goboard6.miz



    begin

    reserve n for Nat,

i,j for Nat,

r,s,r1,s1,r2,s2,r9,s9 for Real,

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

G for Go-board,

x,y for set,

v for Point of ( Euclid 2);

    

     Lm1: ((p + q) `1 ) = ((p `1 ) + (q `1 )) & ((p + q) `2 ) = ((p `2 ) + (q `2 ))

    proof

      (p + q) = |[((p `1 ) + (q `1 )), ((p `2 ) + (q `2 ))]| by EUCLID: 55;

      hence thesis by EUCLID: 52;

    end;

    

     Lm2: ((p - q) `1 ) = ((p `1 ) - (q `1 )) & ((p - q) `2 ) = ((p `2 ) - (q `2 ))

    proof

      (p - q) = |[((p `1 ) - (q `1 )), ((p `2 ) - (q `2 ))]| by EUCLID: 61;

      hence thesis by EUCLID: 52;

    end;

    

     Lm3: ((r * p) `1 ) = (r * (p `1 )) & ((r * p) `2 ) = (r * (p `2 ))

    proof

      (r * p) = |[(r * (p `1 )), (r * (p `2 ))]| by EUCLID: 57;

      hence thesis by EUCLID: 52;

    end;

    theorem :: GOBOARD6:1

    

     Th1: for M be non empty Reflexive MetrStruct, u be Point of M, r be Real holds r > 0 implies u in ( Ball (u,r))

    proof

      let M be non empty Reflexive MetrStruct, u be Point of M, r be Real;

      

       A1: ( Ball (u,r)) = { q where q be Point of M : ( dist (u,q)) < r } & ( dist (u,u)) = 0 by METRIC_1: 1, METRIC_1: 17;

      assume r > 0 ;

      hence thesis by A1;

    end;

    

     Lm4: for M be MetrSpace, B be Subset of ( TopSpaceMetr M), r be Real, u be Point of M st B = ( Ball (u,r)) holds B is open

    proof

      let M be MetrSpace, B be Subset of ( TopSpaceMetr M), r be Real, u be Point of M;

      

       A1: ( TopSpaceMetr M) = TopStruct (# the carrier of M, ( Family_open_set M) #) & ( Ball (u,r)) in ( Family_open_set M) by PCOMPS_1: 29, PCOMPS_1:def 5;

      assume B = ( Ball (u,r));

      hence thesis by A1, PRE_TOPC:def 2;

    end;

    theorem :: GOBOARD6:2

    

     Th2: for p be Point of ( Euclid n), q be Point of ( TOP-REAL n), r be Real st p = q & r > 0 holds ( Ball (p,r)) is a_neighborhood of q

    proof

      let p be Point of ( Euclid n), q be Point of ( TOP-REAL n), r be Real;

      reconsider A = ( Ball (p,r)) as Subset of ( TOP-REAL n) by TOPREAL3: 8;

      

       A1: the TopStruct of ( TOP-REAL n) = ( TopSpaceMetr ( Euclid n)) by EUCLID:def 8;

      then

      reconsider AA = A as Subset of ( TopSpaceMetr ( Euclid n));

      AA is open by TOPMETR: 14;

      then

       A2: A is open by A1, PRE_TOPC: 30;

      assume p = q & r > 0 ;

      hence thesis by A2, Th1, CONNSP_2: 3;

    end;

    theorem :: GOBOARD6:3

    

     Th3: for B be Subset of ( TOP-REAL n), u be Point of ( Euclid n) st B = ( Ball (u,r)) holds B is open

    proof

      let B be Subset of ( TOP-REAL n), u be Point of ( Euclid n);

      

       A1: the TopStruct of ( TOP-REAL n) = ( TopSpaceMetr ( Euclid n)) by EUCLID:def 8;

      then

      reconsider BB = B as Subset of ( TopSpaceMetr ( Euclid n));

      assume B = ( Ball (u,r));

      then BB is open by Lm4;

      hence thesis by A1, PRE_TOPC: 30;

    end;

    theorem :: GOBOARD6:4

    

     Th4: for M be non empty MetrSpace, u be Point of M, P be Subset of ( TopSpaceMetr M) holds u in ( Int P) iff ex r be Real st r > 0 & ( Ball (u,r)) c= P

    proof

      let M be non empty MetrSpace, u be Point of M, P be Subset of ( TopSpaceMetr M);

      hereby

        assume u in ( Int P);

        then

        consider r be Real such that

         A1: r > 0 and

         A2: ( Ball (u,r)) c= ( Int P) by TOPMETR: 15;

        take r;

        thus r > 0 by A1;

        ( Int P) c= P by TOPS_1: 16;

        hence ( Ball (u,r)) c= P by A2;

      end;

      given r be Real such that

       A3: r > 0 and

       A4: ( Ball (u,r)) c= P;

      ( TopSpaceMetr M) = TopStruct (# the carrier of M, ( Family_open_set M) #) by PCOMPS_1:def 5;

      then

      reconsider B = ( Ball (u,r)) as Subset of ( TopSpaceMetr M);

      

       A5: B is open by Lm4;

      u in ( Ball (u,r)) by A3, Th1;

      hence thesis by A4, A5, TOPS_1: 22;

    end;

    

     Lm5: for T be TopSpace, A be Subset of T, B be Subset of the TopStruct of T st A = B holds ( Int A) = ( Int B)

    proof

      let T be TopSpace, A be Subset of T, B be Subset of the TopStruct of T such that

       A1: A = B;

      reconsider AA = ( Int A) as Subset of the TopStruct of T;

      AA is open by PRE_TOPC: 30;

      hence ( Int A) c= ( Int B) by A1, TOPS_1: 16, TOPS_1: 24;

      reconsider BB = ( Int B) as Subset of T;

      BB is open by PRE_TOPC: 30;

      hence ( Int B) c= ( Int A) by A1, TOPS_1: 16, TOPS_1: 24;

    end;

    theorem :: GOBOARD6:5

    

     Th5: for u be Point of ( Euclid n), P be Subset of ( TOP-REAL n) holds u in ( Int P) iff ex r be Real st r > 0 & ( Ball (u,r)) c= P

    proof

      let u be Point of ( Euclid n), P be Subset of ( TOP-REAL n);

      

       A1: the TopStruct of ( TOP-REAL n) = ( TopSpaceMetr ( Euclid n)) by EUCLID:def 8;

      then

      reconsider PP = P as Subset of ( TopSpaceMetr ( Euclid n));

      u in ( Int PP) iff ex r be Real st r > 0 & ( Ball (u,r)) c= PP by Th4;

      hence thesis by A1, Lm5;

    end;

    theorem :: GOBOARD6:6

    

     Th6: for u,v be Point of ( Euclid 2) st u = |[r1, s1]| & v = |[r2, s2]| holds ( dist (u,v)) = ( sqrt (((r1 - r2) ^2 ) + ((s1 - s2) ^2 )))

    proof

      let u,v be Point of ( Euclid 2) such that

       A1: u = |[r1, s1]| & v = |[r2, s2]|;

      

       A2: ( |[r1, s1]| `1 ) = r1 & ( |[r1, s1]| `2 ) = s1 by EUCLID: 52;

      

       A3: ( |[r2, s2]| `1 ) = r2 & ( |[r2, s2]| `2 ) = s2 by EUCLID: 52;

      

      thus ( dist (u,v)) = (( Pitag_dist 2) . (u,v)) by METRIC_1:def 1

      .= ( sqrt (((r1 - r2) ^2 ) + ((s1 - s2) ^2 ))) by A1, A2, A3, TOPREAL3: 7;

    end;

    theorem :: GOBOARD6:7

    

     Th7: for u be Point of ( Euclid 2) st u = |[r, s]| holds 0 <= r2 & r2 < r1 implies |[(r + r2), s]| in ( Ball (u,r1))

    proof

      let u be Point of ( Euclid 2) such that

       A1: u = |[r, s]| and

       A2: 0 <= r2 and

       A3: r2 < r1;

      reconsider v = |[(r + r2), s]| as Point of ( Euclid 2) by TOPREAL3: 8;

      ( dist (u,v)) = ( sqrt (((r - (r + r2)) ^2 ) + ((s - s) ^2 ))) by A1, Th6

      .= ( sqrt (( - (r - (r + r2))) ^2 ))

      .= r2 by A2, SQUARE_1: 22;

      hence thesis by A3, METRIC_1: 11;

    end;

    theorem :: GOBOARD6:8

    

     Th8: for u be Point of ( Euclid 2) st u = |[r, s]| holds 0 <= s2 & s2 < s1 implies |[r, (s + s2)]| in ( Ball (u,s1))

    proof

      let u be Point of ( Euclid 2) such that

       A1: u = |[r, s]| and

       A2: 0 <= s2 and

       A3: s2 < s1;

      reconsider v = |[r, (s + s2)]| as Point of ( Euclid 2) by TOPREAL3: 8;

      ( dist (u,v)) = ( sqrt (((r - r) ^2 ) + ((s - (s + s2)) ^2 ))) by A1, Th6

      .= ( sqrt (( - (s - (s + s2))) ^2 ))

      .= s2 by A2, SQUARE_1: 22;

      hence thesis by A3, METRIC_1: 11;

    end;

    theorem :: GOBOARD6:9

    

     Th9: for u be Point of ( Euclid 2) st u = |[r, s]| holds 0 <= r2 & r2 < r1 implies |[(r - r2), s]| in ( Ball (u,r1))

    proof

      let u be Point of ( Euclid 2) such that

       A1: u = |[r, s]| and

       A2: 0 <= r2 and

       A3: r2 < r1;

      reconsider v = |[(r - r2), s]| as Point of ( Euclid 2) by TOPREAL3: 8;

      ( dist (u,v)) = ( sqrt (((r - (r - r2)) ^2 ) + ((s - s) ^2 ))) by A1, Th6

      .= r2 by A2, SQUARE_1: 22;

      hence thesis by A3, METRIC_1: 11;

    end;

    theorem :: GOBOARD6:10

    

     Th10: for u be Point of ( Euclid 2) st u = |[r, s]| holds 0 <= s2 & s2 < s1 implies |[r, (s - s2)]| in ( Ball (u,s1))

    proof

      let u be Point of ( Euclid 2) such that

       A1: u = |[r, s]| and

       A2: 0 <= s2 and

       A3: s2 < s1;

      reconsider v = |[r, (s - s2)]| as Point of ( Euclid 2) by TOPREAL3: 8;

      ( dist (u,v)) = ( sqrt (((s - (s - s2)) ^2 ) + ((r - r) ^2 ))) by A1, Th6

      .= s2 by A2, SQUARE_1: 22;

      hence thesis by A3, METRIC_1: 11;

    end;

    theorem :: GOBOARD6:11

    

     Th11: 1 <= i & i < ( len G) & 1 <= j & j < ( width G) implies ((G * (i,j)) + (G * ((i + 1),(j + 1)))) = ((G * (i,(j + 1))) + (G * ((i + 1),j)))

    proof

      assume that

       A1: 1 <= i & i < ( len G) and

       A2: 1 <= j & j < ( width G);

      

       A3: 1 <= (j + 1) & (j + 1) <= ( width G) by A2, NAT_1: 13;

      

       A4: 1 <= (i + 1) & (i + 1) <= ( len G) by A1, NAT_1: 13;

      

      then

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

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

      

       A6: ((G * ((i + 1),(j + 1))) `2 ) = ((G * (1,(j + 1))) `2 ) by A4, A3, GOBOARD5: 1

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

      

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

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

      

       A8: (((G * (i,j)) + (G * ((i + 1),(j + 1)))) `2 ) = (((G * (i,j)) `2 ) + ((G * ((i + 1),(j + 1))) `2 )) by Lm1

      .= (((G * (i,(j + 1))) + (G * ((i + 1),j))) `2 ) by A7, A6, Lm1;

      

       A9: ((G * (i,j)) `1 ) = ((G * (i,1)) `1 ) by A1, A2, GOBOARD5: 2

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

      (((G * (i,j)) + (G * ((i + 1),(j + 1)))) `1 ) = (((G * (i,j)) `1 ) + ((G * ((i + 1),(j + 1))) `1 )) by Lm1

      .= (((G * (i,(j + 1))) + (G * ((i + 1),j))) `1 ) by A9, A5, Lm1;

      

      hence ((G * (i,j)) + (G * ((i + 1),(j + 1)))) = |[(((G * (i,(j + 1))) + (G * ((i + 1),j))) `1 ), (((G * (i,(j + 1))) + (G * ((i + 1),j))) `2 )]| by A8, EUCLID: 53

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

    end;

    

     Lm6: the TopStruct of ( TOP-REAL 2) = ( TopSpaceMetr ( Euclid 2)) by EUCLID:def 8

    .= TopStruct (# the carrier of ( Euclid 2), ( Family_open_set ( Euclid 2)) #) by PCOMPS_1:def 5;

    theorem :: GOBOARD6:12

    

     Th12: ( Int ( v_strip (G, 0 ))) = { |[r, s]| : r < ((G * (1,1)) `1 ) }

    proof

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

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

      then

       A1: ( v_strip (G, 0 )) = { |[r, s]| : r <= ((G * (1,1)) `1 ) } by GOBOARD5: 10;

      thus ( Int ( v_strip (G, 0 ))) c= { |[r, s]| : r < ((G * (1,1)) `1 ) }

      proof

        let x be object;

        assume

         A2: x in ( Int ( v_strip (G, 0 )));

        then

        reconsider u = x as Point of ( Euclid 2) by Lm6;

        consider r1 be Real such that

         A3: r1 > 0 and

         A4: ( Ball (u,r1)) c= ( v_strip (G, 0 )) by A2, Th5;

        reconsider p = u as Point of ( TOP-REAL 2) by Lm6;

        

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

        set q = |[((p `1 ) + (r1 / 2)), ((p `2 ) + 0 )]|;

        (r1 / 2) < r1 by A3, XREAL_1: 216;

        then q in ( Ball (u,r1)) by A3, A5, Th7;

        then q in ( v_strip (G, 0 )) by A4;

        then ex r2, s2 st q = |[r2, s2]| & r2 <= ((G * (1,1)) `1 ) by A1;

        then

         A6: ((p `1 ) + (r1 / 2)) <= ((G * (1,1)) `1 ) by SPPOL_2: 1;

        (p `1 ) < ((p `1 ) + (r1 / 2)) by A3, XREAL_1: 29, XREAL_1: 215;

        then (p `1 ) < ((G * (1,1)) `1 ) by A6, XXREAL_0: 2;

        hence thesis by A5;

      end;

      let x be object;

      assume x in { |[r, s]| : r < ((G * (1,1)) `1 ) };

      then

      consider r, s such that

       A7: x = |[r, s]| and

       A8: r < ((G * (1,1)) `1 );

      reconsider u = |[r, s]| as Point of ( Euclid 2) by TOPREAL3: 8;

      

       A9: ( Ball (u,(((G * (1,1)) `1 ) - r))) c= ( v_strip (G, 0 ))

      proof

        let y be object;

        

         A10: ( Ball (u,(((G * (1,1)) `1 ) - r))) = { v : ( dist (u,v)) < (((G * (1,1)) `1 ) - r) } by METRIC_1: 17;

        assume y in ( Ball (u,(((G * (1,1)) `1 ) - r)));

        then

        consider v such that

         A11: v = y and

         A12: ( dist (u,v)) < (((G * (1,1)) `1 ) - r) by A10;

        reconsider q = v as Point of ( TOP-REAL 2) by TOPREAL3: 8;

        ((r - (q `1 )) ^2 ) >= 0 & (((r - (q `1 )) ^2 ) + 0 ) <= (((r - (q `1 )) ^2 ) + ((s - (q `2 )) ^2 )) by XREAL_1: 6, XREAL_1: 63;

        then

         A13: ( sqrt ((r - (q `1 )) ^2 )) <= ( sqrt (((r - (q `1 )) ^2 ) + ((s - (q `2 )) ^2 ))) by SQUARE_1: 26;

        

         A14: q = |[(q `1 ), (q `2 )]| by EUCLID: 53;

        then ( sqrt (((r - (q `1 )) ^2 ) + ((s - (q `2 )) ^2 ))) < (((G * (1,1)) `1 ) - r) by A12, Th6;

        then ( sqrt ((r - (q `1 )) ^2 )) <= (((G * (1,1)) `1 ) - r) by A13, XXREAL_0: 2;

        then

         A15: |.(r - (q `1 )).| <= (((G * (1,1)) `1 ) - r) by COMPLEX1: 72;

        per cases ;

          suppose r <= (q `1 );

          then

           A16: ((q `1 ) - r) >= 0 by XREAL_1: 48;

           |.(r - (q `1 )).| = |.( - (r - (q `1 ))).| by COMPLEX1: 52

          .= ((q `1 ) - r) by A16, ABSVALUE:def 1;

          then (q `1 ) <= ((G * (1,1)) `1 ) by A15, XREAL_1: 9;

          hence thesis by A1, A11, A14;

        end;

          suppose r >= (q `1 );

          then (q `1 ) <= ((G * (1,1)) `1 ) by A8, XXREAL_0: 2;

          hence thesis by A1, A11, A14;

        end;

      end;

      reconsider B = ( Ball (u,(((G * (1,1)) `1 ) - r))) as Subset of ( TOP-REAL 2) by TOPREAL3: 8;

      

       A17: B is open by Th3;

      u in ( Ball (u,(((G * (1,1)) `1 ) - r))) by A8, Th1, XREAL_1: 50;

      hence thesis by A7, A9, A17, TOPS_1: 22;

    end;

    theorem :: GOBOARD6:13

    

     Th13: ( Int ( v_strip (G,( len G)))) = { |[r, s]| : ((G * (( len G),1)) `1 ) < r }

    proof

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

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

      then

       A1: ( v_strip (G,( len G))) = { |[r, s]| : ((G * (( len G),1)) `1 ) <= r } by GOBOARD5: 9;

      thus ( Int ( v_strip (G,( len G)))) c= { |[r, s]| : ((G * (( len G),1)) `1 ) < r }

      proof

        let x be object;

        assume

         A2: x in ( Int ( v_strip (G,( len G))));

        then

        reconsider u = x as Point of ( Euclid 2) by Lm6;

        consider r1 be Real such that

         A3: r1 > 0 and

         A4: ( Ball (u,r1)) c= ( v_strip (G,( len G))) by A2, Th5;

        reconsider p = u as Point of ( TOP-REAL 2) by Lm6;

        

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

        set q = |[((p `1 ) - (r1 / 2)), ((p `2 ) + 0 )]|;

        (r1 / 2) < r1 by A3, XREAL_1: 216;

        then q in ( Ball (u,r1)) by A3, A5, Th9;

        then q in ( v_strip (G,( len G))) by A4;

        then ex r2, s2 st q = |[r2, s2]| & ((G * (( len G),1)) `1 ) <= r2 by A1;

        then ((G * (( len G),1)) `1 ) <= ((p `1 ) - (r1 / 2)) by SPPOL_2: 1;

        then

         A6: (((G * (( len G),1)) `1 ) + (r1 / 2)) <= (p `1 ) by XREAL_1: 19;

        ((G * (( len G),1)) `1 ) < (((G * (( len G),1)) `1 ) + (r1 / 2)) by A3, XREAL_1: 29, XREAL_1: 215;

        then ((G * (( len G),1)) `1 ) < (p `1 ) by A6, XXREAL_0: 2;

        hence thesis by A5;

      end;

      let x be object;

      assume x in { |[r, s]| : ((G * (( len G),1)) `1 ) < r };

      then

      consider r, s such that

       A7: x = |[r, s]| and

       A8: ((G * (( len G),1)) `1 ) < r;

      reconsider u = |[r, s]| as Point of ( Euclid 2) by TOPREAL3: 8;

      

       A9: ( Ball (u,(r - ((G * (( len G),1)) `1 )))) c= ( v_strip (G,( len G)))

      proof

        let y be object;

        

         A10: ( Ball (u,(r - ((G * (( len G),1)) `1 )))) = { v : ( dist (u,v)) < (r - ((G * (( len G),1)) `1 )) } by METRIC_1: 17;

        assume y in ( Ball (u,(r - ((G * (( len G),1)) `1 ))));

        then

        consider v such that

         A11: v = y and

         A12: ( dist (u,v)) < (r - ((G * (( len G),1)) `1 )) by A10;

        reconsider q = v as Point of ( TOP-REAL 2) by TOPREAL3: 8;

        ((r - (q `1 )) ^2 ) >= 0 & (((r - (q `1 )) ^2 ) + 0 ) <= (((r - (q `1 )) ^2 ) + ((s - (q `2 )) ^2 )) by XREAL_1: 6, XREAL_1: 63;

        then

         A13: ( sqrt ((r - (q `1 )) ^2 )) <= ( sqrt (((r - (q `1 )) ^2 ) + ((s - (q `2 )) ^2 ))) by SQUARE_1: 26;

        

         A14: q = |[(q `1 ), (q `2 )]| by EUCLID: 53;

        then ( sqrt (((r - (q `1 )) ^2 ) + ((s - (q `2 )) ^2 ))) < (r - ((G * (( len G),1)) `1 )) by A12, Th6;

        then ( sqrt ((r - (q `1 )) ^2 )) <= (r - ((G * (( len G),1)) `1 )) by A13, XXREAL_0: 2;

        then

         A15: |.(r - (q `1 )).| <= (r - ((G * (( len G),1)) `1 )) by COMPLEX1: 72;

        per cases ;

          suppose r >= (q `1 );

          then (r - (q `1 )) >= 0 by XREAL_1: 48;

          then |.(r - (q `1 )).| = (r - (q `1 )) by ABSVALUE:def 1;

          then ((G * (( len G),1)) `1 ) <= (q `1 ) by A15, XREAL_1: 10;

          hence thesis by A1, A11, A14;

        end;

          suppose r <= (q `1 );

          then ((G * (( len G),1)) `1 ) <= (q `1 ) by A8, XXREAL_0: 2;

          hence thesis by A1, A11, A14;

        end;

      end;

      reconsider B = ( Ball (u,(r - ((G * (( len G),1)) `1 )))) as Subset of ( TOP-REAL 2) by TOPREAL3: 8;

      

       A16: B is open by Th3;

      u in ( Ball (u,(r - ((G * (( len G),1)) `1 )))) by A8, Th1, XREAL_1: 50;

      hence thesis by A7, A9, A16, TOPS_1: 22;

    end;

    theorem :: GOBOARD6:14

    

     Th14: 1 <= i & i < ( len G) implies ( Int ( v_strip (G,i))) = { |[r, s]| : ((G * (i,1)) `1 ) < r & r < ((G * ((i + 1),1)) `1 ) }

    proof

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

      then

       A1: 1 <= ( width G) by NAT_1: 14;

      assume 1 <= i & i < ( len G);

      then

       A2: ( v_strip (G,i)) = { |[r, s]| : ((G * (i,1)) `1 ) <= r & r <= ((G * ((i + 1),1)) `1 ) } by A1, GOBOARD5: 8;

      thus ( Int ( v_strip (G,i))) c= { |[r, s]| : ((G * (i,1)) `1 ) < r & r < ((G * ((i + 1),1)) `1 ) }

      proof

        let x be object;

        assume

         A3: x in ( Int ( v_strip (G,i)));

        then

        reconsider u = x as Point of ( Euclid 2) by Lm6;

        consider r1 be Real such that

         A4: r1 > 0 and

         A5: ( Ball (u,r1)) c= ( v_strip (G,i)) by A3, Th5;

        reconsider p = u as Point of ( TOP-REAL 2) by Lm6;

        

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

        set q2 = |[((p `1 ) - (r1 / 2)), ((p `2 ) + 0 )]|;

        

         A7: (r1 / 2) < r1 by A4, XREAL_1: 216;

        then q2 in ( Ball (u,r1)) by A4, A6, Th9;

        then q2 in ( v_strip (G,i)) by A5;

        then ex r2, s2 st q2 = |[r2, s2]| & ((G * (i,1)) `1 ) <= r2 & r2 <= ((G * ((i + 1),1)) `1 ) by A2;

        then ((G * (i,1)) `1 ) <= ((p `1 ) - (r1 / 2)) by SPPOL_2: 1;

        then

         A8: (((G * (i,1)) `1 ) + (r1 / 2)) <= (p `1 ) by XREAL_1: 19;

        set q1 = |[((p `1 ) + (r1 / 2)), ((p `2 ) + 0 )]|;

        q1 in ( Ball (u,r1)) by A4, A6, A7, Th7;

        then q1 in ( v_strip (G,i)) by A5;

        then ex r2, s2 st q1 = |[r2, s2]| & ((G * (i,1)) `1 ) <= r2 & r2 <= ((G * ((i + 1),1)) `1 ) by A2;

        then

         A9: ((p `1 ) + (r1 / 2)) <= ((G * ((i + 1),1)) `1 ) by SPPOL_2: 1;

        ((G * (i,1)) `1 ) < (((G * (i,1)) `1 ) + (r1 / 2)) by A4, XREAL_1: 29, XREAL_1: 215;

        then

         A10: ((G * (i,1)) `1 ) < (p `1 ) by A8, XXREAL_0: 2;

        (p `1 ) < ((p `1 ) + (r1 / 2)) by A4, XREAL_1: 29, XREAL_1: 215;

        then (p `1 ) < ((G * ((i + 1),1)) `1 ) by A9, XXREAL_0: 2;

        hence thesis by A6, A10;

      end;

      let x be object;

      assume x in { |[r, s]| : ((G * (i,1)) `1 ) < r & r < ((G * ((i + 1),1)) `1 ) };

      then

      consider r, s such that

       A11: x = |[r, s]| and

       A12: ((G * (i,1)) `1 ) < r and

       A13: r < ((G * ((i + 1),1)) `1 );

      reconsider u = |[r, s]| as Point of ( Euclid 2) by TOPREAL3: 8;

      (((G * ((i + 1),1)) `1 ) - r) > 0 & (r - ((G * (i,1)) `1 )) > 0 by A12, A13, XREAL_1: 50;

      then ( min ((r - ((G * (i,1)) `1 )),(((G * ((i + 1),1)) `1 ) - r))) > 0 by XXREAL_0: 15;

      then

       A14: u in ( Ball (u,( min ((r - ((G * (i,1)) `1 )),(((G * ((i + 1),1)) `1 ) - r))))) by Th1;

      

       A15: ( Ball (u,( min ((r - ((G * (i,1)) `1 )),(((G * ((i + 1),1)) `1 ) - r))))) c= ( v_strip (G,i))

      proof

        let y be object;

        

         A16: ( Ball (u,( min ((r - ((G * (i,1)) `1 )),(((G * ((i + 1),1)) `1 ) - r))))) = { v : ( dist (u,v)) < ( min ((r - ((G * (i,1)) `1 )),(((G * ((i + 1),1)) `1 ) - r))) } by METRIC_1: 17;

        assume y in ( Ball (u,( min ((r - ((G * (i,1)) `1 )),(((G * ((i + 1),1)) `1 ) - r)))));

        then

        consider v such that

         A17: v = y and

         A18: ( dist (u,v)) < ( min ((r - ((G * (i,1)) `1 )),(((G * ((i + 1),1)) `1 ) - r))) by A16;

        reconsider q = v as Point of ( TOP-REAL 2) by TOPREAL3: 8;

        ((r - (q `1 )) ^2 ) >= 0 & (((r - (q `1 )) ^2 ) + 0 ) <= (((r - (q `1 )) ^2 ) + ((s - (q `2 )) ^2 )) by XREAL_1: 6, XREAL_1: 63;

        then

         A19: ( sqrt ((r - (q `1 )) ^2 )) <= ( sqrt (((r - (q `1 )) ^2 ) + ((s - (q `2 )) ^2 ))) by SQUARE_1: 26;

        

         A20: q = |[(q `1 ), (q `2 )]| by EUCLID: 53;

        then ( sqrt (((r - (q `1 )) ^2 ) + ((s - (q `2 )) ^2 ))) < ( min ((r - ((G * (i,1)) `1 )),(((G * ((i + 1),1)) `1 ) - r))) by A18, Th6;

        then ( sqrt ((r - (q `1 )) ^2 )) <= ( min ((r - ((G * (i,1)) `1 )),(((G * ((i + 1),1)) `1 ) - r))) by A19, XXREAL_0: 2;

        then

         A21: |.(r - (q `1 )).| <= ( min ((r - ((G * (i,1)) `1 )),(((G * ((i + 1),1)) `1 ) - r))) by COMPLEX1: 72;

        then

         A22: |.(r - (q `1 )).| <= (r - ((G * (i,1)) `1 )) by XXREAL_0: 22;

        

         A23: |.(r - (q `1 )).| <= (((G * ((i + 1),1)) `1 ) - r) by A21, XXREAL_0: 22;

        per cases ;

          suppose

           A24: r <= (q `1 );

          then

           A25: ((q `1 ) - r) >= 0 by XREAL_1: 48;

           |.(r - (q `1 )).| = |.( - (r - (q `1 ))).| by COMPLEX1: 52

          .= ((q `1 ) - r) by A25, ABSVALUE:def 1;

          then

           A26: (q `1 ) <= ((G * ((i + 1),1)) `1 ) by A23, XREAL_1: 9;

          ((G * (i,1)) `1 ) <= (q `1 ) by A12, A24, XXREAL_0: 2;

          hence thesis by A2, A17, A20, A26;

        end;

          suppose

           A27: r >= (q `1 );

          then (r - (q `1 )) >= 0 by XREAL_1: 48;

          then |.(r - (q `1 )).| = (r - (q `1 )) by ABSVALUE:def 1;

          then

           A28: ((G * (i,1)) `1 ) <= (q `1 ) by A22, XREAL_1: 10;

          (q `1 ) <= ((G * ((i + 1),1)) `1 ) by A13, A27, XXREAL_0: 2;

          hence thesis by A2, A17, A20, A28;

        end;

      end;

      reconsider B = ( Ball (u,( min ((r - ((G * (i,1)) `1 )),(((G * ((i + 1),1)) `1 ) - r))))) as Subset of ( TOP-REAL 2) by TOPREAL3: 8;

      B is open by Th3;

      hence thesis by A11, A14, A15, TOPS_1: 22;

    end;

    theorem :: GOBOARD6:15

    

     Th15: ( Int ( h_strip (G, 0 ))) = { |[r, s]| : s < ((G * (1,1)) `2 ) }

    proof

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

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

      then

       A1: ( h_strip (G, 0 )) = { |[r, s]| : s <= ((G * (1,1)) `2 ) } by GOBOARD5: 7;

      thus ( Int ( h_strip (G, 0 ))) c= { |[r, s]| : s < ((G * (1,1)) `2 ) }

      proof

        let x be object;

        assume

         A2: x in ( Int ( h_strip (G, 0 )));

        then

        reconsider u = x as Point of ( Euclid 2) by Lm6;

        consider s1 be Real such that

         A3: s1 > 0 and

         A4: ( Ball (u,s1)) c= ( h_strip (G, 0 )) by A2, Th5;

        reconsider p = u as Point of ( TOP-REAL 2) by Lm6;

        

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

        set q = |[((p `1 ) + 0 ), ((p `2 ) + (s1 / 2))]|;

        (s1 / 2) < s1 by A3, XREAL_1: 216;

        then q in ( Ball (u,s1)) by A3, A5, Th8;

        then q in ( h_strip (G, 0 )) by A4;

        then ex r2, s2 st q = |[r2, s2]| & s2 <= ((G * (1,1)) `2 ) by A1;

        then

         A6: ((p `2 ) + (s1 / 2)) <= ((G * (1,1)) `2 ) by SPPOL_2: 1;

        (p `2 ) < ((p `2 ) + (s1 / 2)) by A3, XREAL_1: 29, XREAL_1: 215;

        then (p `2 ) < ((G * (1,1)) `2 ) by A6, XXREAL_0: 2;

        hence thesis by A5;

      end;

      let x be object;

      assume x in { |[r, s]| : s < ((G * (1,1)) `2 ) };

      then

      consider r, s such that

       A7: x = |[r, s]| and

       A8: s < ((G * (1,1)) `2 );

      reconsider u = |[r, s]| as Point of ( Euclid 2) by TOPREAL3: 8;

      

       A9: ( Ball (u,(((G * (1,1)) `2 ) - s))) c= ( h_strip (G, 0 ))

      proof

        let y be object;

        

         A10: ( Ball (u,(((G * (1,1)) `2 ) - s))) = { v : ( dist (u,v)) < (((G * (1,1)) `2 ) - s) } by METRIC_1: 17;

        assume y in ( Ball (u,(((G * (1,1)) `2 ) - s)));

        then

        consider v such that

         A11: v = y and

         A12: ( dist (u,v)) < (((G * (1,1)) `2 ) - s) by A10;

        reconsider q = v as Point of ( TOP-REAL 2) by TOPREAL3: 8;

        ((s - (q `2 )) ^2 ) >= 0 & (((s - (q `2 )) ^2 ) + 0 ) <= (((r - (q `1 )) ^2 ) + ((s - (q `2 )) ^2 )) by XREAL_1: 6, XREAL_1: 63;

        then

         A13: ( sqrt ((s - (q `2 )) ^2 )) <= ( sqrt (((r - (q `1 )) ^2 ) + ((s - (q `2 )) ^2 ))) by SQUARE_1: 26;

        

         A14: q = |[(q `1 ), (q `2 )]| by EUCLID: 53;

        then ( sqrt (((r - (q `1 )) ^2 ) + ((s - (q `2 )) ^2 ))) < (((G * (1,1)) `2 ) - s) by A12, Th6;

        then ( sqrt ((s - (q `2 )) ^2 )) <= (((G * (1,1)) `2 ) - s) by A13, XXREAL_0: 2;

        then

         A15: |.(s - (q `2 )).| <= (((G * (1,1)) `2 ) - s) by COMPLEX1: 72;

        per cases ;

          suppose s <= (q `2 );

          then

           A16: ((q `2 ) - s) >= 0 by XREAL_1: 48;

           |.(s - (q `2 )).| = |.( - (s - (q `2 ))).| by COMPLEX1: 52

          .= ((q `2 ) - s) by A16, ABSVALUE:def 1;

          then (q `2 ) <= ((G * (1,1)) `2 ) by A15, XREAL_1: 9;

          hence thesis by A1, A11, A14;

        end;

          suppose s >= (q `2 );

          then (q `2 ) <= ((G * (1,1)) `2 ) by A8, XXREAL_0: 2;

          hence thesis by A1, A11, A14;

        end;

      end;

      reconsider B = ( Ball (u,(((G * (1,1)) `2 ) - s))) as Subset of ( TOP-REAL 2) by TOPREAL3: 8;

      

       A17: B is open by Th3;

      u in ( Ball (u,(((G * (1,1)) `2 ) - s))) by A8, Th1, XREAL_1: 50;

      hence thesis by A7, A9, A17, TOPS_1: 22;

    end;

    theorem :: GOBOARD6:16

    

     Th16: ( Int ( h_strip (G,( width G)))) = { |[r, s]| : ((G * (1,( width G))) `2 ) < s }

    proof

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

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

      then

       A1: ( h_strip (G,( width G))) = { |[r, s]| : ((G * (1,( width G))) `2 ) <= s } by GOBOARD5: 6;

      thus ( Int ( h_strip (G,( width G)))) c= { |[r, s]| : ((G * (1,( width G))) `2 ) < s }

      proof

        let x be object;

        assume

         A2: x in ( Int ( h_strip (G,( width G))));

        then

        reconsider u = x as Point of ( Euclid 2) by Lm6;

        consider s1 be Real such that

         A3: s1 > 0 and

         A4: ( Ball (u,s1)) c= ( h_strip (G,( width G))) by A2, Th5;

        reconsider p = u as Point of ( TOP-REAL 2) by Lm6;

        

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

        set q = |[((p `1 ) + 0 ), ((p `2 ) - (s1 / 2))]|;

        (s1 / 2) < s1 by A3, XREAL_1: 216;

        then q in ( Ball (u,s1)) by A3, A5, Th10;

        then q in ( h_strip (G,( width G))) by A4;

        then ex r2, s2 st q = |[r2, s2]| & ((G * (1,( width G))) `2 ) <= s2 by A1;

        then ((G * (1,( width G))) `2 ) <= ((p `2 ) - (s1 / 2)) by SPPOL_2: 1;

        then

         A6: (((G * (1,( width G))) `2 ) + (s1 / 2)) <= (p `2 ) by XREAL_1: 19;

        ((G * (1,( width G))) `2 ) < (((G * (1,( width G))) `2 ) + (s1 / 2)) by A3, XREAL_1: 29, XREAL_1: 215;

        then ((G * (1,( width G))) `2 ) < (p `2 ) by A6, XXREAL_0: 2;

        hence thesis by A5;

      end;

      let x be object;

      assume x in { |[r, s]| : ((G * (1,( width G))) `2 ) < s };

      then

      consider r, s such that

       A7: x = |[r, s]| and

       A8: ((G * (1,( width G))) `2 ) < s;

      reconsider u = |[r, s]| as Point of ( Euclid 2) by TOPREAL3: 8;

      

       A9: ( Ball (u,(s - ((G * (1,( width G))) `2 )))) c= ( h_strip (G,( width G)))

      proof

        let y be object;

        

         A10: ( Ball (u,(s - ((G * (1,( width G))) `2 )))) = { v : ( dist (u,v)) < (s - ((G * (1,( width G))) `2 )) } by METRIC_1: 17;

        assume y in ( Ball (u,(s - ((G * (1,( width G))) `2 ))));

        then

        consider v such that

         A11: v = y and

         A12: ( dist (u,v)) < (s - ((G * (1,( width G))) `2 )) by A10;

        reconsider q = v as Point of ( TOP-REAL 2) by TOPREAL3: 8;

        ((s - (q `2 )) ^2 ) >= 0 & (((s - (q `2 )) ^2 ) + 0 ) <= (((r - (q `1 )) ^2 ) + ((s - (q `2 )) ^2 )) by XREAL_1: 6, XREAL_1: 63;

        then

         A13: ( sqrt ((s - (q `2 )) ^2 )) <= ( sqrt (((r - (q `1 )) ^2 ) + ((s - (q `2 )) ^2 ))) by SQUARE_1: 26;

        

         A14: q = |[(q `1 ), (q `2 )]| by EUCLID: 53;

        then ( sqrt (((r - (q `1 )) ^2 ) + ((s - (q `2 )) ^2 ))) < (s - ((G * (1,( width G))) `2 )) by A12, Th6;

        then ( sqrt ((s - (q `2 )) ^2 )) <= (s - ((G * (1,( width G))) `2 )) by A13, XXREAL_0: 2;

        then

         A15: |.(s - (q `2 )).| <= (s - ((G * (1,( width G))) `2 )) by COMPLEX1: 72;

        per cases ;

          suppose s >= (q `2 );

          then (s - (q `2 )) >= 0 by XREAL_1: 48;

          then |.(s - (q `2 )).| = (s - (q `2 )) by ABSVALUE:def 1;

          then ((G * (1,( width G))) `2 ) <= (q `2 ) by A15, XREAL_1: 10;

          hence thesis by A1, A11, A14;

        end;

          suppose s <= (q `2 );

          then ((G * (1,( width G))) `2 ) <= (q `2 ) by A8, XXREAL_0: 2;

          hence thesis by A1, A11, A14;

        end;

      end;

      reconsider B = ( Ball (u,(s - ((G * (1,( width G))) `2 )))) as Subset of ( TOP-REAL 2) by TOPREAL3: 8;

      

       A16: B is open by Th3;

      u in ( Ball (u,(s - ((G * (1,( width G))) `2 )))) by A8, Th1, XREAL_1: 50;

      hence thesis by A7, A9, A16, TOPS_1: 22;

    end;

    theorem :: GOBOARD6:17

    

     Th17: 1 <= j & j < ( width G) implies ( Int ( h_strip (G,j))) = { |[r, s]| : ((G * (1,j)) `2 ) < s & s < ((G * (1,(j + 1))) `2 ) }

    proof

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

      then

       A1: 1 <= ( len G) by NAT_1: 14;

      assume 1 <= j & j < ( width G);

      then

       A2: ( h_strip (G,j)) = { |[r, s]| : ((G * (1,j)) `2 ) <= s & s <= ((G * (1,(j + 1))) `2 ) } by A1, GOBOARD5: 5;

      thus ( Int ( h_strip (G,j))) c= { |[r, s]| : ((G * (1,j)) `2 ) < s & s < ((G * (1,(j + 1))) `2 ) }

      proof

        let x be object;

        assume

         A3: x in ( Int ( h_strip (G,j)));

        then

        reconsider u = x as Point of ( Euclid 2) by Lm6;

        consider s1 be Real such that

         A4: s1 > 0 and

         A5: ( Ball (u,s1)) c= ( h_strip (G,j)) by A3, Th5;

        reconsider p = u as Point of ( TOP-REAL 2) by Lm6;

        

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

        set q2 = |[((p `1 ) + 0 ), ((p `2 ) - (s1 / 2))]|;

        

         A7: (s1 / 2) < s1 by A4, XREAL_1: 216;

        then q2 in ( Ball (u,s1)) by A4, A6, Th10;

        then q2 in ( h_strip (G,j)) by A5;

        then ex r2, s2 st q2 = |[r2, s2]| & ((G * (1,j)) `2 ) <= s2 & s2 <= ((G * (1,(j + 1))) `2 ) by A2;

        then ((G * (1,j)) `2 ) <= ((p `2 ) - (s1 / 2)) by SPPOL_2: 1;

        then

         A8: (((G * (1,j)) `2 ) + (s1 / 2)) <= (p `2 ) by XREAL_1: 19;

        set q1 = |[((p `1 ) + 0 ), ((p `2 ) + (s1 / 2))]|;

        q1 in ( Ball (u,s1)) by A4, A6, A7, Th8;

        then q1 in ( h_strip (G,j)) by A5;

        then ex r2, s2 st q1 = |[r2, s2]| & ((G * (1,j)) `2 ) <= s2 & s2 <= ((G * (1,(j + 1))) `2 ) by A2;

        then

         A9: ((p `2 ) + (s1 / 2)) <= ((G * (1,(j + 1))) `2 ) by SPPOL_2: 1;

        ((G * (1,j)) `2 ) < (((G * (1,j)) `2 ) + (s1 / 2)) by A4, XREAL_1: 29, XREAL_1: 215;

        then

         A10: ((G * (1,j)) `2 ) < (p `2 ) by A8, XXREAL_0: 2;

        (p `2 ) < ((p `2 ) + (s1 / 2)) by A4, XREAL_1: 29, XREAL_1: 215;

        then (p `2 ) < ((G * (1,(j + 1))) `2 ) by A9, XXREAL_0: 2;

        hence thesis by A6, A10;

      end;

      let x be object;

      assume x in { |[r, s]| : ((G * (1,j)) `2 ) < s & s < ((G * (1,(j + 1))) `2 ) };

      then

      consider r, s such that

       A11: x = |[r, s]| and

       A12: ((G * (1,j)) `2 ) < s and

       A13: s < ((G * (1,(j + 1))) `2 );

      reconsider u = |[r, s]| as Point of ( Euclid 2) by TOPREAL3: 8;

      (((G * (1,(j + 1))) `2 ) - s) > 0 & (s - ((G * (1,j)) `2 )) > 0 by A12, A13, XREAL_1: 50;

      then ( min ((s - ((G * (1,j)) `2 )),(((G * (1,(j + 1))) `2 ) - s))) > 0 by XXREAL_0: 15;

      then

       A14: u in ( Ball (u,( min ((s - ((G * (1,j)) `2 )),(((G * (1,(j + 1))) `2 ) - s))))) by Th1;

      

       A15: ( Ball (u,( min ((s - ((G * (1,j)) `2 )),(((G * (1,(j + 1))) `2 ) - s))))) c= ( h_strip (G,j))

      proof

        let y be object;

        

         A16: ( Ball (u,( min ((s - ((G * (1,j)) `2 )),(((G * (1,(j + 1))) `2 ) - s))))) = { v : ( dist (u,v)) < ( min ((s - ((G * (1,j)) `2 )),(((G * (1,(j + 1))) `2 ) - s))) } by METRIC_1: 17;

        assume y in ( Ball (u,( min ((s - ((G * (1,j)) `2 )),(((G * (1,(j + 1))) `2 ) - s)))));

        then

        consider v such that

         A17: v = y and

         A18: ( dist (u,v)) < ( min ((s - ((G * (1,j)) `2 )),(((G * (1,(j + 1))) `2 ) - s))) by A16;

        reconsider q = v as Point of ( TOP-REAL 2) by TOPREAL3: 8;

        ((s - (q `2 )) ^2 ) >= 0 & (((s - (q `2 )) ^2 ) + 0 ) <= (((r - (q `1 )) ^2 ) + ((s - (q `2 )) ^2 )) by XREAL_1: 6, XREAL_1: 63;

        then

         A19: ( sqrt ((s - (q `2 )) ^2 )) <= ( sqrt (((r - (q `1 )) ^2 ) + ((s - (q `2 )) ^2 ))) by SQUARE_1: 26;

        

         A20: q = |[(q `1 ), (q `2 )]| by EUCLID: 53;

        then ( sqrt (((r - (q `1 )) ^2 ) + ((s - (q `2 )) ^2 ))) < ( min ((s - ((G * (1,j)) `2 )),(((G * (1,(j + 1))) `2 ) - s))) by A18, Th6;

        then ( sqrt ((s - (q `2 )) ^2 )) <= ( min ((s - ((G * (1,j)) `2 )),(((G * (1,(j + 1))) `2 ) - s))) by A19, XXREAL_0: 2;

        then

         A21: |.(s - (q `2 )).| <= ( min ((s - ((G * (1,j)) `2 )),(((G * (1,(j + 1))) `2 ) - s))) by COMPLEX1: 72;

        then

         A22: |.(s - (q `2 )).| <= (s - ((G * (1,j)) `2 )) by XXREAL_0: 22;

        

         A23: |.(s - (q `2 )).| <= (((G * (1,(j + 1))) `2 ) - s) by A21, XXREAL_0: 22;

        per cases ;

          suppose

           A24: s <= (q `2 );

          then

           A25: ((q `2 ) - s) >= 0 by XREAL_1: 48;

           |.(s - (q `2 )).| = |.( - (s - (q `2 ))).| by COMPLEX1: 52

          .= ((q `2 ) - s) by A25, ABSVALUE:def 1;

          then

           A26: (q `2 ) <= ((G * (1,(j + 1))) `2 ) by A23, XREAL_1: 9;

          ((G * (1,j)) `2 ) <= (q `2 ) by A12, A24, XXREAL_0: 2;

          hence thesis by A2, A17, A20, A26;

        end;

          suppose

           A27: s >= (q `2 );

          then (s - (q `2 )) >= 0 by XREAL_1: 48;

          then |.(s - (q `2 )).| = (s - (q `2 )) by ABSVALUE:def 1;

          then

           A28: ((G * (1,j)) `2 ) <= (q `2 ) by A22, XREAL_1: 10;

          (q `2 ) <= ((G * (1,(j + 1))) `2 ) by A13, A27, XXREAL_0: 2;

          hence thesis by A2, A17, A20, A28;

        end;

      end;

      reconsider B = ( Ball (u,( min ((s - ((G * (1,j)) `2 )),(((G * (1,(j + 1))) `2 ) - s))))) as Subset of ( TOP-REAL 2) by TOPREAL3: 8;

      B is open by Th3;

      hence thesis by A11, A14, A15, TOPS_1: 22;

    end;

    theorem :: GOBOARD6:18

    

     Th18: ( Int ( cell (G, 0 , 0 ))) = { |[r, s]| : r < ((G * (1,1)) `1 ) & s < ((G * (1,1)) `2 ) }

    proof

      ( cell (G, 0 , 0 )) = (( v_strip (G, 0 )) /\ ( h_strip (G, 0 ))) by GOBOARD5:def 3;

      then

       A1: ( Int ( cell (G, 0 , 0 ))) = (( Int ( v_strip (G, 0 ))) /\ ( Int ( h_strip (G, 0 )))) by TOPS_1: 17;

      

       A2: ( Int ( h_strip (G, 0 ))) = { |[r, s]| : s < ((G * (1,1)) `2 ) } by Th15;

      

       A3: ( Int ( v_strip (G, 0 ))) = { |[r, s]| : r < ((G * (1,1)) `1 ) } by Th12;

      thus ( Int ( cell (G, 0 , 0 ))) c= { |[r, s]| : r < ((G * (1,1)) `1 ) & s < ((G * (1,1)) `2 ) }

      proof

        let x be object;

        assume

         A4: x in ( Int ( cell (G, 0 , 0 )));

        then x in ( Int ( v_strip (G, 0 ))) by A1, XBOOLE_0:def 4;

        then

        consider r1, s1 such that

         A5: x = |[r1, s1]| and

         A6: r1 < ((G * (1,1)) `1 ) by A3;

        x in ( Int ( h_strip (G, 0 ))) by A1, A4, XBOOLE_0:def 4;

        then

        consider r2, s2 such that

         A7: x = |[r2, s2]| and

         A8: s2 < ((G * (1,1)) `2 ) by A2;

        s1 = s2 by A5, A7, SPPOL_2: 1;

        hence thesis by A5, A6, A8;

      end;

      let x be object;

      assume x in { |[r, s]| : r < ((G * (1,1)) `1 ) & s < ((G * (1,1)) `2 ) };

      then

       A9: ex r, s st x = |[r, s]| & r < ((G * (1,1)) `1 ) & s < ((G * (1,1)) `2 );

      then

       A10: x in ( Int ( h_strip (G, 0 ))) by A2;

      x in ( Int ( v_strip (G, 0 ))) by A3, A9;

      hence thesis by A1, A10, XBOOLE_0:def 4;

    end;

    theorem :: GOBOARD6:19

    

     Th19: ( Int ( cell (G, 0 ,( width G)))) = { |[r, s]| : r < ((G * (1,1)) `1 ) & ((G * (1,( width G))) `2 ) < s }

    proof

      ( cell (G, 0 ,( width G))) = (( v_strip (G, 0 )) /\ ( h_strip (G,( width G)))) by GOBOARD5:def 3;

      then

       A1: ( Int ( cell (G, 0 ,( width G)))) = (( Int ( v_strip (G, 0 ))) /\ ( Int ( h_strip (G,( width G))))) by TOPS_1: 17;

      

       A2: ( Int ( h_strip (G,( width G)))) = { |[r, s]| : ((G * (1,( width G))) `2 ) < s } by Th16;

      

       A3: ( Int ( v_strip (G, 0 ))) = { |[r, s]| : r < ((G * (1,1)) `1 ) } by Th12;

      thus ( Int ( cell (G, 0 ,( width G)))) c= { |[r, s]| : r < ((G * (1,1)) `1 ) & ((G * (1,( width G))) `2 ) < s }

      proof

        let x be object;

        assume

         A4: x in ( Int ( cell (G, 0 ,( width G))));

        then x in ( Int ( v_strip (G, 0 ))) by A1, XBOOLE_0:def 4;

        then

        consider r1, s1 such that

         A5: x = |[r1, s1]| and

         A6: r1 < ((G * (1,1)) `1 ) by A3;

        x in ( Int ( h_strip (G,( width G)))) by A1, A4, XBOOLE_0:def 4;

        then

        consider r2, s2 such that

         A7: x = |[r2, s2]| and

         A8: ((G * (1,( width G))) `2 ) < s2 by A2;

        s1 = s2 by A5, A7, SPPOL_2: 1;

        hence thesis by A5, A6, A8;

      end;

      let x be object;

      assume x in { |[r, s]| : r < ((G * (1,1)) `1 ) & ((G * (1,( width G))) `2 ) < s };

      then

       A9: ex r, s st x = |[r, s]| & r < ((G * (1,1)) `1 ) & ((G * (1,( width G))) `2 ) < s;

      then

       A10: x in ( Int ( h_strip (G,( width G)))) by A2;

      x in ( Int ( v_strip (G, 0 ))) by A3, A9;

      hence thesis by A1, A10, XBOOLE_0:def 4;

    end;

    theorem :: GOBOARD6:20

    

     Th20: 1 <= j & j < ( width G) implies ( Int ( cell (G, 0 ,j))) = { |[r, s]| : r < ((G * (1,1)) `1 ) & ((G * (1,j)) `2 ) < s & s < ((G * (1,(j + 1))) `2 ) }

    proof

      ( cell (G, 0 ,j)) = (( v_strip (G, 0 )) /\ ( h_strip (G,j))) by GOBOARD5:def 3;

      then

       A1: ( Int ( cell (G, 0 ,j))) = (( Int ( v_strip (G, 0 ))) /\ ( Int ( h_strip (G,j)))) by TOPS_1: 17;

      assume 1 <= j & j < ( width G);

      then

       A2: ( Int ( h_strip (G,j))) = { |[r, s]| : ((G * (1,j)) `2 ) < s & s < ((G * (1,(j + 1))) `2 ) } by Th17;

      

       A3: ( Int ( v_strip (G, 0 ))) = { |[r, s]| : r < ((G * (1,1)) `1 ) } by Th12;

      thus ( Int ( cell (G, 0 ,j))) c= { |[r, s]| : r < ((G * (1,1)) `1 ) & ((G * (1,j)) `2 ) < s & s < ((G * (1,(j + 1))) `2 ) }

      proof

        let x be object;

        assume

         A4: x in ( Int ( cell (G, 0 ,j)));

        then x in ( Int ( v_strip (G, 0 ))) by A1, XBOOLE_0:def 4;

        then

        consider r1, s1 such that

         A5: x = |[r1, s1]| and

         A6: r1 < ((G * (1,1)) `1 ) by A3;

        x in ( Int ( h_strip (G,j))) by A1, A4, XBOOLE_0:def 4;

        then

        consider r2, s2 such that

         A7: x = |[r2, s2]| and

         A8: ((G * (1,j)) `2 ) < s2 & s2 < ((G * (1,(j + 1))) `2 ) by A2;

        s1 = s2 by A5, A7, SPPOL_2: 1;

        hence thesis by A5, A6, A8;

      end;

      let x be object;

      assume x in { |[r, s]| : r < ((G * (1,1)) `1 ) & ((G * (1,j)) `2 ) < s & s < ((G * (1,(j + 1))) `2 ) };

      then

       A9: ex r, s st x = |[r, s]| & r < ((G * (1,1)) `1 ) & ((G * (1,j)) `2 ) < s & s < ((G * (1,(j + 1))) `2 );

      then

       A10: x in ( Int ( h_strip (G,j))) by A2;

      x in ( Int ( v_strip (G, 0 ))) by A3, A9;

      hence thesis by A1, A10, XBOOLE_0:def 4;

    end;

    theorem :: GOBOARD6:21

    

     Th21: ( Int ( cell (G,( len G), 0 ))) = { |[r, s]| : ((G * (( len G),1)) `1 ) < r & s < ((G * (1,1)) `2 ) }

    proof

      ( cell (G,( len G), 0 )) = (( v_strip (G,( len G))) /\ ( h_strip (G, 0 ))) by GOBOARD5:def 3;

      then

       A1: ( Int ( cell (G,( len G), 0 ))) = (( Int ( v_strip (G,( len G)))) /\ ( Int ( h_strip (G, 0 )))) by TOPS_1: 17;

      

       A2: ( Int ( h_strip (G, 0 ))) = { |[r, s]| : s < ((G * (1,1)) `2 ) } by Th15;

      

       A3: ( Int ( v_strip (G,( len G)))) = { |[r, s]| : ((G * (( len G),1)) `1 ) < r } by Th13;

      thus ( Int ( cell (G,( len G), 0 ))) c= { |[r, s]| : ((G * (( len G),1)) `1 ) < r & s < ((G * (1,1)) `2 ) }

      proof

        let x be object;

        assume

         A4: x in ( Int ( cell (G,( len G), 0 )));

        then x in ( Int ( v_strip (G,( len G)))) by A1, XBOOLE_0:def 4;

        then

        consider r1, s1 such that

         A5: x = |[r1, s1]| and

         A6: ((G * (( len G),1)) `1 ) < r1 by A3;

        x in ( Int ( h_strip (G, 0 ))) by A1, A4, XBOOLE_0:def 4;

        then

        consider r2, s2 such that

         A7: x = |[r2, s2]| and

         A8: s2 < ((G * (1,1)) `2 ) by A2;

        s1 = s2 by A5, A7, SPPOL_2: 1;

        hence thesis by A5, A6, A8;

      end;

      let x be object;

      assume x in { |[r, s]| : ((G * (( len G),1)) `1 ) < r & s < ((G * (1,1)) `2 ) };

      then

       A9: ex r, s st x = |[r, s]| & ((G * (( len G),1)) `1 ) < r & s < ((G * (1,1)) `2 );

      then

       A10: x in ( Int ( h_strip (G, 0 ))) by A2;

      x in ( Int ( v_strip (G,( len G)))) by A3, A9;

      hence thesis by A1, A10, XBOOLE_0:def 4;

    end;

    theorem :: GOBOARD6:22

    

     Th22: ( Int ( cell (G,( len G),( width G)))) = { |[r, s]| : ((G * (( len G),1)) `1 ) < r & ((G * (1,( width G))) `2 ) < s }

    proof

      ( cell (G,( len G),( width G))) = (( v_strip (G,( len G))) /\ ( h_strip (G,( width G)))) by GOBOARD5:def 3;

      then

       A1: ( Int ( cell (G,( len G),( width G)))) = (( Int ( v_strip (G,( len G)))) /\ ( Int ( h_strip (G,( width G))))) by TOPS_1: 17;

      

       A2: ( Int ( h_strip (G,( width G)))) = { |[r, s]| : ((G * (1,( width G))) `2 ) < s } by Th16;

      

       A3: ( Int ( v_strip (G,( len G)))) = { |[r, s]| : ((G * (( len G),1)) `1 ) < r } by Th13;

      thus ( Int ( cell (G,( len G),( width G)))) c= { |[r, s]| : ((G * (( len G),1)) `1 ) < r & ((G * (1,( width G))) `2 ) < s }

      proof

        let x be object;

        assume

         A4: x in ( Int ( cell (G,( len G),( width G))));

        then x in ( Int ( v_strip (G,( len G)))) by A1, XBOOLE_0:def 4;

        then

        consider r1, s1 such that

         A5: x = |[r1, s1]| and

         A6: ((G * (( len G),1)) `1 ) < r1 by A3;

        x in ( Int ( h_strip (G,( width G)))) by A1, A4, XBOOLE_0:def 4;

        then

        consider r2, s2 such that

         A7: x = |[r2, s2]| and

         A8: ((G * (1,( width G))) `2 ) < s2 by A2;

        s1 = s2 by A5, A7, SPPOL_2: 1;

        hence thesis by A5, A6, A8;

      end;

      let x be object;

      assume x in { |[r, s]| : ((G * (( len G),1)) `1 ) < r & ((G * (1,( width G))) `2 ) < s };

      then

       A9: ex r, s st x = |[r, s]| & ((G * (( len G),1)) `1 ) < r & ((G * (1,( width G))) `2 ) < s;

      then

       A10: x in ( Int ( h_strip (G,( width G)))) by A2;

      x in ( Int ( v_strip (G,( len G)))) by A3, A9;

      hence thesis by A1, A10, XBOOLE_0:def 4;

    end;

    theorem :: GOBOARD6:23

    

     Th23: 1 <= j & j < ( width G) implies ( Int ( cell (G,( len G),j))) = { |[r, s]| : ((G * (( len G),1)) `1 ) < r & ((G * (1,j)) `2 ) < s & s < ((G * (1,(j + 1))) `2 ) }

    proof

      ( cell (G,( len G),j)) = (( v_strip (G,( len G))) /\ ( h_strip (G,j))) by GOBOARD5:def 3;

      then

       A1: ( Int ( cell (G,( len G),j))) = (( Int ( v_strip (G,( len G)))) /\ ( Int ( h_strip (G,j)))) by TOPS_1: 17;

      assume 1 <= j & j < ( width G);

      then

       A2: ( Int ( h_strip (G,j))) = { |[r, s]| : ((G * (1,j)) `2 ) < s & s < ((G * (1,(j + 1))) `2 ) } by Th17;

      

       A3: ( Int ( v_strip (G,( len G)))) = { |[r, s]| : ((G * (( len G),1)) `1 ) < r } by Th13;

      thus ( Int ( cell (G,( len G),j))) c= { |[r, s]| : ((G * (( len G),1)) `1 ) < r & ((G * (1,j)) `2 ) < s & s < ((G * (1,(j + 1))) `2 ) }

      proof

        let x be object;

        assume

         A4: x in ( Int ( cell (G,( len G),j)));

        then x in ( Int ( v_strip (G,( len G)))) by A1, XBOOLE_0:def 4;

        then

        consider r1, s1 such that

         A5: x = |[r1, s1]| and

         A6: ((G * (( len G),1)) `1 ) < r1 by A3;

        x in ( Int ( h_strip (G,j))) by A1, A4, XBOOLE_0:def 4;

        then

        consider r2, s2 such that

         A7: x = |[r2, s2]| and

         A8: ((G * (1,j)) `2 ) < s2 & s2 < ((G * (1,(j + 1))) `2 ) by A2;

        s1 = s2 by A5, A7, SPPOL_2: 1;

        hence thesis by A5, A6, A8;

      end;

      let x be object;

      assume x in { |[r, s]| : ((G * (( len G),1)) `1 ) < r & ((G * (1,j)) `2 ) < s & s < ((G * (1,(j + 1))) `2 ) };

      then

       A9: ex r, s st x = |[r, s]| & ((G * (( len G),1)) `1 ) < r & ((G * (1,j)) `2 ) < s & s < ((G * (1,(j + 1))) `2 );

      then

       A10: x in ( Int ( h_strip (G,j))) by A2;

      x in ( Int ( v_strip (G,( len G)))) by A3, A9;

      hence thesis by A1, A10, XBOOLE_0:def 4;

    end;

    theorem :: GOBOARD6:24

    

     Th24: 1 <= i & i < ( len G) implies ( Int ( cell (G,i, 0 ))) = { |[r, s]| : ((G * (i,1)) `1 ) < r & r < ((G * ((i + 1),1)) `1 ) & s < ((G * (1,1)) `2 ) }

    proof

      ( cell (G,i, 0 )) = (( v_strip (G,i)) /\ ( h_strip (G, 0 ))) by GOBOARD5:def 3;

      then

       A1: ( Int ( cell (G,i, 0 ))) = (( Int ( v_strip (G,i))) /\ ( Int ( h_strip (G, 0 )))) by TOPS_1: 17;

      assume 1 <= i & i < ( len G);

      then

       A2: ( Int ( v_strip (G,i))) = { |[r, s]| : ((G * (i,1)) `1 ) < r & r < ((G * ((i + 1),1)) `1 ) } by Th14;

      

       A3: ( Int ( h_strip (G, 0 ))) = { |[r, s]| : s < ((G * (1,1)) `2 ) } by Th15;

      thus ( Int ( cell (G,i, 0 ))) c= { |[r, s]| : ((G * (i,1)) `1 ) < r & r < ((G * ((i + 1),1)) `1 ) & s < ((G * (1,1)) `2 ) }

      proof

        let x be object;

        assume

         A4: x in ( Int ( cell (G,i, 0 )));

        then x in ( Int ( v_strip (G,i))) by A1, XBOOLE_0:def 4;

        then

        consider r1, s1 such that

         A5: x = |[r1, s1]| and

         A6: ((G * (i,1)) `1 ) < r1 & r1 < ((G * ((i + 1),1)) `1 ) by A2;

        x in ( Int ( h_strip (G, 0 ))) by A1, A4, XBOOLE_0:def 4;

        then

        consider r2, s2 such that

         A7: x = |[r2, s2]| and

         A8: s2 < ((G * (1,1)) `2 ) by A3;

        s1 = s2 by A5, A7, SPPOL_2: 1;

        hence thesis by A5, A6, A8;

      end;

      let x be object;

      assume x in { |[r, s]| : ((G * (i,1)) `1 ) < r & r < ((G * ((i + 1),1)) `1 ) & s < ((G * (1,1)) `2 ) };

      then

       A9: ex r, s st x = |[r, s]| & ((G * (i,1)) `1 ) < r & r < ((G * ((i + 1),1)) `1 ) & s < ((G * (1,1)) `2 );

      then

       A10: x in ( Int ( h_strip (G, 0 ))) by A3;

      x in ( Int ( v_strip (G,i))) by A2, A9;

      hence thesis by A1, A10, XBOOLE_0:def 4;

    end;

    theorem :: GOBOARD6:25

    

     Th25: 1 <= i & i < ( len G) implies ( Int ( cell (G,i,( width G)))) = { |[r, s]| : ((G * (i,1)) `1 ) < r & r < ((G * ((i + 1),1)) `1 ) & ((G * (1,( width G))) `2 ) < s }

    proof

      ( cell (G,i,( width G))) = (( v_strip (G,i)) /\ ( h_strip (G,( width G)))) by GOBOARD5:def 3;

      then

       A1: ( Int ( cell (G,i,( width G)))) = (( Int ( v_strip (G,i))) /\ ( Int ( h_strip (G,( width G))))) by TOPS_1: 17;

      assume 1 <= i & i < ( len G);

      then

       A2: ( Int ( v_strip (G,i))) = { |[r, s]| : ((G * (i,1)) `1 ) < r & r < ((G * ((i + 1),1)) `1 ) } by Th14;

      

       A3: ( Int ( h_strip (G,( width G)))) = { |[r, s]| : ((G * (1,( width G))) `2 ) < s } by Th16;

      thus ( Int ( cell (G,i,( width G)))) c= { |[r, s]| : ((G * (i,1)) `1 ) < r & r < ((G * ((i + 1),1)) `1 ) & ((G * (1,( width G))) `2 ) < s }

      proof

        let x be object;

        assume

         A4: x in ( Int ( cell (G,i,( width G))));

        then x in ( Int ( v_strip (G,i))) by A1, XBOOLE_0:def 4;

        then

        consider r1, s1 such that

         A5: x = |[r1, s1]| and

         A6: ((G * (i,1)) `1 ) < r1 & r1 < ((G * ((i + 1),1)) `1 ) by A2;

        x in ( Int ( h_strip (G,( width G)))) by A1, A4, XBOOLE_0:def 4;

        then

        consider r2, s2 such that

         A7: x = |[r2, s2]| and

         A8: ((G * (1,( width G))) `2 ) < s2 by A3;

        s1 = s2 by A5, A7, SPPOL_2: 1;

        hence thesis by A5, A6, A8;

      end;

      let x be object;

      assume x in { |[r, s]| : ((G * (i,1)) `1 ) < r & r < ((G * ((i + 1),1)) `1 ) & ((G * (1,( width G))) `2 ) < s };

      then

       A9: ex r, s st x = |[r, s]| & ((G * (i,1)) `1 ) < r & r < ((G * ((i + 1),1)) `1 ) & ((G * (1,( width G))) `2 ) < s;

      then

       A10: x in ( Int ( h_strip (G,( width G)))) by A3;

      x in ( Int ( v_strip (G,i))) by A2, A9;

      hence thesis by A1, A10, XBOOLE_0:def 4;

    end;

    theorem :: GOBOARD6:26

    

     Th26: 1 <= i & i < ( len G) & 1 <= j & j < ( width G) implies ( Int ( cell (G,i,j))) = { |[r, s]| : ((G * (i,1)) `1 ) < r & r < ((G * ((i + 1),1)) `1 ) & ((G * (1,j)) `2 ) < s & s < ((G * (1,(j + 1))) `2 ) }

    proof

      assume that

       A1: 1 <= i & i < ( len G) and

       A2: 1 <= j & j < ( width G);

      

       A3: ( Int ( h_strip (G,j))) = { |[r, s]| : ((G * (1,j)) `2 ) < s & s < ((G * (1,(j + 1))) `2 ) } by A2, Th17;

      ( cell (G,i,j)) = (( v_strip (G,i)) /\ ( h_strip (G,j))) by GOBOARD5:def 3;

      then

       A4: ( Int ( cell (G,i,j))) = (( Int ( v_strip (G,i))) /\ ( Int ( h_strip (G,j)))) by TOPS_1: 17;

      

       A5: ( Int ( v_strip (G,i))) = { |[r, s]| : ((G * (i,1)) `1 ) < r & r < ((G * ((i + 1),1)) `1 ) } by A1, Th14;

      thus ( Int ( cell (G,i,j))) c= { |[r, s]| : ((G * (i,1)) `1 ) < r & r < ((G * ((i + 1),1)) `1 ) & ((G * (1,j)) `2 ) < s & s < ((G * (1,(j + 1))) `2 ) }

      proof

        let x be object;

        assume

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

        then x in ( Int ( v_strip (G,i))) by A4, XBOOLE_0:def 4;

        then

        consider r1, s1 such that

         A7: x = |[r1, s1]| and

         A8: ((G * (i,1)) `1 ) < r1 & r1 < ((G * ((i + 1),1)) `1 ) by A5;

        x in ( Int ( h_strip (G,j))) by A4, A6, XBOOLE_0:def 4;

        then

        consider r2, s2 such that

         A9: x = |[r2, s2]| and

         A10: ((G * (1,j)) `2 ) < s2 & s2 < ((G * (1,(j + 1))) `2 ) by A3;

        s1 = s2 by A7, A9, SPPOL_2: 1;

        hence thesis by A7, A8, A10;

      end;

      let x be object;

      assume x in { |[r, s]| : ((G * (i,1)) `1 ) < r & r < ((G * ((i + 1),1)) `1 ) & ((G * (1,j)) `2 ) < s & s < ((G * (1,(j + 1))) `2 ) };

      then

       A11: ex r, s st x = |[r, s]| & ((G * (i,1)) `1 ) < r & r < ((G * ((i + 1),1)) `1 ) & ((G * (1,j)) `2 ) < s & s < ((G * (1,(j + 1))) `2 );

      then

       A12: x in ( Int ( h_strip (G,j))) by A3;

      x in ( Int ( v_strip (G,i))) by A5, A11;

      hence thesis by A4, A12, XBOOLE_0:def 4;

    end;

    theorem :: GOBOARD6:27

    1 <= j & j <= ( width G) & p in ( Int ( h_strip (G,j))) implies (p `2 ) > ((G * (1,j)) `2 )

    proof

      assume that

       A1: 1 <= j and

       A2: j <= ( width G) and

       A3: p in ( Int ( h_strip (G,j)));

      per cases by A2, XXREAL_0: 1;

        suppose j = ( width G);

        then ( Int ( h_strip (G,j))) = { |[r, s]| : ((G * (1,j)) `2 ) < s } by Th16;

        then ex r, s st p = |[r, s]| & ((G * (1,j)) `2 ) < s by A3;

        hence thesis by EUCLID: 52;

      end;

        suppose j < ( width G);

        then ( Int ( h_strip (G,j))) = { |[r, s]| : ((G * (1,j)) `2 ) < s & s < ((G * (1,(j + 1))) `2 ) } by A1, Th17;

        then ex r, s st p = |[r, s]| & ((G * (1,j)) `2 ) < s & s < ((G * (1,(j + 1))) `2 ) by A3;

        hence thesis by EUCLID: 52;

      end;

    end;

    theorem :: GOBOARD6:28

    j < ( width G) & p in ( Int ( h_strip (G,j))) implies (p `2 ) < ((G * (1,(j + 1))) `2 )

    proof

      assume that

       A1: j < ( width G) and

       A2: p in ( Int ( h_strip (G,j)));

      per cases by NAT_1: 14;

        suppose j = 0 ;

        then ( Int ( h_strip (G,j))) = { |[r, s]| : s < ((G * (1,(j + 1))) `2 ) } by Th15;

        then ex r, s st p = |[r, s]| & ((G * (1,(j + 1))) `2 ) > s by A2;

        hence thesis by EUCLID: 52;

      end;

        suppose j >= 1;

        then ( Int ( h_strip (G,j))) = { |[r, s]| : ((G * (1,j)) `2 ) < s & s < ((G * (1,(j + 1))) `2 ) } by A1, Th17;

        then ex r, s st p = |[r, s]| & ((G * (1,j)) `2 ) < s & s < ((G * (1,(j + 1))) `2 ) by A2;

        hence thesis by EUCLID: 52;

      end;

    end;

    theorem :: GOBOARD6:29

    1 <= i & i <= ( len G) & p in ( Int ( v_strip (G,i))) implies (p `1 ) > ((G * (i,1)) `1 )

    proof

      assume that

       A1: 1 <= i and

       A2: i <= ( len G) and

       A3: p in ( Int ( v_strip (G,i)));

      per cases by A2, XXREAL_0: 1;

        suppose i = ( len G);

        then ( Int ( v_strip (G,i))) = { |[r, s]| : ((G * (i,1)) `1 ) < r } by Th13;

        then ex r, s st p = |[r, s]| & ((G * (i,1)) `1 ) < r by A3;

        hence thesis by EUCLID: 52;

      end;

        suppose i < ( len G);

        then ( Int ( v_strip (G,i))) = { |[r, s]| : ((G * (i,1)) `1 ) < r & r < ((G * ((i + 1),1)) `1 ) } by A1, Th14;

        then ex r, s st p = |[r, s]| & ((G * (i,1)) `1 ) < r & r < ((G * ((i + 1),1)) `1 ) by A3;

        hence thesis by EUCLID: 52;

      end;

    end;

    theorem :: GOBOARD6:30

    i < ( len G) & p in ( Int ( v_strip (G,i))) implies (p `1 ) < ((G * ((i + 1),1)) `1 )

    proof

      assume that

       A1: i < ( len G) and

       A2: p in ( Int ( v_strip (G,i)));

      per cases by NAT_1: 14;

        suppose i = 0 ;

        then ( Int ( v_strip (G,i))) = { |[r, s]| : r < ((G * ((i + 1),1)) `1 ) } by Th12;

        then ex r, s st p = |[r, s]| & ((G * ((i + 1),1)) `1 ) > r by A2;

        hence thesis by EUCLID: 52;

      end;

        suppose i >= 1;

        then ( Int ( v_strip (G,i))) = { |[r, s]| : ((G * (i,1)) `1 ) < r & r < ((G * ((i + 1),1)) `1 ) } by A1, Th14;

        then ex r, s st p = |[r, s]| & ((G * (i,1)) `1 ) < r & r < ((G * ((i + 1),1)) `1 ) by A2;

        hence thesis by EUCLID: 52;

      end;

    end;

    theorem :: GOBOARD6:31

    

     Th31: 1 <= i & (i + 1) <= ( len G) & 1 <= j & (j + 1) <= ( width G) implies ((1 / 2) * ((G * (i,j)) + (G * ((i + 1),(j + 1))))) in ( Int ( cell (G,i,j)))

    proof

      assume that

       A1: 1 <= i and

       A2: (i + 1) <= ( len G) and

       A3: 1 <= j and

       A4: (j + 1) <= ( width G);

      

       A5: j < (j + 1) by XREAL_1: 29;

      set r1 = ((G * (i,j)) `1 ), s1 = ((G * (i,j)) `2 ), r2 = ((G * ((i + 1),(j + 1))) `1 ), s2 = ((G * ((i + 1),(j + 1))) `2 );

      

       A6: 1 <= (i + 1) & 1 <= (j + 1) by NAT_1: 11;

      then

       A7: ((G * (1,(j + 1))) `2 ) = s2 by A2, A4, GOBOARD5: 1;

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

      then

       A8: ( Int ( cell (G,i,j))) = { |[r, s]| : ((G * (i,1)) `1 ) < r & r < ((G * ((i + 1),1)) `1 ) & ((G * (1,j)) `2 ) < s & s < ((G * (1,(j + 1))) `2 ) } by A1, A3, Th26;

      (G * (i,j)) = |[r1, s1]| & (G * ((i + 1),(j + 1))) = |[r2, s2]| by EUCLID: 53;

      then ((G * (i,j)) + (G * ((i + 1),(j + 1)))) = |[(r1 + r2), (s1 + s2)]| by EUCLID: 56;

      then

       A9: ((1 / 2) * ((G * (i,j)) + (G * ((i + 1),(j + 1))))) = |[((1 / 2) * (r1 + r2)), ((1 / 2) * (s1 + s2))]| by EUCLID: 58;

      i <= (i + 1) by NAT_1: 11;

      then

       A10: i <= ( len G) by A2, XXREAL_0: 2;

      then

       A11: 1 <= ( len G) by A1, XXREAL_0: 2;

      j <= (j + 1) by NAT_1: 11;

      then

       A12: j <= ( width G) by A4, XXREAL_0: 2;

      then

       A13: 1 <= ( width G) by A3, XXREAL_0: 2;

      

       A14: ((G * (i,1)) `1 ) = r1 by A1, A3, A10, A12, GOBOARD5: 2;

      ((G * (1,j)) `2 ) = s1 by A1, A3, A10, A12, GOBOARD5: 1;

      then

       A15: s1 < s2 by A3, A4, A7, A11, A5, GOBOARD5: 4;

      then (s1 + s1) < (s1 + s2) by XREAL_1: 6;

      then ((1 / 2) * (s1 + s1)) < ((1 / 2) * (s1 + s2)) by XREAL_1: 68;

      then

       A16: ((G * (1,j)) `2 ) < ((1 / 2) * (s1 + s2)) by A1, A3, A10, A12, GOBOARD5: 1;

      

       A17: i < (i + 1) by XREAL_1: 29;

      ((G * ((i + 1),1)) `1 ) = r2 by A2, A4, A6, GOBOARD5: 2;

      then

       A18: r1 < r2 by A1, A2, A14, A13, A17, GOBOARD5: 3;

      then (r1 + r2) < (r2 + r2) by XREAL_1: 6;

      then ((1 / 2) * (r1 + r2)) < ((1 / 2) * (r2 + r2)) by XREAL_1: 68;

      then

       A19: ((1 / 2) * (r1 + r2)) < ((G * ((i + 1),1)) `1 ) by A2, A4, A6, GOBOARD5: 2;

      (s1 + s2) < (s2 + s2) by A15, XREAL_1: 6;

      then ((1 / 2) * (s1 + s2)) < ((1 / 2) * (s2 + s2)) by XREAL_1: 68;

      then

       A20: ((1 / 2) * (s1 + s2)) < ((G * (1,(j + 1))) `2 ) by A2, A4, A6, GOBOARD5: 1;

      (r1 + r1) < (r1 + r2) by A18, XREAL_1: 6;

      then ((1 / 2) * (r1 + r1)) < ((1 / 2) * (r1 + r2)) by XREAL_1: 68;

      hence thesis by A9, A14, A19, A16, A20, A8;

    end;

    theorem :: GOBOARD6:32

    

     Th32: 1 <= i & (i + 1) <= ( len G) implies (((1 / 2) * ((G * (i,( width G))) + (G * ((i + 1),( width G))))) + |[ 0 , 1]|) in ( Int ( cell (G,i,( width G))))

    proof

      assume that

       A1: 1 <= i and

       A2: (i + 1) <= ( len G);

      set r1 = ((G * (i,( width G))) `1 ), s1 = ((G * (i,( width G))) `2 ), r2 = ((G * ((i + 1),( width G))) `1 );

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

      then

       A3: 1 <= ( width G) by NAT_1: 14;

      i < ( len G) by A2, NAT_1: 13;

      then

       A4: ( Int ( cell (G,i,( width G)))) = { |[r, s]| : ((G * (i,1)) `1 ) < r & r < ((G * ((i + 1),1)) `1 ) & ((G * (1,( width G))) `2 ) < s } by A1, Th25;

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

      then

       A5: 1 <= ( width G) by NAT_1: 14;

      i < (i + 1) by XREAL_1: 29;

      then

       A6: r1 < r2 by A1, A2, A5, GOBOARD5: 3;

      then (r1 + r1) < (r1 + r2) by XREAL_1: 6;

      then

       A7: ((1 / 2) * (r1 + r1)) < ((1 / 2) * (r1 + r2)) by XREAL_1: 68;

      

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

      then

       A9: ((G * (1,( width G))) `2 ) = s1 by A1, A3, GOBOARD5: 1;

      then

       A10: ((G * (1,( width G))) `2 ) < (s1 + 1) by XREAL_1: 29;

      

       A11: 1 <= (i + 1) by NAT_1: 11;

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

      then (G * (i,( width G))) = |[r1, s1]| & (G * ((i + 1),( width G))) = |[r2, s1]| by A9, EUCLID: 53;

      then ((1 / 2) * (s1 + s1)) = s1 & ((G * (i,( width G))) + (G * ((i + 1),( width G)))) = |[(r1 + r2), (s1 + s1)]| by EUCLID: 56;

      then ((1 / 2) * ((G * (i,( width G))) + (G * ((i + 1),( width G))))) = |[((1 / 2) * (r1 + r2)), s1]| by EUCLID: 58;

      then

       A12: (((1 / 2) * ((G * (i,( width G))) + (G * ((i + 1),( width G))))) + |[ 0 , 1]|) = |[(((1 / 2) * (r1 + r2)) + 0 ), (s1 + 1)]| by EUCLID: 56;

      (r1 + r2) < (r2 + r2) by A6, XREAL_1: 6;

      then ((1 / 2) * (r1 + r2)) < ((1 / 2) * (r2 + r2)) by XREAL_1: 68;

      then

       A13: ((1 / 2) * (r1 + r2)) < ((G * ((i + 1),1)) `1 ) by A2, A11, A3, GOBOARD5: 2;

      ((G * (i,1)) `1 ) = r1 by A1, A8, A3, GOBOARD5: 2;

      hence thesis by A12, A7, A13, A10, A4;

    end;

    theorem :: GOBOARD6:33

    

     Th33: 1 <= i & (i + 1) <= ( len G) implies (((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1)))) - |[ 0 , 1]|) in ( Int ( cell (G,i, 0 )))

    proof

      assume that

       A1: 1 <= i and

       A2: (i + 1) <= ( len G);

      set r1 = ((G * (i,1)) `1 ), s1 = ((G * (i,1)) `2 ), r2 = ((G * ((i + 1),1)) `1 );

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

      then

       A3: 1 <= ( width G) by NAT_1: 14;

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

      then

       A4: 1 <= ( width G) by NAT_1: 14;

      i < (i + 1) by XREAL_1: 29;

      then

       A5: r1 < r2 by A1, A2, A4, GOBOARD5: 3;

      then (r1 + r1) < (r1 + r2) by XREAL_1: 6;

      then

       A6: ((1 / 2) * (r1 + r1)) < ((1 / 2) * (r1 + r2)) by XREAL_1: 68;

      i < ( len G) by A2, NAT_1: 13;

      then

       A7: ((G * (1,1)) `2 ) = s1 by A1, A3, GOBOARD5: 1;

      then s1 < (((G * (1,1)) `2 ) + 1) by XREAL_1: 29;

      then

       A8: (s1 - 1) < ((G * (1,1)) `2 ) by XREAL_1: 19;

      1 <= (i + 1) by NAT_1: 11;

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

      then (G * (i,1)) = |[r1, s1]| & (G * ((i + 1),1)) = |[r2, s1]| by A7, EUCLID: 53;

      then ((1 / 2) * (s1 + s1)) = s1 & ((G * (i,1)) + (G * ((i + 1),1))) = |[(r1 + r2), (s1 + s1)]| by EUCLID: 56;

      then ((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1)))) = |[((1 / 2) * (r1 + r2)), s1]| by EUCLID: 58;

      

      then

       A9: (((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1)))) - |[ 0 , 1]|) = |[(((1 / 2) * (r1 + r2)) - 0 ), (s1 - 1)]| by EUCLID: 62

      .= |[((1 / 2) * (r1 + r2)), (s1 - 1)]|;

      (r1 + r2) < (r2 + r2) by A5, XREAL_1: 6;

      then

       A10: ((1 / 2) * (r1 + r2)) < ((1 / 2) * (r2 + r2)) by XREAL_1: 68;

      i < ( len G) by A2, NAT_1: 13;

      then ( Int ( cell (G,i, 0 ))) = { |[r, s]| : ((G * (i,1)) `1 ) < r & r < ((G * ((i + 1),1)) `1 ) & s < ((G * (1,1)) `2 ) } by A1, Th24;

      hence thesis by A9, A6, A10, A8;

    end;

    theorem :: GOBOARD6:34

    

     Th34: 1 <= j & (j + 1) <= ( width G) implies (((1 / 2) * ((G * (( len G),j)) + (G * (( len G),(j + 1))))) + |[1, 0 ]|) in ( Int ( cell (G,( len G),j)))

    proof

      assume that

       A1: 1 <= j and

       A2: (j + 1) <= ( width G);

      set s1 = ((G * (( len G),j)) `2 ), r1 = ((G * (( len G),j)) `1 ), s2 = ((G * (( len G),(j + 1))) `2 );

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

      then

       A3: 1 <= ( len G) by NAT_1: 14;

      j < ( width G) by A2, NAT_1: 13;

      then

       A4: ( Int ( cell (G,( len G),j))) = { |[r, s]| : ((G * (( len G),1)) `1 ) < r & ((G * (1,j)) `2 ) < s & s < ((G * (1,(j + 1))) `2 ) } by A1, Th23;

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

      then

       A5: 1 <= ( len G) by NAT_1: 14;

      j < (j + 1) by XREAL_1: 29;

      then

       A6: s1 < s2 by A1, A2, A5, GOBOARD5: 4;

      then (s1 + s1) < (s1 + s2) by XREAL_1: 6;

      then

       A7: ((1 / 2) * (s1 + s1)) < ((1 / 2) * (s1 + s2)) by XREAL_1: 68;

      

       A8: j < ( width G) by A2, NAT_1: 13;

      then

       A9: ((G * (( len G),1)) `1 ) = r1 by A1, A3, GOBOARD5: 2;

      then

       A10: ((G * (( len G),1)) `1 ) < (r1 + 1) by XREAL_1: 29;

      

       A11: 1 <= (j + 1) by NAT_1: 11;

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

      then (G * (( len G),j)) = |[r1, s1]| & (G * (( len G),(j + 1))) = |[r1, s2]| by A9, EUCLID: 53;

      then ((1 / 2) * (r1 + r1)) = r1 & ((G * (( len G),j)) + (G * (( len G),(j + 1)))) = |[(r1 + r1), (s1 + s2)]| by EUCLID: 56;

      then ((1 / 2) * ((G * (( len G),j)) + (G * (( len G),(j + 1))))) = |[r1, ((1 / 2) * (s1 + s2))]| by EUCLID: 58;

      then

       A12: (((1 / 2) * ((G * (( len G),j)) + (G * (( len G),(j + 1))))) + |[1, 0 ]|) = |[(r1 + 1), (((1 / 2) * (s1 + s2)) + 0 )]| by EUCLID: 56;

      (s1 + s2) < (s2 + s2) by A6, XREAL_1: 6;

      then ((1 / 2) * (s1 + s2)) < ((1 / 2) * (s2 + s2)) by XREAL_1: 68;

      then

       A13: ((1 / 2) * (s1 + s2)) < ((G * (1,(j + 1))) `2 ) by A2, A11, A3, GOBOARD5: 1;

      ((G * (1,j)) `2 ) = s1 by A1, A8, A3, GOBOARD5: 1;

      hence thesis by A12, A7, A13, A10, A4;

    end;

    theorem :: GOBOARD6:35

    

     Th35: 1 <= j & (j + 1) <= ( width G) implies (((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1))))) - |[1, 0 ]|) in ( Int ( cell (G, 0 ,j)))

    proof

      assume that

       A1: 1 <= j and

       A2: (j + 1) <= ( width G);

      set s1 = ((G * (1,j)) `2 ), r1 = ((G * (1,j)) `1 ), s2 = ((G * (1,(j + 1))) `2 );

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

      then

       A3: 1 <= ( len G) by NAT_1: 14;

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

      then

       A4: 1 <= ( len G) by NAT_1: 14;

      j < (j + 1) by XREAL_1: 29;

      then

       A5: s1 < s2 by A1, A2, A4, GOBOARD5: 4;

      then (s1 + s1) < (s1 + s2) by XREAL_1: 6;

      then

       A6: ((1 / 2) * (s1 + s1)) < ((1 / 2) * (s1 + s2)) by XREAL_1: 68;

      j < ( width G) by A2, NAT_1: 13;

      then

       A7: ((G * (1,1)) `1 ) = r1 by A1, A3, GOBOARD5: 2;

      then r1 < (((G * (1,1)) `1 ) + 1) by XREAL_1: 29;

      then

       A8: (r1 - 1) < ((G * (1,1)) `1 ) by XREAL_1: 19;

      1 <= (j + 1) by NAT_1: 11;

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

      then (G * (1,j)) = |[r1, s1]| & (G * (1,(j + 1))) = |[r1, s2]| by A7, EUCLID: 53;

      then ((1 / 2) * (r1 + r1)) = r1 & ((G * (1,j)) + (G * (1,(j + 1)))) = |[(r1 + r1), (s1 + s2)]| by EUCLID: 56;

      then ((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1))))) = |[r1, ((1 / 2) * (s1 + s2))]| by EUCLID: 58;

      

      then

       A9: (((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1))))) - |[1, 0 ]|) = |[(r1 - 1), (((1 / 2) * (s1 + s2)) - 0 )]| by EUCLID: 62

      .= |[(r1 - 1), ((1 / 2) * (s1 + s2))]|;

      (s1 + s2) < (s2 + s2) by A5, XREAL_1: 6;

      then

       A10: ((1 / 2) * (s1 + s2)) < ((1 / 2) * (s2 + s2)) by XREAL_1: 68;

      j < ( width G) by A2, NAT_1: 13;

      then ( Int ( cell (G, 0 ,j))) = { |[r, s]| : r < ((G * (1,1)) `1 ) & ((G * (1,j)) `2 ) < s & s < ((G * (1,(j + 1))) `2 ) } by A1, Th20;

      hence thesis by A9, A6, A10, A8;

    end;

    theorem :: GOBOARD6:36

    

     Th36: ((G * (1,1)) - |[1, 1]|) in ( Int ( cell (G, 0 , 0 )))

    proof

      set s1 = ((G * (1,1)) `2 ), r1 = ((G * (1,1)) `1 );

      (G * (1,1)) = |[r1, s1]| by EUCLID: 53;

      then

       A1: ((G * (1,1)) - |[1, 1]|) = |[(r1 - 1), (s1 - 1)]| by EUCLID: 62;

      s1 < (((G * (1,1)) `2 ) + 1) by XREAL_1: 29;

      then

       A2: (s1 - 1) < ((G * (1,1)) `2 ) by XREAL_1: 19;

      r1 < (((G * (1,1)) `1 ) + 1) by XREAL_1: 29;

      then

       A3: (r1 - 1) < ((G * (1,1)) `1 ) by XREAL_1: 19;

      ( Int ( cell (G, 0 , 0 ))) = { |[r, s]| : r < ((G * (1,1)) `1 ) & s < ((G * (1,1)) `2 ) } by Th18;

      hence thesis by A1, A2, A3;

    end;

    theorem :: GOBOARD6:37

    

     Th37: ((G * (( len G),( width G))) + |[1, 1]|) in ( Int ( cell (G,( len G),( width G))))

    proof

      set s1 = ((G * (( len G),( width G))) `2 ), r1 = ((G * (( len G),( width G))) `1 );

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

      then

       A1: 1 <= ( len G) by NAT_1: 14;

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

      then

       A2: 1 <= ( width G) by NAT_1: 14;

      then ((G * (( len G),1)) `1 ) = r1 by A1, GOBOARD5: 2;

      then

       A3: (r1 + 1) > ((G * (( len G),1)) `1 ) by XREAL_1: 29;

      (G * (( len G),( width G))) = |[r1, s1]| by EUCLID: 53;

      then

       A4: ((G * (( len G),( width G))) + |[1, 1]|) = |[(r1 + 1), (s1 + 1)]| by EUCLID: 56;

      ((G * (1,( width G))) `2 ) = s1 by A2, A1, GOBOARD5: 1;

      then

       A5: (s1 + 1) > ((G * (1,( width G))) `2 ) by XREAL_1: 29;

      ( Int ( cell (G,( len G),( width G)))) = { |[r, s]| : ((G * (( len G),1)) `1 ) < r & ((G * (1,( width G))) `2 ) < s } by Th22;

      hence thesis by A4, A5, A3;

    end;

    theorem :: GOBOARD6:38

    

     Th38: ((G * (1,( width G))) + |[( - 1), 1]|) in ( Int ( cell (G, 0 ,( width G))))

    proof

      set s1 = ((G * (1,( width G))) `2 ), r1 = ((G * (1,( width G))) `1 );

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

      then

       A1: 1 <= ( len G) by NAT_1: 14;

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

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

      then ((G * (1,1)) `1 ) = r1 by A1, GOBOARD5: 2;

      then r1 < (((G * (1,1)) `1 ) + 1) by XREAL_1: 29;

      then

       A2: (s1 + 1) > ((G * (1,( width G))) `2 ) & (r1 - 1) < ((G * (1,1)) `1 ) by XREAL_1: 19, XREAL_1: 29;

      (G * (1,( width G))) = |[r1, s1]| by EUCLID: 53;

      

      then

       A3: ((G * (1,( width G))) + |[( - 1), 1]|) = |[(r1 + ( - 1)), (s1 + 1)]| by EUCLID: 56

      .= |[(r1 - 1), (s1 + 1)]|;

      ( Int ( cell (G, 0 ,( width G)))) = { |[r, s]| : r < ((G * (1,1)) `1 ) & ((G * (1,( width G))) `2 ) < s } by Th19;

      hence thesis by A3, A2;

    end;

    theorem :: GOBOARD6:39

    

     Th39: ((G * (( len G),1)) + |[1, ( - 1)]|) in ( Int ( cell (G,( len G), 0 )))

    proof

      set s1 = ((G * (( len G),1)) `2 ), r1 = ((G * (( len G),1)) `1 );

      

       A1: (r1 + 1) > ((G * (( len G),1)) `1 ) by XREAL_1: 29;

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

      then

       A2: 1 <= ( len G) by NAT_1: 14;

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

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

      then ((G * (1,1)) `2 ) = s1 by A2, GOBOARD5: 1;

      then s1 < (((G * (1,1)) `2 ) + 1) by XREAL_1: 29;

      then

       A3: (s1 - 1) < ((G * (1,1)) `2 ) by XREAL_1: 19;

      (G * (( len G),1)) = |[r1, s1]| by EUCLID: 53;

      

      then

       A4: ((G * (( len G),1)) + |[1, ( - 1)]|) = |[(r1 + 1), (s1 + ( - 1))]| by EUCLID: 56

      .= |[(r1 + 1), (s1 - 1)]|;

      ( Int ( cell (G,( len G), 0 ))) = { |[r, s]| : ((G * (( len G),1)) `1 ) < r & s < ((G * (1,1)) `2 ) } by Th21;

      hence thesis by A4, A3, A1;

    end;

    theorem :: GOBOARD6:40

    

     Th40: 1 <= i & i < ( len G) & 1 <= j & j < ( width G) implies ( LSeg (((1 / 2) * ((G * (i,j)) + (G * ((i + 1),(j + 1))))),((1 / 2) * ((G * (i,j)) + (G * (i,(j + 1))))))) c= (( Int ( cell (G,i,j))) \/ {((1 / 2) * ((G * (i,j)) + (G * (i,(j + 1)))))})

    proof

      assume that

       A1: 1 <= i and

       A2: i < ( len G) and

       A3: 1 <= j and

       A4: j < ( width G);

      let x be object;

      assume

       A5: x in ( LSeg (((1 / 2) * ((G * (i,j)) + (G * ((i + 1),(j + 1))))),((1 / 2) * ((G * (i,j)) + (G * (i,(j + 1)))))));

      then

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

      consider r such that

       A6: p = (((1 - r) * ((1 / 2) * ((G * (i,j)) + (G * ((i + 1),(j + 1)))))) + (r * ((1 / 2) * ((G * (i,j)) + (G * (i,(j + 1))))))) and

       A7: 0 <= r and

       A8: r <= 1 by A5;

      now

        per cases by A8, XXREAL_0: 1;

          case r = 1;

          

          then p = (( 0. ( TOP-REAL 2)) + (1 * ((1 / 2) * ((G * (i,j)) + (G * (i,(j + 1))))))) by A6, RLVECT_1: 10

          .= (1 * ((1 / 2) * ((G * (i,j)) + (G * (i,(j + 1)))))) by RLVECT_1: 4

          .= ((1 / 2) * ((G * (i,j)) + (G * (i,(j + 1))))) by RLVECT_1:def 8;

          hence p in {((1 / 2) * ((G * (i,j)) + (G * (i,(j + 1)))))} by TARSKI:def 1;

        end;

          case

           A9: r < 1;

          set r3 = ((1 - r) * (1 / 2)), s3 = (r * (1 / 2));

          set r1 = ((G * (i,1)) `1 ), r2 = ((G * ((i + 1),1)) `1 ), s1 = ((G * (1,j)) `2 ), s2 = ((G * (1,(j + 1))) `2 );

          

           A10: ((r3 * (s1 + s1)) + (s3 * (s1 + s1))) = s1;

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

          then

           A11: 1 <= ( len G) by NAT_1: 14;

          

           A12: (j + 1) <= ( width G) by A4, NAT_1: 13;

          j < (j + 1) by XREAL_1: 29;

          then

           A13: s1 < s2 by A3, A12, A11, GOBOARD5: 4;

          then

           A14: (s1 + s1) < (s1 + s2) by XREAL_1: 6;

          then

           A15: (s3 * (s1 + s1)) <= (s3 * (s1 + s2)) by A7, XREAL_1: 64;

          (1 - r) > 0 by A9, XREAL_1: 50;

          then

           A16: r3 > ((1 / 2) * 0 ) by XREAL_1: 68;

          then (r3 * (s1 + s1)) < (r3 * (s1 + s2)) by A14, XREAL_1: 68;

          then

           A17: s1 < ((r3 * (s1 + s2)) + (s3 * (s1 + s2))) by A15, A10, XREAL_1: 8;

          

           A18: (s1 + s2) < (s2 + s2) by A13, XREAL_1: 6;

          then

           A19: (s3 * (s1 + s2)) <= (s3 * (s2 + s2)) by A7, XREAL_1: 64;

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

          then

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

          

           A21: 1 <= (i + 1) by A1, NAT_1: 13;

          

           A22: ( Int ( cell (G,i,j))) = { |[r9, s9]| : r1 < r9 & r9 < r2 & s1 < s9 & s9 < s2 } by A1, A2, A3, A4, Th26;

          

           A23: 1 <= (j + 1) by A3, NAT_1: 13;

          

           A24: (G * (i,(j + 1))) = |[((G * (i,(j + 1))) `1 ), ((G * (i,(j + 1))) `2 )]| by EUCLID: 53

          .= |[r1, ((G * (i,(j + 1))) `2 )]| by A1, A2, A23, A12, GOBOARD5: 2

          .= |[r1, s2]| by A1, A2, A23, A12, GOBOARD5: 1;

          

           A25: ((r3 * (s2 + s2)) + (s3 * (s2 + s2))) = s2;

          (r3 * (s1 + s2)) < (r3 * (s2 + s2)) by A16, A18, XREAL_1: 68;

          then

           A26: ((r3 * (s1 + s2)) + (s3 * (s1 + s2))) < s2 by A19, A25, XREAL_1: 8;

          

           A27: (i + 1) <= ( len G) by A2, NAT_1: 13;

          i < (i + 1) by XREAL_1: 29;

          then

           A28: r1 < r2 by A1, A27, A20, GOBOARD5: 3;

          then (r1 + r1) < (r2 + r2) by XREAL_1: 8;

          then

           A29: (s3 * (r1 + r1)) <= (s3 * (r2 + r2)) by A7, XREAL_1: 64;

          (r1 + r2) < (r2 + r2) by A28, XREAL_1: 6;

          then

           A30: (r3 * (r1 + r2)) < (r3 * (r2 + r2)) by A16, XREAL_1: 68;

          ((r3 * (r2 + r2)) + (s3 * (r2 + r2))) = r2;

          then

           A31: ((r3 * (r1 + r2)) + (s3 * (r1 + r1))) < r2 by A30, A29, XREAL_1: 8;

          

           A32: (G * (i,j)) = |[((G * (i,j)) `1 ), ((G * (i,j)) `2 )]| by EUCLID: 53

          .= |[r1, ((G * (i,j)) `2 )]| by A1, A2, A3, A4, GOBOARD5: 2

          .= |[r1, s1]| by A1, A2, A3, A4, GOBOARD5: 1;

          

           A33: (G * ((i + 1),(j + 1))) = |[((G * ((i + 1),(j + 1))) `1 ), ((G * ((i + 1),(j + 1))) `2 )]| by EUCLID: 53

          .= |[r2, ((G * ((i + 1),(j + 1))) `2 )]| by A23, A12, A21, A27, GOBOARD5: 2

          .= |[r2, s2]| by A23, A12, A21, A27, GOBOARD5: 1;

          

           A34: ((r3 * (r1 + r1)) + (s3 * (r1 + r1))) = r1;

          (r1 + r1) < (r1 + r2) by A28, XREAL_1: 6;

          then (r3 * (r1 + r1)) < (r3 * (r1 + r2)) by A16, XREAL_1: 68;

          then

           A35: r1 < ((r3 * (r1 + r2)) + (s3 * (r1 + r1))) by A34, XREAL_1: 6;

          p = ((r3 * ((G * (i,j)) + (G * ((i + 1),(j + 1))))) + (r * ((1 / 2) * ((G * (i,j)) + (G * (i,(j + 1))))))) by A6, RLVECT_1:def 7

          .= ((r3 * ((G * (i,j)) + (G * ((i + 1),(j + 1))))) + (s3 * ((G * (i,j)) + (G * (i,(j + 1)))))) by RLVECT_1:def 7

          .= ((r3 * |[(r1 + r2), (s1 + s2)]|) + (s3 * ((G * (i,j)) + (G * (i,(j + 1)))))) by A32, A33, EUCLID: 56

          .= ((r3 * |[(r1 + r2), (s1 + s2)]|) + (s3 * |[(r1 + r1), (s1 + s2)]|)) by A32, A24, EUCLID: 56

          .= ( |[(r3 * (r1 + r2)), (r3 * (s1 + s2))]| + (s3 * |[(r1 + r1), (s1 + s2)]|)) by EUCLID: 58

          .= ( |[(r3 * (r1 + r2)), (r3 * (s1 + s2))]| + |[(s3 * (r1 + r1)), (s3 * (s1 + s2))]|) by EUCLID: 58

          .= |[((r3 * (r1 + r2)) + (s3 * (r1 + r1))), ((r3 * (s1 + s2)) + (s3 * (s1 + s2)))]| by EUCLID: 56;

          hence p in ( Int ( cell (G,i,j))) by A35, A31, A17, A26, A22;

        end;

      end;

      hence thesis by XBOOLE_0:def 3;

    end;

    theorem :: GOBOARD6:41

    

     Th41: 1 <= i & i < ( len G) & 1 <= j & j < ( width G) implies ( LSeg (((1 / 2) * ((G * (i,j)) + (G * ((i + 1),(j + 1))))),((1 / 2) * ((G * (i,(j + 1))) + (G * ((i + 1),(j + 1))))))) c= (( Int ( cell (G,i,j))) \/ {((1 / 2) * ((G * (i,(j + 1))) + (G * ((i + 1),(j + 1)))))})

    proof

      assume that

       A1: 1 <= i and

       A2: i < ( len G) and

       A3: 1 <= j and

       A4: j < ( width G);

      let x be object;

      assume

       A5: x in ( LSeg (((1 / 2) * ((G * (i,j)) + (G * ((i + 1),(j + 1))))),((1 / 2) * ((G * (i,(j + 1))) + (G * ((i + 1),(j + 1)))))));

      then

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

      consider r such that

       A6: p = (((1 - r) * ((1 / 2) * ((G * (i,j)) + (G * ((i + 1),(j + 1)))))) + (r * ((1 / 2) * ((G * (i,(j + 1))) + (G * ((i + 1),(j + 1))))))) and

       A7: 0 <= r and

       A8: r <= 1 by A5;

      now

        per cases by A8, XXREAL_0: 1;

          case r = 1;

          

          then p = (( 0. ( TOP-REAL 2)) + (1 * ((1 / 2) * ((G * (i,(j + 1))) + (G * ((i + 1),(j + 1))))))) by A6, RLVECT_1: 10

          .= (1 * ((1 / 2) * ((G * (i,(j + 1))) + (G * ((i + 1),(j + 1)))))) by RLVECT_1: 4

          .= ((1 / 2) * ((G * (i,(j + 1))) + (G * ((i + 1),(j + 1))))) by RLVECT_1:def 8;

          hence p in {((1 / 2) * ((G * (i,(j + 1))) + (G * ((i + 1),(j + 1)))))} by TARSKI:def 1;

        end;

          case

           A9: r < 1;

          set r3 = ((1 - r) * (1 / 2)), s3 = (r * (1 / 2));

          set r1 = ((G * (i,1)) `1 ), r2 = ((G * ((i + 1),1)) `1 ), s1 = ((G * (1,j)) `2 ), s2 = ((G * (1,(j + 1))) `2 );

          

           A10: ((r3 * (r1 + r1)) + (s3 * (r1 + r1))) = r1;

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

          then

           A11: 1 <= ( width G) by NAT_1: 14;

          

           A12: (i + 1) <= ( len G) by A2, NAT_1: 13;

          i < (i + 1) by XREAL_1: 29;

          then

           A13: r1 < r2 by A1, A12, A11, GOBOARD5: 3;

          then

           A14: (r1 + r1) < (r1 + r2) by XREAL_1: 6;

          then

           A15: (s3 * (r1 + r1)) <= (s3 * (r1 + r2)) by A7, XREAL_1: 64;

          (1 - r) > 0 by A9, XREAL_1: 50;

          then

           A16: r3 > ((1 / 2) * 0 ) by XREAL_1: 68;

          then (r3 * (r1 + r1)) < (r3 * (r1 + r2)) by A14, XREAL_1: 68;

          then

           A17: r1 < ((r3 * (r1 + r2)) + (s3 * (r1 + r2))) by A15, A10, XREAL_1: 8;

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

          then

           A18: 1 <= ( len G) by NAT_1: 14;

          

           A19: 1 <= (i + 1) by A1, NAT_1: 13;

          (r1 + r2) < (r2 + r2) by A13, XREAL_1: 8;

          then

           A20: (s3 * (r1 + r2)) <= (s3 * (r2 + r2)) by A7, XREAL_1: 64;

          

           A21: (j + 1) <= ( width G) by A4, NAT_1: 13;

          (r1 + r2) < (r2 + r2) by A13, XREAL_1: 6;

          then

           A22: (r3 * (r1 + r2)) < (r3 * (r2 + r2)) by A16, XREAL_1: 68;

          ((r3 * (r2 + r2)) + (s3 * (r2 + r2))) = r2;

          then

           A23: ((r3 * (r1 + r2)) + (s3 * (r1 + r2))) < r2 by A22, A20, XREAL_1: 8;

          

           A24: ( Int ( cell (G,i,j))) = { |[r9, s9]| : r1 < r9 & r9 < r2 & s1 < s9 & s9 < s2 } by A1, A2, A3, A4, Th26;

          

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

          j < (j + 1) by XREAL_1: 29;

          then

           A26: s1 < s2 by A3, A21, A18, GOBOARD5: 4;

          then

           A27: (s1 + s1) < (s1 + s2) by XREAL_1: 6;

          

           A28: (G * ((i + 1),(j + 1))) = |[((G * ((i + 1),(j + 1))) `1 ), ((G * ((i + 1),(j + 1))) `2 )]| by EUCLID: 53

          .= |[r2, ((G * ((i + 1),(j + 1))) `2 )]| by A25, A21, A19, A12, GOBOARD5: 2

          .= |[r2, s2]| by A25, A21, A19, A12, GOBOARD5: 1;

          (s1 + s2) < (s2 + s2) by A26, XREAL_1: 6;

          then (s1 + s1) < (s2 + s2) by A27, XXREAL_0: 2;

          then

           A29: (s3 * (s1 + s1)) <= (s3 * (s2 + s2)) by A7, XREAL_1: 64;

          

           A30: (G * (i,j)) = |[((G * (i,j)) `1 ), ((G * (i,j)) `2 )]| by EUCLID: 53

          .= |[r1, ((G * (i,j)) `2 )]| by A1, A2, A3, A4, GOBOARD5: 2

          .= |[r1, s1]| by A1, A2, A3, A4, GOBOARD5: 1;

          

           A31: ((r3 * (s2 + s2)) + (s3 * (s2 + s2))) = s2;

          

           A32: (G * (i,(j + 1))) = |[((G * (i,(j + 1))) `1 ), ((G * (i,(j + 1))) `2 )]| by EUCLID: 53

          .= |[r1, ((G * (i,(j + 1))) `2 )]| by A1, A2, A25, A21, GOBOARD5: 2

          .= |[r1, s2]| by A1, A2, A25, A21, GOBOARD5: 1;

          

           A33: ((r3 * (s1 + s1)) + (s3 * (s1 + s1))) = s1;

          (s1 + s2) < (s2 + s2) by A26, XREAL_1: 6;

          then (r3 * (s1 + s2)) < (r3 * (s2 + s2)) by A16, XREAL_1: 68;

          then

           A34: ((r3 * (s1 + s2)) + (s3 * (s2 + s2))) < s2 by A31, XREAL_1: 8;

          (r3 * (s1 + s1)) < (r3 * (s1 + s2)) by A16, A27, XREAL_1: 68;

          then

           A35: s1 < ((r3 * (s1 + s2)) + (s3 * (s2 + s2))) by A29, A33, XREAL_1: 8;

          p = ((r3 * ((G * (i,j)) + (G * ((i + 1),(j + 1))))) + (r * ((1 / 2) * ((G * (i,(j + 1))) + (G * ((i + 1),(j + 1))))))) by A6, RLVECT_1:def 7

          .= ((r3 * ((G * (i,j)) + (G * ((i + 1),(j + 1))))) + (s3 * ((G * (i,(j + 1))) + (G * ((i + 1),(j + 1)))))) by RLVECT_1:def 7

          .= ((r3 * |[(r1 + r2), (s1 + s2)]|) + (s3 * ((G * (i,(j + 1))) + (G * ((i + 1),(j + 1)))))) by A30, A28, EUCLID: 56

          .= ((r3 * |[(r1 + r2), (s1 + s2)]|) + (s3 * |[(r1 + r2), (s2 + s2)]|)) by A28, A32, EUCLID: 56

          .= ( |[(r3 * (r1 + r2)), (r3 * (s1 + s2))]| + (s3 * |[(r1 + r2), (s2 + s2)]|)) by EUCLID: 58

          .= ( |[(r3 * (r1 + r2)), (r3 * (s1 + s2))]| + |[(s3 * (r1 + r2)), (s3 * (s2 + s2))]|) by EUCLID: 58

          .= |[((r3 * (r1 + r2)) + (s3 * (r1 + r2))), ((r3 * (s1 + s2)) + (s3 * (s2 + s2)))]| by EUCLID: 56;

          hence p in ( Int ( cell (G,i,j))) by A17, A23, A35, A34, A24;

        end;

      end;

      hence thesis by XBOOLE_0:def 3;

    end;

    theorem :: GOBOARD6:42

    

     Th42: 1 <= i & i < ( len G) & 1 <= j & j < ( width G) implies ( LSeg (((1 / 2) * ((G * (i,j)) + (G * ((i + 1),(j + 1))))),((1 / 2) * ((G * ((i + 1),j)) + (G * ((i + 1),(j + 1))))))) c= (( Int ( cell (G,i,j))) \/ {((1 / 2) * ((G * ((i + 1),j)) + (G * ((i + 1),(j + 1)))))})

    proof

      assume that

       A1: 1 <= i and

       A2: i < ( len G) and

       A3: 1 <= j and

       A4: j < ( width G);

      let x be object;

      assume

       A5: x in ( LSeg (((1 / 2) * ((G * (i,j)) + (G * ((i + 1),(j + 1))))),((1 / 2) * ((G * ((i + 1),j)) + (G * ((i + 1),(j + 1)))))));

      then

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

      consider r such that

       A6: p = (((1 - r) * ((1 / 2) * ((G * (i,j)) + (G * ((i + 1),(j + 1)))))) + (r * ((1 / 2) * ((G * ((i + 1),j)) + (G * ((i + 1),(j + 1))))))) and

       A7: 0 <= r and

       A8: r <= 1 by A5;

      now

        per cases by A8, XXREAL_0: 1;

          case r = 1;

          

          then p = (( 0. ( TOP-REAL 2)) + (1 * ((1 / 2) * ((G * ((i + 1),j)) + (G * ((i + 1),(j + 1))))))) by A6, RLVECT_1: 10

          .= (1 * ((1 / 2) * ((G * ((i + 1),j)) + (G * ((i + 1),(j + 1)))))) by RLVECT_1: 4

          .= ((1 / 2) * ((G * ((i + 1),j)) + (G * ((i + 1),(j + 1))))) by RLVECT_1:def 8;

          hence p in {((1 / 2) * ((G * ((i + 1),j)) + (G * ((i + 1),(j + 1)))))} by TARSKI:def 1;

        end;

          case

           A9: r < 1;

          set r3 = ((1 - r) * (1 / 2)), s3 = (r * (1 / 2));

          set r1 = ((G * (i,1)) `1 ), r2 = ((G * ((i + 1),1)) `1 ), s1 = ((G * (1,j)) `2 ), s2 = ((G * (1,(j + 1))) `2 );

          

           A10: ((r3 * (r1 + r1)) + (s3 * (r1 + r1))) = r1;

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

          then

           A11: 1 <= ( width G) by NAT_1: 14;

          

           A12: (i + 1) <= ( len G) by A2, NAT_1: 13;

          i < (i + 1) by XREAL_1: 29;

          then

           A13: r1 < r2 by A1, A12, A11, GOBOARD5: 3;

          then

           A14: (r1 + r1) < (r1 + r2) by XREAL_1: 6;

          (r1 + r2) < (r2 + r2) by A13, XREAL_1: 6;

          then (r1 + r1) < (r2 + r2) by A14, XXREAL_0: 2;

          then

           A15: (s3 * (r1 + r1)) <= (s3 * (r2 + r2)) by A7, XREAL_1: 64;

          (1 - r) > 0 by A9, XREAL_1: 50;

          then

           A16: r3 > ((1 / 2) * 0 ) by XREAL_1: 68;

          then (r3 * (r1 + r1)) < (r3 * (r1 + r2)) by A14, XREAL_1: 68;

          then

           A17: r1 < ((r3 * (r1 + r2)) + (s3 * (r2 + r2))) by A15, A10, XREAL_1: 8;

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

          then

           A18: 1 <= ( len G) by NAT_1: 14;

          

           A19: 1 <= (j + 1) by A3, NAT_1: 13;

          

           A20: ( Int ( cell (G,i,j))) = { |[r9, s9]| : r1 < r9 & r9 < r2 & s1 < s9 & s9 < s2 } by A1, A2, A3, A4, Th26;

          

           A21: ((r3 * (s2 + s2)) + (s3 * (s2 + s2))) = s2;

          

           A22: (G * (i,j)) = |[((G * (i,j)) `1 ), ((G * (i,j)) `2 )]| by EUCLID: 53

          .= |[r1, ((G * (i,j)) `2 )]| by A1, A2, A3, A4, GOBOARD5: 2

          .= |[r1, s1]| by A1, A2, A3, A4, GOBOARD5: 1;

          

           A23: ((r3 * (s1 + s1)) + (s3 * (s1 + s1))) = s1;

          

           A24: 1 <= (i + 1) by A1, NAT_1: 13;

          

           A25: (G * ((i + 1),j)) = |[((G * ((i + 1),j)) `1 ), ((G * ((i + 1),j)) `2 )]| by EUCLID: 53

          .= |[r2, ((G * ((i + 1),j)) `2 )]| by A3, A4, A24, A12, GOBOARD5: 2

          .= |[r2, s1]| by A3, A4, A24, A12, GOBOARD5: 1;

          

           A26: ((r3 * (r2 + r2)) + (s3 * (r2 + r2))) = r2;

          (r1 + r2) < (r2 + r2) by A13, XREAL_1: 6;

          then (r3 * (r1 + r2)) < (r3 * (r2 + r2)) by A16, XREAL_1: 68;

          then

           A27: ((r3 * (r1 + r2)) + (s3 * (r2 + r2))) < r2 by A26, XREAL_1: 8;

          

           A28: (j + 1) <= ( width G) by A4, NAT_1: 13;

          

           A29: (G * ((i + 1),(j + 1))) = |[((G * ((i + 1),(j + 1))) `1 ), ((G * ((i + 1),(j + 1))) `2 )]| by EUCLID: 53

          .= |[r2, ((G * ((i + 1),(j + 1))) `2 )]| by A19, A28, A24, A12, GOBOARD5: 2

          .= |[r2, s2]| by A19, A28, A24, A12, GOBOARD5: 1;

          j < (j + 1) by XREAL_1: 29;

          then

           A30: s1 < s2 by A3, A28, A18, GOBOARD5: 4;

          then

           A31: (s1 + s1) < (s1 + s2) by XREAL_1: 6;

          then

           A32: (s3 * (s1 + s1)) <= (s3 * (s1 + s2)) by A7, XREAL_1: 64;

          (r3 * (s1 + s1)) < (r3 * (s1 + s2)) by A16, A31, XREAL_1: 68;

          then

           A33: s1 < ((r3 * (s1 + s2)) + (s3 * (s1 + s2))) by A32, A23, XREAL_1: 8;

          

           A34: (s1 + s2) < (s2 + s2) by A30, XREAL_1: 6;

          then

           A35: (s3 * (s1 + s2)) <= (s3 * (s2 + s2)) by A7, XREAL_1: 64;

          (r3 * (s1 + s2)) < (r3 * (s2 + s2)) by A16, A34, XREAL_1: 68;

          then

           A36: ((r3 * (s1 + s2)) + (s3 * (s1 + s2))) < s2 by A35, A21, XREAL_1: 8;

          p = ((r3 * ((G * (i,j)) + (G * ((i + 1),(j + 1))))) + (r * ((1 / 2) * ((G * ((i + 1),j)) + (G * ((i + 1),(j + 1))))))) by A6, RLVECT_1:def 7

          .= ((r3 * ((G * (i,j)) + (G * ((i + 1),(j + 1))))) + (s3 * ((G * ((i + 1),j)) + (G * ((i + 1),(j + 1)))))) by RLVECT_1:def 7

          .= ((r3 * |[(r1 + r2), (s1 + s2)]|) + (s3 * ((G * ((i + 1),j)) + (G * ((i + 1),(j + 1)))))) by A22, A29, EUCLID: 56

          .= ((r3 * |[(r1 + r2), (s1 + s2)]|) + (s3 * |[(r2 + r2), (s1 + s2)]|)) by A29, A25, EUCLID: 56

          .= ( |[(r3 * (r1 + r2)), (r3 * (s1 + s2))]| + (s3 * |[(r2 + r2), (s1 + s2)]|)) by EUCLID: 58

          .= ( |[(r3 * (r1 + r2)), (r3 * (s1 + s2))]| + |[(s3 * (r2 + r2)), (s3 * (s1 + s2))]|) by EUCLID: 58

          .= |[((r3 * (r1 + r2)) + (s3 * (r2 + r2))), ((r3 * (s1 + s2)) + (s3 * (s1 + s2)))]| by EUCLID: 56;

          hence p in ( Int ( cell (G,i,j))) by A17, A27, A33, A36, A20;

        end;

      end;

      hence thesis by XBOOLE_0:def 3;

    end;

    theorem :: GOBOARD6:43

    

     Th43: 1 <= i & i < ( len G) & 1 <= j & j < ( width G) implies ( LSeg (((1 / 2) * ((G * (i,j)) + (G * ((i + 1),(j + 1))))),((1 / 2) * ((G * (i,j)) + (G * ((i + 1),j)))))) c= (( Int ( cell (G,i,j))) \/ {((1 / 2) * ((G * (i,j)) + (G * ((i + 1),j))))})

    proof

      assume that

       A1: 1 <= i and

       A2: i < ( len G) and

       A3: 1 <= j and

       A4: j < ( width G);

      let x be object;

      assume

       A5: x in ( LSeg (((1 / 2) * ((G * (i,j)) + (G * ((i + 1),(j + 1))))),((1 / 2) * ((G * (i,j)) + (G * ((i + 1),j))))));

      then

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

      consider r such that

       A6: p = (((1 - r) * ((1 / 2) * ((G * (i,j)) + (G * ((i + 1),(j + 1)))))) + (r * ((1 / 2) * ((G * (i,j)) + (G * ((i + 1),j)))))) and

       A7: 0 <= r and

       A8: r <= 1 by A5;

      now

        per cases by A8, XXREAL_0: 1;

          case r = 1;

          

          then p = (( 0. ( TOP-REAL 2)) + (1 * ((1 / 2) * ((G * (i,j)) + (G * ((i + 1),j)))))) by A6, RLVECT_1: 10

          .= (1 * ((1 / 2) * ((G * (i,j)) + (G * ((i + 1),j))))) by RLVECT_1: 4

          .= ((1 / 2) * ((G * (i,j)) + (G * ((i + 1),j)))) by RLVECT_1:def 8;

          hence p in {((1 / 2) * ((G * (i,j)) + (G * ((i + 1),j))))} by TARSKI:def 1;

        end;

          case

           A9: r < 1;

          set r3 = ((1 - r) * (1 / 2)), s3 = (r * (1 / 2));

          set r1 = ((G * (i,1)) `1 ), r2 = ((G * ((i + 1),1)) `1 ), s1 = ((G * (1,j)) `2 ), s2 = ((G * (1,(j + 1))) `2 );

          

           A10: ((r3 * (r1 + r1)) + (s3 * (r1 + r1))) = r1;

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

          then

           A11: 1 <= ( width G) by NAT_1: 14;

          

           A12: (i + 1) <= ( len G) by A2, NAT_1: 13;

          i < (i + 1) by XREAL_1: 29;

          then

           A13: r1 < r2 by A1, A12, A11, GOBOARD5: 3;

          then

           A14: (r1 + r1) < (r1 + r2) by XREAL_1: 6;

          then

           A15: (s3 * (r1 + r1)) <= (s3 * (r1 + r2)) by A7, XREAL_1: 64;

          (r1 + r2) < (r2 + r2) by A13, XREAL_1: 8;

          then

           A16: (s3 * (r1 + r2)) <= (s3 * (r2 + r2)) by A7, XREAL_1: 64;

          

           A17: 1 <= (i + 1) by A1, NAT_1: 13;

          (1 - r) > 0 by A9, XREAL_1: 50;

          then

           A18: r3 > ((1 / 2) * 0 ) by XREAL_1: 68;

          then (r3 * (r1 + r1)) < (r3 * (r1 + r2)) by A14, XREAL_1: 68;

          then

           A19: r1 < ((r3 * (r1 + r2)) + (s3 * (r1 + r2))) by A15, A10, XREAL_1: 8;

          (r1 + r2) < (r2 + r2) by A13, XREAL_1: 6;

          then

           A20: (r3 * (r1 + r2)) < (r3 * (r2 + r2)) by A18, XREAL_1: 68;

          ((r3 * (r2 + r2)) + (s3 * (r2 + r2))) = r2;

          then

           A21: ((r3 * (r1 + r2)) + (s3 * (r1 + r2))) < r2 by A20, A16, XREAL_1: 8;

          

           A22: ( Int ( cell (G,i,j))) = { |[r9, s9]| : r1 < r9 & r9 < r2 & s1 < s9 & s9 < s2 } by A1, A2, A3, A4, Th26;

          

           A23: (j + 1) <= ( width G) by A4, NAT_1: 13;

          

           A24: (G * ((i + 1),j)) = |[((G * ((i + 1),j)) `1 ), ((G * ((i + 1),j)) `2 )]| by EUCLID: 53

          .= |[r2, ((G * ((i + 1),j)) `2 )]| by A3, A4, A17, A12, GOBOARD5: 2

          .= |[r2, s1]| by A3, A4, A17, A12, GOBOARD5: 1;

          

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

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

          then

           A26: 1 <= ( len G) by NAT_1: 14;

          

           A27: (G * (i,j)) = |[((G * (i,j)) `1 ), ((G * (i,j)) `2 )]| by EUCLID: 53

          .= |[r1, ((G * (i,j)) `2 )]| by A1, A2, A3, A4, GOBOARD5: 2

          .= |[r1, s1]| by A1, A2, A3, A4, GOBOARD5: 1;

          j < (j + 1) by XREAL_1: 29;

          then

           A28: s1 < s2 by A3, A23, A26, GOBOARD5: 4;

          then (s1 + s2) < (s2 + s2) by XREAL_1: 6;

          then

           A29: (r3 * (s1 + s2)) < (r3 * (s2 + s2)) by A18, XREAL_1: 68;

          

           A30: (G * ((i + 1),(j + 1))) = |[((G * ((i + 1),(j + 1))) `1 ), ((G * ((i + 1),(j + 1))) `2 )]| by EUCLID: 53

          .= |[r2, ((G * ((i + 1),(j + 1))) `2 )]| by A25, A23, A17, A12, GOBOARD5: 2

          .= |[r2, s2]| by A25, A23, A17, A12, GOBOARD5: 1;

          

           A31: ((r3 * (s1 + s1)) + (s3 * (s1 + s1))) = s1;

          

           A32: (s1 + s1) < (s1 + s2) by A28, XREAL_1: 6;

          then (r3 * (s1 + s1)) < (r3 * (s1 + s2)) by A18, XREAL_1: 68;

          then

           A33: s1 < ((r3 * (s1 + s2)) + (s3 * (s1 + s1))) by A31, XREAL_1: 8;

          (s1 + s2) < (s2 + s2) by A28, XREAL_1: 6;

          then (s1 + s1) < (s2 + s2) by A32, XXREAL_0: 2;

          then

           A34: (s3 * (s1 + s1)) <= (s3 * (s2 + s2)) by A7, XREAL_1: 64;

          ((r3 * (s2 + s2)) + (s3 * (s2 + s2))) = s2;

          then

           A35: ((r3 * (s1 + s2)) + (s3 * (s1 + s1))) < s2 by A29, A34, XREAL_1: 8;

          p = ((r3 * ((G * (i,j)) + (G * ((i + 1),(j + 1))))) + (r * ((1 / 2) * ((G * (i,j)) + (G * ((i + 1),j)))))) by A6, RLVECT_1:def 7

          .= ((r3 * ((G * (i,j)) + (G * ((i + 1),(j + 1))))) + (s3 * ((G * (i,j)) + (G * ((i + 1),j))))) by RLVECT_1:def 7

          .= ((r3 * |[(r1 + r2), (s1 + s2)]|) + (s3 * ((G * (i,j)) + (G * ((i + 1),j))))) by A27, A30, EUCLID: 56

          .= ((r3 * |[(r1 + r2), (s1 + s2)]|) + (s3 * |[(r1 + r2), (s1 + s1)]|)) by A27, A24, EUCLID: 56

          .= ( |[(r3 * (r1 + r2)), (r3 * (s1 + s2))]| + (s3 * |[(r1 + r2), (s1 + s1)]|)) by EUCLID: 58

          .= ( |[(r3 * (r1 + r2)), (r3 * (s1 + s2))]| + |[(s3 * (r1 + r2)), (s3 * (s1 + s1))]|) by EUCLID: 58

          .= |[((r3 * (r1 + r2)) + (s3 * (r1 + r2))), ((r3 * (s1 + s2)) + (s3 * (s1 + s1)))]| by EUCLID: 56;

          hence p in ( Int ( cell (G,i,j))) by A19, A21, A33, A35, A22;

        end;

      end;

      hence thesis by XBOOLE_0:def 3;

    end;

    theorem :: GOBOARD6:44

    

     Th44: 1 <= j & j < ( width G) implies ( LSeg ((((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1))))) - |[1, 0 ]|),((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1))))))) c= (( Int ( cell (G, 0 ,j))) \/ {((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1)))))})

    proof

      assume that

       A1: 1 <= j and

       A2: j < ( width G);

      let x be object;

      assume

       A3: x in ( LSeg ((((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1))))) - |[1, 0 ]|),((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1)))))));

      then

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

      consider r such that

       A4: p = (((1 - r) * (((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1))))) - |[1, 0 ]|)) + (r * ((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1))))))) and

       A5: 0 <= r and

       A6: r <= 1 by A3;

      now

        per cases by A6, XXREAL_0: 1;

          case r = 1;

          

          then p = (( 0. ( TOP-REAL 2)) + (1 * ((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1))))))) by A4, RLVECT_1: 10

          .= (1 * ((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1)))))) by RLVECT_1: 4

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

          hence p in {((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1)))))} by TARSKI:def 1;

        end;

          case

           A7: r < 1;

          set r3 = ((1 - r) * (1 / 2)), s3 = (r * (1 / 2));

          set r2 = ((G * (1,1)) `1 ), s1 = ((G * (1,j)) `2 ), s2 = ((G * (1,(j + 1))) `2 );

          

           A8: ((r3 * (s1 + s1)) + (s3 * (s1 + s1))) = s1;

          

           A9: (j + 1) <= ( width G) by A2, NAT_1: 13;

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

          then

           A10: 1 <= ( len G) by NAT_1: 14;

          j < (j + 1) by XREAL_1: 29;

          then

           A11: s1 < s2 by A1, A9, A10, GOBOARD5: 4;

          then

           A12: (s1 + s1) < (s1 + s2) by XREAL_1: 6;

          then

           A13: (s3 * (s1 + s1)) <= (s3 * (s1 + s2)) by A5, XREAL_1: 64;

          

           A14: (1 - r) > 0 by A7, XREAL_1: 50;

          then

           A15: r3 > ((1 / 2) * 0 ) by XREAL_1: 68;

          then (r3 * (s1 + s1)) < (r3 * (s1 + s2)) by A12, XREAL_1: 68;

          then

           A16: s1 < ((r3 * (s1 + s2)) + (s3 * (s1 + s2))) by A13, A8, XREAL_1: 8;

          r2 < (r2 + (1 - r)) by A14, XREAL_1: 29;

          then

           A17: (r2 - (1 - r)) < r2 by XREAL_1: 19;

          

           A18: 1 <= (j + 1) by A1, NAT_1: 13;

          

           A19: (G * (1,(j + 1))) = |[((G * (1,(j + 1))) `1 ), ((G * (1,(j + 1))) `2 )]| by EUCLID: 53

          .= |[r2, s2]| by A18, A9, A10, GOBOARD5: 2;

          

           A20: (s1 + s2) < (s2 + s2) by A11, XREAL_1: 6;

          then

           A21: (s3 * (s1 + s2)) <= (s3 * (s2 + s2)) by A5, XREAL_1: 64;

          

           A22: ( Int ( cell (G, 0 ,j))) = { |[r9, s9]| : r9 < ((G * (1,1)) `1 ) & ((G * (1,j)) `2 ) < s9 & s9 < ((G * (1,(j + 1))) `2 ) } by A1, A2, Th20;

          

           A23: ((r3 * (s2 + s2)) + (s3 * (s2 + s2))) = s2;

          (r3 * (s1 + s2)) < (r3 * (s2 + s2)) by A15, A20, XREAL_1: 68;

          then

           A24: ((r3 * (s1 + s2)) + (s3 * (s1 + s2))) < s2 by A21, A23, XREAL_1: 8;

          

           A25: (G * (1,j)) = |[((G * (1,j)) `1 ), ((G * (1,j)) `2 )]| by EUCLID: 53

          .= |[r2, s1]| by A1, A2, A10, GOBOARD5: 2;

          p = ((((1 - r) * ((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1)))))) - ((1 - r) * |[1, 0 ]|)) + (r * ((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1))))))) by A4, RLVECT_1: 34

          .= (((r3 * ((G * (1,j)) + (G * (1,(j + 1))))) - ((1 - r) * |[1, 0 ]|)) + (r * ((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1))))))) by RLVECT_1:def 7

          .= (((r3 * ((G * (1,j)) + (G * (1,(j + 1))))) - |[((1 - r) * 1), ((1 - r) * 0 )]|) + (r * ((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1))))))) by EUCLID: 58

          .= (((r3 * ((G * (1,j)) + (G * (1,(j + 1))))) - |[(1 - r), 0 ]|) + (s3 * ((G * (1,j)) + (G * (1,(j + 1)))))) by RLVECT_1:def 7

          .= (((r3 * |[(r2 + r2), (s1 + s2)]|) - |[(1 - r), 0 ]|) + (s3 * ((G * (1,j)) + (G * (1,(j + 1)))))) by A19, A25, EUCLID: 56

          .= (((r3 * |[(r2 + r2), (s1 + s2)]|) - |[(1 - r), 0 ]|) + (s3 * |[(r2 + r2), (s1 + s2)]|)) by A19, A25, EUCLID: 56

          .= (( |[(r3 * (r2 + r2)), (r3 * (s1 + s2))]| - |[(1 - r), 0 ]|) + (s3 * |[(r2 + r2), (s1 + s2)]|)) by EUCLID: 58

          .= (( |[(r3 * (r2 + r2)), (r3 * (s1 + s2))]| - |[(1 - r), 0 ]|) + |[(s3 * (r2 + r2)), (s3 * (s1 + s2))]|) by EUCLID: 58

          .= ( |[((r3 * (r2 + r2)) - (1 - r)), ((r3 * (s1 + s2)) - 0 )]| + |[(s3 * (r2 + r2)), (s3 * (s1 + s2))]|) by EUCLID: 62

          .= |[(((r3 * (r2 + r2)) - (1 - r)) + (s3 * (r2 + r2))), ((r3 * (s1 + s2)) + (s3 * (s1 + s2)))]| by EUCLID: 56;

          hence p in ( Int ( cell (G, 0 ,j))) by A17, A16, A24, A22;

        end;

      end;

      hence thesis by XBOOLE_0:def 3;

    end;

    theorem :: GOBOARD6:45

    

     Th45: 1 <= j & j < ( width G) implies ( LSeg ((((1 / 2) * ((G * (( len G),j)) + (G * (( len G),(j + 1))))) + |[1, 0 ]|),((1 / 2) * ((G * (( len G),j)) + (G * (( len G),(j + 1))))))) c= (( Int ( cell (G,( len G),j))) \/ {((1 / 2) * ((G * (( len G),j)) + (G * (( len G),(j + 1)))))})

    proof

      assume that

       A1: 1 <= j and

       A2: j < ( width G);

      let x be object;

      assume

       A3: x in ( LSeg ((((1 / 2) * ((G * (( len G),j)) + (G * (( len G),(j + 1))))) + |[1, 0 ]|),((1 / 2) * ((G * (( len G),j)) + (G * (( len G),(j + 1)))))));

      then

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

      consider r such that

       A4: p = (((1 - r) * (((1 / 2) * ((G * (( len G),j)) + (G * (( len G),(j + 1))))) + |[1, 0 ]|)) + (r * ((1 / 2) * ((G * (( len G),j)) + (G * (( len G),(j + 1))))))) and

       A5: 0 <= r and

       A6: r <= 1 by A3;

      now

        per cases by A6, XXREAL_0: 1;

          case r = 1;

          

          then p = (( 0. ( TOP-REAL 2)) + (1 * ((1 / 2) * ((G * (( len G),j)) + (G * (( len G),(j + 1))))))) by A4, RLVECT_1: 10

          .= (1 * ((1 / 2) * ((G * (( len G),j)) + (G * (( len G),(j + 1)))))) by RLVECT_1: 4

          .= ((1 / 2) * ((G * (( len G),j)) + (G * (( len G),(j + 1))))) by RLVECT_1:def 8;

          hence p in {((1 / 2) * ((G * (( len G),j)) + (G * (( len G),(j + 1)))))} by TARSKI:def 1;

        end;

          case

           A7: r < 1;

          set r3 = ((1 - r) * (1 / 2)), s3 = (r * (1 / 2));

          set r2 = ((G * (( len G),1)) `1 ), s1 = ((G * (1,j)) `2 ), s2 = ((G * (1,(j + 1))) `2 );

          

           A8: ((r3 * (s1 + s1)) + (s3 * (s1 + s1))) = s1;

          

           A9: (j + 1) <= ( width G) by A2, NAT_1: 13;

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

          then

           A10: 1 <= ( len G) by NAT_1: 14;

          j < (j + 1) by XREAL_1: 29;

          then

           A11: s1 < s2 by A1, A9, A10, GOBOARD5: 4;

          then

           A12: (s1 + s1) < (s1 + s2) by XREAL_1: 6;

          then

           A13: (s3 * (s1 + s1)) <= (s3 * (s1 + s2)) by A5, XREAL_1: 64;

          

           A14: (1 - r) > 0 by A7, XREAL_1: 50;

          then

           A15: r3 > ((1 / 2) * 0 ) by XREAL_1: 68;

          then (r3 * (s1 + s1)) < (r3 * (s1 + s2)) by A12, XREAL_1: 68;

          then

           A16: s1 < ((r3 * (s1 + s2)) + (s3 * (s1 + s2))) by A13, A8, XREAL_1: 8;

          

           A17: (r2 + (1 - r)) > r2 by A14, XREAL_1: 29;

          

           A18: 1 <= (j + 1) by A1, NAT_1: 13;

          

           A19: (s1 + s2) < (s2 + s2) by A11, XREAL_1: 6;

          then

           A20: (s3 * (s1 + s2)) <= (s3 * (s2 + s2)) by A5, XREAL_1: 64;

          

           A21: ( Int ( cell (G,( len G),j))) = { |[r9, s9]| : ((G * (( len G),1)) `1 ) < r9 & ((G * (1,j)) `2 ) < s9 & s9 < ((G * (1,(j + 1))) `2 ) } by A1, A2, Th23;

          

           A22: ((r3 * (s2 + s2)) + (s3 * (s2 + s2))) = s2;

          (r3 * (s1 + s2)) < (r3 * (s2 + s2)) by A15, A19, XREAL_1: 68;

          then

           A23: ((r3 * (s1 + s2)) + (s3 * (s1 + s2))) < s2 by A20, A22, XREAL_1: 8;

          

           A24: (G * (( len G),j)) = |[((G * (( len G),j)) `1 ), ((G * (( len G),j)) `2 )]| by EUCLID: 53

          .= |[r2, ((G * (( len G),j)) `2 )]| by A1, A2, A10, GOBOARD5: 2

          .= |[r2, s1]| by A1, A2, A10, GOBOARD5: 1;

          

           A25: (G * (( len G),(j + 1))) = |[((G * (( len G),(j + 1))) `1 ), ((G * (( len G),(j + 1))) `2 )]| by EUCLID: 53

          .= |[r2, ((G * (( len G),(j + 1))) `2 )]| by A18, A9, A10, GOBOARD5: 2

          .= |[r2, s2]| by A18, A9, A10, GOBOARD5: 1;

          p = ((((1 - r) * ((1 / 2) * ((G * (( len G),j)) + (G * (( len G),(j + 1)))))) + ((1 - r) * |[1, 0 ]|)) + (r * ((1 / 2) * ((G * (( len G),j)) + (G * (( len G),(j + 1))))))) by A4, RLVECT_1:def 5

          .= (((r3 * ((G * (( len G),j)) + (G * (( len G),(j + 1))))) + ((1 - r) * |[1, 0 ]|)) + (r * ((1 / 2) * ((G * (( len G),j)) + (G * (( len G),(j + 1))))))) by RLVECT_1:def 7

          .= (((r3 * ((G * (( len G),j)) + (G * (( len G),(j + 1))))) + |[((1 - r) * 1), ((1 - r) * 0 )]|) + (r * ((1 / 2) * ((G * (( len G),j)) + (G * (( len G),(j + 1))))))) by EUCLID: 58

          .= (((r3 * ((G * (( len G),j)) + (G * (( len G),(j + 1))))) + |[(1 - r), 0 ]|) + (s3 * ((G * (( len G),j)) + (G * (( len G),(j + 1)))))) by RLVECT_1:def 7

          .= (((r3 * |[(r2 + r2), (s1 + s2)]|) + |[(1 - r), 0 ]|) + (s3 * ((G * (( len G),j)) + (G * (( len G),(j + 1)))))) by A25, A24, EUCLID: 56

          .= (((r3 * |[(r2 + r2), (s1 + s2)]|) + |[(1 - r), 0 ]|) + (s3 * |[(r2 + r2), (s1 + s2)]|)) by A25, A24, EUCLID: 56

          .= (( |[(r3 * (r2 + r2)), (r3 * (s1 + s2))]| + |[(1 - r), 0 ]|) + (s3 * |[(r2 + r2), (s1 + s2)]|)) by EUCLID: 58

          .= (( |[(r3 * (r2 + r2)), (r3 * (s1 + s2))]| + |[(1 - r), 0 ]|) + |[(s3 * (r2 + r2)), (s3 * (s1 + s2))]|) by EUCLID: 58

          .= ( |[((r3 * (r2 + r2)) + (1 - r)), ((r3 * (s1 + s2)) + 0 )]| + |[(s3 * (r2 + r2)), (s3 * (s1 + s2))]|) by EUCLID: 56

          .= |[(((r3 * (r2 + r2)) + (1 - r)) + (s3 * (r2 + r2))), ((r3 * (s1 + s2)) + (s3 * (s1 + s2)))]| by EUCLID: 56;

          hence p in ( Int ( cell (G,( len G),j))) by A17, A16, A23, A21;

        end;

      end;

      hence thesis by XBOOLE_0:def 3;

    end;

    theorem :: GOBOARD6:46

    

     Th46: 1 <= i & i < ( len G) implies ( LSeg ((((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1)))) - |[ 0 , 1]|),((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1)))))) c= (( Int ( cell (G,i, 0 ))) \/ {((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1))))})

    proof

      assume that

       A1: 1 <= i and

       A2: i < ( len G);

      let x be object;

      assume

       A3: x in ( LSeg ((((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1)))) - |[ 0 , 1]|),((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1))))));

      then

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

      consider r such that

       A4: p = (((1 - r) * (((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1)))) - |[ 0 , 1]|)) + (r * ((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1)))))) and

       A5: 0 <= r and

       A6: r <= 1 by A3;

      now

        per cases by A6, XXREAL_0: 1;

          case r = 1;

          

          then p = (( 0. ( TOP-REAL 2)) + (1 * ((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1)))))) by A4, RLVECT_1: 10

          .= (1 * ((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1))))) by RLVECT_1: 4

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

          hence p in {((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1))))} by TARSKI:def 1;

        end;

          case

           A7: r < 1;

          set r3 = ((1 - r) * (1 / 2)), s3 = (r * (1 / 2));

          set s2 = ((G * (1,1)) `2 ), r1 = ((G * (i,1)) `1 ), r2 = ((G * ((i + 1),1)) `1 );

          

           A8: ((r3 * (r1 + r1)) + (s3 * (r1 + r1))) = r1;

          

           A9: (i + 1) <= ( len G) by A2, NAT_1: 13;

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

          then

           A10: 1 <= ( width G) by NAT_1: 14;

          i < (i + 1) by XREAL_1: 29;

          then

           A11: r1 < r2 by A1, A9, A10, GOBOARD5: 3;

          then

           A12: (r1 + r1) < (r1 + r2) by XREAL_1: 6;

          then

           A13: (s3 * (r1 + r1)) <= (s3 * (r1 + r2)) by A5, XREAL_1: 64;

          

           A14: (1 - r) > 0 by A7, XREAL_1: 50;

          then

           A15: r3 > ((1 / 2) * 0 ) by XREAL_1: 68;

          then (r3 * (r1 + r1)) < (r3 * (r1 + r2)) by A12, XREAL_1: 68;

          then

           A16: r1 < ((r3 * (r1 + r2)) + (s3 * (r1 + r2))) by A13, A8, XREAL_1: 8;

          s2 < (s2 + (1 - r)) by A14, XREAL_1: 29;

          then

           A17: (s2 - (1 - r)) < s2 by XREAL_1: 19;

          

           A18: 1 <= (i + 1) by A1, NAT_1: 13;

          

           A19: (G * ((i + 1),1)) = |[((G * ((i + 1),1)) `1 ), ((G * ((i + 1),1)) `2 )]| by EUCLID: 53

          .= |[r2, s2]| by A18, A9, A10, GOBOARD5: 1;

          

           A20: (r1 + r2) < (r2 + r2) by A11, XREAL_1: 6;

          then

           A21: (s3 * (r1 + r2)) <= (s3 * (r2 + r2)) by A5, XREAL_1: 64;

          

           A22: ( Int ( cell (G,i, 0 ))) = { |[r9, s9]| : ((G * (i,1)) `1 ) < r9 & r9 < ((G * ((i + 1),1)) `1 ) & s9 < ((G * (1,1)) `2 ) } by A1, A2, Th24;

          

           A23: ((r3 * (r2 + r2)) + (s3 * (r2 + r2))) = r2;

          (r3 * (r1 + r2)) < (r3 * (r2 + r2)) by A15, A20, XREAL_1: 68;

          then

           A24: ((r3 * (r1 + r2)) + (s3 * (r1 + r2))) < r2 by A21, A23, XREAL_1: 8;

          

           A25: (G * (i,1)) = |[((G * (i,1)) `1 ), ((G * (i,1)) `2 )]| by EUCLID: 53

          .= |[r1, s2]| by A1, A2, A10, GOBOARD5: 1;

          p = ((((1 - r) * ((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1))))) - ((1 - r) * |[ 0 , 1]|)) + (r * ((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1)))))) by A4, RLVECT_1: 34

          .= (((r3 * ((G * (i,1)) + (G * ((i + 1),1)))) - ((1 - r) * |[ 0 , 1]|)) + (r * ((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1)))))) by RLVECT_1:def 7

          .= (((r3 * ((G * (i,1)) + (G * ((i + 1),1)))) - |[((1 - r) * 0 ), ((1 - r) * 1)]|) + (r * ((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1)))))) by EUCLID: 58

          .= (((r3 * ((G * (i,1)) + (G * ((i + 1),1)))) - |[ 0 , (1 - r)]|) + (s3 * ((G * (i,1)) + (G * ((i + 1),1))))) by RLVECT_1:def 7

          .= (((r3 * |[(r1 + r2), (s2 + s2)]|) - |[ 0 , (1 - r)]|) + (s3 * ((G * (i,1)) + (G * ((i + 1),1))))) by A19, A25, EUCLID: 56

          .= (((r3 * |[(r1 + r2), (s2 + s2)]|) - |[ 0 , (1 - r)]|) + (s3 * |[(r1 + r2), (s2 + s2)]|)) by A19, A25, EUCLID: 56

          .= (( |[(r3 * (r1 + r2)), (r3 * (s2 + s2))]| - |[ 0 , (1 - r)]|) + (s3 * |[(r1 + r2), (s2 + s2)]|)) by EUCLID: 58

          .= (( |[(r3 * (r1 + r2)), (r3 * (s2 + s2))]| - |[ 0 , (1 - r)]|) + |[(s3 * (r1 + r2)), (s3 * (s2 + s2))]|) by EUCLID: 58

          .= ( |[((r3 * (r1 + r2)) - 0 ), ((r3 * (s2 + s2)) - (1 - r))]| + |[(s3 * (r1 + r2)), (s3 * (s2 + s2))]|) by EUCLID: 62

          .= |[((r3 * (r1 + r2)) + (s3 * (r1 + r2))), (((r3 * (s2 + s2)) - (1 - r)) + (s3 * (s2 + s2)))]| by EUCLID: 56;

          hence p in ( Int ( cell (G,i, 0 ))) by A17, A16, A24, A22;

        end;

      end;

      hence thesis by XBOOLE_0:def 3;

    end;

    theorem :: GOBOARD6:47

    

     Th47: 1 <= i & i < ( len G) implies ( LSeg ((((1 / 2) * ((G * (i,( width G))) + (G * ((i + 1),( width G))))) + |[ 0 , 1]|),((1 / 2) * ((G * (i,( width G))) + (G * ((i + 1),( width G))))))) c= (( Int ( cell (G,i,( width G)))) \/ {((1 / 2) * ((G * (i,( width G))) + (G * ((i + 1),( width G)))))})

    proof

      assume that

       A1: 1 <= i and

       A2: i < ( len G);

      let x be object;

      assume

       A3: x in ( LSeg ((((1 / 2) * ((G * (i,( width G))) + (G * ((i + 1),( width G))))) + |[ 0 , 1]|),((1 / 2) * ((G * (i,( width G))) + (G * ((i + 1),( width G)))))));

      then

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

      consider r such that

       A4: p = (((1 - r) * (((1 / 2) * ((G * (i,( width G))) + (G * ((i + 1),( width G))))) + |[ 0 , 1]|)) + (r * ((1 / 2) * ((G * (i,( width G))) + (G * ((i + 1),( width G))))))) and

       A5: 0 <= r and

       A6: r <= 1 by A3;

      now

        per cases by A6, XXREAL_0: 1;

          case r = 1;

          

          then p = (( 0. ( TOP-REAL 2)) + (1 * ((1 / 2) * ((G * (i,( width G))) + (G * ((i + 1),( width G))))))) by A4, RLVECT_1: 10

          .= (1 * ((1 / 2) * ((G * (i,( width G))) + (G * ((i + 1),( width G)))))) by RLVECT_1: 4

          .= ((1 / 2) * ((G * (i,( width G))) + (G * ((i + 1),( width G))))) by RLVECT_1:def 8;

          hence p in {((1 / 2) * ((G * (i,( width G))) + (G * ((i + 1),( width G)))))} by TARSKI:def 1;

        end;

          case

           A7: r < 1;

          set r3 = ((1 - r) * (1 / 2)), s3 = (r * (1 / 2));

          set s2 = ((G * (1,( width G))) `2 ), r1 = ((G * (i,1)) `1 ), r2 = ((G * ((i + 1),1)) `1 );

          

           A8: ((r3 * (r1 + r1)) + (s3 * (r1 + r1))) = r1;

          

           A9: (i + 1) <= ( len G) by A2, NAT_1: 13;

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

          then

           A10: 1 <= ( width G) by NAT_1: 14;

          i < (i + 1) by XREAL_1: 29;

          then

           A11: r1 < r2 by A1, A9, A10, GOBOARD5: 3;

          then

           A12: (r1 + r1) < (r1 + r2) by XREAL_1: 6;

          then

           A13: (s3 * (r1 + r1)) <= (s3 * (r1 + r2)) by A5, XREAL_1: 64;

          

           A14: (1 - r) > 0 by A7, XREAL_1: 50;

          then

           A15: r3 > ((1 / 2) * 0 ) by XREAL_1: 68;

          then (r3 * (r1 + r1)) < (r3 * (r1 + r2)) by A12, XREAL_1: 68;

          then

           A16: r1 < ((r3 * (r1 + r2)) + (s3 * (r1 + r2))) by A13, A8, XREAL_1: 8;

          

           A17: (s2 + (1 - r)) > s2 by A14, XREAL_1: 29;

          

           A18: 1 <= (i + 1) by A1, NAT_1: 13;

          

           A19: (r1 + r2) < (r2 + r2) by A11, XREAL_1: 6;

          then

           A20: (s3 * (r1 + r2)) <= (s3 * (r2 + r2)) by A5, XREAL_1: 64;

          

           A21: ( Int ( cell (G,i,( width G)))) = { |[r9, s9]| : ((G * (i,1)) `1 ) < r9 & r9 < ((G * ((i + 1),1)) `1 ) & ((G * (1,( width G))) `2 ) < s9 } by A1, A2, Th25;

          

           A22: ((r3 * (r2 + r2)) + (s3 * (r2 + r2))) = r2;

          (r3 * (r1 + r2)) < (r3 * (r2 + r2)) by A15, A19, XREAL_1: 68;

          then

           A23: ((r3 * (r1 + r2)) + (s3 * (r1 + r2))) < r2 by A20, A22, XREAL_1: 8;

          

           A24: (G * (i,( width G))) = |[((G * (i,( width G))) `1 ), ((G * (i,( width G))) `2 )]| by EUCLID: 53

          .= |[((G * (i,( width G))) `1 ), s2]| by A1, A2, A10, GOBOARD5: 1

          .= |[r1, s2]| by A1, A2, A10, GOBOARD5: 2;

          

           A25: (G * ((i + 1),( width G))) = |[((G * ((i + 1),( width G))) `1 ), ((G * ((i + 1),( width G))) `2 )]| by EUCLID: 53

          .= |[((G * ((i + 1),( width G))) `1 ), s2]| by A18, A9, A10, GOBOARD5: 1

          .= |[r2, s2]| by A18, A9, A10, GOBOARD5: 2;

          p = ((((1 - r) * ((1 / 2) * ((G * (i,( width G))) + (G * ((i + 1),( width G)))))) + ((1 - r) * |[ 0 , 1]|)) + (r * ((1 / 2) * ((G * (i,( width G))) + (G * ((i + 1),( width G))))))) by A4, RLVECT_1:def 5

          .= (((r3 * ((G * (i,( width G))) + (G * ((i + 1),( width G))))) + ((1 - r) * |[ 0 , 1]|)) + (r * ((1 / 2) * ((G * (i,( width G))) + (G * ((i + 1),( width G))))))) by RLVECT_1:def 7

          .= (((r3 * ((G * (i,( width G))) + (G * ((i + 1),( width G))))) + |[((1 - r) * 0 ), ((1 - r) * 1)]|) + (r * ((1 / 2) * ((G * (i,( width G))) + (G * ((i + 1),( width G))))))) by EUCLID: 58

          .= (((r3 * ((G * (i,( width G))) + (G * ((i + 1),( width G))))) + |[ 0 , (1 - r)]|) + (s3 * ((G * (i,( width G))) + (G * ((i + 1),( width G)))))) by RLVECT_1:def 7

          .= (((r3 * |[(r1 + r2), (s2 + s2)]|) + |[ 0 , (1 - r)]|) + (s3 * ((G * (i,( width G))) + (G * ((i + 1),( width G)))))) by A25, A24, EUCLID: 56

          .= (((r3 * |[(r1 + r2), (s2 + s2)]|) + |[ 0 , (1 - r)]|) + (s3 * |[(r1 + r2), (s2 + s2)]|)) by A25, A24, EUCLID: 56

          .= (( |[(r3 * (r1 + r2)), (r3 * (s2 + s2))]| + |[ 0 , (1 - r)]|) + (s3 * |[(r1 + r2), (s2 + s2)]|)) by EUCLID: 58

          .= (( |[(r3 * (r1 + r2)), (r3 * (s2 + s2))]| + |[ 0 , (1 - r)]|) + |[(s3 * (r1 + r2)), (s3 * (s2 + s2))]|) by EUCLID: 58

          .= ( |[((r3 * (r1 + r2)) + 0 ), ((r3 * (s2 + s2)) + (1 - r))]| + |[(s3 * (r1 + r2)), (s3 * (s2 + s2))]|) by EUCLID: 56

          .= |[((r3 * (r1 + r2)) + (s3 * (r1 + r2))), (((r3 * (s2 + s2)) + (1 - r)) + (s3 * (s2 + s2)))]| by EUCLID: 56;

          hence p in ( Int ( cell (G,i,( width G)))) by A17, A16, A23, A21;

        end;

      end;

      hence thesis by XBOOLE_0:def 3;

    end;

    theorem :: GOBOARD6:48

    

     Th48: 1 <= j & j < ( width G) implies ( LSeg ((((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1))))) - |[1, 0 ]|),((G * (1,j)) - |[1, 0 ]|))) c= (( Int ( cell (G, 0 ,j))) \/ {((G * (1,j)) - |[1, 0 ]|)})

    proof

      assume that

       A1: 1 <= j and

       A2: j < ( width G);

      let x be object;

      assume

       A3: x in ( LSeg ((((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1))))) - |[1, 0 ]|),((G * (1,j)) - |[1, 0 ]|)));

      then

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

      consider r such that

       A4: p = (((1 - r) * (((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1))))) - |[1, 0 ]|)) + (r * ((G * (1,j)) - |[1, 0 ]|))) and

       A5: 0 <= r and

       A6: r <= 1 by A3;

      now

        per cases by A6, XXREAL_0: 1;

          case r = 1;

          

          then p = (( 0. ( TOP-REAL 2)) + (1 * ((G * (1,j)) - |[1, 0 ]|))) by A4, RLVECT_1: 10

          .= (1 * ((G * (1,j)) - |[1, 0 ]|)) by RLVECT_1: 4

          .= ((G * (1,j)) - |[1, 0 ]|) by RLVECT_1:def 8;

          hence p in {((G * (1,j)) - |[1, 0 ]|)} by TARSKI:def 1;

        end;

          case

           A7: r < 1;

          set r3 = ((1 - r) * (1 / 2));

          (1 - r) > 0 by A7, XREAL_1: 50;

          then

           A8: r3 > ((1 / 2) * 0 ) by XREAL_1: 68;

          set r2 = ((G * (1,1)) `1 ), s1 = ((G * (1,j)) `2 ), s2 = ((G * (1,(j + 1))) `2 );

          

           A9: ((r3 * (s1 + s1)) + (r * s1)) = s1;

          

           A10: (j + 1) <= ( width G) by A2, NAT_1: 13;

          r2 < (r2 + 1) by XREAL_1: 29;

          then

           A11: (r2 - 1) < r2 by XREAL_1: 19;

          

           A12: ( Int ( cell (G, 0 ,j))) = { |[r9, s9]| : r9 < ((G * (1,1)) `1 ) & ((G * (1,j)) `2 ) < s9 & s9 < ((G * (1,(j + 1))) `2 ) } by A1, A2, Th20;

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

          then

           A13: 1 <= ( len G) by NAT_1: 14;

          j < (j + 1) by XREAL_1: 29;

          then

           A14: s1 < s2 by A1, A10, A13, GOBOARD5: 4;

          then (s1 + s2) < (s2 + s2) by XREAL_1: 6;

          then

           A15: (r3 * (s1 + s2)) < (r3 * (s2 + s2)) by A8, XREAL_1: 68;

          (s1 + s1) < (s1 + s2) by A14, XREAL_1: 6;

          then (r3 * (s1 + s1)) < (r3 * (s1 + s2)) by A8, XREAL_1: 68;

          then

           A16: s1 < ((r3 * (s1 + s2)) + (r * s1)) by A9, XREAL_1: 6;

          

           A17: ((r3 * (s2 + s2)) + (r * s2)) = s2;

          (r * s1) <= (r * s2) by A5, A14, XREAL_1: 64;

          then

           A18: ((r3 * (s1 + s2)) + (r * s1)) < s2 by A15, A17, XREAL_1: 8;

          

           A19: (G * (1,j)) = |[((G * (1,j)) `1 ), ((G * (1,j)) `2 )]| by EUCLID: 53

          .= |[r2, s1]| by A1, A2, A13, GOBOARD5: 2;

          

           A20: 1 <= (j + 1) by A1, NAT_1: 13;

          

           A21: (G * (1,(j + 1))) = |[((G * (1,(j + 1))) `1 ), ((G * (1,(j + 1))) `2 )]| by EUCLID: 53

          .= |[r2, s2]| by A20, A10, A13, GOBOARD5: 2;

          p = ((((1 - r) * ((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1)))))) - ((1 - r) * |[1, 0 ]|)) + (r * ((G * (1,j)) - |[1, 0 ]|))) by A4, RLVECT_1: 34

          .= (((r3 * ((G * (1,j)) + (G * (1,(j + 1))))) - ((1 - r) * |[1, 0 ]|)) + (r * ((G * (1,j)) - |[1, 0 ]|))) by RLVECT_1:def 7

          .= (((r3 * ((G * (1,j)) + (G * (1,(j + 1))))) - |[((1 - r) * 1), ((1 - r) * 0 )]|) + (r * ((G * (1,j)) - |[1, 0 ]|))) by EUCLID: 58

          .= (((r3 * |[(r2 + r2), (s1 + s2)]|) - |[(1 - r), 0 ]|) + (r * ( |[r2, s1]| - |[1, 0 ]|))) by A21, A19, EUCLID: 56

          .= (((r3 * |[(r2 + r2), (s1 + s2)]|) - |[(1 - r), 0 ]|) + ((r * |[r2, s1]|) - (r * |[1, 0 ]|))) by RLVECT_1: 34

          .= (((r3 * |[(r2 + r2), (s1 + s2)]|) - |[(1 - r), 0 ]|) + ( |[(r * r2), (r * s1)]| - (r * |[1, 0 ]|))) by EUCLID: 58

          .= (((r3 * |[(r2 + r2), (s1 + s2)]|) - |[(1 - r), 0 ]|) + ( |[(r * r2), (r * s1)]| - |[(r * 1), (r * 0 )]|)) by EUCLID: 58

          .= (((r3 * |[(r2 + r2), (s1 + s2)]|) - |[(1 - r), 0 ]|) + |[((r * r2) - r), ((r * s1) - 0 )]|) by EUCLID: 62

          .= (( |[(r3 * (r2 + r2)), (r3 * (s1 + s2))]| - |[(1 - r), 0 ]|) + |[((r * r2) - r), ((r * s1) - 0 )]|) by EUCLID: 58

          .= ( |[((r3 * (r2 + r2)) - (1 - r)), ((r3 * (s1 + s2)) - 0 )]| + |[((r * r2) - r), ((r * s1) - 0 )]|) by EUCLID: 62

          .= |[(((r3 * (r2 + r2)) - (1 - r)) + ((r * r2) - r)), ((r3 * (s1 + s2)) + (r * s1))]| by EUCLID: 56;

          hence p in ( Int ( cell (G, 0 ,j))) by A11, A16, A18, A12;

        end;

      end;

      hence thesis by XBOOLE_0:def 3;

    end;

    theorem :: GOBOARD6:49

    

     Th49: 1 <= j & j < ( width G) implies ( LSeg ((((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1))))) - |[1, 0 ]|),((G * (1,(j + 1))) - |[1, 0 ]|))) c= (( Int ( cell (G, 0 ,j))) \/ {((G * (1,(j + 1))) - |[1, 0 ]|)})

    proof

      assume that

       A1: 1 <= j and

       A2: j < ( width G);

      let x be object;

      assume

       A3: x in ( LSeg ((((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1))))) - |[1, 0 ]|),((G * (1,(j + 1))) - |[1, 0 ]|)));

      then

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

      consider r such that

       A4: p = (((1 - r) * (((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1))))) - |[1, 0 ]|)) + (r * ((G * (1,(j + 1))) - |[1, 0 ]|))) and

       A5: 0 <= r and

       A6: r <= 1 by A3;

      now

        per cases by A6, XXREAL_0: 1;

          case r = 1;

          

          then p = (( 0. ( TOP-REAL 2)) + (1 * ((G * (1,(j + 1))) - |[1, 0 ]|))) by A4, RLVECT_1: 10

          .= (1 * ((G * (1,(j + 1))) - |[1, 0 ]|)) by RLVECT_1: 4

          .= ((G * (1,(j + 1))) - |[1, 0 ]|) by RLVECT_1:def 8;

          hence p in {((G * (1,(j + 1))) - |[1, 0 ]|)} by TARSKI:def 1;

        end;

          case

           A7: r < 1;

          set r3 = ((1 - r) * (1 / 2));

          (1 - r) > 0 by A7, XREAL_1: 50;

          then

           A8: r3 > ((1 / 2) * 0 ) by XREAL_1: 68;

          set r2 = ((G * (1,1)) `1 ), s1 = ((G * (1,j)) `2 ), s2 = ((G * (1,(j + 1))) `2 );

          

           A9: ((r3 * (s1 + s1)) + (r * s1)) = s1;

          

           A10: (j + 1) <= ( width G) by A2, NAT_1: 13;

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

          then

           A11: 1 <= ( len G) by NAT_1: 14;

          j < (j + 1) by XREAL_1: 29;

          then

           A12: s1 < s2 by A1, A10, A11, GOBOARD5: 4;

          then (s1 + s1) < (s1 + s2) by XREAL_1: 6;

          then

           A13: (r3 * (s1 + s1)) < (r3 * (s1 + s2)) by A8, XREAL_1: 68;

          (r * s1) <= (r * s2) by A5, A12, XREAL_1: 64;

          then

           A14: s1 < ((r3 * (s1 + s2)) + (r * s2)) by A13, A9, XREAL_1: 8;

          

           A15: 1 <= (j + 1) by A1, NAT_1: 13;

          

           A16: (G * (1,j)) = |[((G * (1,j)) `1 ), ((G * (1,j)) `2 )]| by EUCLID: 53

          .= |[r2, s1]| by A1, A2, A11, GOBOARD5: 2;

          r2 < (r2 + 1) by XREAL_1: 29;

          then

           A17: (r2 - 1) < r2 by XREAL_1: 19;

          

           A18: ((r3 * (s2 + s2)) + (r * s2)) = s2;

          (s1 + s2) < (s2 + s2) by A12, XREAL_1: 6;

          then (r3 * (s1 + s2)) < (r3 * (s2 + s2)) by A8, XREAL_1: 68;

          then

           A19: ((r3 * (s1 + s2)) + (r * s2)) < s2 by A18, XREAL_1: 8;

          

           A20: ( Int ( cell (G, 0 ,j))) = { |[r9, s9]| : r9 < ((G * (1,1)) `1 ) & ((G * (1,j)) `2 ) < s9 & s9 < ((G * (1,(j + 1))) `2 ) } by A1, A2, Th20;

          

           A21: (G * (1,(j + 1))) = |[((G * (1,(j + 1))) `1 ), ((G * (1,(j + 1))) `2 )]| by EUCLID: 53

          .= |[r2, s2]| by A15, A10, A11, GOBOARD5: 2;

          p = ((((1 - r) * ((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1)))))) - ((1 - r) * |[1, 0 ]|)) + (r * ((G * (1,(j + 1))) - |[1, 0 ]|))) by A4, RLVECT_1: 34

          .= (((r3 * ((G * (1,j)) + (G * (1,(j + 1))))) - ((1 - r) * |[1, 0 ]|)) + (r * ((G * (1,(j + 1))) - |[1, 0 ]|))) by RLVECT_1:def 7

          .= (((r3 * ((G * (1,j)) + (G * (1,(j + 1))))) - |[((1 - r) * 1), ((1 - r) * 0 )]|) + (r * ((G * (1,(j + 1))) - |[1, 0 ]|))) by EUCLID: 58

          .= (((r3 * |[(r2 + r2), (s1 + s2)]|) - |[(1 - r), 0 ]|) + (r * ( |[r2, s2]| - |[1, 0 ]|))) by A21, A16, EUCLID: 56

          .= (((r3 * |[(r2 + r2), (s1 + s2)]|) - |[(1 - r), 0 ]|) + ((r * |[r2, s2]|) - (r * |[1, 0 ]|))) by RLVECT_1: 34

          .= (((r3 * |[(r2 + r2), (s1 + s2)]|) - |[(1 - r), 0 ]|) + ( |[(r * r2), (r * s2)]| - (r * |[1, 0 ]|))) by EUCLID: 58

          .= (((r3 * |[(r2 + r2), (s1 + s2)]|) - |[(1 - r), 0 ]|) + ( |[(r * r2), (r * s2)]| - |[(r * 1), (r * 0 )]|)) by EUCLID: 58

          .= (((r3 * |[(r2 + r2), (s1 + s2)]|) - |[(1 - r), 0 ]|) + |[((r * r2) - r), ((r * s2) - 0 )]|) by EUCLID: 62

          .= (( |[(r3 * (r2 + r2)), (r3 * (s1 + s2))]| - |[(1 - r), 0 ]|) + |[((r * r2) - r), ((r * s2) - 0 )]|) by EUCLID: 58

          .= ( |[((r3 * (r2 + r2)) - (1 - r)), ((r3 * (s1 + s2)) - 0 )]| + |[((r * r2) - r), ((r * s2) - 0 )]|) by EUCLID: 62

          .= |[(((r3 * (r2 + r2)) - (1 - r)) + ((r * r2) - r)), ((r3 * (s1 + s2)) + (r * s2))]| by EUCLID: 56;

          hence p in ( Int ( cell (G, 0 ,j))) by A17, A14, A19, A20;

        end;

      end;

      hence thesis by XBOOLE_0:def 3;

    end;

    theorem :: GOBOARD6:50

    

     Th50: 1 <= j & j < ( width G) implies ( LSeg ((((1 / 2) * ((G * (( len G),j)) + (G * (( len G),(j + 1))))) + |[1, 0 ]|),((G * (( len G),j)) + |[1, 0 ]|))) c= (( Int ( cell (G,( len G),j))) \/ {((G * (( len G),j)) + |[1, 0 ]|)})

    proof

      assume that

       A1: 1 <= j and

       A2: j < ( width G);

      let x be object;

      assume

       A3: x in ( LSeg ((((1 / 2) * ((G * (( len G),j)) + (G * (( len G),(j + 1))))) + |[1, 0 ]|),((G * (( len G),j)) + |[1, 0 ]|)));

      then

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

      consider r such that

       A4: p = (((1 - r) * (((1 / 2) * ((G * (( len G),j)) + (G * (( len G),(j + 1))))) + |[1, 0 ]|)) + (r * ((G * (( len G),j)) + |[1, 0 ]|))) and

       A5: 0 <= r and

       A6: r <= 1 by A3;

      now

        per cases by A6, XXREAL_0: 1;

          case r = 1;

          

          then p = (( 0. ( TOP-REAL 2)) + (1 * ((G * (( len G),j)) + |[1, 0 ]|))) by A4, RLVECT_1: 10

          .= (1 * ((G * (( len G),j)) + |[1, 0 ]|)) by RLVECT_1: 4

          .= ((G * (( len G),j)) + |[1, 0 ]|) by RLVECT_1:def 8;

          hence p in {((G * (( len G),j)) + |[1, 0 ]|)} by TARSKI:def 1;

        end;

          case

           A7: r < 1;

          set r3 = ((1 - r) * (1 / 2));

          (1 - r) > 0 by A7, XREAL_1: 50;

          then

           A8: r3 > ((1 / 2) * 0 ) by XREAL_1: 68;

          set r2 = ((G * (( len G),1)) `1 ), s1 = ((G * (1,j)) `2 ), s2 = ((G * (1,(j + 1))) `2 );

          

           A9: ((r3 * (s1 + s1)) + (r * s1)) = s1;

          

           A10: (j + 1) <= ( width G) by A2, NAT_1: 13;

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

          then

           A11: 1 <= ( len G) by NAT_1: 14;

          

           A12: (G * (( len G),j)) = |[((G * (( len G),j)) `1 ), ((G * (( len G),j)) `2 )]| by EUCLID: 53

          .= |[r2, ((G * (( len G),j)) `2 )]| by A1, A2, A11, GOBOARD5: 2

          .= |[r2, s1]| by A1, A2, A11, GOBOARD5: 1;

          

           A13: 1 <= (j + 1) by A1, NAT_1: 13;

          j < (j + 1) by XREAL_1: 29;

          then

           A14: s1 < s2 by A1, A10, A11, GOBOARD5: 4;

          then (s1 + s2) < (s2 + s2) by XREAL_1: 6;

          then

           A15: (r3 * (s1 + s2)) < (r3 * (s2 + s2)) by A8, XREAL_1: 68;

          (s1 + s1) < (s1 + s2) by A14, XREAL_1: 6;

          then (r3 * (s1 + s1)) < (r3 * (s1 + s2)) by A8, XREAL_1: 68;

          then

           A16: r2 < (r2 + 1) & s1 < ((r3 * (s1 + s2)) + (r * s1)) by A9, XREAL_1: 6, XREAL_1: 29;

          

           A17: ( Int ( cell (G,( len G),j))) = { |[r9, s9]| : ((G * (( len G),1)) `1 ) < r9 & ((G * (1,j)) `2 ) < s9 & s9 < ((G * (1,(j + 1))) `2 ) } by A1, A2, Th23;

          

           A18: (G * (( len G),(j + 1))) = |[((G * (( len G),(j + 1))) `1 ), ((G * (( len G),(j + 1))) `2 )]| by EUCLID: 53

          .= |[r2, ((G * (( len G),(j + 1))) `2 )]| by A13, A10, A11, GOBOARD5: 2

          .= |[r2, s2]| by A13, A10, A11, GOBOARD5: 1;

          

           A19: ((r3 * (s2 + s2)) + (r * s2)) = s2;

          (r * s1) <= (r * s2) by A5, A14, XREAL_1: 64;

          then

           A20: ((r3 * (s1 + s2)) + (r * s1)) < s2 by A15, A19, XREAL_1: 8;

          p = ((((1 - r) * ((1 / 2) * ((G * (( len G),j)) + (G * (( len G),(j + 1)))))) + ((1 - r) * |[1, 0 ]|)) + (r * ((G * (( len G),j)) + |[1, 0 ]|))) by A4, RLVECT_1:def 5

          .= (((r3 * ((G * (( len G),j)) + (G * (( len G),(j + 1))))) + ((1 - r) * |[1, 0 ]|)) + (r * ((G * (( len G),j)) + |[1, 0 ]|))) by RLVECT_1:def 7

          .= (((r3 * ((G * (( len G),j)) + (G * (( len G),(j + 1))))) + |[((1 - r) * 1), ((1 - r) * 0 )]|) + (r * ((G * (( len G),j)) + |[1, 0 ]|))) by EUCLID: 58

          .= (((r3 * |[(r2 + r2), (s1 + s2)]|) + |[(1 - r), 0 ]|) + (r * ( |[r2, s1]| + |[1, 0 ]|))) by A18, A12, EUCLID: 56

          .= (((r3 * |[(r2 + r2), (s1 + s2)]|) + |[(1 - r), 0 ]|) + ((r * |[r2, s1]|) + (r * |[1, 0 ]|))) by RLVECT_1:def 5

          .= (((r3 * |[(r2 + r2), (s1 + s2)]|) + |[(1 - r), 0 ]|) + ( |[(r * r2), (r * s1)]| + (r * |[1, 0 ]|))) by EUCLID: 58

          .= (((r3 * |[(r2 + r2), (s1 + s2)]|) + |[(1 - r), 0 ]|) + ( |[(r * r2), (r * s1)]| + |[(r * 1), (r * 0 )]|)) by EUCLID: 58

          .= (((r3 * |[(r2 + r2), (s1 + s2)]|) + |[(1 - r), 0 ]|) + |[((r * r2) + r), ((r * s1) + 0 )]|) by EUCLID: 56

          .= (( |[(r3 * (r2 + r2)), (r3 * (s1 + s2))]| + |[(1 - r), 0 ]|) + |[((r * r2) + r), ((r * s1) + 0 )]|) by EUCLID: 58

          .= ( |[((r3 * (r2 + r2)) + (1 - r)), ((r3 * (s1 + s2)) + 0 )]| + |[((r * r2) + r), ((r * s1) + 0 )]|) by EUCLID: 56

          .= |[(((r3 * (r2 + r2)) + (1 - r)) + ((r * r2) + r)), ((r3 * (s1 + s2)) + (r * s1))]| by EUCLID: 56;

          hence p in ( Int ( cell (G,( len G),j))) by A16, A20, A17;

        end;

      end;

      hence thesis by XBOOLE_0:def 3;

    end;

    theorem :: GOBOARD6:51

    

     Th51: 1 <= j & j < ( width G) implies ( LSeg ((((1 / 2) * ((G * (( len G),j)) + (G * (( len G),(j + 1))))) + |[1, 0 ]|),((G * (( len G),(j + 1))) + |[1, 0 ]|))) c= (( Int ( cell (G,( len G),j))) \/ {((G * (( len G),(j + 1))) + |[1, 0 ]|)})

    proof

      assume that

       A1: 1 <= j and

       A2: j < ( width G);

      let x be object;

      assume

       A3: x in ( LSeg ((((1 / 2) * ((G * (( len G),j)) + (G * (( len G),(j + 1))))) + |[1, 0 ]|),((G * (( len G),(j + 1))) + |[1, 0 ]|)));

      then

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

      consider r such that

       A4: p = (((1 - r) * (((1 / 2) * ((G * (( len G),j)) + (G * (( len G),(j + 1))))) + |[1, 0 ]|)) + (r * ((G * (( len G),(j + 1))) + |[1, 0 ]|))) and

       A5: 0 <= r and

       A6: r <= 1 by A3;

      now

        per cases by A6, XXREAL_0: 1;

          case r = 1;

          

          then p = (( 0. ( TOP-REAL 2)) + (1 * ((G * (( len G),(j + 1))) + |[1, 0 ]|))) by A4, RLVECT_1: 10

          .= (1 * ((G * (( len G),(j + 1))) + |[1, 0 ]|)) by RLVECT_1: 4

          .= ((G * (( len G),(j + 1))) + |[1, 0 ]|) by RLVECT_1:def 8;

          hence p in {((G * (( len G),(j + 1))) + |[1, 0 ]|)} by TARSKI:def 1;

        end;

          case

           A7: r < 1;

          set r3 = ((1 - r) * (1 / 2));

          (1 - r) > 0 by A7, XREAL_1: 50;

          then

           A8: r3 > ((1 / 2) * 0 ) by XREAL_1: 68;

          set r2 = ((G * (( len G),1)) `1 ), s1 = ((G * (1,j)) `2 ), s2 = ((G * (1,(j + 1))) `2 );

          

           A9: ((r3 * (s1 + s1)) + (r * s1)) = s1;

          

           A10: (j + 1) <= ( width G) by A2, NAT_1: 13;

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

          then

           A11: 1 <= ( len G) by NAT_1: 14;

          j < (j + 1) by XREAL_1: 29;

          then

           A12: s1 < s2 by A1, A10, A11, GOBOARD5: 4;

          then (s1 + s1) < (s1 + s2) by XREAL_1: 6;

          then

           A13: (r3 * (s1 + s1)) < (r3 * (s1 + s2)) by A8, XREAL_1: 68;

          

           A14: ((r3 * (s2 + s2)) + (r * s2)) = s2;

          (s1 + s2) < (s2 + s2) by A12, XREAL_1: 6;

          then (r3 * (s1 + s2)) < (r3 * (s2 + s2)) by A8, XREAL_1: 68;

          then

           A15: ((r3 * (s1 + s2)) + (r * s2)) < s2 by A14, XREAL_1: 8;

          

           A16: (G * (( len G),j)) = |[((G * (( len G),j)) `1 ), ((G * (( len G),j)) `2 )]| by EUCLID: 53

          .= |[r2, ((G * (( len G),j)) `2 )]| by A1, A2, A11, GOBOARD5: 2

          .= |[r2, s1]| by A1, A2, A11, GOBOARD5: 1;

          

           A17: 1 <= (j + 1) by A1, NAT_1: 13;

          (r * s1) <= (r * s2) by A5, A12, XREAL_1: 64;

          then

           A18: (r2 + 1) > r2 & s1 < ((r3 * (s1 + s2)) + (r * s2)) by A13, A9, XREAL_1: 8, XREAL_1: 29;

          

           A19: ( Int ( cell (G,( len G),j))) = { |[r9, s9]| : ((G * (( len G),1)) `1 ) < r9 & ((G * (1,j)) `2 ) < s9 & s9 < ((G * (1,(j + 1))) `2 ) } by A1, A2, Th23;

          

           A20: (G * (( len G),(j + 1))) = |[((G * (( len G),(j + 1))) `1 ), ((G * (( len G),(j + 1))) `2 )]| by EUCLID: 53

          .= |[r2, ((G * (( len G),(j + 1))) `2 )]| by A17, A10, A11, GOBOARD5: 2

          .= |[r2, s2]| by A17, A10, A11, GOBOARD5: 1;

          p = ((((1 - r) * ((1 / 2) * ((G * (( len G),j)) + (G * (( len G),(j + 1)))))) + ((1 - r) * |[1, 0 ]|)) + (r * ((G * (( len G),(j + 1))) + |[1, 0 ]|))) by A4, RLVECT_1:def 5

          .= (((r3 * ((G * (( len G),j)) + (G * (( len G),(j + 1))))) + ((1 - r) * |[1, 0 ]|)) + (r * ((G * (( len G),(j + 1))) + |[1, 0 ]|))) by RLVECT_1:def 7

          .= (((r3 * ((G * (( len G),j)) + (G * (( len G),(j + 1))))) + |[((1 - r) * 1), ((1 - r) * 0 )]|) + (r * ((G * (( len G),(j + 1))) + |[1, 0 ]|))) by EUCLID: 58

          .= (((r3 * |[(r2 + r2), (s1 + s2)]|) + |[(1 - r), 0 ]|) + (r * ( |[r2, s2]| + |[1, 0 ]|))) by A20, A16, EUCLID: 56

          .= (((r3 * |[(r2 + r2), (s1 + s2)]|) + |[(1 - r), 0 ]|) + ((r * |[r2, s2]|) + (r * |[1, 0 ]|))) by RLVECT_1:def 5

          .= (((r3 * |[(r2 + r2), (s1 + s2)]|) + |[(1 - r), 0 ]|) + ( |[(r * r2), (r * s2)]| + (r * |[1, 0 ]|))) by EUCLID: 58

          .= (((r3 * |[(r2 + r2), (s1 + s2)]|) + |[(1 - r), 0 ]|) + ( |[(r * r2), (r * s2)]| + |[(r * 1), (r * 0 )]|)) by EUCLID: 58

          .= (((r3 * |[(r2 + r2), (s1 + s2)]|) + |[(1 - r), 0 ]|) + |[((r * r2) + r), ((r * s2) + 0 )]|) by EUCLID: 56

          .= (( |[(r3 * (r2 + r2)), (r3 * (s1 + s2))]| + |[(1 - r), 0 ]|) + |[((r * r2) + r), ((r * s2) + 0 )]|) by EUCLID: 58

          .= ( |[((r3 * (r2 + r2)) + (1 - r)), ((r3 * (s1 + s2)) + 0 )]| + |[((r * r2) + r), ((r * s2) + 0 )]|) by EUCLID: 56

          .= |[(((r3 * (r2 + r2)) + (1 - r)) + ((r * r2) + r)), ((r3 * (s1 + s2)) + (r * s2))]| by EUCLID: 56;

          hence p in ( Int ( cell (G,( len G),j))) by A18, A15, A19;

        end;

      end;

      hence thesis by XBOOLE_0:def 3;

    end;

    theorem :: GOBOARD6:52

    

     Th52: 1 <= i & i < ( len G) implies ( LSeg ((((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1)))) - |[ 0 , 1]|),((G * (i,1)) - |[ 0 , 1]|))) c= (( Int ( cell (G,i, 0 ))) \/ {((G * (i,1)) - |[ 0 , 1]|)})

    proof

      assume that

       A1: 1 <= i and

       A2: i < ( len G);

      let x be object;

      assume

       A3: x in ( LSeg ((((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1)))) - |[ 0 , 1]|),((G * (i,1)) - |[ 0 , 1]|)));

      then

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

      consider r such that

       A4: p = (((1 - r) * (((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1)))) - |[ 0 , 1]|)) + (r * ((G * (i,1)) - |[ 0 , 1]|))) and

       A5: 0 <= r and

       A6: r <= 1 by A3;

      now

        per cases by A6, XXREAL_0: 1;

          case r = 1;

          

          then p = (( 0. ( TOP-REAL 2)) + (1 * ((G * (i,1)) - |[ 0 , 1]|))) by A4, RLVECT_1: 10

          .= (1 * ((G * (i,1)) - |[ 0 , 1]|)) by RLVECT_1: 4

          .= ((G * (i,1)) - |[ 0 , 1]|) by RLVECT_1:def 8;

          hence p in {((G * (i,1)) - |[ 0 , 1]|)} by TARSKI:def 1;

        end;

          case

           A7: r < 1;

          set r3 = ((1 - r) * (1 / 2));

          (1 - r) > 0 by A7, XREAL_1: 50;

          then

           A8: r3 > ((1 / 2) * 0 ) by XREAL_1: 68;

          set s1 = ((G * (1,1)) `2 ), r1 = ((G * (i,1)) `1 ), r2 = ((G * ((i + 1),1)) `1 );

          

           A9: ((r3 * (r1 + r1)) + (r * r1)) = r1;

          

           A10: (i + 1) <= ( len G) by A2, NAT_1: 13;

          s1 < (s1 + 1) by XREAL_1: 29;

          then

           A11: (s1 - 1) < s1 by XREAL_1: 19;

          

           A12: ( Int ( cell (G,i, 0 ))) = { |[r9, s9]| : ((G * (i,1)) `1 ) < r9 & r9 < ((G * ((i + 1),1)) `1 ) & s9 < ((G * (1,1)) `2 ) } by A1, A2, Th24;

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

          then

           A13: 1 <= ( width G) by NAT_1: 14;

          i < (i + 1) by XREAL_1: 29;

          then

           A14: r1 < r2 by A1, A10, A13, GOBOARD5: 3;

          then (r1 + r2) < (r2 + r2) by XREAL_1: 6;

          then

           A15: (r3 * (r1 + r2)) < (r3 * (r2 + r2)) by A8, XREAL_1: 68;

          (r1 + r1) < (r1 + r2) by A14, XREAL_1: 6;

          then (r3 * (r1 + r1)) < (r3 * (r1 + r2)) by A8, XREAL_1: 68;

          then

           A16: r1 < ((r3 * (r1 + r2)) + (r * r1)) by A9, XREAL_1: 6;

          

           A17: ((r3 * (r2 + r2)) + (r * r2)) = r2;

          (r * r1) <= (r * r2) by A5, A14, XREAL_1: 64;

          then

           A18: ((r3 * (r1 + r2)) + (r * r1)) < r2 by A15, A17, XREAL_1: 8;

          

           A19: (G * (i,1)) = |[((G * (i,1)) `1 ), ((G * (i,1)) `2 )]| by EUCLID: 53

          .= |[r1, s1]| by A1, A2, A13, GOBOARD5: 1;

          

           A20: 1 <= (i + 1) by A1, NAT_1: 13;

          

           A21: (G * ((i + 1),1)) = |[((G * ((i + 1),1)) `1 ), ((G * ((i + 1),1)) `2 )]| by EUCLID: 53

          .= |[r2, s1]| by A20, A10, A13, GOBOARD5: 1;

          p = ((((1 - r) * ((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1))))) - ((1 - r) * |[ 0 , 1]|)) + (r * ((G * (i,1)) - |[ 0 , 1]|))) by A4, RLVECT_1: 34

          .= (((r3 * ((G * (i,1)) + (G * ((i + 1),1)))) - ((1 - r) * |[ 0 , 1]|)) + (r * ((G * (i,1)) - |[ 0 , 1]|))) by RLVECT_1:def 7

          .= (((r3 * ((G * (i,1)) + (G * ((i + 1),1)))) - |[((1 - r) * 0 ), ((1 - r) * 1)]|) + (r * ((G * (i,1)) - |[ 0 , 1]|))) by EUCLID: 58

          .= (((r3 * |[(r1 + r2), (s1 + s1)]|) - |[ 0 , (1 - r)]|) + (r * ( |[r1, s1]| - |[ 0 , 1]|))) by A21, A19, EUCLID: 56

          .= (((r3 * |[(r1 + r2), (s1 + s1)]|) - |[ 0 , (1 - r)]|) + ((r * |[r1, s1]|) - (r * |[ 0 , 1]|))) by RLVECT_1: 34

          .= (((r3 * |[(r1 + r2), (s1 + s1)]|) - |[ 0 , (1 - r)]|) + ( |[(r * r1), (r * s1)]| - (r * |[ 0 , 1]|))) by EUCLID: 58

          .= (((r3 * |[(r1 + r2), (s1 + s1)]|) - |[ 0 , (1 - r)]|) + ( |[(r * r1), (r * s1)]| - |[(r * 0 ), (r * 1)]|)) by EUCLID: 58

          .= (((r3 * |[(r1 + r2), (s1 + s1)]|) - |[ 0 , (1 - r)]|) + |[((r * r1) - 0 ), ((r * s1) - r)]|) by EUCLID: 62

          .= (( |[(r3 * (r1 + r2)), (r3 * (s1 + s1))]| - |[ 0 , (1 - r)]|) + |[((r * r1) - 0 ), ((r * s1) - r)]|) by EUCLID: 58

          .= ( |[((r3 * (r1 + r2)) - 0 ), ((r3 * (s1 + s1)) - (1 - r))]| + |[((r * r1) - 0 ), ((r * s1) - r)]|) by EUCLID: 62

          .= |[((r3 * (r1 + r2)) + (r * r1)), (((r3 * (s1 + s1)) - (1 - r)) + ((r * s1) - r))]| by EUCLID: 56;

          hence p in ( Int ( cell (G,i, 0 ))) by A11, A16, A18, A12;

        end;

      end;

      hence thesis by XBOOLE_0:def 3;

    end;

    theorem :: GOBOARD6:53

    

     Th53: 1 <= i & i < ( len G) implies ( LSeg ((((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1)))) - |[ 0 , 1]|),((G * ((i + 1),1)) - |[ 0 , 1]|))) c= (( Int ( cell (G,i, 0 ))) \/ {((G * ((i + 1),1)) - |[ 0 , 1]|)})

    proof

      assume that

       A1: 1 <= i and

       A2: i < ( len G);

      let x be object;

      assume

       A3: x in ( LSeg ((((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1)))) - |[ 0 , 1]|),((G * ((i + 1),1)) - |[ 0 , 1]|)));

      then

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

      consider r such that

       A4: p = (((1 - r) * (((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1)))) - |[ 0 , 1]|)) + (r * ((G * ((i + 1),1)) - |[ 0 , 1]|))) and

       A5: 0 <= r and

       A6: r <= 1 by A3;

      now

        per cases by A6, XXREAL_0: 1;

          case r = 1;

          

          then p = (( 0. ( TOP-REAL 2)) + (1 * ((G * ((i + 1),1)) - |[ 0 , 1]|))) by A4, RLVECT_1: 10

          .= (1 * ((G * ((i + 1),1)) - |[ 0 , 1]|)) by RLVECT_1: 4

          .= ((G * ((i + 1),1)) - |[ 0 , 1]|) by RLVECT_1:def 8;

          hence p in {((G * ((i + 1),1)) - |[ 0 , 1]|)} by TARSKI:def 1;

        end;

          case

           A7: r < 1;

          set r3 = ((1 - r) * (1 / 2));

          (1 - r) > 0 by A7, XREAL_1: 50;

          then

           A8: r3 > ((1 / 2) * 0 ) by XREAL_1: 68;

          set s1 = ((G * (1,1)) `2 ), r1 = ((G * (i,1)) `1 ), r2 = ((G * ((i + 1),1)) `1 );

          

           A9: ((r3 * (r1 + r1)) + (r * r1)) = r1;

          

           A10: (i + 1) <= ( len G) by A2, NAT_1: 13;

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

          then

           A11: 1 <= ( width G) by NAT_1: 14;

          i < (i + 1) by XREAL_1: 29;

          then

           A12: r1 < r2 by A1, A10, A11, GOBOARD5: 3;

          then (r1 + r1) < (r1 + r2) by XREAL_1: 6;

          then

           A13: (r3 * (r1 + r1)) < (r3 * (r1 + r2)) by A8, XREAL_1: 68;

          (r * r1) <= (r * r2) by A5, A12, XREAL_1: 64;

          then

           A14: r1 < ((r3 * (r1 + r2)) + (r * r2)) by A13, A9, XREAL_1: 8;

          

           A15: 1 <= (i + 1) by A1, NAT_1: 13;

          

           A16: (G * (i,1)) = |[((G * (i,1)) `1 ), ((G * (i,1)) `2 )]| by EUCLID: 53

          .= |[r1, s1]| by A1, A2, A11, GOBOARD5: 1;

          s1 < (s1 + 1) by XREAL_1: 29;

          then

           A17: (s1 - 1) < s1 by XREAL_1: 19;

          

           A18: ((r3 * (r2 + r2)) + (r * r2)) = r2;

          (r1 + r2) < (r2 + r2) by A12, XREAL_1: 6;

          then (r3 * (r1 + r2)) < (r3 * (r2 + r2)) by A8, XREAL_1: 68;

          then

           A19: ((r3 * (r1 + r2)) + (r * r2)) < r2 by A18, XREAL_1: 8;

          

           A20: ( Int ( cell (G,i, 0 ))) = { |[r9, s9]| : ((G * (i,1)) `1 ) < r9 & r9 < ((G * ((i + 1),1)) `1 ) & s9 < ((G * (1,1)) `2 ) } by A1, A2, Th24;

          

           A21: (G * ((i + 1),1)) = |[((G * ((i + 1),1)) `1 ), ((G * ((i + 1),1)) `2 )]| by EUCLID: 53

          .= |[r2, s1]| by A15, A10, A11, GOBOARD5: 1;

          p = ((((1 - r) * ((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1))))) - ((1 - r) * |[ 0 , 1]|)) + (r * ((G * ((i + 1),1)) - |[ 0 , 1]|))) by A4, RLVECT_1: 34

          .= (((r3 * ((G * (i,1)) + (G * ((i + 1),1)))) - ((1 - r) * |[ 0 , 1]|)) + (r * ((G * ((i + 1),1)) - |[ 0 , 1]|))) by RLVECT_1:def 7

          .= (((r3 * ((G * (i,1)) + (G * ((i + 1),1)))) - |[((1 - r) * 0 ), ((1 - r) * 1)]|) + (r * ((G * ((i + 1),1)) - |[ 0 , 1]|))) by EUCLID: 58

          .= (((r3 * |[(r1 + r2), (s1 + s1)]|) - |[ 0 , (1 - r)]|) + (r * ( |[r2, s1]| - |[ 0 , 1]|))) by A21, A16, EUCLID: 56

          .= (((r3 * |[(r1 + r2), (s1 + s1)]|) - |[ 0 , (1 - r)]|) + ((r * |[r2, s1]|) - (r * |[ 0 , 1]|))) by RLVECT_1: 34

          .= (((r3 * |[(r1 + r2), (s1 + s1)]|) - |[ 0 , (1 - r)]|) + ( |[(r * r2), (r * s1)]| - (r * |[ 0 , 1]|))) by EUCLID: 58

          .= (((r3 * |[(r1 + r2), (s1 + s1)]|) - |[ 0 , (1 - r)]|) + ( |[(r * r2), (r * s1)]| - |[(r * 0 ), (r * 1)]|)) by EUCLID: 58

          .= (((r3 * |[(r1 + r2), (s1 + s1)]|) - |[ 0 , (1 - r)]|) + |[((r * r2) - 0 ), ((r * s1) - r)]|) by EUCLID: 62

          .= (( |[(r3 * (r1 + r2)), (r3 * (s1 + s1))]| - |[ 0 , (1 - r)]|) + |[((r * r2) - 0 ), ((r * s1) - r)]|) by EUCLID: 58

          .= ( |[((r3 * (r1 + r2)) - 0 ), ((r3 * (s1 + s1)) - (1 - r))]| + |[((r * r2) - 0 ), ((r * s1) - r)]|) by EUCLID: 62

          .= |[((r3 * (r1 + r2)) + (r * r2)), (((r3 * (s1 + s1)) - (1 - r)) + ((r * s1) - r))]| by EUCLID: 56;

          hence p in ( Int ( cell (G,i, 0 ))) by A17, A14, A19, A20;

        end;

      end;

      hence thesis by XBOOLE_0:def 3;

    end;

    theorem :: GOBOARD6:54

    

     Th54: 1 <= i & i < ( len G) implies ( LSeg ((((1 / 2) * ((G * (i,( width G))) + (G * ((i + 1),( width G))))) + |[ 0 , 1]|),((G * (i,( width G))) + |[ 0 , 1]|))) c= (( Int ( cell (G,i,( width G)))) \/ {((G * (i,( width G))) + |[ 0 , 1]|)})

    proof

      assume that

       A1: 1 <= i and

       A2: i < ( len G);

      let x be object;

      assume

       A3: x in ( LSeg ((((1 / 2) * ((G * (i,( width G))) + (G * ((i + 1),( width G))))) + |[ 0 , 1]|),((G * (i,( width G))) + |[ 0 , 1]|)));

      then

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

      consider r such that

       A4: p = (((1 - r) * (((1 / 2) * ((G * (i,( width G))) + (G * ((i + 1),( width G))))) + |[ 0 , 1]|)) + (r * ((G * (i,( width G))) + |[ 0 , 1]|))) and

       A5: 0 <= r and

       A6: r <= 1 by A3;

      now

        per cases by A6, XXREAL_0: 1;

          case r = 1;

          

          then p = (( 0. ( TOP-REAL 2)) + (1 * ((G * (i,( width G))) + |[ 0 , 1]|))) by A4, RLVECT_1: 10

          .= (1 * ((G * (i,( width G))) + |[ 0 , 1]|)) by RLVECT_1: 4

          .= ((G * (i,( width G))) + |[ 0 , 1]|) by RLVECT_1:def 8;

          hence p in {((G * (i,( width G))) + |[ 0 , 1]|)} by TARSKI:def 1;

        end;

          case

           A7: r < 1;

          set r3 = ((1 - r) * (1 / 2));

          (1 - r) > 0 by A7, XREAL_1: 50;

          then

           A8: r3 > ((1 / 2) * 0 ) by XREAL_1: 68;

          set s1 = ((G * (1,( width G))) `2 ), r1 = ((G * (i,1)) `1 ), r2 = ((G * ((i + 1),1)) `1 );

          

           A9: ((r3 * (r1 + r1)) + (r * r1)) = r1;

          

           A10: (i + 1) <= ( len G) by A2, NAT_1: 13;

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

          then

           A11: 1 <= ( width G) by NAT_1: 14;

          

           A12: (G * (i,( width G))) = |[((G * (i,( width G))) `1 ), ((G * (i,( width G))) `2 )]| by EUCLID: 53

          .= |[((G * (i,( width G))) `1 ), s1]| by A1, A2, A11, GOBOARD5: 1

          .= |[r1, s1]| by A1, A2, A11, GOBOARD5: 2;

          

           A13: 1 <= (i + 1) by A1, NAT_1: 13;

          i < (i + 1) by XREAL_1: 29;

          then

           A14: r1 < r2 by A1, A10, A11, GOBOARD5: 3;

          then (r1 + r2) < (r2 + r2) by XREAL_1: 6;

          then

           A15: (r3 * (r1 + r2)) < (r3 * (r2 + r2)) by A8, XREAL_1: 68;

          (r1 + r1) < (r1 + r2) by A14, XREAL_1: 6;

          then (r3 * (r1 + r1)) < (r3 * (r1 + r2)) by A8, XREAL_1: 68;

          then

           A16: s1 < (s1 + 1) & r1 < ((r3 * (r1 + r2)) + (r * r1)) by A9, XREAL_1: 6, XREAL_1: 29;

          

           A17: ( Int ( cell (G,i,( width G)))) = { |[r9, s9]| : ((G * (i,1)) `1 ) < r9 & r9 < ((G * ((i + 1),1)) `1 ) & ((G * (1,( width G))) `2 ) < s9 } by A1, A2, Th25;

          

           A18: (G * ((i + 1),( width G))) = |[((G * ((i + 1),( width G))) `1 ), ((G * ((i + 1),( width G))) `2 )]| by EUCLID: 53

          .= |[((G * ((i + 1),( width G))) `1 ), s1]| by A13, A10, A11, GOBOARD5: 1

          .= |[r2, s1]| by A13, A10, A11, GOBOARD5: 2;

          

           A19: ((r3 * (r2 + r2)) + (r * r2)) = r2;

          (r * r1) <= (r * r2) by A5, A14, XREAL_1: 64;

          then

           A20: ((r3 * (r1 + r2)) + (r * r1)) < r2 by A15, A19, XREAL_1: 8;

          p = ((((1 - r) * ((1 / 2) * ((G * (i,( width G))) + (G * ((i + 1),( width G)))))) + ((1 - r) * |[ 0 , 1]|)) + (r * ((G * (i,( width G))) + |[ 0 , 1]|))) by A4, RLVECT_1:def 5

          .= (((r3 * ((G * (i,( width G))) + (G * ((i + 1),( width G))))) + ((1 - r) * |[ 0 , 1]|)) + (r * ((G * (i,( width G))) + |[ 0 , 1]|))) by RLVECT_1:def 7

          .= (((r3 * ((G * (i,( width G))) + (G * ((i + 1),( width G))))) + |[((1 - r) * 0 ), ((1 - r) * 1)]|) + (r * ((G * (i,( width G))) + |[ 0 , 1]|))) by EUCLID: 58

          .= (((r3 * |[(r1 + r2), (s1 + s1)]|) + |[ 0 , (1 - r)]|) + (r * ( |[r1, s1]| + |[ 0 , 1]|))) by A18, A12, EUCLID: 56

          .= (((r3 * |[(r1 + r2), (s1 + s1)]|) + |[ 0 , (1 - r)]|) + ((r * |[r1, s1]|) + (r * |[ 0 , 1]|))) by RLVECT_1:def 5

          .= (((r3 * |[(r1 + r2), (s1 + s1)]|) + |[ 0 , (1 - r)]|) + ( |[(r * r1), (r * s1)]| + (r * |[ 0 , 1]|))) by EUCLID: 58

          .= (((r3 * |[(r1 + r2), (s1 + s1)]|) + |[ 0 , (1 - r)]|) + ( |[(r * r1), (r * s1)]| + |[(r * 0 ), (r * 1)]|)) by EUCLID: 58

          .= (((r3 * |[(r1 + r2), (s1 + s1)]|) + |[ 0 , (1 - r)]|) + |[((r * r1) + 0 ), ((r * s1) + r)]|) by EUCLID: 56

          .= (( |[(r3 * (r1 + r2)), (r3 * (s1 + s1))]| + |[ 0 , (1 - r)]|) + |[((r * r1) + 0 ), ((r * s1) + r)]|) by EUCLID: 58

          .= ( |[((r3 * (r1 + r2)) + 0 ), ((r3 * (s1 + s1)) + (1 - r))]| + |[((r * r1) + 0 ), ((r * s1) + r)]|) by EUCLID: 56

          .= |[((r3 * (r1 + r2)) + (r * r1)), (((r3 * (s1 + s1)) + (1 - r)) + ((r * s1) + r))]| by EUCLID: 56;

          hence p in ( Int ( cell (G,i,( width G)))) by A16, A20, A17;

        end;

      end;

      hence thesis by XBOOLE_0:def 3;

    end;

    theorem :: GOBOARD6:55

    

     Th55: 1 <= i & i < ( len G) implies ( LSeg ((((1 / 2) * ((G * (i,( width G))) + (G * ((i + 1),( width G))))) + |[ 0 , 1]|),((G * ((i + 1),( width G))) + |[ 0 , 1]|))) c= (( Int ( cell (G,i,( width G)))) \/ {((G * ((i + 1),( width G))) + |[ 0 , 1]|)})

    proof

      assume that

       A1: 1 <= i and

       A2: i < ( len G);

      let x be object;

      assume

       A3: x in ( LSeg ((((1 / 2) * ((G * (i,( width G))) + (G * ((i + 1),( width G))))) + |[ 0 , 1]|),((G * ((i + 1),( width G))) + |[ 0 , 1]|)));

      then

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

      consider r such that

       A4: p = (((1 - r) * (((1 / 2) * ((G * (i,( width G))) + (G * ((i + 1),( width G))))) + |[ 0 , 1]|)) + (r * ((G * ((i + 1),( width G))) + |[ 0 , 1]|))) and

       A5: 0 <= r and

       A6: r <= 1 by A3;

      now

        per cases by A6, XXREAL_0: 1;

          case r = 1;

          

          then p = (( 0. ( TOP-REAL 2)) + (1 * ((G * ((i + 1),( width G))) + |[ 0 , 1]|))) by A4, RLVECT_1: 10

          .= (1 * ((G * ((i + 1),( width G))) + |[ 0 , 1]|)) by RLVECT_1: 4

          .= ((G * ((i + 1),( width G))) + |[ 0 , 1]|) by RLVECT_1:def 8;

          hence p in {((G * ((i + 1),( width G))) + |[ 0 , 1]|)} by TARSKI:def 1;

        end;

          case

           A7: r < 1;

          set r3 = ((1 - r) * (1 / 2));

          (1 - r) > 0 by A7, XREAL_1: 50;

          then

           A8: r3 > ((1 / 2) * 0 ) by XREAL_1: 68;

          set s1 = ((G * (1,( width G))) `2 ), r1 = ((G * (i,1)) `1 ), r2 = ((G * ((i + 1),1)) `1 );

          

           A9: ((r3 * (r1 + r1)) + (r * r1)) = r1;

          

           A10: (i + 1) <= ( len G) by A2, NAT_1: 13;

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

          then

           A11: 1 <= ( width G) by NAT_1: 14;

          i < (i + 1) by XREAL_1: 29;

          then

           A12: r1 < r2 by A1, A10, A11, GOBOARD5: 3;

          then (r1 + r1) < (r1 + r2) by XREAL_1: 6;

          then

           A13: (r3 * (r1 + r1)) < (r3 * (r1 + r2)) by A8, XREAL_1: 68;

          

           A14: ((r3 * (r2 + r2)) + (r * r2)) = r2;

          (r1 + r2) < (r2 + r2) by A12, XREAL_1: 6;

          then (r3 * (r1 + r2)) < (r3 * (r2 + r2)) by A8, XREAL_1: 68;

          then

           A15: ((r3 * (r1 + r2)) + (r * r2)) < r2 by A14, XREAL_1: 8;

          

           A16: (G * (i,( width G))) = |[((G * (i,( width G))) `1 ), ((G * (i,( width G))) `2 )]| by EUCLID: 53

          .= |[((G * (i,( width G))) `1 ), s1]| by A1, A2, A11, GOBOARD5: 1

          .= |[r1, s1]| by A1, A2, A11, GOBOARD5: 2;

          

           A17: 1 <= (i + 1) by A1, NAT_1: 13;

          (r * r1) <= (r * r2) by A5, A12, XREAL_1: 64;

          then

           A18: (s1 + 1) > s1 & r1 < ((r3 * (r1 + r2)) + (r * r2)) by A13, A9, XREAL_1: 8, XREAL_1: 29;

          

           A19: ( Int ( cell (G,i,( width G)))) = { |[r9, s9]| : ((G * (i,1)) `1 ) < r9 & r9 < ((G * ((i + 1),1)) `1 ) & ((G * (1,( width G))) `2 ) < s9 } by A1, A2, Th25;

          

           A20: (G * ((i + 1),( width G))) = |[((G * ((i + 1),( width G))) `1 ), ((G * ((i + 1),( width G))) `2 )]| by EUCLID: 53

          .= |[((G * ((i + 1),( width G))) `1 ), s1]| by A17, A10, A11, GOBOARD5: 1

          .= |[r2, s1]| by A17, A10, A11, GOBOARD5: 2;

          p = ((((1 - r) * ((1 / 2) * ((G * (i,( width G))) + (G * ((i + 1),( width G)))))) + ((1 - r) * |[ 0 , 1]|)) + (r * ((G * ((i + 1),( width G))) + |[ 0 , 1]|))) by A4, RLVECT_1:def 5

          .= (((r3 * ((G * (i,( width G))) + (G * ((i + 1),( width G))))) + ((1 - r) * |[ 0 , 1]|)) + (r * ((G * ((i + 1),( width G))) + |[ 0 , 1]|))) by RLVECT_1:def 7

          .= (((r3 * ((G * (i,( width G))) + (G * ((i + 1),( width G))))) + |[((1 - r) * 0 ), ((1 - r) * 1)]|) + (r * ((G * ((i + 1),( width G))) + |[ 0 , 1]|))) by EUCLID: 58

          .= (((r3 * |[(r1 + r2), (s1 + s1)]|) + |[ 0 , (1 - r)]|) + (r * ( |[r2, s1]| + |[ 0 , 1]|))) by A20, A16, EUCLID: 56

          .= (((r3 * |[(r1 + r2), (s1 + s1)]|) + |[ 0 , (1 - r)]|) + ((r * |[r2, s1]|) + (r * |[ 0 , 1]|))) by RLVECT_1:def 5

          .= (((r3 * |[(r1 + r2), (s1 + s1)]|) + |[ 0 , (1 - r)]|) + ( |[(r * r2), (r * s1)]| + (r * |[ 0 , 1]|))) by EUCLID: 58

          .= (((r3 * |[(r1 + r2), (s1 + s1)]|) + |[ 0 , (1 - r)]|) + ( |[(r * r2), (r * s1)]| + |[(r * 0 ), (r * 1)]|)) by EUCLID: 58

          .= (((r3 * |[(r1 + r2), (s1 + s1)]|) + |[ 0 , (1 - r)]|) + |[((r * r2) + 0 ), ((r * s1) + r)]|) by EUCLID: 56

          .= (( |[(r3 * (r1 + r2)), (r3 * (s1 + s1))]| + |[ 0 , (1 - r)]|) + |[((r * r2) + 0 ), ((r * s1) + r)]|) by EUCLID: 58

          .= ( |[((r3 * (r1 + r2)) + 0 ), ((r3 * (s1 + s1)) + (1 - r))]| + |[((r * r2) + 0 ), ((r * s1) + r)]|) by EUCLID: 56

          .= |[((r3 * (r1 + r2)) + (r * r2)), (((r3 * (s1 + s1)) + (1 - r)) + ((r * s1) + r))]| by EUCLID: 56;

          hence p in ( Int ( cell (G,i,( width G)))) by A18, A15, A19;

        end;

      end;

      hence thesis by XBOOLE_0:def 3;

    end;

    theorem :: GOBOARD6:56

    

     Th56: ( LSeg (((G * (1,1)) - |[1, 1]|),((G * (1,1)) - |[1, 0 ]|))) c= (( Int ( cell (G, 0 , 0 ))) \/ {((G * (1,1)) - |[1, 0 ]|)})

    proof

      let x be object;

      set r1 = ((G * (1,1)) `1 ), s1 = ((G * (1,1)) `2 );

      assume

       A1: x in ( LSeg (((G * (1,1)) - |[1, 1]|),((G * (1,1)) - |[1, 0 ]|)));

      then

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

      consider r such that

       A2: p = (((1 - r) * ((G * (1,1)) - |[1, 1]|)) + (r * ((G * (1,1)) - |[1, 0 ]|))) and 0 <= r and

       A3: r <= 1 by A1;

      now

        per cases by A3, XXREAL_0: 1;

          case r = 1;

          

          then p = (( 0. ( TOP-REAL 2)) + (1 * ((G * (1,1)) - |[1, 0 ]|))) by A2, RLVECT_1: 10

          .= (1 * ((G * (1,1)) - |[1, 0 ]|)) by RLVECT_1: 4

          .= ((G * (1,1)) - |[1, 0 ]|) by RLVECT_1:def 8;

          hence p in {((G * (1,1)) - |[1, 0 ]|)} by TARSKI:def 1;

        end;

          case r < 1;

          then (1 - r) > 0 by XREAL_1: 50;

          then s1 < (s1 + (1 - r)) by XREAL_1: 29;

          then

           A4: (s1 - (1 - r)) < s1 by XREAL_1: 19;

          

           A5: (G * (1,1)) = |[r1, s1]| by EUCLID: 53;

          r1 < (r1 + 1) by XREAL_1: 29;

          then

           A6: (r1 - 1) < r1 by XREAL_1: 19;

          

           A7: ( Int ( cell (G, 0 , 0 ))) = { |[r9, s9]| : r9 < ((G * (1,1)) `1 ) & s9 < ((G * (1,1)) `2 ) } by Th18;

          p = ((((1 - r) * (G * (1,1))) - ((1 - r) * |[1, 1]|)) + (r * ((G * (1,1)) - |[1, 0 ]|))) by A2, RLVECT_1: 34

          .= ((((1 - r) * (G * (1,1))) - ((1 - r) * |[1, 1]|)) + ((r * (G * (1,1))) - (r * |[1, 0 ]|))) by RLVECT_1: 34

          .= (((r * (G * (1,1))) + (((1 - r) * (G * (1,1))) - ((1 - r) * |[1, 1]|))) - (r * |[1, 0 ]|)) by RLVECT_1:def 3

          .= ((((r * (G * (1,1))) + ((1 - r) * (G * (1,1)))) - ((1 - r) * |[1, 1]|)) - (r * |[1, 0 ]|)) by RLVECT_1:def 3

          .= ((((r + (1 - r)) * (G * (1,1))) - ((1 - r) * |[1, 1]|)) - (r * |[1, 0 ]|)) by RLVECT_1:def 6

          .= (((G * (1,1)) - ((1 - r) * |[1, 1]|)) - (r * |[1, 0 ]|)) by RLVECT_1:def 8

          .= (((G * (1,1)) - |[((1 - r) * 1), ((1 - r) * 1)]|) - (r * |[1, 0 ]|)) by EUCLID: 58

          .= (((G * (1,1)) - |[(1 - r), (1 - r)]|) - |[(r * 1), (r * 0 )]|) by EUCLID: 58

          .= ( |[(r1 - (1 - r)), (s1 - (1 - r))]| - |[r, 0 ]|) by A5, EUCLID: 62

          .= |[((r1 - (1 - r)) - r), ((s1 - (1 - r)) - 0 )]| by EUCLID: 62

          .= |[(r1 - 1), (s1 - (1 - r))]|;

          hence p in ( Int ( cell (G, 0 , 0 ))) by A4, A6, A7;

        end;

      end;

      hence thesis by XBOOLE_0:def 3;

    end;

    theorem :: GOBOARD6:57

    

     Th57: ( LSeg (((G * (( len G),1)) + |[1, ( - 1)]|),((G * (( len G),1)) + |[1, 0 ]|))) c= (( Int ( cell (G,( len G), 0 ))) \/ {((G * (( len G),1)) + |[1, 0 ]|)})

    proof

      let x be object;

      set r1 = ((G * (( len G),1)) `1 ), s1 = ((G * (1,1)) `2 );

      assume

       A1: x in ( LSeg (((G * (( len G),1)) + |[1, ( - 1)]|),((G * (( len G),1)) + |[1, 0 ]|)));

      then

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

      consider r such that

       A2: p = (((1 - r) * ((G * (( len G),1)) + |[1, ( - 1)]|)) + (r * ((G * (( len G),1)) + |[1, 0 ]|))) and 0 <= r and

       A3: r <= 1 by A1;

      now

        per cases by A3, XXREAL_0: 1;

          case r = 1;

          

          then p = (( 0. ( TOP-REAL 2)) + (1 * ((G * (( len G),1)) + |[1, 0 ]|))) by A2, RLVECT_1: 10

          .= (1 * ((G * (( len G),1)) + |[1, 0 ]|)) by RLVECT_1: 4

          .= ((G * (( len G),1)) + |[1, 0 ]|) by RLVECT_1:def 8;

          hence p in {((G * (( len G),1)) + |[1, 0 ]|)} by TARSKI:def 1;

        end;

          case r < 1;

          then (1 - r) > 0 by XREAL_1: 50;

          then

           A4: s1 < (s1 + (1 - r)) by XREAL_1: 29;

          (s1 + (r - 1)) = (s1 - (1 - r));

          then

           A5: (s1 + (r - 1)) < s1 by A4, XREAL_1: 19;

          

           A6: r1 < (r1 + 1) by XREAL_1: 29;

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

          then

           A7: 1 <= ( len G) by NAT_1: 14;

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

          then

           A8: 1 <= ( width G) by NAT_1: 14;

          

           A9: (G * (( len G),1)) = |[r1, ((G * (( len G),1)) `2 )]| by EUCLID: 53

          .= |[r1, s1]| by A8, A7, GOBOARD5: 1;

          

           A10: ( Int ( cell (G,( len G), 0 ))) = { |[r9, s9]| : ((G * (( len G),1)) `1 ) < r9 & s9 < ((G * (1,1)) `2 ) } by Th21;

          p = ((((1 - r) * (G * (( len G),1))) + ((1 - r) * |[1, ( - 1)]|)) + (r * ((G * (( len G),1)) + |[1, 0 ]|))) by A2, RLVECT_1:def 5

          .= ((((1 - r) * (G * (( len G),1))) + ((1 - r) * |[1, ( - 1)]|)) + ((r * (G * (( len G),1))) + (r * |[1, 0 ]|))) by RLVECT_1:def 5

          .= (((r * (G * (( len G),1))) + (((1 - r) * (G * (( len G),1))) + ((1 - r) * |[1, ( - 1)]|))) + (r * |[1, 0 ]|)) by RLVECT_1:def 3

          .= ((((r * (G * (( len G),1))) + ((1 - r) * (G * (( len G),1)))) + ((1 - r) * |[1, ( - 1)]|)) + (r * |[1, 0 ]|)) by RLVECT_1:def 3

          .= ((((r + (1 - r)) * (G * (( len G),1))) + ((1 - r) * |[1, ( - 1)]|)) + (r * |[1, 0 ]|)) by RLVECT_1:def 6

          .= (((G * (( len G),1)) + ((1 - r) * |[1, ( - 1)]|)) + (r * |[1, 0 ]|)) by RLVECT_1:def 8

          .= (((G * (( len G),1)) + |[((1 - r) * 1), ((1 - r) * ( - 1))]|) + (r * |[1, 0 ]|)) by EUCLID: 58

          .= (((G * (( len G),1)) + |[(1 - r), (r - 1)]|) + |[(r * 1), (r * 0 )]|) by EUCLID: 58

          .= ( |[(r1 + (1 - r)), (s1 + (r - 1))]| + |[r, 0 ]|) by A9, EUCLID: 56

          .= |[((r1 + (1 - r)) + r), ((s1 + (r - 1)) + 0 )]| by EUCLID: 56

          .= |[(r1 + 1), (s1 + (r - 1))]|;

          hence p in ( Int ( cell (G,( len G), 0 ))) by A5, A6, A10;

        end;

      end;

      hence thesis by XBOOLE_0:def 3;

    end;

    theorem :: GOBOARD6:58

    

     Th58: ( LSeg (((G * (1,( width G))) + |[( - 1), 1]|),((G * (1,( width G))) - |[1, 0 ]|))) c= (( Int ( cell (G, 0 ,( width G)))) \/ {((G * (1,( width G))) - |[1, 0 ]|)})

    proof

      let x be object;

      set r1 = ((G * (1,1)) `1 ), s1 = ((G * (1,( width G))) `2 );

      assume

       A1: x in ( LSeg (((G * (1,( width G))) + |[( - 1), 1]|),((G * (1,( width G))) - |[1, 0 ]|)));

      then

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

      consider r such that

       A2: p = (((1 - r) * ((G * (1,( width G))) + |[( - 1), 1]|)) + (r * ((G * (1,( width G))) - |[1, 0 ]|))) and 0 <= r and

       A3: r <= 1 by A1;

      now

        per cases by A3, XXREAL_0: 1;

          case r = 1;

          

          then p = (( 0. ( TOP-REAL 2)) + (1 * ((G * (1,( width G))) - |[1, 0 ]|))) by A2, RLVECT_1: 10

          .= (1 * ((G * (1,( width G))) - |[1, 0 ]|)) by RLVECT_1: 4

          .= ((G * (1,( width G))) - |[1, 0 ]|) by RLVECT_1:def 8;

          hence p in {((G * (1,( width G))) - |[1, 0 ]|)} by TARSKI:def 1;

        end;

          case r < 1;

          then (1 - r) > 0 by XREAL_1: 50;

          then

           A4: s1 < (s1 + (1 - r)) by XREAL_1: 29;

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

          then

           A5: 1 <= ( width G) by NAT_1: 14;

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

          then

           A6: 1 <= ( len G) by NAT_1: 14;

          

           A7: (G * (1,( width G))) = |[((G * (1,( width G))) `1 ), s1]| by EUCLID: 53

          .= |[r1, s1]| by A5, A6, GOBOARD5: 2;

          

           A8: ( Int ( cell (G, 0 ,( width G)))) = { |[r9, s9]| : r9 < ((G * (1,1)) `1 ) & ((G * (1,( width G))) `2 ) < s9 } by Th19;

          r1 < (r1 + 1) by XREAL_1: 29;

          then

           A9: (r1 - 1) < r1 by XREAL_1: 19;

          p = ((((1 - r) * (G * (1,( width G)))) + ((1 - r) * |[( - 1), 1]|)) + (r * ((G * (1,( width G))) - |[1, 0 ]|))) by A2, RLVECT_1:def 5

          .= ((((1 - r) * (G * (1,( width G)))) + ((1 - r) * |[( - 1), 1]|)) + ((r * (G * (1,( width G)))) - (r * |[1, 0 ]|))) by RLVECT_1: 34

          .= (((r * (G * (1,( width G)))) + (((1 - r) * (G * (1,( width G)))) + ((1 - r) * |[( - 1), 1]|))) - (r * |[1, 0 ]|)) by RLVECT_1:def 3

          .= ((((r * (G * (1,( width G)))) + ((1 - r) * (G * (1,( width G))))) + ((1 - r) * |[( - 1), 1]|)) - (r * |[1, 0 ]|)) by RLVECT_1:def 3

          .= ((((r + (1 - r)) * (G * (1,( width G)))) + ((1 - r) * |[( - 1), 1]|)) - (r * |[1, 0 ]|)) by RLVECT_1:def 6

          .= (((G * (1,( width G))) + ((1 - r) * |[( - 1), 1]|)) - (r * |[1, 0 ]|)) by RLVECT_1:def 8

          .= (((G * (1,( width G))) + |[((1 - r) * ( - 1)), ((1 - r) * 1)]|) - (r * |[1, 0 ]|)) by EUCLID: 58

          .= (((G * (1,( width G))) + |[(r - 1), (1 - r)]|) - |[(r * 1), (r * 0 )]|) by EUCLID: 58

          .= ( |[(r1 + (r - 1)), (s1 + (1 - r))]| - |[r, 0 ]|) by A7, EUCLID: 56

          .= |[((r1 + (r - 1)) - r), ((s1 + (1 - r)) - 0 )]| by EUCLID: 62

          .= |[(r1 - 1), (s1 + (1 - r))]|;

          hence p in ( Int ( cell (G, 0 ,( width G)))) by A4, A9, A8;

        end;

      end;

      hence thesis by XBOOLE_0:def 3;

    end;

    theorem :: GOBOARD6:59

    

     Th59: ( LSeg (((G * (( len G),( width G))) + |[1, 1]|),((G * (( len G),( width G))) + |[1, 0 ]|))) c= (( Int ( cell (G,( len G),( width G)))) \/ {((G * (( len G),( width G))) + |[1, 0 ]|)})

    proof

      let x be object;

      set r1 = ((G * (( len G),1)) `1 ), s1 = ((G * (1,( width G))) `2 );

      assume

       A1: x in ( LSeg (((G * (( len G),( width G))) + |[1, 1]|),((G * (( len G),( width G))) + |[1, 0 ]|)));

      then

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

      consider r such that

       A2: p = (((1 - r) * ((G * (( len G),( width G))) + |[1, 1]|)) + (r * ((G * (( len G),( width G))) + |[1, 0 ]|))) and 0 <= r and

       A3: r <= 1 by A1;

      now

        per cases by A3, XXREAL_0: 1;

          case r = 1;

          

          then p = (( 0. ( TOP-REAL 2)) + (1 * ((G * (( len G),( width G))) + |[1, 0 ]|))) by A2, RLVECT_1: 10

          .= (1 * ((G * (( len G),( width G))) + |[1, 0 ]|)) by RLVECT_1: 4

          .= ((G * (( len G),( width G))) + |[1, 0 ]|) by RLVECT_1:def 8;

          hence p in {((G * (( len G),( width G))) + |[1, 0 ]|)} by TARSKI:def 1;

        end;

          case r < 1;

          then (1 - r) > 0 by XREAL_1: 50;

          then

           A4: s1 < (s1 + (1 - r)) by XREAL_1: 29;

          

           A5: r1 < (r1 + 1) by XREAL_1: 29;

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

          then

           A6: 1 <= ( width G) by NAT_1: 14;

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

          then

           A7: 1 <= ( len G) by NAT_1: 14;

          

           A8: (G * (( len G),( width G))) = |[((G * (( len G),( width G))) `1 ), ((G * (( len G),( width G))) `2 )]| by EUCLID: 53

          .= |[r1, ((G * (( len G),( width G))) `2 )]| by A6, A7, GOBOARD5: 2

          .= |[r1, s1]| by A6, A7, GOBOARD5: 1;

          

           A9: ( Int ( cell (G,( len G),( width G)))) = { |[r9, s9]| : ((G * (( len G),1)) `1 ) < r9 & ((G * (1,( width G))) `2 ) < s9 } by Th22;

          p = ((((1 - r) * (G * (( len G),( width G)))) + ((1 - r) * |[1, 1]|)) + (r * ((G * (( len G),( width G))) + |[1, 0 ]|))) by A2, RLVECT_1:def 5

          .= ((((1 - r) * (G * (( len G),( width G)))) + ((1 - r) * |[1, 1]|)) + ((r * (G * (( len G),( width G)))) + (r * |[1, 0 ]|))) by RLVECT_1:def 5

          .= (((r * (G * (( len G),( width G)))) + (((1 - r) * (G * (( len G),( width G)))) + ((1 - r) * |[1, 1]|))) + (r * |[1, 0 ]|)) by RLVECT_1:def 3

          .= ((((r * (G * (( len G),( width G)))) + ((1 - r) * (G * (( len G),( width G))))) + ((1 - r) * |[1, 1]|)) + (r * |[1, 0 ]|)) by RLVECT_1:def 3

          .= ((((r + (1 - r)) * (G * (( len G),( width G)))) + ((1 - r) * |[1, 1]|)) + (r * |[1, 0 ]|)) by RLVECT_1:def 6

          .= (((G * (( len G),( width G))) + ((1 - r) * |[1, 1]|)) + (r * |[1, 0 ]|)) by RLVECT_1:def 8

          .= (((G * (( len G),( width G))) + |[((1 - r) * 1), ((1 - r) * 1)]|) + (r * |[1, 0 ]|)) by EUCLID: 58

          .= (((G * (( len G),( width G))) + |[(1 - r), (1 - r)]|) + |[(r * 1), (r * 0 )]|) by EUCLID: 58

          .= ( |[(r1 + (1 - r)), (s1 + (1 - r))]| + |[r, 0 ]|) by A8, EUCLID: 56

          .= |[((r1 + (1 - r)) + r), ((s1 + (1 - r)) + 0 )]| by EUCLID: 56

          .= |[(r1 + 1), (s1 + (1 - r))]|;

          hence p in ( Int ( cell (G,( len G),( width G)))) by A4, A5, A9;

        end;

      end;

      hence thesis by XBOOLE_0:def 3;

    end;

    theorem :: GOBOARD6:60

    

     Th60: ( LSeg (((G * (1,1)) - |[1, 1]|),((G * (1,1)) - |[ 0 , 1]|))) c= (( Int ( cell (G, 0 , 0 ))) \/ {((G * (1,1)) - |[ 0 , 1]|)})

    proof

      let x be object;

      set r1 = ((G * (1,1)) `1 ), s1 = ((G * (1,1)) `2 );

      assume

       A1: x in ( LSeg (((G * (1,1)) - |[1, 1]|),((G * (1,1)) - |[ 0 , 1]|)));

      then

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

      consider r such that

       A2: p = (((1 - r) * ((G * (1,1)) - |[1, 1]|)) + (r * ((G * (1,1)) - |[ 0 , 1]|))) and 0 <= r and

       A3: r <= 1 by A1;

      now

        per cases by A3, XXREAL_0: 1;

          case r = 1;

          

          then p = (( 0. ( TOP-REAL 2)) + (1 * ((G * (1,1)) - |[ 0 , 1]|))) by A2, RLVECT_1: 10

          .= (1 * ((G * (1,1)) - |[ 0 , 1]|)) by RLVECT_1: 4

          .= ((G * (1,1)) - |[ 0 , 1]|) by RLVECT_1:def 8;

          hence p in {((G * (1,1)) - |[ 0 , 1]|)} by TARSKI:def 1;

        end;

          case r < 1;

          then (1 - r) > 0 by XREAL_1: 50;

          then r1 < (r1 + (1 - r)) by XREAL_1: 29;

          then

           A4: (r1 - (1 - r)) < r1 by XREAL_1: 19;

          

           A5: (G * (1,1)) = |[r1, s1]| by EUCLID: 53;

          s1 < (s1 + 1) by XREAL_1: 29;

          then

           A6: (s1 - 1) < s1 by XREAL_1: 19;

          

           A7: ( Int ( cell (G, 0 , 0 ))) = { |[r9, s9]| : r9 < ((G * (1,1)) `1 ) & s9 < ((G * (1,1)) `2 ) } by Th18;

          p = ((((1 - r) * (G * (1,1))) - ((1 - r) * |[1, 1]|)) + (r * ((G * (1,1)) - |[ 0 , 1]|))) by A2, RLVECT_1: 34

          .= ((((1 - r) * (G * (1,1))) - ((1 - r) * |[1, 1]|)) + ((r * (G * (1,1))) - (r * |[ 0 , 1]|))) by RLVECT_1: 34

          .= (((r * (G * (1,1))) + (((1 - r) * (G * (1,1))) - ((1 - r) * |[1, 1]|))) - (r * |[ 0 , 1]|)) by RLVECT_1:def 3

          .= ((((r * (G * (1,1))) + ((1 - r) * (G * (1,1)))) - ((1 - r) * |[1, 1]|)) - (r * |[ 0 , 1]|)) by RLVECT_1:def 3

          .= ((((r + (1 - r)) * (G * (1,1))) - ((1 - r) * |[1, 1]|)) - (r * |[ 0 , 1]|)) by RLVECT_1:def 6

          .= (((G * (1,1)) - ((1 - r) * |[1, 1]|)) - (r * |[ 0 , 1]|)) by RLVECT_1:def 8

          .= (((G * (1,1)) - |[((1 - r) * 1), ((1 - r) * 1)]|) - (r * |[ 0 , 1]|)) by EUCLID: 58

          .= (((G * (1,1)) - |[(1 - r), (1 - r)]|) - |[(r * 0 ), (r * 1)]|) by EUCLID: 58

          .= ( |[(r1 - (1 - r)), (s1 - (1 - r))]| - |[ 0 , r]|) by A5, EUCLID: 62

          .= |[((r1 - (1 - r)) - 0 ), ((s1 - (1 - r)) - r)]| by EUCLID: 62

          .= |[(r1 - (1 - r)), (s1 - 1)]|;

          hence p in ( Int ( cell (G, 0 , 0 ))) by A6, A4, A7;

        end;

      end;

      hence thesis by XBOOLE_0:def 3;

    end;

    theorem :: GOBOARD6:61

    

     Th61: ( LSeg (((G * (( len G),1)) + |[1, ( - 1)]|),((G * (( len G),1)) - |[ 0 , 1]|))) c= (( Int ( cell (G,( len G), 0 ))) \/ {((G * (( len G),1)) - |[ 0 , 1]|)})

    proof

      let x be object;

      set r1 = ((G * (( len G),1)) `1 ), s1 = ((G * (1,1)) `2 );

      assume

       A1: x in ( LSeg (((G * (( len G),1)) + |[1, ( - 1)]|),((G * (( len G),1)) - |[ 0 , 1]|)));

      then

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

      consider r such that

       A2: p = (((1 - r) * ((G * (( len G),1)) + |[1, ( - 1)]|)) + (r * ((G * (( len G),1)) - |[ 0 , 1]|))) and 0 <= r and

       A3: r <= 1 by A1;

      now

        per cases by A3, XXREAL_0: 1;

          case r = 1;

          

          then p = (( 0. ( TOP-REAL 2)) + (1 * ((G * (( len G),1)) - |[ 0 , 1]|))) by A2, RLVECT_1: 10

          .= (1 * ((G * (( len G),1)) - |[ 0 , 1]|)) by RLVECT_1: 4

          .= ((G * (( len G),1)) - |[ 0 , 1]|) by RLVECT_1:def 8;

          hence p in {((G * (( len G),1)) - |[ 0 , 1]|)} by TARSKI:def 1;

        end;

          case r < 1;

          then (1 - r) > 0 by XREAL_1: 50;

          then

           A4: r1 < (r1 + (1 - r)) by XREAL_1: 29;

          s1 < (s1 + 1) by XREAL_1: 29;

          then

           A5: (s1 - 1) < s1 by XREAL_1: 19;

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

          then

           A6: 1 <= ( len G) by NAT_1: 14;

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

          then

           A7: 1 <= ( width G) by NAT_1: 14;

          

           A8: (G * (( len G),1)) = |[r1, ((G * (( len G),1)) `2 )]| by EUCLID: 53

          .= |[r1, s1]| by A7, A6, GOBOARD5: 1;

          

           A9: ( Int ( cell (G,( len G), 0 ))) = { |[r9, s9]| : ((G * (( len G),1)) `1 ) < r9 & s9 < ((G * (1,1)) `2 ) } by Th21;

          p = ((((1 - r) * (G * (( len G),1))) + ((1 - r) * |[1, ( - 1)]|)) + (r * ((G * (( len G),1)) - |[ 0 , 1]|))) by A2, RLVECT_1:def 5

          .= ((((1 - r) * (G * (( len G),1))) + ((1 - r) * |[1, ( - 1)]|)) + ((r * (G * (( len G),1))) - (r * |[ 0 , 1]|))) by RLVECT_1: 34

          .= (((r * (G * (( len G),1))) + (((1 - r) * (G * (( len G),1))) + ((1 - r) * |[1, ( - 1)]|))) - (r * |[ 0 , 1]|)) by RLVECT_1:def 3

          .= ((((r * (G * (( len G),1))) + ((1 - r) * (G * (( len G),1)))) + ((1 - r) * |[1, ( - 1)]|)) - (r * |[ 0 , 1]|)) by RLVECT_1:def 3

          .= ((((r + (1 - r)) * (G * (( len G),1))) + ((1 - r) * |[1, ( - 1)]|)) - (r * |[ 0 , 1]|)) by RLVECT_1:def 6

          .= (((G * (( len G),1)) + ((1 - r) * |[1, ( - 1)]|)) - (r * |[ 0 , 1]|)) by RLVECT_1:def 8

          .= (((G * (( len G),1)) + |[((1 - r) * 1), ((1 - r) * ( - 1))]|) - (r * |[ 0 , 1]|)) by EUCLID: 58

          .= (((G * (( len G),1)) + |[(1 - r), (r - 1)]|) - |[(r * 0 ), (r * 1)]|) by EUCLID: 58

          .= ( |[(r1 + (1 - r)), (s1 + (r - 1))]| - |[ 0 , r]|) by A8, EUCLID: 56

          .= |[((r1 + (1 - r)) - 0 ), ((s1 + (r - 1)) - r)]| by EUCLID: 62

          .= |[(r1 + (1 - r)), (s1 - 1)]|;

          hence p in ( Int ( cell (G,( len G), 0 ))) by A5, A4, A9;

        end;

      end;

      hence thesis by XBOOLE_0:def 3;

    end;

    theorem :: GOBOARD6:62

    

     Th62: ( LSeg (((G * (1,( width G))) + |[( - 1), 1]|),((G * (1,( width G))) + |[ 0 , 1]|))) c= (( Int ( cell (G, 0 ,( width G)))) \/ {((G * (1,( width G))) + |[ 0 , 1]|)})

    proof

      let x be object;

      set r1 = ((G * (1,1)) `1 ), s1 = ((G * (1,( width G))) `2 );

      assume

       A1: x in ( LSeg (((G * (1,( width G))) + |[( - 1), 1]|),((G * (1,( width G))) + |[ 0 , 1]|)));

      then

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

      consider r such that

       A2: p = (((1 - r) * ((G * (1,( width G))) + |[( - 1), 1]|)) + (r * ((G * (1,( width G))) + |[ 0 , 1]|))) and 0 <= r and

       A3: r <= 1 by A1;

      now

        per cases by A3, XXREAL_0: 1;

          case r = 1;

          

          then p = (( 0. ( TOP-REAL 2)) + (1 * ((G * (1,( width G))) + |[ 0 , 1]|))) by A2, RLVECT_1: 10

          .= (1 * ((G * (1,( width G))) + |[ 0 , 1]|)) by RLVECT_1: 4

          .= ((G * (1,( width G))) + |[ 0 , 1]|) by RLVECT_1:def 8;

          hence p in {((G * (1,( width G))) + |[ 0 , 1]|)} by TARSKI:def 1;

        end;

          case r < 1;

          then (1 - r) > 0 by XREAL_1: 50;

          then r1 < (r1 + (1 - r)) by XREAL_1: 29;

          then

           A4: s1 < (s1 + 1) & (r1 - (1 - r)) < r1 by XREAL_1: 19, XREAL_1: 29;

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

          then

           A5: 1 <= ( width G) by NAT_1: 14;

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

          then

           A6: 1 <= ( len G) by NAT_1: 14;

          

           A7: (G * (1,( width G))) = |[((G * (1,( width G))) `1 ), s1]| by EUCLID: 53

          .= |[r1, s1]| by A5, A6, GOBOARD5: 2;

          

           A8: ( Int ( cell (G, 0 ,( width G)))) = { |[r9, s9]| : r9 < ((G * (1,1)) `1 ) & ((G * (1,( width G))) `2 ) < s9 } by Th19;

          p = ((((1 - r) * (G * (1,( width G)))) + ((1 - r) * |[( - 1), 1]|)) + (r * ((G * (1,( width G))) + |[ 0 , 1]|))) by A2, RLVECT_1:def 5

          .= ((((1 - r) * (G * (1,( width G)))) + ((1 - r) * |[( - 1), 1]|)) + ((r * (G * (1,( width G)))) + (r * |[ 0 , 1]|))) by RLVECT_1:def 5

          .= (((r * (G * (1,( width G)))) + (((1 - r) * (G * (1,( width G)))) + ((1 - r) * |[( - 1), 1]|))) + (r * |[ 0 , 1]|)) by RLVECT_1:def 3

          .= ((((r * (G * (1,( width G)))) + ((1 - r) * (G * (1,( width G))))) + ((1 - r) * |[( - 1), 1]|)) + (r * |[ 0 , 1]|)) by RLVECT_1:def 3

          .= ((((r + (1 - r)) * (G * (1,( width G)))) + ((1 - r) * |[( - 1), 1]|)) + (r * |[ 0 , 1]|)) by RLVECT_1:def 6

          .= (((G * (1,( width G))) + ((1 - r) * |[( - 1), 1]|)) + (r * |[ 0 , 1]|)) by RLVECT_1:def 8

          .= (((G * (1,( width G))) + |[((1 - r) * ( - 1)), ((1 - r) * 1)]|) + (r * |[ 0 , 1]|)) by EUCLID: 58

          .= (((G * (1,( width G))) + |[(r - 1), (1 - r)]|) + |[(r * 0 ), (r * 1)]|) by EUCLID: 58

          .= ( |[(r1 + (r - 1)), (s1 + (1 - r))]| + |[ 0 , r]|) by A7, EUCLID: 56

          .= |[((r1 + (r - 1)) + 0 ), ((s1 + (1 - r)) + r)]| by EUCLID: 56

          .= |[(r1 - (1 - r)), (s1 + 1)]|;

          hence p in ( Int ( cell (G, 0 ,( width G)))) by A4, A8;

        end;

      end;

      hence thesis by XBOOLE_0:def 3;

    end;

    theorem :: GOBOARD6:63

    

     Th63: ( LSeg (((G * (( len G),( width G))) + |[1, 1]|),((G * (( len G),( width G))) + |[ 0 , 1]|))) c= (( Int ( cell (G,( len G),( width G)))) \/ {((G * (( len G),( width G))) + |[ 0 , 1]|)})

    proof

      let x be object;

      set r1 = ((G * (( len G),1)) `1 ), s1 = ((G * (1,( width G))) `2 );

      assume

       A1: x in ( LSeg (((G * (( len G),( width G))) + |[1, 1]|),((G * (( len G),( width G))) + |[ 0 , 1]|)));

      then

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

      consider r such that

       A2: p = (((1 - r) * ((G * (( len G),( width G))) + |[1, 1]|)) + (r * ((G * (( len G),( width G))) + |[ 0 , 1]|))) and 0 <= r and

       A3: r <= 1 by A1;

      now

        per cases by A3, XXREAL_0: 1;

          case r = 1;

          

          then p = (( 0. ( TOP-REAL 2)) + (1 * ((G * (( len G),( width G))) + |[ 0 , 1]|))) by A2, RLVECT_1: 10

          .= (1 * ((G * (( len G),( width G))) + |[ 0 , 1]|)) by RLVECT_1: 4

          .= ((G * (( len G),( width G))) + |[ 0 , 1]|) by RLVECT_1:def 8;

          hence p in {((G * (( len G),( width G))) + |[ 0 , 1]|)} by TARSKI:def 1;

        end;

          case r < 1;

          then (1 - r) > 0 by XREAL_1: 50;

          then

           A4: s1 < (s1 + 1) & r1 < (r1 + (1 - r)) by XREAL_1: 29;

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

          then

           A5: 1 <= ( width G) by NAT_1: 14;

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

          then

           A6: 1 <= ( len G) by NAT_1: 14;

          

           A7: (G * (( len G),( width G))) = |[((G * (( len G),( width G))) `1 ), ((G * (( len G),( width G))) `2 )]| by EUCLID: 53

          .= |[r1, ((G * (( len G),( width G))) `2 )]| by A5, A6, GOBOARD5: 2

          .= |[r1, s1]| by A5, A6, GOBOARD5: 1;

          

           A8: ( Int ( cell (G,( len G),( width G)))) = { |[r9, s9]| : ((G * (( len G),1)) `1 ) < r9 & ((G * (1,( width G))) `2 ) < s9 } by Th22;

          p = ((((1 - r) * (G * (( len G),( width G)))) + ((1 - r) * |[1, 1]|)) + (r * ((G * (( len G),( width G))) + |[ 0 , 1]|))) by A2, RLVECT_1:def 5

          .= ((((1 - r) * (G * (( len G),( width G)))) + ((1 - r) * |[1, 1]|)) + ((r * (G * (( len G),( width G)))) + (r * |[ 0 , 1]|))) by RLVECT_1:def 5

          .= (((r * (G * (( len G),( width G)))) + (((1 - r) * (G * (( len G),( width G)))) + ((1 - r) * |[1, 1]|))) + (r * |[ 0 , 1]|)) by RLVECT_1:def 3

          .= ((((r * (G * (( len G),( width G)))) + ((1 - r) * (G * (( len G),( width G))))) + ((1 - r) * |[1, 1]|)) + (r * |[ 0 , 1]|)) by RLVECT_1:def 3

          .= ((((r + (1 - r)) * (G * (( len G),( width G)))) + ((1 - r) * |[1, 1]|)) + (r * |[ 0 , 1]|)) by RLVECT_1:def 6

          .= (((G * (( len G),( width G))) + ((1 - r) * |[1, 1]|)) + (r * |[ 0 , 1]|)) by RLVECT_1:def 8

          .= (((G * (( len G),( width G))) + |[((1 - r) * 1), ((1 - r) * 1)]|) + (r * |[ 0 , 1]|)) by EUCLID: 58

          .= (((G * (( len G),( width G))) + |[(1 - r), (1 - r)]|) + |[(r * 0 ), (r * 1)]|) by EUCLID: 58

          .= ( |[(r1 + (1 - r)), (s1 + (1 - r))]| + |[ 0 , r]|) by A7, EUCLID: 56

          .= |[((r1 + (1 - r)) + 0 ), ((s1 + (1 - r)) + r)]| by EUCLID: 56

          .= |[(r1 + (1 - r)), (s1 + 1)]|;

          hence p in ( Int ( cell (G,( len G),( width G)))) by A4, A8;

        end;

      end;

      hence thesis by XBOOLE_0:def 3;

    end;

    theorem :: GOBOARD6:64

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

    proof

      assume that

       A1: 1 <= i and

       A2: i < ( len G) and

       A3: 1 <= j and

       A4: (j + 1) < ( width G);

      set p1 = (G * (i,j)), p2 = (G * (i,(j + 1))), q2 = (G * ((i + 1),(j + 1))), q3 = (G * ((i + 1),(j + 2))), r = (((p2 `2 ) - (p1 `2 )) / ((q3 `2 ) - (p1 `2 )));

      

       A5: (j + 1) >= 1 by NAT_1: 11;

      set I1 = ( Int ( cell (G,i,j))), I2 = ( Int ( cell (G,i,(j + 1))));

      j <= (j + 1) by NAT_1: 11;

      then

       A6: j < ( width G) by A4, XXREAL_0: 2;

      then

       A7: ( LSeg (((1 / 2) * (p1 + q2)),((1 / 2) * (p2 + q2)))) c= (I1 \/ {((1 / 2) * (p2 + q2))}) by A1, A2, A3, Th41;

      j < (j + 1) by XREAL_1: 29;

      then (p1 `2 ) < (p2 `2 ) by A1, A2, A3, A4, GOBOARD5: 4;

      then

       A8: ((p2 `2 ) - (p1 `2 )) > 0 by XREAL_1: 50;

      

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

      then

       A10: (j + 2) >= 1 by NAT_1: 11;

      

       A11: (j + (1 + 1)) <= ( width G) by A4, A9, NAT_1: 13;

      

       A12: (i + 1) >= 1 & (i + 1) <= ( len G) by A2, NAT_1: 11, NAT_1: 13;

      

      then

       A13: (q2 `1 ) = ((G * ((i + 1),1)) `1 ) by A4, A5, GOBOARD5: 2

      .= (q3 `1 ) by A11, A10, A12, GOBOARD5: 2;

      

       A14: (q2 `2 ) = ((G * (1,(j + 1))) `2 ) by A4, A5, A12, GOBOARD5: 1

      .= (p2 `2 ) by A1, A2, A4, A5, GOBOARD5: 1;

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

      then (q2 `2 ) < (q3 `2 ) by A5, A11, A12, GOBOARD5: 4;

      then

       A15: ((p2 `2 ) - (p1 `2 )) < ((q3 `2 ) - (p1 `2 )) by A14, XREAL_1: 9;

      then

       A16: (r * ((q3 `2 ) - (p1 `2 ))) = ((p2 `2 ) - (p1 `2 )) by A8, XCMPLX_1: 87;

      (p1 `1 ) = ((G * (i,1)) `1 ) by A1, A2, A3, A6, GOBOARD5: 2

      .= (p2 `1 ) by A1, A2, A4, A5, GOBOARD5: 2;

      

      then

       A17: ((p2 + q2) `1 ) = (((1 - r) * ((p1 `1 ) + (q2 `1 ))) + (r * ((p2 `1 ) + (q3 `1 )))) by A13, Lm1

      .= (((1 - r) * ((p1 + q2) `1 )) + (r * ((p2 `1 ) + (q3 `1 )))) by Lm1

      .= (((1 - r) * ((p1 + q2) `1 )) + (r * ((p2 + q3) `1 ))) by Lm1

      .= (((1 - r) * ((p1 + q2) `1 )) + ((r * (p2 + q3)) `1 )) by Lm3

      .= ((((1 - r) * (p1 + q2)) `1 ) + ((r * (p2 + q3)) `1 )) by Lm3

      .= ((((1 - r) * (p1 + q2)) + (r * (p2 + q3))) `1 ) by Lm1;

      ((p2 + q2) `2 ) = ((p2 `2 ) + ((r + (1 - r)) * (q2 `2 ))) by Lm1

      .= (((1 - r) * ((p1 `2 ) + (q2 `2 ))) + (r * ((p2 `2 ) + (q3 `2 )))) by A14, A16

      .= (((1 - r) * ((p1 `2 ) + (q2 `2 ))) + (r * ((p2 + q3) `2 ))) by Lm1

      .= (((1 - r) * ((p1 + q2) `2 )) + (r * ((p2 + q3) `2 ))) by Lm1

      .= (((1 - r) * ((p1 + q2) `2 )) + ((r * (p2 + q3)) `2 )) by Lm3

      .= ((((1 - r) * (p1 + q2)) `2 ) + ((r * (p2 + q3)) `2 )) by Lm3

      .= ((((1 - r) * (p1 + q2)) + (r * (p2 + q3))) `2 ) by Lm1;

      

      then (((1 - r) * (p1 + q2)) + (r * (p2 + q3))) = |[((p2 + q2) `1 ), ((p2 + q2) `2 )]| by A17, EUCLID: 53

      .= (p2 + q2) by EUCLID: 53;

      

      then

       A18: ((1 / 2) * (p2 + q2)) = (((1 / 2) * ((1 - r) * (p1 + q2))) + ((1 / 2) * (r * (p2 + q3)))) by RLVECT_1:def 5

      .= ((((1 / 2) * (1 - r)) * (p1 + q2)) + ((1 / 2) * (r * (p2 + q3)))) by RLVECT_1:def 7

      .= (((1 - r) * ((1 / 2) * (p1 + q2))) + ((1 / 2) * (r * (p2 + q3)))) by RLVECT_1:def 7

      .= (((1 - r) * ((1 / 2) * (p1 + q2))) + (((1 / 2) * r) * (p2 + q3))) by RLVECT_1:def 7

      .= (((1 - r) * ((1 / 2) * (p1 + q2))) + (r * ((1 / 2) * (p2 + q3)))) by RLVECT_1:def 7;

      r < 1 by A15, A8, XREAL_1: 189;

      then ((1 / 2) * (p2 + q2)) in ( LSeg (((1 / 2) * (p1 + q2)),((1 / 2) * (p2 + q3)))) by A15, A8, A18;

      then

       A19: ( LSeg (((1 / 2) * (p1 + q2)),((1 / 2) * (p2 + q3)))) = (( LSeg (((1 / 2) * (p1 + q2)),((1 / 2) * (p2 + q2)))) \/ ( LSeg (((1 / 2) * (p2 + q2)),((1 / 2) * (p2 + q3))))) by TOPREAL1: 5;

      

       A20: ((I1 \/ I2) \/ {((1 / 2) * (p2 + q2))}) = (I1 \/ (I2 \/ ( {((1 / 2) * (p2 + q2))} \/ {((1 / 2) * (p2 + q2))}))) by XBOOLE_1: 4

      .= (I1 \/ ((I2 \/ {((1 / 2) * (p2 + q2))}) \/ {((1 / 2) * (p2 + q2))})) by XBOOLE_1: 4

      .= ((I1 \/ {((1 / 2) * (p2 + q2))}) \/ (I2 \/ {((1 / 2) * (p2 + q2))})) by XBOOLE_1: 4;

      ( LSeg (((1 / 2) * (p2 + q2)),((1 / 2) * (p2 + q3)))) c= (I2 \/ {((1 / 2) * (p2 + q2))}) by A1, A2, A4, A5, A9, Th43;

      hence thesis by A19, A7, A20, XBOOLE_1: 13;

    end;

    theorem :: GOBOARD6:65

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

    proof

      assume that

       A1: 1 <= j and

       A2: j < ( width G) and

       A3: 1 <= i and

       A4: (i + 1) < ( len G);

      set p1 = (G * (i,j)), p2 = (G * ((i + 1),j)), q2 = (G * ((i + 1),(j + 1))), q3 = (G * ((i + 2),(j + 1))), r = (((p2 `1 ) - (p1 `1 )) / ((q3 `1 ) - (p1 `1 )));

      

       A5: (i + 1) >= 1 by NAT_1: 11;

      set I1 = ( Int ( cell (G,i,j))), I2 = ( Int ( cell (G,(i + 1),j)));

      i <= (i + 1) by NAT_1: 11;

      then

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

      then

       A7: ( LSeg (((1 / 2) * (p1 + q2)),((1 / 2) * (p2 + q2)))) c= (I1 \/ {((1 / 2) * (p2 + q2))}) by A1, A2, A3, Th42;

      i < (i + 1) by XREAL_1: 29;

      then (p1 `1 ) < (p2 `1 ) by A1, A2, A3, A4, GOBOARD5: 3;

      then

       A8: ((p2 `1 ) - (p1 `1 )) > 0 by XREAL_1: 50;

      

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

      then

       A10: (i + 2) >= 1 by NAT_1: 11;

      

       A11: (i + (1 + 1)) <= ( len G) by A4, A9, NAT_1: 13;

      

       A12: (j + 1) >= 1 & (j + 1) <= ( width G) by A2, NAT_1: 11, NAT_1: 13;

      

      then

       A13: (q2 `2 ) = ((G * (1,(j + 1))) `2 ) by A4, A5, GOBOARD5: 1

      .= (q3 `2 ) by A11, A10, A12, GOBOARD5: 1;

      

       A14: (q2 `1 ) = ((G * ((i + 1),1)) `1 ) by A4, A5, A12, GOBOARD5: 2

      .= (p2 `1 ) by A1, A2, A4, A5, GOBOARD5: 2;

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

      then (q2 `1 ) < (q3 `1 ) by A5, A11, A12, GOBOARD5: 3;

      then

       A15: ((p2 `1 ) - (p1 `1 )) < ((q3 `1 ) - (p1 `1 )) by A14, XREAL_1: 9;

      then

       A16: (r * ((q3 `1 ) - (p1 `1 ))) = ((p2 `1 ) - (p1 `1 )) by A8, XCMPLX_1: 87;

      (p1 `2 ) = ((G * (1,j)) `2 ) by A1, A2, A3, A6, GOBOARD5: 1

      .= (p2 `2 ) by A1, A2, A4, A5, GOBOARD5: 1;

      

      then

       A17: ((p2 + q2) `2 ) = (((1 - r) * ((p1 `2 ) + (q2 `2 ))) + (r * ((p2 `2 ) + (q3 `2 )))) by A13, Lm1

      .= (((1 - r) * ((p1 + q2) `2 )) + (r * ((p2 `2 ) + (q3 `2 )))) by Lm1

      .= (((1 - r) * ((p1 + q2) `2 )) + (r * ((p2 + q3) `2 ))) by Lm1

      .= (((1 - r) * ((p1 + q2) `2 )) + ((r * (p2 + q3)) `2 )) by Lm3

      .= ((((1 - r) * (p1 + q2)) `2 ) + ((r * (p2 + q3)) `2 )) by Lm3

      .= ((((1 - r) * (p1 + q2)) + (r * (p2 + q3))) `2 ) by Lm1;

      ((p2 + q2) `1 ) = ((p2 `1 ) + ((r + (1 - r)) * (q2 `1 ))) by Lm1

      .= (((1 - r) * ((p1 `1 ) + (q2 `1 ))) + (r * ((p2 `1 ) + (q3 `1 )))) by A14, A16

      .= (((1 - r) * ((p1 `1 ) + (q2 `1 ))) + (r * ((p2 + q3) `1 ))) by Lm1

      .= (((1 - r) * ((p1 + q2) `1 )) + (r * ((p2 + q3) `1 ))) by Lm1

      .= (((1 - r) * ((p1 + q2) `1 )) + ((r * (p2 + q3)) `1 )) by Lm3

      .= ((((1 - r) * (p1 + q2)) `1 ) + ((r * (p2 + q3)) `1 )) by Lm3

      .= ((((1 - r) * (p1 + q2)) + (r * (p2 + q3))) `1 ) by Lm1;

      

      then (((1 - r) * (p1 + q2)) + (r * (p2 + q3))) = |[((p2 + q2) `1 ), ((p2 + q2) `2 )]| by A17, EUCLID: 53

      .= (p2 + q2) by EUCLID: 53;

      

      then

       A18: ((1 / 2) * (p2 + q2)) = (((1 / 2) * ((1 - r) * (p1 + q2))) + ((1 / 2) * (r * (p2 + q3)))) by RLVECT_1:def 5

      .= ((((1 / 2) * (1 - r)) * (p1 + q2)) + ((1 / 2) * (r * (p2 + q3)))) by RLVECT_1:def 7

      .= (((1 - r) * ((1 / 2) * (p1 + q2))) + ((1 / 2) * (r * (p2 + q3)))) by RLVECT_1:def 7

      .= (((1 - r) * ((1 / 2) * (p1 + q2))) + (((1 / 2) * r) * (p2 + q3))) by RLVECT_1:def 7

      .= (((1 - r) * ((1 / 2) * (p1 + q2))) + (r * ((1 / 2) * (p2 + q3)))) by RLVECT_1:def 7;

      r < 1 by A15, A8, XREAL_1: 189;

      then ((1 / 2) * (p2 + q2)) in ( LSeg (((1 / 2) * (p1 + q2)),((1 / 2) * (p2 + q3)))) by A15, A8, A18;

      then

       A19: ( LSeg (((1 / 2) * (p1 + q2)),((1 / 2) * (p2 + q3)))) = (( LSeg (((1 / 2) * (p1 + q2)),((1 / 2) * (p2 + q2)))) \/ ( LSeg (((1 / 2) * (p2 + q2)),((1 / 2) * (p2 + q3))))) by TOPREAL1: 5;

      

       A20: ((I1 \/ I2) \/ {((1 / 2) * (p2 + q2))}) = (I1 \/ (I2 \/ ( {((1 / 2) * (p2 + q2))} \/ {((1 / 2) * (p2 + q2))}))) by XBOOLE_1: 4

      .= (I1 \/ ((I2 \/ {((1 / 2) * (p2 + q2))}) \/ {((1 / 2) * (p2 + q2))})) by XBOOLE_1: 4

      .= ((I1 \/ {((1 / 2) * (p2 + q2))}) \/ (I2 \/ {((1 / 2) * (p2 + q2))})) by XBOOLE_1: 4;

      ( LSeg (((1 / 2) * (p2 + q2)),((1 / 2) * (p2 + q3)))) c= (I2 \/ {((1 / 2) * (p2 + q2))}) by A1, A2, A4, A5, A9, Th40;

      hence thesis by A19, A7, A20, XBOOLE_1: 13;

    end;

    theorem :: GOBOARD6:66

    1 <= i & i < ( len G) & 1 < ( width G) implies ( LSeg ((((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1)))) - |[ 0 , 1]|),((1 / 2) * ((G * (i,1)) + (G * ((i + 1),2)))))) c= ((( Int ( cell (G,i, 0 ))) \/ ( Int ( cell (G,i,1)))) \/ {((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1))))})

    proof

      assume that

       A1: 1 <= i and

       A2: i < ( len G) and

       A3: 1 < ( width G);

      set p1 = (G * (i,1)), q2 = (G * ((i + 1),1)), q3 = (G * ((i + 1),2)), r = (1 / (((1 / 2) * ((q3 `2 ) - (p1 `2 ))) + 1));

      

       A4: (i + 1) >= 1 & (i + 1) <= ( len G) by A2, NAT_1: 11, NAT_1: 13;

      

       A5: ( 0 + (1 + 1)) <= ( width G) by A3, NAT_1: 13;

      then

       A6: (q2 `1 ) = (q3 `1 ) by A4, GOBOARD5: 2;

      

       A7: (q2 `2 ) = ((G * (1,( 0 + 1))) `2 ) by A3, A4, GOBOARD5: 1

      .= (p1 `2 ) by A1, A2, A3, GOBOARD5: 1;

      then (p1 `2 ) < (q3 `2 ) by A5, A4, GOBOARD5: 4;

      then

       A8: ((q3 `2 ) - (p1 `2 )) > 0 by XREAL_1: 50;

      then 1 < (((1 / 2) * ((q3 `2 ) - (p1 `2 ))) + 1) by XREAL_1: 29, XREAL_1: 129;

      then

       A9: r < 1 by XREAL_1: 212;

      set I1 = ( Int ( cell (G,i, 0 ))), I2 = ( Int ( cell (G,i,1)));

      

       A10: ( LSeg ((((1 / 2) * (p1 + q2)) - |[ 0 , 1]|),((1 / 2) * (p1 + q2)))) c= (I1 \/ {((1 / 2) * (p1 + q2))}) by A1, A2, Th46;

      

       A11: ((I1 \/ I2) \/ {((1 / 2) * (p1 + q2))}) = (I1 \/ (I2 \/ ( {((1 / 2) * (p1 + q2))} \/ {((1 / 2) * (p1 + q2))}))) by XBOOLE_1: 4

      .= (I1 \/ ((I2 \/ {((1 / 2) * (p1 + q2))}) \/ {((1 / 2) * (p1 + q2))})) by XBOOLE_1: 4

      .= ((I1 \/ {((1 / 2) * (p1 + q2))}) \/ (I2 \/ {((1 / 2) * (p1 + q2))})) by XBOOLE_1: 4;

      

       A12: (((((1 - r) * ((1 / 2) * q2)) + (r * ((1 / 2) * q3))) - ((1 - r) * |[ 0 , 1]|)) `1 ) = (((((1 - r) * ((1 / 2) * q2)) + (r * ((1 / 2) * q3))) `1 ) - (((1 - r) * |[ 0 , 1]|) `1 )) by Lm2

      .= (((((1 - r) * ((1 / 2) * q2)) + (r * ((1 / 2) * q3))) `1 ) - ( |[((1 - r) * 0 ), ((1 - r) * 1)]| `1 )) by EUCLID: 58

      .= (((((1 - r) * ((1 / 2) * q2)) + (r * ((1 / 2) * q3))) `1 ) - 0 ) by EUCLID: 52

      .= ((((1 - r) * ((1 / 2) * q2)) `1 ) + ((r * ((1 / 2) * q3)) `1 )) by Lm1

      .= (((1 - r) * (((1 / 2) * q2) `1 )) + ((r * ((1 / 2) * q3)) `1 )) by Lm3

      .= (((1 - r) * (((1 / 2) * q2) `1 )) + (r * (((1 / 2) * q3) `1 ))) by Lm3

      .= (((1 - r) * ((1 / 2) * (q2 `1 ))) + (r * (((1 / 2) * q3) `1 ))) by Lm3

      .= (((1 - r) * ((1 / 2) * (q2 `1 ))) + (r * ((1 / 2) * (q2 `1 )))) by A6, Lm3

      .= (((1 / 2) * q2) `1 ) by Lm3;

      

       A13: (((((1 - r) * ((1 / 2) * p1)) + (r * ((1 / 2) * p1))) + ((1 - r) * ((1 / 2) * q2))) + (r * ((1 / 2) * q3))) = (((((1 - r) * ((1 / 2) * p1)) + ((1 - r) * ((1 / 2) * q2))) + (r * ((1 / 2) * p1))) + (r * ((1 / 2) * q3))) by RLVECT_1:def 3

      .= ((((1 - r) * ((1 / 2) * p1)) + ((1 - r) * ((1 / 2) * q2))) + ((r * ((1 / 2) * p1)) + (r * ((1 / 2) * q3)))) by RLVECT_1:def 3

      .= ((((1 - r) * ((1 / 2) * p1)) + ((1 - r) * ((1 / 2) * q2))) + (r * (((1 / 2) * p1) + ((1 / 2) * q3)))) by RLVECT_1:def 5

      .= (((1 - r) * (((1 / 2) * p1) + ((1 / 2) * q2))) + (r * (((1 / 2) * p1) + ((1 / 2) * q3)))) by RLVECT_1:def 5

      .= (((1 - r) * (((1 / 2) * p1) + ((1 / 2) * q2))) + (r * ((1 / 2) * (p1 + q3)))) by RLVECT_1:def 5

      .= (((1 - r) * ((1 / 2) * (p1 + q2))) + (r * ((1 / 2) * (p1 + q3)))) by RLVECT_1:def 5;

      

       A14: (((r * ((1 / 2) * (q3 `2 ))) - (r * ((1 / 2) * (q2 `2 )))) + r) = (r * (((1 / 2) * ((q3 `2 ) - (q2 `2 ))) + 1))

      .= 1 by A7, A8, XCMPLX_1: 106;

      (((((1 - r) * ((1 / 2) * q2)) + (r * ((1 / 2) * q3))) - ((1 - r) * |[ 0 , 1]|)) `2 ) = (((((1 - r) * ((1 / 2) * q2)) + (r * ((1 / 2) * q3))) `2 ) - (((1 - r) * |[ 0 , 1]|) `2 )) by Lm2

      .= (((((1 - r) * ((1 / 2) * q2)) + (r * ((1 / 2) * q3))) `2 ) - ( |[((1 - r) * 0 ), ((1 - r) * 1)]| `2 )) by EUCLID: 58

      .= (((((1 - r) * ((1 / 2) * q2)) + (r * ((1 / 2) * q3))) `2 ) - (1 - r)) by EUCLID: 52

      .= (((((1 - r) * ((1 / 2) * q2)) `2 ) + ((r * ((1 / 2) * q3)) `2 )) - (1 - r)) by Lm1

      .= ((((1 - r) * (((1 / 2) * q2) `2 )) + ((r * ((1 / 2) * q3)) `2 )) - (1 - r)) by Lm3

      .= ((((1 - r) * (((1 / 2) * q2) `2 )) + (r * (((1 / 2) * q3) `2 ))) - (1 - r)) by Lm3

      .= ((((1 - r) * ((1 / 2) * (q2 `2 ))) + (r * (((1 / 2) * q3) `2 ))) - (1 - r)) by Lm3

      .= ((((1 - r) * ((1 / 2) * (q2 `2 ))) + (r * ((1 / 2) * (q3 `2 )))) - (1 - r)) by Lm3

      .= (((1 / 2) * q2) `2 ) by A14, Lm3;

      

      then

       A15: ((((1 - r) * ((1 / 2) * q2)) + (r * ((1 / 2) * q3))) - ((1 - r) * |[ 0 , 1]|)) = |[(((1 / 2) * q2) `1 ), (((1 / 2) * q2) `2 )]| by A12, EUCLID: 53

      .= ((1 / 2) * q2) by EUCLID: 53;

      ((1 / 2) * (p1 + q2)) = (((1 / 2) * p1) + ((1 / 2) * q2)) by RLVECT_1:def 5

      .= ((((1 - r) + r) * ((1 / 2) * p1)) + ((1 / 2) * q2)) by RLVECT_1:def 8

      .= ((((1 - r) * ((1 / 2) * p1)) + (r * ((1 / 2) * p1))) + ((1 / 2) * q2)) by RLVECT_1:def 6

      .= (((((1 - r) * ((1 / 2) * p1)) + (r * ((1 / 2) * p1))) + (((1 - r) * ((1 / 2) * q2)) + (r * ((1 / 2) * q3)))) - ((1 - r) * |[ 0 , 1]|)) by A15, RLVECT_1:def 3

      .= ((((((1 - r) * ((1 / 2) * p1)) + (r * ((1 / 2) * p1))) + ((1 - r) * ((1 / 2) * q2))) + (r * ((1 / 2) * q3))) - ((1 - r) * |[ 0 , 1]|)) by RLVECT_1:def 3

      .= (((1 - r) * ((1 / 2) * (p1 + q2))) + ((r * ((1 / 2) * (p1 + q3))) - ((1 - r) * |[ 0 , 1]|))) by A13, RLVECT_1:def 3

      .= (((1 - r) * ((1 / 2) * (p1 + q2))) + ( - (((1 - r) * |[ 0 , 1]|) - (r * ((1 / 2) * (p1 + q3)))))) by RLVECT_1: 33

      .= (((1 - r) * ((1 / 2) * (p1 + q2))) - (((1 - r) * |[ 0 , 1]|) - (r * ((1 / 2) * (p1 + q3)))))

      .= ((((1 - r) * ((1 / 2) * (p1 + q2))) - ((1 - r) * |[ 0 , 1]|)) + (r * ((1 / 2) * (p1 + q3)))) by RLVECT_1: 29

      .= (((1 - r) * (((1 / 2) * (p1 + q2)) - |[ 0 , 1]|)) + (r * ((1 / 2) * (p1 + q3)))) by RLVECT_1: 34;

      then ((1 / 2) * (p1 + q2)) in ( LSeg ((((1 / 2) * (p1 + q2)) - |[ 0 , 1]|),((1 / 2) * (p1 + q3)))) by A8, A9;

      then

       A16: ( LSeg ((((1 / 2) * (p1 + q2)) - |[ 0 , 1]|),((1 / 2) * (p1 + q3)))) = (( LSeg ((((1 / 2) * (p1 + q2)) - |[ 0 , 1]|),((1 / 2) * (p1 + q2)))) \/ ( LSeg (((1 / 2) * (p1 + q2)),((1 / 2) * (p1 + q3))))) by TOPREAL1: 5;

      (( 0 + 1) + 1) = ( 0 + (1 + 1));

      then ( LSeg (((1 / 2) * (p1 + q2)),((1 / 2) * (p1 + q3)))) c= (I2 \/ {((1 / 2) * (p1 + q2))}) by A1, A2, A3, Th43;

      hence thesis by A16, A10, A11, XBOOLE_1: 13;

    end;

    theorem :: GOBOARD6:67

    1 <= i & i < ( len G) & 1 < ( width G) implies ( LSeg ((((1 / 2) * ((G * (i,( width G))) + (G * ((i + 1),( width G))))) + |[ 0 , 1]|),((1 / 2) * ((G * (i,( width G))) + (G * ((i + 1),(( width G) -' 1))))))) c= ((( Int ( cell (G,i,(( width G) -' 1)))) \/ ( Int ( cell (G,i,( width G))))) \/ {((1 / 2) * ((G * (i,( width G))) + (G * ((i + 1),( width G)))))})

    proof

      assume that

       A1: 1 <= i and

       A2: i < ( len G) and

       A3: 1 < ( width G);

      set I1 = ( Int ( cell (G,i,(( width G) -' 1)))), I2 = ( Int ( cell (G,i,( width G))));

      set p1 = (G * (i,( width G))), q2 = (G * ((i + 1),( width G))), q3 = (G * ((i + 1),(( width G) -' 1))), r = (1 / (((1 / 2) * ((p1 `2 ) - (q3 `2 ))) + 1));

      

       A4: ((( width G) -' 1) + 1) = ( width G) by A3, XREAL_1: 235;

      then

       A5: 1 <= (( width G) -' 1) by A3, NAT_1: 13;

      

       A6: (( width G) -' 1) < ( width G) by A4, NAT_1: 13;

      then ((G * (i,( width G))) + (G * ((i + 1),(( width G) -' 1)))) = ((G * (i,(( width G) -' 1))) + (G * ((i + 1),( width G)))) by A1, A2, A4, A5, Th11;

      then

       A7: ( LSeg (((1 / 2) * (p1 + q2)),((1 / 2) * (p1 + q3)))) c= (I1 \/ {((1 / 2) * (p1 + q2))}) by A1, A2, A4, A5, A6, Th41;

      

       A8: (i + 1) >= 1 & (i + 1) <= ( len G) by A2, NAT_1: 11, NAT_1: 13;

      

      then

       A9: (q2 `1 ) = ((G * ((i + 1),1)) `1 ) by A3, GOBOARD5: 2

      .= (q3 `1 ) by A5, A6, A8, GOBOARD5: 2;

      

       A10: (q2 `2 ) = ((G * (1,( width G))) `2 ) by A3, A8, GOBOARD5: 1

      .= (p1 `2 ) by A1, A2, A3, GOBOARD5: 1;

      then (q3 `2 ) < (p1 `2 ) by A5, A6, A8, GOBOARD5: 4;

      then

       A11: ((p1 `2 ) - (q3 `2 )) > 0 by XREAL_1: 50;

      then 1 < (((1 / 2) * ((p1 `2 ) - (q3 `2 ))) + 1) by XREAL_1: 29, XREAL_1: 129;

      then

       A12: r < 1 by XREAL_1: 212;

      

       A13: (((((1 - r) * ((1 / 2) * q2)) + (r * ((1 / 2) * q3))) + ((1 - r) * |[ 0 , 1]|)) `1 ) = (((((1 - r) * ((1 / 2) * q2)) + (r * ((1 / 2) * q3))) `1 ) + (((1 - r) * |[ 0 , 1]|) `1 )) by Lm1

      .= (((((1 - r) * ((1 / 2) * q2)) + (r * ((1 / 2) * q3))) `1 ) + ( |[((1 - r) * 0 ), ((1 - r) * 1)]| `1 )) by EUCLID: 58

      .= (((((1 - r) * ((1 / 2) * q2)) + (r * ((1 / 2) * q3))) `1 ) + 0 ) by EUCLID: 52

      .= ((((1 - r) * ((1 / 2) * q2)) `1 ) + ((r * ((1 / 2) * q3)) `1 )) by Lm1

      .= (((1 - r) * (((1 / 2) * q2) `1 )) + ((r * ((1 / 2) * q3)) `1 )) by Lm3

      .= (((1 - r) * (((1 / 2) * q2) `1 )) + (r * (((1 / 2) * q3) `1 ))) by Lm3

      .= (((1 - r) * ((1 / 2) * (q2 `1 ))) + (r * (((1 / 2) * q3) `1 ))) by Lm3

      .= (((1 - r) * ((1 / 2) * (q2 `1 ))) + (r * ((1 / 2) * (q2 `1 )))) by A9, Lm3

      .= (((1 / 2) * q2) `1 ) by Lm3;

      

       A14: ((I1 \/ I2) \/ {((1 / 2) * (p1 + q2))}) = (I1 \/ (I2 \/ ( {((1 / 2) * (p1 + q2))} \/ {((1 / 2) * (p1 + q2))}))) by XBOOLE_1: 4

      .= (I1 \/ ((I2 \/ {((1 / 2) * (p1 + q2))}) \/ {((1 / 2) * (p1 + q2))})) by XBOOLE_1: 4

      .= ((I1 \/ {((1 / 2) * (p1 + q2))}) \/ (I2 \/ {((1 / 2) * (p1 + q2))})) by XBOOLE_1: 4;

      

       A15: (((((1 - r) * ((1 / 2) * p1)) + (r * ((1 / 2) * p1))) + ((1 - r) * ((1 / 2) * q2))) + (r * ((1 / 2) * q3))) = (((((1 - r) * ((1 / 2) * p1)) + ((1 - r) * ((1 / 2) * q2))) + (r * ((1 / 2) * p1))) + (r * ((1 / 2) * q3))) by RLVECT_1:def 3

      .= ((((1 - r) * ((1 / 2) * p1)) + ((1 - r) * ((1 / 2) * q2))) + ((r * ((1 / 2) * p1)) + (r * ((1 / 2) * q3)))) by RLVECT_1:def 3

      .= ((((1 - r) * ((1 / 2) * p1)) + ((1 - r) * ((1 / 2) * q2))) + (r * (((1 / 2) * p1) + ((1 / 2) * q3)))) by RLVECT_1:def 5

      .= (((1 - r) * (((1 / 2) * p1) + ((1 / 2) * q2))) + (r * (((1 / 2) * p1) + ((1 / 2) * q3)))) by RLVECT_1:def 5

      .= (((1 - r) * (((1 / 2) * p1) + ((1 / 2) * q2))) + (r * ((1 / 2) * (p1 + q3)))) by RLVECT_1:def 5

      .= (((1 - r) * ((1 / 2) * (p1 + q2))) + (r * ((1 / 2) * (p1 + q3)))) by RLVECT_1:def 5;

      

       A16: (((r * ((1 / 2) * (q2 `2 ))) - (r * ((1 / 2) * (q3 `2 )))) + r) = (r * (((1 / 2) * ((q2 `2 ) - (q3 `2 ))) + 1))

      .= 1 by A10, A11, XCMPLX_1: 106;

      (((((1 - r) * ((1 / 2) * q2)) + (r * ((1 / 2) * q3))) + ((1 - r) * |[ 0 , 1]|)) `2 ) = (((((1 - r) * ((1 / 2) * q2)) + (r * ((1 / 2) * q3))) `2 ) + (((1 - r) * |[ 0 , 1]|) `2 )) by Lm1

      .= (((((1 - r) * ((1 / 2) * q2)) + (r * ((1 / 2) * q3))) `2 ) + ( |[((1 - r) * 0 ), ((1 - r) * 1)]| `2 )) by EUCLID: 58

      .= (((((1 - r) * ((1 / 2) * q2)) + (r * ((1 / 2) * q3))) `2 ) + (1 - r)) by EUCLID: 52

      .= (((((1 - r) * ((1 / 2) * q2)) `2 ) + ((r * ((1 / 2) * q3)) `2 )) + (1 - r)) by Lm1

      .= ((((1 - r) * (((1 / 2) * q2) `2 )) + ((r * ((1 / 2) * q3)) `2 )) + (1 - r)) by Lm3

      .= ((((1 - r) * (((1 / 2) * q2) `2 )) + (r * (((1 / 2) * q3) `2 ))) + (1 - r)) by Lm3

      .= ((((1 - r) * ((1 / 2) * (q2 `2 ))) + (r * (((1 / 2) * q3) `2 ))) + (1 - r)) by Lm3

      .= ((((1 - r) * ((1 / 2) * (q2 `2 ))) + (r * ((1 / 2) * (q3 `2 )))) + (1 - r)) by Lm3

      .= (((1 / 2) * q2) `2 ) by A16, Lm3;

      

      then

       A17: ((((1 - r) * ((1 / 2) * q2)) + (r * ((1 / 2) * q3))) + ((1 - r) * |[ 0 , 1]|)) = |[(((1 / 2) * q2) `1 ), (((1 / 2) * q2) `2 )]| by A13, EUCLID: 53

      .= ((1 / 2) * q2) by EUCLID: 53;

      ((1 / 2) * (p1 + q2)) = (((1 / 2) * p1) + ((1 / 2) * q2)) by RLVECT_1:def 5

      .= ((((1 - r) + r) * ((1 / 2) * p1)) + ((1 / 2) * q2)) by RLVECT_1:def 8

      .= ((((1 - r) * ((1 / 2) * p1)) + (r * ((1 / 2) * p1))) + ((1 / 2) * q2)) by RLVECT_1:def 6

      .= (((((1 - r) * ((1 / 2) * p1)) + (r * ((1 / 2) * p1))) + (((1 - r) * ((1 / 2) * q2)) + (r * ((1 / 2) * q3)))) + ((1 - r) * |[ 0 , 1]|)) by A17, RLVECT_1:def 3

      .= ((((((1 - r) * ((1 / 2) * p1)) + (r * ((1 / 2) * p1))) + ((1 - r) * ((1 / 2) * q2))) + (r * ((1 / 2) * q3))) + ((1 - r) * |[ 0 , 1]|)) by RLVECT_1:def 3

      .= ((((1 - r) * ((1 / 2) * (p1 + q2))) + ((1 - r) * |[ 0 , 1]|)) + (r * ((1 / 2) * (p1 + q3)))) by A15, RLVECT_1:def 3

      .= (((1 - r) * (((1 / 2) * (p1 + q2)) + |[ 0 , 1]|)) + (r * ((1 / 2) * (p1 + q3)))) by RLVECT_1:def 5;

      then ((1 / 2) * (p1 + q2)) in ( LSeg ((((1 / 2) * (p1 + q2)) + |[ 0 , 1]|),((1 / 2) * (p1 + q3)))) by A11, A12;

      then

       A18: ( LSeg ((((1 / 2) * (p1 + q2)) + |[ 0 , 1]|),((1 / 2) * (p1 + q3)))) = (( LSeg ((((1 / 2) * (p1 + q2)) + |[ 0 , 1]|),((1 / 2) * (p1 + q2)))) \/ ( LSeg (((1 / 2) * (p1 + q2)),((1 / 2) * (p1 + q3))))) by TOPREAL1: 5;

      ( LSeg ((((1 / 2) * (p1 + q2)) + |[ 0 , 1]|),((1 / 2) * (p1 + q2)))) c= (I2 \/ {((1 / 2) * (p1 + q2))}) by A1, A2, Th47;

      hence thesis by A18, A7, A14, XBOOLE_1: 13;

    end;

    theorem :: GOBOARD6:68

    1 <= j & j < ( width G) & 1 < ( len G) implies ( LSeg ((((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1))))) - |[1, 0 ]|),((1 / 2) * ((G * (1,j)) + (G * (2,(j + 1))))))) c= ((( Int ( cell (G, 0 ,j))) \/ ( Int ( cell (G,1,j)))) \/ {((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1)))))})

    proof

      assume that

       A1: 1 <= j and

       A2: j < ( width G) and

       A3: 1 < ( len G);

      set p1 = (G * (1,j)), q2 = (G * (1,(j + 1))), q3 = (G * (2,(j + 1))), r = (1 / (((1 / 2) * ((q3 `1 ) - (p1 `1 ))) + 1));

      

       A4: (j + 1) >= 1 & (j + 1) <= ( width G) by A2, NAT_1: 11, NAT_1: 13;

      

       A5: ( 0 + (1 + 1)) <= ( len G) by A3, NAT_1: 13;

      then

       A6: (q2 `2 ) = (q3 `2 ) by A4, GOBOARD5: 1;

      

       A7: (q2 `1 ) = ((G * (1,1)) `1 ) by A3, A4, GOBOARD5: 2

      .= (p1 `1 ) by A1, A2, A3, GOBOARD5: 2;

      then (p1 `1 ) < (q3 `1 ) by A5, A4, GOBOARD5: 3;

      then

       A8: ((q3 `1 ) - (p1 `1 )) > 0 by XREAL_1: 50;

      then 1 < (((1 / 2) * ((q3 `1 ) - (p1 `1 ))) + 1) by XREAL_1: 29, XREAL_1: 129;

      then

       A9: r < 1 by XREAL_1: 212;

      set I1 = ( Int ( cell (G, 0 ,j))), I2 = ( Int ( cell (G,1,j)));

      

       A10: ( LSeg ((((1 / 2) * (p1 + q2)) - |[1, 0 ]|),((1 / 2) * (p1 + q2)))) c= (I1 \/ {((1 / 2) * (p1 + q2))}) by A1, A2, Th44;

      

       A11: ((I1 \/ I2) \/ {((1 / 2) * (p1 + q2))}) = (I1 \/ (I2 \/ ( {((1 / 2) * (p1 + q2))} \/ {((1 / 2) * (p1 + q2))}))) by XBOOLE_1: 4

      .= (I1 \/ ((I2 \/ {((1 / 2) * (p1 + q2))}) \/ {((1 / 2) * (p1 + q2))})) by XBOOLE_1: 4

      .= ((I1 \/ {((1 / 2) * (p1 + q2))}) \/ (I2 \/ {((1 / 2) * (p1 + q2))})) by XBOOLE_1: 4;

      

       A12: (((((1 - r) * ((1 / 2) * q2)) + (r * ((1 / 2) * q3))) - ((1 - r) * |[1, 0 ]|)) `2 ) = (((((1 - r) * ((1 / 2) * q2)) + (r * ((1 / 2) * q3))) `2 ) - (((1 - r) * |[1, 0 ]|) `2 )) by Lm2

      .= (((((1 - r) * ((1 / 2) * q2)) + (r * ((1 / 2) * q3))) `2 ) - ( |[((1 - r) * 1), ((1 - r) * 0 )]| `2 )) by EUCLID: 58

      .= (((((1 - r) * ((1 / 2) * q2)) + (r * ((1 / 2) * q3))) `2 ) - 0 ) by EUCLID: 52

      .= ((((1 - r) * ((1 / 2) * q2)) `2 ) + ((r * ((1 / 2) * q3)) `2 )) by Lm1

      .= (((1 - r) * (((1 / 2) * q2) `2 )) + ((r * ((1 / 2) * q3)) `2 )) by Lm3

      .= (((1 - r) * (((1 / 2) * q2) `2 )) + (r * (((1 / 2) * q3) `2 ))) by Lm3

      .= (((1 - r) * ((1 / 2) * (q2 `2 ))) + (r * (((1 / 2) * q3) `2 ))) by Lm3

      .= (((1 - r) * ((1 / 2) * (q2 `2 ))) + (r * ((1 / 2) * (q2 `2 )))) by A6, Lm3

      .= (((1 / 2) * q2) `2 ) by Lm3;

      

       A13: (((((1 - r) * ((1 / 2) * p1)) + (r * ((1 / 2) * p1))) + ((1 - r) * ((1 / 2) * q2))) + (r * ((1 / 2) * q3))) = (((((1 - r) * ((1 / 2) * p1)) + ((1 - r) * ((1 / 2) * q2))) + (r * ((1 / 2) * p1))) + (r * ((1 / 2) * q3))) by RLVECT_1:def 3

      .= ((((1 - r) * ((1 / 2) * p1)) + ((1 - r) * ((1 / 2) * q2))) + ((r * ((1 / 2) * p1)) + (r * ((1 / 2) * q3)))) by RLVECT_1:def 3

      .= ((((1 - r) * ((1 / 2) * p1)) + ((1 - r) * ((1 / 2) * q2))) + (r * (((1 / 2) * p1) + ((1 / 2) * q3)))) by RLVECT_1:def 5

      .= (((1 - r) * (((1 / 2) * p1) + ((1 / 2) * q2))) + (r * (((1 / 2) * p1) + ((1 / 2) * q3)))) by RLVECT_1:def 5

      .= (((1 - r) * (((1 / 2) * p1) + ((1 / 2) * q2))) + (r * ((1 / 2) * (p1 + q3)))) by RLVECT_1:def 5

      .= (((1 - r) * ((1 / 2) * (p1 + q2))) + (r * ((1 / 2) * (p1 + q3)))) by RLVECT_1:def 5;

      

       A14: (((r * ((1 / 2) * (q3 `1 ))) - (r * ((1 / 2) * (q2 `1 )))) + r) = (r * (((1 / 2) * ((q3 `1 ) - (q2 `1 ))) + 1))

      .= 1 by A7, A8, XCMPLX_1: 106;

      (((((1 - r) * ((1 / 2) * q2)) + (r * ((1 / 2) * q3))) - ((1 - r) * |[1, 0 ]|)) `1 ) = (((((1 - r) * ((1 / 2) * q2)) + (r * ((1 / 2) * q3))) `1 ) - (((1 - r) * |[1, 0 ]|) `1 )) by Lm2

      .= (((((1 - r) * ((1 / 2) * q2)) + (r * ((1 / 2) * q3))) `1 ) - ( |[((1 - r) * 1), ((1 - r) * 0 )]| `1 )) by EUCLID: 58

      .= (((((1 - r) * ((1 / 2) * q2)) + (r * ((1 / 2) * q3))) `1 ) - (1 - r)) by EUCLID: 52

      .= (((((1 - r) * ((1 / 2) * q2)) `1 ) + ((r * ((1 / 2) * q3)) `1 )) - (1 - r)) by Lm1

      .= ((((1 - r) * (((1 / 2) * q2) `1 )) + ((r * ((1 / 2) * q3)) `1 )) - (1 - r)) by Lm3

      .= ((((1 - r) * (((1 / 2) * q2) `1 )) + (r * (((1 / 2) * q3) `1 ))) - (1 - r)) by Lm3

      .= ((((1 - r) * ((1 / 2) * (q2 `1 ))) + (r * (((1 / 2) * q3) `1 ))) - (1 - r)) by Lm3

      .= ((((1 - r) * ((1 / 2) * (q2 `1 ))) + (r * ((1 / 2) * (q3 `1 )))) - (1 - r)) by Lm3

      .= (((1 / 2) * q2) `1 ) by A14, Lm3;

      

      then

       A15: ((((1 - r) * ((1 / 2) * q2)) + (r * ((1 / 2) * q3))) - ((1 - r) * |[1, 0 ]|)) = |[(((1 / 2) * q2) `1 ), (((1 / 2) * q2) `2 )]| by A12, EUCLID: 53

      .= ((1 / 2) * q2) by EUCLID: 53;

      ((1 / 2) * (p1 + q2)) = (((1 / 2) * p1) + ((1 / 2) * q2)) by RLVECT_1:def 5

      .= ((((1 - r) + r) * ((1 / 2) * p1)) + ((1 / 2) * q2)) by RLVECT_1:def 8

      .= ((((1 - r) * ((1 / 2) * p1)) + (r * ((1 / 2) * p1))) + ((1 / 2) * q2)) by RLVECT_1:def 6

      .= (((((1 - r) * ((1 / 2) * p1)) + (r * ((1 / 2) * p1))) + (((1 - r) * ((1 / 2) * q2)) + (r * ((1 / 2) * q3)))) - ((1 - r) * |[1, 0 ]|)) by A15, RLVECT_1:def 3

      .= ((((((1 - r) * ((1 / 2) * p1)) + (r * ((1 / 2) * p1))) + ((1 - r) * ((1 / 2) * q2))) + (r * ((1 / 2) * q3))) - ((1 - r) * |[1, 0 ]|)) by RLVECT_1:def 3

      .= (((1 - r) * ((1 / 2) * (p1 + q2))) + ((r * ((1 / 2) * (p1 + q3))) - ((1 - r) * |[1, 0 ]|))) by A13, RLVECT_1:def 3

      .= (((1 - r) * ((1 / 2) * (p1 + q2))) + ( - (((1 - r) * |[1, 0 ]|) - (r * ((1 / 2) * (p1 + q3)))))) by RLVECT_1: 33

      .= (((1 - r) * ((1 / 2) * (p1 + q2))) - (((1 - r) * |[1, 0 ]|) - (r * ((1 / 2) * (p1 + q3)))))

      .= ((((1 - r) * ((1 / 2) * (p1 + q2))) - ((1 - r) * |[1, 0 ]|)) + (r * ((1 / 2) * (p1 + q3)))) by RLVECT_1: 29

      .= (((1 - r) * (((1 / 2) * (p1 + q2)) - |[1, 0 ]|)) + (r * ((1 / 2) * (p1 + q3)))) by RLVECT_1: 34;

      then ((1 / 2) * (p1 + q2)) in ( LSeg ((((1 / 2) * (p1 + q2)) - |[1, 0 ]|),((1 / 2) * (p1 + q3)))) by A8, A9;

      then

       A16: ( LSeg ((((1 / 2) * (p1 + q2)) - |[1, 0 ]|),((1 / 2) * (p1 + q3)))) = (( LSeg ((((1 / 2) * (p1 + q2)) - |[1, 0 ]|),((1 / 2) * (p1 + q2)))) \/ ( LSeg (((1 / 2) * (p1 + q2)),((1 / 2) * (p1 + q3))))) by TOPREAL1: 5;

      (( 0 + 1) + 1) = ( 0 + (1 + 1));

      then ( LSeg (((1 / 2) * (p1 + q2)),((1 / 2) * (p1 + q3)))) c= (I2 \/ {((1 / 2) * (p1 + q2))}) by A1, A2, A3, Th40;

      hence thesis by A16, A10, A11, XBOOLE_1: 13;

    end;

    theorem :: GOBOARD6:69

    1 <= j & j < ( width G) & 1 < ( len G) implies ( LSeg ((((1 / 2) * ((G * (( len G),j)) + (G * (( len G),(j + 1))))) + |[1, 0 ]|),((1 / 2) * ((G * (( len G),j)) + (G * ((( len G) -' 1),(j + 1))))))) c= ((( Int ( cell (G,(( len G) -' 1),j))) \/ ( Int ( cell (G,( len G),j)))) \/ {((1 / 2) * ((G * (( len G),j)) + (G * (( len G),(j + 1)))))})

    proof

      assume that

       A1: 1 <= j and

       A2: j < ( width G) and

       A3: 1 < ( len G);

      set I1 = ( Int ( cell (G,(( len G) -' 1),j))), I2 = ( Int ( cell (G,( len G),j)));

      set p1 = (G * (( len G),j)), q2 = (G * (( len G),(j + 1))), q3 = (G * ((( len G) -' 1),(j + 1))), r = (1 / (((1 / 2) * ((p1 `1 ) - (q3 `1 ))) + 1));

      

       A4: ((( len G) -' 1) + 1) = ( len G) by A3, XREAL_1: 235;

      then

       A5: 1 <= (( len G) -' 1) by A3, NAT_1: 13;

      

       A6: (( len G) -' 1) < ( len G) by A4, NAT_1: 13;

      then ((G * ((( len G) -' 1),j)) + (G * (( len G),(j + 1)))) = ((G * (( len G),j)) + (G * ((( len G) -' 1),(j + 1)))) by A1, A2, A4, A5, Th11;

      then

       A7: ( LSeg (((1 / 2) * (p1 + q2)),((1 / 2) * (p1 + q3)))) c= (I1 \/ {((1 / 2) * (p1 + q2))}) by A1, A2, A4, A5, A6, Th42;

      

       A8: (j + 1) >= 1 & (j + 1) <= ( width G) by A2, NAT_1: 11, NAT_1: 13;

      

      then

       A9: (q2 `2 ) = ((G * (1,(j + 1))) `2 ) by A3, GOBOARD5: 1

      .= (q3 `2 ) by A5, A6, A8, GOBOARD5: 1;

      

       A10: (q2 `1 ) = ((G * (( len G),1)) `1 ) by A3, A8, GOBOARD5: 2

      .= (p1 `1 ) by A1, A2, A3, GOBOARD5: 2;

      then (q3 `1 ) < (p1 `1 ) by A5, A6, A8, GOBOARD5: 3;

      then

       A11: ((p1 `1 ) - (q3 `1 )) > 0 by XREAL_1: 50;

      then 1 < (((1 / 2) * ((p1 `1 ) - (q3 `1 ))) + 1) by XREAL_1: 29, XREAL_1: 129;

      then

       A12: r < 1 by XREAL_1: 212;

      

       A13: (((((1 - r) * ((1 / 2) * q2)) + (r * ((1 / 2) * q3))) + ((1 - r) * |[1, 0 ]|)) `2 ) = (((((1 - r) * ((1 / 2) * q2)) + (r * ((1 / 2) * q3))) `2 ) + (((1 - r) * |[1, 0 ]|) `2 )) by Lm1

      .= (((((1 - r) * ((1 / 2) * q2)) + (r * ((1 / 2) * q3))) `2 ) + ( |[((1 - r) * 1), ((1 - r) * 0 )]| `2 )) by EUCLID: 58

      .= (((((1 - r) * ((1 / 2) * q2)) + (r * ((1 / 2) * q3))) `2 ) + 0 ) by EUCLID: 52

      .= ((((1 - r) * ((1 / 2) * q2)) `2 ) + ((r * ((1 / 2) * q3)) `2 )) by Lm1

      .= (((1 - r) * (((1 / 2) * q2) `2 )) + ((r * ((1 / 2) * q3)) `2 )) by Lm3

      .= (((1 - r) * (((1 / 2) * q2) `2 )) + (r * (((1 / 2) * q3) `2 ))) by Lm3

      .= (((1 - r) * ((1 / 2) * (q2 `2 ))) + (r * (((1 / 2) * q3) `2 ))) by Lm3

      .= (((1 - r) * ((1 / 2) * (q2 `2 ))) + (r * ((1 / 2) * (q2 `2 )))) by A9, Lm3

      .= (((1 / 2) * q2) `2 ) by Lm3;

      

       A14: ((I1 \/ I2) \/ {((1 / 2) * (p1 + q2))}) = (I1 \/ (I2 \/ ( {((1 / 2) * (p1 + q2))} \/ {((1 / 2) * (p1 + q2))}))) by XBOOLE_1: 4

      .= (I1 \/ ((I2 \/ {((1 / 2) * (p1 + q2))}) \/ {((1 / 2) * (p1 + q2))})) by XBOOLE_1: 4

      .= ((I1 \/ {((1 / 2) * (p1 + q2))}) \/ (I2 \/ {((1 / 2) * (p1 + q2))})) by XBOOLE_1: 4;

      

       A15: (((((1 - r) * ((1 / 2) * p1)) + (r * ((1 / 2) * p1))) + ((1 - r) * ((1 / 2) * q2))) + (r * ((1 / 2) * q3))) = (((((1 - r) * ((1 / 2) * p1)) + ((1 - r) * ((1 / 2) * q2))) + (r * ((1 / 2) * p1))) + (r * ((1 / 2) * q3))) by RLVECT_1:def 3

      .= ((((1 - r) * ((1 / 2) * p1)) + ((1 - r) * ((1 / 2) * q2))) + ((r * ((1 / 2) * p1)) + (r * ((1 / 2) * q3)))) by RLVECT_1:def 3

      .= ((((1 - r) * ((1 / 2) * p1)) + ((1 - r) * ((1 / 2) * q2))) + (r * (((1 / 2) * p1) + ((1 / 2) * q3)))) by RLVECT_1:def 5

      .= (((1 - r) * (((1 / 2) * p1) + ((1 / 2) * q2))) + (r * (((1 / 2) * p1) + ((1 / 2) * q3)))) by RLVECT_1:def 5

      .= (((1 - r) * (((1 / 2) * p1) + ((1 / 2) * q2))) + (r * ((1 / 2) * (p1 + q3)))) by RLVECT_1:def 5

      .= (((1 - r) * ((1 / 2) * (p1 + q2))) + (r * ((1 / 2) * (p1 + q3)))) by RLVECT_1:def 5;

      

       A16: (((r * ((1 / 2) * (q2 `1 ))) - (r * ((1 / 2) * (q3 `1 )))) + r) = (r * (((1 / 2) * ((q2 `1 ) - (q3 `1 ))) + 1))

      .= 1 by A10, A11, XCMPLX_1: 106;

      (((((1 - r) * ((1 / 2) * q2)) + (r * ((1 / 2) * q3))) + ((1 - r) * |[1, 0 ]|)) `1 ) = (((((1 - r) * ((1 / 2) * q2)) + (r * ((1 / 2) * q3))) `1 ) + (((1 - r) * |[1, 0 ]|) `1 )) by Lm1

      .= (((((1 - r) * ((1 / 2) * q2)) + (r * ((1 / 2) * q3))) `1 ) + ( |[((1 - r) * 1), ((1 - r) * 0 )]| `1 )) by EUCLID: 58

      .= (((((1 - r) * ((1 / 2) * q2)) + (r * ((1 / 2) * q3))) `1 ) + (1 - r)) by EUCLID: 52

      .= (((((1 - r) * ((1 / 2) * q2)) `1 ) + ((r * ((1 / 2) * q3)) `1 )) + (1 - r)) by Lm1

      .= ((((1 - r) * (((1 / 2) * q2) `1 )) + ((r * ((1 / 2) * q3)) `1 )) + (1 - r)) by Lm3

      .= ((((1 - r) * (((1 / 2) * q2) `1 )) + (r * (((1 / 2) * q3) `1 ))) + (1 - r)) by Lm3

      .= ((((1 - r) * ((1 / 2) * (q2 `1 ))) + (r * (((1 / 2) * q3) `1 ))) + (1 - r)) by Lm3

      .= ((((1 - r) * ((1 / 2) * (q2 `1 ))) + (r * ((1 / 2) * (q3 `1 )))) + (1 - r)) by Lm3

      .= (((1 / 2) * q2) `1 ) by A16, Lm3;

      

      then

       A17: ((((1 - r) * ((1 / 2) * q2)) + (r * ((1 / 2) * q3))) + ((1 - r) * |[1, 0 ]|)) = |[(((1 / 2) * q2) `1 ), (((1 / 2) * q2) `2 )]| by A13, EUCLID: 53

      .= ((1 / 2) * q2) by EUCLID: 53;

      ((1 / 2) * (p1 + q2)) = (((1 / 2) * p1) + ((1 / 2) * q2)) by RLVECT_1:def 5

      .= ((((1 - r) + r) * ((1 / 2) * p1)) + ((1 / 2) * q2)) by RLVECT_1:def 8

      .= ((((1 - r) * ((1 / 2) * p1)) + (r * ((1 / 2) * p1))) + ((1 / 2) * q2)) by RLVECT_1:def 6

      .= (((((1 - r) * ((1 / 2) * p1)) + (r * ((1 / 2) * p1))) + (((1 - r) * ((1 / 2) * q2)) + (r * ((1 / 2) * q3)))) + ((1 - r) * |[1, 0 ]|)) by A17, RLVECT_1:def 3

      .= ((((((1 - r) * ((1 / 2) * p1)) + (r * ((1 / 2) * p1))) + ((1 - r) * ((1 / 2) * q2))) + (r * ((1 / 2) * q3))) + ((1 - r) * |[1, 0 ]|)) by RLVECT_1:def 3

      .= ((((1 - r) * ((1 / 2) * (p1 + q2))) + ((1 - r) * |[1, 0 ]|)) + (r * ((1 / 2) * (p1 + q3)))) by A15, RLVECT_1:def 3

      .= (((1 - r) * (((1 / 2) * (p1 + q2)) + |[1, 0 ]|)) + (r * ((1 / 2) * (p1 + q3)))) by RLVECT_1:def 5;

      then ((1 / 2) * (p1 + q2)) in ( LSeg ((((1 / 2) * (p1 + q2)) + |[1, 0 ]|),((1 / 2) * (p1 + q3)))) by A11, A12;

      then

       A18: ( LSeg ((((1 / 2) * (p1 + q2)) + |[1, 0 ]|),((1 / 2) * (p1 + q3)))) = (( LSeg ((((1 / 2) * (p1 + q2)) + |[1, 0 ]|),((1 / 2) * (p1 + q2)))) \/ ( LSeg (((1 / 2) * (p1 + q2)),((1 / 2) * (p1 + q3))))) by TOPREAL1: 5;

      ( LSeg ((((1 / 2) * (p1 + q2)) + |[1, 0 ]|),((1 / 2) * (p1 + q2)))) c= (I2 \/ {((1 / 2) * (p1 + q2))}) by A1, A2, Th45;

      hence thesis by A18, A7, A14, XBOOLE_1: 13;

    end;

    

     Lm7: ((1 / 2) + (1 / 2)) = 1;

    theorem :: GOBOARD6:70

    1 < ( len G) & 1 <= j & (j + 1) < ( width G) implies ( LSeg ((((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1))))) - |[1, 0 ]|),(((1 / 2) * ((G * (1,(j + 1))) + (G * (1,(j + 2))))) - |[1, 0 ]|))) c= ((( Int ( cell (G, 0 ,j))) \/ ( Int ( cell (G, 0 ,(j + 1))))) \/ {((G * (1,(j + 1))) - |[1, 0 ]|)})

    proof

      assume that

       A1: 1 < ( len G) and

       A2: 1 <= j and

       A3: (j + 1) < ( width G);

      set p1 = (G * (1,j)), p2 = (G * (1,(j + 1))), q3 = (G * (1,(j + 2))), r = (((p2 `2 ) - (p1 `2 )) / ((q3 `2 ) - (p1 `2 )));

      

       A4: ((j + 1) + 1) = (j + (1 + 1));

      then

       A5: (j + 2) >= 1 by NAT_1: 11;

      

       A6: (j + (1 + 1)) <= ( width G) by A3, A4, NAT_1: 13;

      set I1 = ( Int ( cell (G, 0 ,j))), I2 = ( Int ( cell (G, 0 ,(j + 1))));

      

       A7: ((I1 \/ I2) \/ {(p2 - |[1, 0 ]|)}) = (I1 \/ (I2 \/ ( {(p2 - |[1, 0 ]|)} \/ {(p2 - |[1, 0 ]|)}))) by XBOOLE_1: 4

      .= (I1 \/ ((I2 \/ {(p2 - |[1, 0 ]|)}) \/ {(p2 - |[1, 0 ]|)})) by XBOOLE_1: 4

      .= ((I1 \/ {(p2 - |[1, 0 ]|)}) \/ (I2 \/ {(p2 - |[1, 0 ]|)})) by XBOOLE_1: 4;

      

       A8: ( LSeg ((((1 / 2) * (p2 + q3)) - |[1, 0 ]|),(p2 - |[1, 0 ]|))) c= (I2 \/ {(p2 - |[1, 0 ]|)}) by A3, A4, Th48, NAT_1: 11;

      j < (j + 1) by XREAL_1: 29;

      then (p1 `2 ) < (p2 `2 ) by A1, A2, A3, GOBOARD5: 4;

      then

       A9: ((p2 `2 ) - (p1 `2 )) > 0 by XREAL_1: 50;

      

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

      

      then

       A11: (p2 `1 ) = ((G * (1,1)) `1 ) by A1, A3, GOBOARD5: 2

      .= (q3 `1 ) by A1, A6, A5, GOBOARD5: 2;

      j <= (j + 1) by NAT_1: 11;

      then

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

      

      then (p1 `1 ) = ((G * (1,1)) `1 ) by A1, A2, GOBOARD5: 2

      .= (p2 `1 ) by A1, A3, A10, GOBOARD5: 2;

      

      then

       A13: (1 * (p2 `1 )) = (((1 - r) * (p1 `1 )) + (r * (q3 `1 ))) by A11

      .= ((((1 - r) * p1) `1 ) + (r * (q3 `1 ))) by Lm3

      .= ((((1 - r) * p1) `1 ) + ((r * q3) `1 )) by Lm3

      .= ((((1 - r) * p1) + (r * q3)) `1 ) by Lm1;

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

      then (p2 `2 ) < (q3 `2 ) by A1, A10, A6, GOBOARD5: 4;

      then

       A14: ((p2 `2 ) - (p1 `2 )) < ((q3 `2 ) - (p1 `2 )) by XREAL_1: 9;

      then (r * ((q3 `2 ) - (p1 `2 ))) = ((p2 `2 ) - (p1 `2 )) by A9, XCMPLX_1: 87;

      then (p2 `2 ) = (((1 - r) * (p1 `2 )) + (r * (q3 `2 )));

      

      then (1 * (p2 `2 )) = ((((1 - r) * p1) `2 ) + (r * (q3 `2 ))) by Lm3

      .= ((((1 - r) * p1) `2 ) + ((r * q3) `2 )) by Lm3

      .= ((((1 - r) * p1) + (r * q3)) `2 ) by Lm1;

      

      then

       A15: (((1 - r) * p1) + (r * q3)) = |[(p2 `1 ), (p2 `2 )]| by A13, EUCLID: 53

      .= p2 by EUCLID: 53;

      p2 = (1 * p2) by RLVECT_1:def 8

      .= (((1 / 2) * p2) + ((1 / 2) * p2)) by Lm7, RLVECT_1:def 6

      .= (((1 / 2) * (((1 - r) + r) * p2)) + ((1 / 2) * (((1 - r) * p1) + (r * q3)))) by A15, RLVECT_1:def 8

      .= (((1 / 2) * (((1 - r) * p2) + (r * p2))) + ((1 / 2) * (((1 - r) * p1) + (r * q3)))) by RLVECT_1:def 6

      .= ((((1 / 2) * ((1 - r) * p2)) + ((1 / 2) * (r * p2))) + ((1 / 2) * (((1 - r) * p1) + (r * q3)))) by RLVECT_1:def 5

      .= ((((1 / 2) * ((1 - r) * p2)) + ((1 / 2) * (r * p2))) + (((1 / 2) * ((1 - r) * p1)) + ((1 / 2) * (r * q3)))) by RLVECT_1:def 5

      .= (((1 / 2) * ((1 - r) * p2)) + (((1 / 2) * (r * p2)) + (((1 / 2) * ((1 - r) * p1)) + ((1 / 2) * (r * q3))))) by RLVECT_1:def 3

      .= (((1 / 2) * ((1 - r) * p2)) + (((1 / 2) * ((1 - r) * p1)) + (((1 / 2) * (r * p2)) + ((1 / 2) * (r * q3))))) by RLVECT_1:def 3

      .= ((((1 / 2) * ((1 - r) * p2)) + ((1 / 2) * ((1 - r) * p1))) + (((1 / 2) * (r * p2)) + ((1 / 2) * (r * q3)))) by RLVECT_1:def 3

      .= (((((1 / 2) * ((1 - r) * p2)) + ((1 / 2) * ((1 - r) * p1))) + ((1 / 2) * (r * p2))) + ((1 / 2) * (r * q3))) by RLVECT_1:def 3

      .= ((((1 / 2) * (((1 - r) * p2) + ((1 - r) * p1))) + ((1 / 2) * (r * p2))) + ((1 / 2) * (r * q3))) by RLVECT_1:def 5

      .= ((((1 / 2) * ((1 - r) * (p1 + p2))) + ((1 / 2) * (r * p2))) + ((1 / 2) * (r * q3))) by RLVECT_1:def 5

      .= (((((1 / 2) * (1 - r)) * (p1 + p2)) + ((1 / 2) * (r * p2))) + ((1 / 2) * (r * q3))) by RLVECT_1:def 7

      .= ((((1 / 2) * (1 - r)) * (p1 + p2)) + (((1 / 2) * (r * p2)) + ((1 / 2) * (r * q3)))) by RLVECT_1:def 3

      .= ((((1 / 2) * (1 - r)) * (p1 + p2)) + ((1 / 2) * ((r * p2) + (r * q3)))) by RLVECT_1:def 5

      .= ((((1 / 2) * (1 - r)) * (p1 + p2)) + ((1 / 2) * (r * (p2 + q3)))) by RLVECT_1:def 5;

      

      then

       A16: p2 = (((1 - r) * ((1 / 2) * (p1 + p2))) + ((1 / 2) * (r * (p2 + q3)))) by RLVECT_1:def 7

      .= (((1 - r) * ((1 / 2) * (p1 + p2))) + (((1 / 2) * r) * (p2 + q3))) by RLVECT_1:def 7

      .= (((1 - r) * ((1 / 2) * (p1 + p2))) + (r * ((1 / 2) * (p2 + q3)))) by RLVECT_1:def 7;

      

       A17: (((1 - r) * (((1 / 2) * (p1 + p2)) - |[1, 0 ]|)) + (r * (((1 / 2) * (p2 + q3)) - |[1, 0 ]|))) = ((((1 - r) * ((1 / 2) * (p1 + p2))) - ((1 - r) * |[1, 0 ]|)) + (r * (((1 / 2) * (p2 + q3)) - |[1, 0 ]|))) by RLVECT_1: 34

      .= ((((1 - r) * ((1 / 2) * (p1 + p2))) - ((1 - r) * |[1, 0 ]|)) + ((r * ((1 / 2) * (p2 + q3))) - (r * |[1, 0 ]|))) by RLVECT_1: 34

      .= (((r * ((1 / 2) * (p2 + q3))) + (((1 - r) * ((1 / 2) * (p1 + p2))) - ((1 - r) * |[1, 0 ]|))) - (r * |[1, 0 ]|)) by RLVECT_1:def 3

      .= ((((r * ((1 / 2) * (p2 + q3))) + ((1 - r) * ((1 / 2) * (p1 + p2)))) - ((1 - r) * |[1, 0 ]|)) - (r * |[1, 0 ]|)) by RLVECT_1:def 3

      .= (((r * ((1 / 2) * (p2 + q3))) + ((1 - r) * ((1 / 2) * (p1 + p2)))) - (((1 - r) * |[1, 0 ]|) + (r * |[1, 0 ]|))) by RLVECT_1: 27

      .= (((r * ((1 / 2) * (p2 + q3))) + ((1 - r) * ((1 / 2) * (p1 + p2)))) - (((1 - r) + r) * |[1, 0 ]|)) by RLVECT_1:def 6

      .= (p2 - |[1, 0 ]|) by A16, RLVECT_1:def 8;

      r < 1 by A14, A9, XREAL_1: 189;

      then (p2 - |[1, 0 ]|) in ( LSeg ((((1 / 2) * (p1 + p2)) - |[1, 0 ]|),(((1 / 2) * (p2 + q3)) - |[1, 0 ]|))) by A14, A9, A17;

      then

       A18: ( LSeg ((((1 / 2) * (p1 + p2)) - |[1, 0 ]|),(((1 / 2) * (p2 + q3)) - |[1, 0 ]|))) = (( LSeg ((((1 / 2) * (p1 + p2)) - |[1, 0 ]|),(p2 - |[1, 0 ]|))) \/ ( LSeg ((p2 - |[1, 0 ]|),(((1 / 2) * (p2 + q3)) - |[1, 0 ]|)))) by TOPREAL1: 5;

      ( LSeg ((((1 / 2) * (p1 + p2)) - |[1, 0 ]|),(p2 - |[1, 0 ]|))) c= (I1 \/ {(p2 - |[1, 0 ]|)}) by A2, A12, Th49;

      hence thesis by A18, A8, A7, XBOOLE_1: 13;

    end;

    theorem :: GOBOARD6:71

    1 < ( len G) & 1 <= j & (j + 1) < ( width G) implies ( LSeg ((((1 / 2) * ((G * (( len G),j)) + (G * (( len G),(j + 1))))) + |[1, 0 ]|),(((1 / 2) * ((G * (( len G),(j + 1))) + (G * (( len G),(j + 2))))) + |[1, 0 ]|))) c= ((( Int ( cell (G,( len G),j))) \/ ( Int ( cell (G,( len G),(j + 1))))) \/ {((G * (( len G),(j + 1))) + |[1, 0 ]|)})

    proof

      assume that

       A1: 1 < ( len G) and

       A2: 1 <= j and

       A3: (j + 1) < ( width G);

      set p1 = (G * (( len G),j)), p2 = (G * (( len G),(j + 1))), q3 = (G * (( len G),(j + 2))), r = (((p2 `2 ) - (p1 `2 )) / ((q3 `2 ) - (p1 `2 )));

      

       A4: ((j + 1) + 1) = (j + (1 + 1));

      then

       A5: (j + 2) >= 1 by NAT_1: 11;

      

       A6: (j + (1 + 1)) <= ( width G) by A3, A4, NAT_1: 13;

      set I1 = ( Int ( cell (G,( len G),j))), I2 = ( Int ( cell (G,( len G),(j + 1))));

      

       A7: ((I1 \/ I2) \/ {(p2 + |[1, 0 ]|)}) = (I1 \/ (I2 \/ ( {(p2 + |[1, 0 ]|)} \/ {(p2 + |[1, 0 ]|)}))) by XBOOLE_1: 4

      .= (I1 \/ ((I2 \/ {(p2 + |[1, 0 ]|)}) \/ {(p2 + |[1, 0 ]|)})) by XBOOLE_1: 4

      .= ((I1 \/ {(p2 + |[1, 0 ]|)}) \/ (I2 \/ {(p2 + |[1, 0 ]|)})) by XBOOLE_1: 4;

      

       A8: ( LSeg ((((1 / 2) * (p2 + q3)) + |[1, 0 ]|),(p2 + |[1, 0 ]|))) c= (I2 \/ {(p2 + |[1, 0 ]|)}) by A3, A4, Th50, NAT_1: 11;

      j < (j + 1) by XREAL_1: 29;

      then (p1 `2 ) < (p2 `2 ) by A1, A2, A3, GOBOARD5: 4;

      then

       A9: ((p2 `2 ) - (p1 `2 )) > 0 by XREAL_1: 50;

      

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

      

      then

       A11: (p2 `1 ) = ((G * (( len G),1)) `1 ) by A1, A3, GOBOARD5: 2

      .= (q3 `1 ) by A1, A6, A5, GOBOARD5: 2;

      j <= (j + 1) by NAT_1: 11;

      then

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

      

      then (p1 `1 ) = ((G * (( len G),1)) `1 ) by A1, A2, GOBOARD5: 2

      .= (p2 `1 ) by A1, A3, A10, GOBOARD5: 2;

      

      then

       A13: (1 * (p2 `1 )) = (((1 - r) * (p1 `1 )) + (r * (q3 `1 ))) by A11

      .= ((((1 - r) * p1) `1 ) + (r * (q3 `1 ))) by Lm3

      .= ((((1 - r) * p1) `1 ) + ((r * q3) `1 )) by Lm3

      .= ((((1 - r) * p1) + (r * q3)) `1 ) by Lm1;

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

      then (p2 `2 ) < (q3 `2 ) by A1, A10, A6, GOBOARD5: 4;

      then

       A14: ((p2 `2 ) - (p1 `2 )) < ((q3 `2 ) - (p1 `2 )) by XREAL_1: 9;

      then (r * ((q3 `2 ) - (p1 `2 ))) = ((p2 `2 ) - (p1 `2 )) by A9, XCMPLX_1: 87;

      then (p2 `2 ) = (((1 - r) * (p1 `2 )) + (r * (q3 `2 )));

      

      then (1 * (p2 `2 )) = ((((1 - r) * p1) `2 ) + (r * (q3 `2 ))) by Lm3

      .= ((((1 - r) * p1) `2 ) + ((r * q3) `2 )) by Lm3

      .= ((((1 - r) * p1) + (r * q3)) `2 ) by Lm1;

      

      then

       A15: (((1 - r) * p1) + (r * q3)) = |[(p2 `1 ), (p2 `2 )]| by A13, EUCLID: 53

      .= p2 by EUCLID: 53;

      p2 = (1 * p2) by RLVECT_1:def 8

      .= (((1 / 2) * p2) + ((1 / 2) * p2)) by Lm7, RLVECT_1:def 6

      .= (((1 / 2) * (((1 - r) + r) * p2)) + ((1 / 2) * (((1 - r) * p1) + (r * q3)))) by A15, RLVECT_1:def 8

      .= (((1 / 2) * (((1 - r) * p2) + (r * p2))) + ((1 / 2) * (((1 - r) * p1) + (r * q3)))) by RLVECT_1:def 6

      .= ((((1 / 2) * ((1 - r) * p2)) + ((1 / 2) * (r * p2))) + ((1 / 2) * (((1 - r) * p1) + (r * q3)))) by RLVECT_1:def 5

      .= ((((1 / 2) * ((1 - r) * p2)) + ((1 / 2) * (r * p2))) + (((1 / 2) * ((1 - r) * p1)) + ((1 / 2) * (r * q3)))) by RLVECT_1:def 5

      .= (((1 / 2) * ((1 - r) * p2)) + (((1 / 2) * (r * p2)) + (((1 / 2) * ((1 - r) * p1)) + ((1 / 2) * (r * q3))))) by RLVECT_1:def 3

      .= (((1 / 2) * ((1 - r) * p2)) + (((1 / 2) * ((1 - r) * p1)) + (((1 / 2) * (r * p2)) + ((1 / 2) * (r * q3))))) by RLVECT_1:def 3

      .= ((((1 / 2) * ((1 - r) * p2)) + ((1 / 2) * ((1 - r) * p1))) + (((1 / 2) * (r * p2)) + ((1 / 2) * (r * q3)))) by RLVECT_1:def 3

      .= (((((1 / 2) * ((1 - r) * p2)) + ((1 / 2) * ((1 - r) * p1))) + ((1 / 2) * (r * p2))) + ((1 / 2) * (r * q3))) by RLVECT_1:def 3

      .= ((((1 / 2) * (((1 - r) * p2) + ((1 - r) * p1))) + ((1 / 2) * (r * p2))) + ((1 / 2) * (r * q3))) by RLVECT_1:def 5

      .= ((((1 / 2) * ((1 - r) * (p1 + p2))) + ((1 / 2) * (r * p2))) + ((1 / 2) * (r * q3))) by RLVECT_1:def 5

      .= (((((1 / 2) * (1 - r)) * (p1 + p2)) + ((1 / 2) * (r * p2))) + ((1 / 2) * (r * q3))) by RLVECT_1:def 7;

      

      then

       A16: p2 = ((((1 / 2) * (1 - r)) * (p1 + p2)) + (((1 / 2) * (r * p2)) + ((1 / 2) * (r * q3)))) by RLVECT_1:def 3

      .= ((((1 / 2) * (1 - r)) * (p1 + p2)) + ((1 / 2) * ((r * p2) + (r * q3)))) by RLVECT_1:def 5

      .= ((((1 / 2) * (1 - r)) * (p1 + p2)) + ((1 / 2) * (r * (p2 + q3)))) by RLVECT_1:def 5

      .= (((1 - r) * ((1 / 2) * (p1 + p2))) + ((1 / 2) * (r * (p2 + q3)))) by RLVECT_1:def 7

      .= (((1 - r) * ((1 / 2) * (p1 + p2))) + (((1 / 2) * r) * (p2 + q3))) by RLVECT_1:def 7

      .= (((1 - r) * ((1 / 2) * (p1 + p2))) + (r * ((1 / 2) * (p2 + q3)))) by RLVECT_1:def 7;

      

       A17: (((1 - r) * (((1 / 2) * (p1 + p2)) + |[1, 0 ]|)) + (r * (((1 / 2) * (p2 + q3)) + |[1, 0 ]|))) = ((((1 - r) * ((1 / 2) * (p1 + p2))) + ((1 - r) * |[1, 0 ]|)) + (r * (((1 / 2) * (p2 + q3)) + |[1, 0 ]|))) by RLVECT_1:def 5

      .= ((((1 - r) * ((1 / 2) * (p1 + p2))) + ((1 - r) * |[1, 0 ]|)) + ((r * ((1 / 2) * (p2 + q3))) + (r * |[1, 0 ]|))) by RLVECT_1:def 5

      .= (((r * ((1 / 2) * (p2 + q3))) + (((1 - r) * ((1 / 2) * (p1 + p2))) + ((1 - r) * |[1, 0 ]|))) + (r * |[1, 0 ]|)) by RLVECT_1:def 3

      .= ((((r * ((1 / 2) * (p2 + q3))) + ((1 - r) * ((1 / 2) * (p1 + p2)))) + ((1 - r) * |[1, 0 ]|)) + (r * |[1, 0 ]|)) by RLVECT_1:def 3

      .= (((r * ((1 / 2) * (p2 + q3))) + ((1 - r) * ((1 / 2) * (p1 + p2)))) + (((1 - r) * |[1, 0 ]|) + (r * |[1, 0 ]|))) by RLVECT_1:def 3

      .= (((r * ((1 / 2) * (p2 + q3))) + ((1 - r) * ((1 / 2) * (p1 + p2)))) + (((1 - r) + r) * |[1, 0 ]|)) by RLVECT_1:def 6

      .= (p2 + |[1, 0 ]|) by A16, RLVECT_1:def 8;

      r < 1 by A14, A9, XREAL_1: 189;

      then (p2 + |[1, 0 ]|) in ( LSeg ((((1 / 2) * (p1 + p2)) + |[1, 0 ]|),(((1 / 2) * (p2 + q3)) + |[1, 0 ]|))) by A14, A9, A17;

      then

       A18: ( LSeg ((((1 / 2) * (p1 + p2)) + |[1, 0 ]|),(((1 / 2) * (p2 + q3)) + |[1, 0 ]|))) = (( LSeg ((((1 / 2) * (p1 + p2)) + |[1, 0 ]|),(p2 + |[1, 0 ]|))) \/ ( LSeg ((p2 + |[1, 0 ]|),(((1 / 2) * (p2 + q3)) + |[1, 0 ]|)))) by TOPREAL1: 5;

      ( LSeg ((((1 / 2) * (p1 + p2)) + |[1, 0 ]|),(p2 + |[1, 0 ]|))) c= (I1 \/ {(p2 + |[1, 0 ]|)}) by A2, A12, Th51;

      hence thesis by A18, A8, A7, XBOOLE_1: 13;

    end;

    theorem :: GOBOARD6:72

    1 < ( width G) & 1 <= i & (i + 1) < ( len G) implies ( LSeg ((((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1)))) - |[ 0 , 1]|),(((1 / 2) * ((G * ((i + 1),1)) + (G * ((i + 2),1)))) - |[ 0 , 1]|))) c= ((( Int ( cell (G,i, 0 ))) \/ ( Int ( cell (G,(i + 1), 0 )))) \/ {((G * ((i + 1),1)) - |[ 0 , 1]|)})

    proof

      assume that

       A1: 1 < ( width G) and

       A2: 1 <= i and

       A3: (i + 1) < ( len G);

      set p1 = (G * (i,1)), p2 = (G * ((i + 1),1)), q3 = (G * ((i + 2),1)), r = (((p2 `1 ) - (p1 `1 )) / ((q3 `1 ) - (p1 `1 )));

      

       A4: ((i + 1) + 1) = (i + (1 + 1));

      then

       A5: (i + 2) >= 1 by NAT_1: 11;

      

       A6: (i + (1 + 1)) <= ( len G) by A3, A4, NAT_1: 13;

      set I1 = ( Int ( cell (G,i, 0 ))), I2 = ( Int ( cell (G,(i + 1), 0 )));

      

       A7: ((I1 \/ I2) \/ {(p2 - |[ 0 , 1]|)}) = (I1 \/ (I2 \/ ( {(p2 - |[ 0 , 1]|)} \/ {(p2 - |[ 0 , 1]|)}))) by XBOOLE_1: 4

      .= (I1 \/ ((I2 \/ {(p2 - |[ 0 , 1]|)}) \/ {(p2 - |[ 0 , 1]|)})) by XBOOLE_1: 4

      .= ((I1 \/ {(p2 - |[ 0 , 1]|)}) \/ (I2 \/ {(p2 - |[ 0 , 1]|)})) by XBOOLE_1: 4;

      

       A8: ( LSeg ((((1 / 2) * (p2 + q3)) - |[ 0 , 1]|),(p2 - |[ 0 , 1]|))) c= (I2 \/ {(p2 - |[ 0 , 1]|)}) by A3, A4, Th52, NAT_1: 11;

      i < (i + 1) by XREAL_1: 29;

      then (p1 `1 ) < (p2 `1 ) by A1, A2, A3, GOBOARD5: 3;

      then

       A9: ((p2 `1 ) - (p1 `1 )) > 0 by XREAL_1: 50;

      

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

      

      then

       A11: (p2 `2 ) = ((G * (1,1)) `2 ) by A1, A3, GOBOARD5: 1

      .= (q3 `2 ) by A1, A6, A5, GOBOARD5: 1;

      i <= (i + 1) by NAT_1: 11;

      then

       A12: i < ( len G) by A3, XXREAL_0: 2;

      

      then (p1 `2 ) = ((G * (1,1)) `2 ) by A1, A2, GOBOARD5: 1

      .= (p2 `2 ) by A1, A3, A10, GOBOARD5: 1;

      

      then

       A13: (1 * (p2 `2 )) = (((1 - r) * (p1 `2 )) + (r * (q3 `2 ))) by A11

      .= ((((1 - r) * p1) `2 ) + (r * (q3 `2 ))) by Lm3

      .= ((((1 - r) * p1) `2 ) + ((r * q3) `2 )) by Lm3

      .= ((((1 - r) * p1) + (r * q3)) `2 ) by Lm1;

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

      then (p2 `1 ) < (q3 `1 ) by A1, A10, A6, GOBOARD5: 3;

      then

       A14: ((p2 `1 ) - (p1 `1 )) < ((q3 `1 ) - (p1 `1 )) by XREAL_1: 9;

      then (r * ((q3 `1 ) - (p1 `1 ))) = ((p2 `1 ) - (p1 `1 )) by A9, XCMPLX_1: 87;

      then (p2 `1 ) = (((1 - r) * (p1 `1 )) + (r * (q3 `1 )));

      

      then (1 * (p2 `1 )) = ((((1 - r) * p1) `1 ) + (r * (q3 `1 ))) by Lm3

      .= ((((1 - r) * p1) `1 ) + ((r * q3) `1 )) by Lm3

      .= ((((1 - r) * p1) + (r * q3)) `1 ) by Lm1;

      

      then

       A15: (((1 - r) * p1) + (r * q3)) = |[(p2 `1 ), (p2 `2 )]| by A13, EUCLID: 53

      .= p2 by EUCLID: 53;

      p2 = (1 * p2) by RLVECT_1:def 8

      .= (((1 / 2) * p2) + ((1 / 2) * p2)) by Lm7, RLVECT_1:def 6

      .= (((1 / 2) * (((1 - r) + r) * p2)) + ((1 / 2) * (((1 - r) * p1) + (r * q3)))) by A15, RLVECT_1:def 8

      .= (((1 / 2) * (((1 - r) * p2) + (r * p2))) + ((1 / 2) * (((1 - r) * p1) + (r * q3)))) by RLVECT_1:def 6

      .= ((((1 / 2) * ((1 - r) * p2)) + ((1 / 2) * (r * p2))) + ((1 / 2) * (((1 - r) * p1) + (r * q3)))) by RLVECT_1:def 5

      .= ((((1 / 2) * ((1 - r) * p2)) + ((1 / 2) * (r * p2))) + (((1 / 2) * ((1 - r) * p1)) + ((1 / 2) * (r * q3)))) by RLVECT_1:def 5

      .= (((1 / 2) * ((1 - r) * p2)) + (((1 / 2) * (r * p2)) + (((1 / 2) * ((1 - r) * p1)) + ((1 / 2) * (r * q3))))) by RLVECT_1:def 3

      .= (((1 / 2) * ((1 - r) * p2)) + (((1 / 2) * ((1 - r) * p1)) + (((1 / 2) * (r * p2)) + ((1 / 2) * (r * q3))))) by RLVECT_1:def 3

      .= ((((1 / 2) * ((1 - r) * p2)) + ((1 / 2) * ((1 - r) * p1))) + (((1 / 2) * (r * p2)) + ((1 / 2) * (r * q3)))) by RLVECT_1:def 3

      .= (((((1 / 2) * ((1 - r) * p2)) + ((1 / 2) * ((1 - r) * p1))) + ((1 / 2) * (r * p2))) + ((1 / 2) * (r * q3))) by RLVECT_1:def 3

      .= ((((1 / 2) * (((1 - r) * p2) + ((1 - r) * p1))) + ((1 / 2) * (r * p2))) + ((1 / 2) * (r * q3))) by RLVECT_1:def 5

      .= ((((1 / 2) * ((1 - r) * (p1 + p2))) + ((1 / 2) * (r * p2))) + ((1 / 2) * (r * q3))) by RLVECT_1:def 5

      .= (((((1 / 2) * (1 - r)) * (p1 + p2)) + ((1 / 2) * (r * p2))) + ((1 / 2) * (r * q3))) by RLVECT_1:def 7

      .= ((((1 / 2) * (1 - r)) * (p1 + p2)) + (((1 / 2) * (r * p2)) + ((1 / 2) * (r * q3)))) by RLVECT_1:def 3

      .= ((((1 / 2) * (1 - r)) * (p1 + p2)) + ((1 / 2) * ((r * p2) + (r * q3)))) by RLVECT_1:def 5;

      

      then

       A16: p2 = ((((1 / 2) * (1 - r)) * (p1 + p2)) + ((1 / 2) * (r * (p2 + q3)))) by RLVECT_1:def 5

      .= (((1 - r) * ((1 / 2) * (p1 + p2))) + ((1 / 2) * (r * (p2 + q3)))) by RLVECT_1:def 7

      .= (((1 - r) * ((1 / 2) * (p1 + p2))) + (((1 / 2) * r) * (p2 + q3))) by RLVECT_1:def 7

      .= (((1 - r) * ((1 / 2) * (p1 + p2))) + (r * ((1 / 2) * (p2 + q3)))) by RLVECT_1:def 7;

      

       A17: (((1 - r) * (((1 / 2) * (p1 + p2)) - |[ 0 , 1]|)) + (r * (((1 / 2) * (p2 + q3)) - |[ 0 , 1]|))) = ((((1 - r) * ((1 / 2) * (p1 + p2))) - ((1 - r) * |[ 0 , 1]|)) + (r * (((1 / 2) * (p2 + q3)) - |[ 0 , 1]|))) by RLVECT_1: 34

      .= ((((1 - r) * ((1 / 2) * (p1 + p2))) - ((1 - r) * |[ 0 , 1]|)) + ((r * ((1 / 2) * (p2 + q3))) - (r * |[ 0 , 1]|))) by RLVECT_1: 34

      .= (((r * ((1 / 2) * (p2 + q3))) + (((1 - r) * ((1 / 2) * (p1 + p2))) - ((1 - r) * |[ 0 , 1]|))) - (r * |[ 0 , 1]|)) by RLVECT_1:def 3

      .= ((((r * ((1 / 2) * (p2 + q3))) + ((1 - r) * ((1 / 2) * (p1 + p2)))) - ((1 - r) * |[ 0 , 1]|)) - (r * |[ 0 , 1]|)) by RLVECT_1:def 3

      .= (((r * ((1 / 2) * (p2 + q3))) + ((1 - r) * ((1 / 2) * (p1 + p2)))) - (((1 - r) * |[ 0 , 1]|) + (r * |[ 0 , 1]|))) by RLVECT_1: 27

      .= (((r * ((1 / 2) * (p2 + q3))) + ((1 - r) * ((1 / 2) * (p1 + p2)))) - (((1 - r) + r) * |[ 0 , 1]|)) by RLVECT_1:def 6

      .= (p2 - |[ 0 , 1]|) by A16, RLVECT_1:def 8;

      r < 1 by A14, A9, XREAL_1: 189;

      then (p2 - |[ 0 , 1]|) in ( LSeg ((((1 / 2) * (p1 + p2)) - |[ 0 , 1]|),(((1 / 2) * (p2 + q3)) - |[ 0 , 1]|))) by A14, A9, A17;

      then

       A18: ( LSeg ((((1 / 2) * (p1 + p2)) - |[ 0 , 1]|),(((1 / 2) * (p2 + q3)) - |[ 0 , 1]|))) = (( LSeg ((((1 / 2) * (p1 + p2)) - |[ 0 , 1]|),(p2 - |[ 0 , 1]|))) \/ ( LSeg ((p2 - |[ 0 , 1]|),(((1 / 2) * (p2 + q3)) - |[ 0 , 1]|)))) by TOPREAL1: 5;

      ( LSeg ((((1 / 2) * (p1 + p2)) - |[ 0 , 1]|),(p2 - |[ 0 , 1]|))) c= (I1 \/ {(p2 - |[ 0 , 1]|)}) by A2, A12, Th53;

      hence thesis by A18, A8, A7, XBOOLE_1: 13;

    end;

    theorem :: GOBOARD6:73

    1 < ( width G) & 1 <= i & (i + 1) < ( len G) implies ( LSeg ((((1 / 2) * ((G * (i,( width G))) + (G * ((i + 1),( width G))))) + |[ 0 , 1]|),(((1 / 2) * ((G * ((i + 1),( width G))) + (G * ((i + 2),( width G))))) + |[ 0 , 1]|))) c= ((( Int ( cell (G,i,( width G)))) \/ ( Int ( cell (G,(i + 1),( width G))))) \/ {((G * ((i + 1),( width G))) + |[ 0 , 1]|)})

    proof

      assume that

       A1: 1 < ( width G) and

       A2: 1 <= i and

       A3: (i + 1) < ( len G);

      set p1 = (G * (i,( width G))), p2 = (G * ((i + 1),( width G))), q3 = (G * ((i + 2),( width G))), r = (((p2 `1 ) - (p1 `1 )) / ((q3 `1 ) - (p1 `1 )));

      

       A4: ((i + 1) + 1) = (i + (1 + 1));

      then

       A5: (i + 2) >= 1 by NAT_1: 11;

      

       A6: (i + (1 + 1)) <= ( len G) by A3, A4, NAT_1: 13;

      set I1 = ( Int ( cell (G,i,( width G)))), I2 = ( Int ( cell (G,(i + 1),( width G))));

      

       A7: ((I1 \/ I2) \/ {(p2 + |[ 0 , 1]|)}) = (I1 \/ (I2 \/ ( {(p2 + |[ 0 , 1]|)} \/ {(p2 + |[ 0 , 1]|)}))) by XBOOLE_1: 4

      .= (I1 \/ ((I2 \/ {(p2 + |[ 0 , 1]|)}) \/ {(p2 + |[ 0 , 1]|)})) by XBOOLE_1: 4

      .= ((I1 \/ {(p2 + |[ 0 , 1]|)}) \/ (I2 \/ {(p2 + |[ 0 , 1]|)})) by XBOOLE_1: 4;

      

       A8: ( LSeg ((((1 / 2) * (p2 + q3)) + |[ 0 , 1]|),(p2 + |[ 0 , 1]|))) c= (I2 \/ {(p2 + |[ 0 , 1]|)}) by A3, A4, Th54, NAT_1: 11;

      i < (i + 1) by XREAL_1: 29;

      then (p1 `1 ) < (p2 `1 ) by A1, A2, A3, GOBOARD5: 3;

      then

       A9: ((p2 `1 ) - (p1 `1 )) > 0 by XREAL_1: 50;

      

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

      

      then

       A11: (p2 `2 ) = ((G * (1,( width G))) `2 ) by A1, A3, GOBOARD5: 1

      .= (q3 `2 ) by A1, A6, A5, GOBOARD5: 1;

      i <= (i + 1) by NAT_1: 11;

      then

       A12: i < ( len G) by A3, XXREAL_0: 2;

      

      then (p1 `2 ) = ((G * (1,( width G))) `2 ) by A1, A2, GOBOARD5: 1

      .= (p2 `2 ) by A1, A3, A10, GOBOARD5: 1;

      

      then

       A13: (1 * (p2 `2 )) = (((1 - r) * (p1 `2 )) + (r * (q3 `2 ))) by A11

      .= ((((1 - r) * p1) `2 ) + (r * (q3 `2 ))) by Lm3

      .= ((((1 - r) * p1) `2 ) + ((r * q3) `2 )) by Lm3

      .= ((((1 - r) * p1) + (r * q3)) `2 ) by Lm1;

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

      then (p2 `1 ) < (q3 `1 ) by A1, A10, A6, GOBOARD5: 3;

      then

       A14: ((p2 `1 ) - (p1 `1 )) < ((q3 `1 ) - (p1 `1 )) by XREAL_1: 9;

      then (r * ((q3 `1 ) - (p1 `1 ))) = ((p2 `1 ) - (p1 `1 )) by A9, XCMPLX_1: 87;

      then (p2 `1 ) = (((1 - r) * (p1 `1 )) + (r * (q3 `1 )));

      

      then (1 * (p2 `1 )) = ((((1 - r) * p1) `1 ) + (r * (q3 `1 ))) by Lm3

      .= ((((1 - r) * p1) `1 ) + ((r * q3) `1 )) by Lm3

      .= ((((1 - r) * p1) + (r * q3)) `1 ) by Lm1;

      

      then

       A15: (((1 - r) * p1) + (r * q3)) = |[(p2 `1 ), (p2 `2 )]| by A13, EUCLID: 53

      .= p2 by EUCLID: 53;

      p2 = (1 * p2) by RLVECT_1:def 8

      .= (((1 / 2) * p2) + ((1 / 2) * p2)) by Lm7, RLVECT_1:def 6

      .= (((1 / 2) * (((1 - r) + r) * p2)) + ((1 / 2) * (((1 - r) * p1) + (r * q3)))) by A15, RLVECT_1:def 8

      .= (((1 / 2) * (((1 - r) * p2) + (r * p2))) + ((1 / 2) * (((1 - r) * p1) + (r * q3)))) by RLVECT_1:def 6

      .= ((((1 / 2) * ((1 - r) * p2)) + ((1 / 2) * (r * p2))) + ((1 / 2) * (((1 - r) * p1) + (r * q3)))) by RLVECT_1:def 5

      .= ((((1 / 2) * ((1 - r) * p2)) + ((1 / 2) * (r * p2))) + (((1 / 2) * ((1 - r) * p1)) + ((1 / 2) * (r * q3)))) by RLVECT_1:def 5

      .= (((1 / 2) * ((1 - r) * p2)) + (((1 / 2) * (r * p2)) + (((1 / 2) * ((1 - r) * p1)) + ((1 / 2) * (r * q3))))) by RLVECT_1:def 3

      .= (((1 / 2) * ((1 - r) * p2)) + (((1 / 2) * ((1 - r) * p1)) + (((1 / 2) * (r * p2)) + ((1 / 2) * (r * q3))))) by RLVECT_1:def 3

      .= ((((1 / 2) * ((1 - r) * p2)) + ((1 / 2) * ((1 - r) * p1))) + (((1 / 2) * (r * p2)) + ((1 / 2) * (r * q3)))) by RLVECT_1:def 3

      .= (((((1 / 2) * ((1 - r) * p2)) + ((1 / 2) * ((1 - r) * p1))) + ((1 / 2) * (r * p2))) + ((1 / 2) * (r * q3))) by RLVECT_1:def 3

      .= ((((1 / 2) * (((1 - r) * p2) + ((1 - r) * p1))) + ((1 / 2) * (r * p2))) + ((1 / 2) * (r * q3))) by RLVECT_1:def 5

      .= ((((1 / 2) * ((1 - r) * (p1 + p2))) + ((1 / 2) * (r * p2))) + ((1 / 2) * (r * q3))) by RLVECT_1:def 5

      .= (((((1 / 2) * (1 - r)) * (p1 + p2)) + ((1 / 2) * (r * p2))) + ((1 / 2) * (r * q3))) by RLVECT_1:def 7;

      

      then

       A16: p2 = ((((1 / 2) * (1 - r)) * (p1 + p2)) + (((1 / 2) * (r * p2)) + ((1 / 2) * (r * q3)))) by RLVECT_1:def 3

      .= ((((1 / 2) * (1 - r)) * (p1 + p2)) + ((1 / 2) * ((r * p2) + (r * q3)))) by RLVECT_1:def 5

      .= ((((1 / 2) * (1 - r)) * (p1 + p2)) + ((1 / 2) * (r * (p2 + q3)))) by RLVECT_1:def 5

      .= (((1 - r) * ((1 / 2) * (p1 + p2))) + ((1 / 2) * (r * (p2 + q3)))) by RLVECT_1:def 7

      .= (((1 - r) * ((1 / 2) * (p1 + p2))) + (((1 / 2) * r) * (p2 + q3))) by RLVECT_1:def 7

      .= (((1 - r) * ((1 / 2) * (p1 + p2))) + (r * ((1 / 2) * (p2 + q3)))) by RLVECT_1:def 7;

      

       A17: (((1 - r) * (((1 / 2) * (p1 + p2)) + |[ 0 , 1]|)) + (r * (((1 / 2) * (p2 + q3)) + |[ 0 , 1]|))) = ((((1 - r) * ((1 / 2) * (p1 + p2))) + ((1 - r) * |[ 0 , 1]|)) + (r * (((1 / 2) * (p2 + q3)) + |[ 0 , 1]|))) by RLVECT_1:def 5

      .= ((((1 - r) * ((1 / 2) * (p1 + p2))) + ((1 - r) * |[ 0 , 1]|)) + ((r * ((1 / 2) * (p2 + q3))) + (r * |[ 0 , 1]|))) by RLVECT_1:def 5

      .= (((r * ((1 / 2) * (p2 + q3))) + (((1 - r) * ((1 / 2) * (p1 + p2))) + ((1 - r) * |[ 0 , 1]|))) + (r * |[ 0 , 1]|)) by RLVECT_1:def 3

      .= ((((r * ((1 / 2) * (p2 + q3))) + ((1 - r) * ((1 / 2) * (p1 + p2)))) + ((1 - r) * |[ 0 , 1]|)) + (r * |[ 0 , 1]|)) by RLVECT_1:def 3

      .= (((r * ((1 / 2) * (p2 + q3))) + ((1 - r) * ((1 / 2) * (p1 + p2)))) + (((1 - r) * |[ 0 , 1]|) + (r * |[ 0 , 1]|))) by RLVECT_1:def 3

      .= (((r * ((1 / 2) * (p2 + q3))) + ((1 - r) * ((1 / 2) * (p1 + p2)))) + (((1 - r) + r) * |[ 0 , 1]|)) by RLVECT_1:def 6

      .= (p2 + |[ 0 , 1]|) by A16, RLVECT_1:def 8;

      r < 1 by A14, A9, XREAL_1: 189;

      then (p2 + |[ 0 , 1]|) in ( LSeg ((((1 / 2) * (p1 + p2)) + |[ 0 , 1]|),(((1 / 2) * (p2 + q3)) + |[ 0 , 1]|))) by A14, A9, A17;

      then

       A18: ( LSeg ((((1 / 2) * (p1 + p2)) + |[ 0 , 1]|),(((1 / 2) * (p2 + q3)) + |[ 0 , 1]|))) = (( LSeg ((((1 / 2) * (p1 + p2)) + |[ 0 , 1]|),(p2 + |[ 0 , 1]|))) \/ ( LSeg ((p2 + |[ 0 , 1]|),(((1 / 2) * (p2 + q3)) + |[ 0 , 1]|)))) by TOPREAL1: 5;

      ( LSeg ((((1 / 2) * (p1 + p2)) + |[ 0 , 1]|),(p2 + |[ 0 , 1]|))) c= (I1 \/ {(p2 + |[ 0 , 1]|)}) by A2, A12, Th55;

      hence thesis by A18, A8, A7, XBOOLE_1: 13;

    end;

    theorem :: GOBOARD6:74

    1 < ( len G) & 1 < ( width G) implies ( LSeg (((G * (1,1)) - |[1, 1]|),(((1 / 2) * ((G * (1,1)) + (G * (1,2)))) - |[1, 0 ]|))) c= ((( Int ( cell (G, 0 , 0 ))) \/ ( Int ( cell (G, 0 ,1)))) \/ {((G * (1,1)) - |[1, 0 ]|)})

    proof

      assume that

       A1: 1 < ( len G) and

       A2: 1 < ( width G);

      set q2 = (G * (1,1)), q3 = (G * (1,2)), r = (1 / (((1 / 2) * ((q3 `2 ) - (q2 `2 ))) + 1));

      

       A3: ( 0 + (1 + 1)) <= ( width G) by A2, NAT_1: 13;

      then

       A4: (q2 `1 ) = (q3 `1 ) by A1, GOBOARD5: 2;

      (q2 `2 ) < (q3 `2 ) by A1, A3, GOBOARD5: 4;

      then

       A5: ((q3 `2 ) - (q2 `2 )) > 0 by XREAL_1: 50;

      then 1 < (((1 / 2) * ((q3 `2 ) - (q2 `2 ))) + 1) by XREAL_1: 29, XREAL_1: 129;

      then

       A6: r < 1 by XREAL_1: 212;

      

       A7: ((((1 - r) * (q2 - |[1, 1]|)) + (r * (((1 / 2) * (q2 + q3)) - |[1, 0 ]|))) `1 ) = ((((1 - r) * (q2 - |[1, 1]|)) `1 ) + ((r * (((1 / 2) * (q2 + q3)) - |[1, 0 ]|)) `1 )) by Lm1

      .= (((1 - r) * ((q2 - |[1, 1]|) `1 )) + ((r * (((1 / 2) * (q2 + q3)) - |[1, 0 ]|)) `1 )) by Lm3

      .= (((1 - r) * ((q2 - |[1, 1]|) `1 )) + (r * ((((1 / 2) * (q2 + q3)) - |[1, 0 ]|) `1 ))) by Lm3

      .= (((1 - r) * ((q2 `1 ) - ( |[1, 1]| `1 ))) + (r * ((((1 / 2) * (q2 + q3)) - |[1, 0 ]|) `1 ))) by Lm2

      .= (((1 - r) * ((q2 `1 ) - ( |[1, 1]| `1 ))) + (r * ((((1 / 2) * (q2 + q3)) `1 ) - ( |[1, 0 ]| `1 )))) by Lm2

      .= (((1 - r) * ((q2 `1 ) - 1)) + (r * ((((1 / 2) * (q2 + q3)) `1 ) - ( |[1, 0 ]| `1 )))) by EUCLID: 52

      .= ((((1 - r) * (q2 `1 )) - ((1 - r) * 1)) + (r * ((((1 / 2) * (q2 + q3)) `1 ) - 1))) by EUCLID: 52

      .= ((((1 - r) * (q2 `1 )) + (r * (((1 / 2) * (q2 + q3)) `1 ))) - ((1 - r) + r))

      .= ((((1 - r) * (q2 `1 )) + (r * ((1 / 2) * ((q2 + q3) `1 )))) - 1) by Lm3

      .= ((((1 - r) * (q2 `1 )) + (r * ((1 / 2) * ((q2 `1 ) + (q2 `1 ))))) - 1) by A4, Lm1

      .= ((q2 `1 ) - ( |[1, 0 ]| `1 )) by EUCLID: 52

      .= ((q2 - |[1, 0 ]|) `1 ) by Lm2;

      

       A8: (((r * ((1 / 2) * (q3 `2 ))) - (r * ((1 / 2) * (q2 `2 )))) + r) = (r * (((1 / 2) * ((q3 `2 ) - (q2 `2 ))) + 1))

      .= 1 by A5, XCMPLX_1: 106;

      ((((1 - r) * (q2 - |[1, 1]|)) + (r * (((1 / 2) * (q2 + q3)) - |[1, 0 ]|))) `2 ) = ((((1 - r) * (q2 - |[1, 1]|)) `2 ) + ((r * (((1 / 2) * (q2 + q3)) - |[1, 0 ]|)) `2 )) by Lm1

      .= (((((1 - r) * q2) - ((1 - r) * |[1, 1]|)) `2 ) + ((r * (((1 / 2) * (q2 + q3)) - |[1, 0 ]|)) `2 )) by RLVECT_1: 34

      .= (((((1 - r) * q2) `2 ) - (((1 - r) * |[1, 1]|) `2 )) + ((r * (((1 / 2) * (q2 + q3)) - |[1, 0 ]|)) `2 )) by Lm2

      .= (((((1 - r) * q2) `2 ) - ((1 - r) * ( |[1, 1]| `2 ))) + ((r * (((1 / 2) * (q2 + q3)) - |[1, 0 ]|)) `2 )) by Lm3

      .= (((((1 - r) * q2) `2 ) - ((1 - r) * 1)) + ((r * (((1 / 2) * (q2 + q3)) - |[1, 0 ]|)) `2 )) by EUCLID: 52

      .= ((((1 - r) * (q2 `2 )) - ((1 - r) * 1)) + ((r * (((1 / 2) * (q2 + q3)) - |[1, 0 ]|)) `2 )) by Lm3

      .= ((((1 - r) * (q2 `2 )) - (1 - r)) + (r * ((((1 / 2) * (q2 + q3)) - |[1, 0 ]|) `2 ))) by Lm3

      .= ((((1 - r) * (q2 `2 )) - (1 - r)) + (r * ((((1 / 2) * (q2 + q3)) `2 ) - ( |[1, 0 ]| `2 )))) by Lm2

      .= ((((1 - r) * (q2 `2 )) - (1 - r)) + (r * ((((1 / 2) * (q2 + q3)) `2 ) - 0 ))) by EUCLID: 52

      .= ((((1 - r) * (q2 `2 )) - (1 - r)) + (r * ((1 / 2) * ((q2 + q3) `2 )))) by Lm3

      .= ((((1 - r) * (q2 `2 )) - (1 - r)) + (r * ((1 / 2) * ((q2 `2 ) + (q3 `2 ))))) by Lm1

      .= ((q2 `2 ) - 0 ) by A8

      .= ((q2 `2 ) - ( |[1, 0 ]| `2 )) by EUCLID: 52

      .= ((q2 - |[1, 0 ]|) `2 ) by Lm2;

      

      then (((1 - r) * (q2 - |[1, 1]|)) + (r * (((1 / 2) * (q2 + q3)) - |[1, 0 ]|))) = |[((q2 - |[1, 0 ]|) `1 ), ((q2 - |[1, 0 ]|) `2 )]| by A7, EUCLID: 53

      .= (q2 - |[1, 0 ]|) by EUCLID: 53;

      then (q2 - |[1, 0 ]|) in ( LSeg ((q2 - |[1, 1]|),(((1 / 2) * (q2 + q3)) - |[1, 0 ]|))) by A5, A6;

      then

       A9: ( LSeg ((q2 - |[1, 1]|),(((1 / 2) * (q2 + q3)) - |[1, 0 ]|))) = (( LSeg ((q2 - |[1, 1]|),(q2 - |[1, 0 ]|))) \/ ( LSeg ((q2 - |[1, 0 ]|),(((1 / 2) * (q2 + q3)) - |[1, 0 ]|)))) by TOPREAL1: 5;

      set I1 = ( Int ( cell (G, 0 , 0 ))), I2 = ( Int ( cell (G, 0 ,1)));

      (( 0 + 1) + 1) = ( 0 + (1 + 1));

      then

       A10: ( LSeg ((q2 - |[1, 0 ]|),(((1 / 2) * (q2 + q3)) - |[1, 0 ]|))) c= (I2 \/ {(q2 - |[1, 0 ]|)}) by A2, Th48;

      

       A11: ((I1 \/ I2) \/ {(q2 - |[1, 0 ]|)}) = (I1 \/ (I2 \/ ( {(q2 - |[1, 0 ]|)} \/ {(q2 - |[1, 0 ]|)}))) by XBOOLE_1: 4

      .= (I1 \/ ((I2 \/ {(q2 - |[1, 0 ]|)}) \/ {(q2 - |[1, 0 ]|)})) by XBOOLE_1: 4

      .= ((I1 \/ {(q2 - |[1, 0 ]|)}) \/ (I2 \/ {(q2 - |[1, 0 ]|)})) by XBOOLE_1: 4;

      ( LSeg ((q2 - |[1, 1]|),(q2 - |[1, 0 ]|))) c= (I1 \/ {(q2 - |[1, 0 ]|)}) by Th56;

      hence thesis by A9, A10, A11, XBOOLE_1: 13;

    end;

    theorem :: GOBOARD6:75

    1 < ( len G) & 1 < ( width G) implies ( LSeg (((G * (( len G),1)) + |[1, ( - 1)]|),(((1 / 2) * ((G * (( len G),1)) + (G * (( len G),2)))) + |[1, 0 ]|))) c= ((( Int ( cell (G,( len G), 0 ))) \/ ( Int ( cell (G,( len G),1)))) \/ {((G * (( len G),1)) + |[1, 0 ]|)})

    proof

      assume that

       A1: 1 < ( len G) and

       A2: 1 < ( width G);

      set q2 = (G * (( len G),1)), q3 = (G * (( len G),2)), r = (1 / (((1 / 2) * ((q3 `2 ) - (q2 `2 ))) + 1));

      

       A3: ( 0 + (1 + 1)) <= ( width G) by A2, NAT_1: 13;

      then

       A4: (q2 `1 ) = (q3 `1 ) by A1, GOBOARD5: 2;

      (q2 `2 ) < (q3 `2 ) by A1, A3, GOBOARD5: 4;

      then

       A5: ((q3 `2 ) - (q2 `2 )) > 0 by XREAL_1: 50;

      then 1 < (((1 / 2) * ((q3 `2 ) - (q2 `2 ))) + 1) by XREAL_1: 29, XREAL_1: 129;

      then

       A6: r < 1 by XREAL_1: 212;

      

       A7: ((((1 - r) * (q2 + |[1, ( - 1)]|)) + (r * (((1 / 2) * (q2 + q3)) + |[1, 0 ]|))) `1 ) = ((((1 - r) * (q2 + |[1, ( - 1)]|)) `1 ) + ((r * (((1 / 2) * (q2 + q3)) + |[1, 0 ]|)) `1 )) by Lm1

      .= (((1 - r) * ((q2 + |[1, ( - 1)]|) `1 )) + ((r * (((1 / 2) * (q2 + q3)) + |[1, 0 ]|)) `1 )) by Lm3

      .= (((1 - r) * ((q2 + |[1, ( - 1)]|) `1 )) + (r * ((((1 / 2) * (q2 + q3)) + |[1, 0 ]|) `1 ))) by Lm3

      .= (((1 - r) * ((q2 `1 ) + ( |[1, ( - 1)]| `1 ))) + (r * ((((1 / 2) * (q2 + q3)) + |[1, 0 ]|) `1 ))) by Lm1

      .= (((1 - r) * ((q2 `1 ) + ( |[1, ( - 1)]| `1 ))) + (r * ((((1 / 2) * (q2 + q3)) `1 ) + ( |[1, 0 ]| `1 )))) by Lm1

      .= (((1 - r) * ((q2 `1 ) + 1)) + (r * ((((1 / 2) * (q2 + q3)) `1 ) + ( |[1, 0 ]| `1 )))) by EUCLID: 52

      .= ((((1 - r) * (q2 `1 )) + ((1 - r) * 1)) + (r * ((((1 / 2) * (q2 + q3)) `1 ) + 1))) by EUCLID: 52

      .= ((((1 - r) * (q2 `1 )) + (r * (((1 / 2) * (q2 + q3)) `1 ))) + ((1 - r) + r))

      .= ((((1 - r) * (q2 `1 )) + (r * ((1 / 2) * ((q2 + q3) `1 )))) + 1) by Lm3

      .= ((((1 - r) * (q2 `1 )) + (r * ((1 / 2) * ((q2 `1 ) + (q2 `1 ))))) + 1) by A4, Lm1

      .= ((q2 `1 ) + ( |[1, 0 ]| `1 )) by EUCLID: 52

      .= ((q2 + |[1, 0 ]|) `1 ) by Lm1;

      

       A8: (((r * ((1 / 2) * (q3 `2 ))) - (r * ((1 / 2) * (q2 `2 )))) + r) = (r * (((1 / 2) * ((q3 `2 ) - (q2 `2 ))) + 1))

      .= 1 by A5, XCMPLX_1: 106;

      ((((1 - r) * (q2 + |[1, ( - 1)]|)) + (r * (((1 / 2) * (q2 + q3)) + |[1, 0 ]|))) `2 ) = ((((1 - r) * (q2 + |[1, ( - 1)]|)) `2 ) + ((r * (((1 / 2) * (q2 + q3)) + |[1, 0 ]|)) `2 )) by Lm1

      .= (((((1 - r) * q2) + ((1 - r) * |[1, ( - 1)]|)) `2 ) + ((r * (((1 / 2) * (q2 + q3)) + |[1, 0 ]|)) `2 )) by RLVECT_1:def 5

      .= (((((1 - r) * q2) `2 ) + (((1 - r) * |[1, ( - 1)]|) `2 )) + ((r * (((1 / 2) * (q2 + q3)) + |[1, 0 ]|)) `2 )) by Lm1

      .= (((((1 - r) * q2) `2 ) + ((1 - r) * ( |[1, ( - 1)]| `2 ))) + ((r * (((1 / 2) * (q2 + q3)) + |[1, 0 ]|)) `2 )) by Lm3

      .= (((((1 - r) * q2) `2 ) + ((1 - r) * ( - 1))) + ((r * (((1 / 2) * (q2 + q3)) + |[1, 0 ]|)) `2 )) by EUCLID: 52

      .= ((((1 - r) * (q2 `2 )) + ( - (1 - r))) + ((r * (((1 / 2) * (q2 + q3)) + |[1, 0 ]|)) `2 )) by Lm3

      .= ((((1 - r) * (q2 `2 )) - (1 - r)) + (r * ((((1 / 2) * (q2 + q3)) + |[1, 0 ]|) `2 ))) by Lm3

      .= ((((1 - r) * (q2 `2 )) - (1 - r)) + (r * ((((1 / 2) * (q2 + q3)) `2 ) + ( |[1, 0 ]| `2 )))) by Lm1

      .= ((((1 - r) * (q2 `2 )) - (1 - r)) + (r * ((((1 / 2) * (q2 + q3)) `2 ) + 0 ))) by EUCLID: 52

      .= ((((1 - r) * (q2 `2 )) - (1 - r)) + (r * ((1 / 2) * ((q2 + q3) `2 )))) by Lm3

      .= ((((1 - r) * (q2 `2 )) - (1 - r)) + (r * ((1 / 2) * ((q2 `2 ) + (q3 `2 ))))) by Lm1

      .= ((q2 `2 ) + 0 ) by A8

      .= ((q2 `2 ) + ( |[1, 0 ]| `2 )) by EUCLID: 52

      .= ((q2 + |[1, 0 ]|) `2 ) by Lm1;

      

      then (((1 - r) * (q2 + |[1, ( - 1)]|)) + (r * (((1 / 2) * (q2 + q3)) + |[1, 0 ]|))) = |[((q2 + |[1, 0 ]|) `1 ), ((q2 + |[1, 0 ]|) `2 )]| by A7, EUCLID: 53

      .= (q2 + |[1, 0 ]|) by EUCLID: 53;

      then (q2 + |[1, 0 ]|) in ( LSeg ((q2 + |[1, ( - 1)]|),(((1 / 2) * (q2 + q3)) + |[1, 0 ]|))) by A5, A6;

      then

       A9: ( LSeg ((q2 + |[1, ( - 1)]|),(((1 / 2) * (q2 + q3)) + |[1, 0 ]|))) = (( LSeg ((q2 + |[1, ( - 1)]|),(q2 + |[1, 0 ]|))) \/ ( LSeg ((q2 + |[1, 0 ]|),(((1 / 2) * (q2 + q3)) + |[1, 0 ]|)))) by TOPREAL1: 5;

      set I1 = ( Int ( cell (G,( len G), 0 ))), I2 = ( Int ( cell (G,( len G),1)));

      (( 0 + 1) + 1) = ( 0 + (1 + 1));

      then

       A10: ( LSeg ((q2 + |[1, 0 ]|),(((1 / 2) * (q2 + q3)) + |[1, 0 ]|))) c= (I2 \/ {(q2 + |[1, 0 ]|)}) by A2, Th50;

      

       A11: ((I1 \/ I2) \/ {(q2 + |[1, 0 ]|)}) = (I1 \/ (I2 \/ ( {(q2 + |[1, 0 ]|)} \/ {(q2 + |[1, 0 ]|)}))) by XBOOLE_1: 4

      .= (I1 \/ ((I2 \/ {(q2 + |[1, 0 ]|)}) \/ {(q2 + |[1, 0 ]|)})) by XBOOLE_1: 4

      .= ((I1 \/ {(q2 + |[1, 0 ]|)}) \/ (I2 \/ {(q2 + |[1, 0 ]|)})) by XBOOLE_1: 4;

      ( LSeg ((q2 + |[1, ( - 1)]|),(q2 + |[1, 0 ]|))) c= (I1 \/ {(q2 + |[1, 0 ]|)}) by Th57;

      hence thesis by A9, A10, A11, XBOOLE_1: 13;

    end;

    theorem :: GOBOARD6:76

    1 < ( len G) & 1 < ( width G) implies ( LSeg (((G * (1,( width G))) + |[( - 1), 1]|),(((1 / 2) * ((G * (1,( width G))) + (G * (1,(( width G) -' 1))))) - |[1, 0 ]|))) c= ((( Int ( cell (G, 0 ,( width G)))) \/ ( Int ( cell (G, 0 ,(( width G) -' 1))))) \/ {((G * (1,( width G))) - |[1, 0 ]|)})

    proof

      assume that

       A1: 1 < ( len G) and

       A2: 1 < ( width G);

      set q2 = (G * (1,( width G))), q3 = (G * (1,(( width G) -' 1))), r = (1 / (((1 / 2) * ((q2 `2 ) - (q3 `2 ))) + 1));

      

       A3: ((( width G) -' 1) + 1) = ( width G) by A2, XREAL_1: 235;

      then

       A4: (( width G) -' 1) >= 1 by A2, NAT_1: 13;

      

       A5: (( width G) -' 1) < ( width G) by A3, NAT_1: 13;

      then (q3 `2 ) < (q2 `2 ) by A1, A4, GOBOARD5: 4;

      then

       A6: ((q2 `2 ) - (q3 `2 )) > 0 by XREAL_1: 50;

      then 1 < (((1 / 2) * ((q2 `2 ) - (q3 `2 ))) + 1) by XREAL_1: 29, XREAL_1: 129;

      then

       A7: r < 1 by XREAL_1: 212;

      

       A8: (q2 `1 ) = ((G * (1,1)) `1 ) by A1, A2, GOBOARD5: 2

      .= (q3 `1 ) by A1, A4, A5, GOBOARD5: 2;

      

       A9: ((((1 - r) * (q2 + |[( - 1), 1]|)) + (r * (((1 / 2) * (q2 + q3)) - |[1, 0 ]|))) `1 ) = ((((1 - r) * (q2 + |[( - 1), 1]|)) `1 ) + ((r * (((1 / 2) * (q2 + q3)) - |[1, 0 ]|)) `1 )) by Lm1

      .= (((1 - r) * ((q2 + |[( - 1), 1]|) `1 )) + ((r * (((1 / 2) * (q2 + q3)) - |[1, 0 ]|)) `1 )) by Lm3

      .= (((1 - r) * ((q2 + |[( - 1), 1]|) `1 )) + (r * ((((1 / 2) * (q2 + q3)) - |[1, 0 ]|) `1 ))) by Lm3

      .= (((1 - r) * ((q2 `1 ) + ( |[( - 1), 1]| `1 ))) + (r * ((((1 / 2) * (q2 + q3)) - |[1, 0 ]|) `1 ))) by Lm1

      .= (((1 - r) * ((q2 `1 ) + ( |[( - 1), 1]| `1 ))) + (r * ((((1 / 2) * (q2 + q3)) `1 ) - ( |[1, 0 ]| `1 )))) by Lm2

      .= (((1 - r) * ((q2 `1 ) + ( - 1))) + (r * ((((1 / 2) * (q2 + q3)) `1 ) - ( |[1, 0 ]| `1 )))) by EUCLID: 52

      .= (((1 - r) * ((q2 `1 ) - 1)) + (r * ((((1 / 2) * (q2 + q3)) `1 ) - 1))) by EUCLID: 52

      .= ((((1 - r) * (q2 `1 )) + (r * (((1 / 2) * (q2 + q3)) `1 ))) - 1)

      .= ((((1 - r) * (q2 `1 )) + (r * ((1 / 2) * ((q2 + q3) `1 )))) - 1) by Lm3

      .= ((((1 - r) * (q2 `1 )) + (r * ((1 / 2) * ((q2 `1 ) + (q2 `1 ))))) - 1) by A8, Lm1

      .= ((q2 `1 ) - ( |[1, 0 ]| `1 )) by EUCLID: 52

      .= ((q2 - |[1, 0 ]|) `1 ) by Lm2;

      

       A10: (((r * ((1 / 2) * (q2 `2 ))) - (r * ((1 / 2) * (q3 `2 )))) + r) = (r * (((1 / 2) * ((q2 `2 ) - (q3 `2 ))) + 1))

      .= 1 by A6, XCMPLX_1: 106;

      ((((1 - r) * (q2 + |[( - 1), 1]|)) + (r * (((1 / 2) * (q2 + q3)) - |[1, 0 ]|))) `2 ) = ((((1 - r) * (q2 + |[( - 1), 1]|)) `2 ) + ((r * (((1 / 2) * (q2 + q3)) - |[1, 0 ]|)) `2 )) by Lm1

      .= (((((1 - r) * q2) + ((1 - r) * |[( - 1), 1]|)) `2 ) + ((r * (((1 / 2) * (q2 + q3)) - |[1, 0 ]|)) `2 )) by RLVECT_1:def 5

      .= (((((1 - r) * q2) `2 ) + (((1 - r) * |[( - 1), 1]|) `2 )) + ((r * (((1 / 2) * (q2 + q3)) - |[1, 0 ]|)) `2 )) by Lm1

      .= (((((1 - r) * q2) `2 ) + ((1 - r) * ( |[( - 1), 1]| `2 ))) + ((r * (((1 / 2) * (q2 + q3)) - |[1, 0 ]|)) `2 )) by Lm3

      .= (((((1 - r) * q2) `2 ) + ((1 - r) * 1)) + ((r * (((1 / 2) * (q2 + q3)) - |[1, 0 ]|)) `2 )) by EUCLID: 52

      .= ((((1 - r) * (q2 `2 )) + ((1 - r) * 1)) + ((r * (((1 / 2) * (q2 + q3)) - |[1, 0 ]|)) `2 )) by Lm3

      .= ((((1 - r) * (q2 `2 )) + (1 - r)) + (r * ((((1 / 2) * (q2 + q3)) - |[1, 0 ]|) `2 ))) by Lm3

      .= ((((1 - r) * (q2 `2 )) + (1 - r)) + (r * ((((1 / 2) * (q2 + q3)) `2 ) - ( |[1, 0 ]| `2 )))) by Lm2

      .= ((((1 - r) * (q2 `2 )) + (1 - r)) + (r * ((((1 / 2) * (q2 + q3)) `2 ) - 0 ))) by EUCLID: 52

      .= ((((1 - r) * (q2 `2 )) + (1 - r)) + (r * ((1 / 2) * ((q2 + q3) `2 )))) by Lm3

      .= ((((1 - r) * (q2 `2 )) + (1 - r)) + (r * ((1 / 2) * ((q2 `2 ) + (q3 `2 ))))) by Lm1

      .= ((q2 `2 ) - 0 ) by A10

      .= ((q2 `2 ) - ( |[1, 0 ]| `2 )) by EUCLID: 52

      .= ((q2 - |[1, 0 ]|) `2 ) by Lm2;

      

      then (((1 - r) * (q2 + |[( - 1), 1]|)) + (r * (((1 / 2) * (q2 + q3)) - |[1, 0 ]|))) = |[((q2 - |[1, 0 ]|) `1 ), ((q2 - |[1, 0 ]|) `2 )]| by A9, EUCLID: 53

      .= (q2 - |[1, 0 ]|) by EUCLID: 53;

      then (q2 - |[1, 0 ]|) in ( LSeg ((q2 + |[( - 1), 1]|),(((1 / 2) * (q2 + q3)) - |[1, 0 ]|))) by A6, A7;

      then

       A11: ( LSeg ((q2 + |[( - 1), 1]|),(((1 / 2) * (q2 + q3)) - |[1, 0 ]|))) = (( LSeg ((q2 + |[( - 1), 1]|),(q2 - |[1, 0 ]|))) \/ ( LSeg ((q2 - |[1, 0 ]|),(((1 / 2) * (q2 + q3)) - |[1, 0 ]|)))) by TOPREAL1: 5;

      set I1 = ( Int ( cell (G, 0 ,( width G)))), I2 = ( Int ( cell (G, 0 ,(( width G) -' 1))));

      

       A12: ((I1 \/ I2) \/ {(q2 - |[1, 0 ]|)}) = (I1 \/ (I2 \/ ( {(q2 - |[1, 0 ]|)} \/ {(q2 - |[1, 0 ]|)}))) by XBOOLE_1: 4

      .= (I1 \/ ((I2 \/ {(q2 - |[1, 0 ]|)}) \/ {(q2 - |[1, 0 ]|)})) by XBOOLE_1: 4

      .= ((I1 \/ {(q2 - |[1, 0 ]|)}) \/ (I2 \/ {(q2 - |[1, 0 ]|)})) by XBOOLE_1: 4;

      

       A13: ( LSeg ((q2 + |[( - 1), 1]|),(q2 - |[1, 0 ]|))) c= (I1 \/ {(q2 - |[1, 0 ]|)}) by Th58;

      ( LSeg ((q2 - |[1, 0 ]|),(((1 / 2) * (q2 + q3)) - |[1, 0 ]|))) c= (I2 \/ {(q2 - |[1, 0 ]|)}) by A3, A4, A5, Th49;

      hence thesis by A11, A13, A12, XBOOLE_1: 13;

    end;

    theorem :: GOBOARD6:77

    1 < ( len G) & 1 < ( width G) implies ( LSeg (((G * (( len G),( width G))) + |[1, 1]|),(((1 / 2) * ((G * (( len G),( width G))) + (G * (( len G),(( width G) -' 1))))) + |[1, 0 ]|))) c= ((( Int ( cell (G,( len G),( width G)))) \/ ( Int ( cell (G,( len G),(( width G) -' 1))))) \/ {((G * (( len G),( width G))) + |[1, 0 ]|)})

    proof

      assume that

       A1: 1 < ( len G) and

       A2: 1 < ( width G);

      set q2 = (G * (( len G),( width G))), q3 = (G * (( len G),(( width G) -' 1))), r = (1 / (((1 / 2) * ((q2 `2 ) - (q3 `2 ))) + 1));

      

       A3: ((( width G) -' 1) + 1) = ( width G) by A2, XREAL_1: 235;

      then

       A4: (( width G) -' 1) >= 1 by A2, NAT_1: 13;

      

       A5: (( width G) -' 1) < ( width G) by A3, NAT_1: 13;

      then (q3 `2 ) < (q2 `2 ) by A1, A4, GOBOARD5: 4;

      then

       A6: ((q2 `2 ) - (q3 `2 )) > 0 by XREAL_1: 50;

      then 1 < (((1 / 2) * ((q2 `2 ) - (q3 `2 ))) + 1) by XREAL_1: 29, XREAL_1: 129;

      then

       A7: r < 1 by XREAL_1: 212;

      

       A8: (q2 `1 ) = ((G * (( len G),1)) `1 ) by A1, A2, GOBOARD5: 2

      .= (q3 `1 ) by A1, A4, A5, GOBOARD5: 2;

      

       A9: ((((1 - r) * (q2 + |[1, 1]|)) + (r * (((1 / 2) * (q2 + q3)) + |[1, 0 ]|))) `1 ) = ((((1 - r) * (q2 + |[1, 1]|)) `1 ) + ((r * (((1 / 2) * (q2 + q3)) + |[1, 0 ]|)) `1 )) by Lm1

      .= (((1 - r) * ((q2 + |[1, 1]|) `1 )) + ((r * (((1 / 2) * (q2 + q3)) + |[1, 0 ]|)) `1 )) by Lm3

      .= (((1 - r) * ((q2 + |[1, 1]|) `1 )) + (r * ((((1 / 2) * (q2 + q3)) + |[1, 0 ]|) `1 ))) by Lm3

      .= (((1 - r) * ((q2 `1 ) + ( |[1, 1]| `1 ))) + (r * ((((1 / 2) * (q2 + q3)) + |[1, 0 ]|) `1 ))) by Lm1

      .= (((1 - r) * ((q2 `1 ) + ( |[1, 1]| `1 ))) + (r * ((((1 / 2) * (q2 + q3)) `1 ) + ( |[1, 0 ]| `1 )))) by Lm1

      .= (((1 - r) * ((q2 `1 ) + 1)) + (r * ((((1 / 2) * (q2 + q3)) `1 ) + ( |[1, 0 ]| `1 )))) by EUCLID: 52

      .= ((((1 - r) * (q2 `1 )) + ((1 - r) * 1)) + (r * ((((1 / 2) * (q2 + q3)) `1 ) + 1))) by EUCLID: 52

      .= ((((1 - r) * (q2 `1 )) + (r * (((1 / 2) * (q2 + q3)) `1 ))) + ((1 - r) + r))

      .= ((((1 - r) * (q2 `1 )) + (r * ((1 / 2) * ((q2 + q3) `1 )))) + 1) by Lm3

      .= ((((1 - r) * (q2 `1 )) + (r * ((1 / 2) * ((q2 `1 ) + (q2 `1 ))))) + 1) by A8, Lm1

      .= ((q2 `1 ) + ( |[1, 0 ]| `1 )) by EUCLID: 52

      .= ((q2 + |[1, 0 ]|) `1 ) by Lm1;

      

       A10: (((r * ((1 / 2) * (q2 `2 ))) - (r * ((1 / 2) * (q3 `2 )))) + r) = (r * (((1 / 2) * ((q2 `2 ) - (q3 `2 ))) + 1))

      .= 1 by A6, XCMPLX_1: 106;

      ((((1 - r) * (q2 + |[1, 1]|)) + (r * (((1 / 2) * (q2 + q3)) + |[1, 0 ]|))) `2 ) = ((((1 - r) * (q2 + |[1, 1]|)) `2 ) + ((r * (((1 / 2) * (q2 + q3)) + |[1, 0 ]|)) `2 )) by Lm1

      .= (((((1 - r) * q2) + ((1 - r) * |[1, 1]|)) `2 ) + ((r * (((1 / 2) * (q2 + q3)) + |[1, 0 ]|)) `2 )) by RLVECT_1:def 5

      .= (((((1 - r) * q2) `2 ) + (((1 - r) * |[1, 1]|) `2 )) + ((r * (((1 / 2) * (q2 + q3)) + |[1, 0 ]|)) `2 )) by Lm1

      .= (((((1 - r) * q2) `2 ) + ((1 - r) * ( |[1, 1]| `2 ))) + ((r * (((1 / 2) * (q2 + q3)) + |[1, 0 ]|)) `2 )) by Lm3

      .= (((((1 - r) * q2) `2 ) + ((1 - r) * 1)) + ((r * (((1 / 2) * (q2 + q3)) + |[1, 0 ]|)) `2 )) by EUCLID: 52

      .= ((((1 - r) * (q2 `2 )) + (1 - r)) + ((r * (((1 / 2) * (q2 + q3)) + |[1, 0 ]|)) `2 )) by Lm3

      .= ((((1 - r) * (q2 `2 )) + (1 - r)) + (r * ((((1 / 2) * (q2 + q3)) + |[1, 0 ]|) `2 ))) by Lm3

      .= ((((1 - r) * (q2 `2 )) + (1 - r)) + (r * ((((1 / 2) * (q2 + q3)) `2 ) + ( |[1, 0 ]| `2 )))) by Lm1

      .= ((((1 - r) * (q2 `2 )) + (1 - r)) + (r * ((((1 / 2) * (q2 + q3)) `2 ) + 0 ))) by EUCLID: 52

      .= ((((1 - r) * (q2 `2 )) + (1 - r)) + (r * ((1 / 2) * ((q2 + q3) `2 )))) by Lm3

      .= ((((1 - r) * (q2 `2 )) + (1 - r)) + (r * ((1 / 2) * ((q2 `2 ) + (q3 `2 ))))) by Lm1

      .= ((q2 `2 ) + 0 ) by A10

      .= ((q2 `2 ) + ( |[1, 0 ]| `2 )) by EUCLID: 52

      .= ((q2 + |[1, 0 ]|) `2 ) by Lm1;

      

      then (((1 - r) * (q2 + |[1, 1]|)) + (r * (((1 / 2) * (q2 + q3)) + |[1, 0 ]|))) = |[((q2 + |[1, 0 ]|) `1 ), ((q2 + |[1, 0 ]|) `2 )]| by A9, EUCLID: 53

      .= (q2 + |[1, 0 ]|) by EUCLID: 53;

      then (q2 + |[1, 0 ]|) in ( LSeg ((q2 + |[1, 1]|),(((1 / 2) * (q2 + q3)) + |[1, 0 ]|))) by A6, A7;

      then

       A11: ( LSeg ((q2 + |[1, 1]|),(((1 / 2) * (q2 + q3)) + |[1, 0 ]|))) = (( LSeg ((q2 + |[1, 1]|),(q2 + |[1, 0 ]|))) \/ ( LSeg ((q2 + |[1, 0 ]|),(((1 / 2) * (q2 + q3)) + |[1, 0 ]|)))) by TOPREAL1: 5;

      set I1 = ( Int ( cell (G,( len G),( width G)))), I2 = ( Int ( cell (G,( len G),(( width G) -' 1))));

      

       A12: ((I1 \/ I2) \/ {(q2 + |[1, 0 ]|)}) = (I1 \/ (I2 \/ ( {(q2 + |[1, 0 ]|)} \/ {(q2 + |[1, 0 ]|)}))) by XBOOLE_1: 4

      .= (I1 \/ ((I2 \/ {(q2 + |[1, 0 ]|)}) \/ {(q2 + |[1, 0 ]|)})) by XBOOLE_1: 4

      .= ((I1 \/ {(q2 + |[1, 0 ]|)}) \/ (I2 \/ {(q2 + |[1, 0 ]|)})) by XBOOLE_1: 4;

      

       A13: ( LSeg ((q2 + |[1, 1]|),(q2 + |[1, 0 ]|))) c= (I1 \/ {(q2 + |[1, 0 ]|)}) by Th59;

      ( LSeg ((q2 + |[1, 0 ]|),(((1 / 2) * (q2 + q3)) + |[1, 0 ]|))) c= (I2 \/ {(q2 + |[1, 0 ]|)}) by A3, A4, A5, Th51;

      hence thesis by A11, A13, A12, XBOOLE_1: 13;

    end;

    theorem :: GOBOARD6:78

    1 < ( width G) & 1 < ( len G) implies ( LSeg (((G * (1,1)) - |[1, 1]|),(((1 / 2) * ((G * (1,1)) + (G * (2,1)))) - |[ 0 , 1]|))) c= ((( Int ( cell (G, 0 , 0 ))) \/ ( Int ( cell (G,1, 0 )))) \/ {((G * (1,1)) - |[ 0 , 1]|)})

    proof

      assume that

       A1: 1 < ( width G) and

       A2: 1 < ( len G);

      set q2 = (G * (1,1)), q3 = (G * (2,1)), r = (1 / (((1 / 2) * ((q3 `1 ) - (q2 `1 ))) + 1));

      

       A3: ( 0 + (1 + 1)) <= ( len G) by A2, NAT_1: 13;

      then

       A4: (q2 `2 ) = (q3 `2 ) by A1, GOBOARD5: 1;

      (q2 `1 ) < (q3 `1 ) by A1, A3, GOBOARD5: 3;

      then

       A5: ((q3 `1 ) - (q2 `1 )) > 0 by XREAL_1: 50;

      then 1 < (((1 / 2) * ((q3 `1 ) - (q2 `1 ))) + 1) by XREAL_1: 29, XREAL_1: 129;

      then

       A6: r < 1 by XREAL_1: 212;

      

       A7: ((((1 - r) * (q2 - |[1, 1]|)) + (r * (((1 / 2) * (q2 + q3)) - |[ 0 , 1]|))) `2 ) = ((((1 - r) * (q2 - |[1, 1]|)) `2 ) + ((r * (((1 / 2) * (q2 + q3)) - |[ 0 , 1]|)) `2 )) by Lm1

      .= (((1 - r) * ((q2 - |[1, 1]|) `2 )) + ((r * (((1 / 2) * (q2 + q3)) - |[ 0 , 1]|)) `2 )) by Lm3

      .= (((1 - r) * ((q2 - |[1, 1]|) `2 )) + (r * ((((1 / 2) * (q2 + q3)) - |[ 0 , 1]|) `2 ))) by Lm3

      .= (((1 - r) * ((q2 `2 ) - ( |[1, 1]| `2 ))) + (r * ((((1 / 2) * (q2 + q3)) - |[ 0 , 1]|) `2 ))) by Lm2

      .= (((1 - r) * ((q2 `2 ) - ( |[1, 1]| `2 ))) + (r * ((((1 / 2) * (q2 + q3)) `2 ) - ( |[ 0 , 1]| `2 )))) by Lm2

      .= (((1 - r) * ((q2 `2 ) - 1)) + (r * ((((1 / 2) * (q2 + q3)) `2 ) - ( |[ 0 , 1]| `2 )))) by EUCLID: 52

      .= ((((1 - r) * (q2 `2 )) - ((1 - r) * 1)) + (r * ((((1 / 2) * (q2 + q3)) `2 ) - 1))) by EUCLID: 52

      .= ((((1 - r) * (q2 `2 )) + (r * (((1 / 2) * (q2 + q3)) `2 ))) - ((1 - r) + r))

      .= ((((1 - r) * (q2 `2 )) + (r * ((1 / 2) * ((q2 + q3) `2 )))) - 1) by Lm3

      .= ((((1 - r) * (q2 `2 )) + (r * ((1 / 2) * ((q2 `2 ) + (q2 `2 ))))) - 1) by A4, Lm1

      .= ((q2 `2 ) - ( |[ 0 , 1]| `2 )) by EUCLID: 52

      .= ((q2 - |[ 0 , 1]|) `2 ) by Lm2;

      

       A8: (((r * ((1 / 2) * (q3 `1 ))) - (r * ((1 / 2) * (q2 `1 )))) + r) = (r * (((1 / 2) * ((q3 `1 ) - (q2 `1 ))) + 1))

      .= 1 by A5, XCMPLX_1: 106;

      ((((1 - r) * (q2 - |[1, 1]|)) + (r * (((1 / 2) * (q2 + q3)) - |[ 0 , 1]|))) `1 ) = ((((1 - r) * (q2 - |[1, 1]|)) `1 ) + ((r * (((1 / 2) * (q2 + q3)) - |[ 0 , 1]|)) `1 )) by Lm1

      .= (((((1 - r) * q2) - ((1 - r) * |[1, 1]|)) `1 ) + ((r * (((1 / 2) * (q2 + q3)) - |[ 0 , 1]|)) `1 )) by RLVECT_1: 34

      .= (((((1 - r) * q2) `1 ) - (((1 - r) * |[1, 1]|) `1 )) + ((r * (((1 / 2) * (q2 + q3)) - |[ 0 , 1]|)) `1 )) by Lm2

      .= (((((1 - r) * q2) `1 ) - ((1 - r) * ( |[1, 1]| `1 ))) + ((r * (((1 / 2) * (q2 + q3)) - |[ 0 , 1]|)) `1 )) by Lm3

      .= (((((1 - r) * q2) `1 ) - ((1 - r) * 1)) + ((r * (((1 / 2) * (q2 + q3)) - |[ 0 , 1]|)) `1 )) by EUCLID: 52

      .= ((((1 - r) * (q2 `1 )) - ((1 - r) * 1)) + ((r * (((1 / 2) * (q2 + q3)) - |[ 0 , 1]|)) `1 )) by Lm3

      .= ((((1 - r) * (q2 `1 )) - (1 - r)) + (r * ((((1 / 2) * (q2 + q3)) - |[ 0 , 1]|) `1 ))) by Lm3

      .= ((((1 - r) * (q2 `1 )) - (1 - r)) + (r * ((((1 / 2) * (q2 + q3)) `1 ) - ( |[ 0 , 1]| `1 )))) by Lm2

      .= ((((1 - r) * (q2 `1 )) - (1 - r)) + (r * ((((1 / 2) * (q2 + q3)) `1 ) - 0 ))) by EUCLID: 52

      .= ((((1 - r) * (q2 `1 )) - (1 - r)) + (r * ((1 / 2) * ((q2 + q3) `1 )))) by Lm3

      .= ((((1 - r) * (q2 `1 )) - (1 - r)) + (r * ((1 / 2) * ((q2 `1 ) + (q3 `1 ))))) by Lm1

      .= ((q2 `1 ) - 0 ) by A8

      .= ((q2 `1 ) - ( |[ 0 , 1]| `1 )) by EUCLID: 52

      .= ((q2 - |[ 0 , 1]|) `1 ) by Lm2;

      

      then (((1 - r) * (q2 - |[1, 1]|)) + (r * (((1 / 2) * (q2 + q3)) - |[ 0 , 1]|))) = |[((q2 - |[ 0 , 1]|) `1 ), ((q2 - |[ 0 , 1]|) `2 )]| by A7, EUCLID: 53

      .= (q2 - |[ 0 , 1]|) by EUCLID: 53;

      then (q2 - |[ 0 , 1]|) in ( LSeg ((q2 - |[1, 1]|),(((1 / 2) * (q2 + q3)) - |[ 0 , 1]|))) by A5, A6;

      then

       A9: ( LSeg ((q2 - |[1, 1]|),(((1 / 2) * (q2 + q3)) - |[ 0 , 1]|))) = (( LSeg ((q2 - |[1, 1]|),(q2 - |[ 0 , 1]|))) \/ ( LSeg ((q2 - |[ 0 , 1]|),(((1 / 2) * (q2 + q3)) - |[ 0 , 1]|)))) by TOPREAL1: 5;

      set I1 = ( Int ( cell (G, 0 , 0 ))), I2 = ( Int ( cell (G,1, 0 )));

      (( 0 + 1) + 1) = ( 0 + (1 + 1));

      then

       A10: ( LSeg ((q2 - |[ 0 , 1]|),(((1 / 2) * (q2 + q3)) - |[ 0 , 1]|))) c= (I2 \/ {(q2 - |[ 0 , 1]|)}) by A2, Th52;

      

       A11: ((I1 \/ I2) \/ {(q2 - |[ 0 , 1]|)}) = (I1 \/ (I2 \/ ( {(q2 - |[ 0 , 1]|)} \/ {(q2 - |[ 0 , 1]|)}))) by XBOOLE_1: 4

      .= (I1 \/ ((I2 \/ {(q2 - |[ 0 , 1]|)}) \/ {(q2 - |[ 0 , 1]|)})) by XBOOLE_1: 4

      .= ((I1 \/ {(q2 - |[ 0 , 1]|)}) \/ (I2 \/ {(q2 - |[ 0 , 1]|)})) by XBOOLE_1: 4;

      ( LSeg ((q2 - |[1, 1]|),(q2 - |[ 0 , 1]|))) c= (I1 \/ {(q2 - |[ 0 , 1]|)}) by Th60;

      hence thesis by A9, A10, A11, XBOOLE_1: 13;

    end;

    theorem :: GOBOARD6:79

    1 < ( width G) & 1 < ( len G) implies ( LSeg (((G * (1,( width G))) + |[( - 1), 1]|),(((1 / 2) * ((G * (1,( width G))) + (G * (2,( width G))))) + |[ 0 , 1]|))) c= ((( Int ( cell (G, 0 ,( width G)))) \/ ( Int ( cell (G,1,( width G))))) \/ {((G * (1,( width G))) + |[ 0 , 1]|)})

    proof

      assume that

       A1: 1 < ( width G) and

       A2: 1 < ( len G);

      set q2 = (G * (1,( width G))), q3 = (G * (2,( width G))), r = (1 / (((1 / 2) * ((q3 `1 ) - (q2 `1 ))) + 1));

      

       A3: ( 0 + (1 + 1)) <= ( len G) by A2, NAT_1: 13;

      then

       A4: (q2 `2 ) = (q3 `2 ) by A1, GOBOARD5: 1;

      (q2 `1 ) < (q3 `1 ) by A1, A3, GOBOARD5: 3;

      then

       A5: ((q3 `1 ) - (q2 `1 )) > 0 by XREAL_1: 50;

      then 1 < (((1 / 2) * ((q3 `1 ) - (q2 `1 ))) + 1) by XREAL_1: 29, XREAL_1: 129;

      then

       A6: r < 1 by XREAL_1: 212;

      

       A7: ((((1 - r) * (q2 + |[( - 1), 1]|)) + (r * (((1 / 2) * (q2 + q3)) + |[ 0 , 1]|))) `2 ) = ((((1 - r) * (q2 + |[( - 1), 1]|)) `2 ) + ((r * (((1 / 2) * (q2 + q3)) + |[ 0 , 1]|)) `2 )) by Lm1

      .= (((1 - r) * ((q2 + |[( - 1), 1]|) `2 )) + ((r * (((1 / 2) * (q2 + q3)) + |[ 0 , 1]|)) `2 )) by Lm3

      .= (((1 - r) * ((q2 + |[( - 1), 1]|) `2 )) + (r * ((((1 / 2) * (q2 + q3)) + |[ 0 , 1]|) `2 ))) by Lm3

      .= (((1 - r) * ((q2 `2 ) + ( |[( - 1), 1]| `2 ))) + (r * ((((1 / 2) * (q2 + q3)) + |[ 0 , 1]|) `2 ))) by Lm1

      .= (((1 - r) * ((q2 `2 ) + ( |[( - 1), 1]| `2 ))) + (r * ((((1 / 2) * (q2 + q3)) `2 ) + ( |[ 0 , 1]| `2 )))) by Lm1

      .= (((1 - r) * ((q2 `2 ) + 1)) + (r * ((((1 / 2) * (q2 + q3)) `2 ) + ( |[ 0 , 1]| `2 )))) by EUCLID: 52

      .= ((((1 - r) * (q2 `2 )) + ((1 - r) * 1)) + (r * ((((1 / 2) * (q2 + q3)) `2 ) + 1))) by EUCLID: 52

      .= ((((1 - r) * (q2 `2 )) + (r * (((1 / 2) * (q2 + q3)) `2 ))) + ((1 - r) + r))

      .= ((((1 - r) * (q2 `2 )) + (r * ((1 / 2) * ((q2 + q3) `2 )))) + 1) by Lm3

      .= ((((1 - r) * (q2 `2 )) + (r * ((1 / 2) * ((q2 `2 ) + (q2 `2 ))))) + 1) by A4, Lm1

      .= ((q2 `2 ) + ( |[ 0 , 1]| `2 )) by EUCLID: 52

      .= ((q2 + |[ 0 , 1]|) `2 ) by Lm1;

      

       A8: (((r * ((1 / 2) * (q3 `1 ))) - (r * ((1 / 2) * (q2 `1 )))) + r) = (r * (((1 / 2) * ((q3 `1 ) - (q2 `1 ))) + 1))

      .= 1 by A5, XCMPLX_1: 106;

      ((((1 - r) * (q2 + |[( - 1), 1]|)) + (r * (((1 / 2) * (q2 + q3)) + |[ 0 , 1]|))) `1 ) = ((((1 - r) * (q2 + |[( - 1), 1]|)) `1 ) + ((r * (((1 / 2) * (q2 + q3)) + |[ 0 , 1]|)) `1 )) by Lm1

      .= (((((1 - r) * q2) + ((1 - r) * |[( - 1), 1]|)) `1 ) + ((r * (((1 / 2) * (q2 + q3)) + |[ 0 , 1]|)) `1 )) by RLVECT_1:def 5

      .= (((((1 - r) * q2) `1 ) + (((1 - r) * |[( - 1), 1]|) `1 )) + ((r * (((1 / 2) * (q2 + q3)) + |[ 0 , 1]|)) `1 )) by Lm1

      .= (((((1 - r) * q2) `1 ) + ((1 - r) * ( |[( - 1), 1]| `1 ))) + ((r * (((1 / 2) * (q2 + q3)) + |[ 0 , 1]|)) `1 )) by Lm3

      .= (((((1 - r) * q2) `1 ) + ((1 - r) * ( - 1))) + ((r * (((1 / 2) * (q2 + q3)) + |[ 0 , 1]|)) `1 )) by EUCLID: 52

      .= (((((1 - r) * q2) `1 ) - (1 - r)) + ((r * (((1 / 2) * (q2 + q3)) + |[ 0 , 1]|)) `1 ))

      .= ((((1 - r) * (q2 `1 )) - (1 - r)) + ((r * (((1 / 2) * (q2 + q3)) + |[ 0 , 1]|)) `1 )) by Lm3

      .= ((((1 - r) * (q2 `1 )) - (1 - r)) + (r * ((((1 / 2) * (q2 + q3)) + |[ 0 , 1]|) `1 ))) by Lm3

      .= ((((1 - r) * (q2 `1 )) - (1 - r)) + (r * ((((1 / 2) * (q2 + q3)) `1 ) + ( |[ 0 , 1]| `1 )))) by Lm1

      .= ((((1 - r) * (q2 `1 )) - (1 - r)) + (r * ((((1 / 2) * (q2 + q3)) `1 ) + 0 ))) by EUCLID: 52

      .= ((((1 - r) * (q2 `1 )) - (1 - r)) + (r * ((1 / 2) * ((q2 + q3) `1 )))) by Lm3

      .= ((((1 - r) * (q2 `1 )) - (1 - r)) + (r * ((1 / 2) * ((q2 `1 ) + (q3 `1 ))))) by Lm1

      .= ((q2 `1 ) + 0 ) by A8

      .= ((q2 `1 ) + ( |[ 0 , 1]| `1 )) by EUCLID: 52

      .= ((q2 + |[ 0 , 1]|) `1 ) by Lm1;

      

      then (((1 - r) * (q2 + |[( - 1), 1]|)) + (r * (((1 / 2) * (q2 + q3)) + |[ 0 , 1]|))) = |[((q2 + |[ 0 , 1]|) `1 ), ((q2 + |[ 0 , 1]|) `2 )]| by A7, EUCLID: 53

      .= (q2 + |[ 0 , 1]|) by EUCLID: 53;

      then (q2 + |[ 0 , 1]|) in ( LSeg ((q2 + |[( - 1), 1]|),(((1 / 2) * (q2 + q3)) + |[ 0 , 1]|))) by A5, A6;

      then

       A9: ( LSeg ((q2 + |[( - 1), 1]|),(((1 / 2) * (q2 + q3)) + |[ 0 , 1]|))) = (( LSeg ((q2 + |[( - 1), 1]|),(q2 + |[ 0 , 1]|))) \/ ( LSeg ((q2 + |[ 0 , 1]|),(((1 / 2) * (q2 + q3)) + |[ 0 , 1]|)))) by TOPREAL1: 5;

      set I1 = ( Int ( cell (G, 0 ,( width G)))), I2 = ( Int ( cell (G,1,( width G))));

      (( 0 + 1) + 1) = ( 0 + (1 + 1));

      then

       A10: ( LSeg ((q2 + |[ 0 , 1]|),(((1 / 2) * (q2 + q3)) + |[ 0 , 1]|))) c= (I2 \/ {(q2 + |[ 0 , 1]|)}) by A2, Th54;

      

       A11: ((I1 \/ I2) \/ {(q2 + |[ 0 , 1]|)}) = (I1 \/ (I2 \/ ( {(q2 + |[ 0 , 1]|)} \/ {(q2 + |[ 0 , 1]|)}))) by XBOOLE_1: 4

      .= (I1 \/ ((I2 \/ {(q2 + |[ 0 , 1]|)}) \/ {(q2 + |[ 0 , 1]|)})) by XBOOLE_1: 4

      .= ((I1 \/ {(q2 + |[ 0 , 1]|)}) \/ (I2 \/ {(q2 + |[ 0 , 1]|)})) by XBOOLE_1: 4;

      ( LSeg ((q2 + |[( - 1), 1]|),(q2 + |[ 0 , 1]|))) c= (I1 \/ {(q2 + |[ 0 , 1]|)}) by Th62;

      hence thesis by A9, A10, A11, XBOOLE_1: 13;

    end;

    theorem :: GOBOARD6:80

    1 < ( width G) & 1 < ( len G) implies ( LSeg (((G * (( len G),1)) + |[1, ( - 1)]|),(((1 / 2) * ((G * (( len G),1)) + (G * ((( len G) -' 1),1)))) - |[ 0 , 1]|))) c= ((( Int ( cell (G,( len G), 0 ))) \/ ( Int ( cell (G,(( len G) -' 1), 0 )))) \/ {((G * (( len G),1)) - |[ 0 , 1]|)})

    proof

      assume that

       A1: 1 < ( width G) and

       A2: 1 < ( len G);

      set q2 = (G * (( len G),1)), q3 = (G * ((( len G) -' 1),1)), r = (1 / (((1 / 2) * ((q2 `1 ) - (q3 `1 ))) + 1));

      

       A3: ((( len G) -' 1) + 1) = ( len G) by A2, XREAL_1: 235;

      then

       A4: (( len G) -' 1) >= 1 by A2, NAT_1: 13;

      

       A5: (( len G) -' 1) < ( len G) by A3, NAT_1: 13;

      then (q3 `1 ) < (q2 `1 ) by A1, A4, GOBOARD5: 3;

      then

       A6: ((q2 `1 ) - (q3 `1 )) > 0 by XREAL_1: 50;

      then 1 < (((1 / 2) * ((q2 `1 ) - (q3 `1 ))) + 1) by XREAL_1: 29, XREAL_1: 129;

      then

       A7: r < 1 by XREAL_1: 212;

      

       A8: (q2 `2 ) = ((G * (1,1)) `2 ) by A1, A2, GOBOARD5: 1

      .= (q3 `2 ) by A1, A4, A5, GOBOARD5: 1;

      

       A9: ((((1 - r) * (q2 + |[1, ( - 1)]|)) + (r * (((1 / 2) * (q2 + q3)) - |[ 0 , 1]|))) `2 ) = ((((1 - r) * (q2 + |[1, ( - 1)]|)) `2 ) + ((r * (((1 / 2) * (q2 + q3)) - |[ 0 , 1]|)) `2 )) by Lm1

      .= (((1 - r) * ((q2 + |[1, ( - 1)]|) `2 )) + ((r * (((1 / 2) * (q2 + q3)) - |[ 0 , 1]|)) `2 )) by Lm3

      .= (((1 - r) * ((q2 + |[1, ( - 1)]|) `2 )) + (r * ((((1 / 2) * (q2 + q3)) - |[ 0 , 1]|) `2 ))) by Lm3

      .= (((1 - r) * ((q2 `2 ) + ( |[1, ( - 1)]| `2 ))) + (r * ((((1 / 2) * (q2 + q3)) - |[ 0 , 1]|) `2 ))) by Lm1

      .= (((1 - r) * ((q2 `2 ) + ( |[1, ( - 1)]| `2 ))) + (r * ((((1 / 2) * (q2 + q3)) `2 ) - ( |[ 0 , 1]| `2 )))) by Lm2

      .= (((1 - r) * ((q2 `2 ) + ( - 1))) + (r * ((((1 / 2) * (q2 + q3)) `2 ) - ( |[ 0 , 1]| `2 )))) by EUCLID: 52

      .= (((1 - r) * ((q2 `2 ) - 1)) + (r * ((((1 / 2) * (q2 + q3)) `2 ) - 1))) by EUCLID: 52

      .= ((((1 - r) * (q2 `2 )) + (r * (((1 / 2) * (q2 + q3)) `2 ))) - 1)

      .= ((((1 - r) * (q2 `2 )) + (r * ((1 / 2) * ((q2 + q3) `2 )))) - 1) by Lm3

      .= ((((1 - r) * (q2 `2 )) + (r * ((1 / 2) * ((q2 `2 ) + (q2 `2 ))))) - 1) by A8, Lm1

      .= ((q2 `2 ) - ( |[ 0 , 1]| `2 )) by EUCLID: 52

      .= ((q2 - |[ 0 , 1]|) `2 ) by Lm2;

      

       A10: (((r * ((1 / 2) * (q2 `1 ))) - (r * ((1 / 2) * (q3 `1 )))) + r) = (r * (((1 / 2) * ((q2 `1 ) - (q3 `1 ))) + 1))

      .= 1 by A6, XCMPLX_1: 106;

      ((((1 - r) * (q2 + |[1, ( - 1)]|)) + (r * (((1 / 2) * (q2 + q3)) - |[ 0 , 1]|))) `1 ) = ((((1 - r) * (q2 + |[1, ( - 1)]|)) `1 ) + ((r * (((1 / 2) * (q2 + q3)) - |[ 0 , 1]|)) `1 )) by Lm1

      .= (((((1 - r) * q2) + ((1 - r) * |[1, ( - 1)]|)) `1 ) + ((r * (((1 / 2) * (q2 + q3)) - |[ 0 , 1]|)) `1 )) by RLVECT_1:def 5

      .= (((((1 - r) * q2) `1 ) + (((1 - r) * |[1, ( - 1)]|) `1 )) + ((r * (((1 / 2) * (q2 + q3)) - |[ 0 , 1]|)) `1 )) by Lm1

      .= (((((1 - r) * q2) `1 ) + ((1 - r) * ( |[1, ( - 1)]| `1 ))) + ((r * (((1 / 2) * (q2 + q3)) - |[ 0 , 1]|)) `1 )) by Lm3

      .= (((((1 - r) * q2) `1 ) + ((1 - r) * 1)) + ((r * (((1 / 2) * (q2 + q3)) - |[ 0 , 1]|)) `1 )) by EUCLID: 52

      .= ((((1 - r) * (q2 `1 )) + ((1 - r) * 1)) + ((r * (((1 / 2) * (q2 + q3)) - |[ 0 , 1]|)) `1 )) by Lm3

      .= ((((1 - r) * (q2 `1 )) + (1 - r)) + (r * ((((1 / 2) * (q2 + q3)) - |[ 0 , 1]|) `1 ))) by Lm3

      .= ((((1 - r) * (q2 `1 )) + (1 - r)) + (r * ((((1 / 2) * (q2 + q3)) `1 ) - ( |[ 0 , 1]| `1 )))) by Lm2

      .= ((((1 - r) * (q2 `1 )) + (1 - r)) + (r * ((((1 / 2) * (q2 + q3)) `1 ) - 0 ))) by EUCLID: 52

      .= ((((1 - r) * (q2 `1 )) + (1 - r)) + (r * ((1 / 2) * ((q2 + q3) `1 )))) by Lm3

      .= ((((1 - r) * (q2 `1 )) + (1 - r)) + (r * ((1 / 2) * ((q2 `1 ) + (q3 `1 ))))) by Lm1

      .= ((q2 `1 ) - 0 ) by A10

      .= ((q2 `1 ) - ( |[ 0 , 1]| `1 )) by EUCLID: 52

      .= ((q2 - |[ 0 , 1]|) `1 ) by Lm2;

      

      then (((1 - r) * (q2 + |[1, ( - 1)]|)) + (r * (((1 / 2) * (q2 + q3)) - |[ 0 , 1]|))) = |[((q2 - |[ 0 , 1]|) `1 ), ((q2 - |[ 0 , 1]|) `2 )]| by A9, EUCLID: 53

      .= (q2 - |[ 0 , 1]|) by EUCLID: 53;

      then (q2 - |[ 0 , 1]|) in ( LSeg ((q2 + |[1, ( - 1)]|),(((1 / 2) * (q2 + q3)) - |[ 0 , 1]|))) by A6, A7;

      then

       A11: ( LSeg ((q2 + |[1, ( - 1)]|),(((1 / 2) * (q2 + q3)) - |[ 0 , 1]|))) = (( LSeg ((q2 + |[1, ( - 1)]|),(q2 - |[ 0 , 1]|))) \/ ( LSeg ((q2 - |[ 0 , 1]|),(((1 / 2) * (q2 + q3)) - |[ 0 , 1]|)))) by TOPREAL1: 5;

      set I1 = ( Int ( cell (G,( len G), 0 ))), I2 = ( Int ( cell (G,(( len G) -' 1), 0 )));

      

       A12: ((I1 \/ I2) \/ {(q2 - |[ 0 , 1]|)}) = (I1 \/ (I2 \/ ( {(q2 - |[ 0 , 1]|)} \/ {(q2 - |[ 0 , 1]|)}))) by XBOOLE_1: 4

      .= (I1 \/ ((I2 \/ {(q2 - |[ 0 , 1]|)}) \/ {(q2 - |[ 0 , 1]|)})) by XBOOLE_1: 4

      .= ((I1 \/ {(q2 - |[ 0 , 1]|)}) \/ (I2 \/ {(q2 - |[ 0 , 1]|)})) by XBOOLE_1: 4;

      

       A13: ( LSeg ((q2 + |[1, ( - 1)]|),(q2 - |[ 0 , 1]|))) c= (I1 \/ {(q2 - |[ 0 , 1]|)}) by Th61;

      ( LSeg ((q2 - |[ 0 , 1]|),(((1 / 2) * (q2 + q3)) - |[ 0 , 1]|))) c= (I2 \/ {(q2 - |[ 0 , 1]|)}) by A3, A4, A5, Th53;

      hence thesis by A11, A13, A12, XBOOLE_1: 13;

    end;

    theorem :: GOBOARD6:81

    1 < ( width G) & 1 < ( len G) implies ( LSeg (((G * (( len G),( width G))) + |[1, 1]|),(((1 / 2) * ((G * (( len G),( width G))) + (G * ((( len G) -' 1),( width G))))) + |[ 0 , 1]|))) c= ((( Int ( cell (G,( len G),( width G)))) \/ ( Int ( cell (G,(( len G) -' 1),( width G))))) \/ {((G * (( len G),( width G))) + |[ 0 , 1]|)})

    proof

      assume that

       A1: 1 < ( width G) and

       A2: 1 < ( len G);

      set q2 = (G * (( len G),( width G))), q3 = (G * ((( len G) -' 1),( width G))), r = (1 / (((1 / 2) * ((q2 `1 ) - (q3 `1 ))) + 1));

      

       A3: ((( len G) -' 1) + 1) = ( len G) by A2, XREAL_1: 235;

      then

       A4: (( len G) -' 1) >= 1 by A2, NAT_1: 13;

      

       A5: (( len G) -' 1) < ( len G) by A3, NAT_1: 13;

      then (q3 `1 ) < (q2 `1 ) by A1, A4, GOBOARD5: 3;

      then

       A6: ((q2 `1 ) - (q3 `1 )) > 0 by XREAL_1: 50;

      then 1 < (((1 / 2) * ((q2 `1 ) - (q3 `1 ))) + 1) by XREAL_1: 29, XREAL_1: 129;

      then

       A7: r < 1 by XREAL_1: 212;

      

       A8: (q2 `2 ) = ((G * (1,( width G))) `2 ) by A1, A2, GOBOARD5: 1

      .= (q3 `2 ) by A1, A4, A5, GOBOARD5: 1;

      

       A9: ((((1 - r) * (q2 + |[1, 1]|)) + (r * (((1 / 2) * (q2 + q3)) + |[ 0 , 1]|))) `2 ) = ((((1 - r) * (q2 + |[1, 1]|)) `2 ) + ((r * (((1 / 2) * (q2 + q3)) + |[ 0 , 1]|)) `2 )) by Lm1

      .= (((1 - r) * ((q2 + |[1, 1]|) `2 )) + ((r * (((1 / 2) * (q2 + q3)) + |[ 0 , 1]|)) `2 )) by Lm3

      .= (((1 - r) * ((q2 + |[1, 1]|) `2 )) + (r * ((((1 / 2) * (q2 + q3)) + |[ 0 , 1]|) `2 ))) by Lm3

      .= (((1 - r) * ((q2 `2 ) + ( |[1, 1]| `2 ))) + (r * ((((1 / 2) * (q2 + q3)) + |[ 0 , 1]|) `2 ))) by Lm1

      .= (((1 - r) * ((q2 `2 ) + ( |[1, 1]| `2 ))) + (r * ((((1 / 2) * (q2 + q3)) `2 ) + ( |[ 0 , 1]| `2 )))) by Lm1

      .= (((1 - r) * ((q2 `2 ) + 1)) + (r * ((((1 / 2) * (q2 + q3)) `2 ) + ( |[ 0 , 1]| `2 )))) by EUCLID: 52

      .= ((((1 - r) * (q2 `2 )) + ((1 - r) * 1)) + (r * ((((1 / 2) * (q2 + q3)) `2 ) + 1))) by EUCLID: 52

      .= ((((1 - r) * (q2 `2 )) + (r * (((1 / 2) * (q2 + q3)) `2 ))) + ((1 - r) + r))

      .= ((((1 - r) * (q2 `2 )) + (r * ((1 / 2) * ((q2 + q3) `2 )))) + 1) by Lm3

      .= ((((1 - r) * (q2 `2 )) + (r * ((1 / 2) * ((q2 `2 ) + (q2 `2 ))))) + 1) by A8, Lm1

      .= ((q2 `2 ) + ( |[ 0 , 1]| `2 )) by EUCLID: 52

      .= ((q2 + |[ 0 , 1]|) `2 ) by Lm1;

      

       A10: (((r * ((1 / 2) * (q2 `1 ))) - (r * ((1 / 2) * (q3 `1 )))) + r) = (r * (((1 / 2) * ((q2 `1 ) - (q3 `1 ))) + 1))

      .= 1 by A6, XCMPLX_1: 106;

      ((((1 - r) * (q2 + |[1, 1]|)) + (r * (((1 / 2) * (q2 + q3)) + |[ 0 , 1]|))) `1 ) = ((((1 - r) * (q2 + |[1, 1]|)) `1 ) + ((r * (((1 / 2) * (q2 + q3)) + |[ 0 , 1]|)) `1 )) by Lm1

      .= (((((1 - r) * q2) + ((1 - r) * |[1, 1]|)) `1 ) + ((r * (((1 / 2) * (q2 + q3)) + |[ 0 , 1]|)) `1 )) by RLVECT_1:def 5

      .= (((((1 - r) * q2) `1 ) + (((1 - r) * |[1, 1]|) `1 )) + ((r * (((1 / 2) * (q2 + q3)) + |[ 0 , 1]|)) `1 )) by Lm1

      .= (((((1 - r) * q2) `1 ) + ((1 - r) * ( |[1, 1]| `1 ))) + ((r * (((1 / 2) * (q2 + q3)) + |[ 0 , 1]|)) `1 )) by Lm3

      .= (((((1 - r) * q2) `1 ) + ((1 - r) * 1)) + ((r * (((1 / 2) * (q2 + q3)) + |[ 0 , 1]|)) `1 )) by EUCLID: 52

      .= ((((1 - r) * (q2 `1 )) + (1 - r)) + ((r * (((1 / 2) * (q2 + q3)) + |[ 0 , 1]|)) `1 )) by Lm3

      .= ((((1 - r) * (q2 `1 )) + (1 - r)) + (r * ((((1 / 2) * (q2 + q3)) + |[ 0 , 1]|) `1 ))) by Lm3

      .= ((((1 - r) * (q2 `1 )) + (1 - r)) + (r * ((((1 / 2) * (q2 + q3)) `1 ) + ( |[ 0 , 1]| `1 )))) by Lm1

      .= ((((1 - r) * (q2 `1 )) + (1 - r)) + (r * ((((1 / 2) * (q2 + q3)) `1 ) + 0 ))) by EUCLID: 52

      .= ((((1 - r) * (q2 `1 )) + (1 - r)) + (r * ((1 / 2) * ((q2 + q3) `1 )))) by Lm3

      .= ((((1 - r) * (q2 `1 )) + (1 - r)) + (r * ((1 / 2) * ((q2 `1 ) + (q3 `1 ))))) by Lm1

      .= ((q2 `1 ) + 0 ) by A10

      .= ((q2 `1 ) + ( |[ 0 , 1]| `1 )) by EUCLID: 52

      .= ((q2 + |[ 0 , 1]|) `1 ) by Lm1;

      

      then (((1 - r) * (q2 + |[1, 1]|)) + (r * (((1 / 2) * (q2 + q3)) + |[ 0 , 1]|))) = |[((q2 + |[ 0 , 1]|) `1 ), ((q2 + |[ 0 , 1]|) `2 )]| by A9, EUCLID: 53

      .= (q2 + |[ 0 , 1]|) by EUCLID: 53;

      then (q2 + |[ 0 , 1]|) in ( LSeg ((q2 + |[1, 1]|),(((1 / 2) * (q2 + q3)) + |[ 0 , 1]|))) by A6, A7;

      then

       A11: ( LSeg ((q2 + |[1, 1]|),(((1 / 2) * (q2 + q3)) + |[ 0 , 1]|))) = (( LSeg ((q2 + |[1, 1]|),(q2 + |[ 0 , 1]|))) \/ ( LSeg ((q2 + |[ 0 , 1]|),(((1 / 2) * (q2 + q3)) + |[ 0 , 1]|)))) by TOPREAL1: 5;

      set I1 = ( Int ( cell (G,( len G),( width G)))), I2 = ( Int ( cell (G,(( len G) -' 1),( width G))));

      

       A12: ((I1 \/ I2) \/ {(q2 + |[ 0 , 1]|)}) = (I1 \/ (I2 \/ ( {(q2 + |[ 0 , 1]|)} \/ {(q2 + |[ 0 , 1]|)}))) by XBOOLE_1: 4

      .= (I1 \/ ((I2 \/ {(q2 + |[ 0 , 1]|)}) \/ {(q2 + |[ 0 , 1]|)})) by XBOOLE_1: 4

      .= ((I1 \/ {(q2 + |[ 0 , 1]|)}) \/ (I2 \/ {(q2 + |[ 0 , 1]|)})) by XBOOLE_1: 4;

      

       A13: ( LSeg ((q2 + |[1, 1]|),(q2 + |[ 0 , 1]|))) c= (I1 \/ {(q2 + |[ 0 , 1]|)}) by Th63;

      ( LSeg ((q2 + |[ 0 , 1]|),(((1 / 2) * (q2 + q3)) + |[ 0 , 1]|))) c= (I2 \/ {(q2 + |[ 0 , 1]|)}) by A3, A4, A5, Th55;

      hence thesis by A11, A13, A12, XBOOLE_1: 13;

    end;

    theorem :: GOBOARD6:82

    1 <= i & (i + 1) <= ( len G) & 1 <= j & (j + 1) <= ( width G) implies ( LSeg (((1 / 2) * ((G * (i,j)) + (G * ((i + 1),(j + 1))))),p)) meets ( Int ( cell (G,i,j)))

    proof

      assume

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

      now

        take a = ((1 / 2) * ((G * (i,j)) + (G * ((i + 1),(j + 1)))));

        thus a in ( LSeg (((1 / 2) * ((G * (i,j)) + (G * ((i + 1),(j + 1))))),p)) by RLTOPSP1: 68;

        thus a in ( Int ( cell (G,i,j))) by A1, Th31;

      end;

      hence thesis by XBOOLE_0: 3;

    end;

    theorem :: GOBOARD6:83

    1 <= i & (i + 1) <= ( len G) implies ( LSeg (p,(((1 / 2) * ((G * (i,( width G))) + (G * ((i + 1),( width G))))) + |[ 0 , 1]|))) meets ( Int ( cell (G,i,( width G))))

    proof

      assume

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

      now

        take a = (((1 / 2) * ((G * (i,( width G))) + (G * ((i + 1),( width G))))) + |[ 0 , 1]|);

        thus a in ( LSeg (p,(((1 / 2) * ((G * (i,( width G))) + (G * ((i + 1),( width G))))) + |[ 0 , 1]|))) by RLTOPSP1: 68;

        thus a in ( Int ( cell (G,i,( width G)))) by A1, Th32;

      end;

      hence thesis by XBOOLE_0: 3;

    end;

    theorem :: GOBOARD6:84

    1 <= i & (i + 1) <= ( len G) implies ( LSeg ((((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1)))) - |[ 0 , 1]|),p)) meets ( Int ( cell (G,i, 0 )))

    proof

      assume

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

      now

        take a = (((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1)))) - |[ 0 , 1]|);

        thus a in ( LSeg ((((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1)))) - |[ 0 , 1]|),p)) by RLTOPSP1: 68;

        thus a in ( Int ( cell (G,i, 0 ))) by A1, Th33;

      end;

      hence thesis by XBOOLE_0: 3;

    end;

    theorem :: GOBOARD6:85

    1 <= j & (j + 1) <= ( width G) implies ( LSeg ((((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1))))) - |[1, 0 ]|),p)) meets ( Int ( cell (G, 0 ,j)))

    proof

      assume

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

      now

        take a = (((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1))))) - |[1, 0 ]|);

        thus a in ( LSeg ((((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1))))) - |[1, 0 ]|),p)) by RLTOPSP1: 68;

        thus a in ( Int ( cell (G, 0 ,j))) by A1, Th35;

      end;

      hence thesis by XBOOLE_0: 3;

    end;

    theorem :: GOBOARD6:86

    1 <= j & (j + 1) <= ( width G) implies ( LSeg (p,(((1 / 2) * ((G * (( len G),j)) + (G * (( len G),(j + 1))))) + |[1, 0 ]|))) meets ( Int ( cell (G,( len G),j)))

    proof

      assume

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

      now

        take a = (((1 / 2) * ((G * (( len G),j)) + (G * (( len G),(j + 1))))) + |[1, 0 ]|);

        thus a in ( LSeg (p,(((1 / 2) * ((G * (( len G),j)) + (G * (( len G),(j + 1))))) + |[1, 0 ]|))) by RLTOPSP1: 68;

        thus a in ( Int ( cell (G,( len G),j))) by A1, Th34;

      end;

      hence thesis by XBOOLE_0: 3;

    end;

    theorem :: GOBOARD6:87

    ( LSeg (p,((G * (1,1)) - |[1, 1]|))) meets ( Int ( cell (G, 0 , 0 )))

    proof

      now

        take a = ((G * (1,1)) - |[1, 1]|);

        thus a in ( LSeg (p,((G * (1,1)) - |[1, 1]|))) by RLTOPSP1: 68;

        thus a in ( Int ( cell (G, 0 , 0 ))) by Th36;

      end;

      hence thesis by XBOOLE_0: 3;

    end;

    theorem :: GOBOARD6:88

    ( LSeg (p,((G * (( len G),( width G))) + |[1, 1]|))) meets ( Int ( cell (G,( len G),( width G))))

    proof

      now

        take a = ((G * (( len G),( width G))) + |[1, 1]|);

        thus a in ( LSeg (p,((G * (( len G),( width G))) + |[1, 1]|))) by RLTOPSP1: 68;

        thus a in ( Int ( cell (G,( len G),( width G)))) by Th37;

      end;

      hence thesis by XBOOLE_0: 3;

    end;

    theorem :: GOBOARD6:89

    ( LSeg (p,((G * (1,( width G))) + |[( - 1), 1]|))) meets ( Int ( cell (G, 0 ,( width G))))

    proof

      now

        take a = ((G * (1,( width G))) + |[( - 1), 1]|);

        thus a in ( LSeg (p,((G * (1,( width G))) + |[( - 1), 1]|))) by RLTOPSP1: 68;

        thus a in ( Int ( cell (G, 0 ,( width G)))) by Th38;

      end;

      hence thesis by XBOOLE_0: 3;

    end;

    theorem :: GOBOARD6:90

    ( LSeg (p,((G * (( len G),1)) + |[1, ( - 1)]|))) meets ( Int ( cell (G,( len G), 0 )))

    proof

      now

        take a = ((G * (( len G),1)) + |[1, ( - 1)]|);

        thus a in ( LSeg (p,((G * (( len G),1)) + |[1, ( - 1)]|))) by RLTOPSP1: 68;

        thus a in ( Int ( cell (G,( len G), 0 ))) by Th39;

      end;

      hence thesis by XBOOLE_0: 3;

    end;

    theorem :: GOBOARD6:91

    

     Th91: for M be non empty MetrSpace, p be Point of M, q be Point of ( TopSpaceMetr M), r be Real st p = q & r > 0 holds ( Ball (p,r)) is a_neighborhood of q

    proof

      let M be non empty MetrSpace, p be Point of M, q be Point of ( TopSpaceMetr M), r be Real;

      reconsider A = ( Ball (p,r)) as Subset of ( TopSpaceMetr M) by TOPMETR: 12;

      assume p = q & r > 0 ;

      then q in A by Th1;

      hence thesis by CONNSP_2: 3, TOPMETR: 14;

    end;

    theorem :: GOBOARD6:92

    for M be non empty MetrSpace, A be Subset of ( TopSpaceMetr M), p be Point of M holds p in ( Cl A) iff for r be Real st r > 0 holds ( Ball (p,r)) meets A

    proof

      let M be non empty MetrSpace, A be Subset of ( TopSpaceMetr M), p be Point of M;

      reconsider p9 = p as Point of ( TopSpaceMetr M) by TOPMETR: 12;

      hereby

        assume

         A1: p in ( Cl A);

        let r be Real;

        reconsider B = ( Ball (p,r)) as Subset of ( TopSpaceMetr M) by TOPMETR: 12;

        assume r > 0 ;

        then B is a_neighborhood of p9 by Th91;

        hence ( Ball (p,r)) meets A by A1, CONNSP_2: 27;

      end;

      assume

       A2: for r be Real st r > 0 holds ( Ball (p,r)) meets A;

      for G be a_neighborhood of p9 holds G meets A

      proof

        let G be a_neighborhood of p9;

        p in ( Int G) by CONNSP_2:def 1;

        then ex r be Real st r > 0 & ( Ball (p,r)) c= G by Th4;

        hence thesis by A2, XBOOLE_1: 63;

      end;

      hence thesis by CONNSP_2: 27;

    end;

    theorem :: GOBOARD6:93

    for A be Subset of ( TOP-REAL n) holds for p be Point of ( TOP-REAL n) holds for p9 be Point of ( Euclid n) st p = p9 holds for s be Real st s > 0 holds p in ( Cl A) iff for r be Real st 0 < r & r < s holds ( Ball (p9,r)) meets A

    proof

      let A be Subset of ( TOP-REAL n);

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

      let p9 be Point of ( Euclid n);

      assume

       A1: p = p9;

      let s be Real;

      assume

       A2: s > 0 ;

      hereby

        assume

         A3: p in ( Cl A);

        let r be Real;

        assume that

         A4: 0 < r and r < s;

        reconsider B = ( Ball (p9,r)) as Subset of ( TOP-REAL n) by TOPREAL3: 8;

        B is a_neighborhood of p by A1, A4, Th2;

        hence ( Ball (p9,r)) meets A by A3, CONNSP_2: 27;

      end;

      assume

       A5: for r be Real st 0 < r & r < s holds ( Ball (p9,r)) meets A;

      for G be a_neighborhood of p holds G meets A

      proof

        let G be a_neighborhood of p;

        p in ( Int G) by CONNSP_2:def 1;

        then

        consider r9 be Real such that

         A6: r9 > 0 and

         A7: ( Ball (p9,r9)) c= G by A1, Th5;

        set r = ( min (r9,(s / 2)));

        ( Ball (p9,r)) c= ( Ball (p9,r9)) by PCOMPS_1: 1, XXREAL_0: 17;

        then

         A8: ( Ball (p9,r)) c= G by A7;

        (s / 2) < s & r <= (s / 2) by A2, XREAL_1: 216, XXREAL_0: 17;

        then

         A9: r < s by XXREAL_0: 2;

        (s / 2) > 0 by A2, XREAL_1: 215;

        then r > 0 by A6, XXREAL_0: 15;

        hence thesis by A5, A8, A9, XBOOLE_1: 63;

      end;

      hence thesis by CONNSP_2: 27;

    end;