
theorem
  for V being RealUnitarySpace, W being Subspace of V, u,v being VECTOR
  of V holds v - u in v + W iff u in W
proof
  let V be RealUnitarySpace;
  let W be Subspace of V;
  let u,v be VECTOR of V;
A1: v - u = (- u) + v;
A2: - u in W implies - (- u) in W by Th16;
  u in W implies - u in W by Th16;
  hence thesis by A1,A2,Th55,RLVECT_1:17;
end;
