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 Th15: :: Limit Rule, Part 1 (page 84)
  for f,g being eventually-positive Real_Sequence st f/"g is
  convergent & lim( f/"g ) > 0 holds Big_Oh(f) = Big_Oh(g)
proof
  let f,g be eventually-positive Real_Sequence;
  assume
A1: f/"g is convergent & lim( f/"g ) > 0;
  then lim(g/"f) = (lim(f/"g))" by Th2;
  then
A2: g in Big_Oh(f) by A1,Lm3,Th2;
  f in Big_Oh(g) by A1,Lm3;
  hence thesis by A2,Lm2;
end;
