let i1, i2, j1, j2 be Nat; :: thesis: for G1, G2 being Go-board st G1 * ((i1 -' 1),j1) in 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
(G1 * ((i1 -' 1),j1)) `1 <= (G2 * ((i2 -' 1),j2)) `1
let G1, G2 be Go-board; :: thesis: ( G1 * ((i1 -' 1),j1) in 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 (G1 * ((i1 -' 1),j1)) `1 <= (G2 * ((i2 -' 1),j2)) `1 )
assume that
A1: G1 * ((i1 -' 1),j1) in Values G2 and
A2: 1 < i1 and
A3: ( i1 <= len G1 & 1 <= j1 & j1 <= width G1 ) and
A4: 1 < i2 and
A5: i2 <= len G2 and
A6: ( 1 <= j2 & j2 <= width G2 ) and
A7: G1 * (i1,j1) = G2 * (i2,j2) ; :: thesis: (G1 * ((i1 -' 1),j1)) `1 <= (G2 * ((i2 -' 1),j2)) `1
set p = G1 * ((i1 -' 1),j1);
A8: G1 * ((i1 -' 1),j1) in { (G2 * (i,j)) where i, j is Nat : [i,j] in Indices G2 } by A1, MATRIX_0:39;
1 <= i2 -' 1 by A4, NAT_D:49;
then i2 -' 1 < i2 by NAT_D:51;
then A9: i2 -' 1 < len G2 by A5, XXREAL_0:2;
consider i, j being Nat such that
A10: G1 * ((i1 -' 1),j1) = G2 * (i,j) and
A11: [i,j] in Indices G2 by A8;
A12: 1 <= i by A11, MATRIX_0:32;
A13: i <= len G2 by A11, MATRIX_0:32;
( 1 <= j & j <= width G2 ) by A11, MATRIX_0:32;
then A14: (G2 * (i,j)) `1 = (G2 * (i,1)) `1 by A12, A13, GOBOARD5:2
.= (G2 * (i,j2)) `1 by A6, A12, A13, GOBOARD5:2 ;
A15: 1 <= i1 -' 1 by A2, NAT_D:49;
then i1 -' 1 < i1 by NAT_D:51;
then A16: (G2 * (i,j2)) `1 < (G2 * (i2,j2)) `1 by A3, A7, A15, A10, A14, GOBOARD5:3;
A17: now :: thesis: not i2 <= i
assume i2 <= i ; :: thesis: contradiction
then ( i = i2 or i2 < i ) by XXREAL_0:1;
hence contradiction by A4, A6, A13, A16, GOBOARD5:3; :: thesis: verum
end;
assume A18: (G2 * ((i2 -' 1),j2)) `1 < (G1 * ((i1 -' 1),j1)) `1 ; :: thesis: contradiction
now :: thesis: not i <= i2 -' 1
assume i <= i2 -' 1 ; :: thesis: contradiction
then ( i2 -' 1 = i or i < i2 -' 1 ) by XXREAL_0:1;
hence contradiction by A6, A18, A10, A12, A14, A9, GOBOARD5:3; :: thesis: verum
end;
hence contradiction by A17, NAT_D:49; :: thesis: verum