
theorem Th34:
  for S being non empty TopSpace, c being Point of S for N being
net of S, A being Subset of S st c is_a_cluster_point_of N & A is closed & rng
  the mapping of N c= A holds c in A
proof
  let S be non empty TopSpace, c be Point of S, N be net of S, A be Subset of
  S such that
A1: c is_a_cluster_point_of N and
A2: A is closed and
A3: rng the mapping of N c= A;
  consider M being subnet of N such that
A4: c in Lim M by A1,Th32;
  reconsider R = rng the mapping of M as Subset of S;
  ex f being Function of M, N st the mapping of M = (the mapping of N)*f &
for m being Element of N ex n being Element of M st for p being Element of M st
  n <= p holds m <= f.p by YELLOW_6:def 9;
  then R c= rng the mapping of N by RELAT_1:26;
  then R c= A by A3;
  then
A5: Cl R c= Cl A by PRE_TOPC:19;
  c in Cl R by A4,Th24;
  then c in Cl A by A5;
  hence thesis by A2,PRE_TOPC:22;
end;
