let x0 be Real; :: thesis: for f being PartFunc of REAL,REAL st ( f is_left_divergent_to+infty_in x0 or f is_left_divergent_to-infty_in x0 ) & ( 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) = 0 )
let f be PartFunc of REAL,REAL; :: thesis: ( ( f is_left_divergent_to+infty_in x0 or f is_left_divergent_to-infty_in x0 ) & ( 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) = 0 ) )
assume A1: ( f is_left_divergent_to+infty_in x0 or f is_left_divergent_to-infty_in x0 ) ; :: thesis: ( ex r being Real st
( r < x0 & ( for g being Real holds
( not r < g or not g < x0 or not g in dom f or not f . g <> 0 ) ) ) or ( f ^ is_left_convergent_in x0 & lim_left ((f ^),x0) = 0 ) )

A2: now

dom (f ^) = (dom f) \ (f " {0}) by RFUNCT_1:def 8;
then A3: dom (f ^) c= dom f by 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) = 0 ) )
assume that
A4: seq is convergent and
A5: lim seq = x0 and
A6: rng seq c= (dom (f ^)) /\ (left_open_halfline x0) ; :: thesis: ( (f ^) /* seq is convergent & lim ((f ^) /* seq) = 0 )
(dom (f ^)) /\ (left_open_halfline x0) c= left_open_halfline x0 by XBOOLE_1:17;
then A7: rng seq c= left_open_halfline x0 by A6, XBOOLE_1:1;
A8: (dom (f ^)) /\ (left_open_halfline x0) c= dom (f ^) by XBOOLE_1:17;
then rng seq c= dom (f ^) by A6, XBOOLE_1:1;
then rng seq c= dom f by A3, XBOOLE_1:1;
then A9: rng seq c= (dom f) /\ (left_open_halfline x0) by A7, XBOOLE_1:19;

now

per cases ( f is_left_divergent_to+infty_in x0 or f is_left_divergent_to-infty_in x0 ) by A1;

suppose f is_left_divergent_to+infty_in x0 ; :: thesis: ( (f ^) /* seq is convergent & lim ((f ^) /* seq) = 0 )

then A10: f /* seq is divergent_to+infty by A4, A5, A9, Def2;
then A11: lim ((f /* seq) ") = 0 by LIMFUNC1:61;
(f /* seq) " is convergent by A10, LIMFUNC1:61;
hence ( (f ^) /* seq is convergent & lim ((f ^) /* seq) = 0 ) by A6, A8, A11, RFUNCT_2:27, XBOOLE_1:1; :: thesis: verum

end;

suppose f is_left_divergent_to-infty_in x0 ; :: thesis: ( (f ^) /* seq is convergent & lim ((f ^) /* seq) = 0 )

then A12: f /* seq is divergent_to-infty by A4, A5, A9, Def3;
then A13: lim ((f /* seq) ") = 0 by LIMFUNC1:61;
(f /* seq) " is convergent by A12, LIMFUNC1:61;
hence ( (f ^) /* seq is convergent & lim ((f ^) /* seq) = 0 ) by A6, A8, A13, RFUNCT_2:27, XBOOLE_1:1; :: thesis: verum

end;

end;

end;

hence ( (f ^) /* seq is convergent & lim ((f ^) /* seq) = 0 ) ; :: thesis: verum

end;

assume A14: 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) = 0 )

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
A15: r < g and
A16: g < x0 and
A17: g in dom f and
A18: f . g <> 0 by A14;
take g = g; :: thesis: ( r < g & g < x0 & g in dom (f ^) )
thus ( r < g & g < x0 ) by A15, A16; :: thesis: g in dom (f ^)
not f . g in {0} by A18, TARSKI:def 1;
then not g in f " {0} by FUNCT_1:def 13;
then g in (dom f) \ (f " {0}) by A17, XBOOLE_0:def 5;
hence g in dom (f ^) by RFUNCT_1:def 8; :: thesis: verum

end;

hence f ^ is_left_convergent_in x0 by A2, Def1; :: thesis: lim_left ((f ^),x0) = 0
hence lim_left ((f ^),x0) = 0 by A2, Def7; :: thesis: verum