
theorem
  for S, T being up-complete LATTICE, f being Function of S, T holds
  (for N being net of S holds f.(lim_inf N) <= lim_inf (f*N)) implies
  for D being directed non empty Subset of S holds sup (f.:D) = f.(sup D)
proof
  let S, T be up-complete LATTICE, f be Function of S, T;
  assume
A1: for N being net of S holds f.(lim_inf N) <= lim_inf (f*N);
  let D be directed non empty Subset of S;
A2: f is monotone by A1,Th11;
  then
A3: sup (f.:D) <= f.(sup D) by Th16;
  f.(sup D) <= sup (f.:D)
  proof
    sup D = lim_inf Net-Str D by Th10;
    then
A4: f.(sup D) <= lim_inf (f*(Net-Str D)) by A1;
    reconsider fN = f*(Net-Str D) as monotone reflexive net of T by A2;
A5: sup fN = Sup the mapping of fN by WAYBEL_2:def 1
      .= Sup (f * (id the carrier of S)|D) by WAYBEL_9:def 8
      .= sup (rng (f * (id the carrier of S)|D)) by YELLOW_2:def 5;
    rng (f * (id the carrier of S)|D) = rng (f * id D) by FUNCT_3:1
      .= rng (f|D) by RELAT_1:65
      .= f .: D by RELAT_1:115;
    hence thesis by A4,A5,Lm6;
  end;
  hence thesis by A3,ORDERS_2:2;
end;
