reserve i,j for Nat;

theorem
  for K being Ring,M1,M2,M3 being Matrix of K st len M1=len M2 & width
  M1=width M2 holds M1 - M3 = (M1-M2)+(M2 - M3)
proof
  let K be Ring,M1,M2,M3 be Matrix of K;
  assume
A1: len M1=len M2 & width M1=width M2;
  then
A2: len (-M2)=len M1 & width (-M2)=width M1 by MATRIX_3:def 2;
  len (M1+-M2)=len M1 & width (M1+-M2)=width M1 by MATRIX_3:def 3;
  then (M1-M2)+(M2 - M3)=M1+-M2+M2+-M3 by A1,MATRIX_3:3
    .=M1+(-M2+M2)+-M3 by A2,MATRIX_3:3
    .=M1+(M2-M2)+-M3 by A1,A2,MATRIX_3:2
    .=M1+-M3 by A1,Th20;
  hence thesis;
end;
