
theorem Th15: :: THEOREM 2.14.
  for L be continuous complete LATTICE for p be projection
  Function of L,L st p is directed-sups-preserving holds Image p is continuous
  LATTICE
proof
  let L be continuous complete LATTICE;
  let p be projection Function of L,L such that
A1: p is directed-sups-preserving;
  reconsider Lk = {k where k is Element of L: p.k <= k} as non empty Subset of
  L by WAYBEL_1:43;
A2: subrelstr Lk is infs-inheriting by WAYBEL_1:50;
  reconsider pk = p|Lk as Function of subrelstr Lk, subrelstr Lk by WAYBEL_1:46
;
A3: pk is kernel by WAYBEL_1:48;
  now
    let X be Subset of subrelstr Lk;
    reconsider X9 = X as Subset of L by WAYBEL13:3;
    assume
A4: X is non empty directed;
    then reconsider X9 as non empty directed Subset of L by YELLOW_2:7;
A5: p preserves_sup_of X9 by A1,WAYBEL_0:def 37;
    now
      X c= the carrier of subrelstr Lk;
      then X c= Lk by YELLOW_0:def 15;
      then
A6:   pk.:X = p.:X by RELAT_1:129;
      assume ex_sup_of X,subrelstr Lk;
      subrelstr Lk is infs-inheriting by WAYBEL_1:50;
      then subrelstr Lk is complete LATTICE by YELLOW_2:30;
      hence ex_sup_of pk.:X,subrelstr Lk by YELLOW_0:17;
A7:   ex_sup_of X,L by YELLOW_0:17;
A8:   subrelstr Lk is directed-sups-inheriting by A1,WAYBEL_1:52;
      then
A9:   dom pk = the carrier of subrelstr Lk & sup X9 = sup X by A4,A7,
FUNCT_2:def 1,WAYBEL_0:7;
      ex_sup_of p.:X,L & pk.:X is directed Subset of subrelstr Lk by A3,A4,
YELLOW_0:17,YELLOW_2:15;
      hence sup (pk.:X) = sup (p.:X) by A4,A8,A6,WAYBEL_0:7
        .= p.sup X9 by A5,A7,WAYBEL_0:def 31
        .= pk.sup X by A9,FUNCT_1:47;
    end;
    hence pk preserves_sup_of X by WAYBEL_0:def 31;
  end;
  then
A10: pk is directed-sups-preserving by WAYBEL_0:def 37;
  subrelstr Lk is directed-sups-inheriting by A1,WAYBEL_1:52;
  then
A11: subrelstr Lk is continuous LATTICE by A2,WAYBEL_5:28;
A12: the carrier of subrelstr rng p = rng p by YELLOW_0:def 15
    .= rng pk by WAYBEL_1:44
    .= the carrier of subrelstr rng pk by YELLOW_0:def 15;
  subrelstr rng pk is full SubRelStr of L by Th1;
  then
A13: Image p = Image pk by A12,YELLOW_0:57;
  subrelstr Lk is complete by A2,YELLOW_2:30;
  hence thesis by A11,A3,A13,A10,Th14;
end;
