
theorem Th23:
  for X being non empty TopSpace for Y,Z being
  monotone-convergence T_0-TopSpace st Y is_Retract_of Z & oContMaps(X, Z) is
  complete continuous holds oContMaps(X, Y) is complete continuous
proof
  let X be non empty TopSpace;
  let Y,Z be monotone-convergence T_0-TopSpace;
  assume Y is_Retract_of Z;
  then consider S being non empty SubSpace of Z such that
A1: S is_a_retract_of Z and
A2: S,Y are_homeomorphic by YELLOW16:57;
  assume oContMaps(X, Z) is complete continuous;
  then
A3: oContMaps(X,S) is complete continuous by A1,Th22;
  oContMaps(X,S), oContMaps(X,Y) are_isomorphic by A2,Th21;
  hence thesis by A3,WAYBEL15:9,WAYBEL20:18;
end;
