let K be Field; :: thesis: for M1, M2 being Matrix of K st len M1 = len M2 & width M1 = width M2 & M1 - M2 = 0. (K,(len M1),(width M1)) holds
M1 = M2

let M1, M2 be Matrix of K; :: thesis: ( len M1 = len M2 & width M1 = width M2 & M1 - M2 = 0. (K,(len M1),(width M1)) implies M1 = M2 )
assume that
A1: len M1 = len M2 and
A2: width M1 = width M2 and
A3: M1 - M2 = 0. (K,(len M1),(width M1)) ; :: thesis: M1 = M2
per cases ( len M1 > 0 or len M1 = 0 ) by NAT_1:3;
suppose A4: len M1 > 0 ; :: thesis: M1 = M2
then A5: M2 is Matrix of len M1, width M1,K by ;
A6: ( len (- M2) = len M2 & width (- M2) = width M2 ) by MATRIX_3:def 2;
A7: len (0. (K,(len M1),(width M1))) = len M1 by MATRIX_0:def 2;
then width (0. (K,(len M1),(width M1))) = width M1 by ;
then (M1 + (- M2)) + M2 = M2 + (0. (K,(len M1),(width M1))) by
.= M2 by ;
then M1 + ((- M2) + M2) = M2 by ;
then M1 + (M2 + (- M2)) = M2 by ;
then A8: M1 + (0. (K,(len M1),(width M1))) = M2 by ;
M1 is Matrix of len M1, width M1,K by ;
hence M1 = M2 by ; :: thesis: verum
end;
suppose len M1 = 0 ; :: thesis: M1 = M2
hence M1 = M2 by ; :: thesis: verum
end;
end;