
theorem Th17:
  for Z being monotone-convergence T_0-TopSpace for Y being non
  empty SubSpace of Z for f being continuous Function of Z,Y st f is
being_a_retraction holds Omega Y is directed-sups-inheriting SubRelStr of Omega
  Z
proof
  let Z be monotone-convergence T_0-TopSpace;
  let Y be non empty SubSpace of Z;
  reconsider OZ = Omega Z as non empty up-complete non empty Poset;
  reconsider OY = Omega Y as full non empty SubRelStr of Omega Z by WAYBEL25:17
;
  let f be continuous Function of Z,Y;
A1: the RelStr of OZ = the RelStr of Omega Z;
  [#]Y c= [#]Z by PRE_TOPC:def 4;
  then dom f = the carrier of Z & rng f c= the carrier of Z by FUNCT_2:def 1;
  then
A2: f is continuous Function of Z,Z by PRE_TOPC:26,RELSET_1:4;
  the TopStruct of Omega Z = the TopStruct of Z by WAYBEL25:def 2;
  then reconsider f9 = f as continuous Function of Omega Z, Omega Z by A2,
YELLOW12:36;
  reconsider g = f9 as Function of OZ, OZ;
  assume
A3: f is being_a_retraction;
  then g is idempotent directed-sups-preserving by YELLOW16:45;
  then
A4: Image g is directed-sups-inheriting by YELLOW16:6;
  the TopStruct of Omega Y = the TopStruct of Y & rng g = the carrier of
  subrelstr rng g by WAYBEL25:def 2,YELLOW_0:def 15;
  then OY is directed-sups-inheriting by A3,A4,A1,WAYBEL21:22,YELLOW16:44;
  hence thesis;
end;
