reserve
  X for non empty set,
  FX for Filter of X,
  SFX for Subset-Family of X;

theorem
  for X be non empty set, G be Subset-Family of X st
  FinMeetCl G is with_non-empty_elements holds
  FinMeetCl G is filter_base of X &
  ex F be Filter of X st FinMeetCl G c= F
  proof
    let X be non empty set,G be Subset-Family of X;
    assume
A1: FinMeetCl G is with_non-empty_elements;
    reconsider FG=FinMeetCl G as Subset-Family of X;
    now
      let b1,b2 be Element of FG;
      FinMeetCl FG c= FG by CANTOR_1:11;
      then FG is cap-closed by Th02;
      then reconsider b=b1/\b2 as Element of FG;
      b is Element of FG & b c= b1/\b2;
      hence ex b be Element of FG st b c= b1/\b2;
    end;
    then FG is quasi_basis;
    then reconsider FG as filter_base of X by A1;
    FG c= <.FG.) by def3;
    hence thesis;
  end;
