let G be Go-board; :: thesis: 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]|)}
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in LSeg (((G * ((len G),(width G))) + |[1,1]|),((G * ((len G),(width G))) + |[1,0]|)) or x in (Int (cell (G,(len G),(width G)))) \/ {((G * ((len G),(width G))) + |[1,0]|)} )
set r1 = (G * ((len G),1)) `1 ;
set 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]|)) ; :: thesis: x in (Int (cell (G,(len G),(width G)))) \/ {((G * ((len G),(width G))) + |[1,0]|)}
then reconsider p = x as Point of (TOP-REAL 2) ;
consider r being Real 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 :: thesis: ( ( r = 1 & p in {((G * ((len G),(width G))) + |[1,0]|)} ) or ( r < 1 & p in Int (cell (G,(len G),(width G))) ) )

per cases ( r = 1 or r < 1 ) by A3, XXREAL_0:1;

case r = 1 ; :: thesis: p in {((G * ((len G),(width G))) + |[1,0]|)}

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; :: thesis: verum

end;

case r < 1 ; :: thesis: p in Int (cell (G,(len G),(width G)))

then 1 - r > 0 by XREAL_1:50;
then A4: (G * (1,(width G))) `2 < ((G * (1,(width G))) `2) + (1 - r) by XREAL_1:29;
A5: (G * ((len G),1)) `1 < ((G * ((len G),1)) `1) + 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
.= |[((G * ((len G),1)) `1),((G * ((len G),(width G))) `2)]| by A6, A7, GOBOARD5:2
.= |[((G * ((len G),1)) `1),((G * (1,(width G))) `2)]| by A6, A7, GOBOARD5:1 ;
A9: Int (cell (G,(len G),(width G))) = { |[r9,s9]| where r9, s9 is Real : ( (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
.= |[(((G * ((len G),1)) `1) + (1 - r)),(((G * (1,(width G))) `2) + (1 - r))]| + |[r,0]| by A8, EUCLID:56
.= |[((((G * ((len G),1)) `1) + (1 - r)) + r),((((G * (1,(width G))) `2) + (1 - r)) + 0)]| by EUCLID:56
.= |[(((G * ((len G),1)) `1) + 1),(((G * (1,(width G))) `2) + (1 - r))]| ;
hence p in Int (cell (G,(len G),(width G))) by A4, A5, A9; :: thesis: verum

end;

end;

end;

hence x in (Int (cell (G,(len G),(width G)))) \/ {((G * ((len G),(width G))) + |[1,0]|)} by XBOOLE_0:def 3; :: thesis: verum