let i, j be Nat; :: thesis: for G being Go-board st 1 <= i & i + 1 <= len G & 1 <= j & j + 1 <= width G holds
(1 / 2) * ((G * (i,j)) + (G * ((i + 1),(j + 1)))) = (1 / 2) * ((G * (i,(j + 1))) + (G * ((i + 1),j)))

let G be Go-board; :: thesis: ( 1 <= i & i + 1 <= len G & 1 <= j & j + 1 <= width G implies (1 / 2) * ((G * (i,j)) + (G * ((i + 1),(j + 1)))) = (1 / 2) * ((G * (i,(j + 1))) + (G * ((i + 1),j))) )
assume that
A1: 1 <= i and
A2: i + 1 <= len G and
A3: 1 <= j and
A4: j + 1 <= width G ; :: thesis: (1 / 2) * ((G * (i,j)) + (G * ((i + 1),(j + 1)))) = (1 / 2) * ((G * (i,(j + 1))) + (G * ((i + 1),j)))
A5: j < width G by ;
A6: 1 <= j + 1 by NAT_1:11;
A7: 1 <= i + 1 by NAT_1:11;
then A8: (G * ((i + 1),j)) `1 = (G * ((i + 1),1)) `1 by
.= (G * ((i + 1),(j + 1))) `1 by ;
A9: i < len G by ;
then A10: (G * (i,j)) `1 = (G * (i,1)) `1 by
.= (G * (i,(j + 1))) `1 by ;
A11: (G * ((i + 1),(j + 1))) `2 = (G * (1,(j + 1))) `2 by
.= (G * (i,(j + 1))) `2 by ;
A12: (G * (i,j)) `2 = (G * (1,j)) `2 by
.= (G * ((i + 1),j)) `2 by ;
A13: ((1 / 2) * ((G * (i,j)) + (G * ((i + 1),(j + 1))))) `2 = (1 / 2) * (((G * (i,j)) + (G * ((i + 1),(j + 1)))) `2) by TOPREAL3:4
.= (1 / 2) * (((G * (i,j)) `2) + ((G * ((i + 1),(j + 1))) `2)) by TOPREAL3:2
.= (1 / 2) * (((G * (i,(j + 1))) + (G * ((i + 1),j))) `2) by
.= ((1 / 2) * ((G * (i,(j + 1))) + (G * ((i + 1),j)))) `2 by TOPREAL3:4 ;
((1 / 2) * ((G * (i,j)) + (G * ((i + 1),(j + 1))))) `1 = (1 / 2) * (((G * (i,j)) + (G * ((i + 1),(j + 1)))) `1) by TOPREAL3:4
.= (1 / 2) * (((G * (i,j)) `1) + ((G * ((i + 1),(j + 1))) `1)) by TOPREAL3:2
.= (1 / 2) * (((G * (i,(j + 1))) + (G * ((i + 1),j))) `1) by
.= ((1 / 2) * ((G * (i,(j + 1))) + (G * ((i + 1),j)))) `1 by TOPREAL3:4 ;
hence (1 / 2) * ((G * (i,j)) + (G * ((i + 1),(j + 1)))) = |[(((1 / 2) * ((G * (i,(j + 1))) + (G * ((i + 1),j)))) `1),(((1 / 2) * ((G * (i,(j + 1))) + (G * ((i + 1),j)))) `2)]| by
.= (1 / 2) * ((G * (i,(j + 1))) + (G * ((i + 1),j))) by EUCLID:53 ;
:: thesis: verum