
theorem :: COROLLARY 4.24
  for L be complete algebraic LATTICE for p be Element of L holds (ex k
be Element of L st k in the carrier of CompactSublatt L & p is_maximal_in (the
  carrier of L) \ uparrow k) iff p is completely-irreducible
proof
  let L be complete algebraic LATTICE;
  let p be Element of L;
  thus (ex k be Element of L st k in the carrier of CompactSublatt L & p
  is_maximal_in (the carrier of L) \ uparrow k) implies p is
  completely-irreducible by Th26;
  thus p is completely-irreducible implies ex k be Element of L st k in the
  carrier of CompactSublatt L & p is_maximal_in (the carrier of L) \ uparrow k
  proof
    defpred P[Element of L] means $1 in uparrow p & ex k be Element of L st k
    in the carrier of CompactSublatt L & $1 is_maximal_in (the carrier of L) \
    uparrow k;
    reconsider A = { x where x is Element of L : P[x]} as Subset of L from
    DOMAIN_1:sch 7;
    assume
A1: p is completely-irreducible;
    then p = inf A by Th35;
    then p in A by A1,Th34;
    then consider x be Element of L such that
A2: x = p and
    x in uparrow p and
A3: ex k be Element of L st k in the carrier of CompactSublatt L & x
    is_maximal_in (the carrier of L) \ uparrow k;
    consider k be Element of L such that
A4: k in the carrier of CompactSublatt L and
A5: x is_maximal_in (the carrier of L) \ uparrow k by A3;
    take k;
    thus k in the carrier of CompactSublatt L by A4;
    thus thesis by A2,A5;
  end;
end;
