
theorem Th18:
  for X being non empty TopSpace 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 oContMaps(X, f)
  is_a_retraction_of oContMaps(X, Z), oContMaps(X, Y)
proof
  let X be non empty TopSpace;
  let Z be monotone-convergence T_0-TopSpace;
  let Y be non empty SubSpace of Z;
  set XY = oContMaps(X, Y), XZ = oContMaps(X, Z);
  reconsider uXZ = XZ as up-complete non empty Poset by Th7;
  let f be continuous Function of Z,Y;
  set F = oContMaps(X, f);
  reconsider sXY = XY as full non empty SubRelStr of uXZ by Th16;
  assume
A1: f is being_a_retraction;
  then reconsider Y9 = Y as monotone-convergence T_0-TopSpace by Lm1;
  oContMaps(X,Y9) is up-complete by Th7;
  hence F is directed-sups-preserving Function of XZ, XY by Th13;
A2: the carrier of sXY c= the carrier of uXZ by YELLOW_0:def 13;
A3: now
    let x be object;
A4: dom f = the carrier of Z by FUNCT_2:def 1;
A5: rng f = the carrier of Y & f is idempotent by A1,YELLOW16:44,45;
    assume
A6: x in the carrier of XY;
    then reconsider a = x as Element of XZ by A2;
    reconsider a as continuous Function of X, Z by Th2;
    x is Function of X,Y by A6,Th2;
    then rng a c= the carrier of Y by RELAT_1:def 19;
    then f*a = a by A5,A4,YELLOW16:4;
    hence (id XY).x = f*a by A6,FUNCT_1:18
      .= F.a by Def2
      .= (F|the carrier of XY).x by A6,FUNCT_1:49;
  end;
  F|the carrier of XY is Function of the carrier of XY, the carrier of XY
  by A2,FUNCT_2:32;
  then
  dom id XY = the carrier of XY & dom (F|the carrier of XY) = the carrier
  of XY by FUNCT_2:def 1;
  hence F|the carrier of XY = id XY by A3,FUNCT_1:2;
  Omega Y is directed-sups-inheriting full SubRelStr of Omega Z by A1,Th17,
WAYBEL25:17;
  then oContMaps(X, Y9) is directed-sups-inheriting full non empty SubRelStr
  of ( Omega Y)|^the carrier of X & (Omega Y)|^the carrier of X is
  directed-sups-inheriting full SubRelStr of (Omega Z)|^the carrier of X by
WAYBEL24:def 3,WAYBEL25:43,YELLOW16:39,42;
  then
  oContMaps(X, Z) is directed-sups-inheriting full non empty SubRelStr of
( Omega Z)|^the carrier of X & oContMaps(X,Y) is directed-sups-inheriting full
  SubRelStr of (Omega Z)|^the carrier of X by WAYBEL24:def 3,WAYBEL25:43
,YELLOW16:26,27;
  hence thesis by Th16,YELLOW16:28;
end;
