reserve T for TopSpace,
  A, B for Subset of T;

theorem :: Theorem 11
  A is 1st_class & B is 1st_class implies A \/ B is 1st_class & A /\ B
  is 1st_class
proof
  assume that
A1: A is 1st_class and
A2: B is 1st_class;
A3: Cl Int A = Cl Int Cl A by A1,Th41;
A4: Int Cl B = Int Cl Int B by A2,Th41;
A5: Int Cl A = Int Cl Int A by A1,Th41;
A6: Cl Int B = Cl Int Cl B by A2,Th41;
A7: Cl Int (A \/ B) = Cl Int A \/ Cl Int B by A1,A2,Th43
    .= Cl Int Cl (A \/ B) by A3,A6,Th2;
  Int Cl (A /\ B) = Int Cl A /\ Int Cl B by A1,A2,Th43
    .= Int Cl Int (A /\ B) by A5,A4,Th1;
  hence thesis by A7,Th42;
end;
