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

theorem Th42:
  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 closed iff for p being Element of T holds for N being net
  of T st [N,p] in C & N is_often_in S holds p in S
proof
  let T be non empty 1-sorted, C be topological Convergence-Class of T, S be
  Subset of ConvergenceSpace C;
  set CC = ConvergenceSpace C;
A1: the carrier of T = the carrier of CC by Def24;
  hereby
    assume S is closed;
    then
A2: S` is open;
    let p be Element of T;
    let N be net of T such that
A3: [N,p] in C;
    assume N is_often_in S;
    then not N is_eventually_in [#]CC\S by A1,WAYBEL_0:10;
    then not p in S` by A3,A2,Th41;
    hence p in S by A1,XBOOLE_0:def 5;
  end;
  assume
A4: for p being Element of T holds for N being net of T st [N,p] in C &
  N is_often_in S holds p in S;
  now
    let p be Element of T;
    assume p in S`;
    then
A5: not p in S by XBOOLE_0:def 5;
    let N be net of T;
    assume [N,p] in C;
    then not N is_often_in S by A4,A5;
    hence N is_eventually_in S` by A1,WAYBEL_0:10;
  end;
  then S` is open by Th41;
  hence thesis;
end;
