reserve x,y,Y,Z for set,
  L for LATTICE,
  l for Element of L;

theorem Th13: ::3.6, p.69
  for L be LATTICE, p be Element of L, F be Filter of L st p
  is_maximal_in (F`) holds p is irreducible
proof
  let L be LATTICE, p be Element of L, F be Filter of L such that
A1: p is_maximal_in (F`);
  set X = (the carrier of L)\F;
A2: p in X by A1,WAYBEL_4:55;
  now
    let x,y be Element of L;
    assume that
A3: p = x "/\" y and
A4: x <> p and
A5: y <> p;
    p <= y by A3,YELLOW_0:23;
    then
A6: p < y by A5,ORDERS_2:def 6;
    now
      assume x in F & y in F;
      then consider z being Element of L such that
A7:   z in F and
A8:   z <= x & z <= y by WAYBEL_0:def 2;
      p >= z by A3,A8,YELLOW_0:23;
      then p in F by A7,WAYBEL_0:def 20;
      hence contradiction by A2,XBOOLE_0:def 5;
    end;
    then
A9: x in X or y in X by XBOOLE_0:def 5;
    p <= x by A3,YELLOW_0:23;
    then p < x by A4,ORDERS_2:def 6;
    hence contradiction by A1,A9,A6,WAYBEL_4:55;
  end;
  hence thesis;
end;
