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,ExtREAL 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,ExtREAL 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,ExtREAL 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,ExtREAL ; :: 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 A1: ( 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 (- 1) (#) g is_integrable_on M by MESFUNC5:116;
then - g is_integrable_on M by MESFUNC2:11;
then consider E being Element of S such that
A2: ( E = (dom f) /\ (dom (- g)) & Integral M,(f + (- g)) = (Integral M,(f | E)) + (Integral M,((- g) | E)) ) by A1, MESFUNC5:115;
A3: dom g = dom (- g) by MESFUNC1:def 7;
Integral M,(f - g) = (Integral M,(f | E)) + (Integral M,((- g) | E)) by A2, MESFUNC2:9;
hence ex E being Element of S st
( E = (dom f) /\ (dom g) & Integral M,(f - g) = (Integral M,(f | E)) + (Integral M,((- g) | E)) ) by A2, A3; :: thesis: verum