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 A1: ( f is_integrable_on M & B = (dom f) \ A ) ; :: thesis: ( f | A is_integrable_on M & Integral M,f = (Integral M,(f | A)) + (Integral M,(f | B)) )
then A2: ( f | A is_integrable_on M & f | B is_integrable_on M ) by Th91;
A3: ( Re f is_integrable_on M & Im f is_integrable_on M ) by A1, Def4;
( dom f = dom (Re f) & dom f = dom (Im f) ) by Def1, Def2;
then A4: ( Integral M,(Re f) = (Integral M,((Re f) | A)) + (Integral M,((Re f) | B)) & Integral M,(Im f) = (Integral M,((Im f) | A)) + (Integral M,((Im f) | B)) ) by A1, A3, MESFUNC6:93;
( -infty < Integral M,(Re f) & Integral M,(Re f) < +infty & -infty < Integral M,(Im f) & Integral M,(Im f) < +infty ) by A3, MESFUNC6:90;
then reconsider R = Integral M,(Re f), I = Integral M,(Im f) as Real by XXREAL_0:14;
( Re (f | A) is_integrable_on M & Im (f | A) is_integrable_on M ) by A2, Def4;
then ( -infty < Integral M,(Re (f | A)) & Integral M,(Re (f | A)) < +infty & -infty < Integral M,(Im (f | A)) & Integral M,(Im (f | A)) < +infty ) by MESFUNC6:90;
then reconsider R1 = Integral M,(Re (f | A)), I1 = Integral M,(Im (f | A)) as Real by XXREAL_0:14;
( Re (f | B) is_integrable_on M & Im (f | B) is_integrable_on M ) by A2, Def4;
then ( -infty < Integral M,(Re (f | B)) & Integral M,(Re (f | B)) < +infty & -infty < Integral M,(Im (f | B)) & Integral M,(Im (f | B)) < +infty ) by MESFUNC6:90;
then reconsider R2 = Integral M,(Re (f | B)), I2 = Integral M,(Im (f | B)) as Real by XXREAL_0:14;
B1: Integral M,(Re f) = (Integral M,(Re (f | A))) + (Integral M,((Re f) | B)) by A4, COM91
.= (Integral M,(Re (f | A))) + (Integral M,(Re (f | B))) by COM91 ;
B2: Integral M,(Im f) = (Integral M,(Im (f | A))) + (Integral M,((Im f) | B)) by A4, COM91
.= (Integral M,(Im (f | A))) + (Integral M,(Im (f | B))) by COM91 ;
( R_EAL R = (R_EAL R1) + (R_EAL R2) & R_EAL I = (R_EAL I1) + (R_EAL I2) ) by B1, B2;
then C3: ( R = R1 + R2 & I = I1 + I2 ) by SUPINF_2:1;
Integral M,f = R + (I * <i> ) by A1, Def5;
then Integral M,f = (R1 + (I1 * <i> )) + (R2 + (I2 * <i> )) by C3;
then Integral M,f = (Integral M,(f | A)) + (R2 + (I2 * <i> )) by A2, Def5;
hence ( f | A is_integrable_on M & Integral M,f = (Integral M,(f | A)) + (Integral M,(f | B)) ) by A2, Def5; :: thesis: verum