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

theorem Th34: :: Theorem 1.3.4. d)
  for T being non empty TopSpace, F being Subset-Family of T holds
  union Der F c= Der union F
proof
  let T be non empty TopSpace, F be Subset-Family of T;
  let x be object;
  assume x in union Der F;
  then consider Y being set such that
A1: x in Y and
A2: Y in Der F by TARSKI:def 4;
  reconsider Y as Subset of T by A2;
  consider B being Subset of T such that
A3: Y = Der B and
A4: B in F by A2,Def6;
  Der B c= Der union F by A4,Th30,ZFMISC_1:74;
  hence thesis by A1,A3;
end;
