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

theorem
  for X be set, B be Subset-Family of X st B={X} holds
  B is upper
  proof
    let X be set,B be Subset-Family of X;
    assume
A1: B={X};
    now
      let Y1,Y2 be Subset of X;
      assume
A3:   Y1 in B & Y1 c= Y2;
      Y1=X & Y2=X by A1,A3,TARSKI:def 1;
      hence Y2 in B by A3;
    end;
    hence thesis;
  end;
