reserve n,n1,n2,m for Nat;
reserve r,g1,g2,g,g9 for Complex;
reserve R,R2 for Real;
reserve s,s9,s1 for Complex_Sequence;

theorem Th23:
  for s being convergent Complex_Sequence st lim s <> 0c & s is non-zero
   holds s" is convergent
proof
  let s be convergent Complex_Sequence;
  assume that
A1: (lim s)<>0c and
A2: s is non-zero;
  consider n1 such that
A3: for m st n1<=m holds |.(lim s).|/2<|.s.m.| by A1,Th22;
  take(lim s)";
  let p be Real;
  assume
A4: 0<p;
A5: 0<|.(lim s).| by A1,COMPLEX1:47;
  then 0*0<|.(lim s).|*|.(lim s).|;
  then 0<(|.(lim s).|*|.(lim s).|)/2;
  then 0*0<p*((|.(lim s).|*|.(lim s).|)/2) by A4;
  then consider n2 such that
A6: for m st n2<=m holds |.s.m-(lim s).|<p*((|.(lim s).|*|.(lim s).|)/2
  ) by Def6;
  take n=n1+n2;
  let m such that
A7: n<=m;
  n1<=n1+n2 by NAT_1:12;
  then n1<=m by A7,XXREAL_0:2;
  then
A8: |.(lim s).|/2<|.s.m.| by A3;
A9: 0<|.(lim s).|/2 by A5;
  then 0*0<p*(|.(lim s).|/2) by A4;
  then
A10: (p*(|.(lim s).|/2))/|.s.m.|< (p*(|.(lim s).|/2))/(|.(lim s).|/2) by A8,A9,
XREAL_1:76;
A11: 0<>|.(lim s).|/2 by A1,COMPLEX1:47;
A12: (p*(|.(lim s).|/2))/(|.(lim s).|/2 ) =(p*(|.(lim s).|/2))*(|.(lim s).|/
  2 )" by XCMPLX_0:def 9
    .=p*((|.(lim s).|/2)*(|.(lim s).|/2 )")
    .=p*1 by A11,XCMPLX_0:def 7
    .=p;
A13: 0<>|.(lim s).| by A1,COMPLEX1:47;
A14: (p*((|.(lim s).|*|.(lim s).|)/2))/(|.s.m.|*|.(lim s).|) =(p*(2"*(|.(lim
  s).|*|.(lim s).|)))* (|.s.m.|*|.(lim s).|)" by XCMPLX_0:def 9
    .=p*2"*((|.(lim s).|*|.(lim s).|)*(|.(lim s).|*|.s.m.|)")
    .=p*2"*((|.(lim s).|*|.(lim s).|)* (|.(lim s).|"*|.s.m.|")) by XCMPLX_1:204
    .=p*2"*(|.(lim s).|*(|.(lim s).|*|.(lim s).|")*|.s.m.|")
    .=p*2"*(|.(lim s).|*1*|.s.m.|") by A13,XCMPLX_0:def 7
    .=p*(|.(lim s).|/2)*|.s.m.|"
    .=(p*(|.(lim s).|/2))/|.s.m.| by XCMPLX_0:def 9;
 m in NAT by ORDINAL1:def 12;
  then
A15: s.m <> 0 by A2,COMSEQ_1:3;
  then s.m*(lim s)<>0c by A1;
  then 0<|.s.m*(lim s).| by COMPLEX1:47;
  then
A16: 0<|.s.m.|*|.(lim s).| by COMPLEX1:65;
  n2<=n by NAT_1:12;
  then n2<=m by A7,XXREAL_0:2;
  then |.s.m-(lim s).|<p*((|.(lim s).|*|.(lim s).|)/2) by A6;
  then
A17: |.s.m-(lim s).|/(|.s.m.|*|.(lim s).|)< (p*((|.(lim s).|*|.(lim s).|)/2)
  )/(|.s.m.|*|.(lim s).|) by A16,XREAL_1:74;
  |.(s").m-(lim s)".|=|.(s.m)"-(lim s)".| by VALUED_1:10
    .=|.s.m-(lim s).|/(|.s.m.|*|.(lim s).|) by A1,Th1,A15;
  hence thesis by A17,A14,A10,A12,XXREAL_0:2;
end;
