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 Th16:
  seq is convergent implies (lim seq<>0 implies
  ex n st for m st n<=m holds |.lim seq.|/2<|.seq.m.|)
proof
  assume that
A1: seq is convergent and
A2: lim seq<>0;
  0<|.lim seq.| by A2,COMPLEX1:47;
  then 0<|.lim seq.|/2;
  then consider n1 such that
A3: for m st n1<=m holds |.seq.m-lim seq.|<|.lim seq.|/2 by A1,Def6;
  take n=n1;
  let m;
  assume n<=m;
  then
A4: |.seq.m-lim seq.|<|.lim seq.|/2 by A3;
A5: |.lim seq.|-|.seq.m.|<=|.(lim seq)-seq.m.| by COMPLEX1:59;
  |.(lim seq)-seq.m.|=|.-(seq.m-(lim seq)).|
    .=|.seq.m-lim seq.| by COMPLEX1:52;
  then
A6: |.lim seq.|-|.seq.m.|<|.lim seq.|/2 by A4,A5,XXREAL_0:2;
A7: |.lim seq.|/2+(|.seq.m.|-|.lim seq.|/2) = |.seq.m.|;
  |.lim seq.|-|.seq.m.|+(|.seq.m.|-|.lim seq.|/2) = |.lim seq.|/2;
  hence thesis by A6,A7,XREAL_1:6;
end;
