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 + M3 = (M1 + M2) + (M3
  - 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+M2)=len M1 & width (M1+M2)=width M1 by MATRIX_3:def 3;
A6: len (-M2)=len M1 & width (-M2)=width M1 by A1,A3,MATRIX_3:def 2;
  hence (M1 + M2) + (M3 - M2)=M1+M2+(-M2+M3) by A1,A2,A3,A4,MATRIX_3:2
    .=M1+M2+-M2+M3 by A6,A5,MATRIX_3:3
    .=M1+(M2-M2)+M3 by A1,A3,MATRIX_3:3
    .=M1+M3 by A1,A3,Th20;
end;
