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

theorem
  for A,B being Subset of X holds A is a_neighborhood of x & B is
  a_neighborhood of x iff A /\ B is a_neighborhood of x
proof
  let A,B be Subset of X;
  A is a_neighborhood of x & B is a_neighborhood of x iff x in Int A & x
  in Int B by Def1;
  then A is a_neighborhood of x & B is a_neighborhood of x iff x in Int A /\
  Int B by XBOOLE_0:def 4;
  then
  A is a_neighborhood of x & B is a_neighborhood of x iff x in Int (A /\ B
  ) by TOPS_1:17;
  hence thesis by Def1;
end;
