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

theorem Th01:
  for X being non empty set st X is cap-finite-closed holds
  X is cap-closed
  proof
    let X be non empty set;
    assume
A1: X is cap-finite-closed;
    now
      let a,b be set;
      assume a in X & b in X;
      then {a,b} c= X by TARSKI:def 2;
      then meet {a,b} in X by A1;
      hence a/\b in X by SETFAM_1:11;
    end;
    hence thesis;
  end;
