let i, j be Element of NAT ; :: thesis: for G being Go-board st 1 <= i & i < len G & 1 <= j & j < width G holds
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))))}
let G be Go-board; :: thesis: ( 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))))} )
assume that
A1: 1 <= i and
A2: i < len G and
A3: 1 <= j and
A4: j < width G ; :: thesis: 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))))}
let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in LSeg ((1 / 2) * ((G * i,j) + (G * (i + 1),(j + 1)))),((1 / 2) * ((G * i,j) + (G * i,(j + 1)))) or x in (Int (cell G,i,j)) \/ {((1 / 2) * ((G * i,j) + (G * i,(j + 1))))} )
assume A5: x in LSeg ((1 / 2) * ((G * i,j) + (G * (i + 1),(j + 1)))),((1 / 2) * ((G * i,j) + (G * i,(j + 1)))) ; :: thesis: x in (Int (cell G,i,j)) \/ {((1 / 2) * ((G * i,j) + (G * i,(j + 1))))}
then reconsider p = x as Point of (TOP-REAL 2) ;
consider r being Real 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 ( r = 1 or r < 1 ) by A8, XXREAL_0:1;

case r = 1 ; :: thesis: p in {((1 / 2) * ((G * i,j) + (G * i,(j + 1))))}

then p = (0. (TOP-REAL 2)) + (1 * ((1 / 2) * ((G * i,j) + (G * i,(j + 1))))) by A6, EUCLID:33
.= 1 * ((1 / 2) * ((G * i,j) + (G * i,(j + 1)))) by EUCLID:31
.= (1 / 2) * ((G * i,j) + (G * i,(j + 1))) by EUCLID:33 ;
hence p in {((1 / 2) * ((G * i,j) + (G * i,(j + 1))))} by TARSKI:def 1; :: thesis: verum

end;

case A9: r < 1 ; :: thesis: p in Int (cell G,i,j)

set r3 = (1 - r) * (1 / 2);
set s3 = r * (1 / 2);
set r1 = (G * i,1) `1 ;
set r2 = (G * (i + 1),1) `1 ;
set s1 = (G * 1,j) `2 ;
set s2 = (G * 1,(j + 1)) `2 ;
A10: (((1 - r) * (1 / 2)) * (((G * 1,j) `2 ) + ((G * 1,j) `2 ))) + ((r * (1 / 2)) * (((G * 1,j) `2 ) + ((G * 1,j) `2 ))) = (G * 1,j) `2 ;
0 <> len G by GOBOARD1:def 5;
then A11: 1 <= len G by NAT_1:14;
A12: j + 1 <= width G by A4, NAT_1:13;
j < j + 1 by XREAL_1:31;
then A13: (G * 1,j) `2 < (G * 1,(j + 1)) `2 by A3, A12, A11, GOBOARD5:5;
then A14: ((G * 1,j) `2 ) + ((G * 1,j) `2 ) < ((G * 1,j) `2 ) + ((G * 1,(j + 1)) `2 ) by XREAL_1:8;
then A15: (r * (1 / 2)) * (((G * 1,j) `2 ) + ((G * 1,j) `2 )) <= (r * (1 / 2)) * (((G * 1,j) `2 ) + ((G * 1,(j + 1)) `2 )) by A7, XREAL_1:66;
1 - r > 0 by A9, XREAL_1:52;
then A16: (1 - r) * (1 / 2) > (1 / 2) * 0 by XREAL_1:70;
then ((1 - r) * (1 / 2)) * (((G * 1,j) `2 ) + ((G * 1,j) `2 )) < ((1 - r) * (1 / 2)) * (((G * 1,j) `2 ) + ((G * 1,(j + 1)) `2 )) by A14, XREAL_1:70;
then A17: (G * 1,j) `2 < (((1 - r) * (1 / 2)) * (((G * 1,j) `2 ) + ((G * 1,(j + 1)) `2 ))) + ((r * (1 / 2)) * (((G * 1,j) `2 ) + ((G * 1,(j + 1)) `2 ))) by A15, A10, XREAL_1:10;
A18: ((G * 1,j) `2 ) + ((G * 1,(j + 1)) `2 ) < ((G * 1,(j + 1)) `2 ) + ((G * 1,(j + 1)) `2 ) by A13, XREAL_1:8;
then A19: (r * (1 / 2)) * (((G * 1,j) `2 ) + ((G * 1,(j + 1)) `2 )) <= (r * (1 / 2)) * (((G * 1,(j + 1)) `2 ) + ((G * 1,(j + 1)) `2 )) by A7, XREAL_1:66;
0 <> width G by GOBOARD1:def 5;
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]| where r9, s9 is Real : ( (G * i,1) `1 < r9 & r9 < (G * (i + 1),1) `1 & (G * 1,j) `2 < s9 & s9 < (G * 1,(j + 1)) `2 ) } by A1, A2, A3, A4, Th29;
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:57
.= |[((G * i,1) `1 ),((G * i,(j + 1)) `2 )]| by A1, A2, A23, A12, GOBOARD5:3
.= |[((G * i,1) `1 ),((G * 1,(j + 1)) `2 )]| by A1, A2, A23, A12, GOBOARD5:2 ;
A25: (((1 - r) * (1 / 2)) * (((G * 1,(j + 1)) `2 ) + ((G * 1,(j + 1)) `2 ))) + ((r * (1 / 2)) * (((G * 1,(j + 1)) `2 ) + ((G * 1,(j + 1)) `2 ))) = (G * 1,(j + 1)) `2 ;
((1 - r) * (1 / 2)) * (((G * 1,j) `2 ) + ((G * 1,(j + 1)) `2 )) < ((1 - r) * (1 / 2)) * (((G * 1,(j + 1)) `2 ) + ((G * 1,(j + 1)) `2 )) by A16, A18, XREAL_1:70;
then A26: (((1 - r) * (1 / 2)) * (((G * 1,j) `2 ) + ((G * 1,(j + 1)) `2 ))) + ((r * (1 / 2)) * (((G * 1,j) `2 ) + ((G * 1,(j + 1)) `2 ))) < (G * 1,(j + 1)) `2 by A19, A25, XREAL_1:10;
A27: i + 1 <= len G by A2, NAT_1:13;
i < i + 1 by XREAL_1:31;
then A28: (G * i,1) `1 < (G * (i + 1),1) `1 by A1, A27, A20, GOBOARD5:4;
then ((G * i,1) `1 ) + ((G * i,1) `1 ) < ((G * (i + 1),1) `1 ) + ((G * (i + 1),1) `1 ) by XREAL_1:10;
then A29: (r * (1 / 2)) * (((G * i,1) `1 ) + ((G * i,1) `1 )) <= (r * (1 / 2)) * (((G * (i + 1),1) `1 ) + ((G * (i + 1),1) `1 )) by A7, XREAL_1:66;
((G * i,1) `1 ) + ((G * (i + 1),1) `1 ) < ((G * (i + 1),1) `1 ) + ((G * (i + 1),1) `1 ) by A28, XREAL_1:8;
then A30: ((1 - r) * (1 / 2)) * (((G * i,1) `1 ) + ((G * (i + 1),1) `1 )) < ((1 - r) * (1 / 2)) * (((G * (i + 1),1) `1 ) + ((G * (i + 1),1) `1 )) by A16, XREAL_1:70;
(((1 - r) * (1 / 2)) * (((G * (i + 1),1) `1 ) + ((G * (i + 1),1) `1 ))) + ((r * (1 / 2)) * (((G * (i + 1),1) `1 ) + ((G * (i + 1),1) `1 ))) = (G * (i + 1),1) `1 ;
then A31: (((1 - r) * (1 / 2)) * (((G * i,1) `1 ) + ((G * (i + 1),1) `1 ))) + ((r * (1 / 2)) * (((G * i,1) `1 ) + ((G * i,1) `1 ))) < (G * (i + 1),1) `1 by A30, A29, XREAL_1:10;
A32: G * i,j = |[((G * i,j) `1 ),((G * i,j) `2 )]| by EUCLID:57
.= |[((G * i,1) `1 ),((G * i,j) `2 )]| by A1, A2, A3, A4, GOBOARD5:3
.= |[((G * i,1) `1 ),((G * 1,j) `2 )]| by A1, A2, A3, A4, GOBOARD5:2 ;
A33: G * (i + 1),(j + 1) = |[((G * (i + 1),(j + 1)) `1 ),((G * (i + 1),(j + 1)) `2 )]| by EUCLID:57
.= |[((G * (i + 1),1) `1 ),((G * (i + 1),(j + 1)) `2 )]| by A23, A12, A21, A27, GOBOARD5:3
.= |[((G * (i + 1),1) `1 ),((G * 1,(j + 1)) `2 )]| by A23, A12, A21, A27, GOBOARD5:2 ;
A34: (((1 - r) * (1 / 2)) * (((G * i,1) `1 ) + ((G * i,1) `1 ))) + ((r * (1 / 2)) * (((G * i,1) `1 ) + ((G * i,1) `1 ))) = (G * i,1) `1 ;
((G * i,1) `1 ) + ((G * i,1) `1 ) < ((G * i,1) `1 ) + ((G * (i + 1),1) `1 ) by A28, XREAL_1:8;
then ((1 - r) * (1 / 2)) * (((G * i,1) `1 ) + ((G * i,1) `1 )) < ((1 - r) * (1 / 2)) * (((G * i,1) `1 ) + ((G * (i + 1),1) `1 )) by A16, XREAL_1:70;
then A35: (G * i,1) `1 < (((1 - r) * (1 / 2)) * (((G * i,1) `1 ) + ((G * (i + 1),1) `1 ))) + ((r * (1 / 2)) * (((G * i,1) `1 ) + ((G * i,1) `1 ))) by A34, XREAL_1:8;
p = (((1 - r) * (1 / 2)) * ((G * i,j) + (G * (i + 1),(j + 1)))) + (r * ((1 / 2) * ((G * i,j) + (G * i,(j + 1))))) by A6, EUCLID:34
.= (((1 - r) * (1 / 2)) * ((G * i,j) + (G * (i + 1),(j + 1)))) + ((r * (1 / 2)) * ((G * i,j) + (G * i,(j + 1)))) by EUCLID:34
.= (((1 - r) * (1 / 2)) * |[(((G * i,1) `1 ) + ((G * (i + 1),1) `1 )),(((G * 1,j) `2 ) + ((G * 1,(j + 1)) `2 ))]|) + ((r * (1 / 2)) * ((G * i,j) + (G * i,(j + 1)))) by A32, A33, EUCLID:60
.= (((1 - r) * (1 / 2)) * |[(((G * i,1) `1 ) + ((G * (i + 1),1) `1 )),(((G * 1,j) `2 ) + ((G * 1,(j + 1)) `2 ))]|) + ((r * (1 / 2)) * |[(((G * i,1) `1 ) + ((G * i,1) `1 )),(((G * 1,j) `2 ) + ((G * 1,(j + 1)) `2 ))]|) by A32, A24, EUCLID:60
.= |[(((1 - r) * (1 / 2)) * (((G * i,1) `1 ) + ((G * (i + 1),1) `1 ))),(((1 - r) * (1 / 2)) * (((G * 1,j) `2 ) + ((G * 1,(j + 1)) `2 )))]| + ((r * (1 / 2)) * |[(((G * i,1) `1 ) + ((G * i,1) `1 )),(((G * 1,j) `2 ) + ((G * 1,(j + 1)) `2 ))]|) by EUCLID:62
.= |[(((1 - r) * (1 / 2)) * (((G * i,1) `1 ) + ((G * (i + 1),1) `1 ))),(((1 - r) * (1 / 2)) * (((G * 1,j) `2 ) + ((G * 1,(j + 1)) `2 )))]| + |[((r * (1 / 2)) * (((G * i,1) `1 ) + ((G * i,1) `1 ))),((r * (1 / 2)) * (((G * 1,j) `2 ) + ((G * 1,(j + 1)) `2 )))]| by EUCLID:62
.= |[((((1 - r) * (1 / 2)) * (((G * i,1) `1 ) + ((G * (i + 1),1) `1 ))) + ((r * (1 / 2)) * (((G * i,1) `1 ) + ((G * i,1) `1 )))),((((1 - r) * (1 / 2)) * (((G * 1,j) `2 ) + ((G * 1,(j + 1)) `2 ))) + ((r * (1 / 2)) * (((G * 1,j) `2 ) + ((G * 1,(j + 1)) `2 ))))]| by EUCLID:60 ;
hence p in Int (cell G,i,j) by A35, A31, A17, A26, A22; :: thesis: verum

end;

end;

end;

hence x in (Int (cell G,i,j)) \/ {((1 / 2) * ((G * i,j) + (G * i,(j + 1))))} by XBOOLE_0:def 3; :: thesis: verum