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_Omega, Part 1 (page 86)
  for f,g being eventually-positive Real_Sequence st f/"g is convergent
  & lim( f/"g ) > 0 holds Big_Omega(f) = Big_Omega(g)
proof
  let f,g be eventually-positive Real_Sequence;
  assume
A1: f/"g is convergent & lim( f/"g ) > 0;
  now
    let x be object;
    hereby
      g in Big_Oh(g) by Th10;
      then g in Big_Oh(f) by A1,Th15;
      then
A2:   f in Big_Omega(g) by Th19;
      assume x in Big_Omega(f);
      then consider t being Element of Funcs(NAT, REAL) such that
A3:   x = t and
A4:   ex d,N st d > 0 & for n st n >= N holds d*f.n <= t.n & t.n >= 0;
      consider d,N such that
      d > 0 and
A5:   for n st n >= N holds d*f.n <= t.n & t.n >= 0 by A4;
      now
         reconsider N as Nat;
        take N;
        let n be Nat;
A6:      n in NAT by ORDINAL1:def 12;
        assume n >= N;
        hence t.n >= 0 by A5,A6;
      end;
      then
A7:   t is eventually-nonnegative;
      t in Big_Omega(f) by A4;
      hence x in Big_Omega(g) by A3,A7,A2,Th21;
    end;
    assume x in Big_Omega(g);
    then consider t being Element of Funcs(NAT, REAL) such that
A8: x = t and
A9: ex d,N st d > 0 & for n st n >= N holds d*g.n <= t.n & t.n >= 0;
    consider d,N such that
    d > 0 and
A10: for n st n >= N holds d*g.n <= t.n & t.n >= 0 by A9;
    now
       reconsider N as Nat;
      take N;
      let n be Nat;
A11:      n in NAT by ORDINAL1:def 12;
      assume n >= N;
      hence t.n >= 0 by A10,A11;
    end;
    then
A12: t is eventually-nonnegative;
    f in Big_Oh(f) by Th10;
    then f in Big_Oh(g) by A1,Th15;
    then
A13: g in Big_Omega(f) by Th19;
    t in Big_Omega(g) by A9;
    hence x in Big_Omega(f) by A8,A12,A13,Th21;
  end;
  hence thesis by TARSKI:2;
end;
