
theorem Th8:
  for X being non empty TopSpace for L being Scott up-complete
  non empty TopPoset for F being non empty directed Subset of ContMaps(X, L)
  holds "\/"(F, L|^the carrier of X) is continuous Function of X, L
proof
  let X be non empty TopSpace;
  let L be Scott up-complete non empty TopPoset;
  let F be non empty directed Subset of ContMaps(X, L);
  set sF = "\/"(F, L|^the carrier of X);
  Funcs(the carrier of X, the carrier of L) = the carrier of L|^the
  carrier of X by YELLOW_1:28;
  then reconsider sF as Function of X, L by FUNCT_2:66;
  ContMaps(X, L) is full SubRelStr of L|^the carrier of X by WAYBEL24:def 3;
  then reconsider
  aF = F as non empty directed Subset of L|^the carrier of X by YELLOW_2:7;
A1: now
    let A be Subset of L;
    assume
A2: A is open;
    now
      let x be set;
      hereby
        assume
A3:     x in sF"A;
        then
A4:     sF.x in A by FUNCT_1:def 7;
        reconsider y = x as Element of X by A3;
A5:     (the carrier of X)-POS_prod ((the carrier of X) => L) = L|^the
        carrier of X by YELLOW_1:def 5;
A6:     ((the carrier of X) => L).y = L;
        then
A7:     pi(aF,y) is directed non empty by A5,YELLOW16:35;
A8:     ex_sup_of aF, L|^the carrier of X by WAYBEL_0:75;
        then (sup aF).y = sup pi(aF,y) by A6,A5,YELLOW16:33;
        then pi(aF,y) meets A by A2,A4,A7,WAYBEL11:def 1;
        then consider a being object such that
A9:     a in pi(aF,y) and
A10:    a in A by XBOOLE_0:3;
        consider f being Function such that
A11:    f in aF and
A12:    a = f.y by A9,CARD_3:def 6;
        reconsider f as continuous Function of X, L by A11,WAYBEL24:21;
        take Q = f"A;
        [#]L <> {};
        hence Q is open by A2,TOPS_2:43;
A13:    dom sF = the carrier of X by FUNCT_2:def 1;
        thus Q c= sF"A
        proof
          let b be object;
          assume
A14:      b in Q;
          then
A15:      f.b in A by FUNCT_1:def 7;
          reconsider b as Element of X by A14;
A16:      ((the carrier of X) => L).b = L;
          then pi(aF,b) is directed non empty by A5,YELLOW16:35;
          then
A17:      ex_sup_of pi(aF,b), L by WAYBEL_0:75;
          sF.b = sup pi(aF,b) by A8,A5,A16,YELLOW16:33;
          then
A18:      sF.b is_>=_than pi(aF,b) by A17,YELLOW_0:30;
          f.b in pi(aF,b) by A11,CARD_3:def 6;
          then f.b <= sF.b by A18;
          then sF.b in A by A2,A15,WAYBEL_0:def 20;
          hence thesis by A13,FUNCT_1:def 7;
        end;
        dom f = the carrier of X by FUNCT_2:def 1;
        hence x in Q by A10,A12,FUNCT_1:def 7;
      end;
      thus (ex Q being Subset of X st Q is open & Q c= sF"A & x in Q) implies
      x in sF"A;
    end;
    hence sF"A is open by TOPS_1:25;
  end;
  [#]L <> {};
  hence thesis by A1,TOPS_2:43;
end;
