let f be PartFunc of REAL ,REAL ; :: thesis: for r being Real st f is convergent_in-infty holds
( r (#) f is convergent_in-infty & lim_in-infty (r (#) f) = r * (lim_in-infty f) )
let r be Real; :: thesis: ( f is convergent_in-infty implies ( r (#) f is convergent_in-infty & lim_in-infty (r (#) f) = r * (lim_in-infty f) ) )
assume A1: f is convergent_in-infty ; :: thesis: ( r (#) f is convergent_in-infty & lim_in-infty (r (#) f) = r * (lim_in-infty f) )
A2: for r1 being Real ex g being Real st
( g < r1 & g in dom (r (#) f) )

proof

let r1 be Real; :: thesis: ex g being Real st
( g < r1 & g in dom (r (#) f) )
consider g being Real such that
A3: ( g < r1 & g in dom f ) by A1, Def9;
take g ; :: thesis: ( g < r1 & g in dom (r (#) f) )
thus ( g < r1 & g in dom (r (#) f) ) by A3, VALUED_1:def 5; :: thesis: verum

end;

A4: now

let seq be Real_Sequence; :: thesis: ( seq is divergent_to-infty & rng seq c= dom (r (#) f) implies ( (r (#) f) /* seq is convergent & lim ((r (#) f) /* seq) = r * (lim_in-infty f) ) )
A5: lim_in-infty f = lim_in-infty f ;
assume ( seq is divergent_to-infty & rng seq c= dom (r (#) f) ) ; :: thesis: ( (r (#) f) /* seq is convergent & lim ((r (#) f) /* seq) = r * (lim_in-infty f) )
then A6: ( seq is divergent_to-infty & rng seq c= dom f ) by VALUED_1:def 5;
then A7: ( f /* seq is convergent & lim (f /* seq) = lim_in-infty f ) by A1, A5, Def13;
then r (#) (f /* seq) is convergent by SEQ_2:21;
hence (r (#) f) /* seq is convergent by A6, RFUNCT_2:24; :: thesis: lim ((r (#) f) /* seq) = r * (lim_in-infty f)
thus lim ((r (#) f) /* seq) = lim (r (#) (f /* seq)) by A6, RFUNCT_2:24
.= r * (lim_in-infty f) by A7, SEQ_2:22 ; :: thesis: verum

end;

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