let i, j be Element of 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, GOBOARD5:3
.= (G * (i + 1),j) `1 by A2, A4, GOBOARD5:3 ;
A6: (G * (i + 1),(j + 1)) `2 = (G * 1,(j + 1)) `2 by A4, A3, GOBOARD5:2
.= (G * i,(j + 1)) `2 by A1, A3, GOBOARD5:2 ;
A7: (G * i,j) `2 = (G * 1,j) `2 by A1, A2, GOBOARD5:2
.= (G * (i + 1),j) `2 by A2, A4, GOBOARD5:2 ;
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, GOBOARD5:3
.= (G * i,(j + 1)) `1 by A1, A3, GOBOARD5:3 ;
((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:57
.= (G * i,(j + 1)) + (G * (i + 1),j) by EUCLID:57 ;
:: thesis: verum