let j be Element of NAT ; :: thesis: for G being Go-board st 1 <= j & j < width G holds
LSeg (((1 / 2) * ((G * (len G),j) + (G * (len G),(j + 1)))) + |[1,0 ]|),((G * (len G),j) + |[1,0 ]|) c= (Int (cell G,(len G),j)) \/ {((G * (len G),j) + |[1,0 ]|)}
let G be Go-board; :: thesis: ( 1 <= j & j < width G implies LSeg (((1 / 2) * ((G * (len G),j) + (G * (len G),(j + 1)))) + |[1,0 ]|),((G * (len G),j) + |[1,0 ]|) c= (Int (cell G,(len G),j)) \/ {((G * (len G),j) + |[1,0 ]|)} )
assume that
A1: 1 <= j and
A2: j < width G ; :: thesis: LSeg (((1 / 2) * ((G * (len G),j) + (G * (len G),(j + 1)))) + |[1,0 ]|),((G * (len G),j) + |[1,0 ]|) c= (Int (cell G,(len G),j)) \/ {((G * (len G),j) + |[1,0 ]|)}
let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in LSeg (((1 / 2) * ((G * (len G),j) + (G * (len G),(j + 1)))) + |[1,0 ]|),((G * (len G),j) + |[1,0 ]|) or x in (Int (cell G,(len G),j)) \/ {((G * (len G),j) + |[1,0 ]|)} )
assume A3: x in LSeg (((1 / 2) * ((G * (len G),j) + (G * (len G),(j + 1)))) + |[1,0 ]|),((G * (len G),j) + |[1,0 ]|) ; :: thesis: x in (Int (cell G,(len G),j)) \/ {((G * (len G),j) + |[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 * (len G),j) + (G * (len G),(j + 1)))) + |[1,0 ]|)) + (r * ((G * (len G),j) + |[1,0 ]|)) and
A5: 0 <= r and
A6: r <= 1 by A3;

now

per cases ( r = 1 or r < 1 ) by A6, XXREAL_0:1;

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

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

end;

case A7: r < 1 ; :: thesis: p in Int (cell G,(len G),j)

set r3 = (1 - r) * (1 / 2);
1 - r > 0 by A7, XREAL_1:52;
then A8: (1 - r) * (1 / 2) > (1 / 2) * 0 by XREAL_1:70;
set r2 = (G * (len G),1) `1 ;
set s1 = (G * 1,j) `2 ;
set s2 = (G * 1,(j + 1)) `2 ;
A9: (((1 - r) * (1 / 2)) * (((G * 1,j) `2 ) + ((G * 1,j) `2 ))) + (r * ((G * 1,j) `2 )) = (G * 1,j) `2 ;
A10: j + 1 <= width G by A2, NAT_1:13;
0 <> len G by GOBOARD1:def 5;
then A11: 1 <= len G by NAT_1:14;
A12: G * (len G),j = |[((G * (len G),j) `1 ),((G * (len G),j) `2 )]| by EUCLID:57
.= |[((G * (len G),1) `1 ),((G * (len G),j) `2 )]| by A1, A2, A11, GOBOARD5:3
.= |[((G * (len G),1) `1 ),((G * 1,j) `2 )]| by A1, A2, A11, GOBOARD5:2 ;
A13: 1 <= j + 1 by A1, NAT_1:13;
j < j + 1 by XREAL_1:31;
then A14: (G * 1,j) `2 < (G * 1,(j + 1)) `2 by A1, A10, A11, GOBOARD5:5;
then ((G * 1,j) `2 ) + ((G * 1,(j + 1)) `2 ) < ((G * 1,(j + 1)) `2 ) + ((G * 1,(j + 1)) `2 ) by XREAL_1:8;
then A15: ((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 A8, XREAL_1:70;
((G * 1,j) `2 ) + ((G * 1,j) `2 ) < ((G * 1,j) `2 ) + ((G * 1,(j + 1)) `2 ) by A14, XREAL_1:8;
then ((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 A8, XREAL_1:70;
then A16: ( (G * (len G),1) `1 < ((G * (len G),1) `1 ) + 1 & (G * 1,j) `2 < (((1 - r) * (1 / 2)) * (((G * 1,j) `2 ) + ((G * 1,(j + 1)) `2 ))) + (r * ((G * 1,j) `2 )) ) by A9, XREAL_1:8, XREAL_1:31;
A17: Int (cell G,(len G),j) = { |[r9,s9]| where r9, s9 is Real : ( (G * (len G),1) `1 < r9 & (G * 1,j) `2 < s9 & s9 < (G * 1,(j + 1)) `2 ) } by A1, A2, Th26;
A18: G * (len G),(j + 1) = |[((G * (len G),(j + 1)) `1 ),((G * (len G),(j + 1)) `2 )]| by EUCLID:57
.= |[((G * (len G),1) `1 ),((G * (len G),(j + 1)) `2 )]| by A13, A10, A11, GOBOARD5:3
.= |[((G * (len G),1) `1 ),((G * 1,(j + 1)) `2 )]| by A13, A10, A11, GOBOARD5:2 ;
A19: (((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 ;
r * ((G * 1,j) `2 ) <= r * ((G * 1,(j + 1)) `2 ) by A5, A14, XREAL_1:66;
then A20: (((1 - r) * (1 / 2)) * (((G * 1,j) `2 ) + ((G * 1,(j + 1)) `2 ))) + (r * ((G * 1,j) `2 )) < (G * 1,(j + 1)) `2 by A15, A19, XREAL_1:10;
p = (((1 - r) * ((1 / 2) * ((G * (len G),j) + (G * (len G),(j + 1))))) + ((1 - r) * |[1,0 ]|)) + (r * ((G * (len G),j) + |[1,0 ]|)) by A4, EUCLID:36
.= ((((1 - r) * (1 / 2)) * ((G * (len G),j) + (G * (len G),(j + 1)))) + ((1 - r) * |[1,0 ]|)) + (r * ((G * (len G),j) + |[1,0 ]|)) by EUCLID:34
.= ((((1 - r) * (1 / 2)) * ((G * (len G),j) + (G * (len G),(j + 1)))) + |[((1 - r) * 1),((1 - r) * 0 )]|) + (r * ((G * (len G),j) + |[1,0 ]|)) by EUCLID:62
.= ((((1 - r) * (1 / 2)) * |[(((G * (len G),1) `1 ) + ((G * (len G),1) `1 )),(((G * 1,j) `2 ) + ((G * 1,(j + 1)) `2 ))]|) + |[(1 - r),0 ]|) + (r * (|[((G * (len G),1) `1 ),((G * 1,j) `2 )]| + |[1,0 ]|)) by A18, A12, EUCLID:60
.= ((((1 - r) * (1 / 2)) * |[(((G * (len G),1) `1 ) + ((G * (len G),1) `1 )),(((G * 1,j) `2 ) + ((G * 1,(j + 1)) `2 ))]|) + |[(1 - r),0 ]|) + ((r * |[((G * (len G),1) `1 ),((G * 1,j) `2 )]|) + (r * |[1,0 ]|)) by EUCLID:36
.= ((((1 - r) * (1 / 2)) * |[(((G * (len G),1) `1 ) + ((G * (len G),1) `1 )),(((G * 1,j) `2 ) + ((G * 1,(j + 1)) `2 ))]|) + |[(1 - r),0 ]|) + (|[(r * ((G * (len G),1) `1 )),(r * ((G * 1,j) `2 ))]| + (r * |[1,0 ]|)) by EUCLID:62
.= ((((1 - r) * (1 / 2)) * |[(((G * (len G),1) `1 ) + ((G * (len G),1) `1 )),(((G * 1,j) `2 ) + ((G * 1,(j + 1)) `2 ))]|) + |[(1 - r),0 ]|) + (|[(r * ((G * (len G),1) `1 )),(r * ((G * 1,j) `2 ))]| + |[(r * 1),(r * 0 )]|) by EUCLID:62
.= ((((1 - r) * (1 / 2)) * |[(((G * (len G),1) `1 ) + ((G * (len G),1) `1 )),(((G * 1,j) `2 ) + ((G * 1,(j + 1)) `2 ))]|) + |[(1 - r),0 ]|) + |[((r * ((G * (len G),1) `1 )) + r),((r * ((G * 1,j) `2 )) + 0 )]| by EUCLID:60
.= (|[(((1 - r) * (1 / 2)) * (((G * (len G),1) `1 ) + ((G * (len G),1) `1 ))),(((1 - r) * (1 / 2)) * (((G * 1,j) `2 ) + ((G * 1,(j + 1)) `2 )))]| + |[(1 - r),0 ]|) + |[((r * ((G * (len G),1) `1 )) + r),((r * ((G * 1,j) `2 )) + 0 )]| by EUCLID:62
.= |[((((1 - r) * (1 / 2)) * (((G * (len G),1) `1 ) + ((G * (len G),1) `1 ))) + (1 - r)),((((1 - r) * (1 / 2)) * (((G * 1,j) `2 ) + ((G * 1,(j + 1)) `2 ))) + 0 )]| + |[((r * ((G * (len G),1) `1 )) + r),((r * ((G * 1,j) `2 )) + 0 )]| by EUCLID:60
.= |[(((((1 - r) * (1 / 2)) * (((G * (len G),1) `1 ) + ((G * (len G),1) `1 ))) + (1 - r)) + ((r * ((G * (len G),1) `1 )) + r)),((((1 - r) * (1 / 2)) * (((G * 1,j) `2 ) + ((G * 1,(j + 1)) `2 ))) + (r * ((G * 1,j) `2 )))]| by EUCLID:60 ;
hence p in Int (cell G,(len G),j) by A16, A20, A17; :: thesis: verum

end;

end;

end;

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