
theorem
  for S, T being complete lower-bounded LATTICE, f being monotone
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)
proof
  let S, T be complete lower-bounded LATTICE;
  let f be monotone Function of S, T;
  let x be Element of S;
  deffunc A(Element of S) = $1;
  defpred P[Element of S] means $1 <= x;
  defpred R[Element of S] means ex y1 being Element of S st $1 <= y1 & y1 in {
  x};
A1: the carrier of S c= dom f by FUNCT_2:def 1;
A2: f.:{ A(w) where w is Element of S: P[w]} = {f.A(w) where w is Element of
S: P[w]} from FuncFraenkelSL(A1);
A3: for x2 be Element of S holds P[x2] iff R[x2]
  proof
    let x2 be Element of S;
    hereby
A4:   x in {x} by TARSKI:def 1;
      assume x2 <= x;
      hence ex y1 being Element of S st x2 <= y1 & y1 in {x} by A4;
    end;
    given y1 being Element of S such that
A5: x2 <= y1 & y1 in {x};
    thus thesis by A5,TARSKI:def 1;
  end;
  { A(w) where w is Element of S : P[w]} = {A(x1) where x1 is Element of S
  : R[x1]} from Fraenkel6A (A3);
  then
A6: downarrow x = { w where w is Element of S : w <= x } by WAYBEL_0:14;
  sup (f.:downarrow x) = f. x by Th4
    .= f.sup downarrow x by WAYBEL_0:34;
  hence thesis by A2,A6,WAYBEL_0:34;
end;
