let X be non empty set ; :: thesis: for F being Functional_Sequence of X,ExtREAL
for S being SigmaField of X
for E being Element of S
for m being Nat st F is with_the_same_dom & E = dom (F . 0) & ( for n being Nat holds
( F . n is_measurable_on E & F . n is V121() ) ) holds
(Partial_Sums F) . m is_measurable_on E
let F be Functional_Sequence of X,ExtREAL; :: thesis: for S being SigmaField of X
for E being Element of S
for m being Nat st F is with_the_same_dom & E = dom (F . 0) & ( for n being Nat holds
( F . n is_measurable_on E & F . n is V121() ) ) holds
(Partial_Sums F) . m is_measurable_on E
let S be SigmaField of X; :: thesis: for E being Element of S
for m being Nat st F is with_the_same_dom & E = dom (F . 0) & ( for n being Nat holds
( F . n is_measurable_on E & F . n is V121() ) ) holds
(Partial_Sums F) . m is_measurable_on E
let E be Element of S; :: thesis: for m being Nat st F is with_the_same_dom & E = dom (F . 0) & ( for n being Nat holds
( F . n is_measurable_on E & F . n is V121() ) ) holds
(Partial_Sums F) . m is_measurable_on E
let m be Nat; :: thesis: ( F is with_the_same_dom & E = dom (F . 0) & ( for n being Nat holds
( F . n is_measurable_on E & F . n is V121() ) ) implies (Partial_Sums F) . m is_measurable_on E )
assume that
A1: F is with_the_same_dom and
A2: E = dom (F . 0) and
A3: for n being Nat holds
( F . n is_measurable_on E & F . n is V121() ) ; :: thesis: (Partial_Sums F) . m is_measurable_on E

now :: thesis: for n being Nat holds
( (- F) . n is_measurable_on E & (- F) . n is V120() )

let n be Nat; :: thesis: ( (- F) . n is_measurable_on E & (- F) . n is V120() )
E = dom (F . n) by A1, A2, MESFUNC8:def 2;
then - (F . n) is_measurable_on E by A3, MEASUR11:63;
hence (- F) . n is_measurable_on E by Th37; :: thesis: (- F) . n is V120()
F . n is V121() by A3;
then - (F . n) is V120() ;
hence (- F) . n is V120() by Th37; :: thesis: verum

end;

then (Partial_Sums (- F)) . m is_measurable_on E by MESFUNC9:41;
then (- (Partial_Sums F)) . m is_measurable_on E by Th42;
then A5: - ((Partial_Sums F) . m) is_measurable_on E by Th37;
dom ((Partial_Sums F) . m) = E by A1, A2, A3, Th46, MESFUNC9:29;
then dom (- ((Partial_Sums F) . m)) = E by MESFUNC1:def 7;
then - (- ((Partial_Sums F) . m)) is_measurable_on E by A5, MEASUR11:63;
hence (Partial_Sums F) . m is_measurable_on E by DBLSEQ_3:2; :: thesis: verum