reserve x for set,
  K for Ring,
  r for Scalar of K,
  V for LeftMod of K,
  a,b,a1,a2 for Vector of V,
  A,A1,A2 for Subset of V,
  l for Linear_Combination of A,
  W for Subspace of V,
  Li for FinSequence of Submodules(V);

theorem
  A1 <> {} & A1 c= A2 implies for x st x is Vector of A1 holds x is Vector of
  A2
proof
  assume that
A1: A1 <> {} and
A2: A1 c= A2;
  let x;
  assume x is Vector of A1;
  then reconsider a=x as Vector of A1;
  a is Element of A1 by A1,Def11;
  then a in A2 by A1,A2,TARSKI:def 3;
  hence thesis by Def11;
end;
