
theorem
  for V being RealUnitarySpace, W1,W2 being Subspace of V, v being
  VECTOR of V st W1 is Subspace of W2 holds v + W1 c= v + W2
proof
  let V be RealUnitarySpace;
  let W1,W2 being Subspace of V;
  let v being VECTOR of V;
  assume
A1: W1 is Subspace of W2;
  let x be object;
  assume x in v + W1;
  then consider u being VECTOR of V such that
A2: u in W1 and
A3: x = v + u by RUSUB_2:63;
  u in W2 by A1,A2,RUSUB_1:1;
  hence thesis by A3,RUSUB_2:63;
end;
