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,COMPLEX
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,COMPLEX
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,COMPLEX
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,COMPLEX; :: 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)))
A3: f | B is_integrable_on M by ;
then A4: Re (f | B) is_integrable_on M ;
then A5: Integral (M,(Re (f | B))) < +infty by MESFUNC6:90;
A6: Im (f | B) is_integrable_on M by A3;
then A7: -infty < Integral (M,(Im (f | B))) by MESFUNC6:90;
A8: Integral (M,(Im (f | B))) < +infty by ;
-infty < Integral (M,(Re (f | B))) by ;
then reconsider R2 = Integral (M,(Re (f | B))), I2 = Integral (M,(Im (f | B))) as Element of REAL by ;
A9: f | A is_integrable_on M by ;
then A10: Re (f | A) is_integrable_on M ;
then A11: Integral (M,(Re (f | A))) < +infty by MESFUNC6:90;
set C = A \/ B;
A12: f | (A \/ B) is_integrable_on M by ;
then A13: Re (f | (A \/ B)) is_integrable_on M ;
then A14: Integral (M,(Re (f | (A \/ B)))) < +infty by MESFUNC6:90;
A15: Im (f | (A \/ B)) is_integrable_on M by A12;
then A16: -infty < Integral (M,(Im (f | (A \/ B)))) by MESFUNC6:90;
A17: Integral (M,(Im (f | (A \/ B)))) < +infty by ;
-infty < Integral (M,(Re (f | (A \/ B)))) by ;
then reconsider R3 = Integral (M,(Re (f | (A \/ B)))), I3 = Integral (M,(Im (f | (A \/ B)))) as Element of REAL by ;
A18: Integral (M,(f | (A \/ B))) = R3 + (I3 * <i>) by ;
A19: Im (f | A) is_integrable_on M by A9;
then A20: -infty < Integral (M,(Im (f | A))) by MESFUNC6:90;
A21: Integral (M,(Im (f | A))) < +infty by ;
-infty < Integral (M,(Re (f | A))) by ;
then reconsider R1 = Integral (M,(Re (f | A))), I1 = Integral (M,(Im (f | A))) as Element of REAL by ;
Im f is_integrable_on M by A1;
then Integral (M,((Im f) | (A \/ B))) = (Integral (M,((Im f) | A))) + (Integral (M,((Im f) | B))) by ;
then Integral (M,((Im f) | (A \/ B))) = (Integral (M,(Im (f | A)))) + (Integral (M,((Im f) | B))) by Th7
.= (Integral (M,(Im (f | A)))) + (Integral (M,(Im (f | B)))) by Th7 ;
then I3 = I1 + I2 by Th7;
then A22: I3 = I1 + I2 ;
Re f is_integrable_on M by A1;
then Integral (M,((Re f) | (A \/ B))) = (Integral (M,((Re f) | A))) + (Integral (M,((Re f) | B))) by ;
then Integral (M,((Re f) | (A \/ B))) = (Integral (M,(Re (f | A)))) + (Integral (M,((Re f) | B))) by Th7
.= (Integral (M,(Re (f | A)))) + (Integral (M,(Re (f | B)))) by Th7 ;
then R3 = R1 + R2 by Th7;
then R3 = R1 + R2 ;
then Integral (M,(f | (A \/ B))) = (R1 + (I1 * <i>)) + (R2 + (I2 * <i>)) by ;
then Integral (M,(f | (A \/ B))) = (Integral (M,(f | A))) + (R2 + (I2 * <i>)) by ;
hence Integral (M,(f | (A \/ B))) = (Integral (M,(f | A))) + (Integral (M,(f | B))) by ; :: thesis: verum