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 A1, Th23;
then A4: Re (f | B) is_integrable_on M by Def4;
then A5: Integral M,(Re (f | B)) < +infty by MESFUNC6:90;
A6: Im (f | B) is_integrable_on M by A3, Def4;
then A7: -infty < Integral M,(Im (f | B)) by MESFUNC6:90;
A8: Integral M,(Im (f | B)) < +infty by A6, MESFUNC6:90;
-infty < Integral M,(Re (f | B)) by A4, MESFUNC6:90;
then reconsider R2 = Integral M,(Re (f | B)), I2 = Integral M,(Im (f | B)) as Real by A5, A7, A8, XXREAL_0:14;
A9: f | A is_integrable_on M by A1, Th23;
then A10: Re (f | A) is_integrable_on M by Def4;
then A11: Integral M,(Re (f | A)) < +infty by MESFUNC6:90;
set C = A \/ B;
A12: f | (A \/ B) is_integrable_on M by A1, Th23;
then A13: Re (f | (A \/ B)) is_integrable_on M by Def4;
then A14: Integral M,(Re (f | (A \/ B))) < +infty by MESFUNC6:90;
A15: Im (f | (A \/ B)) is_integrable_on M by A12, Def4;
then A16: -infty < Integral M,(Im (f | (A \/ B))) by MESFUNC6:90;
A17: Integral M,(Im (f | (A \/ B))) < +infty by A15, MESFUNC6:90;
-infty < Integral M,(Re (f | (A \/ B))) by A13, MESFUNC6:90;
then reconsider R3 = Integral M,(Re (f | (A \/ B))), I3 = Integral M,(Im (f | (A \/ B))) as Real by A14, A16, A17, XXREAL_0:14;
A18: Integral M,(f | (A \/ B)) = R3 + (I3 * <i> ) by A12, Def5;
A19: Im (f | A) is_integrable_on M by A9, Def4;
then A20: -infty < Integral M,(Im (f | A)) by MESFUNC6:90;
A21: Integral M,(Im (f | A)) < +infty by A19, MESFUNC6:90;
-infty < Integral M,(Re (f | A)) by A10, MESFUNC6:90;
then reconsider R1 = Integral M,(Re (f | A)), I1 = Integral M,(Im (f | A)) as Real by A11, A20, A21, XXREAL_0:14;
Im f is_integrable_on M by A1, Def4;
then Integral M,((Im f) | (A \/ B)) = (Integral M,((Im f) | A)) + (Integral M,((Im f) | B)) by A2, MESFUNC6:92;
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 R_EAL I3 = (R_EAL I1) + (R_EAL I2) by Th7;
then A22: I3 = I1 + I2 by SUPINF_2:1;
Re f is_integrable_on M by A1, Def4;
then Integral M,((Re f) | (A \/ B)) = (Integral M,((Re f) | A)) + (Integral M,((Re f) | B)) by A2, MESFUNC6:92;
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 R_EAL R3 = (R_EAL R1) + (R_EAL R2) by Th7;
then R3 = R1 + R2 by SUPINF_2:1;
then Integral M,(f | (A \/ B)) = (R1 + (I1 * <i> )) + (R2 + (I2 * <i> )) by A22, A18;
then Integral M,(f | (A \/ B)) = (Integral M,(f | A)) + (R2 + (I2 * <i> )) by A9, Def5;
hence Integral M,(f | (A \/ B)) = (Integral M,(f | A)) + (Integral M,(f | B)) by A3, Def5; :: thesis: verum