let X be non empty set ; :: thesis: for S being SigmaField of X
for E being Element of S
for F being Functional_Sequence of X,REAL
for m being Nat st ( for n being Nat holds F . n is E -measurable ) holds
(Partial_Sums F) . m is E -measurable
let S be SigmaField of X; :: thesis: for E being Element of S
for F being Functional_Sequence of X,REAL
for m being Nat st ( for n being Nat holds F . n is E -measurable ) holds
(Partial_Sums F) . m is E -measurable
let E be Element of S; :: thesis: for F being Functional_Sequence of X,REAL
for m being Nat st ( for n being Nat holds F . n is E -measurable ) holds
(Partial_Sums F) . m is E -measurable
let F be Functional_Sequence of X,REAL; :: thesis: for m being Nat st ( for n being Nat holds F . n is E -measurable ) holds
(Partial_Sums F) . m is E -measurable
let m be Nat; :: thesis: ( ( for n being Nat holds F . n is E -measurable ) implies (Partial_Sums F) . m is E -measurable )
set PF = Partial_Sums F;
defpred S₁[ Nat] means (Partial_Sums F) . $1 is E -measurable ;
assume A1: for n being Nat holds F . n is E -measurable ; :: thesis: (Partial_Sums F) . m is E -measurable
A2: for k being Nat st S₁[k] holds
S₁[k + 1]

proof

let k be Nat; :: thesis: ( S₁[k] implies S₁[k + 1] )
assume A3: S₁[k] ; :: thesis: S₁[k + 1]
F . (k + 1) is E -measurable by A1;
then ((Partial_Sums F) . k) + (F . (k + 1)) is E -measurable by A3, MESFUNC6:26;
hence S₁[k + 1] by Def2; :: thesis: verum

end;

(Partial_Sums F) . 0 = F . 0 by Def2;
then A4: S₁[ 0 ] by A1;
for k being Nat holds S₁[k] from NAT_1:sch 2(A4, A2);
hence (Partial_Sums F) . m is E -measurable ; :: thesis: verum