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 Th5:
  for f being Real_Sequence, g being eventually-nonzero
Real_Sequence st f/"g is divergent_to+infty holds g/"f is convergent & lim(g/"f
  )=0
proof
  let f be Real_Sequence, g be eventually-nonzero Real_Sequence;
  consider N1 being Nat such that
A1: for n being Nat st n >= N1 holds g.n <> 0 by Def5;
  assume
A2: f/"g is divergent_to+infty;
A3: now
    let p be Real such that
A4: p>0;
    reconsider w = 1/p as Real;
    consider N being Nat such that
A5: for n being Nat st n>=N holds w<(f/"g).n by A2,LIMFUNC1:def 4;
    reconsider a = max(N, N1) as Nat by TARSKI:1;
    take a;
    let n be Nat;
    assume
A6: n >= a;
    a >= N by XXREAL_0:25;
    then n >= N by A6,XXREAL_0:2;
    then 1/p < (f/"g).n by A5;
    then (1/p)*p < p*(f/"g).n by A4,XREAL_1:68;
    then 1 < p*(f/"g).n by A4,XCMPLX_1:106;
    then
A7: 1 < p*(f.n/g.n) by Lm1;
    then
A8: 1 < p*(f.n*(g.n)") by XCMPLX_0:def 9;
A9: now
      assume g.n*(f.n)" > g.n*0;
      then g.n*f".n > 0 by VALUED_1:10;
      hence (g/"f).n-0 >= 0 by SEQ_1:8;
    end;
    a >= N1 by XXREAL_0:25;
    then
A10: g.n <> 0 by A1,A6,XXREAL_0:2;
A11: f.n <> 0 by A7;
A12: now
      assume g.n*(f.n)" < p*f.n*(f.n)";
      then g.n*f".n < p*f.n*(f.n)" by VALUED_1:10;
      then (g(#)(f")).n < p*(f.n*(f.n)") by SEQ_1:8;
      then (g(#)(f")).n < p*1 by A11,XCMPLX_0:def 7;
      hence (g/"f).n-0 < p;
    end;
    per cases by A10;
    suppose
A13:  g.n > 0;
      then 1*g.n < (p*f.n)*(g.n)"*g.n by A8,XREAL_1:68;
      then g.n < (p*f.n)*((g.n)"*g.n);
      then
A14:  g.n < (p*f.n)*1 by A10,XCMPLX_0:def 7;
      f.n > 0 by A4,A7,A13;
      hence |.(g/"f).n-0.|<p by A12,A9,A13,A14,ABSVALUE:def 1,XREAL_1:68;
    end;
    suppose
A15:  g.n < 0;
      then 1*g.n > (p*f.n)*(g.n)"*g.n by A8,XREAL_1:69;
      then g.n > (p*f.n)*((g.n)"*g.n);
      then
A16:  g.n > (p*f.n)*1 by A10,XCMPLX_0:def 7;
      f.n < 0 by A4,A7,A15;
      hence |.(g/"f).n-0.|<p by A12,A9,A15,A16,ABSVALUE:def 1,XREAL_1:69;
    end;
  end;
  hence g/"f is convergent by SEQ_2:def 6;
  hence thesis by A3,SEQ_2:def 7;
end;
