let X be non empty set ; :: thesis: for S being SigmaField of X
for M being sigma_Measure of S
for F being Functional_Sequence of X,ExtREAL
for E being Element of S st E = dom (F . 0) & F is additive & F is with_the_same_dom & ( for n being Nat holds
( F . n is E -measurable & F . n is nonpositive ) ) holds
ex I being ExtREAL_sequence st
for n being Nat holds
( I . n = Integral (M,(F . n)) & Integral (M,(() . n)) = () . n )

let S be SigmaField of X; :: thesis: for M being sigma_Measure of S
for F being Functional_Sequence of X,ExtREAL
for E being Element of S st E = dom (F . 0) & F is additive & F is with_the_same_dom & ( for n being Nat holds
( F . n is E -measurable & F . n is nonpositive ) ) holds
ex I being ExtREAL_sequence st
for n being Nat holds
( I . n = Integral (M,(F . n)) & Integral (M,(() . n)) = () . n )

let M be sigma_Measure of S; :: thesis: for F being Functional_Sequence of X,ExtREAL
for E being Element of S st E = dom (F . 0) & F is additive & F is with_the_same_dom & ( for n being Nat holds
( F . n is E -measurable & F . n is nonpositive ) ) holds
ex I being ExtREAL_sequence st
for n being Nat holds
( I . n = Integral (M,(F . n)) & Integral (M,(() . n)) = () . n )

let F be Functional_Sequence of X,ExtREAL; :: thesis: for E being Element of S st E = dom (F . 0) & F is additive & F is with_the_same_dom & ( for n being Nat holds
( F . n is E -measurable & F . n is nonpositive ) ) holds
ex I being ExtREAL_sequence st
for n being Nat holds
( I . n = Integral (M,(F . n)) & Integral (M,(() . n)) = () . n )

let E be Element of S; :: thesis: ( E = dom (F . 0) & F is additive & F is with_the_same_dom & ( for n being Nat holds
( F . n is E -measurable & F . n is nonpositive ) ) implies ex I being ExtREAL_sequence st
for n being Nat holds
( I . n = Integral (M,(F . n)) & Integral (M,(() . n)) = () . n ) )

assume that
A1: E = dom (F . 0) and
A3: F is with_the_same_dom and
A4: for n being Nat holds
( F . n is E -measurable & F . n is nonpositive ) ; :: thesis: ex I being ExtREAL_sequence st
for n being Nat holds
( I . n = Integral (M,(F . n)) & Integral (M,(() . n)) = () . n )

set G = - F;
(- F) . 0 = - (F . 0) by Th37;
then A5: E = dom ((- F) . 0) by ;
A7: - F is with_the_same_dom by ;
for n being Nat holds
( (- F) . n is E -measurable & (- F) . n is nonnegative )
proof
let n be Nat; :: thesis: ( (- F) . n is E -measurable & (- F) . n is nonnegative )
( E = dom (F . n) & F . n is E -measurable ) by ;
then - (F . n) is E -measurable by MEASUR11:63;
hence (- F) . n is E -measurable by Th37; :: thesis: (- F) . n is nonnegative
F . n is nonpositive by A4;
then - (F . n) is nonnegative ;
hence (- F) . n is nonnegative by Th37; :: thesis: verum
end;
then consider J being ExtREAL_sequence such that
A8: for n being Nat holds
( J . n = Integral (M,((- F) . n)) & Integral (M,((Partial_Sums (- F)) . n)) = () . n ) by ;
set I = - J;
take - J ; :: thesis: for n being Nat holds
( (- J) . n = Integral (M,(F . n)) & Integral (M,(() . n)) = (Partial_Sums (- J)) . n )

A10: for n being Nat holds
( F . n is E -measurable & F . n is V121() )
proof
let n be Nat; :: thesis: ( F . n is E -measurable & F . n is V121() )
thus F . n is E -measurable by A4; :: thesis: F . n is V121()
F . n is nonpositive by A4;
hence F . n is V121() ; :: thesis: verum
end;
hereby :: thesis: verum
let n be Nat; :: thesis: ( (- J) . n = Integral (M,(F . n)) & Integral (M,(() . n)) = (Partial_Sums (- J)) . n )
dom (- J) = NAT by FUNCT_2:def 1;
then n in dom (- J) by ORDINAL1:def 12;
then (- J) . n = - (J . n) by MESFUNC1:def 7;
then A9: (- J) . n = - (Integral (M,((- F) . n))) by A8;
( E = dom (F . n) & F . n is E -measurable & (- F) . n = - (F . n) ) by ;
then Integral (M,((- F) . n)) = - (Integral (M,(F . n))) by Th52;
hence (- J) . n = Integral (M,(F . n)) by A9; :: thesis: Integral (M,(() . n)) = (Partial_Sums (- J)) . n
A11: E = dom (() . n) by ;
(Partial_Sums (- F)) . n = (- ()) . n by Th42
.= - (() . n) by Th37 ;
then A13: Integral (M,((Partial_Sums (- F)) . n)) = - (Integral (M,(() . n))) by A10, A1, A3, A11, Th52, Th67;
(Partial_Sums (- J)) . n = - (() . n) by Th43
.= - (Integral (M,((Partial_Sums (- F)) . n))) by A8 ;
hence Integral (M,(() . n)) = (Partial_Sums (- J)) . n by A13; :: thesis: verum
end;