let C be non empty set ; :: thesis: for f being PartFunc of C,ExtREAL
for r being Real st f is real-valued holds
r (#) f is real-valued
let f be PartFunc of C,ExtREAL ; :: thesis: for r being Real st f is real-valued holds
r (#) f is real-valued
let r be Real; :: thesis: ( f is real-valued implies r (#) f is real-valued )
assume A1: f is real-valued ; :: thesis: r (#) f is real-valued
for x being Element of C st x in dom (r (#) f) holds
|.((r (#) f) . x).| < +infty

proof

let x be Element of C; :: thesis: ( x in dom (r (#) f) implies |.((r (#) f) . x).| < +infty )
assume A2: x in dom (r (#) f) ; :: thesis: |.((r (#) f) . x).| < +infty
then x in dom f by MESFUNC1:def 6;
then |.(f . x).| < +infty by A1, Def1;
then ( - +infty < f . x & f . x < +infty ) by EXTREAL2:58;
then ( -infty < f . x & f . x < +infty ) by XXREAL_3:def 3;
then reconsider y = f . x as Real by XXREAL_0:14;
( -infty < R_EAL (r * y) & R_EAL (r * y) < +infty ) by XXREAL_0:9, XXREAL_0:12;
then A3: ( -infty < (R_EAL r) * (R_EAL y) & (R_EAL r) * (R_EAL y) < +infty ) by EXTREAL1:38;
(R_EAL r) * (R_EAL y) = (r (#) f) . x by A2, MESFUNC1:def 6;
then ( - +infty < (r (#) f) . x & (r (#) f) . x < +infty ) by A3, XXREAL_3:def 3;
hence |.((r (#) f) . x).| < +infty by EXTREAL2:59; :: thesis: verum

end;

hence r (#) f is real-valued by Def1; :: thesis: verum