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

theorem
  f is_right_convergent_in x0 & f"{0}={} & lim_right(f,x0)<>0 implies f^
  is_right_convergent_in x0 & lim_right(f^,x0)=(lim_right(f,x0))"
proof
  assume that
A1: f is_right_convergent_in x0 and
A2: f"{0}={} and
A3: lim_right(f,x0)<>0;
A4: dom f=dom f\f"{0} by A2
    .=dom(f^) by RFUNCT_1:def 2;
A5: now
    let seq;
    assume that
A6: seq is convergent and
A7: lim seq=x0 and
A8: rng seq c=dom(f^)/\right_open_halfline(x0);
A9: lim(f/*seq)=lim_right(f,x0) by A1,A4,A6,A7,A8,Def8;
A10: dom f/\right_open_halfline(x0) c=dom f by XBOOLE_1:17;
    then
A11: rng seq c=dom f by A4,A8,XBOOLE_1:1;
A12: f/*seq is convergent by A1,A4,A6,A7,A8;
A13: (f/*seq)"=(f^)/*seq by A4,A8,A10,RFUNCT_2:12,XBOOLE_1:1;
    hence (f^)/*seq is convergent by A3,A4,A12,A9,A11,RFUNCT_2:11,SEQ_2:21;
    thus lim((f^)/*seq)=(lim_right(f,x0))" by A3,A4,A12,A9,A11,A13,RFUNCT_2:11
,SEQ_2:22;
  end;
  for r st x0<r ex g st g<r & x0<g & g in dom(f^) by A1,A4;
  hence f^ is_right_convergent_in x0 by A5;
  hence thesis by A5,Def8;
end;
