reserve A, B for non empty set,
  A1, A2, A3 for non empty Subset of A;
reserve X for TopSpace;
reserve X for non empty TopSpace;
reserve X1, X2 for non empty SubSpace of X;

theorem Th16:
  for x being Point of X1 union X2 for x1 being Point of X1, x2
  being Point of X2 st x1 = x & x2 = x for A1 being a_neighborhood of x1, A2
  being a_neighborhood of x2 ex A being a_neighborhood of x st A c= A1 \/ A2
proof
  let x be Point of X1 union X2;
  let x1 be Point of X1, x2 be Point of X2 such that
A1: x1 = x & x2 = x;
  let A1 be a_neighborhood of x1, A2 be a_neighborhood of x2;
  consider V being Subset of X1 union X2 such that
A2: V is open & x in V and
A3: V c= A1 \/ A2 by A1,Th15;
  reconsider W = V as a_neighborhood of x by A2,CONNSP_2:3;
  take W;
  thus thesis by A3;
end;
