reserve i,j for Nat;

theorem Th11:
  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 len M1=len M2 & width M1=width M2;
  then
A1: M1 -(M2 - M2)=M1-0.(K,len M1,width M1) by Th2
    .=M1+0.(K,len M1,width M1) by Th9;
  per cases by NAT_1:3;
  suppose
    len M1 > 0;
    then M1 is Matrix of len M1,width M1,K by MATRIX_0:20;
    hence thesis by A1,MATRIX_3:4;
  end;
  suppose
A2: len M1 = 0;
    then len (M1 - (M2 - M2)) = 0 by MATRIX_3:def 3;
    hence thesis by A2,CARD_2:64;
  end;
end;
