set X = dom f;
A1: dom f c= dom (r (#) f) by VALUED_1:def 5;
A2: f | (dom f) is continuous ;

A3: now :: thesis: for s1 being Real_Sequence st rng s1 c= dom f & s1 is convergent & lim s1 in dom f holds
( (r (#) f) /* s1 is convergent & (r (#) f) . (lim s1) = lim ((r (#) f) /* s1) )

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 that
A4: rng s1 c= dom f and
A5: s1 is convergent and
A6: lim s1 in dom f ; :: thesis: ( (r (#) f) /* s1 is convergent & (r (#) f) . (lim s1) = lim ((r (#) f) /* s1) )
A7: f /* s1 is convergent by A2, A4, A5, A6, Th13;
then A8: r (#) (f /* s1) is convergent ;
f . (lim s1) = lim (f /* s1) by A2, A4, A5, A6, Th13;
then (r (#) f) . (lim s1) = r * (lim (f /* s1)) by A1, A6, VALUED_1:def 5
.= lim (r (#) (f /* s1)) by A7, SEQ_2:8
.= lim ((r (#) f) /* s1) by A4, RFUNCT_2:9 ;
hence ( (r (#) f) /* s1 is convergent & (r (#) f) . (lim s1) = lim ((r (#) f) /* s1) ) by A4, A8, RFUNCT_2:9; :: thesis: verum

end;

dom (r (#) f) = dom f by VALUED_1:def 5;
then (r (#) f) | (dom f) = r (#) f ;
hence for b₁ being PartFunc of REAL,REAL st b₁ = r (#) f holds
b₁ is continuous by A1, A3, Th13; :: thesis: verum