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 Th22:
  seq1 is divergent_to+infty & (ex r st r>0 & 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 r>0 & for n holds seq2.n>=r;
  consider M be Real such that
A3: M>0 and
A4: for n holds seq2.n>=M by A2;
  let r;
A5: 0<=|.r.| by COMPLEX1:46;
  consider n such that
A6: for m st n<=m holds |.r.|/M<seq1.m by A1;
  take n;
  let m;
  assume n<=m;
  then |.r.|/M<seq1.m by A6;
  then |.r.|/M*M<seq1.m*seq2.m by A3,A4,A5,XREAL_1:97;
  then
A7: |.r.|<seq1.m*seq2.m by A3,XCMPLX_1:87;
  r<=|.r.| by ABSVALUE:4;
  then r<seq1.m*seq2.m by A7,XXREAL_0:2;
  hence thesis by SEQ_1:8;
end;
