let X be non empty set ; :: thesis: for S being SigmaField of X
for M being sigma_Measure of S
for f being PartFunc of X,REAL
for r being Real st f is_integrable_on M holds
( r (#) f is_integrable_on M & Integral (M,(r (#) f)) = r * (Integral (M,f)) )
let S be SigmaField of X; :: thesis: for M being sigma_Measure of S
for f being PartFunc of X,REAL
for r being Real st f is_integrable_on M holds
( r (#) f is_integrable_on M & Integral (M,(r (#) f)) = r * (Integral (M,f)) )
let M be sigma_Measure of S; :: thesis: for f being PartFunc of X,REAL
for r being Real st f is_integrable_on M holds
( r (#) f is_integrable_on M & Integral (M,(r (#) f)) = r * (Integral (M,f)) )
let f be PartFunc of X,REAL; :: thesis: for r being Real st f is_integrable_on M holds
( r (#) f is_integrable_on M & Integral (M,(r (#) f)) = r * (Integral (M,f)) )
let r be Real; :: thesis: ( f is_integrable_on M implies ( r (#) f is_integrable_on M & Integral (M,(r (#) f)) = r * (Integral (M,f)) ) )
assume f is_integrable_on M ; :: thesis: ( r (#) f is_integrable_on M & Integral (M,(r (#) f)) = r * (Integral (M,f)) )
then A1: R_EAL f is_integrable_on M ;
then r (#) (R_EAL f) is_integrable_on M by MESFUNC5:110;
then A2: R_EAL (r (#) f) is_integrable_on M by Th20;
Integral (M,(r (#) (R_EAL f))) = r * (Integral (M,(R_EAL f))) by A1, MESFUNC5:110;
hence ( r (#) f is_integrable_on M & Integral (M,(r (#) f)) = r * (Integral (M,f)) ) by A2, Th20; :: thesis: verum