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 ]|),((1 / 2) * ((G * 1,j) + (G * 1,(j + 1)))) c= (Int (cell G,0 ,j)) \/ {((1 / 2) * ((G * 1,j) + (G * 1,(j + 1))))}
let G be Go-board; :: thesis: ( 1 <= j & j < width G implies LSeg (((1 / 2) * ((G * 1,j) + (G * 1,(j + 1)))) - |[1,0 ]|),((1 / 2) * ((G * 1,j) + (G * 1,(j + 1)))) c= (Int (cell G,0 ,j)) \/ {((1 / 2) * ((G * 1,j) + (G * 1,(j + 1))))} )
assume that
A1: 1 <= j and
A2: j < width G ; :: thesis: LSeg (((1 / 2) * ((G * 1,j) + (G * 1,(j + 1)))) - |[1,0 ]|),((1 / 2) * ((G * 1,j) + (G * 1,(j + 1)))) c= (Int (cell G,0 ,j)) \/ {((1 / 2) * ((G * 1,j) + (G * 1,(j + 1))))}
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 ]|),((1 / 2) * ((G * 1,j) + (G * 1,(j + 1)))) or x in (Int (cell G,0 ,j)) \/ {((1 / 2) * ((G * 1,j) + (G * 1,(j + 1))))} )
assume A3: x in LSeg (((1 / 2) * ((G * 1,j) + (G * 1,(j + 1)))) - |[1,0 ]|),((1 / 2) * ((G * 1,j) + (G * 1,(j + 1)))) ; :: thesis: x in (Int (cell G,0 ,j)) \/ {((1 / 2) * ((G * 1,j) + (G * 1,(j + 1))))}
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 * ((1 / 2) * ((G * 1,j) + (G * 1,(j + 1))))) 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 {((1 / 2) * ((G * 1,j) + (G * 1,(j + 1))))}

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

end;

case A7: r < 1 ; :: thesis: p in Int (cell G,0 ,j)

set r3 = (1 - r) * (1 / 2);
set s3 = r * (1 / 2);
set r2 = (G * 1,1) `1 ;
set s1 = (G * 1,j) `2 ;
set s2 = (G * 1,(j + 1)) `2 ;
A8: (((1 - r) * (1 / 2)) * (((G * 1,j) `2 ) + ((G * 1,j) `2 ))) + ((r * (1 / 2)) * (((G * 1,j) `2 ) + ((G * 1,j) `2 ))) = (G * 1,j) `2 ;
A9: j + 1 <= width G by A2, NAT_1:13;
0 <> len G by GOBOARD1:def 5;
then A10: 1 <= len G by NAT_1:14;
j < j + 1 by XREAL_1:31;
then A11: (G * 1,j) `2 < (G * 1,(j + 1)) `2 by A1, A9, A10, GOBOARD5:5;
then A12: ((G * 1,j) `2 ) + ((G * 1,j) `2 ) < ((G * 1,j) `2 ) + ((G * 1,(j + 1)) `2 ) by XREAL_1:8;
then A13: (r * (1 / 2)) * (((G * 1,j) `2 ) + ((G * 1,j) `2 )) <= (r * (1 / 2)) * (((G * 1,j) `2 ) + ((G * 1,(j + 1)) `2 )) by A5, XREAL_1:66;
A14: 1 - r > 0 by A7, XREAL_1:52;
then A15: (1 - r) * (1 / 2) > (1 / 2) * 0 by XREAL_1:70;
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 A12, XREAL_1:70;
then A16: (G * 1,j) `2 < (((1 - r) * (1 / 2)) * (((G * 1,j) `2 ) + ((G * 1,(j + 1)) `2 ))) + ((r * (1 / 2)) * (((G * 1,j) `2 ) + ((G * 1,(j + 1)) `2 ))) by A13, A8, XREAL_1:10;
(G * 1,1) `1 < ((G * 1,1) `1 ) + (1 - r) by A14, XREAL_1:31;
then A17: ((G * 1,1) `1 ) - (1 - r) < (G * 1,1) `1 by XREAL_1:21;
A18: 1 <= j + 1 by A1, NAT_1:13;
A19: 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 A18, A9, A10, GOBOARD5:3 ;
A20: ((G * 1,j) `2 ) + ((G * 1,(j + 1)) `2 ) < ((G * 1,(j + 1)) `2 ) + ((G * 1,(j + 1)) `2 ) by A11, XREAL_1:8;
then A21: (r * (1 / 2)) * (((G * 1,j) `2 ) + ((G * 1,(j + 1)) `2 )) <= (r * (1 / 2)) * (((G * 1,(j + 1)) `2 ) + ((G * 1,(j + 1)) `2 )) by A5, XREAL_1:66;
A22: Int (cell G,0 ,j) = { |[r9,s9]| where r9, s9 is Real : ( r9 < (G * 1,1) `1 & (G * 1,j) `2 < s9 & s9 < (G * 1,(j + 1)) `2 ) } by A1, A2, Th23;
A23: (((1 - r) * (1 / 2)) * (((G * 1,(j + 1)) `2 ) + ((G * 1,(j + 1)) `2 ))) + ((r * (1 / 2)) * (((G * 1,(j + 1)) `2 ) + ((G * 1,(j + 1)) `2 ))) = (G * 1,(j + 1)) `2 ;
((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 A15, A20, XREAL_1:70;
then A24: (((1 - r) * (1 / 2)) * (((G * 1,j) `2 ) + ((G * 1,(j + 1)) `2 ))) + ((r * (1 / 2)) * (((G * 1,j) `2 ) + ((G * 1,(j + 1)) `2 ))) < (G * 1,(j + 1)) `2 by A21, A23, XREAL_1:10;
A25: G * 1,j = |[((G * 1,j) `1 ),((G * 1,j) `2 )]| by EUCLID:57
.= |[((G * 1,1) `1 ),((G * 1,j) `2 )]| by A1, A2, A10, GOBOARD5:3 ;
p = (((1 - r) * ((1 / 2) * ((G * 1,j) + (G * 1,(j + 1))))) - ((1 - r) * |[1,0 ]|)) + (r * ((1 / 2) * ((G * 1,j) + (G * 1,(j + 1))))) by A4, EUCLID:53
.= ((((1 - r) * (1 / 2)) * ((G * 1,j) + (G * 1,(j + 1)))) - ((1 - r) * |[1,0 ]|)) + (r * ((1 / 2) * ((G * 1,j) + (G * 1,(j + 1))))) by EUCLID:34
.= ((((1 - r) * (1 / 2)) * ((G * 1,j) + (G * 1,(j + 1)))) - |[((1 - r) * 1),((1 - r) * 0 )]|) + (r * ((1 / 2) * ((G * 1,j) + (G * 1,(j + 1))))) by EUCLID:62
.= ((((1 - r) * (1 / 2)) * ((G * 1,j) + (G * 1,(j + 1)))) - |[(1 - r),0 ]|) + ((r * (1 / 2)) * ((G * 1,j) + (G * 1,(j + 1)))) by EUCLID:34
.= ((((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 * (1 / 2)) * ((G * 1,j) + (G * 1,(j + 1)))) by A19, A25, 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 * (1 / 2)) * |[(((G * 1,1) `1 ) + ((G * 1,1) `1 )),(((G * 1,j) `2 ) + ((G * 1,(j + 1)) `2 ))]|) by A19, A25, EUCLID:60
.= (|[(((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 * (1 / 2)) * |[(((G * 1,1) `1 ) + ((G * 1,1) `1 )),(((G * 1,j) `2 ) + ((G * 1,(j + 1)) `2 ))]|) by EUCLID:62
.= (|[(((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 * (1 / 2)) * (((G * 1,1) `1 ) + ((G * 1,1) `1 ))),((r * (1 / 2)) * (((G * 1,j) `2 ) + ((G * 1,(j + 1)) `2 )))]| 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 * (1 / 2)) * (((G * 1,1) `1 ) + ((G * 1,1) `1 ))),((r * (1 / 2)) * (((G * 1,j) `2 ) + ((G * 1,(j + 1)) `2 )))]| by EUCLID:66
.= |[(((((1 - r) * (1 / 2)) * (((G * 1,1) `1 ) + ((G * 1,1) `1 ))) - (1 - r)) + ((r * (1 / 2)) * (((G * 1,1) `1 ) + ((G * 1,1) `1 )))),((((1 - r) * (1 / 2)) * (((G * 1,j) `2 ) + ((G * 1,(j + 1)) `2 ))) + ((r * (1 / 2)) * (((G * 1,j) `2 ) + ((G * 1,(j + 1)) `2 ))))]| by EUCLID:60 ;
hence p in Int (cell G,0 ,j) by A17, A16, A24, A22; :: thesis: verum

end;

end;

end;

hence x in (Int (cell G,0 ,j)) \/ {((1 / 2) * ((G * 1,j) + (G * 1,(j + 1))))} by XBOOLE_0:def 3; :: thesis: verum