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

theorem Th53:
  (inferior_setsequence(A1 (\) A)).n = (inferior_setsequence A1).n \ A
proof
  (inferior_setsequence(A1 (\) A)).n = Intersection ((A1 (\) A) ^\n) by Th1
    .= Intersection ((A1 ^\n) (\) A) by Th19
    .= Intersection (A1 ^\n) \ A by Th36
    .= (inferior_setsequence A1).n \ A by Th1;
  hence thesis;
end;
