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

A3: for seq be Real_Sequence st seq is convergent & lim seq=x0 &
      rng seq c= dom f /\ right_open_halfline(x0) holds
        f/*seq is convergent & lim(f/*seq)=f.x0 by A1,FDIFF_3:def 2;
    hence f is_right_convergent_in x0 by A2,LIMFUNC2:def 4;
    hence lim_right(f,x0) = f.x0 by A3,LIMFUNC2:def 8;
end;
