
theorem Th7:
  for V being RealUnitarySpace, W1,W2 being Subspace of V holds W1
  is Subspace of W1 + W2 & W2 is Subspace of W1 + W2
proof
  let V be RealUnitarySpace;
  let W1,W2 be Subspace of V;
  the carrier of W1 c= the carrier of W1 + W2 by Lm2;
  hence W1 is Subspace of W1 + W2 by RUSUB_1:22;
  the carrier of W2 c= the carrier of W2 + W1 by Lm2;
  then the carrier of W2 c= the carrier of W1 + W2 by Lm1;
  hence thesis by RUSUB_1:22;
end;
