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 object ; :: 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]|)} )
set r1 = (G * (1,1)) `1 ;
set s1 = (G * (1,(width G))) `2 ;
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
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 :: thesis: ( ( r = 1 & p in {((G * (1,(width G))) + |[0,1]|)} ) or ( r < 1 & p in Int (cell (G,0,(width G))) ) )

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

end;

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

then 1 - r > 0 by XREAL_1:50;
then (G * (1,1)) `1 < ((G * (1,1)) `1) + (1 - r) by XREAL_1:29;
then A4: ( (G * (1,(width G))) `2 < ((G * (1,(width G))) `2) + 1 & ((G * (1,1)) `1) - (1 - r) < (G * (1,1)) `1 ) 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),((G * (1,(width G))) `2)]| by EUCLID:53
.= |[((G * (1,1)) `1),((G * (1,(width G))) `2)]| by A5, A6, GOBOARD5:2 ;
A8: Int (cell (G,0,(width G))) = { |[r9,s9]| where r9, s9 is Real : ( 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
.= |[(((G * (1,1)) `1) + (r - 1)),(((G * (1,(width G))) `2) + (1 - r))]| + |[0,r]| by A7, EUCLID:56
.= |[((((G * (1,1)) `1) + (r - 1)) + 0),((((G * (1,(width G))) `2) + (1 - r)) + r)]| by EUCLID:56
.= |[(((G * (1,1)) `1) - (1 - r)),(((G * (1,(width G))) `2) + 1)]| ;
hence p in Int (cell (G,0,(width G))) by A4, A8; :: 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