
theorem Th18:
  for X being non empty set, Y being Subset-Family of X holds
  Y is prebasis of TopStruct (#X, UniCl FinMeetCl Y#)
proof
  let X be non empty set, A be Subset-Family of X;
  set T = TopStruct (#X, UniCl FinMeetCl A#);
  reconsider A9 = A as Subset-Family of T;
  now
    A9 c= FinMeetCl A & FinMeetCl A c= the topology of T by Th1,Th4;
    then A9 c= the topology of T;
    hence A9 is open by TOPS_2:64;
    thus A9 is quasi_prebasis
    proof
      reconsider B = FinMeetCl A9 as Basis of T by Th16;
      take B;
      thus thesis;
    end;
  end;
  hence thesis;
end;
