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

theorem Th30: :: Theorem 1.3.4. (b)
  for T being non empty TopSpace, A, B being Subset of T st A c= B
  holds Der A c= Der B
proof
  let T be non empty TopSpace, A, B be Subset of T;
  assume
A1: A c= B;
  let x be object;
  assume
A2: x in Der A;
  then reconsider x9 = x as Point of T;
  for U being open Subset of T st x9 in U ex y being Point of T st y in B
  /\ U & x9 <> y
  proof
    let U be open Subset of T;
    assume x9 in U;
    then
A3: ex y being Point of T st y in A /\ U & x9 <> y by A2,Th17;
    A /\ U c= B /\ U by A1,XBOOLE_1:26;
    hence thesis by A3;
  end;
  hence thesis by Th17;
end;
