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) in cell (G,i,j) & G * (i,(j + 1)) in cell (G,i,j) & G * ((i + 1),(j + 1)) in cell (G,i,j) & G * ((i + 1),j) in cell (G,i,j) )
let G be Go-board; :: thesis: ( 1 <= i & i + 1 <= len G & 1 <= j & j + 1 <= width G implies ( G * (i,j) in cell (G,i,j) & G * (i,(j + 1)) in cell (G,i,j) & G * ((i + 1),(j + 1)) in cell (G,i,j) & G * ((i + 1),j) in cell (G,i,j) ) )
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) in cell (G,i,j) & G * (i,(j + 1)) in cell (G,i,j) & G * ((i + 1),(j + 1)) in cell (G,i,j) & G * ((i + 1),j) in cell (G,i,j) )
A5: ( i < i + 1 & j < width G ) by A4, NAT_1:13;
then A6: (G * (i,j)) `1 <= (G * ((i + 1),j)) `1 by A1, A2, A3, GOBOARD5:3;
A7: (G * (i,j)) `1 <= (G * ((i + 1),j)) `1 by A1, A2, A3, A5, GOBOARD5:3;
A8: ( j < j + 1 & i < len G ) by A2, NAT_1:13;
then A9: (G * (i,j)) `2 <= (G * (i,(j + 1))) `2 by A1, A3, A4, GOBOARD5:4;
A10: (G * ((i + 1),(j + 1))) `1 = (G * ((i + 1),j)) `1 by A1, A2, A3, A4, Th16;
then A11: (G * (i,j)) `1 <= (G * ((i + 1),(j + 1))) `1 by A1, A2, A3, A5, GOBOARD5:3;
(G * (i,j)) `2 <= (G * (i,(j + 1))) `2 by A1, A3, A4, A8, GOBOARD5:4;
hence G * (i,j) in cell (G,i,j) by A1, A2, A3, A4, A6, Th17; :: thesis: ( G * (i,(j + 1)) in cell (G,i,j) & G * ((i + 1),(j + 1)) in cell (G,i,j) & G * ((i + 1),j) in cell (G,i,j) )
A12: (G * (i,j)) `1 = (G * (i,(j + 1))) `1 by A1, A2, A3, A4, Th16;
then (G * (i,(j + 1))) `1 <= (G * ((i + 1),j)) `1 by A1, A2, A3, A5, GOBOARD5:3;
hence G * (i,(j + 1)) in cell (G,i,j) by A1, A2, A3, A4, A12, A9, Th17; :: thesis: ( G * ((i + 1),(j + 1)) in cell (G,i,j) & G * ((i + 1),j) in cell (G,i,j) )
A13: (G * ((i + 1),(j + 1))) `2 = (G * (i,(j + 1))) `2 by A1, A2, A3, A4, Th16;
then (G * (i,j)) `2 <= (G * ((i + 1),(j + 1))) `2 by A1, A3, A4, A8, GOBOARD5:4;
hence G * ((i + 1),(j + 1)) in cell (G,i,j) by A1, A2, A3, A4, A10, A11, A13, Th17; :: thesis: G * ((i + 1),j) in cell (G,i,j)
A14: (G * (i,j)) `2 = (G * ((i + 1),j)) `2 by A1, A2, A3, A4, Th16;
then (G * ((i + 1),j)) `2 <= (G * (i,(j + 1))) `2 by A1, A3, A4, A8, GOBOARD5:4;
hence G * ((i + 1),j) in cell (G,i,j) by A1, A2, A3, A4, A7, A14, Th17; :: thesis: verum