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 Th29:
  f is_convergent_in x0 implies f is_left_convergent_in x0 & f
is_right_convergent_in x0 & lim_left(f,x0)=lim_right(f,x0) & lim(f,x0)=lim_left
  (f,x0) & lim(f,x0)=lim_right(f,x0)
proof
  assume
A1: f is_convergent_in x0;
A2: now
    let s be Real_Sequence;
    assume that
A3: s is convergent and
A4: lim s=x0 and
A5: rng s c=dom f/\ right_open_halfline(x0);
    rng s c=dom f\{x0} by A5,Th1;
    hence f/*s is convergent & lim(f/*s)=lim(f,x0) by A1,A3,A4,Def4;
  end;
A6: now
    let s be Real_Sequence;
    assume that
A7: s is convergent and
A8: lim s=x0 and
A9: rng s c=dom f/\left_open_halfline(x0);
    rng s c=dom f\{x0} by A9,Th1;
    hence f/*s is convergent & lim(f/*s)=lim(f,x0) by A1,A7,A8,Def4;
  end;
A10: for r1,r2 st r1<x0 & x0<r2 ex g1,g2 st r1<g1 & g1<x0 & g1 in dom f & g2<
  r2 & x0<g2 & g2 in dom f by A1;
  then for r st r<x0 ex g st r<g & g<x0 & g in dom f by Th8;
  hence f is_left_convergent_in x0 by A6,LIMFUNC2:def 1;
  then
A11: lim_left(f,x0)=lim(f,x0) by A6,LIMFUNC2:def 7;
  for r st x0<r ex g st g<r & x0<g & g in dom f by A10,Th8;
  hence f is_right_convergent_in x0 by A2,LIMFUNC2:def 4;
  hence thesis by A11,A2,LIMFUNC2:def 8;
end;
