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 B, A being Element of S st f is_integrable_on M & B = (dom f) \ A holds
( f | A is_integrable_on M & Integral (M,f) = (Integral (M,(f | A))) + (Integral (M,(f | B))) )
let S be SigmaField of X; :: thesis: for M being sigma_Measure of S
for f being PartFunc of X,REAL
for B, A being Element of S st f is_integrable_on M & B = (dom f) \ A holds
( f | A is_integrable_on M & Integral (M,f) = (Integral (M,(f | A))) + (Integral (M,(f | B))) )
let M be sigma_Measure of S; :: thesis: for f being PartFunc of X,REAL
for B, A being Element of S st f is_integrable_on M & B = (dom f) \ A holds
( f | A is_integrable_on M & Integral (M,f) = (Integral (M,(f | A))) + (Integral (M,(f | B))) )
let f be PartFunc of X,REAL; :: thesis: for B, A being Element of S st f is_integrable_on M & B = (dom f) \ A holds
( f | A is_integrable_on M & Integral (M,f) = (Integral (M,(f | A))) + (Integral (M,(f | B))) )
let B, A be Element of S; :: thesis: ( f is_integrable_on M & B = (dom f) \ A implies ( f | A is_integrable_on M & Integral (M,f) = (Integral (M,(f | A))) + (Integral (M,(f | B))) ) )
assume that
A1: f is_integrable_on M and
A2: B = (dom f) \ A ; :: thesis: ( f | A is_integrable_on M & Integral (M,f) = (Integral (M,(f | A))) + (Integral (M,(f | B))) )
A3: R_EAL f is_integrable_on M by A1, Def9;
then R_EAL (f | A) is_integrable_on M by A2, MESFUNC5:105;
hence f | A is_integrable_on M by Def9; :: thesis: Integral (M,f) = (Integral (M,(f | A))) + (Integral (M,(f | B)))
thus Integral (M,f) = (Integral (M,(f | A))) + (Integral (M,(f | B))) by A2, A3, MESFUNC5:105; :: thesis: verum