let x0 be Real; :: thesis: for f being PartFunc of REAL ,REAL st f is_left_convergent_in x0 holds
( abs f is_left_convergent_in x0 & lim_left (abs f),x0 = abs (lim_left f,x0) )
let f be PartFunc of REAL ,REAL ; :: thesis: ( f is_left_convergent_in x0 implies ( abs f is_left_convergent_in x0 & lim_left (abs f),x0 = abs (lim_left f,x0) ) )
assume A1: f is_left_convergent_in x0 ; :: thesis: ( abs f is_left_convergent_in x0 & lim_left (abs f),x0 = abs (lim_left f,x0) )

A2: now

let r be Real; :: thesis: ( r < x0 implies ex g being Real st
( r < g & g < x0 & g in dom (abs f) ) )
assume r < x0 ; :: thesis: ex g being Real st
( r < g & g < x0 & g in dom (abs f) )
then consider g being Real such that
A3: ( r < g & g < x0 & g in dom f ) by A1, Def1;
take g = g; :: thesis: ( r < g & g < x0 & g in dom (abs f) )
thus ( r < g & g < x0 & g in dom (abs f) ) by A3, VALUED_1:def 11; :: thesis: verum

end;

A4: now

A5: lim_left f,x0 = lim_left f,x0 ;
let seq be Real_Sequence; :: thesis: ( seq is convergent & lim seq = x0 & rng seq c= (dom (abs f)) /\ (left_open_halfline x0) implies ( (abs f) /* seq is convergent & lim ((abs f) /* seq) = abs (lim_left f,x0) ) )
assume A6: ( seq is convergent & lim seq = x0 & rng seq c= (dom (abs f)) /\ (left_open_halfline x0) ) ; :: thesis: ( (abs f) /* seq is convergent & lim ((abs f) /* seq) = abs (lim_left f,x0) )
then A7: rng seq c= (dom f) /\ (left_open_halfline x0) by VALUED_1:def 11;
then A8: ( f /* seq is convergent & lim (f /* seq) = lim_left f,x0 ) by A1, A5, A6, Def7;
(dom f) /\ (left_open_halfline x0) c= dom f by XBOOLE_1:17;
then rng seq c= dom f by A7, XBOOLE_1:1;
then A9: abs (f /* seq) = (abs f) /* seq by RFUNCT_2:25;
hence (abs f) /* seq is convergent by A8, SEQ_4:26; :: thesis: lim ((abs f) /* seq) = abs (lim_left f,x0)
thus lim ((abs f) /* seq) = abs (lim_left f,x0) by A8, A9, SEQ_4:27; :: thesis: verum

end;

hence abs f is_left_convergent_in x0 by A2, Def1; :: thesis: lim_left (abs f),x0 = abs (lim_left f,x0)
hence lim_left (abs f),x0 = abs (lim_left f,x0) by A4, Def7; :: thesis: verum