theorem
  s is convergent & s1 is bounded & (lim s)=0c implies lim (s(#)s1)*' = 0c
proof
  assume
A1: s is convergent & s1 is bounded & (lim s)=0c;
  then s(#)s1 is convergent by Th29;
  hence lim (s(#)s1)*' = (lim (s(#)s1))*' by Th11
    .= 0c by A1,Th30,COMPLEX1:28;
end;
