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 that
A1: f is_integrable_on M and
A2: 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))) )
R_EAL g is_integrable_on M by A2;
then (- jj) (#) (R_EAL g) is_integrable_on M by MESFUNC5:110;
then - (R_EAL g) is_integrable_on M by MESFUNC2:9;
then A3: R_EAL (- g) is_integrable_on M by MESFUNC6:28;
R_EAL f is_integrable_on M by A1;
then consider E being Element of S such that
A4: E = (dom (R_EAL f)) /\ (dom (R_EAL (- g))) and
A5: Integral (M,((R_EAL f) + (R_EAL (- g)))) = (Integral (M,((R_EAL f) | E))) + (Integral (M,((R_EAL (- g)) | E))) by A3, MESFUNC5:109;
take E ; :: thesis: ( E = (dom f) /\ (dom g) & Integral (M,(f - g)) = (Integral (M,(f | E))) + (Integral (M,((- g) | E))) )
dom (R_EAL (- g)) = dom (- (R_EAL g)) by MESFUNC6:28;
hence ( E = (dom f) /\ (dom g) & Integral (M,(f - g)) = (Integral (M,(f | E))) + (Integral (M,((- g) | E))) ) by A4, A5, MESFUNC1:def 7, MESFUNC6:23; :: thesis: verum