reserve c, c1, d for Real,
  k for Nat,
  n, m, N, n1, N1, N2, N3, N4, N5, M for Element of NAT,
  x for set;

theorem :: also Problem 3.29:: Smoothness Rule (page 90)
  for f being eventually-nonnegative Real_Sequence, t being
eventually-nonnegative eventually-nondecreasing Real_Sequence, b being Element
of NAT st f is smooth & b >= 2 & t in Big_Theta(f, the set of all
 b|^n where n is Element of
  NAT  ) holds t in Big_Theta(f)
proof
  let f be eventually-nonnegative Real_Sequence, t be eventually-nonnegative
  eventually-nondecreasing Real_Sequence, b be Element of NAT;
  assume that
A1: f is smooth & b >= 2 and
A2: t in Big_Theta(f, the set of all  b|^n where n is Element of NAT  );
  set X = the set of all  b|^n where n is Element of NAT ;
A3: t in Big_Oh(f,X) /\ Big_Omega(f,X) by A2,Th36;
  then t in Big_Omega(f,X) by XBOOLE_0:def 4;
  then
A4: t in Big_Omega(f) by A1,Th39;
  t in Big_Oh(f,X) by A3,XBOOLE_0:def 4;
  then t in Big_Oh(f) by A1,Th38;
  hence thesis by A4,XBOOLE_0:def 4;
end;
