reserve X1, X2, Y for non empty RelStr,
  f for Function of [:X1, X2:], Y,
  x for Element of X1,
  y for Element of X2;
reserve S for non empty RelStr,
  T for complete LATTICE;

theorem Th29:
  for S, T being complete Scott TopLattice, F being non empty
  Subset of ContMaps (S, T) holds "\/" (F, (T |^ the carrier of S)) is monotone
  Function of S, T
proof
  let S, T be complete Scott TopLattice, F be non empty Subset of ContMaps (S,
  T);
  ContMaps (S, T) is full SubRelStr of (T |^ the carrier of S) by Def3;
  then
  the carrier of ContMaps (S, T) c= the carrier of (T |^ the carrier of S
  ) by YELLOW_0:def 13;
  then reconsider F9 = F as Subset of (T |^ the carrier of S) by XBOOLE_1:1;
  reconsider sF = sup F9 as Function of S, T by Th6;
  now
    let x, y be Element of S;
    set G1 = { f.x where f is Element of (T |^ the carrier of S) : f in F9 };
    assume
A1: x <= y;
A2: G1 is_<=_than sF.y
    proof
      let a be Element of T;
      assume
      a in { f.x where f is Element of (T |^ the carrier of S) : f in F9 };
      then consider f9 being Element of (T |^ the carrier of S) such that
A3:   a = f9.x and
A4:   f9 in F9;
      reconsider f1 = f9 as continuous Function of S, T by A4,Th21;
      reconsider f1 as monotone Function of S, T;
      f9 <= sup F9 by A4,YELLOW_2:22;
      then f1 <= sF by WAYBEL10:11;
      then
A5:   f1.y <= sF.y by YELLOW_2:9;
      f1.x <= f1.y by A1,WAYBEL_1:def 2;
      hence thesis by A3,A5,YELLOW_0:def 2;
    end;
    sF.x = "\/" (G1, T) by Th25;
    hence sF.x <= sF.y by A2,YELLOW_0:32;
  end;
  hence thesis by WAYBEL_1:def 2;
end;
