let f be PartFunc of REAL,REAL; :: thesis:
thus ( f is divergent_in-infty_to-infty implies ( ( for r being Real ex g being Real st
( g < r & g in dom f ) ) & ( for g being Real ex r being Real st
for r1 being Real st r1 < r & r1 in dom f holds
f . r1 < g ) ) ) :: thesis:

deffunc H₁( Nat) -> object = - $1;
assume A1: f is divergent_in-infty_to-infty ; :: thesis:
assume ( ex r being Real st
for g being Real holds
( not g < r or not g in dom f ) or ex g being Real st
for r being Real ex r1 being Real st
( r1 < r & r1 in dom f & f . r1 >= g ) ) ; :: thesis:
then consider g being Real such that
A2: for r being Real ex r1 being Real st
( r1 < r & r1 in dom f & f . r1 >= g ) by A1;
defpred S₁[ Nat, Real] means ( $2 < - $1 & $2 in dom f & g <= f . $2 );
A3: for n being Element of NAT ex r being Element of REAL st S₁[n,r]

let n be Element of NAT ; :: thesis:
consider r being Real such that
A4: S₁[n,r] by A2;
reconsider r = r as Real ;
S₁[n,r] by A4;
hence ex r being Element of REAL st S₁[n,r] ; :: thesis:

end;

consider s being Real_Sequence such that
A5: for n being Element of NAT holds S₁[n,s . n] from FUNCT_2:sch 3(A3);

end;

then A6: rng s c= dom f ;
consider s1 being Real_Sequence such that
A7: for n being Nat holds s1 . n = H₁(n) from SEQ_1:sch 1();

end;

end;

assume that
A11: for r being Real ex g being Real st
( g < r & g in dom f ) and
A12: for g being Real ex r being Real st
for r1 being Real st r1 < r & r1 in dom f holds
f . r1 < g ; :: thesis:

let g be Real; :: thesis:
consider r being Real such that
A15: for r1 being Real st r1 < r & r1 in dom f holds
f . r1 < g by A12;
consider n being Nat such that
A16: for m being Nat st n <= m holds
s . m < r by A13;
take n = n; :: thesis:
let m be Nat; :: thesis:
A17: s . m in rng s by VALUED_0:28;
A18: m in NAT by ORDINAL1:def 12;
assume n <= m ; :: thesis:
then f . (s . m) < g by A14, A15, A16, A17;
hence (f /* s) . m < g by A14, FUNCT_2:108, A18; :: thesis:

end;

end;