let X be non empty set ; :: thesis: for F being Functional_Sequence of X,REAL st ( for m being Nat holds F . m is nonnegative ) holds
for m being Nat holds (Partial_Sums F) . m is nonnegative
let F be Functional_Sequence of X,REAL; :: thesis: ( ( for m being Nat holds F . m is nonnegative ) implies for m being Nat holds (Partial_Sums F) . m is nonnegative )
assume A1: for m being Nat holds F . m is nonnegative ; :: thesis: for m being Nat holds (Partial_Sums F) . m is nonnegative
defpred S₁[ Nat] means (Partial_Sums F) . $1 is nonnegative ;
(Partial_Sums F) . 0 = F . 0 by MESFUN9C:def 2;
then A2: S₁[ 0 ] by A1;

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

let k be Nat; :: thesis: ( S₁[k] implies S₁[k + 1] )
assume S₁[k] ; :: thesis: S₁[k + 1]
then A4: ( (Partial_Sums F) . k is nonnegative & F . (k + 1) is nonnegative ) by A1;
(Partial_Sums F) . (k + 1) = ((Partial_Sums F) . k) + (F . (k + 1)) by MESFUN9C:def 2;
hence S₁[k + 1] by A4, MESFUNC6:56; :: thesis: verum

end;

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