reserve a,b,r for Real;

theorem Th3:
  for V be Abelian add-associative right_zeroed
  vector-distributive scalar-distributive scalar-associative scalar-unital
   non empty RLSStruct, V1 be add-closed multi-closed non
empty Subset of V st 0.V in V1 holds RLSStruct (# V1,In (0.V, V1), add|(V1,V),
    Mult_ V1 #) is Abelian add-associative right_zeroed vector-distributive
    scalar-distributive scalar-associative scalar-unital
proof
  let V be Abelian add-associative right_zeroed vector-distributive
  scalar-distributive scalar-associative scalar-unital non
  empty RLSStruct, V1 be add-closed multi-closed non empty Subset of V;
  assume 0.V in V1;
  then In(0.V,V1) = 0.V by SUBSET_1:def 8;
  hence thesis by Th2;
end;
