let G be Go-board; :: thesis: for i1, i2, j1, j2 being Nat st 1 <= j1 & j1 <= width G & 1 <= j2 & j2 <= width G & 1 <= i1 & i1 < i2 & i2 <= len G holds
(G * (i1,j1)) `1 < (G * (i2,j2)) `1
let i1, i2, j1, j2 be Nat; :: thesis: ( 1 <= j1 & j1 <= width G & 1 <= j2 & j2 <= width G & 1 <= i1 & i1 < i2 & i2 <= len G implies (G * (i1,j1)) `1 < (G * (i2,j2)) `1 )
assume that
A1: 1 <= j1 and
A2: j1 <= width G and
A3: 1 <= j2 and
A4: j2 <= width G and
A5: 1 <= i1 and
A6: i1 < i2 and
A7: i2 <= len G ; :: thesis: (G * (i1,j1)) `1 < (G * (i2,j2)) `1
A8: 1 <= i2 by A5, A6, XXREAL_0:2;
then (G * (i2,j1)) `1 = (G * (i2,1)) `1 by A1, A2, A7, GOBOARD5:2
.= (G * (i2,j2)) `1 by A3, A4, A7, A8, GOBOARD5:2 ;
hence (G * (i1,j1)) `1 < (G * (i2,j2)) `1 by A1, A2, A5, A6, A7, GOBOARD5:3; :: thesis: verum