
theorem Th36:
  for T being monotone-convergence T_0-TopSpace, R being
  T_0-TopSpace st R is_Retract_of T holds R is monotone-convergence
proof
  let T be monotone-convergence T_0-TopSpace, R be T_0-TopSpace;
  given c being continuous Function of R, T, r being continuous Function of T,
  R such that
A1: r * c = id R;
  let D be non empty directed Subset of Omega R;
A2: the TopStruct of R = the TopStruct of Omega R by Def2;
  then reconsider DR = D as non empty Subset of R;
A3: the TopStruct of T = the TopStruct of Omega T by Def2;
  then reconsider f = c*r as Function of Omega T, Omega T;
  reconsider rr = r as Function of Omega T, Omega R by A3,A2;
A4: r.:(c.:D) = (r*c).:DR by RELAT_1:126
    .= D by A1,FUNCT_1:92;
  reconsider cc = c as Function of Omega R, Omega T by A3,A2;
  cc is continuous by A3,A2,YELLOW12:36;
  then
A5: cc.:D is directed by YELLOW_2:15;
  then
A6: ex_sup_of cc.:D, Omega T by Def4;
  f is continuous by A3,YELLOW12:36;
  then
A7: f preserves_sup_of cc.:D by A5,WAYBEL_0:def 37;
  rr is continuous by A3,A2,YELLOW12:36;
  then
A8: rr preserves_sup_of cc.:D by A5,WAYBEL_0:def 37;
  hence ex_sup_of D, Omega R by A6,A4;
A9: c.sup D = c.(r.sup(cc.:D)) by A6,A4,A8
    .= f.sup(cc.:D) by A3,FUNCT_2:15
    .= sup(f.:(cc.:D)) by A6,A7
    .= sup (cc.:D) by A4,RELAT_1:126;
  let V be open Subset of R;
  assume sup D in V;
  then
A10: c.sup D in c.:V by FUNCT_2:35;
A11: c.:V c= r"V
  proof
    let a be object;
    assume a in c.:V;
    then consider x being object such that
A12: x in the carrier of R and
A13: x in V and
A14: a = c.x by FUNCT_2:64;
    reconsider x as Point of R by A12;
A15: a = c.x by A14;
    r.a = (r*c).x by A14,FUNCT_2:15
      .= x by A1;
    hence thesis by A13,A15,FUNCT_2:38;
  end;
  [#]R <> {};
  then r"V is open by TOPS_2:43;
  then c.:D meets r"V by A5,A11,A9,A10,Def4;
  then consider d being object such that
A16: d in cc.:D and
A17: d in r"V by XBOOLE_0:3;
  now
    take a = r.d;
    thus a in D by A4,A16,FUNCT_2:35;
    thus a in V by A17,FUNCT_2:38;
  end;
  hence thesis by XBOOLE_0:3;
end;
