
theorem Th31: :: THEOREM 4.22
  for L be lower-bounded algebraic LATTICE for x,y be Element of L
holds not y <= x implies ex p be Element of L st p is completely-irreducible &
  x <= p & not y <= p
proof
  let L be lower-bounded algebraic LATTICE;
  let x,y be Element of L;
  assume
A1: not y <= x;
  for z be Element of L holds waybelow z is non empty directed;
  then consider k1 be Element of L such that
A2: k1 << y and
A3: not k1 <= x by A1,WAYBEL_3:24;
  consider k be Element of L such that
A4: k in the carrier of CompactSublatt L and
A5: k1 <= k and
A6: k <= y by A2,WAYBEL_8:7;
  not k <= x by A3,A5,ORDERS_2:3;
  then not x in uparrow k by WAYBEL_0:18;
  then x in (the carrier of L) \ (uparrow k) by XBOOLE_0:def 5;
  then
A7: x in (uparrow k)` by SUBSET_1:def 4;
  k is compact by A4,WAYBEL_8:def 1;
  then uparrow k is Open Filter of L by WAYBEL_8:2;
  then consider p be Element of L such that
A8: x <= p and
A9: p is_maximal_in ((uparrow k)`) by A7,WAYBEL_6:9;
  take p;
  (the carrier of L) \ uparrow k = (uparrow k)` by SUBSET_1:def 4;
  hence p is completely-irreducible by A9,Th26;
  thus x <= p by A8;
  thus not y <= p
  proof
    p in (uparrow k)` by A9,WAYBEL_4:55;
    then p in (the carrier of L) \ uparrow k by SUBSET_1:def 4;
    then
A10: not p in uparrow k by XBOOLE_0:def 5;
    assume y <= p;
    then k <= p by A6,ORDERS_2:3;
    hence contradiction by A10,WAYBEL_0:18;
  end;
end;
