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, E being Element of S st E = dom f & f is E -measurable & f is nonpositive & 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,ExtREAL
for A, B, E being Element of S st E = dom f & f is E -measurable & f is nonpositive & 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,ExtREAL
for A, B, E being Element of S st E = dom f & f is E -measurable & f is nonpositive & A misses B holds
Integral (M,(f | (A \/ B))) = (Integral (M,(f | A))) + (Integral (M,(f | B)))

let f be PartFunc of X,ExtREAL; :: thesis: for A, B, E being Element of S st E = dom f & f is E -measurable & f is nonpositive & A misses B holds
Integral (M,(f | (A \/ B))) = (Integral (M,(f | A))) + (Integral (M,(f | B)))

let A, B, E be Element of S; :: thesis: ( E = dom f & f is E -measurable & f is nonpositive & A misses B implies Integral (M,(f | (A \/ B))) = (Integral (M,(f | A))) + (Integral (M,(f | B))) )
assume that
A1: ( E = dom f & f is E -measurable ) and
A2: f is nonpositive and
A3: A misses B ; :: thesis: Integral (M,(f | (A \/ B))) = (Integral (M,(f | A))) + (Integral (M,(f | B)))
set f1 = f | (A \/ B);
set f2 = f | A;
set f3 = f | B;
set g = - f;
reconsider E1 = E /\ (A \/ B) as Element of S ;
A4: dom (f | (A \/ B)) = E1 by ;
A5: E1 = (dom f) /\ E1 by ;
A6: f is E1 -measurable by ;
A7: f | E1 = (f | E) | (A \/ B) by RELAT_1:71;
(- f) | (A \/ B) = - (f | (A \/ B)) by Th3;
then A8: Integral (M,((- f) | (A \/ B))) = - (Integral (M,(f | (A \/ B)))) by ;
reconsider E2 = E /\ A as Element of S ;
A9: dom (f | A) = E2 by ;
A10: E2 = (dom f) /\ E2 by ;
A11: f is E2 -measurable by ;
A12: f | E2 = (f | E) | A by RELAT_1:71;
(- f) | A = - (f | A) by Th3;
then A13: Integral (M,((- f) | A)) = - (Integral (M,(f | A))) by ;
reconsider E3 = E /\ B as Element of S ;
A14: dom (f | B) = E3 by ;
A15: E3 = (dom f) /\ E3 by ;
A16: f is E3 -measurable by ;
A17: f | E3 = (f | E) | B by RELAT_1:71;
(- f) | B = - (f | B) by Th3;
then A18: Integral (M,((- f) | B)) = - (Integral (M,(f | B))) by ;
E = dom (- f) by ;
then Integral (M,((- f) | (A \/ B))) = (Integral (M,((- f) | A))) + (Integral (M,((- f) | B))) by ;
then - (Integral (M,(f | (A \/ B)))) = - ((Integral (M,(f | A))) + (Integral (M,(f | B)))) by ;
hence Integral (M,(f | (A \/ B))) = (Integral (M,(f | A))) + (Integral (M,(f | B))) by XXREAL_3:10; :: thesis: verum