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 implies lim abs seq = |.lim seq.|
proof
  assume
A1: seq is convergent;
  now
    let p;
    assume 0<p;
    then consider n1 such that
A2: for m st n1<=m holds |.(seq.m)-(lim seq).|<p by A1,SEQ_2:def 7;
    take n=n1;
    let m;
    |.(lim seq)-seq.m.|=|.-((seq.m)-(lim seq)).|
      .=|.(seq.m)-(lim seq).| by COMPLEX1:52;
    then |.lim seq.|-|.seq.m.|<=|.seq.m-lim seq.| by COMPLEX1:59;
    then |.seq.m.|-|.lim seq.|<=|.seq.m-lim seq.| & -|.(seq.m)-(lim
    seq).|<=- (|.lim seq.|-|.seq.m.|) by COMPLEX1:59,XREAL_1:24;
    then |.|.seq.m.|-|.lim seq.|.|<=|.seq.m-lim seq.| by ABSVALUE:5;
    then
A3: |.((abs seq).m)-|.lim seq.|.| <=|.seq.m-lim seq.| by SEQ_1:12;
    assume n<=m;
    then |.seq.m-lim seq.|<p by A2;
    hence |.((abs seq).m)-|.lim seq.|.|<p by A3,XXREAL_0:2;
  end;
  hence thesis by A1,SEQ_2:def 7;
end;
