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 implies A \/ B is a_neighborhood of x
proof
  let A,B be Subset of X;
  reconsider A1 = A, B1 = B as Subset of X;
  A1 is a_neighborhood of x & B1 is a_neighborhood of x implies A1 \/ B1
  is a_neighborhood of x
  proof
    assume that
A1: x in Int A1 and
    x in Int B1;
A2: Int A1 \/ Int B1 c= Int (A1 \/ B1) by TOPS_1:20;
    x in Int A1 \/ Int B1 by A1,XBOOLE_0:def 3;
    hence thesis by A2,Def1;
  end;
  hence thesis;
end;
