let i, j be Nat; :: thesis: for G being Go-board st 1 <= i & i < len G & 1 <= j & j < width G holds
(G * (i,j)) + (G * ((i + 1),(j + 1))) = (G * (i,(j + 1))) + (G * ((i + 1),j))
let G be Go-board; :: thesis: ( 1 <= i & i < len G & 1 <= j & j < width G implies (G * (i,j)) + (G * ((i + 1),(j + 1))) = (G * (i,(j + 1))) + (G * ((i + 1),j)) )
assume that
A1: ( 1 <= i & i < len G ) and
A2: ( 1 <= j & j < width G ) ; :: thesis: (G * (i,j)) + (G * ((i + 1),(j + 1))) = (G * (i,(j + 1))) + (G * ((i + 1),j))
A3: ( 1 <= j + 1 & j + 1 <= width G ) by A2, NAT_1:13;
A4: ( 1 <= i + 1 & i + 1 <= len G ) by A1, NAT_1:13;
then A5: (G * ((i + 1),(j + 1))) `1 = (G * ((i + 1),1)) `1 by A3, GOBOARD5:2
.= (G * ((i + 1),j)) `1 by A2, A4, GOBOARD5:2 ;
A6: (G * ((i + 1),(j + 1))) `2 = (G * (1,(j + 1))) `2 by A4, A3, GOBOARD5:1
.= (G * (i,(j + 1))) `2 by A1, A3, GOBOARD5:1 ;
A7: (G * (i,j)) `2 = (G * (1,j)) `2 by A1, A2, GOBOARD5:1
.= (G * ((i + 1),j)) `2 by A2, A4, GOBOARD5:1 ;
A8: ((G * (i,j)) + (G * ((i + 1),(j + 1)))) `2 = ((G * (i,j)) `2) + ((G * ((i + 1),(j + 1))) `2) by Lm1
.= ((G * (i,(j + 1))) + (G * ((i + 1),j))) `2 by A7, A6, Lm1 ;
A9: (G * (i,j)) `1 = (G * (i,1)) `1 by A1, A2, GOBOARD5:2
.= (G * (i,(j + 1))) `1 by A1, A3, GOBOARD5:2 ;
((G * (i,j)) + (G * ((i + 1),(j + 1)))) `1 = ((G * (i,j)) `1) + ((G * ((i + 1),(j + 1))) `1) by Lm1
.= ((G * (i,(j + 1))) + (G * ((i + 1),j))) `1 by A9, A5, Lm1 ;
hence (G * (i,j)) + (G * ((i + 1),(j + 1))) = |[(((G * (i,(j + 1))) + (G * ((i + 1),j))) `1),(((G * (i,(j + 1))) + (G * ((i + 1),j))) `2)]| by A8, EUCLID:53
.= (G * (i,(j + 1))) + (G * ((i + 1),j)) by EUCLID:53 ;
:: thesis: verum