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 A1: ( f is_integrable_on M & B = (dom f) \ A ) ; :: thesis: ( f | A is_integrable_on M & Integral M,f = (Integral M,(f | A)) + (Integral M,(f | B)) )
then A2: R_EAL f is_integrable_on M by Def9;
then ( R_EAL (f | A) is_integrable_on M & Integral M,f = (Integral M,(R_EAL (f | A))) + (Integral M,(R_EAL (f | B))) ) by A1, 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 A1, A2, MESFUNC5:105; :: thesis: verum