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 A1: ( f is_integrable_on M & A misses B ) ; :: thesis: Integral M,(f | (A \/ B)) = (Integral M,(f | A)) + (Integral M,(f | B))
then ( Re f is_integrable_on M & Im f is_integrable_on M ) by Def4;
then A2: ( Integral M,((Re f) | (A \/ B)) = (Integral M,((Re f) | A)) + (Integral M,((Re f) | B)) & Integral M,((Im f) | (A \/ B)) = (Integral M,((Im f) | A)) + (Integral M,((Im f) | B)) ) by A1, MESFUNC6:92;
A3: ( f | A is_integrable_on M & f | B is_integrable_on M & f | (A \/ B) is_integrable_on M ) by A1, Th91;
( Re (f | A) is_integrable_on M & Im (f | A) is_integrable_on M ) by A3, 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 A3, 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;
set C = A \/ B;
( Re (f | (A \/ B)) is_integrable_on M & Im (f | (A \/ B)) is_integrable_on M ) by A3, Def4;
then ( -infty < Integral M,(Re (f | (A \/ B))) & Integral M,(Re (f | (A \/ B))) < +infty & -infty < Integral M,(Im (f | (A \/ B))) & Integral M,(Im (f | (A \/ B))) < +infty ) by MESFUNC6:90;
then reconsider R3 = Integral M,(Re (f | (A \/ B))), I3 = Integral M,(Im (f | (A \/ B))) as Real by XXREAL_0:14;
B2: Integral M,((Re f) | (A \/ B)) = (Integral M,(Re (f | A))) + (Integral M,((Re f) | B)) by A2, COM91
.= (Integral M,(Re (f | A))) + (Integral M,(Re (f | B))) by COM91 ;
B4: Integral M,((Im f) | (A \/ B)) = (Integral M,(Im (f | A))) + (Integral M,((Im f) | B)) by A2, COM91
.= (Integral M,(Im (f | A))) + (Integral M,(Im (f | B))) by COM91 ;
( R_EAL R3 = (R_EAL R1) + (R_EAL R2) & R_EAL I3 = (R_EAL I1) + (R_EAL I2) ) by B2, B4, COM91;
then C3: ( R3 = R1 + R2 & I3 = I1 + I2 ) by SUPINF_2:1;
Integral M,(f | (A \/ B)) = R3 + (I3 * <i> ) by A3, Def5;
then Integral M,(f | (A \/ B)) = (R1 + (I1 * <i> )) + (R2 + (I2 * <i> )) by C3;
then Integral M,(f | (A \/ B)) = (Integral M,(f | A)) + (R2 + (I2 * <i> )) by A3, Def5;
hence Integral M,(f | (A \/ B)) = (Integral M,(f | A)) + (Integral M,(f | B)) by A3, Def5; :: thesis: verum