set X = dom f;
dom (r (#) f) = dom f by VALUED_1:def 5;
then HA: (r (#) f) | (dom f) = r (#) f by RELAT_1:98;
A3: dom f c= dom (r (#) f) by VALUED_1:def 5;
A1: f | (dom f) is continuous ;

now

let s1 be Real_Sequence; :: thesis: ( rng s1 c= dom f & s1 is convergent & lim s1 in dom f implies ( (r (#) f) /* s1 is convergent & (r (#) f) . (lim s1) = lim ((r (#) f) /* s1) ) )
assume A4: ( rng s1 c= dom f & s1 is convergent & lim s1 in dom f ) ; :: thesis: ( (r (#) f) /* s1 is convergent & (r (#) f) . (lim s1) = lim ((r (#) f) /* s1) )
then A5: ( f /* s1 is convergent & f . (lim s1) = lim (f /* s1) ) by A1, Th14;
then A6: r (#) (f /* s1) is convergent by SEQ_2:21;
(r (#) f) . (lim s1) = r * (lim (f /* s1)) by A3, A4, A5, VALUED_1:def 5
.= lim (r (#) (f /* s1)) by A5, SEQ_2:22
.= lim ((r (#) f) /* s1) by A4, RFUNCT_2:24 ;
hence ( (r (#) f) /* s1 is convergent & (r (#) f) . (lim s1) = lim ((r (#) f) /* s1) ) by A4, A6, RFUNCT_2:24; :: thesis: verum

end;

hence r (#) f is continuous PartFunc of REAL , REAL by HA, A3, Th14; :: thesis: verum