
theorem Th25:
  for S being LATTICE, T being complete LATTICE,
  f being Function of S, T holds ( for x being Element of S holds
  f.x = "\/"({ f.w where w is Element of S : w <= x & w is compact },T ) )
  implies f is monotone
proof
  let S be LATTICE, T be complete LATTICE, f be Function of S, T;
  assume
A1: for x being Element of S holds
  f.x = "\/"({ f.w where w is Element of S : w <= x & w is compact },T );
  thus f is monotone
  proof
    let X,Y be Element of S;
    assume X <= Y;
    then
A2: compactbelow X c= compactbelow Y by WAYBEL13:1;
A3: f.X = "\/"
    ({ f.w where w is Element of S : w <= X & w is compact },T) by A1;
A4: f.Y = "\/"
    ({ f.w where w is Element of S : w <= Y & w is compact },T) by A1;
A5: the carrier of S c= dom f by FUNCT_2:def 1;
    defpred P[Element of S] means $1 <= X & $1 is compact;
    defpred Q[Element of S] means $1 <= Y & $1 is compact;
    deffunc A(Element of S) = $1;
    f.:{ A(w) where w is Element of S : P[w]} =
    { f.A(w) where w is Element of S : P[w]} from FuncFraenkelSL(A5);
    then
A6: f.X = "\/"(f.:compactbelow X,T) by A3,WAYBEL_8:def 2;
    f.:{ A(w) where w is Element of S : Q[w]} =
    { f.A(w) where w is Element of S : Q[w]} from FuncFraenkelSL(A5);
    then
A7: f.Y = "\/"(f.:compactbelow Y,T) by A4,WAYBEL_8:def 2;
A8: ex_sup_of f.:compactbelow X,T by YELLOW_0:17;
    ex_sup_of f.:compactbelow Y,T by YELLOW_0:17;
    hence thesis by A2,A6,A7,A8,RELAT_1:123,YELLOW_0:34;
  end;
end;
