
theorem Th9:
for f be PartFunc of REAL,REAL, x0 be Real
 st f is_right_divergent_to-infty_in x0 holds
   not f is_right_divergent_to+infty_in x0 & not f is_right_convergent_in x0
proof
    let f be PartFunc of REAL,REAL, x0 be Real;
    assume A1: f is_right_divergent_to-infty_in x0; then
    consider seq be Real_Sequence such that
A2:  seq is convergent & lim seq = x0
   & rng seq c= dom f /\ right_open_halfline x0 by Th5;
A3: f/*seq is divergent_to-infty by A1,A2,LIMFUNC2:def 6;
    hence not f is_right_divergent_to+infty_in x0 by A2,Th3,LIMFUNC2:def 5;

    not ex g be Real st for seq be Real_Sequence
     st seq is convergent & lim seq=x0
      & rng seq c= dom f /\ right_open_halfline x0
      holds f/*seq is convergent & lim(f/*seq)=g by A2,A3,Th3;
    hence not f is_right_convergent_in x0 by LIMFUNC2:def 4;
end;
