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
  s is convergent & s1 is bounded & (lim s)=0c implies lim |.s(#)s1.| = 0
proof
  assume
A1: s is convergent & s1 is bounded & (lim s)=0c;
  then s(#)s1 is convergent by COMSEQ_2:42;
  hence lim |.s(#)s1.| = |.(lim (s(#)s1)).| by Th27
    .= 0 by A1,COMPLEX1:44,COMSEQ_2:43;
end;
