theorem Th8:
  for W2 being strict Subspace of M holds W1 is Subspace of W2 iff W1 + W2 = W2
proof
  let W2 be strict Subspace of M;
  thus W1 is Subspace of W2 implies W1 + W2 = W2
  proof
    assume W1 is Subspace of W2;
    then the carrier of W1 c= the carrier of W2 by VECTSP_4:def 2;
    hence thesis by Lm3;
  end;
  thus thesis by Th7;
end;
