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

theorem
  (A \/ B)^0 = (A^0) \/ (B^0)
proof
  thus (A \/ B)^0 c= (A^0) \/ (B^0)
  proof
    let x be object;
    assume
A1: x in (A \/ B)^0;
    then reconsider x9 = x as Point of T;
    x9 is_a_condensation_point_of A \/ B by A1,Def10;
    then
    x9 is_a_condensation_point_of A or x9 is_a_condensation_point_of B by Th53;
    then x9 in A^0 or x9 in B^0 by Def10;
    hence thesis by XBOOLE_0:def 3;
  end;
  A^0 c= (A \/ B)^0 & B^0 c= (A \/ B)^0 by Th52,XBOOLE_1:7;
  hence thesis by XBOOLE_1:8;
end;
