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 (/\) A2)).n = (inferior_setsequence A1).n /\
  (inferior_setsequence A2).n
proof
  (inferior_setsequence(A1 (/\) A2)).n = Intersection ((A1 (/\) A2) ^\n) by Th1
    .= Intersection ((A1 ^\n) (/\) (A2 ^\n)) by Th4
    .= Intersection (A1 ^\n) /\ Intersection (A2 ^\n) by Th12
    .= (inferior_setsequence A1).n /\ Intersection (A2 ^\n) by Th1
    .= (inferior_setsequence A1).n /\ (inferior_setsequence A2).n by Th1;
  hence thesis;
end;
