reserve i,j for Nat;

theorem Th2:
  for K being Ring,M being Matrix of K holds M+(-M)=0.(K,len M, width M)
proof
  let K be Ring,M be Matrix of K;
  per cases by NAT_1:3;
  suppose
    len M>0;
    then M is Matrix of len M,width M,K by MATRIX_0:20;
    hence thesis by MATRIX_3:5;
  end;
  suppose
A1: len M = 0;
A2: len (M+(-M)) = len M by MATRIX_3:def 3;
    len (0.(K,len M,width M)) = 0 by A1;
    hence thesis by A1,A2,CARD_2:64;
  end;
end;
