let K be Ring; :: thesis: for M1, M2 being Matrix of K st len M1 = len M2 & width M1 = width M2 & M1 = M1 + M2 holds
M2 = 0. (K,(len M1),(width M1))
let M1, M2 be Matrix of K; :: thesis: ( len M1 = len M2 & width M1 = width M2 & M1 = M1 + M2 implies M2 = 0. (K,(len M1),(width M1)) )
assume that
A1: len M1 = len M2 and
A2: width M1 = width M2 and
A3: M1 = M1 + M2 ; :: thesis: M2 = 0. (K,(len M1),(width M1))
0. (K,(len M1),(width M1)) = (M1 + M2) + (- M1) by A3, Th2;
then 0. (K,(len M1),(width M1)) = (M2 + M1) + (- M1) by A1, A2, MATRIX_3:2;
then 0. (K,(len M1),(width M1)) = M2 + (M1 + (- M1)) by A1, A2, MATRIX_3:3;
then A4: 0. (K,(len M1),(width M1)) = M2 + (0. (K,(len M1),(width M1))) by Th2;

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

suppose len M1 > 0 ; :: thesis: M2 = 0. (K,(len M1),(width M1))

then M2 is Matrix of len M1, width M1,K by A1, A2, MATRIX_0:20;
hence M2 = 0. (K,(len M1),(width M1)) by A4, MATRIX_3:4; :: thesis: verum

end;

suppose A5: len M1 = 0 ; :: thesis: M2 = 0. (K,(len M1),(width M1))

then len (0. (K,(len M1),(width M1))) = 0 ;
hence M2 = 0. (K,(len M1),(width M1)) by A1, A5, CARD_2:64; :: thesis: verum

end;

end;