
theorem Th33:
  for S,T being up-complete non empty Poset
  for f being monotone Function of S,T
  for D being non empty directed Subset of S
  holds lim_inf (f*Net-Str D) = sup (f.:D)
proof
  let S,T be up-complete non empty Poset, f be monotone Function of S,T;
  let X be non empty directed Subset of S;
  set M = Net-Str X, N = f*M;
  deffunc up(Element of N) = {N.i where i is Element of N: i >= $1};
  deffunc infy(Element of N) = "/\"(up($1), T);
  set NT = the set of all infy(j) where j is Element of N;
A1: the RelStr of N = the RelStr of M by WAYBEL_9:def 8;
A2: the mapping of N = f*the mapping of M by WAYBEL_9:def 8;
A3: the carrier of M = X by Th32;
A4: the mapping of M = id X by Th32;
A5: now
    let i be Element of N;
    thus N.i = f.((id X).i) by A1,A2,A4,FUNCT_2:15
      .= f.i by A1,A3;
  end;
A6: for i being Element of N holds infy(i) = f.i
  proof
    let i be Element of N;
    i in X by A1,A3;
    then reconsider x = i as Element of S;
A7: i <= i;
    N.i = f.x by A5;
    then f.x in up(i) by A7;
    then
A8: for z being Element of T st z is_<=_than up(i) holds z <= f.x;
    f.x is_<=_than up(i)
    proof
      let z be Element of T;
      assume z in up(i);
      then consider j being Element of N such that
A9:   z = N.j and
A10:  j >= i;
      reconsider j as Element of N;
      j in X by A1,A3;
      then reconsider y = j as Element of S;
A11:  M is full SubRelStr of S by Th32;
      the RelStr of S = the RelStr of S;
      then N is full SubRelStr of S by A1,A11,Th12;
      then y >= x by A10,YELLOW_0:59;
      then f.y >= f.x by WAYBEL_1:def 2;
      hence thesis by A5,A9;
    end;
    hence thesis by A8,YELLOW_0:31;
  end;
  NT = f.:X
  proof
    thus NT c= f.:X
    proof
      let x be object;
      assume x in NT;
      then consider j being Element of N such that
A12:  x = infy(j);
      reconsider j as Element of N;
A13:  x = f.j by A6,A12;
      j in X by A1,A3;
      hence thesis by A13,FUNCT_2:35;
    end;
    let y be object;
    assume y in f.:X;
    then consider x being object such that
A14: x in the carrier of S and
A15: x in X and
A16: y = f.x by FUNCT_2:64;
    reconsider x as Element of S by A14;
    reconsider i = x as Element of N by A1,A15,Th32;
    f.x = infy(i) by A6;
    hence thesis by A16;
  end;
  hence thesis by WAYBEL11:def 6;
end;
