
theorem
  for S,T being algebraic complete LATTICE holds UPS(S,T) is algebraic
proof
  let S, T be algebraic complete LATTICE;
  consider X being non empty set, p being closure Function of BoolePoset X,
  BoolePoset X such that
A1: p is directed-sups-preserving and
A2: T, Image p are_isomorphic by WAYBEL13:35;
  now
    let i be Element of UPS(S, BoolePoset X);
    reconsider f = i as directed-sups-preserving Function of S, BoolePoset X
    by Def4;
A3: UPS(id S, p).f = p*f*(id the carrier of S) by A1,Def5
      .= p*f by FUNCT_2:17;
A4: now
      let s be Element of S;
A5:   id BoolePoset X <= p by WAYBEL_1:def 14;
A6:   (id BoolePoset X).(i.s) = i.s;
      (p*f).s = p.(f.s) by FUNCT_2:15;
      hence i.s <= UPS(id S, p).i.s by A5,A6,A3,YELLOW_2:9;
    end;
    thus (id UPS(S, BoolePoset X)).i <= UPS(id S, p).i by A4,Th23;
  end;
  then
A7: (id UPS(S, BoolePoset X)) <= UPS(id S, p) by YELLOW_2:9;
A8: (id S)*id S = id S by QUANTAL1:def 9;
  p*p = p by QUANTAL1:def 9;
  then UPS(id S, p) * UPS(id S, p) = UPS(id S, p) by A8,A1,Th28;
  then UPS(id S, p) is directed-sups-preserving idempotent Function of UPS(S,
  BoolePoset X), UPS(S, BoolePoset X) by A1,Th30,QUANTAL1:def 9;
  then UPS(id S, p) is projection;
  then reconsider
  Sp = UPS(id S, p) as directed-sups-preserving closure Function of
  UPS(S, BoolePoset X), UPS(S, BoolePoset X) by A7,A1,Th30,WAYBEL_1:def 14;
  Image p is complete non empty Poset by A2,WAYBEL20:18;
  then UPS(S, T), UPS(S, Image p) are_isomorphic by A2,Th36;
  then
A9: UPS(S, T), Image Sp are_isomorphic by A1,Th37;
  BoolePoset X, (BoolePoset{0})|^X are_isomorphic by Th18;
  then
A10: UPS(S, BoolePoset X), UPS(S, (BoolePoset{0})|^X) are_isomorphic by Th36;
  set L = the Scott TopAugmentation of S;
A11: InclPoset sigma S = InclPoset the topology of L by YELLOW_9:51;
A12: the RelStr of L = the RelStr of S by YELLOW_9:def 4;
  then L is algebraic by WAYBEL13:26,32;
  then ex B being Basis of L st B = {uparrow x where x is Element of L: x in
  the carrier of CompactSublatt L} by WAYBEL14:42;
  then InclPoset sigma L is algebraic by WAYBEL14:43;
  then
A13: InclPoset sigma S is algebraic by A12,YELLOW_9:52;
  UPS(S, BoolePoset{0}), InclPoset sigma S are_isomorphic by Th34;
  then UPS(S, BoolePoset{0}) is algebraic lower-bounded by A13,A11,WAYBEL13:32
,WAYBEL_1:6;
  then X-POS_prod(X --> UPS(S, BoolePoset{0})) is lower-bounded algebraic;
  then
A14: UPS(S, BoolePoset{0})|^X is algebraic lower-bounded by YELLOW_1:def 5;
  UPS(S, (BoolePoset{0})|^X), UPS(S, BoolePoset{0})|^X are_isomorphic by Th42;
  then UPS(S, BoolePoset X), UPS(S, BoolePoset{0})|^X are_isomorphic by A10,
WAYBEL_1:7;
  then UPS(S, BoolePoset X) is algebraic lower-bounded LATTICE by A14,
WAYBEL13:32,WAYBEL_1:6;
  then
A15: Image Sp is algebraic by WAYBEL_8:24;
  Image Sp is complete LATTICE by YELLOW_2:30;
  hence thesis by A9,A15,WAYBEL13:32,WAYBEL_1:6;
end;
