reserve x for set;

theorem Th15:
  for L being complete LATTICE, F being ultra Filter of BoolePoset
  [#]L for p being greater_or_equal_to_id Function of a_net F, a_net F holds
  lim_inf F >= inf ((a_net F) * p)
proof
  let L be complete LATTICE, F be ultra Filter of BoolePoset [#]L, p be
  greater_or_equal_to_id Function of a_net F, a_net F;
  set M = (a_net F)*p;
  set rM = rng the mapping of M;
  set C = the Element of F;
A1: inf M = Inf the mapping of M by WAYBEL_9:def 2
    .= "/\"(rM,L) by YELLOW_2:def 6;
  F c= the carrier of BoolePoset [#]L;
  then F c= bool [#]L by WAYBEL_7:2;
  then C in bool [#]L;
  then
A2: C \ rM c= [#]L by XBOOLE_1:1;
  then reconsider D=C \ rM, A=C /\ rM as Element of BoolePoset [#] L by
WAYBEL_7:2;
A3: the carrier of M = the carrier of a_net F by WAYBEL28:6;
  then reconsider g = p as Function of M, a_net F;
A4: now
    set d = the Element of D;
    assume
A5: D in F;
    not Bottom (BoolePoset [#]L) in F by WAYBEL_7:4;
    then
A6: D <> {} by A5,YELLOW_1:18;
    then
A7: d in D;
    reconsider D as Element of F by A5;
    reconsider d as Element of L by A2,A7;
    [d,D] in {[a, f] where a is Element of L, f is Element of F: a in f} by A6;
    then reconsider dD= [d,D] as Element of a_net F by YELLOW19:def 4;
    reconsider dD9 = dD as Element of M by WAYBEL28:6;
A8: dom p = the carrier of a_net F by FUNCT_2:52;
    ex i being Element of M st for j being Element of M st j >= i holds g
    .j >= dD
    proof
      consider i being Element of M such that
A9:   i = dD9;
      take i;
      for j being Element of M st j >= i holds g.j >= dD
      proof
        reconsider i9=i as Element of a_net F by WAYBEL28:6;
        let j be Element of M;
        reconsider j9=j as Element of a_net F by WAYBEL28:6;
A10:    j9 <= g.j by WAYBEL28:def 1;
        reconsider i9 as Element of a_net F;
        reconsider j9 as Element of a_net F;
A11:    the RelStr of M = the RelStr of a_net F by WAYBEL28:def 2;
        assume j >= i;
        then i9 <= j9 by A11,YELLOW_0:1;
        hence thesis by A9,A10,YELLOW_0:def 2;
      end;
      hence thesis;
    end;
    then consider i being Element of M such that
A12: for j being Element of M st j >= i holds g.j >= dD;
    the RelStr of M = the RelStr of a_net F by WAYBEL28:def 2;
    then M is reflexive by WAYBEL_8:12;
    then i >= i by YELLOW_0:def 1;
    then
A13: g.i >= dD by A12;
    [d,D]`2 = D;
    then
A14: (p.i)`2 c= D by A13,YELLOW19:def 4;
    reconsider G = g.i as Element of a_net F;
    g.i in the carrier of a_net F;
    then
    g.i in {[a, f] where a is Element of L, f is Element of F: a in f} by
YELLOW19:def 4;
    then consider a being Element of L, f being Element of F such that
A15: g.i = [a, f] and
A16: a in f;
A17: (p.i)`1 in (p.i)`2 by A15,A16;
    M.i = ((the mapping of a_net F)*p).i by WAYBEL28:def 2
      .= (a_net F).G by A3,A8,FUNCT_1:13
      .= (p.i)`1 by YELLOW19:def 4;
    then not M.i in rM by A14,A17,XBOOLE_0:def 5;
    hence contradiction by FUNCT_2:4;
  end;
  set Y = {inf B where B is Subset of L: B in F};
  reconsider A9 = A as Subset of L;
A18: D"\/"A = D \/ A by YELLOW_1:17
    .= C by XBOOLE_1:51;
  F is prime by WAYBEL_7:22;
  then A in F by A18,A4;
  then inf A9 in Y;
  then
A19: inf A9 <= lim_inf F by YELLOW_0:17,YELLOW_4:1;
  A c= rM by XBOOLE_1:17;
  then inf M <= inf A9 by A1,WAYBEL_7:1;
  hence thesis by A19,YELLOW_0:def 2;
end;
