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 M1 - (M2 - M3) = M1 -
  M2 + M3
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;
  len (-M3)=len M1 & width (-M3)=width M1 by A1,A2,A3,A4,MATRIX_3:def 2;
  then M1-(M2-M3)= M1-M2--M3 by A1,A3,Th31
    .=M1+-M2+M3 by Th1;
  hence thesis;
end;
