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 A2: S₁[ 0 ] by A1;

A3: now :: thesis: for j being Nat st S₁[j] holds
S₁[j + 1]

let j be Nat; :: thesis: ( S₁[j] implies S₁[j + 1] )
assume S₁[j] ; :: thesis: S₁[j + 1]
then A4: ( (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 A4, Th25; :: thesis: verum

end;

for j being Nat holds S₁[j] from NAT_1:sch 2(A2, A3);
hence for m being Nat holds (Partial_Sums F) . m in Lp_Functions (M,k) ; :: thesis: verum