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
  (ex r st f|left_open_halfline r is increasing & not f|
  left_open_halfline r is bounded_below) & (for r ex g st g<r & g in dom f)
  implies f is divergent_in-infty_to-infty by Th68;
