
theorem
  for T being non empty TopSpace, A being Subset of T, p being set holds
  p in A \ Der A iff p is_isolated_in A
proof
  let T be non empty TopSpace, A be Subset of T, p be set;
  hereby
    assume
A1: p in A \ Der A;
    then not p in Der A by XBOOLE_0:def 5;
    then
A2: not p is_an_accumulation_point_of A by Th16;
    p in A by A1,XBOOLE_0:def 5;
    hence p is_isolated_in A by A2;
  end;
  assume
A3: p is_isolated_in A;
  then not p is_an_accumulation_point_of A;
  then
A4: not p in Der A by Th16;
  p in A by A3;
  hence thesis by A4,XBOOLE_0:def 5;
end;
