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 Th12: :: Transitivity of Big_Oh (page 83; Problem 3.10)
  for f,g,h being eventually-nonnegative Real_Sequence holds f in
  Big_Oh(g) & g in Big_Oh(h) implies f in Big_Oh(h)
proof
  let f,g,h be eventually-nonnegative Real_Sequence;
  assume that
A1: f in Big_Oh(g) and
A2: g in Big_Oh(h);
  Big_Oh(g) c= Big_Oh(h) by A2,Th11;
  hence thesis by A1;
end;
