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

theorem Th37:
  for F being Subset-Family of T st F is dense-in-itself holds
  union F c= union Der F
proof
  let F be Subset-Family of T;
  assume
A1: F is dense-in-itself;
  thus union F c= union Der F
  proof
    let x be object;
    assume x in union F;
    then consider A being set such that
A2: x in A and
A3: A in F by TARSKI:def 4;
    reconsider A as Subset of T by A3;
    A is dense-in-itself by A1,A3;
    then
A4: A c= Der A;
    Der A in Der F by A3,Def6;
    hence thesis by A2,A4,TARSKI:def 4;
  end;
end;
