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 A1: ( 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 ) ) ) ; :: thesis: ( f ^ is_left_convergent_in x0 & lim_left (f ^ ),x0 = (lim_left f,x0) " )
A2: (dom f) \ (f " {0 }) = dom (f ^ ) by RFUNCT_1:def 8;

A3: now

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
A4: ( r < g & g < x0 & g in dom f & f . g <> 0 ) by A1;
take g = g; :: thesis: ( r < g & g < x0 & g in dom (f ^ ) )
not f . g in {0 } by A4, TARSKI:def 1;
then not g in f " {0 } by FUNCT_1:def 13;
hence ( r < g & g < x0 & g in dom (f ^ ) ) by A2, A4, XBOOLE_0:def 5; :: thesis: verum

end;

A5: now

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 A6: ( seq is convergent & lim seq = x0 & rng seq c= (dom (f ^ )) /\ (left_open_halfline x0) ) ; :: thesis: ( (f ^ ) /* seq is convergent & lim ((f ^ ) /* seq) = (lim_left f,x0) " )
A7: ( (dom (f ^ )) /\ (left_open_halfline x0) c= dom (f ^ ) & (dom (f ^ )) /\ (left_open_halfline x0) c= left_open_halfline x0 ) by XBOOLE_1:17;
then A8: ( rng seq c= dom (f ^ ) & rng seq c= left_open_halfline x0 ) by A6, XBOOLE_1:1;
dom (f ^ ) c= dom f by A2, XBOOLE_1:36;
then rng seq c= dom f by A8, XBOOLE_1:1;
then rng seq c= (dom f) /\ (left_open_halfline x0) by A8, XBOOLE_1:19;
then A9: ( f /* seq is convergent & lim (f /* seq) = lim_left f,x0 ) by A1, A6, Def7;
A10: f /* seq is non-zero by A6, A7, RFUNCT_2:26, XBOOLE_1:1;
A11: (f /* seq) " = (f ^ ) /* seq by A6, A7, RFUNCT_2:27, XBOOLE_1:1;
hence (f ^ ) /* seq is convergent by A1, A9, A10, SEQ_2:35; :: thesis: lim ((f ^ ) /* seq) = (lim_left f,x0) "
thus lim ((f ^ ) /* seq) = (lim_left f,x0) " by A1, A9, A10, A11, SEQ_2:36; :: thesis: verum

end;

hence f ^ is_left_convergent_in x0 by A3, Def1; :: thesis: lim_left (f ^ ),x0 = (lim_left f,x0) "
hence lim_left (f ^ ),x0 = (lim_left f,x0) " by A5, Def7; :: thesis: verum