
theorem Th17:
  for V being RealUnitarySpace, W2 being Subspace of V, W1 being
  strict Subspace of V holds W1 is Subspace of W2 iff W1 /\ W2 = W1
proof
  let V be RealUnitarySpace;
  let W2 be Subspace of V;
  let W1 be strict Subspace of V;
  thus W1 is Subspace of W2 implies W1 /\ W2 = W1
  proof
    assume W1 is Subspace of W2;
    then
A1: the carrier of W1 c= the carrier of W2 by RUSUB_1:def 1;
    the carrier of W1 /\ W2 = (the carrier of W1) /\ (the carrier of W2)
    by Def2;
    hence thesis by A1,RUSUB_1:24,XBOOLE_1:28;
  end;
  thus thesis by Th16;
end;
