let K be Field; :: 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 A1: ( len M1 = len M2 & width M1 = width M2 & M1 = M1 + M2 ) ; :: thesis: M2 = 0. K,(len M1),(width M1)
A2: ( len (- M1) = len M1 & width (- M1) = width M1 ) by MATRIX_3:def 2;
0. K,(len M1),(width M1) = (M1 + M2) + (- M1) by A1, Th2;
then 0. K,(len M1),(width M1) = (M2 + M1) + (- M1) by A1, MATRIX_3:4;
then 0. K,(len M1),(width M1) = M2 + (M1 + (- M1)) by A1, A2, MATRIX_3:5;
then A3: 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, MATRIX_1:20;
hence M2 = 0. K,(len M1),(width M1) by A3, MATRIX_3:6; :: thesis: verum

end;

suppose A4: len M1 = 0 ; :: thesis: M2 = 0. K,(len M1),(width M1)

then len (0. K,(len M1),(width M1)) = 0 by FINSEQ_1:def 18;
hence M2 = 0. K,(len M1),(width M1) by A1, A4, CARD_2:83; :: thesis: verum

end;

end;