
theorem Th36:
  for X being non empty TopSpace st InclPoset the topology of X is
continuous for Y being injective T_0-TopSpace holds oContMaps(X, Y) is complete
  continuous
proof
  let X be non empty TopSpace such that
A1: InclPoset the topology of X is continuous;
  InclPoset the topology of X, oContMaps(X, Sierpinski_Space)
  are_isomorphic by Th6;
  then reconsider
  XS = oContMaps(X, Sierpinski_Space) as complete continuous non
  empty Poset by A1,WAYBEL15:9,WAYBEL20:18;
  let Y be injective T_0-TopSpace;
  consider M being non empty set such that
A2: Y is_Retract_of M-TOP_prod (M --> Sierpinski_Space) by WAYBEL18:19;
  for i be Element of M holds (M --> Sierpinski_Space).i is injective by
FUNCOP_1:7;
  then reconsider MS = M-TOP_prod (M --> Sierpinski_Space) as injective
  T_0-TopSpace by WAYBEL18:7;
  for i be Element of M holds (M --> XS).i is complete continuous LATTICE
  by FUNCOP_1:7;
  then
A3: M-POS_prod (M --> XS) is complete continuous by WAYBEL20:33;
  M-POS_prod (M --> oContMaps(X, Sierpinski_Space)), oContMaps(X, M
  -TOP_prod (M --> Sierpinski_Space)) are_isomorphic by Th35,WAYBEL_1:6;
  then oContMaps(X, MS) is complete continuous by A3,WAYBEL15:9,WAYBEL20:18;
  hence thesis by A2,Th23;
end;
