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 Th73:
  f1 is divergent_in-infty_to-infty & (for r ex g st g<r & g in
dom f) & (ex r st dom f/\left_open_halfline(r) c= dom f1 /\ left_open_halfline(
r) & for g st g in dom f /\ left_open_halfline(r) holds f.g<=f1.g) implies f 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 f;
  given r1 such that
A3: dom f/\left_open_halfline(r1)c= dom f1/\left_open_halfline(r1) and
A4: for g st g in dom f/\left_open_halfline(r1) holds f.g<=f1.g;
  now
    let seq;
    assume that
A5: seq is divergent_to-infty and
A6: rng seq c=dom f;
    consider k such that
A7: for n st k<=n holds seq.n<r1 by A5;
    now
      let x be object;
      assume x in rng(seq^\k);
      then consider n being Element of NAT such that
A8:   (seq^\k).n=x by FUNCT_2:113;
      seq.(n+k)<r1 by A7,NAT_1:12;
      then (seq^\k).n<r1 by NAT_1:def 3;
      then x in {g2: g2<r1} by A8;
      hence x in left_open_halfline(r1) by XXREAL_1:229;
    end;
    then
A9: rng(seq^\k)c=left_open_halfline(r1);
A10: rng(seq^\k)c=rng seq by VALUED_0:21;
    then rng(seq^\k)c=dom f by A6;
    then
A11: rng(seq^\k)c=dom f/\left_open_halfline(r1) by A9,XBOOLE_1:19;
    then
A12: rng(seq^\k)c=dom f1/\left_open_halfline(r1) by A3;
A13: dom f1/\left_open_halfline(r1)c=dom f1 by XBOOLE_1:17;
A14: now
      let n;
A15: n in NAT by ORDINAL1:def 12;
      (seq^\k).n in rng(seq^\k) by VALUED_0:28;
      then f.((seq^\k).n)<=f1.((seq^\k).n) by A4,A11;
      then (f/*(seq^\k)).n<=f1.((seq^\k).n)
by A6,A10,FUNCT_2:108,XBOOLE_1:1,A15;
      hence (f/*(seq^\k)).n<=(f1/*(seq^\k)).n
              by A12,A13,FUNCT_2:108,XBOOLE_1:1,A15;
    end;
A16: seq^\k is divergent_to-infty by A5,Th27;
    rng(seq^\k)c=dom f1 by A12,A13;
    then f1/*(seq^\k) is divergent_to-infty by A1,A16;
    then
A17: f/*(seq^\k) is divergent_to-infty by A14,Th43;
    f/*(seq^\k)=(f/*seq)^\k by A6,VALUED_0:27;
    hence f/*seq is divergent_to-infty by A17,Th7;
  end;
  hence thesis by A2;
end;
