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 Th24:
  s is convergent & (lim s)<>0c & s is non-zero implies lim s"=( lim s)"
proof
  assume that
A1: s is convergent and
A2: (lim s)<>0c and
A3: s is non-zero;
  consider n1 such that
A4: for m st n1<=m holds |.(lim s).|/2<|.s.m.| by A1,A2,Th22;
A5: 0<|.(lim s).| by A2,COMPLEX1:47;
  then 0*0<|.(lim s).|*|.(lim s).|;
  then
A6: 0<(|.(lim s).|*|.(lim s).|)/2;
A7: 0<>|.(lim s).| by A2,COMPLEX1:47;
A8: now
    let p be Real;
A9: 0<>|.(lim s).|/2 by A2,COMPLEX1:47;
A10: (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 A9,XCMPLX_0:def 7
      .=p;
    assume
A11: 0<p;
    then 0*0<p*((|.(lim s).|*|.(lim s).|)/2) by A6;
    then consider n2 such that
A12: for m st n2<=m holds |.s.m-(lim s).|<p*((|.(lim s).|*|.(lim s).|)
    /2) by A1,Def6;
    take n=n1+n2;
    let m such that
A13: n<=m;
    n1<=n1+n2 by NAT_1:12;
    then n1<=m by A13,XXREAL_0:2;
    then
A14: |.(lim s).|/2<|.s.m.| by A4;
A15: 0<|.(lim s).|/2 by A5;
    then 0*0<p*(|.(lim s).|/2) by A11;
    then
A16: (p*(|.(lim s).|/2))/|.s.m.|< (p*(|.(lim s).|/2))/(|.(lim s).|/2) by A14
,A15,XREAL_1:76;
A17: (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 A7,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
A18: s.m <> 0 by A3,COMSEQ_1:3;
    then s.m*(lim s)<>0c by A2;
    then 0<|.s.m*(lim s).| by COMPLEX1:47;
    then
A19: 0<|.s.m.|*|.(lim s).| by COMPLEX1:65;
    n2<=n by NAT_1:12;
    then n2<=m by A13,XXREAL_0:2;
    then |.s.m-(lim s).|<p*((|.(lim s).|*|.(lim s).|)/2) by A12;
    then
A20: |.s.m-(lim s).|/(|.s.m.|*|.(lim s).|)< (p*((|.(lim s).|*|.(lim s).|)/
    2))/(|.s.m.|*|.(lim s).|) by A19,XREAL_1:74;
    |.(s").m-(lim s)".|=|.(s.m)"-(lim s)".| by VALUED_1:10
      .=|.s.m-(lim s).|/(|.s.m.|*|.(lim s).|) by A2,Th1,A18;
    hence |.(s").m-(lim s)".|<p by A20,A17,A16,A10,XXREAL_0:2;
  end;
  s" is convergent by A1,A2,A3,Th23;
  hence thesis by A8,Def6;
end;
