let x0 be Real; :: thesis: for f being PartFunc of REAL,REAL st f is_left_convergent_in x0 & lim_left (f,x0) <> 0 & ( for r being Real st r < x0 holds
ex g being Real st
( r < g & g < x0 & g in dom f & f . g <> 0 ) ) holds
( f ^ is_left_convergent_in x0 & lim_left ((f ^),x0) = (lim_left (f,x0)) " )
let f be PartFunc of REAL,REAL; :: thesis: ( f is_left_convergent_in x0 & lim_left (f,x0) <> 0 & ( for r being Real st r < x0 holds
ex g being Real st
( r < g & g < x0 & g in dom f & f . g <> 0 ) ) implies ( f ^ is_left_convergent_in x0 & lim_left ((f ^),x0) = (lim_left (f,x0)) " ) )
assume that
A1: f is_left_convergent_in x0 and
A2: lim_left (f,x0) <> 0 and
A3: for r being Real st r < x0 holds
ex g being Real st
( r < g & g < x0 & g in dom f & f . g <> 0 ) ; :: thesis: ( f ^ is_left_convergent_in x0 & lim_left ((f ^),x0) = (lim_left (f,x0)) " )
A4: (dom f) \ (f " {0}) = dom (f ^) by RFUNCT_1:def 2;

A5: now :: thesis: for seq being Real_Sequence st seq is convergent & lim seq = x0 & rng seq c= (dom (f ^)) /\ (left_open_halfline x0) holds
( (f ^) /* seq is convergent & lim ((f ^) /* seq) = (lim_left (f,x0)) " )

A6: dom (f ^) c= dom f by A4, XBOOLE_1:36;
let seq be Real_Sequence; :: thesis: ( seq is convergent & lim seq = x0 & rng seq c= (dom (f ^)) /\ (left_open_halfline x0) implies ( (f ^) /* seq is convergent & lim ((f ^) /* seq) = (lim_left (f,x0)) " ) )
assume that
A7: seq is convergent and
A8: lim seq = x0 and
A9: rng seq c= (dom (f ^)) /\ (left_open_halfline x0) ; :: thesis: ( (f ^) /* seq is convergent & lim ((f ^) /* seq) = (lim_left (f,x0)) " )
A10: (dom (f ^)) /\ (left_open_halfline x0) c= dom (f ^) by XBOOLE_1:17;
then A11: f /* seq is non-zero by A9, RFUNCT_2:11, XBOOLE_1:1;
rng seq c= dom (f ^) by A9, A10, XBOOLE_1:1;
then A12: rng seq c= dom f by A6, XBOOLE_1:1;
(dom (f ^)) /\ (left_open_halfline x0) c= left_open_halfline x0 by XBOOLE_1:17;
then rng seq c= left_open_halfline x0 by A9, XBOOLE_1:1;
then A13: rng seq c= (dom f) /\ (left_open_halfline x0) by A12, XBOOLE_1:19;
then A14: lim (f /* seq) = lim_left (f,x0) by A1, A7, A8, Def7;
A15: (f /* seq) " = (f ^) /* seq by A9, A10, RFUNCT_2:12, XBOOLE_1:1;
A16: f /* seq is convergent by A1, A7, A8, A13;
hence (f ^) /* seq is convergent by A2, A14, A11, A15, SEQ_2:21; :: thesis: lim ((f ^) /* seq) = (lim_left (f,x0)) "
thus lim ((f ^) /* seq) = (lim_left (f,x0)) " by A2, A16, A14, A11, A15, SEQ_2:22; :: thesis: verum

end;

now :: thesis: for r being Real st r < x0 holds
ex g being Real st
( r < g & g < x0 & g in dom (f ^) )

let r be Real; :: thesis: ( r < x0 implies ex g being Real st
( r < g & g < x0 & g in dom (f ^) ) )
assume r < x0 ; :: thesis: ex g being Real st
( r < g & g < x0 & g in dom (f ^) )
then consider g being Real such that
A17: r < g and
A18: g < x0 and
A19: g in dom f and
A20: f . g <> 0 by A3;
take g = g; :: thesis: ( r < g & g < x0 & g in dom (f ^) )
not f . g in {0} by A20, TARSKI:def 1;
then not g in f " {0} by FUNCT_1:def 7;
hence ( r < g & g < x0 & g in dom (f ^) ) by A4, A17, A18, A19, XBOOLE_0:def 5; :: thesis: verum

end;

hence f ^ is_left_convergent_in x0 by A5; :: thesis: lim_left ((f ^),x0) = (lim_left (f,x0)) "
hence lim_left ((f ^),x0) = (lim_left (f,x0)) " by A5, Def7; :: thesis: verum