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_EAL 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_EAL 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_EAL 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_EAL 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_EAL r) * (Integral M,f) ) )
assume f is_integrable_on M ; :: thesis: ( r (#) f is_integrable_on M & Integral M,(r (#) f) = (R_EAL r) * (Integral M,f) )
then A1: R_EAL f is_integrable_on M by Def9;
then r (#) (R_EAL f) is_integrable_on M by MESFUNC5:116;
then A2: R_EAL (r (#) f) is_integrable_on M by Th20;
Integral M,(r (#) (R_EAL f)) = (R_EAL r) * (Integral M,(R_EAL f)) by A1, MESFUNC5:116;
hence ( r (#) f is_integrable_on M & Integral M,(r (#) f) = (R_EAL r) * (Integral M,f) ) by A2, Def9, Th20; :: thesis: verum