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 Th12:
  X1 meets X2 implies for x being Point of X1 meet X2 holds (ex x1
  being Point of X1 st x1 = x) & ex x2 being Point of X2 st x2 = x
proof
  assume
A1: X1 meets X2;
  reconsider A2 = the carrier of X2 as Subset of X by TSEP_1:1;
  reconsider A1 = the carrier of X1 as Subset of X by TSEP_1:1;
  reconsider A0 = the carrier of X1 meet X2 as Subset of X by TSEP_1:1;
  let x be Point of X1 meet X2;
  A0 = A1 /\ A2 by A1,TSEP_1:def 4;
  then x in A1 & x in A2 by XBOOLE_0:def 4;
  hence thesis;
end;
