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
  for x being Point of X1 union X2 for F1 being Subset of X1, F2 being
  Subset of X2 st F1 is closed & x in F1 & F2 is closed & x in F2 ex H being
  Subset of X1 union X2 st H is closed & x in H & H c= F1 \/ F2
proof
  let x be Point of X1 union X2;
  let F1 be Subset of X1, F2 be Subset of X2 such that
A1: F1 is closed and
A2: x in F1 and
A3: F2 is closed and
A4: x in F2;
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 H1 being Subset of X1 union X2 such that
A6: H1 is closed and
A7: H1 /\ [#]X1 = F1 by A1,A5,PRE_TOPC:13;
A8: x in H1 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 H2 being Subset of X1 union X2 such that
A10: H2 is closed and
A11: H2 /\ [#]X2 = F2 by A3,A9,PRE_TOPC:13;
A12: x in H2 by A4,A11,XBOOLE_0:def 4;
  take H = H1 /\ H2;
A13: H /\ C1 c= H1 /\ C1 & H /\ C2 c= H2 /\ C2 by XBOOLE_1:17,26;
  the carrier of X1 union X2 = C1 \/ C2 by TSEP_1:def 2;
  then H = H /\ (C1 \/ C2) by XBOOLE_1:28
    .= (H /\ C1) \/ (H /\ C2) by XBOOLE_1:23;
  hence thesis by A6,A7,A10,A11,A13,A8,A12,XBOOLE_0:def 4,XBOOLE_1:13;
end;
