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

theorem Th41:
  for T being non empty 1-sorted, C being topological
  Convergence-Class of T, S being Subset of (ConvergenceSpace C qua non empty
  TopSpace) holds S is open iff for p being Element of T st p in S for N being
  net of T st [N,p] in C holds N is_eventually_in S
proof
  let T be non empty 1-sorted, C be topological Convergence-Class of T, S be
  Subset of ConvergenceSpace C;
A1: the topology of ConvergenceSpace C = { V where V is Subset of T: for p
  being Element of T st p in V for N being net of T st [N,p] in C holds N
  is_eventually_in V} by Def24;
  then
A2: S in the topology of ConvergenceSpace C implies ex V be Subset of T st S
  = V & for p being Element of T st p in V for N being net of T st [N,p] in C
  holds N is_eventually_in V;
  the carrier of ConvergenceSpace C = the carrier of T by Def24;
  then (for p being Element of T st p in S for N being net of T st [N,p] in C
holds N is_eventually_in S) implies S in the topology of ConvergenceSpace C by
A1;
  hence thesis by A2;
end;
