let D be non empty set ; :: thesis: for M1, M2 being Matrix of D st width M1 > 0 & width M2 > 0 & M1 @ = M2 @ & width (M1 @ ) = width (M2 @ ) holds
M1 = M2
let M1, M2 be Matrix of D; :: thesis: ( width M1 > 0 & width M2 > 0 & M1 @ = M2 @ & width (M1 @ ) = width (M2 @ ) implies M1 = M2 )
assume A1: ( width M1 > 0 & width M2 > 0 ) ; :: thesis: ( not M1 @ = M2 @ or not width (M1 @ ) = width (M2 @ ) or M1 = M2 )
assume A2: ( M1 @ = M2 @ & width (M1 @ ) = width (M2 @ ) ) ; :: thesis: M1 = M2
( width (M1 @ ) = len M1 & width (M2 @ ) = len M2 ) by A1, Th12;
hence M1 = M2 by A2, Th11; :: thesis: verum