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

theorem
  for X be set,S be Subset-Family of X st
  FinMeetCl S is with_non-empty_elements holds
  S is with_non-empty_elements
  proof
    let X be set,S be Subset-Family of X;
    assume
A1: FinMeetCl S is with_non-empty_elements;
    assume not S is with_non-empty_elements;
    then {} in S & S c= FinMeetCl S by CANTOR_1:4;
    hence contradiction by A1;
  end;
