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 ;

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 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 Element of 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 ;

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 ;

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 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 Element of REAL by A14, A16, A17, XXREAL_0:14;

A18: Integral (M,(f | (A \/ B))) = R3 + (I3 * <i>) by A12, Def3;

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 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 Element of REAL by A11, A20, A21, XXREAL_0:14;

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 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 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 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 R3 = R1 + R2 by Th7;

then R3 = R1 + R2 ;

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, Def3;

hence Integral (M,(f | (A \/ B))) = (Integral (M,(f | A))) + (Integral (M,(f | B))) by A3, Def3; :: thesis: verum

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 ;

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 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 Element of 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 ;

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 ;

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 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 Element of REAL by A14, A16, A17, XXREAL_0:14;

A18: Integral (M,(f | (A \/ B))) = R3 + (I3 * <i>) by A12, Def3;

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 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 Element of REAL by A11, A20, A21, XXREAL_0:14;

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 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 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 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 R3 = R1 + R2 by Th7;

then R3 = R1 + R2 ;

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, Def3;

hence Integral (M,(f | (A \/ B))) = (Integral (M,(f | A))) + (Integral (M,(f | B))) by A3, Def3; :: thesis: verum