
theorem Th35:
  for L be complete algebraic LATTICE for p be Element of L st p
  is completely-irreducible holds p = "/\" ({ x where x is Element of L : x in
  uparrow p & ex k be Element of L st k in the carrier of CompactSublatt L & x
  is_maximal_in (the carrier of L) \ uparrow k },L)
proof
  let L be complete algebraic LATTICE;
  let p be Element of L;
  set A = { x where x is Element of L : x in uparrow p & ex k be Element of L
st k in the carrier of CompactSublatt L & x is_maximal_in (the carrier of L) \
  uparrow k };
  p <= p;
  then
A1: p in uparrow p by WAYBEL_0:18;
  now
    let a be Element of L;
    assume a in A;
    then
    ex a1 be Element of L st a1 = a & a1 in uparrow p & ex k be Element of
L st k in the carrier of CompactSublatt L & a1 is_maximal_in ( the carrier of L
    ) \ uparrow k;
    hence p <= a by WAYBEL_0:18;
  end;
  then
A2: p is_<=_than A by LATTICE3:def 8;
  assume p is completely-irreducible;
  then consider q be Element of L such that
A3: p < q and
A4: for s be Element of L st p < s holds q <= s and
  uparrow p = {p} \/ uparrow q by Th20;
A5: compactbelow p <> compactbelow q
  proof
    assume compactbelow p = compactbelow q;
    then p = sup compactbelow q by WAYBEL_8:def 3
      .= q by WAYBEL_8:def 3;
    hence contradiction by A3;
  end;
  p <= q by A3,ORDERS_2:def 6;
  then compactbelow p c= compactbelow q by WAYBEL13:1;
  then not compactbelow q c= compactbelow p by A5;
  then consider k1 be object such that
A6: k1 in compactbelow q and
A7: not k1 in compactbelow p;
  k1 in { y where y is Element of L: q >= y & y is compact } by A6,
WAYBEL_8:def 2;
  then consider k be Element of L such that
A8: k = k1 and
A9: q >= k and
A10: k is compact;
A11: not ex y be Element of L st y in (the carrier of L) \ uparrow k & p < y
  proof
    given y be Element of L such that
A12: y in (the carrier of L) \ uparrow k and
A13: p < y;
    q <= y by A4,A13;
    then k <= y by A9,ORDERS_2:3;
    then y in uparrow k by WAYBEL_0:18;
    hence contradiction by A12,XBOOLE_0:def 5;
  end;
  not k <= p by A7,A8,A10,WAYBEL_8:4;
  then not p in uparrow k by WAYBEL_0:18;
  then p in (the carrier of L) \ uparrow k by XBOOLE_0:def 5;
  then
A14: p is_maximal_in (the carrier of L) \ uparrow k by A11,WAYBEL_4:55;
  k in the carrier of CompactSublatt L by A10,WAYBEL_8:def 1;
  then p in A by A1,A14;
  then for u be Element of L st u is_<=_than A holds p >= u by LATTICE3:def 8;
  hence thesis by A2,YELLOW_0:33;
end;
