
theorem Th12:
  for S, T being continuous lower-bounded LATTICE,
  f being Function of S, T holds ( f is directed-sups-preserving implies
  for x being Element of S holds
  f.x = "\/"({ f.w where w is Element of S : w << x },T) )
proof
  let S, T be continuous lower-bounded LATTICE;
  let f be Function of S, T;
  assume
A1: f is directed-sups-preserving;
  let x be Element of S;
  defpred P[Element of S] means $1 << x;
  deffunc A(Element of S) = $1;
A2: f preserves_sup_of waybelow x by A1;
A3: ex_sup_of waybelow x,S by YELLOW_0:17;
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.x = f.(sup waybelow x) by WAYBEL_3:def 5
    .= "\/"({f.w where w is Element of S : w << x },T) by A2,A3,A5;
  hence thesis;
end;
