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 E -measurable & F . n is V121() ) ) holds
() . m is E -measurable

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 E -measurable & F . n is V121() ) ) holds
() . m is E -measurable

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 E -measurable & F . n is V121() ) ) holds
() . m is E -measurable

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 E -measurable & F . n is V121() ) ) holds
() . m is E -measurable

let m be Nat; :: thesis: ( F is with_the_same_dom & E = dom (F . 0) & ( for n being Nat holds
( F . n is E -measurable & F . n is V121() ) ) implies () . m is E -measurable )

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 E -measurable & F . n is V121() ) ; :: thesis: () . m is E -measurable
now :: thesis: for n being Nat holds
( (- F) . n is E -measurable & (- F) . n is V120() )
let n be Nat; :: thesis: ( (- F) . n is E -measurable & (- F) . n is V120() )
E = dom (F . n) by ;
then - (F . n) is E -measurable by ;
hence (- F) . n is E -measurable 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 E -measurable by MESFUNC9:41;
then (- ()) . m is E -measurable by Th42;
then A5: - (() . m) is E -measurable by Th37;
dom (() . m) = E by ;
then dom (- (() . m)) = E by MESFUNC1:def 7;
then - (- (() . m)) is E -measurable by ;
hence (Partial_Sums F) . m is E -measurable by DBLSEQ_3:2; :: thesis: verum