
theorem Th13:
  for X being non empty TopSpace for Y,Z being
monotone-convergence T_0-TopSpace for f being continuous Function of Y,Z holds
  oContMaps(X, f) is directed-sups-preserving
proof
  let X be non empty TopSpace;
  let Y,Z be monotone-convergence T_0-TopSpace;
  let f be continuous Function of Y,Z;
  let A be Subset of oContMaps(X, Y);
  reconsider sA = sup A as continuous Function of X,Y by Th2;
  set Xf = oContMaps(X, f);
  reconsider sfA = sup (Xf.:A), XfsA = Xf.sup A as Function of X, Omega Z by
Th1;
  reconsider XZ = oContMaps(X, Z) as directed-sups-inheriting non empty full
  SubRelStr of (Omega Z) |^ the carrier of X by WAYBEL24:def 3,WAYBEL25:43;
  assume A is non empty directed;
  then reconsider A9 = A as non empty directed Subset of oContMaps(X, Y);
  reconsider fA9 = Xf.:A9 as non empty directed Subset of oContMaps(X, Z) by
Th8,YELLOW_2:15;
  reconsider XY = oContMaps(X, Y) as directed-sups-inheriting non empty full
  SubRelStr of (Omega Y) |^ the carrier of X by WAYBEL24:def 3,WAYBEL25:43;
  reconsider B = A9 as non empty directed Subset of XY;
  reconsider B9 = B as non empty directed Subset of (Omega Y) |^ the carrier
  of X by YELLOW_2:7;
  reconsider fB = fA9 as non empty directed Subset of XZ;
  reconsider fB9 = fB as non empty directed Subset of (Omega Z) |^ the carrier
  of X by YELLOW_2:7;
  assume ex_sup_of A, oContMaps(X, Y);
  set I = the carrier of X;
  set J1 = I --> Omega Y;
  set J2 = I --> Omega Z;
  the TopStruct of Y = the TopStruct of Omega Y & the TopStruct of Z = the
  TopStruct of Omega Z by WAYBEL25:def 2;
  then reconsider f9 = f as continuous Function of Omega Y, Omega Z by
YELLOW12:36;
  ex_sup_of fB9, (Omega Z) |^ the carrier of X by WAYBEL_0:75;
  then
A1: sup fB9 = sup (oContMaps(X, f).:A) by WAYBEL_0:7;
  oContMaps(X, Z) is up-complete & fA9 is directed by Th7;
  hence ex_sup_of oContMaps(X, f).:A, oContMaps(X, Z) by WAYBEL_0:75;
A2: (Omega Z) |^ I = I-POS_prod J2 by YELLOW_1:def 5;
  then reconsider fB99 = fB9 as non empty directed Subset of I-POS_prod J2;
  now
    let x be Element of X;
    J2.x = Omega Z & 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, I-POS_prod J2 by YELLOW16:31;
A4: (Omega Y) |^ the carrier of X = I-POS_prod J1 by YELLOW_1:def 5;
  then reconsider B99 = B9 as non empty directed Subset of I-POS_prod J1;
A5: ex_sup_of B9, (Omega Y) |^ the carrier of X by WAYBEL_0:75;
  then
A6: sup B9 = sup A by WAYBEL_0:7;
  now
    let x be Element of X;
A7: J1.x = Omega Y by FUNCOP_1:7;
    then reconsider Bx = pi(B99,x) as directed non empty Subset of Omega Y by
YELLOW16:35;
A8: J2.x = Omega Z & ex_sup_of Bx, Omega Y by FUNCOP_1:7,WAYBEL_0:75;
A9: (sup B99).x = sup pi(B99,x) by A5,A4,YELLOW16:33;
A10: f9 preserves_sup_of Bx & pi(fB99,x) = f9.:Bx by Th12,WAYBEL_0:def 37;
    thus sfA.x = sup pi(fB99,x) by A1,A2,A3,YELLOW16:33
      .= f.sup Bx by A8,A10
      .= (f*sA).x by A6,A4,A9,A7,FUNCT_2:15
      .= XfsA.x by Def2;
  end;
  hence thesis by FUNCT_2:63;
end;
