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 A3: |.(f . x).| < +infty by A1, Def1;
then - +infty < f . x by EXTREAL2:10;
then A4: -infty < f . x by XXREAL_3:def 3;
f . x < +infty by A3, EXTREAL2:10;
then reconsider y = f . x as Real by A4, XXREAL_0:14;
A5: -infty < R_EAL (r * y) by XXREAL_0:12;
A6: R_EAL (r * y) < +infty by XXREAL_0:9;
A7: -infty < (R_EAL r) * (R_EAL y) by A5, EXTREAL1:5;
A8: (R_EAL r) * (R_EAL y) = (r (#) f) . x by A2, MESFUNC1:def 6;
then A9: - +infty < (r (#) f) . x by A7, XXREAL_3:def 3;
(r (#) f) . x < +infty by A6, A8, EXTREAL1:5;
hence |.((r (#) f) . x).| < +infty by A9, EXTREAL2:11; :: thesis: verum

end;

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