reserve i,j for Nat;

theorem
  for K being Ring, M1, M2, M3 being Matrix of K st len M1 = len M2 &
len M2 = len M3 & width M1 = width M2 & width M2 = width M3 holds (M3 - M1) - (
  M3 - M2) = - (M1 - M2)
proof
  let K be Ring, M1, M2, M3 be Matrix of K;
  assume that
A1: len M1 = len M2 and
A2: len M2 = len M3 and
A3: width M1 = width M2 and
A4: width M2 = width M3;
A5: len (-M1) = len M1 & width (-M1) = width M1 by MATRIX_3:def 2;
A6: len (-M2) = len M2 & width (-M2) = width M2 by MATRIX_3:def 2;
  (M3 - M1) - (M3 - M2) = M2 - M1 by A1,A2,A3,A4,Th18
    .= (- M1) + M2 by A1,A3,A5,MATRIX_3:2
    .= (- M1) +--M2 by Th1
    .= - (M1 + -M2) by A1,A3,A6,Th12;
  hence thesis;
end;
