let G be Go-board; :: thesis: 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]|)}
let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in LSeg ((G * 1,(width G)) + |[(- 1),1]|),((G * 1,(width G)) + |[0 ,1]|) or x in (Int (cell G,0 ,(width G))) \/ {((G * 1,(width G)) + |[0 ,1]|)} )
assume A1: x in LSeg ((G * 1,(width G)) + |[(- 1),1]|),((G * 1,(width G)) + |[0 ,1]|) ; :: thesis: x in (Int (cell G,0 ,(width G))) \/ {((G * 1,(width G)) + |[0 ,1]|)}
then reconsider p = x as Point of (TOP-REAL 2) ;
consider r being Real such that
A3: p = ((1 - r) * ((G * 1,(width G)) + |[(- 1),1]|)) + (r * ((G * 1,(width G)) + |[0 ,1]|)) and
A2: ( 0 <= r & r <= 1 ) by A1;
set r1 = (G * 1,1) `1 ;
set s1 = (G * 1,(width G)) `2 ;

now

per cases ( r = 1 or r < 1 ) by A2, XXREAL_0:1;

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

then p = (0. (TOP-REAL 2)) + (1 * ((G * 1,(width G)) + |[0 ,1]|)) by A3, EUCLID:33
.= 1 * ((G * 1,(width G)) + |[0 ,1]|) by EUCLID:31
.= (G * 1,(width G)) + |[0 ,1]| by EUCLID:33 ;
hence p in {((G * 1,(width G)) + |[0 ,1]|)} by TARSKI:def 1; :: thesis: verum

end;

case r < 1 ; :: thesis: p in Int (cell G,0 ,(width G))

then A4: 1 - r > 0 by XREAL_1:52;
A5: (G * 1,(width G)) `2 < ((G * 1,(width G)) `2 ) + 1 by XREAL_1:31;
0 <> width G by GOBOARD1:def 5;
then A6: 1 <= width G by NAT_1:14;
0 <> len G by GOBOARD1:def 5;
then A7: 1 <= len G by NAT_1:14;
A8: G * 1,(width G) = |[((G * 1,(width G)) `1 ),((G * 1,(width G)) `2 )]| by EUCLID:57
.= |[((G * 1,1) `1 ),((G * 1,(width G)) `2 )]| by A6, A7, GOBOARD5:3 ;
A9: p = (((1 - r) * (G * 1,(width G))) + ((1 - r) * |[(- 1),1]|)) + (r * ((G * 1,(width G)) + |[0 ,1]|)) by A3, EUCLID:36
.= (((1 - r) * (G * 1,(width G))) + ((1 - r) * |[(- 1),1]|)) + ((r * (G * 1,(width G))) + (r * |[0 ,1]|)) by EUCLID:36
.= ((r * (G * 1,(width G))) + (((1 - r) * (G * 1,(width G))) + ((1 - r) * |[(- 1),1]|))) + (r * |[0 ,1]|) by EUCLID:30
.= (((r * (G * 1,(width G))) + ((1 - r) * (G * 1,(width G)))) + ((1 - r) * |[(- 1),1]|)) + (r * |[0 ,1]|) by EUCLID:30
.= (((r + (1 - r)) * (G * 1,(width G))) + ((1 - r) * |[(- 1),1]|)) + (r * |[0 ,1]|) by EUCLID:37
.= ((G * 1,(width G)) + ((1 - r) * |[(- 1),1]|)) + (r * |[0 ,1]|) by EUCLID:33
.= ((G * 1,(width G)) + |[((1 - r) * (- 1)),((1 - r) * 1)]|) + (r * |[0 ,1]|) by EUCLID:62
.= ((G * 1,(width G)) + |[(r - 1),(1 - r)]|) + |[(r * 0 ),(r * 1)]| by EUCLID:62
.= |[(((G * 1,1) `1 ) + (r - 1)),(((G * 1,(width G)) `2 ) + (1 - r))]| + |[0 ,r]| by A8, EUCLID:60
.= |[((((G * 1,1) `1 ) + (r - 1)) + 0 ),((((G * 1,(width G)) `2 ) + (1 - r)) + r)]| by EUCLID:60
.= |[(((G * 1,1) `1 ) - (1 - r)),(((G * 1,(width G)) `2 ) + 1)]| ;
(G * 1,1) `1 < ((G * 1,1) `1 ) + (1 - r) by A4, XREAL_1:31;
then A10: ((G * 1,1) `1 ) - (1 - r) < (G * 1,1) `1 by XREAL_1:21;
Int (cell G,0 ,(width G)) = { |[r',s']| where r', s' is Real : ( r' < (G * 1,1) `1 & (G * 1,(width G)) `2 < s' ) } by Th22;
hence p in Int (cell G,0 ,(width G)) by A5, A9, A10; :: thesis: verum

end;

end;

end;

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