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 c= (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 Th4;
  then
  (superior_setsequence(A1 (/\) A2)).n c= Union (A1 ^\n) /\ Union (A2 ^\n)
  by Th8;
  then (superior_setsequence(A1 (/\) A2)).n c= (superior_setsequence A1).n /\
  Union (A2 ^\n) by Th2;
  hence thesis by Th2;
end;
