theorem
  Intersection F1 = meet rng F1
proof
  per cases;
  suppose
A1: F1 <> {};
    now
      let x be object;
      x in Intersection F1 iff for n st n in dom F1 holds x in F1.n by A1,Th51;
      hence x in Intersection F1 iff x in meet rng F1 by A1,Th52;
    end;
    hence thesis by TARSKI:2;
  end;
  suppose
    F1 = {};
    hence thesis by Def6,RELAT_1:38,SETFAM_1:1;
  end;
end;
