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,ExtREAL
for A, B 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,ExtREAL
for A, B 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,ExtREAL
for A, B 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,ExtREAL; :: thesis: for A, B 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 A, B 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))) )
A \/ B = A \/ (dom f) by A2, XBOOLE_1:39;
then A3: (dom f) /\ (A \/ B) = dom f by XBOOLE_1:7, XBOOLE_1:28;
A4: f | (A \/ B) = (f | (dom f)) | (A \/ B) by GRFUNC_1:23
.= f | ((dom f) /\ (A \/ B)) by RELAT_1:71
.= f by A3, GRFUNC_1:23 ;
A misses B by A2, XBOOLE_1:79;
hence ( f | A is_integrable_on M & Integral (M,f) = (Integral (M,(f | A))) + (Integral (M,(f | B))) ) by A1, A4, Th97, Th98; :: thesis: verum