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

theorem
  (superior_setsequence(A1 (\/) A2)).n = (superior_setsequence A1).n \/
  (superior_setsequence A2).n
proof
  (superior_setsequence(A1 (\/) A2)).n = Union ((A1 (\/) A2) ^\n) by Th2
    .= Union ((A1 ^\n) (\/) (A2 ^\n)) by Th5
    .= Union (A1 ^\n) \/ Union (A2 ^\n) by Th9
    .= (superior_setsequence A1).n \/ Union (A2 ^\n) by Th2;
  hence thesis by Th2;
end;
