set X = dom f;
A1: dom f = dom (r (#) f) by VFUNCT_1:def 4;
then A3: (r (#) f) | (dom f) = r (#) f by RELAT_1:98;

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 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) )
f | (dom f) is continuous ;
then A7: ( f /* s1 is convergent & f /. (lim s1) = lim (f /* s1) ) by A4, A5, A6, Th14;
then A8: r * (f /* s1) is convergent by NORMSP_1:37;
(r (#) f) /. (lim s1) = r * (lim (f /* s1)) by A1, A6, A7, VFUNCT_1:def 4
.= lim (r * (f /* s1)) by A7, NORMSP_1:45
.= lim ((r (#) f) /* s1) by A4, XTh20 ;
hence ( (r (#) f) /* s1 is convergent & (r (#) f) /. (lim s1) = lim ((r (#) f) /* s1) ) by A4, A8, XTh20; :: thesis: verum

end;

hence for b₁ being PartFunc of REAL, the carrier of S st b₁ = r (#) f holds
b₁ is continuous by A1, A3, Th14; :: thesis: verum