
theorem Th21:
  for L being up-complete non empty Poset, x,y being Element of L st
  for I being Ideal of L st y <= sup I holds x in I holds x << y
proof
  let L be up-complete non empty Poset;
  let x,y be Element of L;
  assume
A1: for I being Ideal of L st y <= sup I holds x in I;
  let D be non empty directed Subset of L;
  assume
A2: y <= sup D;
  ex_sup_of D,L by WAYBEL_0:75;
  then sup D = sup downarrow D by WAYBEL_0:33;
  then x in downarrow D by A1,A2;
  then ex d being Element of L st x <= d & d in D by WAYBEL_0:def 15;
  hence thesis;
end;
