let i1, j1, i2, j2 be Element of NAT ; :: thesis: for G1, G2 being Go-board st Values G1 c= Values G2 & 1 <= i1 & i1 < len G1 & 1 <= j1 & j1 <= width G1 & 1 <= i2 & i2 < len G2 & 1 <= j2 & j2 <= width G2 & G1 * (i1,j1) = G2 * (i2,j2) holds
(G2 * ((i2 + 1),j2)) `1 <= (G1 * ((i1 + 1),j1)) `1
let G1, G2 be Go-board; :: thesis: ( Values G1 c= Values G2 & 1 <= i1 & i1 < len G1 & 1 <= j1 & j1 <= width G1 & 1 <= i2 & i2 < len G2 & 1 <= j2 & j2 <= width G2 & G1 * (i1,j1) = G2 * (i2,j2) implies (G2 * ((i2 + 1),j2)) `1 <= (G1 * ((i1 + 1),j1)) `1 )
assume that
A1: Values G1 c= Values G2 and
A2: 1 <= i1 and
A3: i1 < len G1 and
A4: ( 1 <= j1 & j1 <= width G1 ) and
A5: 1 <= i2 and
A6: i2 < len G2 and
A7: ( 1 <= j2 & j2 <= width G2 ) and
A8: G1 * (i1,j1) = G2 * (i2,j2) ; :: thesis: (G2 * ((i2 + 1),j2)) `1 <= (G1 * ((i1 + 1),j1)) `1
set p = G1 * ((i1 + 1),j1);
A9: i1 + 1 <= len G1 by A3, NAT_1:13;
1 <= i1 + 1 by A2, NAT_1:13;
then [(i1 + 1),j1] in Indices G1 by A4, A9, MATRIX_1:37;
then G1 * ((i1 + 1),j1) in { (G1 * (i,j)) where i, j is Element of NAT : [i,j] in Indices G1 } ;
then G1 * ((i1 + 1),j1) in Values G1 by Th7;
then G1 * ((i1 + 1),j1) in Values G2 by A1;
then G1 * ((i1 + 1),j1) in { (G2 * (i,j)) where i, j is Element of NAT : [i,j] in Indices G2 } by Th7;
then consider i, j being Element of NAT such that
A10: G1 * ((i1 + 1),j1) = G2 * (i,j) and
A11: [i,j] in Indices G2 ;
A12: 1 <= i by A11, MATRIX_1:39;
A13: i <= len G2 by A11, MATRIX_1:39;
( 1 <= j & j <= width G2 ) by A11, MATRIX_1:39;
then A14: (G2 * (i,j)) `1 = (G2 * (i,1)) `1 by A12, A13, GOBOARD5:3
.= (G2 * (i,j2)) `1 by A7, A12, A13, GOBOARD5:3 ;
i1 < i1 + 1 by NAT_1:13;
then A15: (G2 * (i2,j2)) `1 < (G2 * (i,j2)) `1 by A2, A4, A8, A9, A10, A14, GOBOARD5:4;

A16: now

assume i <= i2 ; :: thesis: contradiction
then ( i = i2 or i < i2 ) by XXREAL_0:1;
hence contradiction by A6, A7, A12, A15, GOBOARD5:4; :: thesis: verum

end;

assume A17: (G1 * ((i1 + 1),j1)) `1 < (G2 * ((i2 + 1),j2)) `1 ; :: thesis: contradiction
A18: 1 <= i2 + 1 by A5, NAT_1:13;

now

assume i2 + 1 <= i ; :: thesis: contradiction
then ( i2 + 1 = i or i2 + 1 < i ) by XXREAL_0:1;
hence contradiction by A7, A17, A10, A13, A14, A18, GOBOARD5:4; :: thesis: verum

end;

hence contradiction by A16, NAT_1:13; :: thesis: verum