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 Th15:
  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 V being Subset of X1 union X2 st V is open & x in
  V & V 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 U1 being Subset of X1 such that
A2: U1 is open and
A3: U1 c= A1 and
A4: x1 in U1 by CONNSP_2:6;
  consider U2 being Subset of X2 such that
A5: U2 is open and
A6: U2 c= A2 and
A7: x2 in U2 by CONNSP_2:6;
  consider V being Subset of X1 union X2 such that
A8: V is open & x in V & V c= U1 \/ U2 by A1,A2,A4,A5,A7,Th14;
  take V;
  U1 \/ U2 c= A1 \/ A2 by A3,A6,XBOOLE_1:13;
  hence thesis by A8,XBOOLE_1:1;
end;
