
theorem Th6:
for f be PartFunc of REAL,REAL, x0 be Real
 st f is_left_divergent_to+infty_in x0  holds
   not f is_left_divergent_to-infty_in x0 & not f is_left_convergent_in x0
proof
    let f be PartFunc of REAL,REAL, x0 be Real;
    assume A1: f is_left_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 /\ left_open_halfline x0 by Th4;
A3: f/*seq is divergent_to+infty by A1,A2,LIMFUNC2:def 2;
    hence not f is_left_divergent_to-infty_in x0 by A2,Th2,LIMFUNC2:def 3;

    not ex g be Real st for seq be Real_Sequence
     st seq is convergent & lim seq=x0
      & rng seq c= dom f /\ left_open_halfline x0
      holds f/*seq is convergent & lim(f/*seq)=g by A2,A3,Th2;
    hence not f is_left_convergent_in x0 by LIMFUNC2:def 1;
end;
