reserve r,r1,r2,g,g1,g2,x0,t for Real;
reserve n,k for Nat;
reserve seq for Real_Sequence;
reserve f,f1,f2 for PartFunc of REAL,REAL;

theorem
  f is_right_divergent_to-infty_in x0 iff (for r st x0<r ex g st g<r &
x0<g & g in dom f) & for g1 ex r st x0<r & for r1 st r1<r & x0<r1 & r1 in dom f
  holds f.r1<g1
proof
  thus f is_right_divergent_to-infty_in x0 implies (for r st x0<r ex g st g<r
& x0<g & g in dom f) & for g1 ex r st x0<r & for r1 st r1<r & x0<r1 & r1 in dom
  f holds f.r1<g1
  proof
    assume that
A1: f is_right_divergent_to-infty_in x0 and
A2: (not for r st x0<r ex g st g<r & x0<g & g in dom f) or ex g1 st
    for r st x0<r ex r1 st r1<r & x0<r1 & r1 in dom f & g1<=f.r1;
    consider g1 such that
A3: for r st x0<r ex r1 st r1<r & x0<r1 & r1 in dom f & g1<=f.r1 by A1,A2;
    defpred X[Nat,Real] means x0<$2 & $2<x0+1/($1+1) & $2 in
    dom f & g1<=f.($2);
A4: now
      let n be Element of NAT;
      x0<x0+1/(n+1) by Lm3;
      then consider r1 such that
A5:   r1<x0+(1/(n+1)) and
A6:   x0<r1 and
A7:   r1 in dom f and
A8:   g1<=f.r1 by A3;
       reconsider r1 as Element of REAL by XREAL_0:def 1;
      take r1;
      thus X[n,r1] by A5,A6,A7,A8;
    end;
    consider s be Real_Sequence such that
A9: for n being Element of NAT holds X[n,s.n] from FUNCT_2:sch 3(A4);
A10: for n being Nat holds X[n,s.n]
     proof let n;
      n in NAT by ORDINAL1:def 12;
      hence thesis by A9;
     end;
A11: rng s c=dom f/\right_open_halfline(x0) by A10,Th6;
A12: lim s=x0 by A10,Th6;
    s is convergent by A10,Th6;
    then f/*s is divergent_to-infty by A1,A12,A11;
    then consider n such that
A13: for k st n<=k holds (f/*s).k<g1;
A14: (f/*s).n<g1 by A13;
A15: n in NAT by ORDINAL1:def 12;
    rng s c=dom f by A10,Th6;
    then f.(s.n)<g1 by A14,FUNCT_2:108,A15;
    hence contradiction by A10;
  end;
  assume that
A16: for r st x0<r ex g st g<r & x0<g & g in dom f and
A17: for g1 ex r st x0<r & for r1 st r1<r & x0<r1 & r1 in dom f holds f. r1<g1;
  for s be Real_Sequence holds s is convergent & lim s=x0 & rng s c=dom f
  /\right_open_halfline(x0) implies f/*s is divergent_to-infty by A17,Lm5;
  hence thesis by A16;
end;
