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 & (for r ex g st g<r & g in dom(f1+
  f2)) & (ex r st f2|left_open_halfline r is bounded_below) implies f1+f2 is
  divergent_in-infty_to+infty
proof
  assume that
A1: f1 is divergent_in-infty_to+infty and
A2: for r ex g st g<r & g in dom(f1+f2);
  given r1 such that
A3: f2|left_open_halfline r1 is bounded_below;
  now
    let seq;
    assume that
A4: seq is divergent_to-infty and
A5: rng seq c=dom(f1+f2);
    consider k such that
A6: for n st k<=n holds seq.n<r1 by A4;
A7: rng(seq^\k)c=rng seq by VALUED_0:21;
    dom(f1+f2)=dom f1/\dom f2 by A5,Lm2;
    then rng(seq^\k)c=dom f1/\dom f2 by A5,A7;
    then
A8: f1/*(seq^\k)+f2/*(seq^\k)=(f1+f2)/*(seq^\k) by RFUNCT_2:8
      .=((f1+f2)/*seq)^\k by A5,VALUED_0:27;
    consider r2 be Real such that
A9: for g being object st g in left_open_halfline(r1)/\dom f2 holds r2<=
    f2.g by A3,RFUNCT_1:71;
A10: rng seq c=dom f2 by A5,Lm2;
    then
A11: rng(seq^\k)c=dom f2 by A7;
    now
      let n;
A12: n in NAT by ORDINAL1:def 12;
      seq.(n+k)<r1 by A6,NAT_1:12;
      then (seq^\k).n<r1 by NAT_1:def 3;
      then (seq^\k).n in {g2: g2<r1};
      then
      (seq^\k).n in rng(seq^\k) & (seq^\k).n in left_open_halfline(r1) by
VALUED_0:28,XXREAL_1:229;
      then (seq^\k).n in left_open_halfline(r1)/\dom f2 by A11,XBOOLE_0:def 4;
      then r2<=f2.((seq^\k).n) by A9;
      then
A13:  r2<=(f2/*(seq^\k)).n by A10,A7,FUNCT_2:108,XBOOLE_1:1,A12;
      -|.r2.|<=r2 by ABSVALUE:4;
      then -|.r2.|-1<r2-0 by XREAL_1:15;
      hence -|.r2.|-1<(f2/*(seq^\k)).n by A13,XXREAL_0:2;
    end;
    then
A14: f2/*(seq^\k) is bounded_below;
    rng seq c=dom f1 by A5,Lm2;
    then
A15: rng(seq^\k)c=dom f1 by A7;
    seq^\k is divergent_to-infty by A4,Th27;
    then f1/*(seq^\k) is divergent_to+infty by A1,A15;
    then f1/*(seq^\k)+f2/*(seq^\k) is divergent_to+infty by A14,Th9;
    hence (f1+f2)/*seq is divergent_to+infty by A8,Th7;
  end;
  hence thesis by A2;
end;
