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 & (ex r st 0<r & for n holds seq2.n>=r)
  implies seq1(#)seq2 is divergent_to-infty
proof
  assume that
A1: seq1 is divergent_to-infty and
A2: ex r st 0<r & for n holds seq2.n>=r;
  (-jj)(#)seq1 is divergent_to+infty by A1,Th14;
  then
A3: (-jj)(#)seq1(#)seq2 is divergent_to+infty by A2,Th22;
  (-1)(#)((-1)(#)seq1(#)seq2)=(-1)(#)((-1)(#)(seq1(#)seq2)) by SEQ_1:18
    .=(-1)*(-1)(#)(seq1(#)seq2) by SEQ_1:23
    .=seq1(#)seq2 by SEQ_1:27;
  hence thesis by A3,Th13;
end;
