reserve n,n1,n2,m for Nat,
  r,r1,r2,p,g1,g2,g for Real,
  seq,seq9,seq1 for Real_Sequence,
  y for set;

theorem Th22:
  seq is convergent & lim seq <> 0 & seq is non-zero
  implies lim seq"=(lim seq)"
proof
  assume that
A1: seq is convergent and
A2: (lim seq)<>0 and
A3: seq is non-zero;
A4: seq" is convergent by A1,A2,A3,Th21;
A5: 0<|.lim seq.| by A2,COMPLEX1:47;
A6: 0<>|.lim seq.| by A2,COMPLEX1:47;
  consider n1 such that
A7: for m st n1<=m holds |.lim seq.|/2<|.seq.m.| by A1,A2,Th16;
  0*0<|.lim seq.|*|.lim seq.| by A5;
  then
A8: 0<(|.lim seq.|*|.lim seq.|)/2;
  now
    let p;
    assume
A9: 0<p;
    then 0*0<p*((|.lim seq.|*|.lim seq.|)/2) by A8;
    then consider n2 such that
A10: for m st n2<=m holds
    |.seq.m-lim seq.|<p*((|.lim seq.|*|.lim seq.|)/2) by A1,Def6;
    take n=n1+n2;
    let m such that
A11: n<=m;
    n2<=n by NAT_1:12;
    then n2<=m by A11,XXREAL_0:2;
    then
A12: |.seq.m-lim seq.|<p*((|.lim seq.|*|.lim seq.|)/2) by A10;
    n1<=n1+n2 by NAT_1:12;
    then n1<=m by A11,XXREAL_0:2;
    then
A13: |.lim seq.|/2<|.seq.m.| by A7;
A14: seq.m<>0 by A3,SEQ_1:5;
    then seq.m*(lim seq)<>0 by A2;
    then 0<|.seq.m*(lim seq).| by COMPLEX1:47;
    then 0<|.seq.m.|*|.lim seq.| by COMPLEX1:65;
    then
A15: |.seq.m-lim seq.|/(|.seq.m.|*|.lim seq.|)<
    (p*((|.lim seq.|*|.lim seq.|)/2))/(|.seq.m.|*|.lim seq.|)
    by A12,XREAL_1:74;
A16: (p*((|.lim seq.|*|.lim seq.|)/2))/(|.seq.m.|*|.lim seq.|)
    =(p*(2"*(|.lim seq.|*|.lim seq.|)))*
    (|.seq.m.|*|.lim seq.|)" by XCMPLX_0:def 9
      .=p*2"*((|.lim seq.|*|.lim seq.|)*(|.lim seq.|*|.seq.m.|)")
      .=p*2"*((|.lim seq.|*|.lim seq.|)*
    ((|.lim seq.|)"*(|.seq.m.|)")) by XCMPLX_1:204
      .=p*2"*(|.lim seq.|*(|.lim seq.|*|.lim seq.|")*|.seq.m.|")
      .=p*2"*(|.lim seq.|*1*|.seq.m.|") by A6,XCMPLX_0:def 7
      .=p*(|.lim seq.|/2)*|.seq.m.|"
      .=(p*(|.lim seq.|/2))/|.seq.m.| by XCMPLX_0:def 9;
A17: |.(seq").m-(lim seq)".|=|.(seq.m)"-(lim seq)".| by VALUED_1:10
      .=|.seq.m-lim seq.|/(|.seq.m.|*|.lim seq.|) by A2,A14,Th2;
A18: 0<|.lim seq.|/2 by A5;
A19: 0<>|.lim seq.|/2 by A2,COMPLEX1:47;
    0*0<p*(|.lim seq.|/2) by A9,A18;
    then
A20: (p*(|.lim seq.|/2))/|.seq.m.|<
    (p*(|.lim seq.|/2))/(|.lim seq.|/2) by A13,A18,XREAL_1:76;
    (p*(|.lim seq.|/2))/(|.lim seq.|/2 )
    =(p*(|.lim seq.|/2))*(|.lim seq.|/2 )" by XCMPLX_0:def 9
      .=p*((|.lim seq.|/2)*(|.lim seq.|/2 )")
      .=p*1 by A19,XCMPLX_0:def 7
      .=p;
    hence |.(seq").m-(lim seq)".|<p by A15,A16,A17,A20,XXREAL_0:2;
  end;
  hence thesis by A4,Def6;
end;
