reserve X for non empty TopSpace;
reserve x for Point of X;
reserve U1 for Subset of X;

theorem Th5:
  U1 is a_neighborhood of x implies ex V being non empty Subset of
  X st V is a_neighborhood of x & V is open & V c= U1
proof
  assume U1 is a_neighborhood of x;
  then x in Int(U1) by Def1;
  then consider V being Subset of X such that
A1: V is open & V c= U1 and
A2: x in V by TOPS_1:22;
  reconsider V as non empty Subset of X by A2;
  take V;
  thus thesis by A1,A2,Th3;
end;
