reserve x for set;

theorem Th12:
  for L being complete LATTICE, F being proper Filter of
  BoolePoset [#]L, f being Subset of L st f in F for i being Element of a_net F
  st i`2 = f holds inf f = inf ((a_net F)|i)
proof
  let L be complete LATTICE;
  let F be proper Filter of BoolePoset [#]L;
  let f be Subset of L;
  assume
A1: f in F;
  let i be Element of a_net F;
  assume
A2: i`2 = f;
  for b being object st b in f holds b in rng the mapping of ((a_net F)|i)
  proof
    let b being object;
    assume
A3: b in f;
    then reconsider b as Element of L;
    reconsider f as Element of F by A1;
    [b,f] in {[a, g] where a is Element of L, g is Element of F: a in g} by A3;
    then reconsider k = [b,f] as Element of (a_net F) by YELLOW19:def 4;
    reconsider l = k as Element of (a_net F);
    [b,f]`1 = b;
    then
A4: b = (a_net F).k by YELLOW19:def 4;
    k`2 c= i`2 by A2;
    then i <= k by YELLOW19:def 4;
    then l in { y where y is Element of (a_net F) : i <= y};
    then reconsider k as Element of (a_net F)|i by WAYBEL_9:12;
    reconsider k as Element of (a_net F)|i;
    (a_net F).l = ((a_net F)|i).k by WAYBEL_9:16;
    then
    b in the set of all ((a_net F)|i).m where m is Element of (a_net F)|
i by A4;
    hence thesis by WAYBEL11:19;
  end;
  then
A5: f c= rng the mapping of ((a_net F)|i);
  for b being object st b in rng the mapping of ((a_net F)|i) holds b in f
  proof
    let b being object;
    assume b in rng the mapping of ((a_net F)|i);
    then
    b in the set of all ((a_net F)|i).k where k is Element of (a_net F)|i
     by WAYBEL11:19;
    then consider k being Element of (a_net F)|i such that
A6: b = ((a_net F)|i).k;
A7: the carrier of (a_net F)|i c= the carrier of (a_net F) by WAYBEL_9:13;
    then reconsider l = k as Element of a_net F;
    k in the carrier of (a_net F) by A7;
    then
A8: k in {[c, g] where c is Element of L, g is Element of F: c in g} by
YELLOW19:def 4;
    k in the carrier of (a_net F)|i;
    then k in { y where y is Element of a_net F : i <= y} by WAYBEL_9:12;
    then ex y being Element of a_net F st k = y & i <= y;
    then
A9: l`2 c= f by A2,YELLOW19:def 4;
    consider c being Element of L, g being Element of F such that
A10: k = [c,g] and
A11: c in g by A8;
    reconsider k as Element of (a_net F)|i;
    (a_net F).l = ((a_net F)|i).k by WAYBEL_9:16;
    then b = l`1 by A6,YELLOW19:def 4;
    hence thesis by A11,A9,A10;
  end;
  then
A12: rng the mapping of ((a_net F)|i) c= f;
  inf ((a_net F)|i) = Inf the mapping of ((a_net F)|i) by WAYBEL_9:def 2
    .= "/\"(rng the mapping of ((a_net F)|i),L) by YELLOW_2:def 6;
  hence thesis by A12,A5,XBOOLE_0:def 10;
end;
