theorem
  for V being strict RealLinearSpace, W being strict Subspace of V holds
  (for v being VECTOR of V holds v in W iff v in V) implies W = V
proof
  let V be strict RealLinearSpace, W be strict Subspace of V;
  assume
A1: for v being VECTOR of V holds v in W iff v in V;
  V is Subspace of V by Th25;
  hence thesis by A1,Th31;
end;
