
theorem Th24:
  for L be non empty Poset for x be Element of L holds x is
completely-irreducible implies for X be Subset of L st ex_inf_of X,L & x = inf
  X holds x in X
proof
  let L be non empty Poset;
  let x be Element of L;
  assume x is completely-irreducible;
  then consider q be Element of L such that
A1: x < q and
A2: for s be Element of L st x < s holds q <= s and
  uparrow x = {x} \/ uparrow q by Th20;
  let X be Subset of L;
  assume that
A3: ex_inf_of X,L and
A4: x = inf X and
A5: not x in X;
A6: X c= uparrow q
  proof
    let y be object;
    assume
A7: y in X;
    then reconsider y1 = y as Element of L;
    inf X is_<=_than X by A3,YELLOW_0:31;
    then x <= y1 by A4,A7,LATTICE3:def 8;
    then x < y1 by A5,A7,ORDERS_2:def 6;
    then q <= y1 by A2;
    hence thesis by WAYBEL_0:18;
  end;
  ex_inf_of uparrow q,L by WAYBEL_0:39;
  then inf (uparrow q) <= inf X by A3,A6,YELLOW_0:35;
  then q <= x by A4,WAYBEL_0:39;
  hence contradiction by A1,ORDERS_2:6;
end;
