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