
theorem Th33:
  for S being complete LATTICE ex F being Function of UPS(S,
  BoolePoset{0}), InclPoset sigma S st F is isomorphic & for f being
  directed-sups-preserving Function of S, BoolePoset{0} holds F.f = f"{1}
proof
  set T = BoolePoset{0};
  reconsider T9 = Omega Sierpinski_Space as Scott TopAugmentation of T by Th31,
WAYBEL26:4;
  let S be complete LATTICE;
  set S9 = the Scott TopAugmentation of S;
A1: T = the RelStr of T9 by YELLOW_9:def 4;
A2: the topology of S9 = sigma S by YELLOW_9:51;
  the RelStr of S = the RelStr of S9 by YELLOW_9:def 4;
  then UPS(S, T) = UPS(S9, T9) by A1,Th25
    .= SCMaps(S9, T9) by Th24
    .= ContMaps(S9, T9) by WAYBEL24:38
    .= oContMaps(S9, Sierpinski_Space) by WAYBEL26:def 1;
  then consider G being Function of InclPoset sigma S, UPS(S, T) such that
A3: G is isomorphic and
A4: for V being open Subset of S9 holds G.V = chi(V, the carrier of S9)
  by A2,WAYBEL26:5;
  take F = G";
A5: rng G = [#]UPS(S,T) by A3,WAYBEL_0:66;
  then G is onto by FUNCT_2:def 3;
  then
A6: F = G qua Function" by A3,TOPS_2:def 4;
  hence F is isomorphic by A3,WAYBEL_0:68;
  let f be directed-sups-preserving Function of S, T;
  f in the carrier of UPS(S, T) by Def4;
  then consider V being object such that
A7: V in dom G and
A8: f = G.V by A5,FUNCT_1:def 3;
  dom G = the carrier of InclPoset sigma S by FUNCT_2:def 1
    .= sigma S by YELLOW_1:1;
  then reconsider V as open Subset of S9 by A2,A7,PRE_TOPC:def 2;
  thus F.f = V by A3,A6,A7,A8,FUNCT_1:34
    .= chi(V, the carrier of S9)"{1} by Th13
    .= f"{1} by A4,A8;
end;
