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

theorem Th31: :: Theorem 1.3.4. (c)
  for T being non empty TopSpace, A, B being Subset of T holds Der
  (A \/ B) = Der A \/ Der B
proof
  let T be non empty TopSpace, A, B be Subset of T;
  thus Der (A \/ B) c= Der A \/ Der B
  proof
    let x be object;
    assume x in Der (A \/ B);
    then x is_an_accumulation_point_of A \/ B by Th16;
    then
A1: x in Cl ((A \/ B) \ {x});
    (A \/ B) \ {x} = (A \ {x}) \/ (B \ {x}) by XBOOLE_1:42;
    then Cl ((A \/ B) \ {x}) = Cl (A \ {x}) \/ Cl (B \ {x}) by PRE_TOPC:20;
    then x in Cl (A \ {x}) or x in Cl (B \ {x}) by A1,XBOOLE_0:def 3;
    then
    x is_an_accumulation_point_of A or x is_an_accumulation_point_of B;
    then x in Der A or x in Der B by Th16;
    hence thesis by XBOOLE_0:def 3;
  end;
  let x be object;
  assume x in Der A \/ Der B;
  then x in Der A or x in Der B by XBOOLE_0:def 3;
  then
A2: x is_an_accumulation_point_of A or x is_an_accumulation_point_of B by Th16;
  x in Cl ((A \/ B) \ {x})
  proof
    B \ {x} c= (A \/ B) \ {x} by XBOOLE_1:7,33;
    then
A3: Cl (B \ {x}) c= Cl ((A \/ B) \ {x}) by PRE_TOPC:19;
    A \ {x} c= (A \/ B) \ {x} by XBOOLE_1:7,33;
    then
A4: Cl (A \ {x}) c= Cl ((A \/ B) \ {x}) by PRE_TOPC:19;
    per cases by A2;
    suppose
      x in Cl (A \ {x});
      hence thesis by A4;
    end;
    suppose
      x in Cl (B \ {x});
      hence thesis by A3;
    end;
  end;
  then x is_an_accumulation_point_of A \/ B;
  hence thesis by Th16;
end;
