
theorem Th45:

:: 1.8. THEOREM, (1) <=> (2), generalized, p. 145
  for S,T being Lawson complete TopLattice
  for f being meet-preserving Function of S,T holds
  f is continuous iff f is directed-sups-preserving &
  for X being non empty Subset of S holds f preserves_inf_of X
proof
  let S,T be Lawson complete TopLattice;
A1: [#]T <> {};

set Ss = the Scott TopAugmentation of S,Ts = the Scott TopAugmentation of T,Sl
= the lower correct TopAugmentation of S,Tl = the lower correct TopAugmentation
of T;
A2: S is TopAugmentation of S by YELLOW_9:44;
A3: T is TopAugmentation of T by YELLOW_9:44;
A4: S is Refinement of Ss,Sl by A2,WAYBEL19:29;
A5: T is Refinement of Ts,Tl by A3,WAYBEL19:29;
A6: T is TopAugmentation of Ts by YELLOW_9:45;
A7: S is TopAugmentation of Ss by YELLOW_9:45;
A8: the RelStr of Ss = the RelStr of S by YELLOW_9:def 4;
A9: the RelStr of Sl = the RelStr of S by YELLOW_9:def 4;
A10: the RelStr of Ts = the RelStr of T by YELLOW_9:def 4;
A11: the RelStr of Tl = the RelStr of T by YELLOW_9:def 4;
  let f be meet-preserving Function of S,T;
  reconsider g = f as Function of Sl,Tl by A9,A11;
  reconsider h = f as Function of Ss,Ts by A8,A10;
A12: [#]Ts <> {};
  hereby
    assume
A13: f is continuous;
    now
      let P be Subset of Ts;
      reconsider A = P as Subset of Ts;
      reconsider C = A as Subset of T by A10;
      assume
A14:  P is open;
      then C is open by A6,WAYBEL19:37;
      then
A15:  f"C is open by A1,A13,TOPS_2:43;
      A is upper by A14,WAYBEL11:def 4;
      then h"A is upper by A8,A10,WAYBEL17:2,WAYBEL_9:1;
      then f"C is upper by A8,WAYBEL_0:25;
      hence h"P is open by A7,A15,WAYBEL19:41;
    end;
    then h is continuous by A12,TOPS_2:43;
    hence f is directed-sups-preserving by A8,A10,Th6;
    for X being non empty filtered Subset of S holds f preserves_inf_of X
    proof
      let F be non empty filtered Subset of S;
      assume ex_inf_of F,S;
      thus ex_inf_of f.:F,T by YELLOW_0:17;
      {inf F} = Lim (F opp+id) by WAYBEL19:43;
      then Im(f,inf F) c= Lim (f*(F opp+id)) by A13,Th24;
      then {f.inf F} c= Lim (f*(F opp+id)) by SETWISEO:8;
      then
A16:  f.inf F in Lim (f*(F opp+id)) by ZFMISC_1:31;
      reconsider G = f.:F as filtered non empty Subset of T by WAYBEL20:24;
A17:  rng the mapping of f*(F opp+id)
      = rng (f*the mapping of F opp+id) by WAYBEL_9:def 8
        .= rng (f * id F) by WAYBEL19:27
        .= rng (f|F) by RELAT_1:65
        .= G by RELAT_1:115;
      Lim (f*(F opp+id)) = {inf (f*(F opp+id))} by Th44
        .= {Inf the mapping of f*(F opp+id)} by WAYBEL_9:def 2
        .= {inf G} by A17,YELLOW_2:def 6;
      hence thesis by A16,TARSKI:def 1;
    end;
    hence for X being non empty Subset of S holds f preserves_inf_of X by Th4;
  end;
  assume f is directed-sups-preserving;
  then
A18: h is directed-sups-preserving by A8,A10,Th6;
  assume for X being non empty Subset of S holds f preserves_inf_of X;
  then for X being non empty Subset of Sl
  holds g preserves_inf_of X by A9,A11,WAYBEL_0:65;
  then g is continuous by WAYBEL19:8;
  hence thesis by A4,A5,A18,WAYBEL19:24;
end;
