let X be non empty set ; :: thesis: for F being Functional_Sequence of X,ExtREAL
for n being Nat st ( for m being Nat holds F . m is nonnegative ) holds
(Partial_Sums F) . n is nonnegative
let F be Functional_Sequence of X,ExtREAL; :: thesis: for n being Nat st ( for m being Nat holds F . m is nonnegative ) holds
(Partial_Sums F) . n is nonnegative
let n be Nat; :: thesis: ( ( for m being Nat holds F . m is nonnegative ) implies (Partial_Sums F) . n is nonnegative )
defpred S₁[ Nat] means (Partial_Sums F) . $1 is nonnegative ;
assume A1: for m being Nat holds F . m is nonnegative ; :: thesis: (Partial_Sums F) . n is nonnegative

A2: 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 A3: S₁[k] ; :: thesis: S₁[k + 1]
A4: (Partial_Sums F) . (k + 1) = ((Partial_Sums F) . k) + (F . (k + 1)) by Def4;
F . (k + 1) is nonnegative by A1;
hence S₁[k + 1] by A3, A4, MESFUNC5:22; :: thesis: verum

end;

(Partial_Sums F) . 0 = F . 0 by Def4;
then A5: S₁[ 0 ] by A1;
for k being Nat holds S₁[k] from NAT_1:sch 2(A5, A2);
hence (Partial_Sums F) . n is nonnegative ; :: thesis: verum