
theorem Th24:
  for S, T being continuous complete Scott TopLattice,
  f being Function of S, T holds ( f is continuous iff
  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 complete Scott TopLattice, f be Function of S, T;
  thus f is continuous implies for x being Element of S holds
  f.x = "\/"({ f.w where w is Element of S : w << x },T) by Th12;
  assume for x being Element of S holds
  f.x = "\/"({ f.w where w is Element of S : w << x },T);
  then f is directed-sups-preserving by Lm16;
  hence thesis;
end;
