let j be Element of NAT ; :: thesis: for G being Go-board st 1 <= j & j + 1 <= width G holds
((1 / 2) * ((G * (len G),j) + (G * (len G),(j + 1)))) + |[1,0 ]| in Int (cell G,(len G),j)
let G be Go-board; :: thesis: ( 1 <= j & j + 1 <= width G implies ((1 / 2) * ((G * (len G),j) + (G * (len G),(j + 1)))) + |[1,0 ]| in Int (cell G,(len G),j) )
assume A1: ( 1 <= j & j + 1 <= width G ) ; :: thesis: ((1 / 2) * ((G * (len G),j) + (G * (len G),(j + 1)))) + |[1,0 ]| in Int (cell G,(len G),j)
set s1 = (G * (len G),j) `2 ;
set r1 = (G * (len G),j) `1 ;
set s2 = (G * (len G),(j + 1)) `2 ;
A2: j < width G by A1, NAT_1:13;
len G <> 0 by GOBOARD1:def 5;
then A3: ( 1 <= j + 1 & 1 <= len G ) by NAT_1:11, NAT_1:14;
A4: (1 / 2) * (((G * (len G),j) `1 ) + ((G * (len G),j) `1 )) = (G * (len G),j) `1 ;
A5: G * (len G),j = |[((G * (len G),j) `1 ),((G * (len G),j) `2 )]| by EUCLID:57;
A6: (G * (len G),1) `1 = (G * (len G),j) `1 by A1, A2, A3, GOBOARD5:3;
(G * (len G),1) `1 = (G * (len G),(j + 1)) `1 by A1, A3, GOBOARD5:3;
then G * (len G),(j + 1) = |[((G * (len G),j) `1 ),((G * (len G),(j + 1)) `2 )]| by A6, EUCLID:57;
then (G * (len G),j) + (G * (len G),(j + 1)) = |[(((G * (len G),j) `1 ) + ((G * (len G),j) `1 )),(((G * (len G),j) `2 ) + ((G * (len G),(j + 1)) `2 ))]| by A5, EUCLID:60;
then (1 / 2) * ((G * (len G),j) + (G * (len G),(j + 1))) = |[((G * (len G),j) `1 ),((1 / 2) * (((G * (len G),j) `2 ) + ((G * (len G),(j + 1)) `2 )))]| by A4, EUCLID:62;
then A7: ((1 / 2) * ((G * (len G),j) + (G * (len G),(j + 1)))) + |[1,0 ]| = |[(((G * (len G),j) `1 ) + 1),(((1 / 2) * (((G * (len G),j) `2 ) + ((G * (len G),(j + 1)) `2 ))) + 0 )]| by EUCLID:60;
A8: (G * 1,j) `2 = (G * (len G),j) `2 by A1, A2, A3, GOBOARD5:2;
len G <> 0 by GOBOARD1:def 5;
then ( 1 <= len G & j < j + 1 ) by NAT_1:14, XREAL_1:31;
then A9: (G * (len G),j) `2 < (G * (len G),(j + 1)) `2 by A1, GOBOARD5:5;
then ((G * (len G),j) `2 ) + ((G * (len G),j) `2 ) < ((G * (len G),j) `2 ) + ((G * (len G),(j + 1)) `2 ) by XREAL_1:8;
then A10: (1 / 2) * (((G * (len G),j) `2 ) + ((G * (len G),j) `2 )) < (1 / 2) * (((G * (len G),j) `2 ) + ((G * (len G),(j + 1)) `2 )) by XREAL_1:70;
((G * (len G),j) `2 ) + ((G * (len G),(j + 1)) `2 ) < ((G * (len G),(j + 1)) `2 ) + ((G * (len G),(j + 1)) `2 ) by A9, XREAL_1:8;
then (1 / 2) * (((G * (len G),j) `2 ) + ((G * (len G),(j + 1)) `2 )) < (1 / 2) * (((G * (len G),(j + 1)) `2 ) + ((G * (len G),(j + 1)) `2 )) by XREAL_1:70;
then A11: (1 / 2) * (((G * (len G),j) `2 ) + ((G * (len G),(j + 1)) `2 )) < (G * 1,(j + 1)) `2 by A1, A3, GOBOARD5:2;
A12: (G * (len G),1) `1 < ((G * (len G),j) `1 ) + 1 by A6, XREAL_1:31;
( 1 <= j & j < width G ) by A1, NAT_1:13;
then Int (cell G,(len G),j) = { |[r,s]| where r, s is Real : ( (G * (len G),1) `1 < r & (G * 1,j) `2 < s & s < (G * 1,(j + 1)) `2 ) } by Th26;
hence ((1 / 2) * ((G * (len G),j) + (G * (len G),(j + 1)))) + |[1,0 ]| in Int (cell G,(len G),j) by A7, A8, A10, A11, A12; :: thesis: verum