let f be PartFunc of REAL ,REAL ; :: thesis: ( f is convergent_in-infty implies ( abs f is convergent_in-infty & lim_in-infty (abs f) = abs (lim_in-infty f) ) )
assume A1: f is convergent_in-infty ; :: thesis: ( abs f is convergent_in-infty & lim_in-infty (abs f) = abs (lim_in-infty f) )

A2: now

let r be Real; :: thesis: ex g being Real st
( g < r & g in dom (abs f) )
consider g being Real such that
A3: ( g < r & g in dom f ) by A1, Def9;
take g = g; :: thesis: ( g < r & g in dom (abs f) )
thus ( g < r & g in dom (abs f) ) by A3, VALUED_1:def 11; :: thesis: verum

end;

A4: now

let seq be Real_Sequence; :: thesis: ( seq is divergent_to-infty & rng seq c= dom (abs f) implies ( (abs f) /* seq is convergent & lim ((abs f) /* seq) = abs (lim_in-infty f) ) )
A5: lim_in-infty f = lim_in-infty f ;
assume ( seq is divergent_to-infty & rng seq c= dom (abs f) ) ; :: thesis: ( (abs f) /* seq is convergent & lim ((abs f) /* seq) = abs (lim_in-infty f) )
then A6: ( seq is divergent_to-infty & rng seq c= dom f ) by VALUED_1:def 11;
then A7: ( f /* seq is convergent & lim (f /* seq) = lim_in-infty f ) by A1, A5, Def13;
then abs (f /* seq) is convergent by SEQ_4:26;
hence (abs f) /* seq is convergent by A6, RFUNCT_2:25; :: thesis: lim ((abs f) /* seq) = abs (lim_in-infty f)
thus lim ((abs f) /* seq) = lim (abs (f /* seq)) by A6, RFUNCT_2:25
.= abs (lim_in-infty f) by A7, SEQ_4:27 ; :: thesis: verum

end;

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