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 B, A 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 B, A 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 B, A 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 B, A 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 B, A 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, Def4;
then A4: Integral M,(Re f) < +infty by MESFUNC6:90;
A5: Im f is_integrable_on M by A1, Def4;
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 Real by A4, A6, A7, XXREAL_0:14;
A8: Integral M,f = R + (I * <i> ) by A1, Def5;
A9: f | B is_integrable_on M by A1, Th23;
then A10: Re (f | B) is_integrable_on M by Def4;
then A11: Integral M,(Re (f | B)) < +infty by MESFUNC6:90;
A12: Im (f | B) is_integrable_on M by A9, Def4;
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 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 by Def4;
then A17: Integral M,(Re (f | A)) < +infty by MESFUNC6:90;
A18: Im (f | A) is_integrable_on M by A15, Def4;
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 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, Def5;
hence ( f | A is_integrable_on M & Integral M,f = (Integral M,(f | A)) + (Integral M,(f | B)) ) by A1, A9, Def5, Th23; :: thesis: verum