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

now

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

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

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

end;

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

then A4: 1 - r > 0 by XREAL_1:52;
(G * 1,1) `2 < ((G * 1,1) `2 ) + 1 by XREAL_1:31;
then A5: ((G * 1,1) `2 ) - 1 < (G * 1,1) `2 by XREAL_1:21;
A6: G * 1,1 = |[((G * 1,1) `1 ),((G * 1,1) `2 )]| by EUCLID:57;
A7: p = (((1 - r) * (G * 1,1)) - ((1 - r) * |[1,1]|)) + (r * ((G * 1,1) - |[0 ,1]|)) by A3, EUCLID:53
.= (((1 - r) * (G * 1,1)) - ((1 - r) * |[1,1]|)) + ((r * (G * 1,1)) - (r * |[0 ,1]|)) by EUCLID:53
.= ((r * (G * 1,1)) + (((1 - r) * (G * 1,1)) - ((1 - r) * |[1,1]|))) - (r * |[0 ,1]|) by EUCLID:49
.= (((r * (G * 1,1)) + ((1 - r) * (G * 1,1))) - ((1 - r) * |[1,1]|)) - (r * |[0 ,1]|) by EUCLID:49
.= (((r + (1 - r)) * (G * 1,1)) - ((1 - r) * |[1,1]|)) - (r * |[0 ,1]|) by EUCLID:37
.= ((G * 1,1) - ((1 - r) * |[1,1]|)) - (r * |[0 ,1]|) by EUCLID:33
.= ((G * 1,1) - |[((1 - r) * 1),((1 - r) * 1)]|) - (r * |[0 ,1]|) by EUCLID:62
.= ((G * 1,1) - |[(1 - r),(1 - r)]|) - |[(r * 0 ),(r * 1)]| by EUCLID:62
.= |[(((G * 1,1) `1 ) - (1 - r)),(((G * 1,1) `2 ) - (1 - r))]| - |[0 ,r]| by A6, EUCLID:66
.= |[((((G * 1,1) `1 ) - (1 - r)) - 0 ),((((G * 1,1) `2 ) - (1 - r)) - r)]| by EUCLID:66
.= |[(((G * 1,1) `1 ) - (1 - r)),(((G * 1,1) `2 ) - 1)]| ;
(G * 1,1) `1 < ((G * 1,1) `1 ) + (1 - r) by A4, XREAL_1:31;
then A8: ((G * 1,1) `1 ) - (1 - r) < (G * 1,1) `1 by XREAL_1:21;
Int (cell G,0 ,0 ) = { |[r',s']| where r', s' is Real : ( r' < (G * 1,1) `1 & s' < (G * 1,1) `2 ) } by Th21;
hence p in Int (cell G,0 ,0 ) by A5, A7, A8; :: thesis: verum

end;

end;

end;

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