reserve x,y,z,X for set,
  T for Universe;

theorem Th39:
  for T being non empty TopSpace, N being net of T, p being Point
  of T st p in Lim N for J being net_set of the carrier of N, T st for i being
  Element of N holds N.i in Lim(J.i) holds p in Lim Iterated J
proof
  let T be non empty TopSpace, N be net of T, p be Point of T such that
A1: p in Lim N;
  let J be net_set of the carrier of N, T such that
A2: for i being Element of N holds N.i in Lim(J.i);
  for V being a_neighborhood of p holds Iterated J is_eventually_in V
  proof
    let V be a_neighborhood of p;
    p in Int Int V by CONNSP_2:def 1;
    then reconsider W = Int V as a_neighborhood of p by CONNSP_2:def 1;
    N is_eventually_in W by A1,Def15;
    then consider i being Element of N such that
A3: for j being Element of N st i <= j holds N.j in W;
    defpred P[Element of N,object] means
ex m being Element of J.$1 st m = $2 & (
    i <= $1 implies for n being Element of J.$1 st m <= n holds J.$1.n in W);
A4: Int V = Int Int V;
A5: for j being Element of N ex e being object st P[j, e]
    proof
      let j be Element of N;
      reconsider j9 = j as Element of N;
      per cases;
      suppose
        i <= j;
        then N.j9 in W by A3;
        then
A6:     W is a_neighborhood of N.j by A4,CONNSP_2:def 1;
        N.j in Lim(J.j) by A2;
        then J.j is_eventually_in W by A6,Def15;
        then consider e being Element of J.j such that
A7:     for u being Element of J.j st e <= u holds J.j.u in W;
        take e,e;
        thus e = e;
        assume i <= j;
        thus thesis by A7;
      end;
      suppose
A8:     not i <= j;
        set e = the Element of J.j;
        take e,e;
        thus thesis by A8;
      end;
    end;
    consider f being ManySortedSet of the carrier of N such that
A9: for j being Element of N holds P[j, f.j] from PBOOLE:sch 6(A5);
A10: for x being object st x in dom Carrier J holds f.x in (Carrier J).x
    proof
      let x be object;
      assume x in dom Carrier J;
      then reconsider j = x as Element of N;
      ex m being Element of J.j st m = f.j & (i <= j implies for n being
      Element of J.j st m <= n holds J.j.n in W) by A9;
      then f.x in the carrier of J.j;
      hence thesis by Th2;
    end;
    dom Carrier J = the carrier of N by PARTFUN1:def 2;
    then dom f = dom Carrier J by PARTFUN1:def 2;
    then
A11: f in product Carrier J by A10,CARD_3:9;
A12: the carrier of Iterated J = [:the carrier of N, product Carrier J:]
    by Th26;
    then reconsider x = [i,f] as Element of Iterated J by A11,ZFMISC_1:87;
    take x;
    let j be Element of Iterated J such that
A13: x <= j;
    consider j1 being Element of N, j2 being Element of product Carrier J such
    that
A14: j = [j1,j2] by A12,DOMAIN_1:1;
    reconsider j2, i2 = f as Element of product J by A11,YELLOW_1:def 4;
    reconsider i1 = i, j1 as Element of N;
    i2 in the carrier of product J;
    then
A15: i2 in product Carrier J by YELLOW_1:def 4;
    the RelStr of Iterated J = [:N, product J:] by Def13;
    then
A16: [i1,i2] <= [j1,j2] by A13,A14,YELLOW_0:1;
    then i2 <= j2 by YELLOW_3:11;
    then ex f,g being Function st f = i2 & g = j2 &
for i be object st i in the
carrier of N ex R being RelStr, xi,yi being Element of R st R = J.i & xi = f.i
    & yi = g.i & xi <= yi by A15,YELLOW_1:def 4;
    then
A17: ex R being RelStr, xi,yi being Element of R st R = J.j1 & xi = i2.j1
    & yi = j2.j1 & xi <= yi;
    ex m being Element of J.j1 st m = f.j1 &( i <= j1 implies for n being
    Element of J.j1 st m <= n holds J.j1.n in W) by A9;
    then (J.j1).(j2.j1) in W by A16,A17,YELLOW_3:11;
    then
A18: (Iterated J).j in W by A14,Th27;
    W c= V by TOPS_1:16;
    hence thesis by A18;
  end;
  hence thesis by Def15;
end;
