reserve n,k,k1,m,m1,n1,n2,l for Nat;
reserve r,r1,r2,p,p1,g,g1,g2,s,s1,s2,t for Real;
reserve seq,seq1,seq2 for Real_Sequence;
reserve Nseq for increasing sequence of NAT;
reserve x for set;
reserve X,Y for Subset of REAL;

theorem
  seq is convergent & (ex k st for n st k<=n holds seq1.n=seq.n) implies
  lim seq=lim seq1
proof
  assume that
A1: seq is convergent;
   given k such that
A2: for n st k<=n holds seq1.n=seq.n;
A3: now
    let p;
    assume 0<p;
    then consider n1 such that
A4: for m st n1<=m holds |.seq.m-lim seq.|<p by A1,SEQ_2:def 7;
A5: now
      assume
A6:   n1<=k;
      take n=k;
      let m;
      assume
A7:   n<=m;
      then n1<=m by A6,XXREAL_0:2;
      then |.seq.m-lim seq.|<p by A4;
      hence |.(seq1.m)-(lim seq).|<p by A2,A7;
    end;
    now
      assume
A8:   k<=n1;
      take n=n1;
      let m;
      assume
A9:  n<=m;
      then seq1.m=seq.m by A2,A8,XXREAL_0:2;
      hence |.(seq1.m)-(lim seq).|<p by A4,A9;
    end;
    hence ex n st for m st n<=m holds |.seq1.m-lim seq.|<p by A5;
  end;
  seq1 is convergent by A1,A2,Th18;
  hence thesis by A3,SEQ_2:def 7;
end;
