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
  f is divergent_in+infty_to+infty iff (for r ex g st r<g & g in dom f)
  & for g ex r st for r1 st r<r1 & r1 in dom f holds g<f.r1
proof
  thus f is divergent_in+infty_to+infty implies (for r ex g st r<g & g in dom
  f) & for g ex r st for r1 st r<r1 & r1 in dom f holds g<f.r1
  proof
    assume
A1: f is divergent_in+infty_to+infty;
    assume not (for r ex g st r<g & g in dom f) or ex g st for r ex r1 st r<
    r1 & r1 in dom f & g>=f.r1;
    then consider g such that
A2: for r ex r1 st r<r1 & r1 in dom f & g>=f.r1 by A1;
    defpred X[Nat,Real] means $1<$2 & $2 in dom f & g >= f.
    $2;
A3: for n being Element of NAT ex r being Element of REAL st X[n,r]
     proof let n be Element of NAT;
       consider r such that
A4:      X[n,r] by A2;
       reconsider r as Real;
       X[n,r] by A4;
      hence thesis;
     end;
    consider s be Real_Sequence such that
A5: for n being Element of NAT holds X[n,s.n] from FUNCT_2:sch 3(A3);
    now
      let x be object;
      assume x in rng s;
      then ex n being Element of NAT st s.n=x by FUNCT_2:113;
      hence x in dom f by A5;
    end;
    then
A6: rng s c=dom f;
    now
      let n;
A7: n in NAT by ORDINAL1:def 12;
      n<s.n by A5,A7;
      hence s1.n<=s.n by FUNCT_1:18,A7;
    end;
    then s is divergent_to+infty by Lm5,Th20,Th42;
    then f/*s is divergent_to+infty by A1,A6;
    then consider n such that
A8: for m st n<=m holds g<(f/*s).m;
A9: n in NAT by ORDINAL1:def 12;
    g<(f/*s).n by A8;
    then g<f.(s.n) by A6,FUNCT_2:108,A9;
    hence contradiction by A5,A9;
  end;
  assume that
A10: for r ex g st r<g & g in dom f and
A11: for g ex r st for r1 st r<r1 & r1 in dom f holds g<f.r1;
  now
    let s be Real_Sequence such that
A12: s is divergent_to+infty and
A13: rng s c=dom f;
    now
      let g;
      consider r such that
A14:  for r1 st r<r1 & r1 in dom f holds g<f.r1 by A11;
      consider n such that
A15:  for m st n<=m holds r<s.m by A12;
      take n;
      let m;
A16:  s.m in rng s by VALUED_0:28;
A17: m in NAT by ORDINAL1:def 12;
      assume n<=m;
      then g<f.(s.m) by A13,A14,A15,A16;
      hence g<(f/*s).m by A13,FUNCT_2:108,A17;
    end;
    hence f/*s is divergent_to+infty;
  end;
  hence thesis by A10;
end;
