let r be Real; :: thesis: for X being set
for f being PartFunc of REAL,REAL st X c= dom f & f | X is continuous holds
(r (#) f) | X is continuous
let X be set ; :: thesis: for f being PartFunc of REAL,REAL st X c= dom f & f | X is continuous holds
(r (#) f) | X is continuous
let f be PartFunc of REAL,REAL; :: thesis: ( X c= dom f & f | X is continuous implies (r (#) f) | X is continuous )
assume A1: X c= dom f ; :: thesis: ( not f | X is continuous or (r (#) f) | X is continuous )
assume A2: f | X is continuous ; :: thesis: (r (#) f) | X is continuous
A3: X c= dom (r (#) f) by A1, VALUED_1:def 5;

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

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 and
A5: 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, A5, A6, Th13;
then A8: r (#) (f /* s1) is convergent ;
f . (lim s1) = lim (f /* s1) by A1, A2, A4, A5, A6, Th13;
then (r (#) f) . (lim s1) = r * (lim (f /* s1)) by A3, A6, VALUED_1:def 5
.= lim (r (#) (f /* s1)) by A7, SEQ_2:8
.= lim ((r (#) f) /* s1) by A1, A4, RFUNCT_2:9, XBOOLE_1:1 ;
hence ( (r (#) f) /* s1 is convergent & (r (#) f) . (lim s1) = lim ((r (#) f) /* s1) ) by A1, A4, A8, RFUNCT_2:9, XBOOLE_1:1; :: thesis: verum

end;

hence (r (#) f) | X is continuous by A3, Th13; :: thesis: verum