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 :: Limit Rule for Big_Theta, Part 1 (page 88)
  for f,g being eventually-positive Real_Sequence st f/"g is convergent
  & lim( f/"g ) > 0 holds f in Big_Theta(g)
proof
  let f,g be eventually-positive Real_Sequence;
  assume f/"g is convergent & lim( f/"g ) > 0;
  then
A1: Big_Oh(f) = Big_Oh(g) by Th15;
  then g in Big_Oh(f) by Th10;
  then
A2: f in Big_Omega(g) by Th19;
  f in Big_Oh(g) by A1,Th10;
  hence thesis by A2,XBOOLE_0:def 4;
end;
