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

theorem
  (inferior_setsequence(A (\) A1)).n c= A \ (inferior_setsequence A1).n
proof
  (inferior_setsequence(A (\) A1)).n = Intersection ((A (\) A1) ^\n) by Th1
    .= Intersection (A (\) (A1 ^\n)) by Th18;
  then (inferior_setsequence(A (\) A1)).n c= A \ Intersection (A1 ^\n) by Th35;
  hence thesis by Th1;
end;
