let X be non empty set ; :: thesis: for S being SigmaField of X
for M being sigma_Measure of S
for F being Functional_Sequence of X,ExtREAL st ( for n being Nat holds F . n is_integrable_on M ) holds
for m being Nat holds (Partial_Sums F) . m is_integrable_on M
let S be SigmaField of X; :: thesis: for M being sigma_Measure of S
for F being Functional_Sequence of X,ExtREAL st ( for n being Nat holds F . n is_integrable_on M ) holds
for m being Nat holds (Partial_Sums F) . m is_integrable_on M
let M be sigma_Measure of S; :: thesis: for F being Functional_Sequence of X,ExtREAL st ( for n being Nat holds F . n is_integrable_on M ) holds
for m being Nat holds (Partial_Sums F) . m is_integrable_on M
let F be Functional_Sequence of X,ExtREAL ; :: thesis: ( ( for n being Nat holds F . n is_integrable_on M ) implies for m being Nat holds (Partial_Sums F) . m is_integrable_on M )
assume A2: for n being Nat holds F . n is_integrable_on M ; :: thesis: for m being Nat holds (Partial_Sums F) . m is_integrable_on M
set PF = Partial_Sums F;
defpred S₁[ Nat] means (Partial_Sums F) . $1 is_integrable_on M;
(Partial_Sums F) . 0 = F . 0 by Def0;
then A3: S₁[ 0 ] by A2;
A4: for k being Nat st S₁[k] holds
S₁[k + 1]

proof

let k be Nat; :: thesis: ( S₁[k] implies S₁[k + 1] )
assume S₁[k] ; :: thesis: S₁[k + 1]
then ( (Partial_Sums F) . k is_integrable_on M & F . (k + 1) is_integrable_on M ) by A2;
then ((Partial_Sums F) . k) + (F . (k + 1)) is_integrable_on M by MESFUNC5:114;
hence S₁[k + 1] by Def0; :: thesis: verum

end;

thus for m being Nat holds S₁[m] from NAT_1:sch 2(A3, A4); :: thesis: verum