
theorem Th14:
  for L being complete LATTICE, N being net of L, x being Element
  of L holds (x=lim_inf N & for p being greater_or_equal_to_id Function of N,N
  holds x >= inf (N * p) ) implies for M being subnet of N holds x = lim_inf M
proof
  let L be complete LATTICE, N be net of L, x be Element of L;
  assume that
A1: x=lim_inf N and
A2: for p being greater_or_equal_to_id Function of N,N holds x >= inf (N * p);
  let M be subnet of N;
  consider f being Function of M, N such that
A3: the mapping of M = (the mapping of N)*f and
A4: for j being Element of N ex k being Element of M st for m being
  Element of M st k <= m holds j <= f.m by YELLOW_6:def 9;
A5: x <= lim_inf M by A1,WAYBEL21:37;
A6: for k0 being Element of M holds "/\"({M.k where k is Element of M:k >=
  k0},L)<=x
  proof
    let k0 be Element of M;
    defpred P[object,object] means
    for j being Element of N, v,v9 being Element of M
    st $1=j & $2=v & v9>= v holds v >= k0 & f.(v9)>=j & f.v >= j;
A7: for j being Element of N ex v being Element of M st v >= k0 & for v9
    being Element of M st v9>= v holds f.(v9)>=j & f.v >= j
    proof
      let j be Element of N;
      consider w being Element of M such that
A8:   for w9 being Element of M st w <= w9 holds j <= f.(w9) by A4;
      consider v being Element of M such that
A9:   v >= k0 and
A10:  v >= w by Th5;
      take v;
      thus v >= k0 by A9;
      let v9 be Element of M;
      assume v9>= v;
      then v9 >= w by A10,YELLOW_0:def 2;
      hence f.(v9)>=j by A8;
      thus thesis by A8,A10;
    end;
A11: for e being object st e in the carrier of N
    ex u being object st u in the
    carrier of M & P[e,u]
    proof
      let e be object;
      assume e in the carrier of N;
      then reconsider e9=e as Element of N;
      consider u being Element of M such that
A12:  u >= k0 and
A13:  for v9 being Element of M st v9>= u holds f.(v9)>=e9 & f.u >= e9 by A7;
      take u;
      thus u in the carrier of M;
      let j be Element of N, v,v9 be Element of M;
      assume that
A14:  e=j and
A15:  u=v and
A16:  v9>= v;
      thus v >= k0 by A12,A15;
      thus f.(v9)>=j by A13,A14,A15,A16;
      thus thesis by A13,A14,A15,A16;
    end;
    consider g being Function such that
A17: dom g = the carrier of N and
A18: rng g c= the carrier of M and
A19: for e being object st e in the carrier of N holds P[e,g.e] from
    FUNCT_1:sch 6(A11);
    reconsider g as Function of the carrier of N,the carrier of M by A17,A18,
FUNCT_2:2;
A20: for j being Element of N holds g.j >= k0
    proof
      let j be Element of N;
      reconsider v=g.j as Element of M;
      ex v9 being Element of M st v9>= v & v9>= v by Th5;
      hence thesis by A19;
    end;
    reconsider g as Function of N,M;
    reconsider p=f*g as Function of N,N;
    for j being Element of N holds j <= p.j
    proof
      let j be Element of N;
      reconsider v=g.j as Element of M;
      [#]M is directed by WAYBEL_0:def 6;
      then ex v9 being Element of M st v9 in [#]M & v <= v9 & v <= v9 by
WAYBEL_0:def 1;
      then j <= f.(g.j) by A19;
      hence thesis by A17,FUNCT_1:13;
    end;
    then reconsider p as greater_or_equal_to_id Function of N,N by Def1;
A21: the set of all (N*p).j where j is Element of (N*p) c= {M.k where
    k is Element of M:k >= k0}
    proof
      let y be object;
      assume y in the set of all (N*p).j where j is Element of (N*p);
      then consider j being Element of (N*p) such that
A22:  y = (N*p).j;
      reconsider j9=j as Element of N by Th6;
A23:  the carrier of (N*p)= the carrier of N by Th6;
      y = (the mapping of (N*p)).j by A22,WAYBEL_0:def 8
        .= ((the mapping of N)*(f*g)).j by Def2
        .= ((the mapping of M)*g).j by A3,RELAT_1:36
        .= (the mapping of M).(g.j) by A17,A23,FUNCT_1:13;
      then
A24:  y = M.(g.j9) by WAYBEL_0:def 8;
      g.j9 >= k0 by A20;
      hence thesis by A24;
    end;
A25: ex_inf_of (the set of all (N*p).j where j is Element of (N*p)),L
    & ex_inf_of {M.k where k is Element of M:k >= k0},L by YELLOW_0:17;
A26: dom (the mapping of (N*p)) = the carrier of (N*p) by FUNCT_2:def 1;
A27: rng the mapping of (N*p) = the set of all
(N*p).j where j is Element of (N*p)
    proof
      thus rng the mapping of (N*p) c= the set of all
(N*p).j where j is Element of (N*p)
      proof
        let y be object;
        assume y in rng the mapping of (N*p);
        then consider j1 being object such that
A28:    j1 in dom (the mapping of (N*p)) and
A29:    (the mapping of (N*p)).j1 = y by FUNCT_1:def 3;
        reconsider j1 as Element of (N*p) by A28;
        y = (N*p).j1 by A29,WAYBEL_0:def 8;
        hence thesis;
      end;
      let y be object;
      assume y in the set of all (N*p).j where j is Element of (N*p);
      then consider j being Element of (N*p) such that
A30:  y = (N*p).j;
      y = (the mapping of (N*p)).j by A30,WAYBEL_0:def 8;
      hence thesis by A26,FUNCT_1:def 3;
    end;
A31: inf (N * p) <= x by A2;
    inf (N*p) = Inf the mapping of (N*p) by WAYBEL_9:def 2
      .= "/\"((the set of all (N*p).j where j is Element of (N*p)),L)
      by A27,YELLOW_2:def 6;
    then
    "/\"({M.k where k is Element of M:k >= k0},L)<= inf (N*p) by A25,A21,
YELLOW_0:35;
    hence thesis by A31,YELLOW_0:def 2;
  end;
  for b being Element of L st b in the set of all
 "/\"({M.k where k is Element of M:k
  >= k0},L) where k0 is Element of M holds b <= x
  proof
    let b be Element of L;
    assume b in the set of all
 "/\"({M.k where k is Element of M:k >= k0},L) where k0 is
    Element of M;
    then
    ex k0 being Element of M st b = "/\"({M.k where k is Element of M:k >=
    k0},L);
    hence thesis by A6;
  end;
  then
A32: x is_>=_than the set of all
 "/\" ({M.k where k is Element of M:k >= k0},L) where k0
  is Element of M by LATTICE3:def 9;
  ex_sup_of (the set of all "/\"({M.k where k is Element of M:k >= k0},L)
  where k0 is Element of M),L by YELLOW_0:17;
  then "\/"((the set of all "/\"({M.k where k is Element of M:k >= k0},L)
  where k0 is Element of M),L)<=x by A32,YELLOW_0:def 9;
  hence thesis by A5,ORDERS_2:2;
end;
