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 divergent_in+infty_to-infty & (ex r st right_open_halfline(r) c=
dom f /\ dom f1 & for g st g in right_open_halfline(r) holds f.g<=f1.g) implies
  f is divergent_in+infty_to-infty
proof
  assume
A1: f1 is divergent_in+infty_to-infty;
  given r1 such that
A2: right_open_halfline(r1)c=dom f/\dom f1 and
A3: for g st g in right_open_halfline(r1) holds f.g<=f1.g;
A4: now
    let r;
    consider g being Real such that
A5: |.r.|+|.r1.|<g by XREAL_1:1;
    take g;
    0<=|.r1.| & r<=|.r.| by ABSVALUE:4,COMPLEX1:46;
    then 0+r<=|.r.|+|.r1.| by XREAL_1:7;
    hence r<g by A5,XXREAL_0:2;
    0<=|.r.| & r1<=|.r1.| by ABSVALUE:4,COMPLEX1:46;
    then 0+r1<=|.r.|+|.r1.| by XREAL_1:7;
    then r1<g by A5,XXREAL_0:2;
    then g in {g2: r1<g2};
    then g in right_open_halfline(r1) by XXREAL_1:230;
    hence g in dom f by A2,XBOOLE_0:def 4;
  end;
  now
    dom f/\dom f1 c=dom f by XBOOLE_1:17;
    then
A6: dom f/\right_open_halfline(r1)=right_open_halfline(r1) by A2,XBOOLE_1:1,28;
    dom f/\dom f1 c=dom f1 by XBOOLE_1:17;
    hence dom f/\right_open_halfline(r1) c=dom f1/\right_open_halfline(r1) by
A2,A6,XBOOLE_1:1,28;
    let g;
    assume g in dom f/\right_open_halfline(r1);
    hence f.g<=f1.g by A3,A6;
  end;
  hence thesis by A1,A4,Th71;
end;
