reserve r,r1,r2,g,g1,g2,x0,t for Real;
reserve n,k,m for Element of NAT;
reserve seq for Real_Sequence;
reserve f,f1,f2 for PartFunc of REAL,REAL;

theorem Th3:
  seq is convergent & lim seq=x0 & rng seq c= dom f \ {x0} implies
  for r st 0<r ex n st for k st n<=k holds 0<|.x0-seq.k.| & |.x0-seq.k.|<r &
  seq.k in dom f
proof
  assume that
A1: seq is convergent and
A2: lim seq=x0 and
A3: rng seq c=dom f\{x0};
  let r;
  assume 0<r;
  then consider n being Nat such that
A4: for k being Nat st n<=k holds |.seq.k-x0.|<r by A1,A2,SEQ_2:def 7;
   reconsider n as Element of NAT by ORDINAL1:def 12;
  take n;
  let k;
  assume n<=k;
  then |.seq.k-x0.|<r by A4;
  then
A5: |.-(x0-seq.k).|<r;
  now
    let n;
    seq.n in rng seq by VALUED_0:28;
    then not seq.n in {x0} by A3,XBOOLE_0:def 5;
    hence seq.n-x0<>0 by TARSKI:def 1;
  end;
  then seq.k-x0<>0;
  then 0<|.-(x0-seq.k).| by COMPLEX1:47;
  hence 0<|.x0-seq.k.| by COMPLEX1:52;
  thus |.x0-seq.k.|<r by A5,COMPLEX1:52;
  seq.k in rng seq by VALUED_0:28;
  hence thesis by A3,XBOOLE_0:def 5;
end;
