
theorem
  for S,T being continuous complete LATTICE holds UPS(S,T) is continuous
proof
  let S, T be continuous complete LATTICE;
  consider X being non empty set, p being projection Function of BoolePoset X,
  BoolePoset X such that
A1: p is directed-sups-preserving and
A2: T, Image p are_isomorphic by WAYBEL15:18;
A3: (id S)*id S = id S by QUANTAL1:def 9;
  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
A4: UPS(S, T), Image UPS(id S, p) are_isomorphic by A1,Th37;
  set L = the Scott TopAugmentation of S;
A5: InclPoset sigma L is continuous by WAYBEL14:36;
A6: UPS(S, BoolePoset{0}), InclPoset sigma S are_isomorphic by Th34;
  p*p = p by QUANTAL1:def 9;
  then UPS(id S, p) * UPS(id S, p) = UPS(id S, p) by A3,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
A7: UPS(id S, p) is directed-sups-preserving projection Function of UPS(S,
  BoolePoset X), UPS(S, BoolePoset X) by WAYBEL_1:def 13;
  BoolePoset X, (BoolePoset{0})|^X are_isomorphic by Th18;
  then
A8: UPS(S, BoolePoset X), UPS(S, (BoolePoset{0})|^X) are_isomorphic by Th36;
  the RelStr of L = the RelStr of S by YELLOW_9:def 4;
  then InclPoset sigma S is continuous by A5,YELLOW_9:52;
  then UPS(S, BoolePoset{0}) is continuous complete
      by A6,WAYBEL15:9,WAYBEL_1:6;
  then for x being Element of X holds (X --> UPS(S, BoolePoset{0})).x is
  continuous complete LATTICE;
  then X-POS_prod(X --> UPS(S, BoolePoset{0})) is continuous by WAYBEL20:33;
  then
A9: UPS(S, BoolePoset{0})|^X is continuous 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 A8,
WAYBEL_1:7;
  then UPS(S, BoolePoset X) is continuous LATTICE by A9,WAYBEL15:9,WAYBEL_1:6;
  then Image UPS(id S, p) is continuous by A7,WAYBEL15:15;
  hence thesis by A4,WAYBEL15:9,WAYBEL_1:6;
end;
