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 & 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,COMPLEX
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,COMPLEX
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,COMPLEX; :: 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))) )
A3: Re f is_integrable_on M by A1;
then A4: Integral (M,(Re f)) < +infty by MESFUNC6:90;
A5: Im f is_integrable_on M by A1;
then A6: -infty < Integral (M,(Im f)) by MESFUNC6:90;
A7: Integral (M,(Im f)) < +infty by A5, MESFUNC6:90;
-infty < Integral (M,(Re f)) by A3, MESFUNC6:90;
then reconsider R = Integral (M,(Re f)), I = Integral (M,(Im f)) as Element of REAL by A4, A6, A7, XXREAL_0:14;
A8: Integral (M,f) = R + (I * <i>) by A1, Def3;
A9: f | B is_integrable_on M by A1, Th23;
then A10: Re (f | B) is_integrable_on M ;
then A11: Integral (M,(Re (f | B))) < +infty by MESFUNC6:90;
A12: Im (f | B) is_integrable_on M by A9;
then A13: -infty < Integral (M,(Im (f | B))) by MESFUNC6:90;
A14: Integral (M,(Im (f | B))) < +infty by A12, MESFUNC6:90;
-infty < Integral (M,(Re (f | B))) by A10, MESFUNC6:90;
then reconsider R2 = Integral (M,(Re (f | B))), I2 = Integral (M,(Im (f | B))) as Element of REAL by A11, A13, A14, XXREAL_0:14;
A15: f | A is_integrable_on M by A1, Th23;
then A16: Re (f | A) is_integrable_on M ;
then A17: Integral (M,(Re (f | A))) < +infty by MESFUNC6:90;
A18: Im (f | A) is_integrable_on M by A15;
then A19: -infty < Integral (M,(Im (f | A))) by MESFUNC6:90;
A20: Integral (M,(Im (f | A))) < +infty by A18, MESFUNC6:90;
-infty < Integral (M,(Re (f | A))) by A16, MESFUNC6:90;
then reconsider R1 = Integral (M,(Re (f | A))), I1 = Integral (M,(Im (f | A))) as Element of REAL by A17, A19, A20, XXREAL_0:14;
dom f = dom (Im f) by COMSEQ_3:def 4;
then Integral (M,(Im f)) = (Integral (M,((Im f) | A))) + (Integral (M,((Im f) | B))) by A2, A5, MESFUNC6:93;
then Integral (M,(Im f)) = (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 A21: I = I1 + I2 by SUPINF_2:1;
dom f = dom (Re f) by COMSEQ_3:def 3;
then Integral (M,(Re f)) = (Integral (M,((Re f) | A))) + (Integral (M,((Re f) | B))) by A2, A3, MESFUNC6:93;
then Integral (M,(Re f)) = (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 = R1 + R2 by SUPINF_2:1;
then Integral (M,f) = (R1 + (I1 * <i>)) + (R2 + (I2 * <i>)) by A21, A8;
then Integral (M,f) = (Integral (M,(f | A))) + (R2 + (I2 * <i>)) by A15, Def3;
hence ( f | A is_integrable_on M & Integral (M,f) = (Integral (M,(f | A))) + (Integral (M,(f | B))) ) by A1, A9, Def3, Th23; :: thesis: verum