
theorem Th22:
  for S, T being complete Scott TopLattice, f being Function of S, T holds
  f is continuous iff f is directed-sups-preserving
proof
  let S, T be complete Scott TopLattice, f be Function of S, T;
  thus f is continuous implies f is directed-sups-preserving
  proof
    assume f is continuous;
    then for N be net of S holds f.(lim_inf N) <= lim_inf (f*N) by Lm9;
    hence thesis by Lm8;
  end;
  thus thesis by Lm4;
end;
