deffunc H₁( Element of NAT ) -> Element of REAL = - $1;
defpred S₁[ Element of NAT , real number ] means ( $2 < - $1 & $2 in dom f );
let g1, g2 be Real; :: thesis: ( ( for seq being Real_Sequence st seq is divergent_to-infty & rng seq c= dom f holds
( f /* seq is convergent & lim (f /* seq) = g1 ) ) & ( for seq being Real_Sequence st seq is divergent_to-infty & rng seq c= dom f holds
( f /* seq is convergent & lim (f /* seq) = g2 ) ) implies g1 = g2 )
assume that
A2: for seq being Real_Sequence st seq is divergent_to-infty & rng seq c= dom f holds
( f /* seq is convergent & lim (f /* seq) = g1 ) and
A3: for seq being Real_Sequence st seq is divergent_to-infty & rng seq c= dom f holds
( f /* seq is convergent & lim (f /* seq) = g2 ) ; :: thesis: g1 = g2
consider s2 being Real_Sequence such that
A4: for n being Element of NAT holds s2 . n = H₁(n) from SEQ_1:sch 1();
A5: for n being Element of NAT ex r being Real st S₁[n,r] by A1, Def9;
consider s1 being Real_Sequence such that
A6: for n being Element of NAT holds S₁[n,s1 . n] from FUNCT_2:sch 3(A5);
A7: rng s1 c= dom f

proof

let x be real number ; :: according to MEMBERED:def 9 :: thesis: ( not x in rng s1 or x in dom f )
assume x in rng s1 ; :: thesis: x in dom f
then ex n being Element of NAT st x = s1 . n by FUNCT_2:113;
hence x in dom f by A6; :: thesis: verum

end;

now

let n be Element of NAT ; :: thesis: s1 . n <= s2 . n
s1 . n < - n by A6;
hence s1 . n <= s2 . n by A4; :: thesis: verum

end;

then A8: s1 is divergent_to-infty by A4, Th48, Th70;
then lim (f /* s1) = g1 by A2, A7;
hence g1 = g2 by A3, A8, A7; :: thesis: verum