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
  f is_convergent_in x0 & f"{0}={} & lim(f,x0)<>0 implies f^
  is_convergent_in x0 & lim(f^,x0)=(lim(f,x0))"
proof
  assume that
A1: f is_convergent_in x0 and
A2: f"{0}={} and
A3: lim(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^)\{x0};
A9: lim(f/*seq)=lim(f,x0) by A1,A4,A6,A7,A8,Def4;
A10: (f/*seq)"=(f^)/*seq by A8,RFUNCT_2:12,XBOOLE_1:1;
A11: rng seq c=dom f by A4,A8,XBOOLE_1:1;
A12: f/*seq is convergent by A1,A4,A6,A7,A8;
    hence (f^)/*seq is convergent by A3,A4,A9,A11,A10,RFUNCT_2:11,SEQ_2:21;
    thus lim((f^)/*seq)=(lim(f,x0))" by A3,A4,A12,A9,A11,A10,RFUNCT_2:11
,SEQ_2:22;
  end;
  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,A4;
  hence f^ is_convergent_in x0 by A5;
  hence thesis by A5,Def4;
end;
