let i, j be Element of 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, GOBOARD5:4;
A7: (G * i,j) `1 <= (G * (i + 1),j) `1 by A1, A2, A3, A5, GOBOARD5:4;
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, GOBOARD5:5;
A10: (G * (i + 1),(j + 1)) `1 = (G * (i + 1),j) `1 by A1, A2, A3, A4, Th18;
then A11: (G * i,j) `1 <= (G * (i + 1),(j + 1)) `1 by A1, A2, A3, A5, GOBOARD5:4;
(G * i,j) `2 <= (G * i,(j + 1)) `2 by A1, A3, A4, A8, GOBOARD5:5;
hence G * i,j in cell G,i,j by A1, A2, A3, A4, A6, Th19; :: 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, Th18;
then (G * i,(j + 1)) `1 <= (G * (i + 1),j) `1 by A1, A2, A3, A5, GOBOARD5:4;
hence G * i,(j + 1) in cell G,i,j by A1, A2, A3, A4, A12, A9, Th19; :: 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, Th18;
then (G * i,j) `2 <= (G * (i + 1),(j + 1)) `2 by A1, A3, A4, A8, GOBOARD5:5;
hence G * (i + 1),(j + 1) in cell G,i,j by A1, A2, A3, A4, A10, A11, A13, Th19; :: 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, Th18;
then (G * (i + 1),j) `2 <= (G * i,(j + 1)) `2 by A1, A3, A4, A8, GOBOARD5:5;
hence G * (i + 1),j in cell G,i,j by A1, A2, A3, A4, A7, A14, Th19; :: thesis: verum