let x0 be Real; for f being PartFunc of REAL,REAL st f is_left_convergent_in x0 & f " {0} = {} & lim_left (f,x0) <> 0 holds
( f ^ is_left_convergent_in x0 & lim_left ((f ^),x0) = (lim_left (f,x0)) " )
let f be PartFunc of REAL,REAL; ( f is_left_convergent_in x0 & f " {0} = {} & lim_left (f,x0) <> 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:
f " {0} = {}
and
A3:
lim_left (f,x0) <> 0
; ( f ^ is_left_convergent_in x0 & lim_left ((f ^),x0) = (lim_left (f,x0)) " )
A4: dom f =
(dom f) \ (f " {0})
by A2
.=
dom (f ^)
by RFUNCT_1:def 2
;
A5:
now 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)) " )let seq be
Real_Sequence;
( 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 A6:
seq is
convergent
and A7:
lim seq = x0
and A8:
rng seq c= (dom (f ^)) /\ (left_open_halfline x0)
;
( (f ^) /* seq is convergent & lim ((f ^) /* seq) = (lim_left (f,x0)) " )A9:
lim (f /* seq) = lim_left (
f,
x0)
by A1, A4, A6, A7, A8, Def7;
A10:
(dom f) /\ (left_open_halfline x0) c= dom f
by XBOOLE_1:17;
then A11:
rng seq c= dom f
by A4, A8, XBOOLE_1:1;
A12:
f /* seq is
convergent
by A1, A4, A6, A7, A8;
A13:
(f /* seq) " = (f ^) /* seq
by A4, A8, A10, RFUNCT_2:12, XBOOLE_1:1;
hence
(f ^) /* seq is
convergent
by A3, A4, A12, A9, A11, RFUNCT_2:11, SEQ_2:21;
lim ((f ^) /* seq) = (lim_left (f,x0)) " thus
lim ((f ^) /* seq) = (lim_left (f,x0)) "
by A3, A4, A12, A9, A11, A13, RFUNCT_2:11, SEQ_2:22;
verum end;
for r being Real st r < x0 holds
ex g being Real st
( r < g & g < x0 & g in dom (f ^) )
by A1, A4;
hence
f ^ is_left_convergent_in x0
by A5; lim_left ((f ^),x0) = (lim_left (f,x0)) "
hence
lim_left ((f ^),x0) = (lim_left (f,x0)) "
by A5, Def7; verum