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 ) and
A6: f . g <> 0 by A2;
take g = g; :: thesis:
not f . g in {0 } by A6, 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 A7: f is divergent_in-infty_to+infty ; :: thesis:

A8: now

let seq be Real_Sequence; :: thesis:
assume that
A9: seq is divergent_to-infty and
A10: rng seq c= dom (f ^ ) ; :: thesis:
dom (f ^ ) c= dom f by A3, XBOOLE_1:36;
then rng seq c= dom f by A10, XBOOLE_1:1;
then f /* seq is divergent_to+infty by A7, A9, Def10;
then ( (f /* seq) " is convergent & lim ((f /* seq) " ) = 0 ) by Th61;
hence ( (f ^ ) /* seq is convergent & lim ((f ^ ) /* seq) = 0 ) by A10, 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 A8, Def13; :: thesis:

end;

suppose A11: f is divergent_in-infty_to-infty ; :: thesis:

A12: now

let seq be Real_Sequence; :: thesis:
assume that
A13: seq is divergent_to-infty and
A14: rng seq c= dom (f ^ ) ; :: thesis:
dom (f ^ ) c= dom f by A3, XBOOLE_1:36;
then rng seq c= dom f by A14, XBOOLE_1:1;
then f /* seq is divergent_to-infty by A11, A13, Def11;
then ( (f /* seq) " is convergent & lim ((f /* seq) " ) = 0 ) by Th61;
hence ( (f ^ ) /* seq is convergent & lim ((f ^ ) /* seq) = 0 ) by A14, 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 A12, Def13; :: thesis:

end;

end;

end;

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