let f be PartFunc of REAL , REAL ; :: thesis:
assume that
A1: ( f is divergent_in-infty_to+infty or f is divergent_in-infty_to-infty ) and
A2: for r being Real ex g being Real st
( g < r & g in dom f & f . g <> 0 ) ; :: thesis:
A3: dom (f ^ ) = (dom f) \ (f " {0 }) by RFUNCT_1:def 8;

A4: now

let r be Real; :: thesis:
consider g being Real such that
A5: ( g < r & g in dom f & f . g <> 0 ) by A2;
take g = g; :: thesis:
not f . g in {0 } by A5, TARSKI:def 1;
then not g in f " {0 } by FUNCT_1:def 13;
hence ( g < r & g in dom (f ^ ) ) by A3, A5, XBOOLE_0:def 5; :: thesis:

end;

now

per cases by A1;

suppose A6: f is divergent_in-infty_to+infty ; :: thesis:

A7: now

let seq be Real_Sequence; :: thesis:
assume A8: ( seq is divergent_to-infty & rng seq c= dom (f ^ ) ) ; :: thesis:
dom (f ^ ) c= dom f by A3, XBOOLE_1:36;
then rng seq c= dom f by A8, XBOOLE_1:1;
then f /* seq is divergent_to+infty by A6, A8, Def10;
then ( (f /* seq) " is convergent & lim ((f /* seq) " ) = 0 ) by Th61;
hence ( (f ^ ) /* seq is convergent & lim ((f ^ ) /* seq) = 0 ) by A8, RFUNCT_2:27; :: thesis:

end;

hence f ^ is convergent_in-infty by A4, Def9; :: thesis:
hence ( f ^ is convergent_in-infty & lim_in-infty (f ^ ) = 0 ) by A7, Def13; :: thesis:

end;

suppose A9: f is divergent_in-infty_to-infty ; :: thesis:

A10: now

let seq be Real_Sequence; :: thesis:
assume A11: ( seq is divergent_to-infty & rng seq c= dom (f ^ ) ) ; :: thesis:
dom (f ^ ) c= dom f by A3, XBOOLE_1:36;
then rng seq c= dom f by A11, XBOOLE_1:1;
then f /* seq is divergent_to-infty by A9, A11, Def11;
then ( (f /* seq) " is convergent & lim ((f /* seq) " ) = 0 ) by Th61;
hence ( (f ^ ) /* seq is convergent & lim ((f ^ ) /* seq) = 0 ) by A11, RFUNCT_2:27; :: thesis:

end;

hence f ^ is convergent_in-infty by A4, Def9; :: thesis:
hence ( f ^ is convergent_in-infty & lim_in-infty (f ^ ) = 0 ) by A10, Def13; :: thesis:

end;

hence ( f ^ is convergent_in-infty & lim_in-infty (f ^ ) = 0 ) ; :: thesis: