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

theorem Th43: :: Theorem 10
  A is 1st_class & B is 1st_class implies Int Cl A /\ Int Cl B =
  Int Cl (A /\ B) & Cl Int A \/ Cl Int B = Cl Int (A \/ B)
proof
  assume that
A1: A is 1st_class and
A2: B is 1st_class;
A3: Cl Int B = Cl Int Cl B by A2,Th41;
  Cl Int A = Cl Int Cl A by A1,Th41;
  then
A4: Cl Int A \/ Cl Int B = Cl Int Cl (A \/ B) by A3,Th2;
  Int Cl (A /\ B) c= Int (Cl A /\ Cl B) by PRE_TOPC:21,TOPS_1:19;
  then
A5: Int Cl (A /\ B) c= Int Cl A /\ Int Cl B by TOPS_1:17;
  Int (A \/ B) c= Int Cl (A \/ B) by PRE_TOPC:18,TOPS_1:19;
  then
A6: Cl Int (A \/ B) c= Cl Int Cl (A \/ B) by PRE_TOPC:19;
  Cl (Int A \/ Int B) c= Cl Int (A \/ B) by PRE_TOPC:19,TOPS_1:20;
  then
A7: Cl Int A \/ Cl Int B c= Cl Int (A \/ B) by PRE_TOPC:20;
A8: Int Cl B = Int Cl Int B by A2,Th41;
  Cl Int (A /\ B) c= Cl (A /\ B) by PRE_TOPC:19,TOPS_1:16;
  then
A9: Int Cl Int (A /\ B) c= Int Cl (A /\ B) by TOPS_1:19;
  Int Cl A = Int Cl Int A by A1,Th41;
  then Int Cl A /\ Int Cl B = Int Cl Int (A /\ B) by A8,Th1;
  hence thesis by A5,A9,A7,A4,A6,XBOOLE_0:def 10;
end;
