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

theorem
  for L be LATTICE,l be Element of L st (uparrow l \ {l}) is Filter of L
  holds l is irreducible
proof
  let L be LATTICE, l be Element of L;
  set F = (uparrow l \ {l});
  assume
A1: (uparrow l \ {l}) is Filter of L;
  now
    let x,y be Element of L;
    assume that
A2: l = x "/\" y and
A3: x <> l and
A4: y <> l;
    l <= y by A2,YELLOW_0:23;
    then y in uparrow l by WAYBEL_0:18;
    then
A5: y in F by A4,ZFMISC_1:56;
    l <= x by A2,YELLOW_0:23;
    then x in uparrow l by WAYBEL_0:18;
    then x in F by A3,ZFMISC_1:56;
    then consider z being Element of L such that
A6: z in F and
A7: z <= x & z <= y by A1,A5,WAYBEL_0:def 2;
    l >= z by A2,A7,YELLOW_0:23;
    then l in F by A1,A6,WAYBEL_0:def 20;
    hence contradiction by ZFMISC_1:56;
  end;
  hence thesis;
end;
