let X be non empty set ; :: thesis: for S being SigmaField of X
for M being sigma_Measure of S
for f, g being PartFunc of X,REAL st f is_integrable_on M & g is_integrable_on M holds
ex E being Element of S st
( E = (dom f) /\ (dom g) & Integral M,(f + g) = (Integral M,(f | E)) + (Integral M,(g | E)) )
let S be SigmaField of X; :: thesis: for M being sigma_Measure of S
for f, g being PartFunc of X,REAL st f is_integrable_on M & g is_integrable_on M holds
ex E being Element of S st
( E = (dom f) /\ (dom g) & Integral M,(f + g) = (Integral M,(f | E)) + (Integral M,(g | E)) )
let M be sigma_Measure of S; :: thesis: for f, g being PartFunc of X,REAL st f is_integrable_on M & g is_integrable_on M holds
ex E being Element of S st
( E = (dom f) /\ (dom g) & Integral M,(f + g) = (Integral M,(f | E)) + (Integral M,(g | E)) )
let f, g be PartFunc of X,REAL ; :: thesis: ( f is_integrable_on M & g is_integrable_on M implies ex E being Element of S st
( E = (dom f) /\ (dom g) & Integral M,(f + g) = (Integral M,(f | E)) + (Integral M,(g | E)) ) )
assume ( f is_integrable_on M & g is_integrable_on M ) ; :: thesis: ex E being Element of S st
( E = (dom f) /\ (dom g) & Integral M,(f + g) = (Integral M,(f | E)) + (Integral M,(g | E)) )
then ( R_EAL f is_integrable_on M & R_EAL g is_integrable_on M ) by Def9;
then consider E being Element of S such that
A1: ( E = (dom (R_EAL f)) /\ (dom (R_EAL g)) & Integral M,((R_EAL f) + (R_EAL g)) = (Integral M,((R_EAL f) | E)) + (Integral M,((R_EAL g) | E)) ) by MESFUNC5:115;
take E ; :: thesis: ( E = (dom f) /\ (dom g) & Integral M,(f + g) = (Integral M,(f | E)) + (Integral M,(g | E)) )
thus ( E = (dom f) /\ (dom g) & Integral M,(f + g) = (Integral M,(f | E)) + (Integral M,(g | E)) ) by A1, Th23; :: thesis: verum