
theorem Th13:
  for S being LATTICE, T being up-complete lower-bounded 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 },T) ) implies
  f is monotone)
proof
  let S be LATTICE, T be up-complete lower-bounded LATTICE;
  let 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 },T);
  let X,Y be Element of S;
  deffunc A(Element of S) = $1;
  defpred P[Element of S] means $1 << X;
  defpred Q[Element of S] means $1 << Y;
  assume X <= Y;
  then
A2: waybelow X c= waybelow Y by WAYBEL_3:12;
A3: f.X = "\/"({ f.w where w is Element of S : w << X },T) by A1;
A4: the carrier of S c= dom f by FUNCT_2:def 1;
A5: f.:{ A(w) where w is Element of S : P[w] } =
  { f.A(w) where w is Element of S : P[w] } from FuncFraenkelSL(A4);
  f.:{ A(w) where w is Element of S : Q[w] } =
  { f.A(w) where w is Element of S : Q[w] } from FuncFraenkelSL(A4);
  then
A6: f.Y = "\/"(f.:waybelow Y,T) by A1;
A7: ex_sup_of f.:waybelow X,T by YELLOW_0:17;
  ex_sup_of f.:waybelow Y,T by YELLOW_0:17;
  hence thesis by A2,A3,A5,A6,A7,RELAT_1:123,YELLOW_0:34;
end;
