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 Th70:
  f1 is divergent_in+infty_to+infty &
    (for r ex g st r<g & g in dom f) &
    (ex r st dom f /\ right_open_halfline(r)
        c= dom f1 /\ right_open_halfline(r) &
   for g st g in dom f /\ right_open_halfline(r)
  holds f1.g<=f.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 r<g & g in dom f;
  given r1 such that
A3: dom f/\right_open_halfline(r1)c= dom f1/\right_open_halfline(r1) and
A4: for g st g in dom f/\right_open_halfline(r1) holds f1.g<=f.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 r1<seq.n 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;
      r1<seq.(n+k) by A7,NAT_1:12;
      then r1<(seq^\k).n by NAT_1:def 3;
      then x in {g2: r1<g2} by A8;
      hence x in right_open_halfline(r1) by XXREAL_1:230;
    end;
    then
A9: rng(seq^\k)c=right_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/\right_open_halfline(r1) by A9,XBOOLE_1:19;
    then
A12: rng(seq^\k)c=dom f1/\right_open_halfline(r1) by A3;
A13: dom f1/\right_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 f1.((seq^\k).n)<=f.((seq^\k).n) by A4,A11;
      then (f1/*(seq^\k)).n<=f.((seq^\k).n)
         by A12,A13,FUNCT_2:108,XBOOLE_1:1,A15;
      hence (f1/*(seq^\k)).n<=(f/*(seq^\k)).n
      by A6,A10,FUNCT_2:108,XBOOLE_1:1,A15;
    end;
A16: seq^\k is divergent_to+infty by A5,Th26;
    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,Th42;
    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;
