reserve n,m,k for Nat,
  x,X for set,
  A for Subset of X,
  A1,A2 for SetSequence of X;

theorem
  (inferior_setsequence A1).n \/ (inferior_setsequence A2).n c= (
  inferior_setsequence(A1 (\/) A2)).n
proof
A1: Intersection ((A1 ^\n) (\/) (A2 ^\n)) = Intersection ((A1 (\/) A2) ^\n)
  by Th5
    .= (inferior_setsequence(A1 (\/) A2)).n by Th1;
  (inferior_setsequence A1).n \/ (inferior_setsequence A2).n =
  Intersection (A1 ^\n) \/ (inferior_setsequence A2).n by Th1
    .= Intersection (A1 ^\n) \/ Intersection (A2 ^\n) by Th1;
  hence thesis by A1,Th13;
end;
