defpred S₁[ Nat, Real] means ( $1 < $2 & $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
A4: for n being Element of NAT ex r being Element of REAL st S₁[n,r]

proof

let n be Element of NAT ; :: thesis: ex r being Element of REAL st S₁[n,r]
consider r being Real such that
A5: S₁[n,r] by A1;
reconsider r = r as Real ;
S₁[n,r] by A5;
hence ex r being Element of REAL st S₁[n,r] ; :: thesis: verum

end;

consider s2 being Real_Sequence such that
A6: for n being Element of NAT holds S₁[n,s2 . n] from FUNCT_2:sch 3(A4);
A7: rng s2 c= dom f

proof

let x be Real; :: according to MEMBERED:def 9 :: thesis: ( not x in rng s2 or x in dom f )
assume x in rng s2 ; :: thesis: x in dom f
then ex n being Element of NAT st x = s2 . n by FUNCT_2:113;
hence x in dom f by A6; :: thesis: verum

end;

now :: thesis: for n being Nat holds s1 . n <= s2 . n

let n be Nat; :: thesis: s1 . n <= s2 . n
A8: n in NAT by ORDINAL1:def 12;
then n < s2 . n by A6;
hence s1 . n <= s2 . n by FUNCT_1:18, A8; :: thesis: verum

end;

then A9: s2 is divergent_to+infty by Lm5, Th20, Th42;
then lim (f /* s2) = g1 by A2, A7;
hence g1 = g2 by A3, A9, A7; :: thesis: verum