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

theorem Th45:
  for T being discrete TopSpace, A being Subset of T holds Der A = {}
proof
  let T be discrete TopSpace, A be Subset of T;
  per cases;
  suppose
A1: T is non empty;
    assume Der A <> {};
    then consider x being object such that
A2: x in Der A by XBOOLE_0:def 1;
    x is_an_accumulation_point_of A by A1,A2,Th16;
    then x in Cl (A \ {x});
    then x in A \ {x} by PRE_TOPC:22;
    hence thesis by ZFMISC_1:56;
  end;
  suppose
    T is empty;
    then the carrier of T is empty;
    hence thesis;
  end;
end;
