reserve T for non empty TopSpace,
  A, B for Subset of T,
  F, G for Subset-Family of T,
  x for set;

theorem Th32:
  for T being non empty TopSpace, A being Subset of T st T is T_1
  holds Der Der A c= Der A
proof
  let T be non empty TopSpace, A be Subset of T;
  assume
A1: T is T_1;
  let x be object;
  assume
A2: x in Der Der A;
  then reconsider x9 = x as Point of T;
  assume not x in Der A;
  then consider G being open Subset of T such that
A3: x9 in G and
A4: not ex y being Point of T st y in A /\ G & x9 <> y by Th17;
  Cl {x9} = {x9} by A1,YELLOW_8:26;
  then
A5: G \ {x9} is open by FRECHET:4;
  consider y being Point of T such that
A6: y in Der A /\ G and
A7: x <> y by A2,A3,Th17;
  y in Der A by A6,XBOOLE_0:def 4;
  then
A8: y in Der A \ {x} by A7,ZFMISC_1:56;
  y in G by A6,XBOOLE_0:def 4;
  then consider q being set such that
A9: q in G and
A10: q in Der A \ {x} by A8;
  reconsider q as Point of T by A9;
  not q in {x} by A10,XBOOLE_0:def 5;
  then
A11: q in G \ {x} by A9,XBOOLE_0:def 5;
  set U = G \ {x9};
A12: G misses A \ {x}
  proof
    assume G meets A \ {x};
    then consider g being object such that
A13: g in G and
A14: g in A \ {x} by XBOOLE_0:3;
    g in A by A14,XBOOLE_0:def 5;
    then g in A /\ G by A13,XBOOLE_0:def 4;
    then x9 = g by A4;
    hence thesis by A14,ZFMISC_1:56;
  end;
  q in Der A by A10,XBOOLE_0:def 5;
  then consider y being Point of T such that
A15: y in A /\ U and
A16: q <> y by A11,A5,Th17;
  y in A by A15,XBOOLE_0:def 4;
  then
A17: y in A \ {q} by A16,ZFMISC_1:56;
  y in U by A15,XBOOLE_0:def 4;
  then consider f being set such that
A18: f in G \ {x9} and
A19: f in A \ {q} by A17;
  f <> x9 & f in A by A18,A19,ZFMISC_1:56;
  then
A20: f in A \ {x9} by ZFMISC_1:56;
  f in G by A18,ZFMISC_1:56;
  hence thesis by A12,A20,XBOOLE_0:3;
end;
