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
for B being Element of S st f is_integrable_on M holds
( f | B is_integrable_on M & Integral_on (M,B,(r (#) f)) = r * (Integral_on (M,B,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
for B being Element of S st f is_integrable_on M holds
( f | B is_integrable_on M & Integral_on (M,B,(r (#) f)) = r * (Integral_on (M,B,f)) )
let M be sigma_Measure of S; :: thesis: for f being PartFunc of X,REAL
for r being Real
for B being Element of S st f is_integrable_on M holds
( f | B is_integrable_on M & Integral_on (M,B,(r (#) f)) = r * (Integral_on (M,B,f)) )
let f be PartFunc of X,REAL; :: thesis: for r being Real
for B being Element of S st f is_integrable_on M holds
( f | B is_integrable_on M & Integral_on (M,B,(r (#) f)) = r * (Integral_on (M,B,f)) )
let r be Real; :: thesis: for B being Element of S st f is_integrable_on M holds
( f | B is_integrable_on M & Integral_on (M,B,(r (#) f)) = r * (Integral_on (M,B,f)) )
let B be Element of S; :: thesis: ( f is_integrable_on M implies ( f | B is_integrable_on M & Integral_on (M,B,(r (#) f)) = r * (Integral_on (M,B,f)) ) )
assume f is_integrable_on M ; :: thesis: ( f | B is_integrable_on M & Integral_on (M,B,(r (#) f)) = r * (Integral_on (M,B,f)) )
then A1: R_EAL f is_integrable_on M ;
then Integral_on (M,B,(r (#) (R_EAL f))) = r * (Integral_on (M,B,(R_EAL f))) by MESFUNC5:112;
then A2: Integral_on (M,B,(R_EAL (r (#) f))) = r * (Integral_on (M,B,(R_EAL f))) by Th20;
R_EAL (f | B) is_integrable_on M by A1, MESFUNC5:112;
hence ( f | B is_integrable_on M & Integral_on (M,B,(r (#) f)) = r * (Integral_on (M,B,f)) ) by A2; :: thesis: verum