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
( r < g & g in dom f & f . g <> 0 ) ; :: thesis:
A3: dom (f ^) = (dom f) \ (f " {0}) by RFUNCT_1:def 2;

A4: now :: thesis:

let r be Real; :: thesis:
consider g being Real such that
A5: ( r < g & 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 7;
hence ( r < g & g in dom (f ^) ) by A3, A5, XBOOLE_0:def 5; :: thesis:

end;

per cases by A1;

suppose A7: f is divergent_in+infty_to+infty ; :: thesis:

A8: now :: thesis:

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;
then f /* seq is divergent_to+infty by A7, A9;
then ( (f /* seq) " is convergent & lim ((f /* seq) ") = 0 ) by Th34;
hence ( (f ^) /* seq is convergent & lim ((f ^) /* seq) = 0 ) by A10, RFUNCT_2:12; :: thesis:

end;

hence f ^ is convergent_in+infty by A4; :: thesis:
hence ( f ^ is convergent_in+infty & lim_in+infty (f ^) = 0 ) by A8, Def12; :: thesis:

end;

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

A12: now :: thesis:

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;
then f /* seq is divergent_to-infty by A11, A13;
then ( (f /* seq) " is convergent & lim ((f /* seq) ") = 0 ) by Th34;
hence ( (f ^) /* seq is convergent & lim ((f ^) /* seq) = 0 ) by A14, RFUNCT_2:12; :: thesis:

end;

hence f ^ is convergent_in+infty by A4; :: thesis:
hence ( f ^ is convergent_in+infty & lim_in+infty (f ^) = 0 ) by A12, Def12; :: thesis:

end;

end;

end;

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