reserve X for set,
  F for Filter of BoolePoset X,
  x for Element of BoolePoset X ,
  z for Element of X;

theorem
  for L being continuous complete non empty Poset, f being Function of
  FixedUltraFilters X, the carrier of L holds f-extension_to_hom is monotone
proof
  let L be continuous complete non empty Poset, f be Function of
  FixedUltraFilters X, the carrier of L;
  let F1, F2 be Element of InclPoset Filt BoolePoset X;
  set F = f-extension_to_hom;
  set F1s = {"/\"({f.(uparrow x) : ex z st x = {z} & z in Y }, L) where Y is
  Subset of X : Y in F1};
  set F2s = {"/\"({f.(uparrow x) : ex z st x = {z} & z in Y }, L) where Y is
  Subset of X : Y in F2 };
A1: ex_sup_of F1s, L & ex_sup_of F2s, L by YELLOW_0:17;
A2: F.F1 = "\/"({"/\"({f.(uparrow x) : ex z st x = {z} & z in Y }, L) where
  Y is Subset of X : Y in F1}, L) by Def3;
  assume F1 <= F2;
  then
A3: F1 c= F2 by YELLOW_1:3;
A4: F1s c= F2s
  proof
    let s be object;
    assume s in F1s;
    then
    ex Y being Subset of X st s = "/\"({f.(uparrow x) : ex z st x = {z} & z
    in Y}, L) & Y in F1;
    hence thesis by A3;
  end;
A5: F.F2 = "\/"({"/\"({f.(uparrow x) : ex z st x = {z} & z in Y }, L) where
  Y is Subset of X : Y in F2}, L) by Def3;
  let FF1, FF2 be Element of L;
  assume FF1 = F.F1 & FF2 = F.F2;
  hence FF1 <= FF2 by A2,A5,A1,A4,YELLOW_0:34;
end;
