reserve i,j for Nat;

theorem
  for K being Ring, M being Matrix of K holds 0.(K, len M, width M) - M = - M
proof
  let K be Ring, M be Matrix of K;
A1: len (-M) = len M by MATRIX_3:def 2;
A2: width (-M) = width M by MATRIX_3:def 2;
A3: len (0.(K, len M, width M)) = len M by MATRIX_0:def 2;
  per cases by NAT_1:3;
  suppose
A4: len M>0;
    then width (0.(K, len M, width M)) = width M by A3,MATRIX_0:20;
    then
A5: 0.(K, len M, width M) - M = (-M) + 0.(K, len M, width M) by A1,A2,A3,
MATRIX_3:2;
    -M is Matrix of len M, width M, K by A1,A2,A4,MATRIX_0:20;
    hence thesis by A5,MATRIX_3:4;
  end;
  suppose
A6: len M = 0;
    len (0.(K, len M, width M) - M) = len (0.(K, len M, width M)) by
MATRIX_3:def 3;
    hence thesis by A1,A3,A6,CARD_2:64;
  end;
end;
