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
for I being ExtREAL_sequence
for m being Nat 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 & I . n = Integral (M,(F . n)) ) ) holds
Integral (M,(() . m)) = () . m

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
for I being ExtREAL_sequence
for m being Nat 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 & I . n = Integral (M,(F . n)) ) ) holds
Integral (M,(() . m)) = () . m

let M be sigma_Measure of S; :: thesis: for F being Functional_Sequence of X,ExtREAL
for E being Element of S
for I being ExtREAL_sequence
for m being Nat 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 & I . n = Integral (M,(F . n)) ) ) holds
Integral (M,(() . m)) = () . m

let F be Functional_Sequence of X,ExtREAL; :: thesis: for E being Element of S
for I being ExtREAL_sequence
for m being Nat 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 & I . n = Integral (M,(F . n)) ) ) holds
Integral (M,(() . m)) = () . m

let E be Element of S; :: thesis: for I being ExtREAL_sequence
for m being Nat 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 & I . n = Integral (M,(F . n)) ) ) holds
Integral (M,(() . m)) = () . m

let I be ExtREAL_sequence; :: thesis: for m being Nat 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 & I . n = Integral (M,(F . n)) ) ) holds
Integral (M,(() . m)) = () . m

let m be Nat; :: 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 & I . n = Integral (M,(F . n)) ) ) implies Integral (M,(() . m)) = () . m )

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 & I . n = Integral (M,(F . n)) ) ; :: thesis: Integral (M,(() . m)) = () . m
set G = - F;
set J = - I;
(- F) . 0 = - (F . 0) by Th37;
then A5: E = dom ((- F) . 0) by ;
A6: - F is with_the_same_dom by ;
A7: E = dom (() . m) by ;
A8: 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;
for n being Nat holds
( (- F) . n is E -measurable & (- F) . n is nonnegative & (- I) . n = Integral (M,((- F) . n)) )
proof
let n be Nat; :: thesis: ( (- F) . n is E -measurable & (- F) . n is nonnegative & (- I) . n = Integral (M,((- F) . n)) )
A9: ( F . n is nonpositive & I . n = Integral (M,(F . n)) ) by A4;
A10: (- F) . n = - (F . n) by Th37;
dom (- I) = NAT by FUNCT_2:def 1;
then A11: n in dom (- I) by ORDINAL1:def 12;
A12: dom (F . n) = E by ;
hence (- F) . n is E -measurable by ; :: thesis: ( (- F) . n is nonnegative & (- I) . n = Integral (M,((- F) . n)) )
thus (- F) . n is nonnegative by ; :: thesis: (- I) . n = Integral (M,((- F) . n))
Integral (M,((- F) . n)) = - (Integral (M,(F . n))) by A4, A10, A12, Th52;
hence (- I) . n = Integral (M,((- F) . n)) by ; :: thesis: verum
end;
then Integral (M,((Partial_Sums (- F)) . m)) = (Partial_Sums (- I)) . m by ;
then Integral (M,((- ()) . m)) = (Partial_Sums (- I)) . m by Th42;
then Integral (M,((- ()) . m)) = - (() . m) by Th43;
then Integral (M,(- (() . m))) = - (() . m) by Th37;
then - (Integral (M,(() . m))) = - (() . m) by A1, A3, A7, A8, Th67, Th52;
hence Integral (M,(() . m)) = () . m by XXREAL_3:10; :: thesis: verum