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 Th2:
  for f,g being eventually-positive Real_Sequence st f/"g is
convergent & lim(f/"g) <> 0 holds g/"f is convergent & lim(g/"f) = (lim(f/"g))"
proof
  let f,g be eventually-positive Real_Sequence;
  set s = f/"g;
  set als = |.lim s.|;
  set ls = lim s;
  assume that
A1: f/"g is convergent and
A2: lim(f/"g) <> 0;
  consider n1 being Nat such that
A3: for m being Nat st n1<=m holds als/2<|.s.m.| by A1,A2,SEQ_2:16;
A4: 0<als by A2,COMPLEX1:47;
  then 0*0<als*als by XREAL_1:96;
  then
A5: 0<(als*als)/2 by XREAL_1:215;
  consider N2 being Nat such that
A6: for n being Nat st n >= N2 holds g.n > 0 by Def4;
  consider N1 being Nat such that
A7: for n being Nat st n >= N1 holds f.n > 0 by Def4;
A8: 0<>als by A2,COMPLEX1:47;
A9: now
    set N3=N1+N2;
    let p be Real;
    set N4=N3+n1;
A10: 0<>als/2 by A2,COMPLEX1:47;
A11: (p*(als/2))/(als/2 ) = (p*(als/2))*(als/2 )" by XCMPLX_0:def 9
      .=p*((als/2)*(als/2 )")
      .=p*1 by A10,XCMPLX_0:def 7
      .=p;
    assume
A12: 0 < p;
    then 0*0<p*((als*als)/2) by A5,XREAL_1:96;
    then consider n2 being Nat such that
A13: for m being Nat st n2<=m holds |.s.m-ls.|<p*((als*als)/2)
         by A1,SEQ_2:def 7;
    take n=N4+n2;
    let m being Nat such that
A14: n<=m;
    set asm = |.s.m.|;
A15: (p*((als*als)/2))/(asm*als) =(p*(2"*(als*als)))*(asm*als)" by
XCMPLX_0:def 9
      .=p*2"*((als*als)*(als*asm)")
      .=p*2"*((als*als)*(als"*asm")) by XCMPLX_1:204
      .=p*2"*(als*(als*als")*asm")
      .=p*2"*(als*1*asm") by A8,XCMPLX_0:def 7
      .=p*(als/2)*asm"
      .=(p*(als/2))/asm by XCMPLX_0:def 9;
    n1 <= N4 by NAT_1:12;
    then n1<=n by NAT_1:12;
    then n1<=m by A14,XXREAL_0:2;
    then
A16: als/2<asm by A3;
A17: 0<als/2 by A4,XREAL_1:215;
    then 0*0<p*(als/2) by A12,XREAL_1:96;
    then
A18: (p*(als/2))/asm < (p*(als/2))/(als/2) by A16,A17,XREAL_1:76;
    N2 <= N3 by NAT_1:12;
    then N2 <= N4 by NAT_1:12;
    then N2<=n by NAT_1:12;
    then N2<=m by A14,XXREAL_0:2;
    then
A19: g.m<>0 by A6;
    N1 <= N3 by NAT_1:12;
    then N1 <= N4 by NAT_1:12;
    then N1<=n by NAT_1:12;
    then N1<=m by A14,XXREAL_0:2;
    then f.m<>0 by A7;
    then f.m/g.m<>0 by A19,XCMPLX_1:50;
    then
A20: s.m<>0 by Lm1;
    then s.m*ls<>0 by A2,XCMPLX_1:6;
    then 0<|.s.m*ls.| by COMPLEX1:47;
    then
A21: 0<asm*als by COMPLEX1:65;
    n2<=n by NAT_1:12;
    then n2<=m by A14,XXREAL_0:2;
    then |.s.m-ls.|<p*((als*als)/2) by A13;
    then
A22: |.s.m-ls.|/(asm*als) < (p*((als*als)/2))/(asm*als) by A21,XREAL_1:74;
    |.(g/"f).m-ls".| = |.(g.m/f.m)-ls".| by Lm1
      .=|.(f.m/g.m)"-ls".| by XCMPLX_1:213
      .=|.(s.m)"-ls".| by Lm1
      .=|.s.m-ls.|/(asm*als) by A2,A20,SEQ_2:2;
    hence |.(g/"f).m-ls".|<p by A22,A15,A18,A11,XXREAL_0:2;
  end;
  hence g/"f is convergent by SEQ_2:def 6;
  hence thesis by A9,SEQ_2:def 7;
end;
