let i1, j1, i2, j2 be Element of NAT ; :: thesis: for G being Go-board st [i1,j1] in Indices G & [i2,j2] in Indices G & G * i1,j1 = G * i2,j2 holds
( i1 = i2 & j1 = j2 )
let G be Go-board; :: thesis: ( [i1,j1] in Indices G & [i2,j2] in Indices G & G * i1,j1 = G * i2,j2 implies ( i1 = i2 & j1 = j2 ) )
assume that
A1: [i1,j1] in Indices G and
A2: [i2,j2] in Indices G and
A3: G * i1,j1 = G * i2,j2 ; :: thesis: ( i1 = i2 & j1 = j2 )
A4: ( 1 <= i1 & i1 <= len G ) by A1, MATRIX_1:39;
A5: ( 1 <= j1 & j1 <= width G ) by A1, MATRIX_1:39;
A6: ( 1 <= i2 & i2 <= len G ) by A2, MATRIX_1:39;
A7: ( 1 <= j2 & j2 <= width G ) by A2, MATRIX_1:39;
then A8: (G * i1,j2) `1 = (G * i1,1) `1 by A4, GOBOARD5:3
.= (G * i1,j1) `1 by A4, A5, GOBOARD5:3 ;
A9: (G * i1,j2) `2 = (G * 1,j2) `2 by A4, A7, GOBOARD5:2
.= (G * i1,j1) `2 by A3, A6, A7, GOBOARD5:2 ;
assume A10: ( not i1 = i2 or not j1 = j2 ) ; :: thesis: contradiction

per cases ( i1 < i2 or i1 > i2 or j1 < j2 or j1 > j2 ) by A10, XXREAL_0:1;

suppose i1 < i2 ; :: thesis: contradiction

hence contradiction by A3, A4, A6, A7, A8, GOBOARD5:4; :: thesis: verum

end;

suppose i1 > i2 ; :: thesis: contradiction

hence contradiction by A3, A4, A6, A7, A8, GOBOARD5:4; :: thesis: verum

end;

suppose j1 < j2 ; :: thesis: contradiction

hence contradiction by A4, A5, A7, A9, GOBOARD5:5; :: thesis: verum

end;

suppose j1 > j2 ; :: thesis: contradiction

hence contradiction by A4, A5, A7, A9, GOBOARD5:5; :: thesis: verum

end;

end;