let r be Real; :: thesis: for CNS being ComplexNormSpace
for RNS being RealNormSpace
for X being set
for f being PartFunc of CNS,RNS st f is_continuous_on X holds
r (#) f is_continuous_on X
let CNS be ComplexNormSpace; :: thesis: for RNS being RealNormSpace
for X being set
for f being PartFunc of CNS,RNS st f is_continuous_on X holds
r (#) f is_continuous_on X
let RNS be RealNormSpace; :: thesis: for X being set
for f being PartFunc of CNS,RNS st f is_continuous_on X holds
r (#) f is_continuous_on X
let X be set ; :: thesis: for f being PartFunc of CNS,RNS st f is_continuous_on X holds
r (#) f is_continuous_on X
let f be PartFunc of CNS,RNS; :: thesis: ( f is_continuous_on X implies r (#) f is_continuous_on X )
assume A1: f is_continuous_on X ; :: thesis: r (#) f is_continuous_on X
then A2: X c= dom f by Th63;
then A3: X c= dom (r (#) f) by VFUNCT_1:def 4;

now

let s1 be sequence of CNS; :: 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 A4: ( rng s1 c= X & s1 is convergent & lim s1 in X ) ; :: 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, Th63;
then A6: r * (f /* s1) is convergent by NORMSP_1:37;
(r (#) f) /. (lim s1) = r * (lim (f /* s1)) by A3, A4, A5, VFUNCT_1:def 4
.= lim (r * (f /* s1)) by A5, NORMSP_1:45
.= lim ((r (#) f) /* s1) by A2, A4, Th48, XBOOLE_1:1 ;
hence ( (r (#) f) /* s1 is convergent & (r (#) f) /. (lim s1) = lim ((r (#) f) /* s1) ) by A2, A4, A6, Th48, XBOOLE_1:1; :: thesis: verum

end;

hence r (#) f is_continuous_on X by A3, Th63; :: thesis: verum