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 c= (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 Th6;
  then (inferior_setsequence(A1 (\) A2)).n c= Intersection (A1 ^\n) \
  Intersection (A2 ^\n) by Th14;
  then (inferior_setsequence(A1 (\) A2)).n c= (inferior_setsequence A1).n \
  Intersection (A2 ^\n) by Th1;
  hence thesis by Th1;
end;
