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 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 ;
then 0. (K,(len M1),(width M1)) = (M2 + M1) + (- M1) by ;
then 0. (K,(len M1),(width M1)) = M2 + (M1 + (- M1)) by ;
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 ;
hence M2 = 0. (K,(len M1),(width M1)) by ; :: 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 ; :: thesis: verum
end;
end;