reserve x for set;

theorem Th13:
  for L being complete LATTICE, F being proper Filter of
  BoolePoset [#]L holds lim_inf F = lim_inf a_net F
proof
  let L be complete LATTICE;
  let F be proper Filter of BoolePoset [#]L;
  set X =the set of all inf ((a_net F)|i) where i is Element of a_net F;
  set Y = {inf B where B is Subset of L: B in F};
  for x being object st x in X holds x in Y
  proof
    let x be object;
    assume x in X;
    then consider i being Element of a_net F such that
A1: x = inf ((a_net F)|i);
    reconsider i as Element of a_net F;
    i in the carrier of a_net F;
    then i in {[b, g] where b is Element of L, g is Element of F: b in g} by
YELLOW19:def 4;
    then consider a being Element of L, f being Element of F such that
A2: i = [a,f] and
    a in f;
    reconsider i as Element of a_net F;
    reconsider f as Element of BoolePoset [#]L;
    reconsider f as Subset of L by WAYBEL_7:2;
    [a,f]`2 = f;
    then inf f = inf ((a_net F)|i) by Th12,A2;
    hence thesis by A1;
  end;
  then
A3: X c= Y;
  for x being object st x in Y holds x in X
  proof
    let x be object;
    assume x in Y;
    then consider B being Subset of L such that
A4: x = inf B and
A5: B in F;
    not Bottom (BoolePoset [#]L) in F by WAYBEL_7:4;
    then B <> {} by A5,YELLOW_1:18;
    then consider a being Element of L such that
A6: a in B by SUBSET_1:4;
    reconsider B as Element of F by A5;
    [a,B] in {[b,f] where b is Element of L, f is Element of F: b in f} by A6;
    then reconsider i = [a,B] as Element of a_net F by YELLOW19:def 4;
    [a,B]`2 = B;
    then x = inf ((a_net F)|i) by A4,Th12;
    hence thesis;
  end;
  then
A7: Y c= X;
  lim_inf a_net F = "\/"(X,L) by Th11;
  hence thesis by A3,A7,XBOOLE_0:def 10;
end;
