let K be Ring; :: thesis: for M1, M2 being Matrix of K st len M1 = len M2 & width M1 = width M2 holds
M1 = M1 + (M2 - M2)
let M1, M2 be Matrix of K; :: thesis: ( len M1 = len M2 & width M1 = width M2 implies M1 = M1 + (M2 - M2) )
assume A1: ( len M1 = len M2 & width M1 = width M2 ) ; :: thesis: M1 = M1 + (M2 - M2)

per cases ( len M1 > 0 or len M1 = 0 ) by NAT_1:3;

suppose A2: len M1 > 0 ; :: thesis: M1 = M1 + (M2 - M2)

then A3: M1 is Matrix of len M1, width M1,K by MATRIX_0:20;
M2 is Matrix of len M1, width M1,K by A1, A2, MATRIX_0:20;
hence M1 + (M2 - M2) = M1 + (0. (K,(len M1),(width M1))) by MATRIX_3:5
.= M1 by A3, MATRIX_3:4 ;
:: thesis: verum

end;

suppose A4: len M1 = 0 ; :: thesis: M1 = M1 + (M2 - M2)

len (M1 + (M2 - M2)) = len M1 by MATRIX_3:def 3;
hence M1 = M1 + (M2 - M2) by A4, CARD_2:64; :: thesis: verum

end;

end;