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

theorem
  Der (F \/ G) = Der F \/ Der G
proof
  thus Der (F \/ G) c= Der F \/ Der G
  proof
    let x be object;
    assume
A1: x in Der (F \/ G);
    then reconsider A = x as Subset of T;
    consider B being Subset of T such that
A2: A = Der B and
A3: B in F \/ G by A1,Def6;
    per cases by A3,XBOOLE_0:def 3;
    suppose
      B in F;
      then A in Der F by A2,Def6;
      hence thesis by XBOOLE_0:def 3;
    end;
    suppose
      B in G;
      then A in Der G by A2,Def6;
      hence thesis by XBOOLE_0:def 3;
    end;
  end;
  Der F c= Der (F \/ G) & Der G c= Der (F \/ G) by Th26,XBOOLE_1:7;
  hence thesis by XBOOLE_1:8;
end;
