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 :: Transitivity of Big_Theta (Problem 3.18 Part 3)
  for f,g,h being eventually-nonnegative Real_Sequence st f in Big_Theta
  (g) & g in Big_Theta(h) holds f in Big_Theta(h)
proof
  let f,g,h be eventually-nonnegative Real_Sequence;
  assume
A1: f in Big_Theta(g) & g in Big_Theta(h);
  then f in Big_Omega(g) & g in Big_Omega(h) by XBOOLE_0:def 4;
  then
A2: f in Big_Omega(h) by Th21;
  f in Big_Oh(g) & g in Big_Oh(h) by A1,XBOOLE_0:def 4;
  then f in Big_Oh(h) by Th12;
  hence thesis by A2,XBOOLE_0:def 4;
end;
