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 = M3 - M2
  + M1
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;
  hence M1 - M2 + M3 =-M2+M1+M3 by MATRIX_3:2
    .=-M2+(M1+M3) by A5,MATRIX_3:3
    .=-M2+(M3+M1) by A1,A2,A3,A4,MATRIX_3:2
    .=-M2+M3+M1 by A1,A2,A3,A4,A5,MATRIX_3:3
    .=M3 - M2 + M1 by A1,A2,A3,A4,A5,MATRIX_3:2;
end;
