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

theorem Th43:
  for T being non empty 1-sorted, C being topological
Convergence-Class of T, S being Subset of ConvergenceSpace C, p being Point of
  ConvergenceSpace C st p in Cl S ex N being net of ConvergenceSpace C st [N,p]
  in C & rng the mapping of N c= S & p in Lim N
proof
  let T be non empty 1-sorted, C be topological Convergence-Class of T, S be
  Subset of (ConvergenceSpace C qua non empty TopSpace), p be Point of
  ConvergenceSpace C such that
A1: p in Cl S;
  set CC = ConvergenceSpace C;
  defpred P[Point of CC] means ex N being net of ConvergenceSpace C st [N,$1]
  in C & rng the mapping of N c= S & $1 in Lim N;
  set F = { q where q is Point of CC: P[q]};
  F is Subset of CC from DOMAIN_1:sch 7;
  then reconsider F as Subset of CC;
  for p being Element of T holds for N being net of T st [N,p] in C & N
  is_often_in F holds p in F
  proof
    let p be Element of T;
    reconsider q = p as Point of CC by Def24;
    let N be net of T such that
A2: [N,p] in C and
A3: N is_often_in F;
    C c= [:NetUniv T, the carrier of T:] by Def18;
    then N in NetUniv T by A2,ZFMISC_1:87;
    then
A4: ex N0 being strict net of T st N0 = N & the carrier of N0 in
    the_universe_of the carrier of T by Def11;
    reconsider M = N"F as subnet of N by A3,Th22;
    defpred P[Element of M, object] means
[$2,M.$1] in C & ex X being net of T st
    X = $2 & rng the mapping of X c= S;
A5: now
      let i be Element of M;
      i in the carrier of M;
      then i in (the mapping of N)"F by Def10;
      then
A6:   (the mapping of N).i in F by FUNCT_2:38;
      the mapping of M = (the mapping of N)|the carrier of M by Def6;
      then (the mapping of M).i in F by A6,FUNCT_1:49;
      then consider q being Point of CC such that
A7:   M.i = q and
A8:   ex N being net of ConvergenceSpace C st [N,q] in C & rng the
      mapping of N c= S & q in Lim N;
      consider N being net of CC such that
A9:   [N,q] in C and
A10:  rng the mapping of N c= S and
      q in Lim N by A8;
      reconsider x = N as object;
      take x;
      thus P[i, x]
      proof
        thus [x,M.i] in C by A7,A9;
        reconsider X = N as net of T by Def24;
        take X;
        thus X = x;
        thus thesis by A10;
      end;
    end;
    consider J being ManySortedSet of the carrier of M such that
A11: for i being Element of M holds P[i, J.i] from PBOOLE:sch 6(A5);
    for i being set st i in the carrier of M holds J.i is net of T
    proof
      let i be set;
      assume i in the carrier of M;
      then
      ex X being net of T st X = J.i & rng the mapping of X c= S by A11;
      hence thesis;
    end;
    then reconsider J as net_set of the carrier of M,T by Th24;
    set XX = the set of all  rng the mapping of J.i where i is Element of M;
A12: for i being Element of M holds [J.i,M.i] in C & rng the mapping of J.
    i c= S
    proof
      let i be Element of M;
      thus [J.i,M.i] in C by A11;
      ex X being net of T st X = J.i & rng the mapping of X c= S by A11;
      hence thesis;
    end;
    for x st x in XX holds x c= S
    proof
      let x;
      assume x in XX;
      then ex i being Element of M st x = rng the mapping of J.i;
      hence thesis by A12;
    end;
    then
A13: union XX c= S by ZFMISC_1:76;
    reconsider I = Iterated J as net of CC by Def24;
    rng the mapping of I c= union XX by Th28;
    then
A14: rng the mapping of I c= S by A13;
    the carrier of M = (the mapping of N)"F by Def10;
    then the carrier of M in the_universe_of the carrier of T by A4,
CLASSES1:def 1;
    then M in NetUniv T by Def11;
    then [M,p] in C by A2,Def21;
    then
A15: [I,p] in C by A12,Def23;
    C c= Convergence CC by Th40;
    then q in Lim I by A15,Def19;
    hence thesis by A15,A14;
  end;
  then
A16: F is closed by Th42;
  S c= F
  proof
    {} in {{}} by TARSKI:def 1;
    then reconsider r = {[{},{}]} as Relation of {{}} by RELSET_1:3;
    set R = RelStr(#{{}},r#);
A17: now
      let x,y be Element of R;
      x = {} & y = {} by TARSKI:def 1;
      then [x,y] in {[{},{}]} by TARSKI:def 1;
      hence x <= y by ORDERS_2:def 5;
    end;
A18: R is transitive
    by A17;
    R is directed
    proof
      let x,y be Element of R;
      take x;
      thus thesis by A17;
    end;
    then reconsider R as transitive directed non empty RelStr by A18;
    let x be object;
    set V = the_universe_of the carrier of T;
    assume
A19: x in S;
    then reconsider q = x as Point of CC;
    set N = ConstantNet(R,q);
    the mapping of N = (the carrier of N) --> q by Def5;
    then rng the mapping of N = {q} by FUNCOP_1:8;
    then
A20: rng the mapping of N c= S by A19,ZFMISC_1:31;
    {} in V by CLASSES2:56;
    then
A21: the carrier of R in V by CLASSES2:2;
    the carrier of CC = the carrier of T by Def24;
    then reconsider M = N as constant strict net of T by Lm6;
A22: the_value_of N = q by Th13;
    then the_value_of the mapping of M = q by Def8;
    then
A23: the_value_of M = q by Def8;
    the RelStr of N = the RelStr of R by Def5;
    then M in NetUniv T by A21,Def11;
    then
A24: [M,q] in C by A23,Def20;
    q in Lim N by A22,Th33;
    hence thesis by A24,A20;
  end;
  then Cl S c= F by A16,TOPS_1:5;
  then p in F by A1;
  then ex q being Point of CC st p = q & ex N being net of ConvergenceSpace C
  st [N,q] in C & rng the mapping of N c= S & q in Lim N;
  hence thesis;
end;
