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 S_{1}[ Nat] means (Partial_Sums F) . $1 is nonnegative ;

(Partial_Sums F) . 0 = F . 0 by MESFUN9C:def 2;

then A2: S_{1}[ 0 ]
by A1;

_{1}[k]
from NAT_1:sch 2(A2, A3);

hence for m being Nat holds (Partial_Sums F) . m is nonnegative ; :: thesis: verum

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 S

(Partial_Sums F) . 0 = F . 0 by MESFUN9C:def 2;

then A2: S

A3: now :: thesis: for k being Nat st S_{1}[k] holds

S_{1}[k + 1]

for k being Nat holds SS

let k be Nat; :: thesis: ( S_{1}[k] implies S_{1}[k + 1] )

assume S_{1}[k]
; :: thesis: S_{1}[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 S_{1}[k + 1]
by A4, MESFUNC6:56; :: thesis: verum

end;assume S

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 S

hence for m being Nat holds (Partial_Sums F) . m is nonnegative ; :: thesis: verum