let X be set ; :: thesis: for r being Real
for S being RealNormSpace
for f being PartFunc of REAL, the carrier of S st X c= dom f & f | X is continuous holds
(r (#) f) | X is continuous
let r be Real; :: thesis: for S being RealNormSpace
for f being PartFunc of REAL, the carrier of S st X c= dom f & f | X is continuous holds
(r (#) f) | X is continuous
let S be RealNormSpace; :: thesis: for f being PartFunc of REAL, the carrier of S st X c= dom f & f | X is continuous holds
(r (#) f) | X is continuous
let f be PartFunc of REAL, the carrier of S; :: thesis: ( X c= dom f & f | X is continuous implies (r (#) f) | X is continuous )
assume that
A1: X c= dom f and
A2: f | X is continuous ; :: thesis: (r (#) f) | X is continuous
A3: X c= dom (r (#) f) by A1, VFUNCT_1:def 4;

now

let s1 be Real_Sequence; :: thesis: ( rng s1 c= X & s1 is convergent & lim s1 in X implies ( (r (#) f) /* s1 is convergent & (r (#) f) /. (lim s1) = lim ((r (#) f) /* s1) ) )
assume that
A4: ( rng s1 c= X & s1 is convergent ) and
A6: lim s1 in X ; :: thesis: ( (r (#) f) /* s1 is convergent & (r (#) f) /. (lim s1) = lim ((r (#) f) /* s1) )
A7: f /* s1 is convergent by A1, A2, A4, A6, Th14;
then A8: r * (f /* s1) is convergent by NORMSP_1:37;
f /. (lim s1) = lim (f /* s1) by A1, A2, A4, A6, Th14;
then (r (#) f) /. (lim s1) = r * (lim (f /* s1)) by A3, A6, VFUNCT_1:def 4
.= lim (r * (f /* s1)) by A7, NORMSP_1:45
.= lim ((r (#) f) /* s1) by A1, A4, XTh20, XBOOLE_1:1 ;
hence ( (r (#) f) /* s1 is convergent & (r (#) f) /. (lim s1) = lim ((r (#) f) /* s1) ) by A1, A4, A8, XTh20, XBOOLE_1:1; :: thesis: verum

end;

hence (r (#) f) | X is continuous by A3, Th14; :: thesis: verum