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 Th11:
  for x being Point of X1 union X2 holds (ex x1 being Point of X1
  st x1 = x) or ex x2 being Point of X2 st x2 = x
proof
  let x be Point of X1 union X2;
  reconsider A0 = the carrier of X1 union X2 as Subset of X by TSEP_1:1;
  reconsider A1 = the carrier of X1 as Subset of X by TSEP_1:1;
  reconsider A2 = the carrier of X2 as Subset of X by TSEP_1:1;
  assume
A1: not ex x1 being Point of X1 st x1 = x;
  ex x2 being Point of X2 st x2 = x
  proof
    A0 = A1 \/ A2 & not x in A1 by A1,TSEP_1:def 2;
    then reconsider x2 = x as Point of X2 by XBOOLE_0:def 3;
    take x2;
    thus thesis;
  end;
  hence thesis;
end;
