reserve x,y,X for set;

theorem Th22:
  for T being non empty TopSpace, A being Subset of T for x being
  Point of T holds x in Cl A iff ex N being convergent net of T st N
  is_eventually_in A & x in Lim N
proof
  let T be non empty TopSpace, A be Subset of T;
  let x be Point of T;
  hereby
    assume x in Cl A;
    then consider N being net of T such that
A1: N is_eventually_in A and
A2: x is_a_cluster_point_of N by Th21;
    consider S being subnet of N such that
A3: x in Lim S by A2,WAYBEL_9:32;
    reconsider S as convergent net of T by A3,YELLOW_6:def 16;
    take S;
    thus S is_eventually_in A by A1,Th19;
    thus x in Lim S by A3;
  end;
  given N being convergent net of T such that
A4: N is_eventually_in A and
A5: x in Lim N;
  x is_a_cluster_point_of N by A5,WAYBEL_9:29;
  hence thesis by A4,Th21;
end;
