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,((Partial_Sums F) . m)) = (Partial_Sums I) . 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,((Partial_Sums F) . m)) = (Partial_Sums I) . 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,((Partial_Sums F) . m)) = (Partial_Sums I) . 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,((Partial_Sums F) . m)) = (Partial_Sums I) . 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,((Partial_Sums F) . m)) = (Partial_Sums I) . 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,((Partial_Sums F) . m)) = (Partial_Sums I) . 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,((Partial_Sums F) . m)) = (Partial_Sums I) . m )
assume that
A1: E = dom (F . 0) and
A2: F is additive 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,((Partial_Sums F) . m)) = (Partial_Sums I) . m
set G = - F;
set J = - I;
(- F) . 0 = - (F . 0) by Th37;
then A5: E = dom ((- F) . 0) by A1, MESFUNC1:def 7;
A6: - F is with_the_same_dom by A3, Th40;
A7: E = dom ((Partial_Sums F) . m) by A1, A2, A3, MESFUNC9:29;
A8: for n being Nat holds
( F . n is E -measurable & F . n is without+infty )

proof

let n be Nat; :: thesis: ( F . n is E -measurable & F . n is without+infty )
thus F . n is E -measurable by A4; :: thesis: F . n is without+infty
F . n is nonpositive by A4;
hence F . n is without+infty ; :: 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 A1, A3, MESFUNC8:def 2;
hence (- F) . n is E -measurable by A4, A10, MEASUR11:63; :: thesis: ( (- F) . n is nonnegative & (- I) . n = Integral (M,((- F) . n)) )
thus (- F) . n is nonnegative by A9, A10; :: 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 A9, A11, MESFUNC1:def 7; :: thesis: verum

end;

then Integral (M,((Partial_Sums (- F)) . m)) = (Partial_Sums (- I)) . m by A5, A2, A6, Th41, MESFUNC9:46;
then Integral (M,((- (Partial_Sums F)) . m)) = (Partial_Sums (- I)) . m by Th42;
then Integral (M,((- (Partial_Sums F)) . m)) = - ((Partial_Sums I) . m) by Th43;
then Integral (M,(- ((Partial_Sums F) . m))) = - ((Partial_Sums I) . m) by Th37;
then - (Integral (M,((Partial_Sums F) . m))) = - ((Partial_Sums I) . m) by A1, A3, A7, A8, Th67, Th52;
hence Integral (M,((Partial_Sums F) . m)) = (Partial_Sums I) . m by XXREAL_3:10; :: thesis: verum