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

A2: now :: thesis: for seq being Real_Sequence st seq is convergent & lim seq = x0 & rng seq c= (dom (r (#) f)) /\ (left_open_halfline x0) holds
( (r (#) f) /* seq is convergent & lim ((r (#) f) /* seq) = r * (lim_left (f,x0)) )

let seq be Real_Sequence; :: thesis: ( seq is convergent & lim seq = x0 & rng seq c= (dom (r (#) f)) /\ (left_open_halfline x0) implies ( (r (#) f) /* seq is convergent & lim ((r (#) f) /* seq) = r * (lim_left (f,x0)) ) )
assume that
A3: seq is convergent and
A4: lim seq = x0 and
A5: rng seq c= (dom (r (#) f)) /\ (left_open_halfline x0) ; :: thesis: ( (r (#) f) /* seq is convergent & lim ((r (#) f) /* seq) = r * (lim_left (f,x0)) )
A6: rng seq c= (dom f) /\ (left_open_halfline x0) by A5, VALUED_1:def 5;
A7: (dom f) /\ (left_open_halfline x0) c= dom f by XBOOLE_1:17;
then A8: r (#) (f /* seq) = (r (#) f) /* seq by A6, RFUNCT_2:9, XBOOLE_1:1;
A9: f /* seq is convergent by A1, A3, A4, A6;
then r (#) (f /* seq) is convergent ;
hence (r (#) f) /* seq is convergent by A6, A7, RFUNCT_2:9, XBOOLE_1:1; :: thesis: lim ((r (#) f) /* seq) = r * (lim_left (f,x0))
lim (f /* seq) = lim_left (f,x0) by A1, A3, A4, A6, Def7;
hence lim ((r (#) f) /* seq) = r * (lim_left (f,x0)) by A9, A8, SEQ_2:8; :: thesis: verum

end;

now :: thesis: for r1 being Real st r1 < x0 holds
ex g being Real st
( r1 < g & g < x0 & g in dom (r (#) f) )

let r1 be Real; :: thesis: ( r1 < x0 implies ex g being Real st
( r1 < g & g < x0 & g in dom (r (#) f) ) )
assume r1 < x0 ; :: thesis: ex g being Real st
( r1 < g & g < x0 & g in dom (r (#) f) )
then consider g being Real such that
A10: r1 < g and
A11: g < x0 and
A12: g in dom f by A1;
take g = g; :: thesis: ( r1 < g & g < x0 & g in dom (r (#) f) )
thus ( r1 < g & g < x0 & g in dom (r (#) f) ) by A10, A11, A12, VALUED_1:def 5; :: thesis: verum

end;

hence r (#) f is_left_convergent_in x0 by A2; :: thesis: lim_left ((r (#) f),x0) = r * (lim_left (f,x0))
hence lim_left ((r (#) f),x0) = r * (lim_left (f,x0)) by A2, Def7; :: thesis: verum