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 & f is_measurable_on B holds
( f | B is_integrable_on M & Integral_on M,B,(r (#) f) = (R_EAL 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 & f is_measurable_on B holds
( f | B is_integrable_on M & Integral_on M,B,(r (#) f) = (R_EAL 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 & f is_measurable_on B holds
( f | B is_integrable_on M & Integral_on M,B,(r (#) f) = (R_EAL 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 & f is_measurable_on B holds
( f | B is_integrable_on M & Integral_on M,B,(r (#) f) = (R_EAL r) * (Integral_on M,B,f) )
let r be Real; :: thesis: for B being Element of S st f is_integrable_on M & f is_measurable_on B holds
( f | B is_integrable_on M & Integral_on M,B,(r (#) f) = (R_EAL r) * (Integral_on M,B,f) )
let B be Element of S; :: thesis: ( f is_integrable_on M & f is_measurable_on B implies ( f | B is_integrable_on M & Integral_on M,B,(r (#) f) = (R_EAL r) * (Integral_on M,B,f) ) )
assume ( f is_integrable_on M & f is_measurable_on B ) ; :: thesis: ( f | B is_integrable_on M & Integral_on M,B,(r (#) f) = (R_EAL r) * (Integral_on M,B,f) )
then ( R_EAL f is_integrable_on M & R_EAL f is_measurable_on B ) by Def6, Def9;
then ( (R_EAL f) | B is_integrable_on M & Integral_on M,B,(r (#) (R_EAL f)) = (R_EAL r) * (Integral_on M,B,(R_EAL f)) ) by MESFUNC5:118;
then ( R_EAL (f | B) is_integrable_on M & Integral_on M,B,(R_EAL (r (#) f)) = (R_EAL r) * (Integral_on M,B,(R_EAL f)) ) by Th20;
hence ( f | B is_integrable_on M & Integral_on M,B,(r (#) f) = (R_EAL r) * (Integral_on M,B,f) ) by Def9; :: thesis: verum