reserve i,j for Nat;

theorem Th20:
  for K being Ring,M1,M2 being Matrix of K st len M1=len M2 &
  width M1=width M2 holds M1 = M1 + (M2 - M2)
proof
  let K be Ring,M1,M2 be Matrix of K;
  assume
A1: len M1=len M2 & width M1=width M2;
  per cases by NAT_1:3;
  suppose
A2: len M1>0;
    then
A3: M1 is Matrix of len M1,width M1,K by MATRIX_0:20;
    M2 is Matrix of len M1,width M1,K by A1,A2,MATRIX_0:20;
    hence M1+(M2-M2)=M1+0.(K,len M1,width M1) by MATRIX_3:5
      .=M1 by A3,MATRIX_3:4;
  end;
  suppose
A4: len M1 = 0;
    len (M1 + (M2 - M2)) = len M1 by MATRIX_3:def 3;
    hence thesis by A4,CARD_2:64;
  end;
end;
