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;

theorem
  seq is convergent & seq1 is bounded & lim seq=0 implies lim(seq(#)seq1)=0
proof
  assume that
A1: seq is convergent and
A2: seq1 is bounded and
A3: lim seq=0;
A4: seq(#)seq1 is convergent by A1,A2,A3,Th25;
  now
    let p such that
A5: 0<p;
    consider r such that
A6: 0<r and
A7: for m holds |.seq1.m.|<r by A2,Th3;
A8: 0<p/r by A5,A6;
    then consider n1 such that
A9: for m st n1<=m holds |.seq.m-0.|<p/r by A1,A3,Def6;
    take n=n1;
    let m;
    assume n<=m;
    then
A10: |.seq.m-0.|<p/r by A9;
A11: |.((seq(#)seq1).m)-0.|=|.seq.m*seq1.m-0.| by SEQ_1:8
      .=|.seq.m.|*|.seq1.m.| by COMPLEX1:65;
    now
      assume
A12:  |.seq1.m.|<>0;
      (p/r)*r=p*r"*r by XCMPLX_0:def 9
        .=p*(r"*r)
        .=p*1 by A6,XCMPLX_0:def 7;
      then
A13:  (p/r)*|.seq1.m.|<p by A7,A8,XREAL_1:68;
      0<=|.seq1.m.| by COMPLEX1:46;
      then |.((seq(#)seq1).m)-0.|<(p/r)*|.seq1.m.| by A10,A11,A12,XREAL_1:68;
      hence |.((seq(#)seq1).m)-0.|<p by A13,XXREAL_0:2;
    end;
    hence |.((seq(#)seq1).m)-0.|<p by A5,A11;
  end;
  hence thesis by A4,Def6;
end;
