
theorem Th25:
  for L being complete LATTICE, D being directed non empty Subset
  of L holds for M being subnet of Net-Str D holds lim_inf M = sup D
proof
  let L be complete LATTICE;
  let D be directed non empty Subset of L;
  for M being subnet of Net-Str D holds sup D >= inf M
  proof
    let M be subnet of Net-Str D;
    set i = the Element of M;
    set f=the mapping of M;
    consider g being Function of M, Net-Str D such that
A1: the mapping of M = (the mapping of Net-Str D)*g and
    for m being Element of Net-Str D ex n being Element of M st for p
    being Element of M st n <= p holds m <= g.p by YELLOW_6:def 9;
A2: dom f = the carrier of M by FUNCT_2:def 1;
    then f.i in rng f by FUNCT_1:def 3;
    then
A3: "/\"(rng f,L) <= f.i by YELLOW_0:17,YELLOW_4:2;
    g.i in the carrier of Net-Str D;
    then
A4: g.i in D by WAYBEL21:32;
    then g.i = (id D).(g.i) by FUNCT_1:18
      .= (the mapping of Net-Str D).(g.i) by WAYBEL21:32
      .= f.i by A1,A2,FUNCT_1:12;
    then f.i <= sup D by A4,YELLOW_2:22;
    then sup D >= "/\"(rng f,L) by A3,YELLOW_0:def 2;
    then sup D >= Inf the mapping of M by YELLOW_2:def 6;
    hence thesis by WAYBEL_9:def 2;
  end;
  then lim_inf Net-Str D =sup D & for p being greater_or_equal_to_id Function
  of Net-Str(D),Net-Str(D) holds sup D >= inf (Net-Str(D)*p) by WAYBEL17:10;
  hence thesis by Th14;
end;
