let f be PartFunc of REAL ,REAL ; :: thesis: ( f is convergent_in-infty & lim_in-infty f <> 0 & ( for r being Real ex g being Real st
( g < r & g in dom f & f . g <> 0 ) ) implies ( f ^ is convergent_in-infty & lim_in-infty (f ^ ) = (lim_in-infty f) " ) )
assume A1: ( f is convergent_in-infty & lim_in-infty f <> 0 & ( for r being Real ex g being Real st
( g < r & g in dom f & f . g <> 0 ) ) ) ; :: thesis: ( f ^ is convergent_in-infty & lim_in-infty (f ^ ) = (lim_in-infty f) " )

A2: now

let r be Real; :: thesis: ex g being Real st
( g < r & g in dom (f ^ ) )
consider g being Real such that
A3: ( g < r & g in dom f & f . g <> 0 ) by A1;
take g = g; :: thesis: ( g < r & g in dom (f ^ ) )
not f . g in {0 } by A3, TARSKI:def 1;
then not g in f " {0 } by FUNCT_1:def 13;
then g in (dom f) \ (f " {0 }) by A3, XBOOLE_0:def 5;
hence ( g < r & g in dom (f ^ ) ) by A3, RFUNCT_1:def 8; :: thesis: verum

end;

A4: now

let seq be Real_Sequence; :: thesis: ( seq is divergent_to-infty & rng seq c= dom (f ^ ) implies ( (f ^ ) /* seq is convergent & lim ((f ^ ) /* seq) = (lim_in-infty f) " ) )
assume A5: ( seq is divergent_to-infty & rng seq c= dom (f ^ ) ) ; :: thesis: ( (f ^ ) /* seq is convergent & lim ((f ^ ) /* seq) = (lim_in-infty f) " )
A6: dom (f ^ ) = (dom f) \ (f " {0 }) by RFUNCT_1:def 8;
(dom f) \ (f " {0 }) c= dom f by XBOOLE_1:36;
then rng seq c= dom f by A5, A6, XBOOLE_1:1;
then A7: ( f /* seq is convergent & lim (f /* seq) = lim_in-infty f ) by A1, A5, Def13;
(f /* seq) " is convergent by A1, A7, A5, RFUNCT_2:26, SEQ_2:35;
hence (f ^ ) /* seq is convergent by A5, RFUNCT_2:27; :: thesis: lim ((f ^ ) /* seq) = (lim_in-infty f) "
thus lim ((f ^ ) /* seq) = lim ((f /* seq) " ) by A5, RFUNCT_2:27
.= (lim_in-infty f) " by A1, A7, A5, RFUNCT_2:26, SEQ_2:36 ; :: thesis: verum

end;

hence f ^ is convergent_in-infty by A2, Def9; :: thesis: lim_in-infty (f ^ ) = (lim_in-infty f) "
hence lim_in-infty (f ^ ) = (lim_in-infty f) " by A4, Def13; :: thesis: verum