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
  f1 is_convergent_in x0 & f2 is_convergent_in x0 & lim(f1,x0)=lim(f2,x0
  ) & (ex r st 0<r & ].x0-r,x0.[ \/ ].x0,x0+r.[ c= dom f1 /\ dom f2 /\ dom f &
for g st g in ].x0-r,x0.[ \/ ].x0,x0+r.[ holds f1.g<=f.g & f.g<=f2.g) implies f
  is_convergent_in x0 & lim(f,x0)=lim(f1,x0)
proof
  assume that
A1: f1 is_convergent_in x0 and
A2: f2 is_convergent_in x0 and
A3: lim(f1,x0)=lim(f2,x0);
  given r such that
A4: 0<r and
A5: ].x0-r,x0.[\/].x0,x0+r.[c=dom f1/\dom f2/\dom f and
A6: for g st g in ].x0-r,x0.[\/].x0,x0+r.[ holds f1.g<=f.g & f.g<=f2.g;
A7: dom f/\(].x0-r,x0.[\/].x0,x0+r.[)=].x0-r,x0.[\/].x0,x0+r.[ by A5,
XBOOLE_1:18,28;
A8: ].x0-r,x0.[\/].x0,x0+r.[c=dom f1/\dom f2 by A5,XBOOLE_1:18;
  then
A9: dom f1/\(].x0-r,x0.[\/].x0,x0+r.[)=].x0-r,x0.[\/].x0,x0+r.[ by XBOOLE_1:18
,28;
A10: dom f2/\(].x0-r,x0.[\/].x0,x0+r.[)=].x0-r,x0.[\/].x0,x0+r.[ by A8,
XBOOLE_1:18,28;
  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 A4,A5,Th5,XBOOLE_1:18;
  hence thesis by A1,A2,A3,A4,A6,A7,A9,A10,Th41;
end;
