theorem
  for V being strict Abelian add-associative right_zeroed
right_complementable vector-distributive scalar-distributive
scalar-associative scalar-unital non empty ModuleStr over GF, W being strict
  Subspace of V holds (for v being Element of V holds v in W) implies W = V
proof
  let V be strict Abelian add-associative right_zeroed right_complementable
  vector-distributive scalar-distributive scalar-associative scalar-unital
   non empty ModuleStr over GF, W be strict Subspace of V;
  assume for v being Element of V holds v in W;
  then
A1: for v be Element of V holds ( v in W iff v in V);
  V is Subspace of V by Th24;
  hence thesis by A1,Th30;
end;
