reserve r,r1,r2,g,g1,g2,x0,t for Real;
reserve n,k,m for Element of NAT;
reserve seq for Real_Sequence;
reserve f,f1,f2 for PartFunc of REAL,REAL;

theorem
  f is_convergent_in x0 & lim(f,x0)=0 & (ex r st 0<r & for g st g in dom
  f /\ (].x0-r,x0.[ \/ ].x0,x0+r.[) holds 0<f.g) implies f^
  is_divergent_to+infty_in x0
proof
  assume that
A1: f is_convergent_in x0 and
A2: lim(f,x0)=0;
A3: f is_right_convergent_in x0 by A1,Th29;
  given r such that
A4: 0<r and
A5: for g st g in dom f/\(].x0-r,x0.[\/].x0,x0+r.[) holds 0<f.g;
A6: now
    let g;
    assume
A7: g in dom f/\].x0,x0+r.[;
    then g in ].x0,x0+r.[ by XBOOLE_0:def 4;
    then
A8: g in ].x0-r,x0.[\/].x0,x0+r.[ by XBOOLE_0:def 3;
    g in dom f by A7,XBOOLE_0:def 4;
    then g in dom f/\(].x0-r,x0.[\/].x0,x0+r.[) by A8,XBOOLE_0:def 4;
    hence 0<f.g by A5;
  end;
A9: now
    let g;
    assume
A10: g in dom f/\].x0-r,x0.[;
    then g in ].x0-r,x0.[ by XBOOLE_0:def 4;
    then
A11: g in ].x0-r,x0.[\/].x0,x0+r.[ by XBOOLE_0:def 3;
    g in dom f by A10,XBOOLE_0:def 4;
    then g in dom f/\(].x0-r,x0.[\/].x0,x0+r.[) by A11,XBOOLE_0:def 4;
    hence 0<f.g by A5;
  end;
  lim_right(f,x0)=0 by A1,A2,Th29;
  then
A12: f^ is_right_divergent_to+infty_in x0 by A3,A4,A6,LIMFUNC2:73;
A13: f is_left_convergent_in x0 by A1,Th29;
  lim_left(f,x0)=0 by A1,A2,Th29;
  then f^ is_left_divergent_to+infty_in x0 by A13,A4,A9,LIMFUNC2:71;
  hence thesis by A12,Th12;
end;
