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

theorem Th40:
  for S be non empty 1-sorted, C be Convergence-Class of S holds C
  c= Convergence ConvergenceSpace C
proof
  let S be non empty 1-sorted, C be Convergence-Class of S;
  set T = ConvergenceSpace C;
  let x,y be object;
  assume
A1: [x,y] in C;
  C c= [:NetUniv S,the carrier of S:] by Def18;
  then consider M being Element of NetUniv S, p being Element of S such that
A2: [x,y] = [M,p] by A1,DOMAIN_1:1;
  reconsider q = p as Point of T by Def24;
A3: the carrier of S = the carrier of T & M in NetUniv S by Def24;
  ex N being strict net of S st N = M & the carrier of N in
  the_universe_of the carrier of S by Def11;
  then reconsider M as net of S;
  reconsider N = M as net of T by Def24;
A4: the topology of T = { V where V is Subset of S: for p being Element of S
  st p in V for N being net of S st [N,p] in C holds N is_eventually_in V} by
Def24;
  now
    let V be a_neighborhood of q;
    Int V in the topology of T by PRE_TOPC:def 2;
    then p in Int V & ex W being Subset of S st W = Int V & for p being
    Element of S st p in W for N being net of S st [N,p] in C holds N
    is_eventually_in W by A4,CONNSP_2:def 1;
    then M is_eventually_in Int V by A1,A2;
    then consider ii being Element of M such that
A5: for j being Element of M st ii <= j holds M.j in Int V;
    reconsider i = ii as Element of N;
    now
      let j be Element of N such that
A6:   i <= j;
      reconsider jj = j as Element of M;
      M.jj = N.j;
      hence N.j in Int V by A5,A6;
    end;
    then Int V c= V & N is_eventually_in Int V by TOPS_1:16;
    hence N is_eventually_in V by WAYBEL_0:8;
  end;
  then
A7: p in Lim N by Def15;
  N in NetUniv T by A3,Lm7;
  hence thesis by A2,A7,Def19;
end;
