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 ,F be Filter of X st
  G c= F holds FinMeetCl G c= F & FinMeetCl G is with_non-empty_elements
  proof
    let X be non empty set,G be Subset-Family of X, F be Filter of X;
    assume
A1: G c= F;
A2: FinMeetCl G c= FinMeetCl F by A1,CANTOR_1:14;
    FinMeetCl F c= F by Th02,CARD_FIL:5;
    hence FinMeetCl G c= F by A2;
    hence FinMeetCl G is with_non-empty_elements by CARD_FIL:def 1;
  end;
