
theorem Th26: :: PROPOSITION 4.21 (i)
  for L be complete LATTICE for p be Element of L holds (ex k be
  Element of L st p is_maximal_in (the carrier of L) \ uparrow k) implies p is
  completely-irreducible
proof
  let L be complete LATTICE;
  let p be Element of L;
  given k be Element of L such that
A1: p is_maximal_in (the carrier of L) \ uparrow k;
  k <= p "\/" k by YELLOW_0:22;
  then
A2: p "\/" k in uparrow k by WAYBEL_0:18;
  p <= p "\/" k by YELLOW_0:22;
  then p "\/" k in uparrow p by WAYBEL_0:18;
  then
A3: ex_inf_of (uparrow p) \ {p},L & p "\/" k in (uparrow p) /\ (uparrow k)
  by A2,XBOOLE_0:def 4,YELLOW_0:17;
A4: (uparrow p) \ {p} = (uparrow p) /\ (uparrow k) by A1,Th3;
  then "/\" ((uparrow p) \ {p},L) = p "\/" k by Th1;
  then ex_min_of (uparrow p) \ {p},L by A4,A3,WAYBEL_1:def 4;
  hence thesis;
end;
