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 Th14:
  for x being Point of X1 union X2 for U1 being Subset of X1, U2
  being Subset of X2 st U1 is open & x in U1 & U2 is open & x in U2 ex V being
  Subset of X1 union X2 st V is open & x in V & V c= U1 \/ U2
proof
  let x be Point of X1 union X2;
  let U1 be Subset of X1, U2 be Subset of X2 such that
A1: U1 is open and
A2: x in U1 and
A3: U2 is open and
A4: x in U2;
A5: X1 is SubSpace of X1 union X2 by TSEP_1:22;
  then reconsider C1 = the carrier of X1 as Subset of X1 union X2 by TSEP_1:1;
  consider V1 being Subset of X1 union X2 such that
A6: V1 is open and
A7: V1 /\ [#]X1 = U1 by A1,A5,TOPS_2:24;
A8: x in V1 by A2,A7,XBOOLE_0:def 4;
A9: X2 is SubSpace of X1 union X2 by TSEP_1:22;
  then reconsider C2 = the carrier of X2 as Subset of X1 union X2 by TSEP_1:1;
  consider V2 being Subset of X1 union X2 such that
A10: V2 is open and
A11: V2 /\ [#]X2 = U2 by A3,A9,TOPS_2:24;
A12: x in V2 by A4,A11,XBOOLE_0:def 4;
  take V = V1 /\ V2;
A13: V /\ C1 c= V1 /\ C1 & V /\ C2 c= V2 /\ C2 by XBOOLE_1:17,26;
  the carrier of X1 union X2 = C1 \/ C2 by TSEP_1:def 2;
  then V = V /\ (C1 \/ C2) by XBOOLE_1:28
    .= (V /\ C1) \/ (V /\ C2) by XBOOLE_1:23;
  hence thesis by A6,A7,A10,A11,A13,A8,A12,XBOOLE_0:def 4,XBOOLE_1:13;
end;
