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 A, B being Element of S st f is_integrable_on M & A misses B holds
Integral (M,(f | (A \/ B))) = (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 A, B being Element of S st f is_integrable_on M & A misses B holds
Integral (M,(f | (A \/ B))) = (Integral (M,(f | A))) + (Integral (M,(f | B)))
let M be sigma_Measure of S; :: thesis: for f being PartFunc of X,REAL
for A, B being Element of S st f is_integrable_on M & A misses B holds
Integral (M,(f | (A \/ B))) = (Integral (M,(f | A))) + (Integral (M,(f | B)))
let f be PartFunc of X,REAL; :: thesis: for A, B being Element of S st f is_integrable_on M & A misses B holds
Integral (M,(f | (A \/ B))) = (Integral (M,(f | A))) + (Integral (M,(f | B)))
let A, B be Element of S; :: thesis: ( f is_integrable_on M & A misses B implies Integral (M,(f | (A \/ B))) = (Integral (M,(f | A))) + (Integral (M,(f | B))) )
assume that
A1: f is_integrable_on M and
A2: A misses B ; :: thesis: Integral (M,(f | (A \/ B))) = (Integral (M,(f | A))) + (Integral (M,(f | B)))
R_EAL f is_integrable_on M by A1, Def9;
hence Integral (M,(f | (A \/ B))) = (Integral (M,(f | A))) + (Integral (M,(f | B))) by A2, MESFUNC5:98; :: thesis: verum