
theorem
for x0 be Real, f be PartFunc of REAL,REAL st f is_Lcontinuous_in x0 &
 (for r be Real st r<x0 ex g be Real st r<g & g<x0 & g in dom f)
 holds f is_left_convergent_in x0 & lim_left(f,x0) = f.x0
proof
    let x0 be Real, f be PartFunc of REAL,REAL;
    assume that
A1: f is_Lcontinuous_in x0 and
A2: for r be Real st r<x0 ex g be Real st r<g & g<x0 & g in dom f;

A3: 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)=f.x0 by A1,FDIFF_3:def 1;
    hence f is_left_convergent_in x0 by A2,LIMFUNC2:def 1;
    hence lim_left(f,x0) = f.x0 by A3,LIMFUNC2:def 7;
end;
