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
( (G * (i,j)) `1 = (G * (i,(j + 1))) `1 & (G * (i,j)) `2 = (G * ((i + 1),j)) `2 & (G * ((i + 1),(j + 1))) `1 = (G * ((i + 1),j)) `1 & (G * ((i + 1),(j + 1))) `2 = (G * (i,(j + 1))) `2 )
let G be Go-board; :: thesis: ( 1 <= i & i + 1 <= len G & 1 <= j & j + 1 <= width G implies ( (G * (i,j)) `1 = (G * (i,(j + 1))) `1 & (G * (i,j)) `2 = (G * ((i + 1),j)) `2 & (G * ((i + 1),(j + 1))) `1 = (G * ((i + 1),j)) `1 & (G * ((i + 1),(j + 1))) `2 = (G * (i,(j + 1))) `2 ) )
assume that
A1: 1 <= i and
A2: i + 1 <= len G and
A3: 1 <= j and
A4: j + 1 <= width G ; :: thesis: ( (G * (i,j)) `1 = (G * (i,(j + 1))) `1 & (G * (i,j)) `2 = (G * ((i + 1),j)) `2 & (G * ((i + 1),(j + 1))) `1 = (G * ((i + 1),j)) `1 & (G * ((i + 1),(j + 1))) `2 = (G * (i,(j + 1))) `2 )
A5: j < width G by A4, NAT_1:13;
A6: 1 <= j + 1 by NAT_1:11;
A7: i < len G by A2, NAT_1:13;
hence (G * (i,j)) `1 = (G * (i,1)) `1 by A1, A3, A5, GOBOARD5:2
.= (G * (i,(j + 1))) `1 by A1, A4, A7, A6, GOBOARD5:2 ;
:: thesis: ( (G * (i,j)) `2 = (G * ((i + 1),j)) `2 & (G * ((i + 1),(j + 1))) `1 = (G * ((i + 1),j)) `1 & (G * ((i + 1),(j + 1))) `2 = (G * (i,(j + 1))) `2 )
A8: 1 <= i + 1 by NAT_1:11;
thus (G * (i,j)) `2 = (G * (1,j)) `2 by A1, A3, A7, A5, GOBOARD5:1
.= (G * ((i + 1),j)) `2 by A2, A3, A5, A8, GOBOARD5:1 ; :: thesis: ( (G * ((i + 1),(j + 1))) `1 = (G * ((i + 1),j)) `1 & (G * ((i + 1),(j + 1))) `2 = (G * (i,(j + 1))) `2 )
thus (G * ((i + 1),(j + 1))) `1 = (G * ((i + 1),1)) `1 by A2, A4, A8, A6, GOBOARD5:2
.= (G * ((i + 1),j)) `1 by A2, A3, A5, A8, GOBOARD5:2 ; :: thesis: (G * ((i + 1),(j + 1))) `2 = (G * (i,(j + 1))) `2
thus (G * ((i + 1),(j + 1))) `2 = (G * (1,(j + 1))) `2 by A2, A4, A8, A6, GOBOARD5:1
.= (G * (i,(j + 1))) `2 by A1, A4, A7, A6, GOBOARD5:1 ; :: thesis: verum