reserve i,j for Nat;

theorem Th31:
  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;
A5: len (-M2)=len M1 & width (-M2)=width M1 by A1,A3,MATRIX_3:def 2;
  M1-(M2+M3)=M1+(-M2+-M3) by A2,A4,Th12
    .=M1-M2+-M3 by A5,MATRIX_3:3;
  hence thesis;
end;
