let j be Element of NAT ; :: thesis: for G being Go-board st 1 <= j & j < width G holds
LSeg (((1 / 2) * ((G * 1,j) + (G * 1,(j + 1)))) - |[1,0 ]|),((G * 1,(j + 1)) - |[1,0 ]|) c= (Int (cell G,0 ,j)) \/ {((G * 1,(j + 1)) - |[1,0 ]|)}
let G be Go-board; :: thesis: ( 1 <= j & j < width G implies LSeg (((1 / 2) * ((G * 1,j) + (G * 1,(j + 1)))) - |[1,0 ]|),((G * 1,(j + 1)) - |[1,0 ]|) c= (Int (cell G,0 ,j)) \/ {((G * 1,(j + 1)) - |[1,0 ]|)} )
assume A1: ( 1 <= j & j < width G ) ; :: thesis: LSeg (((1 / 2) * ((G * 1,j) + (G * 1,(j + 1)))) - |[1,0 ]|),((G * 1,(j + 1)) - |[1,0 ]|) c= (Int (cell G,0 ,j)) \/ {((G * 1,(j + 1)) - |[1,0 ]|)}
let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in LSeg (((1 / 2) * ((G * 1,j) + (G * 1,(j + 1)))) - |[1,0 ]|),((G * 1,(j + 1)) - |[1,0 ]|) or x in (Int (cell G,0 ,j)) \/ {((G * 1,(j + 1)) - |[1,0 ]|)} )
assume A2: x in LSeg (((1 / 2) * ((G * 1,j) + (G * 1,(j + 1)))) - |[1,0 ]|),((G * 1,(j + 1)) - |[1,0 ]|) ; :: thesis: x in (Int (cell G,0 ,j)) \/ {((G * 1,(j + 1)) - |[1,0 ]|)}
then reconsider p = x as Point of (TOP-REAL 2) ;
consider r being Real such that
A4: p = ((1 - r) * (((1 / 2) * ((G * 1,j) + (G * 1,(j + 1)))) - |[1,0 ]|)) + (r * ((G * 1,(j + 1)) - |[1,0 ]|)) and
A3: ( 0 <= r & r <= 1 ) by A2;

now

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

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

then p = (0. (TOP-REAL 2)) + (1 * ((G * 1,(j + 1)) - |[1,0 ]|)) by A4, EUCLID:33
.= 1 * ((G * 1,(j + 1)) - |[1,0 ]|) by EUCLID:31
.= (G * 1,(j + 1)) - |[1,0 ]| by EUCLID:33 ;
hence p in {((G * 1,(j + 1)) - |[1,0 ]|)} by TARSKI:def 1; :: thesis: verum

end;

case A5: r < 1 ; :: thesis: p in Int (cell G,0 ,j)

set r2 = (G * 1,1) `1 ;
set s1 = (G * 1,j) `2 ;
set s2 = (G * 1,(j + 1)) `2 ;
A6: ( 1 <= j + 1 & j + 1 <= width G ) by A1, NAT_1:13;
0 <> len G by GOBOARD1:def 5;
then A7: 1 <= len G by NAT_1:14;
A8: G * 1,(j + 1) = |[((G * 1,(j + 1)) `1 ),((G * 1,(j + 1)) `2 )]| by EUCLID:57
.= |[((G * 1,1) `1 ),((G * 1,(j + 1)) `2 )]| by A6, A7, GOBOARD5:3 ;
A9: G * 1,j = |[((G * 1,j) `1 ),((G * 1,j) `2 )]| by EUCLID:57
.= |[((G * 1,1) `1 ),((G * 1,j) `2 )]| by A1, A7, GOBOARD5:3 ;
set r3 = (1 - r) * (1 / 2);
A10: p = (((1 - r) * ((1 / 2) * ((G * 1,j) + (G * 1,(j + 1))))) - ((1 - r) * |[1,0 ]|)) + (r * ((G * 1,(j + 1)) - |[1,0 ]|)) by A4, EUCLID:53
.= ((((1 - r) * (1 / 2)) * ((G * 1,j) + (G * 1,(j + 1)))) - ((1 - r) * |[1,0 ]|)) + (r * ((G * 1,(j + 1)) - |[1,0 ]|)) by EUCLID:34
.= ((((1 - r) * (1 / 2)) * ((G * 1,j) + (G * 1,(j + 1)))) - |[((1 - r) * 1),((1 - r) * 0 )]|) + (r * ((G * 1,(j + 1)) - |[1,0 ]|)) by EUCLID:62
.= ((((1 - r) * (1 / 2)) * |[(((G * 1,1) `1 ) + ((G * 1,1) `1 )),(((G * 1,j) `2 ) + ((G * 1,(j + 1)) `2 ))]|) - |[(1 - r),0 ]|) + (r * (|[((G * 1,1) `1 ),((G * 1,(j + 1)) `2 )]| - |[1,0 ]|)) by A8, A9, EUCLID:60
.= ((((1 - r) * (1 / 2)) * |[(((G * 1,1) `1 ) + ((G * 1,1) `1 )),(((G * 1,j) `2 ) + ((G * 1,(j + 1)) `2 ))]|) - |[(1 - r),0 ]|) + ((r * |[((G * 1,1) `1 ),((G * 1,(j + 1)) `2 )]|) - (r * |[1,0 ]|)) by EUCLID:53
.= ((((1 - r) * (1 / 2)) * |[(((G * 1,1) `1 ) + ((G * 1,1) `1 )),(((G * 1,j) `2 ) + ((G * 1,(j + 1)) `2 ))]|) - |[(1 - r),0 ]|) + (|[(r * ((G * 1,1) `1 )),(r * ((G * 1,(j + 1)) `2 ))]| - (r * |[1,0 ]|)) by EUCLID:62
.= ((((1 - r) * (1 / 2)) * |[(((G * 1,1) `1 ) + ((G * 1,1) `1 )),(((G * 1,j) `2 ) + ((G * 1,(j + 1)) `2 ))]|) - |[(1 - r),0 ]|) + (|[(r * ((G * 1,1) `1 )),(r * ((G * 1,(j + 1)) `2 ))]| - |[(r * 1),(r * 0 )]|) by EUCLID:62
.= ((((1 - r) * (1 / 2)) * |[(((G * 1,1) `1 ) + ((G * 1,1) `1 )),(((G * 1,j) `2 ) + ((G * 1,(j + 1)) `2 ))]|) - |[(1 - r),0 ]|) + |[((r * ((G * 1,1) `1 )) - r),((r * ((G * 1,(j + 1)) `2 )) - 0 )]| by EUCLID:66
.= (|[(((1 - r) * (1 / 2)) * (((G * 1,1) `1 ) + ((G * 1,1) `1 ))),(((1 - r) * (1 / 2)) * (((G * 1,j) `2 ) + ((G * 1,(j + 1)) `2 )))]| - |[(1 - r),0 ]|) + |[((r * ((G * 1,1) `1 )) - r),((r * ((G * 1,(j + 1)) `2 )) - 0 )]| by EUCLID:62
.= |[((((1 - r) * (1 / 2)) * (((G * 1,1) `1 ) + ((G * 1,1) `1 ))) - (1 - r)),((((1 - r) * (1 / 2)) * (((G * 1,j) `2 ) + ((G * 1,(j + 1)) `2 ))) - 0 )]| + |[((r * ((G * 1,1) `1 )) - r),((r * ((G * 1,(j + 1)) `2 )) - 0 )]| by EUCLID:66
.= |[(((((1 - r) * (1 / 2)) * (((G * 1,1) `1 ) + ((G * 1,1) `1 ))) - (1 - r)) + ((r * ((G * 1,1) `1 )) - r)),((((1 - r) * (1 / 2)) * (((G * 1,j) `2 ) + ((G * 1,(j + 1)) `2 ))) + (r * ((G * 1,(j + 1)) `2 )))]| by EUCLID:60 ;
1 - r > 0 by A5, XREAL_1:52;
then A11: (1 - r) * (1 / 2) > (1 / 2) * 0 by XREAL_1:70;
j < j + 1 by XREAL_1:31;
then A12: (G * 1,j) `2 < (G * 1,(j + 1)) `2 by A1, A6, A7, GOBOARD5:5;
(G * 1,1) `1 < ((G * 1,1) `1 ) + 1 by XREAL_1:31;
then A13: ((G * 1,1) `1 ) - 1 < (G * 1,1) `1 by XREAL_1:21;
((G * 1,j) `2 ) + ((G * 1,j) `2 ) < ((G * 1,j) `2 ) + ((G * 1,(j + 1)) `2 ) by A12, XREAL_1:8;
then A14: ((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 A11, XREAL_1:70;
A15: r * ((G * 1,j) `2 ) <= r * ((G * 1,(j + 1)) `2 ) by A3, A12, XREAL_1:66;
(((1 - r) * (1 / 2)) * (((G * 1,j) `2 ) + ((G * 1,j) `2 ))) + (r * ((G * 1,j) `2 )) = (G * 1,j) `2 ;
then A16: (G * 1,j) `2 < (((1 - r) * (1 / 2)) * (((G * 1,j) `2 ) + ((G * 1,(j + 1)) `2 ))) + (r * ((G * 1,(j + 1)) `2 )) by A14, A15, XREAL_1:10;
((G * 1,j) `2 ) + ((G * 1,(j + 1)) `2 ) < ((G * 1,(j + 1)) `2 ) + ((G * 1,(j + 1)) `2 ) by A12, XREAL_1:8;
then A17: ((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 A11, XREAL_1:70;
(((1 - r) * (1 / 2)) * (((G * 1,(j + 1)) `2 ) + ((G * 1,(j + 1)) `2 ))) + (r * ((G * 1,(j + 1)) `2 )) = (G * 1,(j + 1)) `2 ;
then A18: (((1 - r) * (1 / 2)) * (((G * 1,j) `2 ) + ((G * 1,(j + 1)) `2 ))) + (r * ((G * 1,(j + 1)) `2 )) < (G * 1,(j + 1)) `2 by A17, XREAL_1:10;
Int (cell G,0 ,j) = { |[r',s']| where r', s' is Real : ( r' < (G * 1,1) `1 & (G * 1,j) `2 < s' & s' < (G * 1,(j + 1)) `2 ) } by A1, Th23;
hence p in Int (cell G,0 ,j) by A10, A13, A16, A18; :: thesis: verum

end;

end;

end;

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