
theorem Th15:
  for Y,Z being non empty TopSpace for X being
monotone-convergence T_0-TopSpace for f being continuous Function of Y,Z holds
  oContMaps(f, X) is directed-sups-preserving
proof
  let Y,Z be non empty TopSpace;
  let X be monotone-convergence T_0-TopSpace;
  let f be continuous Function of Y,Z;
  let A be Subset of oContMaps(Z, X);
  reconsider sA = sup A as continuous Function of Z,X by Th2;
  set fX = oContMaps(f, X);
  reconsider sfA = sup (fX.:A), XfsA = fX.sup A as Function of Y, Omega X by
Th1;
  reconsider YX = oContMaps(Y, X) as directed-sups-inheriting non empty full
  SubRelStr of (Omega X) |^ the carrier of Y by WAYBEL24:def 3,WAYBEL25:43;
  assume A is non empty directed;
  then reconsider A9 = A as non empty directed Subset of oContMaps(Z, X);
  reconsider fA9 = fX.:A9 as non empty directed Subset of oContMaps(Y, X) by
Th10,YELLOW_2:15;
  reconsider ZX = oContMaps(Z, X) as directed-sups-inheriting non empty full
  SubRelStr of (Omega X) |^ the carrier of Z by WAYBEL24:def 3,WAYBEL25:43;
  reconsider B = A9 as non empty directed Subset of ZX;
  reconsider B9 = B as non empty directed Subset of (Omega X) |^ the carrier
  of Z by YELLOW_2:7;
  reconsider fB = fA9 as non empty directed Subset of YX;
  reconsider fB9 = fB as non empty directed Subset of (Omega X) |^ the carrier
  of Y by YELLOW_2:7;
  assume ex_sup_of A, oContMaps(Z, X);
  set I1 = the carrier of Z, I2 = the carrier of Y;
  set J1 = I1 --> Omega X;
  set J2 = I2 --> Omega X;
  ex_sup_of fB9, (Omega X) |^ the carrier of Y by WAYBEL_0:75;
  then
A1: sup fB9 = sup (oContMaps(f, X).:A) by WAYBEL_0:7;
  oContMaps(Y, X) is up-complete & fA9 is directed by Th7;
  hence ex_sup_of oContMaps(f, X).:A, oContMaps(Y, X) by WAYBEL_0:75;
A2: (Omega X) |^ I2 = I2-POS_prod J2 by YELLOW_1:def 5;
  then reconsider fB99 = fB9 as non empty directed Subset of I2-POS_prod J2;
  now
    let x be Element of Y;
    J2.x = Omega X & pi(fB99,x) is directed by FUNCOP_1:7,YELLOW16:35;
    hence ex_sup_of pi(fB99,x), J2.x by WAYBEL_0:75;
  end;
  then
A3: ex_sup_of fB99, I2-POS_prod J2 by YELLOW16:31;
A4: (Omega X) |^ I1 = I1-POS_prod J1 by YELLOW_1:def 5;
  then reconsider B99 = B9 as non empty directed Subset of I1-POS_prod J1;
A5: ex_sup_of B9, (Omega X) |^ the carrier of Z by WAYBEL_0:75;
  then
A6: sup B9 = sup A by WAYBEL_0:7;
  now
    let x be Element of Y;
A7: J1.(f.x) = Omega X & J2.x = Omega X by FUNCOP_1:7;
A8: pi(fB99,x) = pi(B99,f.x) by Th14;
    thus sfA.x = sup pi(fB99,x) by A1,A2,A3,YELLOW16:33
      .= (sup B99).(f.x) by A5,A4,A7,A8,YELLOW16:33
      .= (sA*f).x by A6,A4,FUNCT_2:15
      .= XfsA.x by Def3;
  end;
  hence thesis by FUNCT_2:63;
end;
