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 Th21: :: Transitivity of Big_Omega (Problem 3.12)
  for f,g,h being eventually-nonnegative Real_Sequence holds f in
  Big_Omega(g) & g in Big_Omega(h) implies f in Big_Omega(h)
proof
  let f,g,h be eventually-nonnegative Real_Sequence;
  assume f in Big_Omega(g) & g in Big_Omega(h);
  then h in Big_Oh(g) & g in Big_Oh(f) by Th19;
  then h in Big_Oh(f) by Th12;
  hence thesis by Th19;
end;
