let X be non empty set ; :: thesis: for S being SigmaField of X
for M being sigma_Measure of S
for k being positive Real
for F being Functional_Sequence of X,REAL st ( for m being Nat holds F . m in Lp_Functions M,k ) holds
for m being Nat holds (Partial_Sums F) . m in Lp_Functions M,k
let S be SigmaField of X; :: thesis: for M being sigma_Measure of S
for k being positive Real
for F being Functional_Sequence of X,REAL st ( for m being Nat holds F . m in Lp_Functions M,k ) holds
for m being Nat holds (Partial_Sums F) . m in Lp_Functions M,k
let M be sigma_Measure of S; :: thesis: for k being positive Real
for F being Functional_Sequence of X,REAL st ( for m being Nat holds F . m in Lp_Functions M,k ) holds
for m being Nat holds (Partial_Sums F) . m in Lp_Functions M,k
let k be positive Real; :: thesis: for F being Functional_Sequence of X,REAL st ( for m being Nat holds F . m in Lp_Functions M,k ) holds
for m being Nat holds (Partial_Sums F) . m in Lp_Functions M,k
let F be Functional_Sequence of X,REAL ; :: thesis: ( ( for m being Nat holds F . m in Lp_Functions M,k ) implies for m being Nat holds (Partial_Sums F) . m in Lp_Functions M,k )
assume A1: for m being Nat holds F . m in Lp_Functions M,k ; :: thesis: for m being Nat holds (Partial_Sums F) . m in Lp_Functions M,k
defpred S₁[ Nat] means (Partial_Sums F) . $1 in Lp_Functions M,k;
(Partial_Sums F) . 0 = F . 0 by MESFUN9C:def 2;
then B1: S₁[ 0 ] by A1;

B2: now

let j be Nat; :: thesis: ( S₁[j] implies S₁[j + 1] )
assume S₁[j] ; :: thesis: S₁[j + 1]
then B01: ( (Partial_Sums F) . j in Lp_Functions M,k & F . (j + 1) in Lp_Functions M,k ) by A1;
(Partial_Sums F) . (j + 1) = ((Partial_Sums F) . j) + (F . (j + 1)) by MESFUN9C:def 2;
hence S₁[j + 1] by B01, Th01aLp; :: thesis: verum

end;

for j being Nat holds S₁[j] from NAT_1:sch 2(B1, B2);
hence for m being Nat holds (Partial_Sums F) . m in Lp_Functions M,k ; :: thesis: verum