
theorem Th8:
  for V being RealUnitarySpace, W1 being Subspace of V, W2 being
  strict Subspace of V holds W1 is Subspace of W2 iff W1 + W2 = W2
proof
  let V be RealUnitarySpace;
  let W1 be Subspace of V;
  let W2 be strict Subspace of V;
  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 RUSUB_1:def 1;
    hence thesis by Lm3;
  end;
  thus thesis by Th7;
end;
