reserve f,g for PartFunc of REAL,REAL,
  r,r1,r2,g1,g2,g3,g4,g5,g6,x,x0,t,c for Real,
  a,b,s for Real_Sequence,
  n,k for Element of NAT;

theorem Th3:
  f is_left_convergent_in x0 & lim_left(f,x0) = t iff (for r st r<
  x0 ex t st r<t & t<x0 & t in dom f) & for a st a is convergent & lim a = x0 &
rng a c= dom f /\ left_open_halfline(x0) holds f/*a is convergent & lim(f/*a) =
  t
proof
  thus f is_left_convergent_in x0 & lim_left(f,x0) = t implies (for r st r<x0
ex t st r<t & t<x0 & t in dom f) & for a st a is convergent & lim a = x0 & rng
  a c= dom f /\ left_open_halfline(x0) holds f/*a is convergent & lim(f/*a) = t
  by LIMFUNC2:def 1,def 7;
  reconsider t as Real;
  (for r st r<x0 ex t st r<t & t<x0 & t in dom f) & (for a st a is
  convergent & lim a = x0 & rng a c= dom f /\ left_open_halfline(x0) holds f/*a
is convergent & lim(f/*a) = t) implies f is_left_convergent_in x0 & lim_left(f,
  x0) = t
  proof
    assume that
A1: for r st r<x0 ex t st r<t & t<x0 & t in dom f and
A2: for a st a is convergent & lim a = x0 & rng a c= dom f /\
    left_open_halfline(x0) holds f/*a is convergent & lim(f/*a) = t;
    thus f is_left_convergent_in x0 by A1,A2,LIMFUNC2:def 1;
    hence thesis by A2,LIMFUNC2:def 7;
  end;
 hence thesis;
end;
