
theorem Th16:
  for L1,L2 be up-complete non empty reflexive antisymmetric
RelStr st the RelStr of L1 = the RelStr of L2 & L1 is satisfying_axiom_K & for
  x be Element of L1 holds compactbelow x is non empty directed holds L2 is
  satisfying_axiom_K
proof
  let L1,L2 be up-complete non empty reflexive antisymmetric RelStr;
  assume that
A1: the RelStr of L1 = the RelStr of L2 and
A2: L1 is satisfying_axiom_K and
A3: for x be Element of L1 holds compactbelow x is non empty directed;
  now
    let x be Element of L2;
    reconsider x9 = x as Element of L1 by A1;
    compactbelow x9 is non empty directed by A3;
    then
A4: ex_sup_of compactbelow x9,L1 by WAYBEL_0:75;
    x9 = sup compactbelow x9 & compactbelow x = compactbelow x9 by A1,A2,Th10;
    hence x = sup compactbelow x by A1,A4,YELLOW_0:26;
  end;
  hence thesis;
end;
