let G be Go-board; :: thesis: for i1, i2, j1, j2 being Nat st 1 <= i1 & i1 <= len G & 1 <= i2 & i2 <= len G & 1 <= j1 & j1 < j2 & j2 <= width G holds
(G * (i1,j1)) `2 < (G * (i2,j2)) `2
let i1, i2, j1, j2 be Nat; :: thesis: ( 1 <= i1 & i1 <= len G & 1 <= i2 & i2 <= len G & 1 <= j1 & j1 < j2 & j2 <= width G implies (G * (i1,j1)) `2 < (G * (i2,j2)) `2 )
assume that
A1: 1 <= i1 and
A2: i1 <= len G and
A3: 1 <= i2 and
A4: i2 <= len G and
A5: 1 <= j1 and
A6: j1 < j2 and
A7: j2 <= width G ; :: thesis: (G * (i1,j1)) `2 < (G * (i2,j2)) `2
A8: 1 <= j2 by A5, A6, XXREAL_0:2;
then (G * (i1,j2)) `2 = (G * (1,j2)) `2 by A1, A2, A7, GOBOARD5:1
.= (G * (i2,j2)) `2 by A3, A4, A7, A8, GOBOARD5:1 ;
hence (G * (i1,j1)) `2 < (G * (i2,j2)) `2 by A1, A2, A5, A6, A7, GOBOARD5:4; :: thesis: verum