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;
reserve g for Complex;
reserve s,s1,s9 for Complex_Sequence;

theorem Th27:
  s is convergent implies lim |.s.| = |.lim s.|
proof
  set g = lim s;
  assume
A1: s is convergent;
  then reconsider s1 = s as convergent Complex_Sequence;
  reconsider w = |.s1.| as Real_Sequence;
A2: w is convergent;
  reconsider w = |.lim s.| as Real;
  now
    let p be Real;
    assume
A3: 0<p;
    consider n such that
A4: for m st n<=m holds |.s.m-g.|<p by A1,A3,COMSEQ_2:def 6;
    take n;
    let m;
    assume n<=m;
    then |.s.m-g.|<p by A4;
    then |.|.s.m.|- |.g.|.| + |.s.m-g.| < p+|.s.m-g.|
      by COMPLEX1:64,XREAL_1:8;
    then |.|.s.m.|- |.g.|.| + |.s.m-g.|- |.s.m-g.| < p+|.s.m-g.|- |.s.m-g.|
    by XREAL_1:9;
    hence |.|.s.|.m - w.| < p by VALUED_1:18;
  end;
  hence thesis by A2,Def6;
end;
