      assume that

       A1: 1 <= i and

       A2: (i + 1) <= ( len ( GoB f)) and

       A3: 1 <= j and

       A4: (j + 1) <= ( width ( GoB f)) and

       A5: 1 <= k and

       A6: (k + 1) < ( len f) and

       A7: ( LSeg ((( GoB f) * ((i + 1),j)),(( GoB f) * ((i + 1),(j + 1))))) = ( LSeg (f,k)) and

       A8: ( LSeg ((( GoB f) * (i,(j + 1))),(( GoB f) * ((i + 1),(j + 1))))) = ( LSeg (f,(k + 1)));

      

       A9: 1 <= (j + 1) by NAT_1: 11;

      

       A10: i < (i + 1) by NAT_1: 13;

      

       A11: 1 <= (i + 1) by NAT_1: 11;

      j < ( width ( GoB f)) by A4, NAT_1: 13;

      

      then ((( GoB f) * ((i + 1),j)) `1 ) = ((( GoB f) * ((i + 1),1)) `1 ) by A2, A3, A11, GOBOARD5: 2

      .= ((( GoB f) * ((i + 1),(j + 1))) `1 ) by A2, A4, A9, A11, GOBOARD5: 2;

      then

       A12: (( GoB f) * (i,(j + 1))) <> (( GoB f) * ((i + 1),j)) by A1, A2, A4, A9, A10, GOBOARD5: 3;

      

       A13: 1 <= (k + 1) by NAT_1: 11;

      

       A14: (k + (1 + 1)) = ((k + 1) + 1);

      then (k + 2) <= ( len f) by A6, NAT_1: 13;

      then

       A15: ( LSeg ((( GoB f) * (i,(j + 1))),(( GoB f) * ((i + 1),(j + 1))))) = ( LSeg ((f /. (k + 1)),(f /. (k + 2)))) by A8, A14, A13, TOPREAL1:def 3;

      then

       A16: (( GoB f) * ((i + 1),(j + 1))) = (f /. (k + 1)) & (( GoB f) * (i,(j + 1))) = (f /. (k + 2)) or (( GoB f) * ((i + 1),(j + 1))) = (f /. (k + 2)) & (( GoB f) * (i,(j + 1))) = (f /. (k + 1)) by SPPOL_1: 8;

      

       A17: ( LSeg ((( GoB f) * ((i + 1),j)),(( GoB f) * ((i + 1),(j + 1))))) = ( LSeg ((f /. k),(f /. (k + 1)))) by A5, A6, A7, TOPREAL1:def 3;

      then (( GoB f) * ((i + 1),j)) = (f /. k) & (( GoB f) * ((i + 1),(j + 1))) = (f /. (k + 1)) or (( GoB f) * ((i + 1),j)) = (f /. (k + 1)) & (( GoB f) * ((i + 1),(j + 1))) = (f /. k) by SPPOL_1: 8;

      hence (f /. k) = (( GoB f) * ((i + 1),j)) by A15, A12, SPPOL_1: 8;

      thus (f /. (k + 1)) = (( GoB f) * ((i + 1),(j + 1))) by A17, A16, A12, SPPOL_1: 8;

      thus thesis by A17, A16, A12, SPPOL_1: 8;

    end;

    theorem :: GOBOARD7:48

    1 <= i & i <= ( len ( GoB f)) & 1 <= j & (j + 1) < ( width ( GoB f)) & 1 <= k & (k + 1) < ( len f) & ( LSeg ((( GoB f) * (i,j)),(( GoB f) * (i,(j + 1))))) = ( LSeg (f,k)) & ( LSeg ((( GoB f) * (i,(j + 1))),(( GoB f) * (i,(j + 2))))) = ( LSeg (f,(k + 1))) implies (f /. k) = (( GoB f) * (i,j)) & (f /. (k + 1)) = (( GoB f) * (i,(j + 1))) & (f /. (k + 2)) = (( GoB f) * (i,(j + 2)))

    proof

      assume that

       A1: 1 <= i & i <= ( len ( GoB f)) & 1 <= j and

       A2: (j + 1) < ( width ( GoB f)) and

       A3: 1 <= k and

       A4: (k + 1) < ( len f) and

       A5: ( LSeg ((( GoB f) * (i,j)),(( GoB f) * (i,(j + 1))))) = ( LSeg (f,k)) and

       A6: ( LSeg ((( GoB f) * (i,(j + 1))),(( GoB f) * (i,(j + 2))))) = ( LSeg (f,(k + 1)));

      

       A7: 1 <= (k + 1) by NAT_1: 11;

      

       A8: (k + (1 + 1)) = ((k + 1) + 1);

      then (k + 2) <= ( len f) by A4, NAT_1: 13;

      then

       A9: ( LSeg ((( GoB f) * (i,(j + 1))),(( GoB f) * (i,(j + 2))))) = ( LSeg ((f /. (k + 1)),(f /. (k + 2)))) by A6, A8, A7, TOPREAL1:def 3;

      then

       A10: (( GoB f) * (i,(j + 1))) = (f /. (k + 1)) & (( GoB f) * (i,(j + 2))) = (f /. (k + 2)) or (( GoB f) * (i,(j + 1))) = (f /. (k + 2)) & (( GoB f) * (i,(j + 2))) = (f /. (k + 1)) by SPPOL_1: 8;

      

       A11: j < (j + 2) by XREAL_1: 29;

      (j + (1 + 1)) = ((j + 1) + 1);

      then (j + 2) <= ( width ( GoB f)) by A2, NAT_1: 13;

      then

       A12: ((( GoB f) * (i,j)) `2 ) < ((( GoB f) * (i,(j + 2))) `2 ) by A1, A11, GOBOARD5: 4;

      

       A13: ( LSeg ((( GoB f) * (i,j)),(( GoB f) * (i,(j + 1))))) = ( LSeg ((f /. k),(f /. (k + 1)))) by A3, A4, A5, TOPREAL1:def 3;

      then (( GoB f) * (i,j)) = (f /. k) & (( GoB f) * (i,(j + 1))) = (f /. (k + 1)) or (( GoB f) * (i,j)) = (f /. (k + 1)) & (( GoB f) * (i,(j + 1))) = (f /. k) by SPPOL_1: 8;

      hence (f /. k) = (( GoB f) * (i,j)) by A9, A12, SPPOL_1: 8;

      thus (f /. (k + 1)) = (( GoB f) * (i,(j + 1))) by A13, A10, A12, SPPOL_1: 8;

      thus thesis by A13, A10, A12, SPPOL_1: 8;

    end;

    theorem :: GOBOARD7:49

    1 <= i & (i + 1) <= ( len ( GoB f)) & 1 <= j & (j + 1) <= ( width ( GoB f)) & 1 <= k & (k + 1) < ( len f) & ( LSeg ((( GoB f) * (i,j)),(( GoB f) * (i,(j + 1))))) = ( LSeg (f,k)) & ( LSeg ((( GoB f) * (i,(j + 1))),(( GoB f) * ((i + 1),(j + 1))))) = ( LSeg (f,(k + 1))) implies (f /. k) = (( GoB f) * (i,j)) & (f /. (k + 1)) = (( GoB f) * (i,(j + 1))) & (f /. (k + 2)) = (( GoB f) * ((i + 1),(j + 1)))

    proof

      assume that

       A1: 1 <= i and

       A2: (i + 1) <= ( len ( GoB f)) and

       A3: 1 <= j and

       A4: (j + 1) <= ( width ( GoB f)) and

       A5: 1 <= k and

       A6: (k + 1) < ( len f) and

       A7: ( LSeg ((( GoB f) * (i,j)),(( GoB f) * (i,(j + 1))))) = ( LSeg (f,k)) and

       A8: ( LSeg ((( GoB f) * (i,(j + 1))),(( GoB f) * ((i + 1),(j + 1))))) = ( LSeg (f,(k + 1)));

      

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

      

       A10: j < (j + 1) by NAT_1: 13;

      

       A11: 1 <= (i + 1) by NAT_1: 11;

      i < ( len ( GoB f)) by A2, NAT_1: 13;

      

      then ((( GoB f) * (i,j)) `2 ) = ((( GoB f) * (1,j)) `2 ) by A1, A3, A9, GOBOARD5: 1

      .= ((( GoB f) * ((i + 1),j)) `2 ) by A2, A3, A11, A9, GOBOARD5: 1;

      then

       A12: (( GoB f) * (i,j)) <> (( GoB f) * ((i + 1),(j + 1))) by A2, A3, A4, A11, A10, GOBOARD5: 4;

      

       A13: 1 <= (k + 1) by NAT_1: 11;

      

       A14: (k + (1 + 1)) = ((k + 1) + 1);

      then (k + 2) <= ( len f) by A6, NAT_1: 13;

      then

       A15: ( LSeg ((( GoB f) * (i,(j + 1))),(( GoB f) * ((i + 1),(j + 1))))) = ( LSeg ((f /. (k + 1)),(f /. (k + 2)))) by A8, A14, A13, TOPREAL1:def 3;

      then

       A16: (( GoB f) * ((i + 1),(j + 1))) = (f /. (k + 1)) & (( GoB f) * (i,(j + 1))) = (f /. (k + 2)) or (( GoB f) * ((i + 1),(j + 1))) = (f /. (k + 2)) & (( GoB f) * (i,(j + 1))) = (f /. (k + 1)) by SPPOL_1: 8;

      

       A17: ( LSeg ((( GoB f) * (i,j)),(( GoB f) * (i,(j + 1))))) = ( LSeg ((f /. k),(f /. (k + 1)))) by A5, A6, A7, TOPREAL1:def 3;

      then (( GoB f) * (i,j)) = (f /. k) & (( GoB f) * (i,(j + 1))) = (f /. (k + 1)) or (( GoB f) * (i,j)) = (f /. (k + 1)) & (( GoB f) * (i,(j + 1))) = (f /. k) by SPPOL_1: 8;

      hence (f /. k) = (( GoB f) * (i,j)) by A15, A12, SPPOL_1: 8;

      thus (f /. (k + 1)) = (( GoB f) * (i,(j + 1))) by A17, A16, A12, SPPOL_1: 8;

      thus thesis by A17, A16, A12, SPPOL_1: 8;

    end;

    theorem :: GOBOARD7:50

    1 <= i & (i + 1) <= ( len ( GoB f)) & 1 <= j & (j + 1) <= ( width ( GoB f)) & 1 <= k & (k + 1) < ( len f) & ( LSeg ((( GoB f) * (i,(j + 1))),(( GoB f) * ((i + 1),(j + 1))))) = ( LSeg (f,k)) & ( LSeg ((( GoB f) * ((i + 1),j)),(( GoB f) * ((i + 1),(j + 1))))) = ( LSeg (f,(k + 1))) implies (f /. k) = (( GoB f) * (i,(j + 1))) & (f /. (k + 1)) = (( GoB f) * ((i + 1),(j + 1))) & (f /. (k + 2)) = (( GoB f) * ((i + 1),j))

    proof

      assume that

       A1: 1 <= i and

       A2: (i + 1) <= ( len ( GoB f)) and

       A3: 1 <= j and

       A4: (j + 1) <= ( width ( GoB f)) and

       A5: 1 <= k and

       A6: (k + 1) < ( len f) and

       A7: ( LSeg ((( GoB f) * (i,(j + 1))),(( GoB f) * ((i + 1),(j + 1))))) = ( LSeg (f,k)) and

       A8: ( LSeg ((( GoB f) * ((i + 1),j)),(( GoB f) * ((i + 1),(j + 1))))) = ( LSeg (f,(k + 1)));

      

       A9: 1 <= (i + 1) by NAT_1: 11;

      

       A10: j < (j + 1) by NAT_1: 13;

      

       A11: 1 <= (j + 1) by NAT_1: 11;

      i < ( len ( GoB f)) by A2, NAT_1: 13;

      

      then ((( GoB f) * (i,(j + 1))) `2 ) = ((( GoB f) * (1,(j + 1))) `2 ) by A1, A4, A11, GOBOARD5: 1

      .= ((( GoB f) * ((i + 1),(j + 1))) `2 ) by A2, A4, A9, A11, GOBOARD5: 1;

      then

       A12: (( GoB f) * ((i + 1),j)) <> (( GoB f) * (i,(j + 1))) by A2, A3, A4, A9, A10, GOBOARD5: 4;

      

       A13: 1 <= (k + 1) by NAT_1: 11;

      

       A14: (k + (1 + 1)) = ((k + 1) + 1);

      then (k + 2) <= ( len f) by A6, NAT_1: 13;

      then

       A15: ( LSeg ((( GoB f) * ((i + 1),j)),(( GoB f) * ((i + 1),(j + 1))))) = ( LSeg ((f /. (k + 1)),(f /. (k + 2)))) by A8, A14, A13, TOPREAL1:def 3;

      then

       A16: (( GoB f) * ((i + 1),(j + 1))) = (f /. (k + 1)) & (( GoB f) * ((i + 1),j)) = (f /. (k + 2)) or (( GoB f) * ((i + 1),(j + 1))) = (f /. (k + 2)) & (( GoB f) * ((i + 1),j)) = (f /. (k + 1)) by SPPOL_1: 8;

      

       A17: ( LSeg ((( GoB f) * (i,(j + 1))),(( GoB f) * ((i + 1),(j + 1))))) = ( LSeg ((f /. k),(f /. (k + 1)))) by A5, A6, A7, TOPREAL1:def 3;

      then (( GoB f) * (i,(j + 1))) = (f /. k) & (( GoB f) * ((i + 1),(j + 1))) = (f /. (k + 1)) or (( GoB f) * (i,(j + 1))) = (f /. (k + 1)) & (( GoB f) * ((i + 1),(j + 1))) = (f /. k) by SPPOL_1: 8;

      hence (f /. k) = (( GoB f) * (i,(j + 1))) by A15, A12, SPPOL_1: 8;

      thus (f /. (k + 1)) = (( GoB f) * ((i + 1),(j + 1))) by A17, A16, A12, SPPOL_1: 8;

      thus thesis by A17, A16, A12, SPPOL_1: 8;

    end;

    theorem :: GOBOARD7:51

    1 <= i & (i + 1) < ( len ( GoB f)) & 1 <= j & j <= ( width ( GoB f)) & 1 <= k & (k + 1) < ( len f) & ( LSeg ((( GoB f) * (i,j)),(( GoB f) * ((i + 1),j)))) = ( LSeg (f,k)) & ( LSeg ((( GoB f) * ((i + 1),j)),(( GoB f) * ((i + 2),j)))) = ( LSeg (f,(k + 1))) implies (f /. k) = (( GoB f) * (i,j)) & (f /. (k + 1)) = (( GoB f) * ((i + 1),j)) & (f /. (k + 2)) = (( GoB f) * ((i + 2),j))

    proof

      assume that

       A1: 1 <= i and

       A2: (i + 1) < ( len ( GoB f)) and

       A3: 1 <= j & j <= ( width ( GoB f)) and

       A4: 1 <= k and

       A5: (k + 1) < ( len f) and

       A6: ( LSeg ((( GoB f) * (i,j)),(( GoB f) * ((i + 1),j)))) = ( LSeg (f,k)) and

       A7: ( LSeg ((( GoB f) * ((i + 1),j)),(( GoB f) * ((i + 2),j)))) = ( LSeg (f,(k + 1)));

      

       A8: i < (i + 2) by XREAL_1: 29;

      (i + (1 + 1)) = ((i + 1) + 1);

      then (i + 2) <= ( len ( GoB f)) by A2, NAT_1: 13;

      then

       A9: ((( GoB f) * (i,j)) `1 ) < ((( GoB f) * ((i + 2),j)) `1 ) by A1, A3, A8, GOBOARD5: 3;

      

       A10: 1 <= (k + 1) by NAT_1: 11;

      

       A11: (k + (1 + 1)) = ((k + 1) + 1);

      then (k + 2) <= ( len f) by A5, NAT_1: 13;

      then

       A12: ( LSeg ((( GoB f) * ((i + 1),j)),(( GoB f) * ((i + 2),j)))) = ( LSeg ((f /. (k + 1)),(f /. (k + 2)))) by A7, A11, A10, TOPREAL1:def 3;

      then

       A13: (( GoB f) * ((i + 1),j)) = (f /. (k + 1)) & (( GoB f) * ((i + 2),j)) = (f /. (k + 2)) or (( GoB f) * ((i + 1),j)) = (f /. (k + 2)) & (( GoB f) * ((i + 2),j)) = (f /. (k + 1)) by SPPOL_1: 8;

      

       A14: ( LSeg ((( GoB f) * (i,j)),(( GoB f) * ((i + 1),j)))) = ( LSeg ((f /. k),(f /. (k + 1)))) by A4, A5, A6, TOPREAL1:def 3;

      then (( GoB f) * (i,j)) = (f /. k) & (( GoB f) * ((i + 1),j)) = (f /. (k + 1)) or (( GoB f) * (i,j)) = (f /. (k + 1)) & (( GoB f) * ((i + 1),j)) = (f /. k) by SPPOL_1: 8;

      hence (f /. k) = (( GoB f) * (i,j)) by A12, A9, SPPOL_1: 8;

      thus (f /. (k + 1)) = (( GoB f) * ((i + 1),j)) by A14, A13, A9, SPPOL_1: 8;

      thus thesis by A14, A13, A9, SPPOL_1: 8;

    end;

    theorem :: GOBOARD7:52

    1 <= i & (i + 1) <= ( len ( GoB f)) & 1 <= j & (j + 1) <= ( width ( GoB f)) & 1 <= k & (k + 1) < ( len f) & ( LSeg ((( GoB f) * (i,j)),(( GoB f) * ((i + 1),j)))) = ( LSeg (f,k)) & ( LSeg ((( GoB f) * ((i + 1),j)),(( GoB f) * ((i + 1),(j + 1))))) = ( LSeg (f,(k + 1))) implies (f /. k) = (( GoB f) * (i,j)) & (f /. (k + 1)) = (( GoB f) * ((i + 1),j)) & (f /. (k + 2)) = (( GoB f) * ((i + 1),(j + 1)))

    proof

      assume that

       A1: 1 <= i and

       A2: (i + 1) <= ( len ( GoB f)) and

       A3: 1 <= j and

       A4: (j + 1) <= ( width ( GoB f)) and

       A5: 1 <= k and

       A6: (k + 1) < ( len f) and

       A7: ( LSeg ((( GoB f) * (i,j)),(( GoB f) * ((i + 1),j)))) = ( LSeg (f,k)) and

       A8: ( LSeg ((( GoB f) * ((i + 1),j)),(( GoB f) * ((i + 1),(j + 1))))) = ( LSeg (f,(k + 1)));

      

       A9: i < ( len ( GoB f)) by A2, NAT_1: 13;

      

       A10: i < (i + 1) by NAT_1: 13;

      

       A11: 1 <= (j + 1) by NAT_1: 11;

      j < ( width ( GoB f)) by A4, NAT_1: 13;

      

      then ((( GoB f) * (i,j)) `1 ) = ((( GoB f) * (i,1)) `1 ) by A1, A3, A9, GOBOARD5: 2

      .= ((( GoB f) * (i,(j + 1))) `1 ) by A1, A4, A11, A9, GOBOARD5: 2;

      then

       A12: (( GoB f) * (i,j)) <> (( GoB f) * ((i + 1),(j + 1))) by A1, A2, A4, A11, A10, GOBOARD5: 3;

      

       A13: 1 <= (k + 1) by NAT_1: 11;

      

       A14: (k + (1 + 1)) = ((k + 1) + 1);

      then (k + 2) <= ( len f) by A6, NAT_1: 13;

      then

       A15: ( LSeg ((( GoB f) * ((i + 1),j)),(( GoB f) * ((i + 1),(j + 1))))) = ( LSeg ((f /. (k + 1)),(f /. (k + 2)))) by A8, A14, A13, TOPREAL1:def 3;

      then

       A16: (( GoB f) * ((i + 1),(j + 1))) = (f /. (k + 1)) & (( GoB f) * ((i + 1),j)) = (f /. (k + 2)) or (( GoB f) * ((i + 1),(j + 1))) = (f /. (k + 2)) & (( GoB f) * ((i + 1),j)) = (f /. (k + 1)) by SPPOL_1: 8;

      

       A17: ( LSeg ((( GoB f) * (i,j)),(( GoB f) * ((i + 1),j)))) = ( LSeg ((f /. k),(f /. (k + 1)))) by A5, A6, A7, TOPREAL1:def 3;

      then (( GoB f) * (i,j)) = (f /. k) & (( GoB f) * ((i + 1),j)) = (f /. (k + 1)) or (( GoB f) * (i,j)) = (f /. (k + 1)) & (( GoB f) * ((i + 1),j)) = (f /. k) by SPPOL_1: 8;

      hence (f /. k) = (( GoB f) * (i,j)) by A15, A12, SPPOL_1: 8;

      thus (f /. (k + 1)) = (( GoB f) * ((i + 1),j)) by A17, A16, A12, SPPOL_1: 8;

      thus thesis by A17, A16, A12, SPPOL_1: 8;

    end;

    theorem :: GOBOARD7:53

    

     Th53: 1 <= i & i <= ( len ( GoB f)) & 1 <= j & (j + 1) < ( width ( GoB f)) & ( LSeg ((( GoB f) * (i,j)),(( GoB f) * (i,(j + 1))))) c= ( L~ f) & ( LSeg ((( GoB f) * (i,(j + 1))),(( GoB f) * (i,(j + 2))))) c= ( L~ f) implies (f /. 1) = (( GoB f) * (i,(j + 1))) & ((f /. 2) = (( GoB f) * (i,j)) & (f /. (( len f) -' 1)) = (( GoB f) * (i,(j + 2))) or (f /. 2) = (( GoB f) * (i,(j + 2))) & (f /. (( len f) -' 1)) = (( GoB f) * (i,j))) or ex k st 1 <= k & (k + 1) < ( len f) & (f /. (k + 1)) = (( GoB f) * (i,(j + 1))) & ((f /. k) = (( GoB f) * (i,j)) & (f /. (k + 2)) = (( GoB f) * (i,(j + 2))) or (f /. k) = (( GoB f) * (i,(j + 2))) & (f /. (k + 2)) = (( GoB f) * (i,j)))

    proof

      assume that

       A1: 1 <= i & i <= ( len ( GoB f)) and

       A2: 1 <= j and

       A3: (j + 1) < ( width ( GoB f)) and

       A4: ( LSeg ((( GoB f) * (i,j)),(( GoB f) * (i,(j + 1))))) c= ( L~ f) and

       A5: ( LSeg ((( GoB f) * (i,(j + 1))),(( GoB f) * (i,(j + 2))))) c= ( L~ f);

      

       A6: 1 <= (j + 1) by NAT_1: 11;

      ((1 / 2) * ((( GoB f) * (i,j)) + (( GoB f) * (i,(j + 1))))) in ( LSeg ((( GoB f) * (i,j)),(( GoB f) * (i,(j + 1))))) by RLTOPSP1: 69;

      then

      consider k1 such that

       A7: 1 <= k1 and

       A8: (k1 + 1) <= ( len f) and

       A9: ( LSeg ((( GoB f) * (i,j)),(( GoB f) * (i,(j + 1))))) = ( LSeg (f,k1)) by A1, A2, A3, A4, Th39;

      

       A10: k1 < ( len f) by A8, NAT_1: 13;

       A11:

      now

        assume k1 > 1;

        then k1 >= (1 + 1) by NAT_1: 13;

        hence k1 = 2 or k1 > 2 by XXREAL_0: 1;

      end;

      

       A12: j < ( width ( GoB f)) by A3, NAT_1: 13;

      

       A13: (j + (1 + 1)) = ((j + 1) + 1);

      then

       A14: 1 <= (j + 2) by NAT_1: 11;

      

       A15: (j + 2) <= ( width ( GoB f)) by A3, A13, NAT_1: 13;

      ((1 / 2) * ((( GoB f) * (i,(j + 1))) + (( GoB f) * (i,(j + 2))))) in ( LSeg ((( GoB f) * (i,(j + 1))),(( GoB f) * (i,(j + 2))))) by RLTOPSP1: 69;

      then

      consider k2 such that

       A16: 1 <= k2 and

       A17: (k2 + 1) <= ( len f) and

       A18: ( LSeg ((( GoB f) * (i,(j + 1))),(( GoB f) * (i,(j + 2))))) = ( LSeg (f,k2)) by A1, A5, A6, A13, A15, Th39;

      

       A19: k2 < ( len f) by A17, NAT_1: 13;

       A20:

      now

        assume k2 > 1;

        then k2 >= (1 + 1) by NAT_1: 13;

        hence k2 = 2 or k2 > 2 by XXREAL_0: 1;

      end;

      

       A21: k1 = 1 or k1 > 1 by A7, XXREAL_0: 1;

      now

        per cases by A16, A11, A20, A21, XXREAL_0: 1;

          case that

           A22: k1 = 1 and

           A23: k2 = 2;

          

           A24: ( LSeg (f,2)) = ( LSeg ((f /. 2),(f /. (2 + 1)))) by A17, A23, TOPREAL1:def 3;

          then

           A25: (( GoB f) * (i,(j + 1))) = (f /. 2) & (( GoB f) * (i,(j + 2))) = (f /. (2 + 1)) or (( GoB f) * (i,(j + 1))) = (f /. (2 + 1)) & (( GoB f) * (i,(j + 2))) = (f /. 2) by A18, A23, SPPOL_1: 8;

          thus 1 <= 1 & (1 + 1) < ( len f) by A17, A23, NAT_1: 13;

          

           A26: 3 < ( len f) by Th34, XXREAL_0: 2;

          then

           A27: (f /. 1) <> (f /. 3) by Th36;

          

           A28: ( LSeg (f,1)) = ( LSeg ((f /. 1),(f /. (1 + 1)))) by A8, A22, TOPREAL1:def 3;

          then

           A29: (( GoB f) * (i,j)) = (f /. 1) & (( GoB f) * (i,(j + 1))) = (f /. 2) or (( GoB f) * (i,j)) = (f /. 2) & (( GoB f) * (i,(j + 1))) = (f /. 1) by A9, A22, SPPOL_1: 8;

          hence (f /. (1 + 1)) = (( GoB f) * (i,(j + 1))) by A25, A26, Th36;

          thus (f /. 1) = (( GoB f) * (i,j)) by A18, A23, A29, A24, A27, SPPOL_1: 8;

          thus (f /. (1 + 2)) = (( GoB f) * (i,(j + 2))) by A9, A22, A28, A25, A27, SPPOL_1: 8;

        end;

          case that

           A30: k1 = 1 and

           A31: k2 > 2;

          

           A32: ( LSeg (f,1)) = ( LSeg ((f /. 1),(f /. (1 + 1)))) by A8, A30, TOPREAL1:def 3;

          then

           A33: (( GoB f) * (i,j)) = (f /. 1) & (( GoB f) * (i,(j + 1))) = (f /. 2) or (( GoB f) * (i,j)) = (f /. 2) & (( GoB f) * (i,(j + 1))) = (f /. 1) by A9, A30, SPPOL_1: 8;

          

           A34: 2 < (k2 + 1) by A31, NAT_1: 13;

          then

           A35: (f /. (k2 + 1)) <> (f /. 2) by A17, Th37;

          ( LSeg (f,k2)) = ( LSeg ((f /. k2),(f /. (k2 + 1)))) by A16, A17, TOPREAL1:def 3;

          then

           A36: (( GoB f) * (i,(j + 1))) = (f /. k2) & (( GoB f) * (i,(j + 2))) = (f /. (k2 + 1)) or (( GoB f) * (i,(j + 1))) = (f /. (k2 + 1)) & (( GoB f) * (i,(j + 2))) = (f /. k2) by A18, SPPOL_1: 8;

          

           A37: (f /. k2) <> (f /. 2) by A19, A31, Th36;

          hence (f /. 1) = (( GoB f) * (i,(j + 1))) by A9, A30, A32, A36, A35, SPPOL_1: 8;

          thus (f /. 2) = (( GoB f) * (i,j)) by A9, A30, A32, A36, A37, A35, SPPOL_1: 8;

          

           A38: k2 > 1 by A31, XXREAL_0: 2;

          then

           A39: (k2 + 1) > 1 by NAT_1: 13;

          then (k2 + 1) = ( len f) by A17, A19, A31, A33, A36, A38, A34, Th37, Th38;

          then (k2 + 1) = ((( len f) -' 1) + 1) by A39, XREAL_1: 235;

          hence (f /. (( len f) -' 1)) = (( GoB f) * (i,(j + 2))) by A19, A31, A33, A36, A38, Th36;

        end;

          case that

           A40: k2 = 1 and

           A41: k1 = 2;

          

           A42: ( LSeg (f,2)) = ( LSeg ((f /. 2),(f /. (2 + 1)))) by A8, A41, TOPREAL1:def 3;

          then

           A43: (( GoB f) * (i,(j + 1))) = (f /. 2) & (( GoB f) * (i,j)) = (f /. (2 + 1)) or (( GoB f) * (i,(j + 1))) = (f /. (2 + 1)) & (( GoB f) * (i,j)) = (f /. 2) by A9, A41, SPPOL_1: 8;

          thus 1 <= 1 & (1 + 1) < ( len f) by A8, A41, NAT_1: 13;

          

           A44: 3 < ( len f) by Th34, XXREAL_0: 2;

          then

           A45: (f /. 1) <> (f /. 3) by Th36;

          

           A46: ( LSeg (f,1)) = ( LSeg ((f /. 1),(f /. (1 + 1)))) by A17, A40, TOPREAL1:def 3;

          then

           A47: (( GoB f) * (i,(j + 2))) = (f /. 1) & (( GoB f) * (i,(j + 1))) = (f /. 2) or (( GoB f) * (i,(j + 2))) = (f /. 2) & (( GoB f) * (i,(j + 1))) = (f /. 1) by A18, A40, SPPOL_1: 8;

          hence (f /. (1 + 1)) = (( GoB f) * (i,(j + 1))) by A43, A44, Th36;

          thus (f /. 1) = (( GoB f) * (i,(j + 2))) by A9, A41, A47, A42, A45, SPPOL_1: 8;

          thus (f /. (1 + 2)) = (( GoB f) * (i,j)) by A18, A40, A46, A43, A45, SPPOL_1: 8;

        end;

          case that

           A48: k2 = 1 and

           A49: k1 > 2;

          

           A50: ( LSeg (f,1)) = ( LSeg ((f /. 1),(f /. (1 + 1)))) by A17, A48, TOPREAL1:def 3;

          then

           A51: (( GoB f) * (i,(j + 2))) = (f /. 1) & (( GoB f) * (i,(j + 1))) = (f /. 2) or (( GoB f) * (i,(j + 2))) = (f /. 2) & (( GoB f) * (i,(j + 1))) = (f /. 1) by A18, A48, SPPOL_1: 8;

          

           A52: 2 < (k1 + 1) by A49, NAT_1: 13;

          then

           A53: (f /. (k1 + 1)) <> (f /. 2) by A8, Th37;

          ( LSeg (f,k1)) = ( LSeg ((f /. k1),(f /. (k1 + 1)))) by A7, A8, TOPREAL1:def 3;

          then

           A54: (( GoB f) * (i,(j + 1))) = (f /. k1) & (( GoB f) * (i,j)) = (f /. (k1 + 1)) or (( GoB f) * (i,(j + 1))) = (f /. (k1 + 1)) & (( GoB f) * (i,j)) = (f /. k1) by A9, SPPOL_1: 8;

          

           A55: (f /. k1) <> (f /. 2) by A10, A49, Th36;

          hence (f /. 1) = (( GoB f) * (i,(j + 1))) by A18, A48, A50, A54, A53, SPPOL_1: 8;

          thus (f /. 2) = (( GoB f) * (i,(j + 2))) by A18, A48, A50, A54, A55, A53, SPPOL_1: 8;

          

           A56: k1 > 1 by A49, XXREAL_0: 2;

          then

           A57: (k1 + 1) > 1 by NAT_1: 13;

          then (k1 + 1) = ( len f) by A8, A10, A49, A51, A54, A56, A52, Th37, Th38;

          then (k1 + 1) = ((( len f) -' 1) + 1) by A57, XREAL_1: 235;

          hence (f /. (( len f) -' 1)) = (( GoB f) * (i,j)) by A10, A49, A51, A54, A56, Th36;

        end;

          case k1 = k2;

          then

           A58: (( GoB f) * (i,j)) = (( GoB f) * (i,(j + 2))) or (( GoB f) * (i,j)) = (( GoB f) * (i,(j + 1))) by A9, A18, SPPOL_1: 8;

          

           A59: [i, (j + 2)] in ( Indices ( GoB f)) by A1, A15, A14, MATRIX_0: 30;

           [i, j] in ( Indices ( GoB f)) & [i, (j + 1)] in ( Indices ( GoB f)) by A1, A2, A3, A6, A12, MATRIX_0: 30;

          then j = (j + 1) or j = (j + 2) by A58, A59, GOBOARD1: 5;

          hence contradiction;

        end;

          case that

           A60: k1 > 1 and

           A61: k2 > k1;

          

           A62: 1 < (k1 + 1) & (k1 + 1) < (k2 + 1) by A60, A61, NAT_1: 13, XREAL_1: 6;

          

           A63: k1 < (k2 + 1) by A61, NAT_1: 13;

          then

           A64: (f /. k1) <> (f /. (k2 + 1)) by A17, A60, Th37;

          

           A65: (k1 + 1) <= k2 by A61, NAT_1: 13;

          ( LSeg (f,k2)) = ( LSeg ((f /. k2),(f /. (k2 + 1)))) by A16, A17, TOPREAL1:def 3;

          then

           A66: (( GoB f) * (i,(j + 1))) = (f /. k2) & (( GoB f) * (i,(j + 2))) = (f /. (k2 + 1)) or (( GoB f) * (i,(j + 1))) = (f /. (k2 + 1)) & (( GoB f) * (i,(j + 2))) = (f /. k2) by A18, SPPOL_1: 8;

          

           A67: k2 < ( len f) by A17, NAT_1: 13;

          then

           A68: (f /. k1) <> (f /. k2) by A60, A61, Th37;

          

           A69: ( LSeg (f,k1)) = ( LSeg ((f /. k1),(f /. (k1 + 1)))) by A7, A8, TOPREAL1:def 3;

          then (( GoB f) * (i,j)) = (f /. k1) & (( GoB f) * (i,(j + 1))) = (f /. (k1 + 1)) or (( GoB f) * (i,j)) = (f /. (k1 + 1)) & (( GoB f) * (i,(j + 1))) = (f /. k1) by A9, SPPOL_1: 8;

          then (k1 + 1) >= k2 by A17, A60, A61, A66, A63, A67, A62, Th37;

          then

           A70: (k1 + 1) = k2 by A65, XXREAL_0: 1;

          hence 1 <= k1 & (k1 + 1) < ( len f) by A17, A60, NAT_1: 13;

          thus (f /. (k1 + 1)) = (( GoB f) * (i,(j + 1))) by A9, A69, A66, A64, A68, SPPOL_1: 8;

          thus (f /. k1) = (( GoB f) * (i,j)) by A9, A69, A66, A64, A68, SPPOL_1: 8;

          thus (f /. (k1 + 2)) = (( GoB f) * (i,(j + 2))) by A9, A69, A66, A64, A70, SPPOL_1: 8;

        end;

          case that

           A71: k2 > 1 and

           A72: k1 > k2;

          

           A73: 1 < (k2 + 1) & (k2 + 1) < (k1 + 1) by A71, A72, NAT_1: 13, XREAL_1: 6;

          

           A74: k2 < (k1 + 1) by A72, NAT_1: 13;

          then

           A75: (f /. k2) <> (f /. (k1 + 1)) by A8, A71, Th37;

          

           A76: (k2 + 1) <= k1 by A72, NAT_1: 13;

          ( LSeg (f,k1)) = ( LSeg ((f /. k1),(f /. (k1 + 1)))) by A7, A8, TOPREAL1:def 3;

          then

           A77: (( GoB f) * (i,(j + 1))) = (f /. k1) & (( GoB f) * (i,j)) = (f /. (k1 + 1)) or (( GoB f) * (i,(j + 1))) = (f /. (k1 + 1)) & (( GoB f) * (i,j)) = (f /. k1) by A9, SPPOL_1: 8;

          

           A78: k1 < ( len f) by A8, NAT_1: 13;

          then

           A79: (f /. k2) <> (f /. k1) by A71, A72, Th37;

          

           A80: ( LSeg (f,k2)) = ( LSeg ((f /. k2),(f /. (k2 + 1)))) by A16, A17, TOPREAL1:def 3;

          then (( GoB f) * (i,(j + 2))) = (f /. k2) & (( GoB f) * (i,(j + 1))) = (f /. (k2 + 1)) or (( GoB f) * (i,(j + 2))) = (f /. (k2 + 1)) & (( GoB f) * (i,(j + 1))) = (f /. k2) by A18, SPPOL_1: 8;

          then (k2 + 1) >= k1 by A8, A71, A72, A77, A74, A78, A73, Th37;

          then

           A81: (k2 + 1) = k1 by A76, XXREAL_0: 1;

          hence 1 <= k2 & (k2 + 1) < ( len f) by A8, A71, NAT_1: 13;

          thus (f /. (k2 + 1)) = (( GoB f) * (i,(j + 1))) by A18, A80, A77, A75, A79, SPPOL_1: 8;

          thus (f /. k2) = (( GoB f) * (i,(j + 2))) by A18, A80, A77, A75, A79, SPPOL_1: 8;

          thus (f /. (k2 + 2)) = (( GoB f) * (i,j)) by A18, A80, A77, A75, A81, SPPOL_1: 8;

        end;

      end;

      hence thesis;

    end;

    theorem :: GOBOARD7:54

    

     Th54: 1 <= i & (i + 1) <= ( len ( GoB f)) & 1 <= j & (j + 1) <= ( width ( GoB f)) & ( LSeg ((( GoB f) * (i,j)),(( GoB f) * (i,(j + 1))))) c= ( L~ f) & ( LSeg ((( GoB f) * (i,(j + 1))),(( GoB f) * ((i + 1),(j + 1))))) c= ( L~ f) implies (f /. 1) = (( GoB f) * (i,(j + 1))) & ((f /. 2) = (( GoB f) * (i,j)) & (f /. (( len f) -' 1)) = (( GoB f) * ((i + 1),(j + 1))) or (f /. 2) = (( GoB f) * ((i + 1),(j + 1))) & (f /. (( len f) -' 1)) = (( GoB f) * (i,j))) or ex k st 1 <= k & (k + 1) < ( len f) & (f /. (k + 1)) = (( GoB f) * (i,(j + 1))) & ((f /. k) = (( GoB f) * (i,j)) & (f /. (k + 2)) = (( GoB f) * ((i + 1),(j + 1))) or (f /. k) = (( GoB f) * ((i + 1),(j + 1))) & (f /. (k + 2)) = (( GoB f) * (i,j)))

    proof

      assume that

       A1: 1 <= i and

       A2: (i + 1) <= ( len ( GoB f)) and

       A3: 1 <= j and

       A4: (j + 1) <= ( width ( GoB f)) and

       A5: ( LSeg ((( GoB f) * (i,j)),(( GoB f) * (i,(j + 1))))) c= ( L~ f) and

       A6: ( LSeg ((( GoB f) * (i,(j + 1))),(( GoB f) * ((i + 1),(j + 1))))) c= ( L~ f);

      

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

      

       A8: j < ( width ( GoB f)) by A4, NAT_1: 13;

      

       A9: 1 <= (i + 1) by NAT_1: 11;

      ((1 / 2) * ((( GoB f) * (i,j)) + (( GoB f) * (i,(j + 1))))) in ( LSeg ((( GoB f) * (i,j)),(( GoB f) * (i,(j + 1))))) by RLTOPSP1: 69;

      then

      consider k1 such that

       A10: 1 <= k1 and

       A11: (k1 + 1) <= ( len f) and

       A12: ( LSeg ((( GoB f) * (i,j)),(( GoB f) * (i,(j + 1))))) = ( LSeg (f,k1)) by A1, A3, A4, A5, A7, Th39;

      

       A13: k1 < ( len f) by A11, NAT_1: 13;

       A14:

      now

        assume k1 > 1;

        then k1 >= (1 + 1) by NAT_1: 13;

        hence k1 = 2 or k1 > 2 by XXREAL_0: 1;

      end;

      

       A15: 1 <= (j + 1) by NAT_1: 11;

      ((1 / 2) * ((( GoB f) * (i,(j + 1))) + (( GoB f) * ((i + 1),(j + 1))))) in ( LSeg ((( GoB f) * (i,(j + 1))),(( GoB f) * ((i + 1),(j + 1))))) by RLTOPSP1: 69;

      then

      consider k2 such that

       A16: 1 <= k2 and

       A17: (k2 + 1) <= ( len f) and

       A18: ( LSeg ((( GoB f) * (i,(j + 1))),(( GoB f) * ((i + 1),(j + 1))))) = ( LSeg (f,k2)) by A1, A2, A4, A6, A15, Th40;

      

       A19: k2 < ( len f) by A17, NAT_1: 13;

       A20:

      now

        assume k2 > 1;

        then k2 >= (1 + 1) by NAT_1: 13;

        hence k2 = 2 or k2 > 2 by XXREAL_0: 1;

      end;

      

       A21: k1 = 1 or k1 > 1 by A10, XXREAL_0: 1;

      now

        per cases by A16, A14, A20, A21, XXREAL_0: 1;

          case that

           A22: k1 = 1 and

           A23: k2 = 2;

          

           A24: ( LSeg (f,2)) = ( LSeg ((f /. 2),(f /. (2 + 1)))) by A17, A23, TOPREAL1:def 3;

          then

           A25: (( GoB f) * (i,(j + 1))) = (f /. 2) & (( GoB f) * ((i + 1),(j + 1))) = (f /. (2 + 1)) or (( GoB f) * (i,(j + 1))) = (f /. (2 + 1)) & (( GoB f) * ((i + 1),(j + 1))) = (f /. 2) by A18, A23, SPPOL_1: 8;

          thus 1 <= 1 & (1 + 1) < ( len f) by A17, A23, NAT_1: 13;

          

           A26: 3 < ( len f) by Th34, XXREAL_0: 2;

          then

           A27: (f /. 1) <> (f /. 3) by Th36;

          

           A28: ( LSeg (f,1)) = ( LSeg ((f /. 1),(f /. (1 + 1)))) by A11, A22, TOPREAL1:def 3;

          then

           A29: (( GoB f) * (i,j)) = (f /. 1) & (( GoB f) * (i,(j + 1))) = (f /. 2) or (( GoB f) * (i,j)) = (f /. 2) & (( GoB f) * (i,(j + 1))) = (f /. 1) by A12, A22, SPPOL_1: 8;

          hence (f /. (1 + 1)) = (( GoB f) * (i,(j + 1))) by A25, A26, Th36;

          thus (f /. 1) = (( GoB f) * (i,j)) by A18, A23, A29, A24, A27, SPPOL_1: 8;

          thus (f /. (1 + 2)) = (( GoB f) * ((i + 1),(j + 1))) by A12, A22, A28, A25, A27, SPPOL_1: 8;

        end;

          case that

           A30: k1 = 1 and

           A31: k2 > 2;

          

           A32: ( LSeg (f,1)) = ( LSeg ((f /. 1),(f /. (1 + 1)))) by A11, A30, TOPREAL1:def 3;

          then

           A33: (( GoB f) * (i,j)) = (f /. 1) & (( GoB f) * (i,(j + 1))) = (f /. 2) or (( GoB f) * (i,j)) = (f /. 2) & (( GoB f) * (i,(j + 1))) = (f /. 1) by A12, A30, SPPOL_1: 8;

          

           A34: 2 < (k2 + 1) by A31, NAT_1: 13;

          then

           A35: (f /. (k2 + 1)) <> (f /. 2) by A17, Th37;

          ( LSeg (f,k2)) = ( LSeg ((f /. k2),(f /. (k2 + 1)))) by A16, A17, TOPREAL1:def 3;

          then

           A36: (( GoB f) * (i,(j + 1))) = (f /. k2) & (( GoB f) * ((i + 1),(j + 1))) = (f /. (k2 + 1)) or (( GoB f) * (i,(j + 1))) = (f /. (k2 + 1)) & (( GoB f) * ((i + 1),(j + 1))) = (f /. k2) by A18, SPPOL_1: 8;

          

           A37: (f /. k2) <> (f /. 2) by A19, A31, Th36;

          hence (f /. 1) = (( GoB f) * (i,(j + 1))) by A12, A30, A32, A36, A35, SPPOL_1: 8;

          thus (f /. 2) = (( GoB f) * (i,j)) by A12, A30, A32, A36, A37, A35, SPPOL_1: 8;

          

           A38: k2 > 1 by A31, XXREAL_0: 2;

          then

           A39: (k2 + 1) > 1 by NAT_1: 13;

          then (k2 + 1) = ( len f) by A17, A19, A31, A33, A36, A38, A34, Th37, Th38;

          then (k2 + 1) = ((( len f) -' 1) + 1) by A39, XREAL_1: 235;

          hence (f /. (( len f) -' 1)) = (( GoB f) * ((i + 1),(j + 1))) by A19, A31, A33, A36, A38, Th36;

        end;

          case that

           A40: k2 = 1 and

           A41: k1 = 2;

          

           A42: ( LSeg (f,2)) = ( LSeg ((f /. 2),(f /. (2 + 1)))) by A11, A41, TOPREAL1:def 3;

          then

           A43: (( GoB f) * (i,(j + 1))) = (f /. 2) & (( GoB f) * (i,j)) = (f /. (2 + 1)) or (( GoB f) * (i,(j + 1))) = (f /. (2 + 1)) & (( GoB f) * (i,j)) = (f /. 2) by A12, A41, SPPOL_1: 8;

          thus 1 <= 1 & (1 + 1) < ( len f) by A11, A41, NAT_1: 13;

          

           A44: 3 < ( len f) by Th34, XXREAL_0: 2;

          then

           A45: (f /. 1) <> (f /. 3) by Th36;

          

           A46: ( LSeg (f,1)) = ( LSeg ((f /. 1),(f /. (1 + 1)))) by A17, A40, TOPREAL1:def 3;

          then

           A47: (( GoB f) * ((i + 1),(j + 1))) = (f /. 1) & (( GoB f) * (i,(j + 1))) = (f /. 2) or (( GoB f) * ((i + 1),(j + 1))) = (f /. 2) & (( GoB f) * (i,(j + 1))) = (f /. 1) by A18, A40, SPPOL_1: 8;

          hence (f /. (1 + 1)) = (( GoB f) * (i,(j + 1))) by A43, A44, Th36;

          thus (f /. 1) = (( GoB f) * ((i + 1),(j + 1))) by A12, A41, A47, A42, A45, SPPOL_1: 8;

          thus (f /. (1 + 2)) = (( GoB f) * (i,j)) by A18, A40, A46, A43, A45, SPPOL_1: 8;

        end;

          case that

           A48: k2 = 1 and

           A49: k1 > 2;

          

           A50: ( LSeg (f,1)) = ( LSeg ((f /. 1),(f /. (1 + 1)))) by A17, A48, TOPREAL1:def 3;

          then

           A51: (( GoB f) * ((i + 1),(j + 1))) = (f /. 1) & (( GoB f) * (i,(j + 1))) = (f /. 2) or (( GoB f) * ((i + 1),(j + 1))) = (f /. 2) & (( GoB f) * (i,(j + 1))) = (f /. 1) by A18, A48, SPPOL_1: 8;

          

           A52: 2 < (k1 + 1) by A49, NAT_1: 13;

          then

           A53: (f /. (k1 + 1)) <> (f /. 2) by A11, Th37;

          ( LSeg (f,k1)) = ( LSeg ((f /. k1),(f /. (k1 + 1)))) by A10, A11, TOPREAL1:def 3;

          then

           A54: (( GoB f) * (i,(j + 1))) = (f /. k1) & (( GoB f) * (i,j)) = (f /. (k1 + 1)) or (( GoB f) * (i,(j + 1))) = (f /. (k1 + 1)) & (( GoB f) * (i,j)) = (f /. k1) by A12, SPPOL_1: 8;

          

           A55: (f /. k1) <> (f /. 2) by A13, A49, Th36;

          hence (f /. 1) = (( GoB f) * (i,(j + 1))) by A18, A48, A50, A54, A53, SPPOL_1: 8;

          thus (f /. 2) = (( GoB f) * ((i + 1),(j + 1))) by A18, A48, A50, A54, A55, A53, SPPOL_1: 8;

          

           A56: k1 > 1 by A49, XXREAL_0: 2;

          then

           A57: (k1 + 1) > 1 by NAT_1: 13;

          then (k1 + 1) = ( len f) by A11, A13, A49, A51, A54, A56, A52, Th37, Th38;

          then (k1 + 1) = ((( len f) -' 1) + 1) by A57, XREAL_1: 235;

          hence (f /. (( len f) -' 1)) = (( GoB f) * (i,j)) by A13, A49, A51, A54, A56, Th36;

        end;

          case k1 = k2;

          then

           A58: (( GoB f) * (i,j)) = (( GoB f) * ((i + 1),(j + 1))) or (( GoB f) * (i,j)) = (( GoB f) * (i,(j + 1))) by A12, A18, SPPOL_1: 8;

          

           A59: [(i + 1), (j + 1)] in ( Indices ( GoB f)) by A2, A4, A15, A9, MATRIX_0: 30;

           [i, j] in ( Indices ( GoB f)) & [i, (j + 1)] in ( Indices ( GoB f)) by A1, A3, A4, A15, A8, A7, MATRIX_0: 30;

          then j = (j + 1) by A58, A59, GOBOARD1: 5;

          hence contradiction;

        end;

          case that

           A60: k1 > 1 and

           A61: k2 > k1;

          

           A62: 1 < (k1 + 1) & (k1 + 1) < (k2 + 1) by A60, A61, NAT_1: 13, XREAL_1: 6;

          

           A63: k1 < (k2 + 1) by A61, NAT_1: 13;

          then

           A64: (f /. k1) <> (f /. (k2 + 1)) by A17, A60, Th37;

          

           A65: (k1 + 1) <= k2 by A61, NAT_1: 13;

          ( LSeg (f,k2)) = ( LSeg ((f /. k2),(f /. (k2 + 1)))) by A16, A17, TOPREAL1:def 3;

          then

           A66: (( GoB f) * (i,(j + 1))) = (f /. k2) & (( GoB f) * ((i + 1),(j + 1))) = (f /. (k2 + 1)) or (( GoB f) * (i,(j + 1))) = (f /. (k2 + 1)) & (( GoB f) * ((i + 1),(j + 1))) = (f /. k2) by A18, SPPOL_1: 8;

          

           A67: k2 < ( len f) by A17, NAT_1: 13;

          then

           A68: (f /. k1) <> (f /. k2) by A60, A61, Th37;

          

           A69: ( LSeg (f,k1)) = ( LSeg ((f /. k1),(f /. (k1 + 1)))) by A10, A11, TOPREAL1:def 3;

          then (( GoB f) * (i,j)) = (f /. k1) & (( GoB f) * (i,(j + 1))) = (f /. (k1 + 1)) or (( GoB f) * (i,j)) = (f /. (k1 + 1)) & (( GoB f) * (i,(j + 1))) = (f /. k1) by A12, SPPOL_1: 8;

          then (k1 + 1) >= k2 by A17, A60, A61, A66, A63, A67, A62, Th37;

          then

           A70: (k1 + 1) = k2 by A65, XXREAL_0: 1;

          hence 1 <= k1 & (k1 + 1) < ( len f) by A17, A60, NAT_1: 13;

          thus (f /. (k1 + 1)) = (( GoB f) * (i,(j + 1))) by A12, A69, A66, A64, A68, SPPOL_1: 8;

          thus (f /. k1) = (( GoB f) * (i,j)) by A12, A69, A66, A64, A68, SPPOL_1: 8;

          thus (f /. (k1 + 2)) = (( GoB f) * ((i + 1),(j + 1))) by A12, A69, A66, A64, A70, SPPOL_1: 8;

        end;

          case that

           A71: k2 > 1 and

           A72: k1 > k2;

          

           A73: 1 < (k2 + 1) & (k2 + 1) < (k1 + 1) by A71, A72, NAT_1: 13, XREAL_1: 6;

          

           A74: k2 < (k1 + 1) by A72, NAT_1: 13;

          then

           A75: (f /. k2) <> (f /. (k1 + 1)) by A11, A71, Th37;

          

           A76: (k2 + 1) <= k1 by A72, NAT_1: 13;

          ( LSeg (f,k1)) = ( LSeg ((f /. k1),(f /. (k1 + 1)))) by A10, A11, TOPREAL1:def 3;

          then

           A77: (( GoB f) * (i,(j + 1))) = (f /. k1) & (( GoB f) * (i,j)) = (f /. (k1 + 1)) or (( GoB f) * (i,(j + 1))) = (f /. (k1 + 1)) & (( GoB f) * (i,j)) = (f /. k1) by A12, SPPOL_1: 8;

          

           A78: k1 < ( len f) by A11, NAT_1: 13;

          then

           A79: (f /. k2) <> (f /. k1) by A71, A72, Th37;

          

           A80: ( LSeg (f,k2)) = ( LSeg ((f /. k2),(f /. (k2 + 1)))) by A16, A17, TOPREAL1:def 3;

          then (( GoB f) * ((i + 1),(j + 1))) = (f /. k2) & (( GoB f) * (i,(j + 1))) = (f /. (k2 + 1)) or (( GoB f) * ((i + 1),(j + 1))) = (f /. (k2 + 1)) & (( GoB f) * (i,(j + 1))) = (f /. k2) by A18, SPPOL_1: 8;

          then (k2 + 1) >= k1 by A11, A71, A72, A77, A74, A78, A73, Th37;

          then

           A81: (k2 + 1) = k1 by A76, XXREAL_0: 1;

          hence 1 <= k2 & (k2 + 1) < ( len f) by A11, A71, NAT_1: 13;

          thus (f /. (k2 + 1)) = (( GoB f) * (i,(j + 1))) by A18, A80, A77, A75, A79, SPPOL_1: 8;

          thus (f /. k2) = (( GoB f) * ((i + 1),(j + 1))) by A18, A80, A77, A75, A79, SPPOL_1: 8;

          thus (f /. (k2 + 2)) = (( GoB f) * (i,j)) by A18, A80, A77, A75, A81, SPPOL_1: 8;

        end;

      end;

      hence thesis;

    end;

    theorem :: GOBOARD7:55

    

     Th55: 1 <= i & (i + 1) <= ( len ( GoB f)) & 1 <= j & (j + 1) <= ( width ( GoB f)) & ( LSeg ((( GoB f) * (i,(j + 1))),(( GoB f) * ((i + 1),(j + 1))))) c= ( L~ f) & ( LSeg ((( GoB f) * ((i + 1),(j + 1))),(( GoB f) * ((i + 1),j)))) c= ( L~ f) implies (f /. 1) = (( GoB f) * ((i + 1),(j + 1))) & ((f /. 2) = (( GoB f) * (i,(j + 1))) & (f /. (( len f) -' 1)) = (( GoB f) * ((i + 1),j)) or (f /. 2) = (( GoB f) * ((i + 1),j)) & (f /. (( len f) -' 1)) = (( GoB f) * (i,(j + 1)))) or ex k st 1 <= k & (k + 1) < ( len f) & (f /. (k + 1)) = (( GoB f) * ((i + 1),(j + 1))) & ((f /. k) = (( GoB f) * (i,(j + 1))) & (f /. (k + 2)) = (( GoB f) * ((i + 1),j)) or (f /. k) = (( GoB f) * ((i + 1),j)) & (f /. (k + 2)) = (( GoB f) * (i,(j + 1))))

    proof

      assume that

       A1: 1 <= i and

       A2: (i + 1) <= ( len ( GoB f)) and

       A3: 1 <= j and

       A4: (j + 1) <= ( width ( GoB f)) and

       A5: ( LSeg ((( GoB f) * (i,(j + 1))),(( GoB f) * ((i + 1),(j + 1))))) c= ( L~ f) and

       A6: ( LSeg ((( GoB f) * ((i + 1),(j + 1))),(( GoB f) * ((i + 1),j)))) c= ( L~ f);

      

       A7: 1 <= (j + 1) by NAT_1: 11;

      ((1 / 2) * ((( GoB f) * (i,(j + 1))) + (( GoB f) * ((i + 1),(j + 1))))) in ( LSeg ((( GoB f) * (i,(j + 1))),(( GoB f) * ((i + 1),(j + 1))))) by RLTOPSP1: 69;

      then

      consider k1 such that

       A8: 1 <= k1 and

       A9: (k1 + 1) <= ( len f) and

       A10: ( LSeg ((( GoB f) * (i,(j + 1))),(( GoB f) * ((i + 1),(j + 1))))) = ( LSeg (f,k1)) by A1, A2, A4, A5, A7, Th40;

      

       A11: k1 < ( len f) by A9, NAT_1: 13;

       A12:

      now

        assume k1 > 1;

        then k1 >= (1 + 1) by NAT_1: 13;

        hence k1 = 2 or k1 > 2 by XXREAL_0: 1;

      end;

      

       A13: j < ( width ( GoB f)) & i < ( len ( GoB f)) by A2, A4, NAT_1: 13;

      

       A14: 1 <= (i + 1) by NAT_1: 11;

      ((1 / 2) * ((( GoB f) * ((i + 1),j)) + (( GoB f) * ((i + 1),(j + 1))))) in ( LSeg ((( GoB f) * ((i + 1),(j + 1))),(( GoB f) * ((i + 1),j)))) by RLTOPSP1: 69;

      then

      consider k2 such that

       A15: 1 <= k2 and

       A16: (k2 + 1) <= ( len f) and

       A17: ( LSeg ((( GoB f) * ((i + 1),j)),(( GoB f) * ((i + 1),(j + 1))))) = ( LSeg (f,k2)) by A2, A3, A4, A6, A14, Th39;

      

       A18: k2 < ( len f) by A16, NAT_1: 13;

       A19:

      now

        assume k2 > 1;

        then k2 >= (1 + 1) by NAT_1: 13;

        hence k2 = 2 or k2 > 2 by XXREAL_0: 1;

      end;

      

       A20: k1 = 1 or k1 > 1 by A8, XXREAL_0: 1;

      now

        per cases by A15, A12, A19, A20, XXREAL_0: 1;

          case that

           A21: k1 = 1 and

           A22: k2 = 2;

          

           A23: ( LSeg (f,2)) = ( LSeg ((f /. 2),(f /. (2 + 1)))) by A16, A22, TOPREAL1:def 3;

          then

           A24: (( GoB f) * ((i + 1),(j + 1))) = (f /. 2) & (( GoB f) * ((i + 1),j)) = (f /. (2 + 1)) or (( GoB f) * ((i + 1),(j + 1))) = (f /. (2 + 1)) & (( GoB f) * ((i + 1),j)) = (f /. 2) by A17, A22, SPPOL_1: 8;

          thus 1 <= 1 & (1 + 1) < ( len f) by A16, A22, NAT_1: 13;

          

           A25: 3 < ( len f) by Th34, XXREAL_0: 2;

          then

           A26: (f /. 1) <> (f /. 3) by Th36;

          

           A27: ( LSeg (f,1)) = ( LSeg ((f /. 1),(f /. (1 + 1)))) by A9, A21, TOPREAL1:def 3;

          then

           A28: (( GoB f) * (i,(j + 1))) = (f /. 1) & (( GoB f) * ((i + 1),(j + 1))) = (f /. 2) or (( GoB f) * (i,(j + 1))) = (f /. 2) & (( GoB f) * ((i + 1),(j + 1))) = (f /. 1) by A10, A21, SPPOL_1: 8;

          hence (f /. (1 + 1)) = (( GoB f) * ((i + 1),(j + 1))) by A24, A25, Th36;

          thus (f /. 1) = (( GoB f) * (i,(j + 1))) by A17, A22, A28, A23, A26, SPPOL_1: 8;

          thus (f /. (1 + 2)) = (( GoB f) * ((i + 1),j)) by A10, A21, A27, A24, A26, SPPOL_1: 8;

        end;

          case that

           A29: k1 = 1 and

           A30: k2 > 2;

          

           A31: ( LSeg (f,1)) = ( LSeg ((f /. 1),(f /. (1 + 1)))) by A9, A29, TOPREAL1:def 3;

          then

           A32: (( GoB f) * (i,(j + 1))) = (f /. 1) & (( GoB f) * ((i + 1),(j + 1))) = (f /. 2) or (( GoB f) * (i,(j + 1))) = (f /. 2) & (( GoB f) * ((i + 1),(j + 1))) = (f /. 1) by A10, A29, SPPOL_1: 8;

          

           A33: 2 < (k2 + 1) by A30, NAT_1: 13;

          then

           A34: (f /. (k2 + 1)) <> (f /. 2) by A16, Th37;

          ( LSeg (f,k2)) = ( LSeg ((f /. k2),(f /. (k2 + 1)))) by A15, A16, TOPREAL1:def 3;

          then

           A35: (( GoB f) * ((i + 1),(j + 1))) = (f /. k2) & (( GoB f) * ((i + 1),j)) = (f /. (k2 + 1)) or (( GoB f) * ((i + 1),(j + 1))) = (f /. (k2 + 1)) & (( GoB f) * ((i + 1),j)) = (f /. k2) by A17, SPPOL_1: 8;

          

           A36: (f /. k2) <> (f /. 2) by A18, A30, Th36;

          hence (f /. 1) = (( GoB f) * ((i + 1),(j + 1))) by A10, A29, A31, A35, A34, SPPOL_1: 8;

          thus (f /. 2) = (( GoB f) * (i,(j + 1))) by A10, A29, A31, A35, A36, A34, SPPOL_1: 8;

          

           A37: k2 > 1 by A30, XXREAL_0: 2;

          then

           A38: (k2 + 1) > 1 by NAT_1: 13;

          then (k2 + 1) = ( len f) by A16, A18, A30, A32, A35, A37, A33, Th37, Th38;

          then (k2 + 1) = ((( len f) -' 1) + 1) by A38, XREAL_1: 235;

          hence (f /. (( len f) -' 1)) = (( GoB f) * ((i + 1),j)) by A18, A30, A32, A35, A37, Th36;

        end;

          case that

           A39: k2 = 1 and

           A40: k1 = 2;

          

           A41: ( LSeg (f,2)) = ( LSeg ((f /. 2),(f /. (2 + 1)))) by A9, A40, TOPREAL1:def 3;

          then

           A42: (( GoB f) * ((i + 1),(j + 1))) = (f /. 2) & (( GoB f) * (i,(j + 1))) = (f /. (2 + 1)) or (( GoB f) * ((i + 1),(j + 1))) = (f /. (2 + 1)) & (( GoB f) * (i,(j + 1))) = (f /. 2) by A10, A40, SPPOL_1: 8;

          thus 1 <= 1 & (1 + 1) < ( len f) by A9, A40, NAT_1: 13;

          

           A43: 3 < ( len f) by Th34, XXREAL_0: 2;

          then

           A44: (f /. 1) <> (f /. 3) by Th36;

          

           A45: ( LSeg (f,1)) = ( LSeg ((f /. 1),(f /. (1 + 1)))) by A16, A39, TOPREAL1:def 3;

          then

           A46: (( GoB f) * ((i + 1),j)) = (f /. 1) & (( GoB f) * ((i + 1),(j + 1))) = (f /. 2) or (( GoB f) * ((i + 1),j)) = (f /. 2) & (( GoB f) * ((i + 1),(j + 1))) = (f /. 1) by A17, A39, SPPOL_1: 8;

          hence (f /. (1 + 1)) = (( GoB f) * ((i + 1),(j + 1))) by A42, A43, Th36;

          thus (f /. 1) = (( GoB f) * ((i + 1),j)) by A10, A40, A46, A41, A44, SPPOL_1: 8;

          thus (f /. (1 + 2)) = (( GoB f) * (i,(j + 1))) by A17, A39, A45, A42, A44, SPPOL_1: 8;

        end;

          case that

           A47: k2 = 1 and

           A48: k1 > 2;

          

           A49: ( LSeg (f,1)) = ( LSeg ((f /. 1),(f /. (1 + 1)))) by A16, A47, TOPREAL1:def 3;

          then

           A50: (( GoB f) * ((i + 1),j)) = (f /. 1) & (( GoB f) * ((i + 1),(j + 1))) = (f /. 2) or (( GoB f) * ((i + 1),j)) = (f /. 2) & (( GoB f) * ((i + 1),(j + 1))) = (f /. 1) by A17, A47, SPPOL_1: 8;

          

           A51: 2 < (k1 + 1) by A48, NAT_1: 13;

          then

           A52: (f /. (k1 + 1)) <> (f /. 2) by A9, Th37;

          ( LSeg (f,k1)) = ( LSeg ((f /. k1),(f /. (k1 + 1)))) by A8, A9, TOPREAL1:def 3;

          then

           A53: (( GoB f) * ((i + 1),(j + 1))) = (f /. k1) & (( GoB f) * (i,(j + 1))) = (f /. (k1 + 1)) or (( GoB f) * ((i + 1),(j + 1))) = (f /. (k1 + 1)) & (( GoB f) * (i,(j + 1))) = (f /. k1) by A10, SPPOL_1: 8;

          

           A54: (f /. k1) <> (f /. 2) by A11, A48, Th36;

          hence (f /. 1) = (( GoB f) * ((i + 1),(j + 1))) by A17, A47, A49, A53, A52, SPPOL_1: 8;

          thus (f /. 2) = (( GoB f) * ((i + 1),j)) by A17, A47, A49, A53, A54, A52, SPPOL_1: 8;

          

           A55: k1 > 1 by A48, XXREAL_0: 2;

          then

           A56: (k1 + 1) > 1 by NAT_1: 13;

          then (k1 + 1) = ( len f) by A9, A11, A48, A50, A53, A55, A51, Th37, Th38;

          then (k1 + 1) = ((( len f) -' 1) + 1) by A56, XREAL_1: 235;

          hence (f /. (( len f) -' 1)) = (( GoB f) * (i,(j + 1))) by A11, A48, A50, A53, A55, Th36;

        end;

          case k1 = k2;

          then

           A57: (( GoB f) * (i,(j + 1))) = (( GoB f) * ((i + 1),j)) or (( GoB f) * (i,(j + 1))) = (( GoB f) * ((i + 1),(j + 1))) by A10, A17, SPPOL_1: 8;

          

           A58: [(i + 1), (j + 1)] in ( Indices ( GoB f)) by A2, A4, A7, A14, MATRIX_0: 30;

           [(i + 1), j] in ( Indices ( GoB f)) & [i, (j + 1)] in ( Indices ( GoB f)) by A1, A2, A3, A4, A7, A14, A13, MATRIX_0: 30;

          then i = (i + 1) or j = (j + 1) by A57, A58, GOBOARD1: 5;

          hence contradiction;

        end;

          case that

           A59: k1 > 1 and

           A60: k2 > k1;

          

           A61: 1 < (k1 + 1) & (k1 + 1) < (k2 + 1) by A59, A60, NAT_1: 13, XREAL_1: 6;

          

           A62: k1 < (k2 + 1) by A60, NAT_1: 13;

          then

           A63: (f /. k1) <> (f /. (k2 + 1)) by A16, A59, Th37;

          

           A64: (k1 + 1) <= k2 by A60, NAT_1: 13;

          ( LSeg (f,k2)) = ( LSeg ((f /. k2),(f /. (k2 + 1)))) by A15, A16, TOPREAL1:def 3;

          then

           A65: (( GoB f) * ((i + 1),(j + 1))) = (f /. k2) & (( GoB f) * ((i + 1),j)) = (f /. (k2 + 1)) or (( GoB f) * ((i + 1),(j + 1))) = (f /. (k2 + 1)) & (( GoB f) * ((i + 1),j)) = (f /. k2) by A17, SPPOL_1: 8;

          

           A66: k2 < ( len f) by A16, NAT_1: 13;

          then

           A67: (f /. k1) <> (f /. k2) by A59, A60, Th37;

          

           A68: ( LSeg (f,k1)) = ( LSeg ((f /. k1),(f /. (k1 + 1)))) by A8, A9, TOPREAL1:def 3;

          then (( GoB f) * (i,(j + 1))) = (f /. k1) & (( GoB f) * ((i + 1),(j + 1))) = (f /. (k1 + 1)) or (( GoB f) * (i,(j + 1))) = (f /. (k1 + 1)) & (( GoB f) * ((i + 1),(j + 1))) = (f /. k1) by A10, SPPOL_1: 8;

          then (k1 + 1) >= k2 by A16, A59, A60, A65, A62, A66, A61, Th37;

          then

           A69: (k1 + 1) = k2 by A64, XXREAL_0: 1;

          hence 1 <= k1 & (k1 + 1) < ( len f) by A16, A59, NAT_1: 13;

          thus (f /. (k1 + 1)) = (( GoB f) * ((i + 1),(j + 1))) by A10, A68, A65, A63, A67, SPPOL_1: 8;

          thus (f /. k1) = (( GoB f) * (i,(j + 1))) by A10, A68, A65, A63, A67, SPPOL_1: 8;

          thus (f /. (k1 + 2)) = (( GoB f) * ((i + 1),j)) by A10, A68, A65, A63, A69, SPPOL_1: 8;

        end;

          case that

           A70: k2 > 1 and

           A71: k1 > k2;

          

           A72: 1 < (k2 + 1) & (k2 + 1) < (k1 + 1) by A70, A71, NAT_1: 13, XREAL_1: 6;

          

           A73: k2 < (k1 + 1) by A71, NAT_1: 13;

          then

           A74: (f /. k2) <> (f /. (k1 + 1)) by A9, A70, Th37;

          

           A75: (k2 + 1) <= k1 by A71, NAT_1: 13;

          ( LSeg (f,k1)) = ( LSeg ((f /. k1),(f /. (k1 + 1)))) by A8, A9, TOPREAL1:def 3;

          then

           A76: (( GoB f) * ((i + 1),(j + 1))) = (f /. k1) & (( GoB f) * (i,(j + 1))) = (f /. (k1 + 1)) or (( GoB f) * ((i + 1),(j + 1))) = (f /. (k1 + 1)) & (( GoB f) * (i,(j + 1))) = (f /. k1) by A10, SPPOL_1: 8;

          

           A77: k1 < ( len f) by A9, NAT_1: 13;

          then

           A78: (f /. k2) <> (f /. k1) by A70, A71, Th37;

          

           A79: ( LSeg (f,k2)) = ( LSeg ((f /. k2),(f /. (k2 + 1)))) by A15, A16, TOPREAL1:def 3;

          then (( GoB f) * ((i + 1),j)) = (f /. k2) & (( GoB f) * ((i + 1),(j + 1))) = (f /. (k2 + 1)) or (( GoB f) * ((i + 1),j)) = (f /. (k2 + 1)) & (( GoB f) * ((i + 1),(j + 1))) = (f /. k2) by A17, SPPOL_1: 8;

          then (k2 + 1) >= k1 by A9, A70, A71, A76, A73, A77, A72, Th37;

          then

           A80: (k2 + 1) = k1 by A75, XXREAL_0: 1;

          hence 1 <= k2 & (k2 + 1) < ( len f) by A9, A70, NAT_1: 13;

          thus (f /. (k2 + 1)) = (( GoB f) * ((i + 1),(j + 1))) by A17, A79, A76, A74, A78, SPPOL_1: 8;

          thus (f /. k2) = (( GoB f) * ((i + 1),j)) by A17, A79, A76, A74, A78, SPPOL_1: 8;

          thus (f /. (k2 + 2)) = (( GoB f) * (i,(j + 1))) by A17, A79, A76, A74, A80, SPPOL_1: 8;

        end;

      end;

      hence thesis;

    end;

    theorem :: GOBOARD7:56

    

     Th56: 1 <= i & (i + 1) < ( len ( GoB f)) & 1 <= j & j <= ( width ( GoB f)) & ( LSeg ((( GoB f) * (i,j)),(( GoB f) * ((i + 1),j)))) c= ( L~ f) & ( LSeg ((( GoB f) * ((i + 1),j)),(( GoB f) * ((i + 2),j)))) c= ( L~ f) implies (f /. 1) = (( GoB f) * ((i + 1),j)) & ((f /. 2) = (( GoB f) * (i,j)) & (f /. (( len f) -' 1)) = (( GoB f) * ((i + 2),j)) or (f /. 2) = (( GoB f) * ((i + 2),j)) & (f /. (( len f) -' 1)) = (( GoB f) * (i,j))) or ex k st 1 <= k & (k + 1) < ( len f) & (f /. (k + 1)) = (( GoB f) * ((i + 1),j)) & ((f /. k) = (( GoB f) * (i,j)) & (f /. (k + 2)) = (( GoB f) * ((i + 2),j)) or (f /. k) = (( GoB f) * ((i + 2),j)) & (f /. (k + 2)) = (( GoB f) * (i,j)))

    proof

      assume that

       A1: 1 <= i and

       A2: (i + 1) < ( len ( GoB f)) and

       A3: 1 <= j & j <= ( width ( GoB f)) and

       A4: ( LSeg ((( GoB f) * (i,j)),(( GoB f) * ((i + 1),j)))) c= ( L~ f) and

       A5: ( LSeg ((( GoB f) * ((i + 1),j)),(( GoB f) * ((i + 2),j)))) c= ( L~ f);

      

       A6: 1 <= (i + 1) by NAT_1: 11;

      ((1 / 2) * ((( GoB f) * (i,j)) + (( GoB f) * ((i + 1),j)))) in ( LSeg ((( GoB f) * (i,j)),(( GoB f) * ((i + 1),j)))) by RLTOPSP1: 69;

      then

      consider k1 such that

       A7: 1 <= k1 and

       A8: (k1 + 1) <= ( len f) and

       A9: ( LSeg ((( GoB f) * (i,j)),(( GoB f) * ((i + 1),j)))) = ( LSeg (f,k1)) by A1, A2, A3, A4, Th40;

      

       A10: k1 < ( len f) by A8, NAT_1: 13;

       A11:

      now

        assume k1 > 1;

        then k1 >= (1 + 1) by NAT_1: 13;

        hence k1 = 2 or k1 > 2 by XXREAL_0: 1;

      end;

      

       A12: i < ( len ( GoB f)) by A2, NAT_1: 13;

      

       A13: (i + (1 + 1)) = ((i + 1) + 1);

      then

       A14: 1 <= (i + 2) by NAT_1: 11;

      

       A15: (i + 2) <= ( len ( GoB f)) by A2, A13, NAT_1: 13;

      ((1 / 2) * ((( GoB f) * ((i + 1),j)) + (( GoB f) * ((i + 2),j)))) in ( LSeg ((( GoB f) * ((i + 1),j)),(( GoB f) * ((i + 2),j)))) by RLTOPSP1: 69;

      then

      consider k2 such that

       A16: 1 <= k2 and

       A17: (k2 + 1) <= ( len f) and

       A18: ( LSeg ((( GoB f) * ((i + 1),j)),(( GoB f) * ((i + 2),j)))) = ( LSeg (f,k2)) by A3, A5, A6, A13, A15, Th40;

      

       A19: k2 < ( len f) by A17, NAT_1: 13;

       A20:

      now

        assume k2 > 1;

        then k2 >= (1 + 1) by NAT_1: 13;

        hence k2 = 2 or k2 > 2 by XXREAL_0: 1;

      end;

      

       A21: k1 = 1 or k1 > 1 by A7, XXREAL_0: 1;

      now

        per cases by A16, A11, A20, A21, XXREAL_0: 1;

          case that

           A22: k1 = 1 and

           A23: k2 = 2;

          

           A24: ( LSeg (f,2)) = ( LSeg ((f /. 2),(f /. (2 + 1)))) by A17, A23, TOPREAL1:def 3;

          then

           A25: (( GoB f) * ((i + 1),j)) = (f /. 2) & (( GoB f) * ((i + 2),j)) = (f /. (2 + 1)) or (( GoB f) * ((i + 1),j)) = (f /. (2 + 1)) & (( GoB f) * ((i + 2),j)) = (f /. 2) by A18, A23, SPPOL_1: 8;

          thus 1 <= 1 & (1 + 1) < ( len f) by A17, A23, NAT_1: 13;

          

           A26: 3 < ( len f) by Th34, XXREAL_0: 2;

          then

           A27: (f /. 1) <> (f /. 3) by Th36;

          

           A28: ( LSeg (f,1)) = ( LSeg ((f /. 1),(f /. (1 + 1)))) by A8, A22, TOPREAL1:def 3;

          then

           A29: (( GoB f) * (i,j)) = (f /. 1) & (( GoB f) * ((i + 1),j)) = (f /. 2) or (( GoB f) * (i,j)) = (f /. 2) & (( GoB f) * ((i + 1),j)) = (f /. 1) by A9, A22, SPPOL_1: 8;

          hence (f /. (1 + 1)) = (( GoB f) * ((i + 1),j)) by A25, A26, Th36;

          thus (f /. 1) = (( GoB f) * (i,j)) by A18, A23, A29, A24, A27, SPPOL_1: 8;

          thus (f /. (1 + 2)) = (( GoB f) * ((i + 2),j)) by A9, A22, A28, A25, A27, SPPOL_1: 8;

        end;

          case that

           A30: k1 = 1 and

           A31: k2 > 2;

          

           A32: ( LSeg (f,1)) = ( LSeg ((f /. 1),(f /. (1 + 1)))) by A8, A30, TOPREAL1:def 3;

          then

           A33: (( GoB f) * (i,j)) = (f /. 1) & (( GoB f) * ((i + 1),j)) = (f /. 2) or (( GoB f) * (i,j)) = (f /. 2) & (( GoB f) * ((i + 1),j)) = (f /. 1) by A9, A30, SPPOL_1: 8;

          

           A34: 2 < (k2 + 1) by A31, NAT_1: 13;

          then

           A35: (f /. (k2 + 1)) <> (f /. 2) by A17, Th37;

          ( LSeg (f,k2)) = ( LSeg ((f /. k2),(f /. (k2 + 1)))) by A16, A17, TOPREAL1:def 3;

          then

           A36: (( GoB f) * ((i + 1),j)) = (f /. k2) & (( GoB f) * ((i + 2),j)) = (f /. (k2 + 1)) or (( GoB f) * ((i + 1),j)) = (f /. (k2 + 1)) & (( GoB f) * ((i + 2),j)) = (f /. k2) by A18, SPPOL_1: 8;

          

           A37: (f /. k2) <> (f /. 2) by A19, A31, Th36;

          hence (f /. 1) = (( GoB f) * ((i + 1),j)) by A9, A30, A32, A36, A35, SPPOL_1: 8;

          thus (f /. 2) = (( GoB f) * (i,j)) by A9, A30, A32, A36, A37, A35, SPPOL_1: 8;

          

           A38: k2 > 1 by A31, XXREAL_0: 2;

          then

           A39: (k2 + 1) > 1 by NAT_1: 13;

          then (k2 + 1) = ( len f) by A17, A19, A31, A33, A36, A38, A34, Th37, Th38;

          then (k2 + 1) = ((( len f) -' 1) + 1) by A39, XREAL_1: 235;

          hence (f /. (( len f) -' 1)) = (( GoB f) * ((i + 2),j)) by A19, A31, A33, A36, A38, Th36;

        end;

          case that

           A40: k2 = 1 and

           A41: k1 = 2;

          

           A42: ( LSeg (f,2)) = ( LSeg ((f /. 2),(f /. (2 + 1)))) by A8, A41, TOPREAL1:def 3;

          then

           A43: (( GoB f) * ((i + 1),j)) = (f /. 2) & (( GoB f) * (i,j)) = (f /. (2 + 1)) or (( GoB f) * ((i + 1),j)) = (f /. (2 + 1)) & (( GoB f) * (i,j)) = (f /. 2) by A9, A41, SPPOL_1: 8;

          thus 1 <= 1 & (1 + 1) < ( len f) by A8, A41, NAT_1: 13;

          

           A44: 3 < ( len f) by Th34, XXREAL_0: 2;

          then

           A45: (f /. 1) <> (f /. 3) by Th36;

          

           A46: ( LSeg (f,1)) = ( LSeg ((f /. 1),(f /. (1 + 1)))) by A17, A40, TOPREAL1:def 3;

          then

           A47: (( GoB f) * ((i + 2),j)) = (f /. 1) & (( GoB f) * ((i + 1),j)) = (f /. 2) or (( GoB f) * ((i + 2),j)) = (f /. 2) & (( GoB f) * ((i + 1),j)) = (f /. 1) by A18, A40, SPPOL_1: 8;

          hence (f /. (1 + 1)) = (( GoB f) * ((i + 1),j)) by A43, A44, Th36;

          thus (f /. 1) = (( GoB f) * ((i + 2),j)) by A9, A41, A47, A42, A45, SPPOL_1: 8;

          thus (f /. (1 + 2)) = (( GoB f) * (i,j)) by A18, A40, A46, A43, A45, SPPOL_1: 8;

        end;

          case that

           A48: k2 = 1 and

           A49: k1 > 2;

          

           A50: ( LSeg (f,1)) = ( LSeg ((f /. 1),(f /. (1 + 1)))) by A17, A48, TOPREAL1:def 3;

          then

           A51: (( GoB f) * ((i + 2),j)) = (f /. 1) & (( GoB f) * ((i + 1),j)) = (f /. 2) or (( GoB f) * ((i + 2),j)) = (f /. 2) & (( GoB f) * ((i + 1),j)) = (f /. 1) by A18, A48, SPPOL_1: 8;

          

           A52: 2 < (k1 + 1) by A49, NAT_1: 13;

          then

           A53: (f /. (k1 + 1)) <> (f /. 2) by A8, Th37;

          ( LSeg (f,k1)) = ( LSeg ((f /. k1),(f /. (k1 + 1)))) by A7, A8, TOPREAL1:def 3;

          then

           A54: (( GoB f) * ((i + 1),j)) = (f /. k1) & (( GoB f) * (i,j)) = (f /. (k1 + 1)) or (( GoB f) * ((i + 1),j)) = (f /. (k1 + 1)) & (( GoB f) * (i,j)) = (f /. k1) by A9, SPPOL_1: 8;

          

           A55: (f /. k1) <> (f /. 2) by A10, A49, Th36;

          hence (f /. 1) = (( GoB f) * ((i + 1),j)) by A18, A48, A50, A54, A53, SPPOL_1: 8;

          thus (f /. 2) = (( GoB f) * ((i + 2),j)) by A18, A48, A50, A54, A55, A53, SPPOL_1: 8;

          

           A56: k1 > 1 by A49, XXREAL_0: 2;

          then

           A57: (k1 + 1) > 1 by NAT_1: 13;

          then (k1 + 1) = ( len f) by A8, A10, A49, A51, A54, A56, A52, Th37, Th38;

          then (k1 + 1) = ((( len f) -' 1) + 1) by A57, XREAL_1: 235;

          hence (f /. (( len f) -' 1)) = (( GoB f) * (i,j)) by A10, A49, A51, A54, A56, Th36;

        end;

          case k1 = k2;

          then

           A58: (( GoB f) * (i,j)) = (( GoB f) * ((i + 2),j)) or (( GoB f) * (i,j)) = (( GoB f) * ((i + 1),j)) by A9, A18, SPPOL_1: 8;

          

           A59: [(i + 2), j] in ( Indices ( GoB f)) by A3, A15, A14, MATRIX_0: 30;

           [i, j] in ( Indices ( GoB f)) & [(i + 1), j] in ( Indices ( GoB f)) by A1, A2, A3, A6, A12, MATRIX_0: 30;

          then i = (i + 1) or i = (i + 2) by A58, A59, GOBOARD1: 5;

          hence contradiction;

        end;

          case that

           A60: k1 > 1 and

           A61: k2 > k1;

          

           A62: 1 < (k1 + 1) & (k1 + 1) < (k2 + 1) by A60, A61, NAT_1: 13, XREAL_1: 6;

          

           A63: k1 < (k2 + 1) by A61, NAT_1: 13;

          then

           A64: (f /. k1) <> (f /. (k2 + 1)) by A17, A60, Th37;

          

           A65: (k1 + 1) <= k2 by A61, NAT_1: 13;

          ( LSeg (f,k2)) = ( LSeg ((f /. k2),(f /. (k2 + 1)))) by A16, A17, TOPREAL1:def 3;

          then

           A66: (( GoB f) * ((i + 1),j)) = (f /. k2) & (( GoB f) * ((i + 2),j)) = (f /. (k2 + 1)) or (( GoB f) * ((i + 1),j)) = (f /. (k2 + 1)) & (( GoB f) * ((i + 2),j)) = (f /. k2) by A18, SPPOL_1: 8;

          

           A67: k2 < ( len f) by A17, NAT_1: 13;

          then

           A68: (f /. k1) <> (f /. k2) by A60, A61, Th37;

          

           A69: ( LSeg (f,k1)) = ( LSeg ((f /. k1),(f /. (k1 + 1)))) by A7, A8, TOPREAL1:def 3;

          then (( GoB f) * (i,j)) = (f /. k1) & (( GoB f) * ((i + 1),j)) = (f /. (k1 + 1)) or (( GoB f) * (i,j)) = (f /. (k1 + 1)) & (( GoB f) * ((i + 1),j)) = (f /. k1) by A9, SPPOL_1: 8;

          then (k1 + 1) >= k2 by A17, A60, A61, A66, A63, A67, A62, Th37;

          then

           A70: (k1 + 1) = k2 by A65, XXREAL_0: 1;

          hence 1 <= k1 & (k1 + 1) < ( len f) by A17, A60, NAT_1: 13;

          thus (f /. (k1 + 1)) = (( GoB f) * ((i + 1),j)) by A9, A69, A66, A64, A68, SPPOL_1: 8;

          thus (f /. k1) = (( GoB f) * (i,j)) by A9, A69, A66, A64, A68, SPPOL_1: 8;

          thus (f /. (k1 + 2)) = (( GoB f) * ((i + 2),j)) by A9, A69, A66, A64, A70, SPPOL_1: 8;

        end;

          case that

           A71: k2 > 1 and

           A72: k1 > k2;

          

           A73: 1 < (k2 + 1) & (k2 + 1) < (k1 + 1) by A71, A72, NAT_1: 13, XREAL_1: 6;

          

           A74: k2 < (k1 + 1) by A72, NAT_1: 13;

          then

           A75: (f /. k2) <> (f /. (k1 + 1)) by A8, A71, Th37;

          

           A76: (k2 + 1) <= k1 by A72, NAT_1: 13;

          ( LSeg (f,k1)) = ( LSeg ((f /. k1),(f /. (k1 + 1)))) by A7, A8, TOPREAL1:def 3;

          then

           A77: (( GoB f) * ((i + 1),j)) = (f /. k1) & (( GoB f) * (i,j)) = (f /. (k1 + 1)) or (( GoB f) * ((i + 1),j)) = (f /. (k1 + 1)) & (( GoB f) * (i,j)) = (f /. k1) by A9, SPPOL_1: 8;

          

           A78: k1 < ( len f) by A8, NAT_1: 13;

          then

           A79: (f /. k2) <> (f /. k1) by A71, A72, Th37;

          

           A80: ( LSeg (f,k2)) = ( LSeg ((f /. k2),(f /. (k2 + 1)))) by A16, A17, TOPREAL1:def 3;

          then (( GoB f) * ((i + 2),j)) = (f /. k2) & (( GoB f) * ((i + 1),j)) = (f /. (k2 + 1)) or (( GoB f) * ((i + 2),j)) = (f /. (k2 + 1)) & (( GoB f) * ((i + 1),j)) = (f /. k2) by A18, SPPOL_1: 8;

          then (k2 + 1) >= k1 by A8, A71, A72, A77, A74, A78, A73, Th37;

          then

           A81: (k2 + 1) = k1 by A76, XXREAL_0: 1;

          hence 1 <= k2 & (k2 + 1) < ( len f) by A8, A71, NAT_1: 13;

          thus (f /. (k2 + 1)) = (( GoB f) * ((i + 1),j)) by A18, A80, A77, A75, A79, SPPOL_1: 8;

          thus (f /. k2) = (( GoB f) * ((i + 2),j)) by A18, A80, A77, A75, A79, SPPOL_1: 8;

          thus (f /. (k2 + 2)) = (( GoB f) * (i,j)) by A18, A80, A77, A75, A81, SPPOL_1: 8;

        end;

      end;

      hence thesis;

    end;

    theorem :: GOBOARD7:57

    

     Th57: 1 <= i & (i + 1) <= ( len ( GoB f)) & 1 <= j & (j + 1) <= ( width ( GoB f)) & ( LSeg ((( GoB f) * (i,j)),(( GoB f) * ((i + 1),j)))) c= ( L~ f) & ( LSeg ((( GoB f) * ((i + 1),j)),(( GoB f) * ((i + 1),(j + 1))))) c= ( L~ f) implies (f /. 1) = (( GoB f) * ((i + 1),j)) & ((f /. 2) = (( GoB f) * (i,j)) & (f /. (( len f) -' 1)) = (( GoB f) * ((i + 1),(j + 1))) or (f /. 2) = (( GoB f) * ((i + 1),(j + 1))) & (f /. (( len f) -' 1)) = (( GoB f) * (i,j))) or ex k st 1 <= k & (k + 1) < ( len f) & (f /. (k + 1)) = (( GoB f) * ((i + 1),j)) & ((f /. k) = (( GoB f) * (i,j)) & (f /. (k + 2)) = (( GoB f) * ((i + 1),(j + 1))) or (f /. k) = (( GoB f) * ((i + 1),(j + 1))) & (f /. (k + 2)) = (( GoB f) * (i,j)))

    proof

      assume that

       A1: 1 <= i and

       A2: (i + 1) <= ( len ( GoB f)) and

       A3: 1 <= j and

       A4: (j + 1) <= ( width ( GoB f)) and

       A5: ( LSeg ((( GoB f) * (i,j)),(( GoB f) * ((i + 1),j)))) c= ( L~ f) and

       A6: ( LSeg ((( GoB f) * ((i + 1),j)),(( GoB f) * ((i + 1),(j + 1))))) c= ( L~ f);

      

       A7: j < ( width ( GoB f)) by A4, NAT_1: 13;

      

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

      

       A9: 1 <= (j + 1) by NAT_1: 11;

      ((1 / 2) * ((( GoB f) * (i,j)) + (( GoB f) * ((i + 1),j)))) in ( LSeg ((( GoB f) * (i,j)),(( GoB f) * ((i + 1),j)))) by RLTOPSP1: 69;

      then

      consider k1 such that

       A10: 1 <= k1 and

       A11: (k1 + 1) <= ( len f) and

       A12: ( LSeg ((( GoB f) * (i,j)),(( GoB f) * ((i + 1),j)))) = ( LSeg (f,k1)) by A1, A2, A3, A5, A7, Th40;

      

       A13: k1 < ( len f) by A11, NAT_1: 13;

       A14:

      now

        assume k1 > 1;

        then k1 >= (1 + 1) by NAT_1: 13;

        hence k1 = 2 or k1 > 2 by XXREAL_0: 1;

      end;

      

       A15: 1 <= (i + 1) by NAT_1: 11;

      ((1 / 2) * ((( GoB f) * ((i + 1),j)) + (( GoB f) * ((i + 1),(j + 1))))) in ( LSeg ((( GoB f) * ((i + 1),j)),(( GoB f) * ((i + 1),(j + 1))))) by RLTOPSP1: 69;

      then

      consider k2 such that

       A16: 1 <= k2 and

       A17: (k2 + 1) <= ( len f) and

       A18: ( LSeg ((( GoB f) * ((i + 1),j)),(( GoB f) * ((i + 1),(j + 1))))) = ( LSeg (f,k2)) by A2, A3, A4, A6, A15, Th39;

      

       A19: k2 < ( len f) by A17, NAT_1: 13;

       A20:

      now

        assume k2 > 1;

        then k2 >= (1 + 1) by NAT_1: 13;

        hence k2 = 2 or k2 > 2 by XXREAL_0: 1;

      end;

      

       A21: k1 = 1 or k1 > 1 by A10, XXREAL_0: 1;

      now

        per cases by A16, A14, A20, A21, XXREAL_0: 1;

          case that

           A22: k1 = 1 and

           A23: k2 = 2;

          

           A24: ( LSeg (f,2)) = ( LSeg ((f /. 2),(f /. (2 + 1)))) by A17, A23, TOPREAL1:def 3;

          then

           A25: (( GoB f) * ((i + 1),j)) = (f /. 2) & (( GoB f) * ((i + 1),(j + 1))) = (f /. (2 + 1)) or (( GoB f) * ((i + 1),j)) = (f /. (2 + 1)) & (( GoB f) * ((i + 1),(j + 1))) = (f /. 2) by A18, A23, SPPOL_1: 8;

          thus 1 <= 1 & (1 + 1) < ( len f) by A17, A23, NAT_1: 13;

          

           A26: 3 < ( len f) by Th34, XXREAL_0: 2;

          then

           A27: (f /. 1) <> (f /. 3) by Th36;

          

           A28: ( LSeg (f,1)) = ( LSeg ((f /. 1),(f /. (1 + 1)))) by A11, A22, TOPREAL1:def 3;

          then

           A29: (( GoB f) * (i,j)) = (f /. 1) & (( GoB f) * ((i + 1),j)) = (f /. 2) or (( GoB f) * (i,j)) = (f /. 2) & (( GoB f) * ((i + 1),j)) = (f /. 1) by A12, A22, SPPOL_1: 8;

          hence (f /. (1 + 1)) = (( GoB f) * ((i + 1),j)) by A25, A26, Th36;

          thus (f /. 1) = (( GoB f) * (i,j)) by A18, A23, A29, A24, A27, SPPOL_1: 8;

          thus (f /. (1 + 2)) = (( GoB f) * ((i + 1),(j + 1))) by A12, A22, A28, A25, A27, SPPOL_1: 8;

        end;

          case that

           A30: k1 = 1 and

           A31: k2 > 2;

          

           A32: ( LSeg (f,1)) = ( LSeg ((f /. 1),(f /. (1 + 1)))) by A11, A30, TOPREAL1:def 3;

          then

           A33: (( GoB f) * (i,j)) = (f /. 1) & (( GoB f) * ((i + 1),j)) = (f /. 2) or (( GoB f) * (i,j)) = (f /. 2) & (( GoB f) * ((i + 1),j)) = (f /. 1) by A12, A30, SPPOL_1: 8;

          

           A34: 2 < (k2 + 1) by A31, NAT_1: 13;

          then

           A35: (f /. (k2 + 1)) <> (f /. 2) by A17, Th37;

          ( LSeg (f,k2)) = ( LSeg ((f /. k2),(f /. (k2 + 1)))) by A16, A17, TOPREAL1:def 3;

          then

           A36: (( GoB f) * ((i + 1),j)) = (f /. k2) & (( GoB f) * ((i + 1),(j + 1))) = (f /. (k2 + 1)) or (( GoB f) * ((i + 1),j)) = (f /. (k2 + 1)) & (( GoB f) * ((i + 1),(j + 1))) = (f /. k2) by A18, SPPOL_1: 8;

          

           A37: (f /. k2) <> (f /. 2) by A19, A31, Th36;

          hence (f /. 1) = (( GoB f) * ((i + 1),j)) by A12, A30, A32, A36, A35, SPPOL_1: 8;

          thus (f /. 2) = (( GoB f) * (i,j)) by A12, A30, A32, A36, A37, A35, SPPOL_1: 8;

          

           A38: k2 > 1 by A31, XXREAL_0: 2;

          then

           A39: (k2 + 1) > 1 by NAT_1: 13;

          then (k2 + 1) = ( len f) by A17, A19, A31, A33, A36, A38, A34, Th37, Th38;

          then (k2 + 1) = ((( len f) -' 1) + 1) by A39, XREAL_1: 235;

          hence (f /. (( len f) -' 1)) = (( GoB f) * ((i + 1),(j + 1))) by A19, A31, A33, A36, A38, Th36;

        end;

          case that

           A40: k2 = 1 and

           A41: k1 = 2;

          

           A42: ( LSeg (f,2)) = ( LSeg ((f /. 2),(f /. (2 + 1)))) by A11, A41, TOPREAL1:def 3;

          then

           A43: (( GoB f) * ((i + 1),j)) = (f /. 2) & (( GoB f) * (i,j)) = (f /. (2 + 1)) or (( GoB f) * ((i + 1),j)) = (f /. (2 + 1)) & (( GoB f) * (i,j)) = (f /. 2) by A12, A41, SPPOL_1: 8;

          thus 1 <= 1 & (1 + 1) < ( len f) by A11, A41, NAT_1: 13;

          

           A44: 3 < ( len f) by Th34, XXREAL_0: 2;

          then

           A45: (f /. 1) <> (f /. 3) by Th36;

          

           A46: ( LSeg (f,1)) = ( LSeg ((f /. 1),(f /. (1 + 1)))) by A17, A40, TOPREAL1:def 3;

          then

           A47: (( GoB f) * ((i + 1),(j + 1))) = (f /. 1) & (( GoB f) * ((i + 1),j)) = (f /. 2) or (( GoB f) * ((i + 1),(j + 1))) = (f /. 2) & (( GoB f) * ((i + 1),j)) = (f /. 1) by A18, A40, SPPOL_1: 8;

          hence (f /. (1 + 1)) = (( GoB f) * ((i + 1),j)) by A43, A44, Th36;

          thus (f /. 1) = (( GoB f) * ((i + 1),(j + 1))) by A12, A41, A47, A42, A45, SPPOL_1: 8;

          thus (f /. (1 + 2)) = (( GoB f) * (i,j)) by A18, A40, A46, A43, A45, SPPOL_1: 8;

        end;

          case that

           A48: k2 = 1 and

           A49: k1 > 2;

          

           A50: ( LSeg (f,1)) = ( LSeg ((f /. 1),(f /. (1 + 1)))) by A17, A48, TOPREAL1:def 3;

          then

           A51: (( GoB f) * ((i + 1),(j + 1))) = (f /. 1) & (( GoB f) * ((i + 1),j)) = (f /. 2) or (( GoB f) * ((i + 1),(j + 1))) = (f /. 2) & (( GoB f) * ((i + 1),j)) = (f /. 1) by A18, A48, SPPOL_1: 8;

          

           A52: 2 < (k1 + 1) by A49, NAT_1: 13;

          then

           A53: (f /. (k1 + 1)) <> (f /. 2) by A11, Th37;

          ( LSeg (f,k1)) = ( LSeg ((f /. k1),(f /. (k1 + 1)))) by A10, A11, TOPREAL1:def 3;

          then

           A54: (( GoB f) * ((i + 1),j)) = (f /. k1) & (( GoB f) * (i,j)) = (f /. (k1 + 1)) or (( GoB f) * ((i + 1),j)) = (f /. (k1 + 1)) & (( GoB f) * (i,j)) = (f /. k1) by A12, SPPOL_1: 8;

          

           A55: (f /. k1) <> (f /. 2) by A13, A49, Th36;

          hence (f /. 1) = (( GoB f) * ((i + 1),j)) by A18, A48, A50, A54, A53, SPPOL_1: 8;

          thus (f /. 2) = (( GoB f) * ((i + 1),(j + 1))) by A18, A48, A50, A54, A55, A53, SPPOL_1: 8;

          

           A56: k1 > 1 by A49, XXREAL_0: 2;

          then

           A57: (k1 + 1) > 1 by NAT_1: 13;

          then (k1 + 1) = ( len f) by A11, A13, A49, A51, A54, A56, A52, Th37, Th38;

          then (k1 + 1) = ((( len f) -' 1) + 1) by A57, XREAL_1: 235;

          hence (f /. (( len f) -' 1)) = (( GoB f) * (i,j)) by A13, A49, A51, A54, A56, Th36;

        end;

          case k1 = k2;

          then

           A58: (( GoB f) * (i,j)) = (( GoB f) * ((i + 1),(j + 1))) or (( GoB f) * (i,j)) = (( GoB f) * ((i + 1),j)) by A12, A18, SPPOL_1: 8;

          

           A59: [(i + 1), (j + 1)] in ( Indices ( GoB f)) by A2, A4, A15, A9, MATRIX_0: 30;

           [i, j] in ( Indices ( GoB f)) & [(i + 1), j] in ( Indices ( GoB f)) by A1, A2, A3, A15, A8, A7, MATRIX_0: 30;

          then i = (i + 1) by A58, A59, GOBOARD1: 5;

          hence contradiction;

        end;

          case that

           A60: k1 > 1 and

           A61: k2 > k1;

          

           A62: 1 < (k1 + 1) & (k1 + 1) < (k2 + 1) by A60, A61, NAT_1: 13, XREAL_1: 6;

          

           A63: k1 < (k2 + 1) by A61, NAT_1: 13;

          then

           A64: (f /. k1) <> (f /. (k2 + 1)) by A17, A60, Th37;

          

           A65: (k1 + 1) <= k2 by A61, NAT_1: 13;

          ( LSeg (f,k2)) = ( LSeg ((f /. k2),(f /. (k2 + 1)))) by A16, A17, TOPREAL1:def 3;

          then

           A66: (( GoB f) * ((i + 1),j)) = (f /. k2) & (( GoB f) * ((i + 1),(j + 1))) = (f /. (k2 + 1)) or (( GoB f) * ((i + 1),j)) = (f /. (k2 + 1)) & (( GoB f) * ((i + 1),(j + 1))) = (f /. k2) by A18, SPPOL_1: 8;

          

           A67: k2 < ( len f) by A17, NAT_1: 13;

          then

           A68: (f /. k1) <> (f /. k2) by A60, A61, Th37;

          

           A69: ( LSeg (f,k1)) = ( LSeg ((f /. k1),(f /. (k1 + 1)))) by A10, A11, TOPREAL1:def 3;

          then (( GoB f) * (i,j)) = (f /. k1) & (( GoB f) * ((i + 1),j)) = (f /. (k1 + 1)) or (( GoB f) * (i,j)) = (f /. (k1 + 1)) & (( GoB f) * ((i + 1),j)) = (f /. k1) by A12, SPPOL_1: 8;

          then (k1 + 1) >= k2 by A17, A60, A61, A66, A63, A67, A62, Th37;

          then

           A70: (k1 + 1) = k2 by A65, XXREAL_0: 1;

          hence 1 <= k1 & (k1 + 1) < ( len f) by A17, A60, NAT_1: 13;

          thus (f /. (k1 + 1)) = (( GoB f) * ((i + 1),j)) by A12, A69, A66, A64, A68, SPPOL_1: 8;

          thus (f /. k1) = (( GoB f) * (i,j)) by A12, A69, A66, A64, A68, SPPOL_1: 8;

          thus (f /. (k1 + 2)) = (( GoB f) * ((i + 1),(j + 1))) by A12, A69, A66, A64, A70, SPPOL_1: 8;

        end;

          case that

           A71: k2 > 1 and

           A72: k1 > k2;

          

           A73: 1 < (k2 + 1) & (k2 + 1) < (k1 + 1) by A71, A72, NAT_1: 13, XREAL_1: 6;

          

           A74: k2 < (k1 + 1) by A72, NAT_1: 13;

          then

           A75: (f /. k2) <> (f /. (k1 + 1)) by A11, A71, Th37;

          

           A76: (k2 + 1) <= k1 by A72, NAT_1: 13;

          ( LSeg (f,k1)) = ( LSeg ((f /. k1),(f /. (k1 + 1)))) by A10, A11, TOPREAL1:def 3;

          then

           A77: (( GoB f) * ((i + 1),j)) = (f /. k1) & (( GoB f) * (i,j)) = (f /. (k1 + 1)) or (( GoB f) * ((i + 1),j)) = (f /. (k1 + 1)) & (( GoB f) * (i,j)) = (f /. k1) by A12, SPPOL_1: 8;

          

           A78: k1 < ( len f) by A11, NAT_1: 13;

          then

           A79: (f /. k2) <> (f /. k1) by A71, A72, Th37;

          

           A80: ( LSeg (f,k2)) = ( LSeg ((f /. k2),(f /. (k2 + 1)))) by A16, A17, TOPREAL1:def 3;

          then (( GoB f) * ((i + 1),(j + 1))) = (f /. k2) & (( GoB f) * ((i + 1),j)) = (f /. (k2 + 1)) or (( GoB f) * ((i + 1),(j + 1))) = (f /. (k2 + 1)) & (( GoB f) * ((i + 1),j)) = (f /. k2) by A18, SPPOL_1: 8;

          then (k2 + 1) >= k1 by A11, A71, A72, A77, A74, A78, A73, Th37;

          then

           A81: (k2 + 1) = k1 by A76, XXREAL_0: 1;

          hence 1 <= k2 & (k2 + 1) < ( len f) by A11, A71, NAT_1: 13;

          thus (f /. (k2 + 1)) = (( GoB f) * ((i + 1),j)) by A18, A80, A77, A75, A79, SPPOL_1: 8;

          thus (f /. k2) = (( GoB f) * ((i + 1),(j + 1))) by A18, A80, A77, A75, A79, SPPOL_1: 8;

          thus (f /. (k2 + 2)) = (( GoB f) * (i,j)) by A18, A80, A77, A75, A81, SPPOL_1: 8;

        end;

      end;

      hence thesis;

    end;

    theorem :: GOBOARD7:58

    

     Th58: 1 <= i & (i + 1) <= ( len ( GoB f)) & 1 <= j & (j + 1) <= ( width ( GoB f)) & ( LSeg ((( GoB f) * ((i + 1),j)),(( GoB f) * ((i + 1),(j + 1))))) c= ( L~ f) & ( LSeg ((( GoB f) * ((i + 1),(j + 1))),(( GoB f) * (i,(j + 1))))) c= ( L~ f) implies (f /. 1) = (( GoB f) * ((i + 1),(j + 1))) & ((f /. 2) = (( GoB f) * ((i + 1),j)) & (f /. (( len f) -' 1)) = (( GoB f) * (i,(j + 1))) or (f /. 2) = (( GoB f) * (i,(j + 1))) & (f /. (( len f) -' 1)) = (( GoB f) * ((i + 1),j))) or ex k st 1 <= k & (k + 1) < ( len f) & (f /. (k + 1)) = (( GoB f) * ((i + 1),(j + 1))) & ((f /. k) = (( GoB f) * ((i + 1),j)) & (f /. (k + 2)) = (( GoB f) * (i,(j + 1))) or (f /. k) = (( GoB f) * (i,(j + 1))) & (f /. (k + 2)) = (( GoB f) * ((i + 1),j)))

    proof

      assume that

       A1: 1 <= i and

       A2: (i + 1) <= ( len ( GoB f)) and

       A3: 1 <= j and

       A4: (j + 1) <= ( width ( GoB f)) and

       A5: ( LSeg ((( GoB f) * ((i + 1),j)),(( GoB f) * ((i + 1),(j + 1))))) c= ( L~ f) and

       A6: ( LSeg ((( GoB f) * ((i + 1),(j + 1))),(( GoB f) * (i,(j + 1))))) c= ( L~ f);

      

       A7: 1 <= (i + 1) by NAT_1: 11;

      ((1 / 2) * ((( GoB f) * ((i + 1),j)) + (( GoB f) * ((i + 1),(j + 1))))) in ( LSeg ((( GoB f) * ((i + 1),j)),(( GoB f) * ((i + 1),(j + 1))))) by RLTOPSP1: 69;

      then

      consider k1 such that

       A8: 1 <= k1 and

       A9: (k1 + 1) <= ( len f) and

       A10: ( LSeg ((( GoB f) * ((i + 1),j)),(( GoB f) * ((i + 1),(j + 1))))) = ( LSeg (f,k1)) by A2, A3, A4, A5, A7, Th39;

      

       A11: k1 < ( len f) by A9, NAT_1: 13;

       A12:

      now

        assume k1 > 1;

        then k1 >= (1 + 1) by NAT_1: 13;

        hence k1 = 2 or k1 > 2 by XXREAL_0: 1;

      end;

      

       A13: i < ( len ( GoB f)) & j < ( width ( GoB f)) by A2, A4, NAT_1: 13;

      

       A14: 1 <= (j + 1) by NAT_1: 11;

      ((1 / 2) * ((( GoB f) * (i,(j + 1))) + (( GoB f) * ((i + 1),(j + 1))))) in ( LSeg ((( GoB f) * ((i + 1),(j + 1))),(( GoB f) * (i,(j + 1))))) by RLTOPSP1: 69;

      then

      consider k2 such that

       A15: 1 <= k2 and

       A16: (k2 + 1) <= ( len f) and

       A17: ( LSeg ((( GoB f) * (i,(j + 1))),(( GoB f) * ((i + 1),(j + 1))))) = ( LSeg (f,k2)) by A1, A2, A4, A6, A14, Th40;

      

       A18: k2 < ( len f) by A16, NAT_1: 13;

       A19:

      now

        assume k2 > 1;

        then k2 >= (1 + 1) by NAT_1: 13;

        hence k2 = 2 or k2 > 2 by XXREAL_0: 1;

      end;

      

       A20: k1 = 1 or k1 > 1 by A8, XXREAL_0: 1;

      now

        per cases by A15, A12, A19, A20, XXREAL_0: 1;

          case that

           A21: k1 = 1 and

           A22: k2 = 2;

          

           A23: ( LSeg (f,2)) = ( LSeg ((f /. 2),(f /. (2 + 1)))) by A16, A22, TOPREAL1:def 3;

          then

           A24: (( GoB f) * ((i + 1),(j + 1))) = (f /. 2) & (( GoB f) * (i,(j + 1))) = (f /. (2 + 1)) or (( GoB f) * ((i + 1),(j + 1))) = (f /. (2 + 1)) & (( GoB f) * (i,(j + 1))) = (f /. 2) by A17, A22, SPPOL_1: 8;

          thus 1 <= 1 & (1 + 1) < ( len f) by A16, A22, NAT_1: 13;

          

           A25: 3 < ( len f) by Th34, XXREAL_0: 2;

          then

           A26: (f /. 1) <> (f /. 3) by Th36;

          

           A27: ( LSeg (f,1)) = ( LSeg ((f /. 1),(f /. (1 + 1)))) by A9, A21, TOPREAL1:def 3;

          then

           A28: (( GoB f) * ((i + 1),j)) = (f /. 1) & (( GoB f) * ((i + 1),(j + 1))) = (f /. 2) or (( GoB f) * ((i + 1),j)) = (f /. 2) & (( GoB f) * ((i + 1),(j + 1))) = (f /. 1) by A10, A21, SPPOL_1: 8;

          hence (f /. (1 + 1)) = (( GoB f) * ((i + 1),(j + 1))) by A24, A25, Th36;

          thus (f /. 1) = (( GoB f) * ((i + 1),j)) by A17, A22, A28, A23, A26, SPPOL_1: 8;

          thus (f /. (1 + 2)) = (( GoB f) * (i,(j + 1))) by A10, A21, A27, A24, A26, SPPOL_1: 8;

        end;

          case that

           A29: k1 = 1 and

           A30: k2 > 2;

          

           A31: ( LSeg (f,1)) = ( LSeg ((f /. 1),(f /. (1 + 1)))) by A9, A29, TOPREAL1:def 3;

          then

           A32: (( GoB f) * ((i + 1),j)) = (f /. 1) & (( GoB f) * ((i + 1),(j + 1))) = (f /. 2) or (( GoB f) * ((i + 1),j)) = (f /. 2) & (( GoB f) * ((i + 1),(j + 1))) = (f /. 1) by A10, A29, SPPOL_1: 8;

          

           A33: 2 < (k2 + 1) by A30, NAT_1: 13;

          then

           A34: (f /. (k2 + 1)) <> (f /. 2) by A16, Th37;

          ( LSeg (f,k2)) = ( LSeg ((f /. k2),(f /. (k2 + 1)))) by A15, A16, TOPREAL1:def 3;

          then

           A35: (( GoB f) * ((i + 1),(j + 1))) = (f /. k2) & (( GoB f) * (i,(j + 1))) = (f /. (k2 + 1)) or (( GoB f) * ((i + 1),(j + 1))) = (f /. (k2 + 1)) & (( GoB f) * (i,(j + 1))) = (f /. k2) by A17, SPPOL_1: 8;

          

           A36: (f /. k2) <> (f /. 2) by A18, A30, Th36;

          hence (f /. 1) = (( GoB f) * ((i + 1),(j + 1))) by A10, A29, A31, A35, A34, SPPOL_1: 8;

          thus (f /. 2) = (( GoB f) * ((i + 1),j)) by A10, A29, A31, A35, A36, A34, SPPOL_1: 8;

          

           A37: k2 > 1 by A30, XXREAL_0: 2;

          then

           A38: (k2 + 1) > 1 by NAT_1: 13;

          then (k2 + 1) = ( len f) by A16, A18, A30, A32, A35, A37, A33, Th37, Th38;

          then (k2 + 1) = ((( len f) -' 1) + 1) by A38, XREAL_1: 235;

          hence (f /. (( len f) -' 1)) = (( GoB f) * (i,(j + 1))) by A18, A30, A32, A35, A37, Th36;

        end;

          case that

           A39: k2 = 1 and

           A40: k1 = 2;

          

           A41: ( LSeg (f,2)) = ( LSeg ((f /. 2),(f /. (2 + 1)))) by A9, A40, TOPREAL1:def 3;

          then

           A42: (( GoB f) * ((i + 1),(j + 1))) = (f /. 2) & (( GoB f) * ((i + 1),j)) = (f /. (2 + 1)) or (( GoB f) * ((i + 1),(j + 1))) = (f /. (2 + 1)) & (( GoB f) * ((i + 1),j)) = (f /. 2) by A10, A40, SPPOL_1: 8;

          thus 1 <= 1 & (1 + 1) < ( len f) by A9, A40, NAT_1: 13;

          

           A43: 3 < ( len f) by Th34, XXREAL_0: 2;

          then

           A44: (f /. 1) <> (f /. 3) by Th36;

          

           A45: ( LSeg (f,1)) = ( LSeg ((f /. 1),(f /. (1 + 1)))) by A16, A39, TOPREAL1:def 3;

          then

           A46: (( GoB f) * (i,(j + 1))) = (f /. 1) & (( GoB f) * ((i + 1),(j + 1))) = (f /. 2) or (( GoB f) * (i,(j + 1))) = (f /. 2) & (( GoB f) * ((i + 1),(j + 1))) = (f /. 1) by A17, A39, SPPOL_1: 8;

          hence (f /. (1 + 1)) = (( GoB f) * ((i + 1),(j + 1))) by A42, A43, Th36;

          thus (f /. 1) = (( GoB f) * (i,(j + 1))) by A10, A40, A46, A41, A44, SPPOL_1: 8;

          thus (f /. (1 + 2)) = (( GoB f) * ((i + 1),j)) by A17, A39, A45, A42, A44, SPPOL_1: 8;

        end;

          case that

           A47: k2 = 1 and

           A48: k1 > 2;

          

           A49: ( LSeg (f,1)) = ( LSeg ((f /. 1),(f /. (1 + 1)))) by A16, A47, TOPREAL1:def 3;

          then

           A50: (( GoB f) * (i,(j + 1))) = (f /. 1) & (( GoB f) * ((i + 1),(j + 1))) = (f /. 2) or (( GoB f) * (i,(j + 1))) = (f /. 2) & (( GoB f) * ((i + 1),(j + 1))) = (f /. 1) by A17, A47, SPPOL_1: 8;

          

           A51: 2 < (k1 + 1) by A48, NAT_1: 13;

          then

           A52: (f /. (k1 + 1)) <> (f /. 2) by A9, Th37;

          ( LSeg (f,k1)) = ( LSeg ((f /. k1),(f /. (k1 + 1)))) by A8, A9, TOPREAL1:def 3;

          then

           A53: (( GoB f) * ((i + 1),(j + 1))) = (f /. k1) & (( GoB f) * ((i + 1),j)) = (f /. (k1 + 1)) or (( GoB f) * ((i + 1),(j + 1))) = (f /. (k1 + 1)) & (( GoB f) * ((i + 1),j)) = (f /. k1) by A10, SPPOL_1: 8;

          

           A54: (f /. k1) <> (f /. 2) by A11, A48, Th36;

          hence (f /. 1) = (( GoB f) * ((i + 1),(j + 1))) by A17, A47, A49, A53, A52, SPPOL_1: 8;

          thus (f /. 2) = (( GoB f) * (i,(j + 1))) by A17, A47, A49, A53, A54, A52, SPPOL_1: 8;

          

           A55: k1 > 1 by A48, XXREAL_0: 2;

          then

           A56: (k1 + 1) > 1 by NAT_1: 13;

          then (k1 + 1) = ( len f) by A9, A11, A48, A50, A53, A55, A51, Th37, Th38;

          then (k1 + 1) = ((( len f) -' 1) + 1) by A56, XREAL_1: 235;

          hence (f /. (( len f) -' 1)) = (( GoB f) * ((i + 1),j)) by A11, A48, A50, A53, A55, Th36;

        end;

          case k1 = k2;

          then

           A57: (( GoB f) * ((i + 1),j)) = (( GoB f) * (i,(j + 1))) or (( GoB f) * ((i + 1),j)) = (( GoB f) * ((i + 1),(j + 1))) by A10, A17, SPPOL_1: 8;

          

           A58: [(i + 1), (j + 1)] in ( Indices ( GoB f)) by A2, A4, A7, A14, MATRIX_0: 30;

           [i, (j + 1)] in ( Indices ( GoB f)) & [(i + 1), j] in ( Indices ( GoB f)) by A1, A2, A3, A4, A7, A14, A13, MATRIX_0: 30;

          then j = (j + 1) or i = (i + 1) by A57, A58, GOBOARD1: 5;

          hence contradiction;

        end;

          case that

           A59: k1 > 1 and

           A60: k2 > k1;

          

           A61: 1 < (k1 + 1) & (k1 + 1) < (k2 + 1) by A59, A60, NAT_1: 13, XREAL_1: 6;

          

           A62: k1 < (k2 + 1) by A60, NAT_1: 13;

          then

           A63: (f /. k1) <> (f /. (k2 + 1)) by A16, A59, Th37;

          

           A64: (k1 + 1) <= k2 by A60, NAT_1: 13;

          ( LSeg (f,k2)) = ( LSeg ((f /. k2),(f /. (k2 + 1)))) by A15, A16, TOPREAL1:def 3;

          then

           A65: (( GoB f) * ((i + 1),(j + 1))) = (f /. k2) & (( GoB f) * (i,(j + 1))) = (f /. (k2 + 1)) or (( GoB f) * ((i + 1),(j + 1))) = (f /. (k2 + 1)) & (( GoB f) * (i,(j + 1))) = (f /. k2) by A17, SPPOL_1: 8;

          

           A66: k2 < ( len f) by A16, NAT_1: 13;

          then

           A67: (f /. k1) <> (f /. k2) by A59, A60, Th37;

          

           A68: ( LSeg (f,k1)) = ( LSeg ((f /. k1),(f /. (k1 + 1)))) by A8, A9, TOPREAL1:def 3;

          then (( GoB f) * ((i + 1),j)) = (f /. k1) & (( GoB f) * ((i + 1),(j + 1))) = (f /. (k1 + 1)) or (( GoB f) * ((i + 1),j)) = (f /. (k1 + 1)) & (( GoB f) * ((i + 1),(j + 1))) = (f /. k1) by A10, SPPOL_1: 8;

          then (k1 + 1) >= k2 by A16, A59, A60, A65, A62, A66, A61, Th37;

          then

           A69: (k1 + 1) = k2 by A64, XXREAL_0: 1;

          hence 1 <= k1 & (k1 + 1) < ( len f) by A16, A59, NAT_1: 13;

          thus (f /. (k1 + 1)) = (( GoB f) * ((i + 1),(j + 1))) by A10, A68, A65, A63, A67, SPPOL_1: 8;

          thus (f /. k1) = (( GoB f) * ((i + 1),j)) by A10, A68, A65, A63, A67, SPPOL_1: 8;

          thus (f /. (k1 + 2)) = (( GoB f) * (i,(j + 1))) by A10, A68, A65, A63, A69, SPPOL_1: 8;

        end;

          case that

           A70: k2 > 1 and

           A71: k1 > k2;

          

           A72: 1 < (k2 + 1) & (k2 + 1) < (k1 + 1) by A70, A71, NAT_1: 13, XREAL_1: 6;

          

           A73: k2 < (k1 + 1) by A71, NAT_1: 13;

          then

           A74: (f /. k2) <> (f /. (k1 + 1)) by A9, A70, Th37;

          

           A75: (k2 + 1) <= k1 by A71, NAT_1: 13;

          ( LSeg (f,k1)) = ( LSeg ((f /. k1),(f /. (k1 + 1)))) by A8, A9, TOPREAL1:def 3;

          then

           A76: (( GoB f) * ((i + 1),(j + 1))) = (f /. k1) & (( GoB f) * ((i + 1),j)) = (f /. (k1 + 1)) or (( GoB f) * ((i + 1),(j + 1))) = (f /. (k1 + 1)) & (( GoB f) * ((i + 1),j)) = (f /. k1) by A10, SPPOL_1: 8;

          

           A77: k1 < ( len f) by A9, NAT_1: 13;

          then

           A78: (f /. k2) <> (f /. k1) by A70, A71, Th37;

          

           A79: ( LSeg (f,k2)) = ( LSeg ((f /. k2),(f /. (k2 + 1)))) by A15, A16, TOPREAL1:def 3;

          then (( GoB f) * (i,(j + 1))) = (f /. k2) & (( GoB f) * ((i + 1),(j + 1))) = (f /. (k2 + 1)) or (( GoB f) * (i,(j + 1))) = (f /. (k2 + 1)) & (( GoB f) * ((i + 1),(j + 1))) = (f /. k2) by A17, SPPOL_1: 8;

          then (k2 + 1) >= k1 by A9, A70, A71, A76, A73, A77, A72, Th37;

          then

           A80: (k2 + 1) = k1 by A75, XXREAL_0: 1;

          hence 1 <= k2 & (k2 + 1) < ( len f) by A9, A70, NAT_1: 13;

          thus (f /. (k2 + 1)) = (( GoB f) * ((i + 1),(j + 1))) by A17, A79, A76, A74, A78, SPPOL_1: 8;

          thus (f /. k2) = (( GoB f) * (i,(j + 1))) by A17, A79, A76, A74, A78, SPPOL_1: 8;

          thus (f /. (k2 + 2)) = (( GoB f) * ((i + 1),j)) by A17, A79, A76, A74, A80, SPPOL_1: 8;

        end;

      end;

      hence thesis;

    end;

    theorem :: GOBOARD7:59

    1 <= i & i < ( len ( GoB f)) & 1 <= j & (j + 1) < ( width ( GoB f)) implies not (( LSeg ((( GoB f) * (i,j)),(( GoB f) * (i,(j + 1))))) c= ( L~ f) & ( LSeg ((( GoB f) * (i,(j + 1))),(( GoB f) * (i,(j + 2))))) c= ( L~ f) & ( LSeg ((( GoB f) * (i,(j + 1))),(( GoB f) * ((i + 1),(j + 1))))) c= ( L~ f))

    proof

      assume that

       A1: 1 <= i and

       A2: i < ( len ( GoB f)) and

       A3: 1 <= j and

       A4: (j + 1) < ( width ( GoB f)) and

       A5: ( LSeg ((( GoB f) * (i,j)),(( GoB f) * (i,(j + 1))))) c= ( L~ f) & ( LSeg ((( GoB f) * (i,(j + 1))),(( GoB f) * (i,(j + 2))))) c= ( L~ f) & ( LSeg ((( GoB f) * (i,(j + 1))),(( GoB f) * ((i + 1),(j + 1))))) c= ( L~ f);

      

       A6: (i + 1) <= ( len ( GoB f)) by A2, NAT_1: 13;

      (j + (1 + 1)) = ((j + 1) + 1);

      then

       A7: (j + 2) <= ( width ( GoB f)) by A4, NAT_1: 13;

      

       A8: 1 <= (j + 1) by NAT_1: 11;

      

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

      

       A10: 1 <= (i + 1) by NAT_1: 11;

      (j + 1) <= (j + 2) by XREAL_1: 6;

      then

       A11: 1 <= (j + 2) by A8, XXREAL_0: 2;

      per cases by A1, A2, A3, A4, A5, A6, Th53, Th54;

        suppose

         A12: (f /. (( len f) -' 1)) = (( GoB f) * (i,(j + 2))) & (f /. (( len f) -' 1)) = (( GoB f) * ((i + 1),(j + 1)));

         [i, (j + 2)] in ( Indices ( GoB f)) & [(i + 1), (j + 1)] in ( Indices ( GoB f)) by A1, A2, A4, A6, A10, A8, A7, A11, MATRIX_0: 30;

        then i = (i + 1) by A12, GOBOARD1: 5;

        hence contradiction;

      end;

        suppose

         A13: (f /. 2) = (( GoB f) * (i,j)) & (f /. 2) = (( GoB f) * ((i + 1),(j + 1)));

         [i, j] in ( Indices ( GoB f)) & [(i + 1), (j + 1)] in ( Indices ( GoB f)) by A1, A2, A3, A4, A6, A10, A9, A8, MATRIX_0: 30;

        then i = (i + 1) by A13, GOBOARD1: 5;

        hence contradiction;

      end;

        suppose

         A14: (f /. 2) = (( GoB f) * (i,(j + 2))) & (f /. 2) = (( GoB f) * (i,j));

         [i, (j + 2)] in ( Indices ( GoB f)) & [i, j] in ( Indices ( GoB f)) by A1, A2, A3, A9, A7, A11, MATRIX_0: 30;

        then j = (j + 2) by A14, GOBOARD1: 5;

        hence contradiction;

      end;

        suppose

         A15: (f /. 2) = (( GoB f) * (i,(j + 2))) & (f /. 2) = (( GoB f) * ((i + 1),(j + 1)));

         [i, (j + 2)] in ( Indices ( GoB f)) & [(i + 1), (j + 1)] in ( Indices ( GoB f)) by A1, A2, A4, A6, A10, A8, A7, A11, MATRIX_0: 30;

        then i = (i + 1) by A15, GOBOARD1: 5;

        hence contradiction;

      end;

        suppose that

         A16: (f /. 1) = (( GoB f) * (i,(j + 1))) and

         A17: ex k st 1 <= k & (k + 1) < ( len f) & (f /. (k + 1)) = (( GoB f) * (i,(j + 1))) & ((f /. k) = (( GoB f) * (i,j)) & (f /. (k + 2)) = (( GoB f) * ((i + 1),(j + 1))) or (f /. k) = (( GoB f) * ((i + 1),(j + 1))) & (f /. (k + 2)) = (( GoB f) * (i,j)));

        consider k such that

         A18: 1 <= k and

         A19: (k + 1) < ( len f) & (f /. (k + 1)) = (( GoB f) * (i,(j + 1))) and (f /. k) = (( GoB f) * (i,j)) & (f /. (k + 2)) = (( GoB f) * ((i + 1),(j + 1))) or (f /. k) = (( GoB f) * ((i + 1),(j + 1))) & (f /. (k + 2)) = (( GoB f) * (i,j)) by A17;

        1 < (k + 1) by A18, NAT_1: 13;

        hence contradiction by A16, A19, Th36;

      end;

        suppose that

         A20: ex k st 1 <= k & (k + 1) < ( len f) & (f /. (k + 1)) = (( GoB f) * (i,(j + 1))) & ((f /. k) = (( GoB f) * (i,j)) & (f /. (k + 2)) = (( GoB f) * (i,(j + 2))) or (f /. k) = (( GoB f) * (i,(j + 2))) & (f /. (k + 2)) = (( GoB f) * (i,j))) and

         A21: (f /. 1) = (( GoB f) * (i,(j + 1)));

        consider k such that

         A22: 1 <= k and

         A23: (k + 1) < ( len f) & (f /. (k + 1)) = (( GoB f) * (i,(j + 1))) and (f /. k) = (( GoB f) * (i,j)) & (f /. (k + 2)) = (( GoB f) * (i,(j + 2))) or (f /. k) = (( GoB f) * (i,(j + 2))) & (f /. (k + 2)) = (( GoB f) * (i,j)) by A20;

        1 < (k + 1) by A22, NAT_1: 13;

        hence contradiction by A21, A23, Th36;

      end;

        suppose that

         A24: ex k st 1 <= k & (k + 1) < ( len f) & (f /. (k + 1)) = (( GoB f) * (i,(j + 1))) & ((f /. k) = (( GoB f) * (i,j)) & (f /. (k + 2)) = (( GoB f) * (i,(j + 2))) or (f /. k) = (( GoB f) * (i,(j + 2))) & (f /. (k + 2)) = (( GoB f) * (i,j))) and

         A25: ex k st 1 <= k & (k + 1) < ( len f) & (f /. (k + 1)) = (( GoB f) * (i,(j + 1))) & ((f /. k) = (( GoB f) * (i,j)) & (f /. (k + 2)) = (( GoB f) * ((i + 1),(j + 1))) or (f /. k) = (( GoB f) * ((i + 1),(j + 1))) & (f /. (k + 2)) = (( GoB f) * (i,j)));

        consider k1 such that 1 <= k1 and

         A26: (k1 + 1) < ( len f) and

         A27: (f /. (k1 + 1)) = (( GoB f) * (i,(j + 1))) and

         A28: (f /. k1) = (( GoB f) * (i,j)) & (f /. (k1 + 2)) = (( GoB f) * (i,(j + 2))) or (f /. k1) = (( GoB f) * (i,(j + 2))) & (f /. (k1 + 2)) = (( GoB f) * (i,j)) by A24;

        consider k2 such that 1 <= k2 and

         A29: (k2 + 1) < ( len f) and

         A30: (f /. (k2 + 1)) = (( GoB f) * (i,(j + 1))) and

         A31: (f /. k2) = (( GoB f) * (i,j)) & (f /. (k2 + 2)) = (( GoB f) * ((i + 1),(j + 1))) or (f /. k2) = (( GoB f) * ((i + 1),(j + 1))) & (f /. (k2 + 2)) = (( GoB f) * (i,j)) by A25;

         A32:

        now

          assume

           A33: k1 <> k2;

          per cases by A33, XXREAL_0: 1;

            suppose k1 < k2;

            then (k1 + 1) < (k2 + 1) by XREAL_1: 6;

            hence contradiction by A27, A29, A30, Th36, NAT_1: 11;

          end;

            suppose k2 < k1;

            then (k2 + 1) < (k1 + 1) by XREAL_1: 6;

            hence contradiction by A26, A27, A30, Th36, NAT_1: 11;

          end;

        end;

        now

          per cases by A28, A31;

            suppose

             A34: (f /. (k1 + 2)) = (( GoB f) * (i,(j + 2))) & (f /. (k2 + 2)) = (( GoB f) * ((i + 1),(j + 1)));

             [i, (j + 2)] in ( Indices ( GoB f)) & [(i + 1), (j + 1)] in ( Indices ( GoB f)) by A1, A2, A4, A6, A10, A8, A7, A11, MATRIX_0: 30;

            then i = (i + 1) by A32, A34, GOBOARD1: 5;

            hence contradiction;

          end;

            suppose

             A35: (f /. k1) = (( GoB f) * (i,j)) & (f /. k2) = (( GoB f) * ((i + 1),(j + 1)));

             [i, j] in ( Indices ( GoB f)) & [(i + 1), (j + 1)] in ( Indices ( GoB f)) by A1, A2, A3, A4, A6, A10, A9, A8, MATRIX_0: 30;

            then i = (i + 1) by A32, A35, GOBOARD1: 5;

            hence contradiction;

          end;

            suppose

             A36: (f /. k1) = (( GoB f) * (i,(j + 2))) & (f /. k2) = (( GoB f) * (i,j));

             [i, (j + 2)] in ( Indices ( GoB f)) & [i, j] in ( Indices ( GoB f)) by A1, A2, A3, A9, A7, A11, MATRIX_0: 30;

            then j = (j + 2) by A32, A36, GOBOARD1: 5;

            hence contradiction;

          end;

            suppose

             A37: (f /. k1) = (( GoB f) * (i,(j + 2))) & (f /. k2) = (( GoB f) * ((i + 1),(j + 1)));

             [i, (j + 2)] in ( Indices ( GoB f)) & [(i + 1), (j + 1)] in ( Indices ( GoB f)) by A1, A2, A4, A6, A10, A8, A7, A11, MATRIX_0: 30;

            then i = (i + 1) by A32, A37, GOBOARD1: 5;

            hence contradiction;

          end;

        end;

        hence contradiction;

      end;

    end;

    theorem :: GOBOARD7:60

    1 <= i & i < ( len ( GoB f)) & 1 <= j & (j + 1) < ( width ( GoB f)) implies not (( LSeg ((( GoB f) * ((i + 1),j)),(( GoB f) * ((i + 1),(j + 1))))) c= ( L~ f) & ( LSeg ((( GoB f) * ((i + 1),(j + 1))),(( GoB f) * ((i + 1),(j + 2))))) c= ( L~ f) & ( LSeg ((( GoB f) * (i,(j + 1))),(( GoB f) * ((i + 1),(j + 1))))) c= ( L~ f))

    proof

      assume that

       A1: 1 <= i and

       A2: i < ( len ( GoB f)) and

       A3: 1 <= j and

       A4: (j + 1) < ( width ( GoB f)) and

       A5: ( LSeg ((( GoB f) * ((i + 1),j)),(( GoB f) * ((i + 1),(j + 1))))) c= ( L~ f) & ( LSeg ((( GoB f) * ((i + 1),(j + 1))),(( GoB f) * ((i + 1),(j + 2))))) c= ( L~ f) & ( LSeg ((( GoB f) * (i,(j + 1))),(( GoB f) * ((i + 1),(j + 1))))) c= ( L~ f);

      

       A6: (i + 1) <= ( len ( GoB f)) by A2, NAT_1: 13;

      (j + (1 + 1)) = ((j + 1) + 1);

      then

       A7: (j + 2) <= ( width ( GoB f)) by A4, NAT_1: 13;

      

       A8: 1 <= (j + 1) by NAT_1: 11;

      

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

      

       A10: 1 <= (i + 1) by NAT_1: 11;

      (j + 1) <= (j + 2) by XREAL_1: 6;

      then

       A11: 1 <= (j + 2) by A8, XXREAL_0: 2;

      per cases by A1, A3, A4, A5, A6, A10, Th53, Th55;

        suppose

         A12: (f /. 2) = (( GoB f) * ((i + 1),j)) & (f /. 2) = (( GoB f) * (i,(j + 1)));

         [(i + 1), j] in ( Indices ( GoB f)) & [i, (j + 1)] in ( Indices ( GoB f)) by A1, A2, A3, A4, A6, A10, A9, A8, MATRIX_0: 30;

        then i = (i + 1) by A12, GOBOARD1: 5;

        hence contradiction;

      end;

        suppose

         A13: (f /. (( len f) -' 1)) = (( GoB f) * ((i + 1),(j + 2))) & (f /. (( len f) -' 1)) = (( GoB f) * (i,(j + 1)));

         [(i + 1), (j + 2)] in ( Indices ( GoB f)) & [i, (j + 1)] in ( Indices ( GoB f)) by A1, A2, A4, A6, A10, A8, A7, A11, MATRIX_0: 30;

        then i = (i + 1) by A13, GOBOARD1: 5;

        hence contradiction;

      end;

        suppose

         A14: (f /. 2) = (( GoB f) * ((i + 1),(j + 2))) & (f /. 2) = (( GoB f) * (i,(j + 1)));

         [(i + 1), (j + 2)] in ( Indices ( GoB f)) & [i, (j + 1)] in ( Indices ( GoB f)) by A1, A2, A4, A6, A10, A8, A7, A11, MATRIX_0: 30;

        then i = (i + 1) by A14, GOBOARD1: 5;

        hence contradiction;

      end;

        suppose

         A15: (f /. 2) = (( GoB f) * ((i + 1),(j + 2))) & (f /. 2) = (( GoB f) * ((i + 1),j));

         [(i + 1), (j + 2)] in ( Indices ( GoB f)) & [(i + 1), j] in ( Indices ( GoB f)) by A3, A6, A10, A9, A7, A11, MATRIX_0: 30;

        then j = (j + 2) by A15, GOBOARD1: 5;

        hence contradiction;

      end;

        suppose that

         A16: (f /. 1) = (( GoB f) * ((i + 1),(j + 1))) and

         A17: ex k st 1 <= k & (k + 1) < ( len f) & (f /. (k + 1)) = (( GoB f) * ((i + 1),(j + 1))) & ((f /. k) = (( GoB f) * (i,(j + 1))) & (f /. (k + 2)) = (( GoB f) * ((i + 1),j)) or (f /. k) = (( GoB f) * ((i + 1),j)) & (f /. (k + 2)) = (( GoB f) * (i,(j + 1))));

        consider k such that

         A18: 1 <= k and

         A19: (k + 1) < ( len f) & (f /. (k + 1)) = (( GoB f) * ((i + 1),(j + 1))) and (f /. k) = (( GoB f) * (i,(j + 1))) & (f /. (k + 2)) = (( GoB f) * ((i + 1),j)) or (f /. k) = (( GoB f) * ((i + 1),j)) & (f /. (k + 2)) = (( GoB f) * (i,(j + 1))) by A17;

        1 < (k + 1) by A18, NAT_1: 13;

        hence contradiction by A16, A19, Th36;

      end;

        suppose that

         A20: ex k st 1 <= k & (k + 1) < ( len f) & (f /. (k + 1)) = (( GoB f) * ((i + 1),(j + 1))) & ((f /. k) = (( GoB f) * ((i + 1),j)) & (f /. (k + 2)) = (( GoB f) * ((i + 1),(j + 2))) or (f /. k) = (( GoB f) * ((i + 1),(j + 2))) & (f /. (k + 2)) = (( GoB f) * ((i + 1),j))) and

         A21: (f /. 1) = (( GoB f) * ((i + 1),(j + 1)));

        consider k such that

         A22: 1 <= k and

         A23: (k + 1) < ( len f) & (f /. (k + 1)) = (( GoB f) * ((i + 1),(j + 1))) and (f /. k) = (( GoB f) * ((i + 1),j)) & (f /. (k + 2)) = (( GoB f) * ((i + 1),(j + 2))) or (f /. k) = (( GoB f) * ((i + 1),(j + 2))) & (f /. (k + 2)) = (( GoB f) * ((i + 1),j)) by A20;

        1 < (k + 1) by A22, NAT_1: 13;

        hence contradiction by A21, A23, Th36;

      end;

        suppose that

         A24: ex k st 1 <= k & (k + 1) < ( len f) & (f /. (k + 1)) = (( GoB f) * ((i + 1),(j + 1))) & ((f /. k) = (( GoB f) * ((i + 1),j)) & (f /. (k + 2)) = (( GoB f) * ((i + 1),(j + 2))) or (f /. k) = (( GoB f) * ((i + 1),(j + 2))) & (f /. (k + 2)) = (( GoB f) * ((i + 1),j))) and

         A25: ex k st 1 <= k & (k + 1) < ( len f) & (f /. (k + 1)) = (( GoB f) * ((i + 1),(j + 1))) & ((f /. k) = (( GoB f) * (i,(j + 1))) & (f /. (k + 2)) = (( GoB f) * ((i + 1),j)) or (f /. k) = (( GoB f) * ((i + 1),j)) & (f /. (k + 2)) = (( GoB f) * (i,(j + 1))));

        consider k1 such that 1 <= k1 and

         A26: (k1 + 1) < ( len f) and

         A27: (f /. (k1 + 1)) = (( GoB f) * ((i + 1),(j + 1))) and

         A28: (f /. k1) = (( GoB f) * ((i + 1),j)) & (f /. (k1 + 2)) = (( GoB f) * ((i + 1),(j + 2))) or (f /. k1) = (( GoB f) * ((i + 1),(j + 2))) & (f /. (k1 + 2)) = (( GoB f) * ((i + 1),j)) by A24;

        consider k2 such that 1 <= k2 and

         A29: (k2 + 1) < ( len f) and

         A30: (f /. (k2 + 1)) = (( GoB f) * ((i + 1),(j + 1))) and

         A31: (f /. k2) = (( GoB f) * (i,(j + 1))) & (f /. (k2 + 2)) = (( GoB f) * ((i + 1),j)) or (f /. k2) = (( GoB f) * ((i + 1),j)) & (f /. (k2 + 2)) = (( GoB f) * (i,(j + 1))) by A25;

         A32:

        now

          assume

           A33: k1 <> k2;

          per cases by A33, XXREAL_0: 1;

            suppose k1 < k2;

            then (k1 + 1) < (k2 + 1) by XREAL_1: 6;

            hence contradiction by A27, A29, A30, Th36, NAT_1: 11;

          end;

            suppose k2 < k1;

            then (k2 + 1) < (k1 + 1) by XREAL_1: 6;

            hence contradiction by A26, A27, A30, Th36, NAT_1: 11;

          end;

        end;

        now

          per cases by A28, A31;

            suppose

             A34: (f /. k1) = (( GoB f) * ((i + 1),j)) & (f /. k2) = (( GoB f) * (i,(j + 1)));

             [(i + 1), j] in ( Indices ( GoB f)) & [i, (j + 1)] in ( Indices ( GoB f)) by A1, A2, A3, A4, A6, A10, A9, A8, MATRIX_0: 30;

            then i = (i + 1) by A32, A34, GOBOARD1: 5;

            hence contradiction;

          end;

            suppose

             A35: (f /. (k1 + 2)) = (( GoB f) * ((i + 1),(j + 2))) & (f /. (k2 + 2)) = (( GoB f) * (i,(j + 1)));

             [(i + 1), (j + 2)] in ( Indices ( GoB f)) & [i, (j + 1)] in ( Indices ( GoB f)) by A1, A2, A4, A6, A10, A8, A7, A11, MATRIX_0: 30;

            then i = (i + 1) by A32, A35, GOBOARD1: 5;

            hence contradiction;

          end;

            suppose

             A36: (f /. k1) = (( GoB f) * ((i + 1),(j + 2))) & (f /. k2) = (( GoB f) * (i,(j + 1)));

             [(i + 1), (j + 2)] in ( Indices ( GoB f)) & [i, (j + 1)] in ( Indices ( GoB f)) by A1, A2, A4, A6, A10, A8, A7, A11, MATRIX_0: 30;

            then i = (i + 1) by A32, A36, GOBOARD1: 5;

            hence contradiction;

          end;

            suppose

             A37: (f /. k1) = (( GoB f) * ((i + 1),(j + 2))) & (f /. k2) = (( GoB f) * ((i + 1),j));

             [(i + 1), (j + 2)] in ( Indices ( GoB f)) & [(i + 1), j] in ( Indices ( GoB f)) by A3, A6, A10, A9, A7, A11, MATRIX_0: 30;

            then j = (j + 2) by A32, A37, GOBOARD1: 5;

            hence contradiction;

          end;

        end;

        hence contradiction;

      end;

    end;

    theorem :: GOBOARD7:61

    1 <= j & j < ( width ( GoB f)) & 1 <= i & (i + 1) < ( len ( GoB f)) implies not (( LSeg ((( GoB f) * (i,j)),(( GoB f) * ((i + 1),j)))) c= ( L~ f) & ( LSeg ((( GoB f) * ((i + 1),j)),(( GoB f) * ((i + 2),j)))) c= ( L~ f) & ( LSeg ((( GoB f) * ((i + 1),j)),(( GoB f) * ((i + 1),(j + 1))))) c= ( L~ f))

    proof

      assume that

       A1: 1 <= j and

       A2: j < ( width ( GoB f)) and

       A3: 1 <= i and

       A4: (i + 1) < ( len ( GoB f)) and

       A5: ( LSeg ((( GoB f) * (i,j)),(( GoB f) * ((i + 1),j)))) c= ( L~ f) & ( LSeg ((( GoB f) * ((i + 1),j)),(( GoB f) * ((i + 2),j)))) c= ( L~ f) & ( LSeg ((( GoB f) * ((i + 1),j)),(( GoB f) * ((i + 1),(j + 1))))) c= ( L~ f);

      

       A6: (j + 1) <= ( width ( GoB f)) by A2, NAT_1: 13;

      (i + (1 + 1)) = ((i + 1) + 1);

      then

       A7: (i + 2) <= ( len ( GoB f)) by A4, NAT_1: 13;

      

       A8: 1 <= (i + 1) by NAT_1: 11;

      

       A9: i < ( len ( GoB f)) by A4, NAT_1: 13;

      

       A10: 1 <= (j + 1) by NAT_1: 11;

      (i + 1) <= (i + 2) by XREAL_1: 6;

      then

       A11: 1 <= (i + 2) by A8, XXREAL_0: 2;

      per cases by A1, A2, A3, A4, A5, A6, Th56, Th57;

        suppose

         A12: (f /. (( len f) -' 1)) = (( GoB f) * ((i + 2),j)) & (f /. (( len f) -' 1)) = (( GoB f) * ((i + 1),(j + 1)));

         [(i + 2), j] in ( Indices ( GoB f)) & [(i + 1), (j + 1)] in ( Indices ( GoB f)) by A1, A2, A4, A6, A10, A8, A7, A11, MATRIX_0: 30;

        then j = (j + 1) by A12, GOBOARD1: 5;

        hence contradiction;

      end;

        suppose

         A13: (f /. 2) = (( GoB f) * (i,j)) & (f /. 2) = (( GoB f) * ((i + 1),(j + 1)));

         [i, j] in ( Indices ( GoB f)) & [(i + 1), (j + 1)] in ( Indices ( GoB f)) by A1, A2, A3, A4, A6, A10, A9, A8, MATRIX_0: 30;

        then j = (j + 1) by A13, GOBOARD1: 5;

        hence contradiction;

      end;

        suppose

         A14: (f /. 2) = (( GoB f) * ((i + 2),j)) & (f /. 2) = (( GoB f) * (i,j));

         [(i + 2), j] in ( Indices ( GoB f)) & [i, j] in ( Indices ( GoB f)) by A1, A2, A3, A9, A7, A11, MATRIX_0: 30;

        then i = (i + 2) by A14, GOBOARD1: 5;

        hence contradiction;

      end;

        suppose

         A15: (f /. 2) = (( GoB f) * ((i + 2),j)) & (f /. 2) = (( GoB f) * ((i + 1),(j + 1)));

         [(i + 2), j] in ( Indices ( GoB f)) & [(i + 1), (j + 1)] in ( Indices ( GoB f)) by A1, A2, A4, A6, A10, A8, A7, A11, MATRIX_0: 30;

        then j = (j + 1) by A15, GOBOARD1: 5;

        hence contradiction;

      end;

        suppose that

         A16: (f /. 1) = (( GoB f) * ((i + 1),j)) and

         A17: ex k st 1 <= k & (k + 1) < ( len f) & (f /. (k + 1)) = (( GoB f) * ((i + 1),j)) & ((f /. k) = (( GoB f) * (i,j)) & (f /. (k + 2)) = (( GoB f) * ((i + 1),(j + 1))) or (f /. k) = (( GoB f) * ((i + 1),(j + 1))) & (f /. (k + 2)) = (( GoB f) * (i,j)));

        consider k such that

         A18: 1 <= k and

         A19: (k + 1) < ( len f) & (f /. (k + 1)) = (( GoB f) * ((i + 1),j)) and (f /. k) = (( GoB f) * (i,j)) & (f /. (k + 2)) = (( GoB f) * ((i + 1),(j + 1))) or (f /. k) = (( GoB f) * ((i + 1),(j + 1))) & (f /. (k + 2)) = (( GoB f) * (i,j)) by A17;

        1 < (k + 1) by A18, NAT_1: 13;

        hence contradiction by A16, A19, Th36;

      end;

        suppose that

         A20: ex k st 1 <= k & (k + 1) < ( len f) & (f /. (k + 1)) = (( GoB f) * ((i + 1),j)) & ((f /. k) = (( GoB f) * (i,j)) & (f /. (k + 2)) = (( GoB f) * ((i + 2),j)) or (f /. k) = (( GoB f) * ((i + 2),j)) & (f /. (k + 2)) = (( GoB f) * (i,j))) and

         A21: (f /. 1) = (( GoB f) * ((i + 1),j));

        consider k such that

         A22: 1 <= k and

         A23: (k + 1) < ( len f) & (f /. (k + 1)) = (( GoB f) * ((i + 1),j)) and (f /. k) = (( GoB f) * (i,j)) & (f /. (k + 2)) = (( GoB f) * ((i + 2),j)) or (f /. k) = (( GoB f) * ((i + 2),j)) & (f /. (k + 2)) = (( GoB f) * (i,j)) by A20;

        1 < (k + 1) by A22, NAT_1: 13;

        hence contradiction by A21, A23, Th36;

      end;

        suppose that

         A24: ex k st 1 <= k & (k + 1) < ( len f) & (f /. (k + 1)) = (( GoB f) * ((i + 1),j)) & ((f /. k) = (( GoB f) * (i,j)) & (f /. (k + 2)) = (( GoB f) * ((i + 2),j)) or (f /. k) = (( GoB f) * ((i + 2),j)) & (f /. (k + 2)) = (( GoB f) * (i,j))) and

         A25: ex k st 1 <= k & (k + 1) < ( len f) & (f /. (k + 1)) = (( GoB f) * ((i + 1),j)) & ((f /. k) = (( GoB f) * (i,j)) & (f /. (k + 2)) = (( GoB f) * ((i + 1),(j + 1))) or (f /. k) = (( GoB f) * ((i + 1),(j + 1))) & (f /. (k + 2)) = (( GoB f) * (i,j)));

        consider k1 such that 1 <= k1 and

         A26: (k1 + 1) < ( len f) and

         A27: (f /. (k1 + 1)) = (( GoB f) * ((i + 1),j)) and

         A28: (f /. k1) = (( GoB f) * (i,j)) & (f /. (k1 + 2)) = (( GoB f) * ((i + 2),j)) or (f /. k1) = (( GoB f) * ((i + 2),j)) & (f /. (k1 + 2)) = (( GoB f) * (i,j)) by A24;

        consider k2 such that 1 <= k2 and

         A29: (k2 + 1) < ( len f) and

         A30: (f /. (k2 + 1)) = (( GoB f) * ((i + 1),j)) and

         A31: (f /. k2) = (( GoB f) * (i,j)) & (f /. (k2 + 2)) = (( GoB f) * ((i + 1),(j + 1))) or (f /. k2) = (( GoB f) * ((i + 1),(j + 1))) & (f /. (k2 + 2)) = (( GoB f) * (i,j)) by A25;

         A32:

        now

          assume

           A33: k1 <> k2;

          per cases by A33, XXREAL_0: 1;

            suppose k1 < k2;

            then (k1 + 1) < (k2 + 1) by XREAL_1: 6;

            hence contradiction by A27, A29, A30, Th36, NAT_1: 11;

          end;

            suppose k2 < k1;

            then (k2 + 1) < (k1 + 1) by XREAL_1: 6;

            hence contradiction by A26, A27, A30, Th36, NAT_1: 11;

          end;

        end;

        now

          per cases by A28, A31;

            suppose

             A34: (f /. (k1 + 2)) = (( GoB f) * ((i + 2),j)) & (f /. (k2 + 2)) = (( GoB f) * ((i + 1),(j + 1)));

             [(i + 2), j] in ( Indices ( GoB f)) & [(i + 1), (j + 1)] in ( Indices ( GoB f)) by A1, A2, A4, A6, A10, A8, A7, A11, MATRIX_0: 30;

            then j = (j + 1) by A32, A34, GOBOARD1: 5;

            hence contradiction;

          end;

            suppose

             A35: (f /. k1) = (( GoB f) * (i,j)) & (f /. k2) = (( GoB f) * ((i + 1),(j + 1)));

             [i, j] in ( Indices ( GoB f)) & [(i + 1), (j + 1)] in ( Indices ( GoB f)) by A1, A2, A3, A4, A6, A10, A9, A8, MATRIX_0: 30;

            then j = (j + 1) by A32, A35, GOBOARD1: 5;

            hence contradiction;

          end;

            suppose

             A36: (f /. k1) = (( GoB f) * ((i + 2),j)) & (f /. k2) = (( GoB f) * (i,j));

             [(i + 2), j] in ( Indices ( GoB f)) & [i, j] in ( Indices ( GoB f)) by A1, A2, A3, A9, A7, A11, MATRIX_0: 30;

            then i = (i + 2) by A32, A36, GOBOARD1: 5;

            hence contradiction;

          end;

            suppose

             A37: (f /. k1) = (( GoB f) * ((i + 2),j)) & (f /. k2) = (( GoB f) * ((i + 1),(j + 1)));

             [(i + 2), j] in ( Indices ( GoB f)) & [(i + 1), (j + 1)] in ( Indices ( GoB f)) by A1, A2, A4, A6, A10, A8, A7, A11, MATRIX_0: 30;

            then j = (j + 1) by A32, A37, GOBOARD1: 5;

            hence contradiction;

          end;

        end;

        hence contradiction;

      end;

    end;

    theorem :: GOBOARD7:62

    1 <= j & j < ( width ( GoB f)) & 1 <= i & (i + 1) < ( len ( GoB f)) implies not (( LSeg ((( GoB f) * (i,(j + 1))),(( GoB f) * ((i + 1),(j + 1))))) c= ( L~ f) & ( LSeg ((( GoB f) * ((i + 1),(j + 1))),(( GoB f) * ((i + 2),(j + 1))))) c= ( L~ f) & ( LSeg ((( GoB f) * ((i + 1),j)),(( GoB f) * ((i + 1),(j + 1))))) c= ( L~ f))

    proof

      assume that

       A1: 1 <= j and

       A2: j < ( width ( GoB f)) and

       A3: 1 <= i and

       A4: (i + 1) < ( len ( GoB f)) and

       A5: ( LSeg ((( GoB f) * (i,(j + 1))),(( GoB f) * ((i + 1),(j + 1))))) c= ( L~ f) & ( LSeg ((( GoB f) * ((i + 1),(j + 1))),(( GoB f) * ((i + 2),(j + 1))))) c= ( L~ f) & ( LSeg ((( GoB f) * ((i + 1),j)),(( GoB f) * ((i + 1),(j + 1))))) c= ( L~ f);

      

       A6: (j + 1) <= ( width ( GoB f)) by A2, NAT_1: 13;

      (i + (1 + 1)) = ((i + 1) + 1);

      then

       A7: (i + 2) <= ( len ( GoB f)) by A4, NAT_1: 13;

      

       A8: 1 <= (i + 1) by NAT_1: 11;

      

       A9: i < ( len ( GoB f)) by A4, NAT_1: 13;

      

       A10: 1 <= (j + 1) by NAT_1: 11;

      (i + 1) <= (i + 2) by XREAL_1: 6;

      then

       A11: 1 <= (i + 2) by A8, XXREAL_0: 2;

      per cases by A1, A3, A4, A5, A6, A10, Th56, Th58;

        suppose

         A12: (f /. 2) = (( GoB f) * (i,(j + 1))) & (f /. 2) = (( GoB f) * ((i + 1),j));

         [i, (j + 1)] in ( Indices ( GoB f)) & [(i + 1), j] in ( Indices ( GoB f)) by A1, A2, A3, A4, A6, A10, A9, A8, MATRIX_0: 30;

        then j = (j + 1) by A12, GOBOARD1: 5;

        hence contradiction;

      end;

        suppose

         A13: (f /. (( len f) -' 1)) = (( GoB f) * ((i + 2),(j + 1))) & (f /. (( len f) -' 1)) = (( GoB f) * ((i + 1),j));

         [(i + 2), (j + 1)] in ( Indices ( GoB f)) & [(i + 1), j] in ( Indices ( GoB f)) by A1, A2, A4, A6, A10, A8, A7, A11, MATRIX_0: 30;

        then j = (j + 1) by A13, GOBOARD1: 5;

        hence contradiction;

      end;

        suppose

         A14: (f /. 2) = (( GoB f) * ((i + 2),(j + 1))) & (f /. 2) = (( GoB f) * ((i + 1),j));

         [(i + 2), (j + 1)] in ( Indices ( GoB f)) & [(i + 1), j] in ( Indices ( GoB f)) by A1, A2, A4, A6, A10, A8, A7, A11, MATRIX_0: 30;

        then j = (j + 1) by A14, GOBOARD1: 5;

        hence contradiction;

      end;

        suppose

         A15: (f /. 2) = (( GoB f) * ((i + 2),(j + 1))) & (f /. 2) = (( GoB f) * (i,(j + 1)));

         [(i + 2), (j + 1)] in ( Indices ( GoB f)) & [i, (j + 1)] in ( Indices ( GoB f)) by A3, A6, A10, A9, A7, A11, MATRIX_0: 30;

        then i = (i + 2) by A15, GOBOARD1: 5;

        hence contradiction;

      end;

        suppose that

         A16: (f /. 1) = (( GoB f) * ((i + 1),(j + 1))) and

         A17: ex k st 1 <= k & (k + 1) < ( len f) & (f /. (k + 1)) = (( GoB f) * ((i + 1),(j + 1))) & ((f /. k) = (( GoB f) * ((i + 1),j)) & (f /. (k + 2)) = (( GoB f) * (i,(j + 1))) or (f /. k) = (( GoB f) * (i,(j + 1))) & (f /. (k + 2)) = (( GoB f) * ((i + 1),j)));

        consider k such that

         A18: 1 <= k and

         A19: (k + 1) < ( len f) & (f /. (k + 1)) = (( GoB f) * ((i + 1),(j + 1))) and (f /. k) = (( GoB f) * ((i + 1),j)) & (f /. (k + 2)) = (( GoB f) * (i,(j + 1))) or (f /. k) = (( GoB f) * (i,(j + 1))) & (f /. (k + 2)) = (( GoB f) * ((i + 1),j)) by A17;

        1 < (k + 1) by A18, NAT_1: 13;

        hence contradiction by A16, A19, Th36;

      end;

        suppose that

         A20: ex k st 1 <= k & (k + 1) < ( len f) & (f /. (k + 1)) = (( GoB f) * ((i + 1),(j + 1))) & ((f /. k) = (( GoB f) * (i,(j + 1))) & (f /. (k + 2)) = (( GoB f) * ((i + 2),(j + 1))) or (f /. k) = (( GoB f) * ((i + 2),(j + 1))) & (f /. (k + 2)) = (( GoB f) * (i,(j + 1)))) and

         A21: (f /. 1) = (( GoB f) * ((i + 1),(j + 1)));

        consider k such that

         A22: 1 <= k and

         A23: (k + 1) < ( len f) & (f /. (k + 1)) = (( GoB f) * ((i + 1),(j + 1))) and (f /. k) = (( GoB f) * (i,(j + 1))) & (f /. (k + 2)) = (( GoB f) * ((i + 2),(j + 1))) or (f /. k) = (( GoB f) * ((i + 2),(j + 1))) & (f /. (k + 2)) = (( GoB f) * (i,(j + 1))) by A20;

        1 < (k + 1) by A22, NAT_1: 13;

        hence contradiction by A21, A23, Th36;

      end;

        suppose that

         A24: ex k st 1 <= k & (k + 1) < ( len f) & (f /. (k + 1)) = (( GoB f) * ((i + 1),(j + 1))) & ((f /. k) = (( GoB f) * (i,(j + 1))) & (f /. (k + 2)) = (( GoB f) * ((i + 2),(j + 1))) or (f /. k) = (( GoB f) * ((i + 2),(j + 1))) & (f /. (k + 2)) = (( GoB f) * (i,(j + 1)))) and

         A25: ex k st 1 <= k & (k + 1) < ( len f) & (f /. (k + 1)) = (( GoB f) * ((i + 1),(j + 1))) & ((f /. k) = (( GoB f) * ((i + 1),j)) & (f /. (k + 2)) = (( GoB f) * (i,(j + 1))) or (f /. k) = (( GoB f) * (i,(j + 1))) & (f /. (k + 2)) = (( GoB f) * ((i + 1),j)));

        consider k1 such that 1 <= k1 and

         A26: (k1 + 1) < ( len f) and

         A27: (f /. (k1 + 1)) = (( GoB f) * ((i + 1),(j + 1))) and

         A28: (f /. k1) = (( GoB f) * (i,(j + 1))) & (f /. (k1 + 2)) = (( GoB f) * ((i + 2),(j + 1))) or (f /. k1) = (( GoB f) * ((i + 2),(j + 1))) & (f /. (k1 + 2)) = (( GoB f) * (i,(j + 1))) by A24;

        consider k2 such that 1 <= k2 and

         A29: (k2 + 1) < ( len f) and

         A30: (f /. (k2 + 1)) = (( GoB f) * ((i + 1),(j + 1))) and

         A31: (f /. k2) = (( GoB f) * ((i + 1),j)) & (f /. (k2 + 2)) = (( GoB f) * (i,(j + 1))) or (f /. k2) = (( GoB f) * (i,(j + 1))) & (f /. (k2 + 2)) = (( GoB f) * ((i + 1),j)) by A25;

         A32:

        now

          assume

           A33: k1 <> k2;

          per cases by A33, XXREAL_0: 1;

            suppose k1 < k2;

            then (k1 + 1) < (k2 + 1) by XREAL_1: 6;

            hence contradiction by A27, A29, A30, Th36, NAT_1: 11;

          end;

            suppose k2 < k1;

            then (k2 + 1) < (k1 + 1) by XREAL_1: 6;

            hence contradiction by A26, A27, A30, Th36, NAT_1: 11;

          end;

        end;

        now

          per cases by A28, A31;

            suppose

             A34: (f /. k1) = (( GoB f) * (i,(j + 1))) & (f /. k2) = (( GoB f) * ((i + 1),j));

             [i, (j + 1)] in ( Indices ( GoB f)) & [(i + 1), j] in ( Indices ( GoB f)) by A1, A2, A3, A4, A6, A10, A9, A8, MATRIX_0: 30;

            then j = (j + 1) by A32, A34, GOBOARD1: 5;

            hence contradiction;

          end;

            suppose

             A35: (f /. (k1 + 2)) = (( GoB f) * ((i + 2),(j + 1))) & (f /. (k2 + 2)) = (( GoB f) * ((i + 1),j));

             [(i + 2), (j + 1)] in ( Indices ( GoB f)) & [(i + 1), j] in ( Indices ( GoB f)) by A1, A2, A4, A6, A10, A8, A7, A11, MATRIX_0: 30;

            then j = (j + 1) by A32, A35, GOBOARD1: 5;

            hence contradiction;

          end;

            suppose

             A36: (f /. k1) = (( GoB f) * ((i + 2),(j + 1))) & (f /. k2) = (( GoB f) * ((i + 1),j));

             [(i + 2), (j + 1)] in ( Indices ( GoB f)) & [(i + 1), j] in ( Indices ( GoB f)) by A1, A2, A4, A6, A10, A8, A7, A11, MATRIX_0: 30;

            then j = (j + 1) by A32, A36, GOBOARD1: 5;

            hence contradiction;

          end;

            suppose

             A37: (f /. k1) = (( GoB f) * ((i + 2),(j + 1))) & (f /. k2) = (( GoB f) * (i,(j + 1)));

             [(i + 2), (j + 1)] in ( Indices ( GoB f)) & [i, (j + 1)] in ( Indices ( GoB f)) by A3, A6, A10, A9, A7, A11, MATRIX_0: 30;

            then i = (i + 2) by A32, A37, GOBOARD1: 5;

            hence contradiction;

          end;

        end;

        hence contradiction;

      end;

    end;

    theorem :: GOBOARD7:63

    for p,q,p1,q1 be Point of ( TOP-REAL 2) st ( LSeg (p,q)) is vertical & ( LSeg (p1,q1)) is vertical & (p `1 ) = (p1 `1 ) & (p `2 ) <= (p1 `2 ) & (p1 `2 ) <= (q1 `2 ) & (q1 `2 ) <= (q `2 ) holds ( LSeg (p1,q1)) c= ( LSeg (p,q))

    proof

      let p,q,p1,q1 be Point of ( TOP-REAL 2);

      assume that

       A1: ( LSeg (p,q)) is vertical and

       A2: ( LSeg (p1,q1)) is vertical and

       A3: (p `1 ) = (p1 `1 ) and

       A4: (p `2 ) <= (p1 `2 ) and

       A5: (p1 `2 ) <= (q1 `2 ) and

       A6: (q1 `2 ) <= (q `2 );

      

       A7: (p `1 ) = (q `1 ) by A1, SPPOL_1: 16;

      let x be object;

      assume

       A8: x in ( LSeg (p1,q1));

      then

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

      (p1 `2 ) <= (r `2 ) by A5, A8, TOPREAL1: 4;

      then

       A9: (p `2 ) <= (r `2 ) by A4, XXREAL_0: 2;

      (r `2 ) <= (q1 `2 ) by A5, A8, TOPREAL1: 4;

      then

       A10: (r `2 ) <= (q `2 ) by A6, XXREAL_0: 2;

      (p1 `1 ) = (r `1 ) by A2, A8, SPPOL_1: 41;

      hence thesis by A3, A7, A9, A10, Th7;

    end;

    theorem :: GOBOARD7:64

    for p,q,p1,q1 be Point of ( TOP-REAL 2) st ( LSeg (p,q)) is horizontal & ( LSeg (p1,q1)) is horizontal & (p `2 ) = (p1 `2 ) & (p `1 ) <= (p1 `1 ) & (p1 `1 ) <= (q1 `1 ) & (q1 `1 ) <= (q `1 ) holds ( LSeg (p1,q1)) c= ( LSeg (p,q))

    proof

      let p,q,p1,q1 be Point of ( TOP-REAL 2);

      assume that

       A1: ( LSeg (p,q)) is horizontal and

       A2: ( LSeg (p1,q1)) is horizontal and

       A3: (p `2 ) = (p1 `2 ) and

       A4: (p `1 ) <= (p1 `1 ) and

       A5: (p1 `1 ) <= (q1 `1 ) and

       A6: (q1 `1 ) <= (q `1 );

      

       A7: (p `2 ) = (q `2 ) by A1, SPPOL_1: 15;

      let x be object;

      assume

       A8: x in ( LSeg (p1,q1));

      then

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

      (p1 `1 ) <= (r `1 ) by A5, A8, TOPREAL1: 3;

      then

       A9: (p `1 ) <= (r `1 ) by A4, XXREAL_0: 2;

      (r `1 ) <= (q1 `1 ) by A5, A8, TOPREAL1: 3;

      then

       A10: (r `1 ) <= (q `1 ) by A6, XXREAL_0: 2;

      (p1 `2 ) = (r `2 ) by A2, A8, SPPOL_1: 40;

      hence thesis by A3, A7, A9, A10, Th8;

    end;