reserve r,r1,g for Real,
  n,m,k for Nat,
  seq,seq1, seq2 for Real_Sequence,
  f,f1,f2 for PartFunc of REAL,REAL,
  x for set;
reserve r,r1,r2,g,g1,g2 for Real;

theorem
  seq1 is divergent_to+infty & seq2 is convergent & 0<lim seq2 implies
  seq1(#)seq2 is divergent_to+infty
proof
  assume that
A1: seq1 is divergent_to+infty and
A2: seq2 is convergent and
A3: 0<lim seq2;
  consider n1 be Nat such that
A4: for m st n1<=m holds (lim seq2)/2<seq2.m by A2,A3,Th5;
  now
    thus 0<(lim seq2)/2 by A3,XREAL_1:215;
    let n;
    (lim seq2)/2<seq2.(n+n1) by A4,NAT_1:12;
    hence (lim seq2)/2<=(seq2^\n1).n by NAT_1:def 3;
  end;
  then (seq1^\n1)(#)(seq2^\n1) is divergent_to+infty by A1,Th22,Th26;
  then (seq1(#)seq2)^\n1 is divergent_to+infty by SEQM_3:19;
  hence thesis by Th7;
end;
