reserve r,r1,g for Real,
  n,m,k for Nat,
  seq,seq1, seq2 for Real_Sequence,
  f,f1,f2 for PartFunc of REAL,REAL,
  x for set;
reserve r,r1,r2,g,g1,g2 for Real;

theorem
  f1 is convergent_in-infty & f2 is convergent_in-infty & lim_in-infty
f1=lim_in-infty f2 & (ex r st left_open_halfline(r) c= dom f1 /\ dom f2 /\ dom
f & for g st g in left_open_halfline(r) holds f1.g<=f.g & f.g<=f2.g) implies f
  is convergent_in-infty & lim_in-infty f=lim_in-infty f1
proof
  assume
A1: f1 is convergent_in-infty & f2 is convergent_in-infty & lim_in-infty
  f1= lim_in-infty f2;
  given r1 such that
A2: left_open_halfline(r1)c=dom f1/\dom f2/\dom f and
A3: for g st g in left_open_halfline(r1) holds f1.g<=f.g & f.g<=f2.g;
  dom f1/\dom f2/\dom f c=dom f by XBOOLE_1:17;
  then
A4: left_open_halfline(r1) c=dom f by A2;
A5: now
    let r;
    consider g being Real such that
A6: g<-|.r.|-|.r1.| by XREAL_1:2;
    take g;
    -|.r.|<=r & 0<=|.r1.| by ABSVALUE:4,COMPLEX1:46;
    then -|.r.|-|.r1.|<=r-0 by XREAL_1:13;
    hence g<r by A6,XXREAL_0:2;
    -|.r1.|<=r1 & 0<=|.r.| by ABSVALUE:4,COMPLEX1:46;
    then -|.r1.|-|.r.|<=r1-0 by XREAL_1:13;
    then g<r1 by A6,XXREAL_0:2;
    then g in {g1: g1<r1};
    then g in left_open_halfline(r1) by XXREAL_1:229;
    hence g in dom f by A4;
  end;
A7: dom f1/\dom f2/\dom f c=dom f1/\dom f2 by XBOOLE_1:17;
  now
    dom f1/\dom f2 c=dom f1 by XBOOLE_1:17;
    then dom f1/\dom f2/\dom f c=dom f1 by A7;
    then
A8: dom f1/\left_open_halfline(r1)=left_open_halfline(r1) by A2,XBOOLE_1:1,28;
    dom f1/\dom f2 c=dom f2 by XBOOLE_1:17;
    then dom f1/\dom f2/\ dom f c=dom f2 by A7;
    hence
    dom f1/\left_open_halfline(r1) c=dom f2/\left_open_halfline(r1) by A2,A8,
XBOOLE_1:1,28;
    thus dom f/\left_open_halfline(r1)c=dom f1/\left_open_halfline(r1) by A8,
XBOOLE_1:17;
    let g;
    assume g in dom f/\left_open_halfline(r1);
    then g in left_open_halfline(r1) by XBOOLE_0:def 4;
    hence f1.g<=f.g & f.g<=f2.g by A3;
  end;
  hence thesis by A1,A5,Th102;
end;
