
theorem Th36:
  for V being RealUnitarySpace, W being Subspace of V, v being
  VECTOR of V holds 0.V in v + W iff v in W
proof
  let V be RealUnitarySpace;
  let W be Subspace of V;
  let v be VECTOR of V;
  thus 0.V in v + W implies v in W
  proof
    assume 0.V in v + W;
    then consider u being VECTOR of V such that
A1: 0.V = v + u and
A2: u in W;
    v = - u by A1,RLVECT_1:def 10;
    hence thesis by A2,Th16;
  end;
  assume v in W;
  then
A3: - v in W by Th16;
  0.V = v - v by RLVECT_1:15
    .= v + (- v);
  hence thesis by A3;
end;
