for z being set st z in dom (Partial_Sums_in_cod2 f) holds
0. >= (Partial_Sums_in_cod2 f) . z

proof

let z be set ; :: thesis: ( z in dom (Partial_Sums_in_cod2 f) implies 0. >= (Partial_Sums_in_cod2 f) . z )
assume z in dom (Partial_Sums_in_cod2 f) ; :: thesis: 0. >= (Partial_Sums_in_cod2 f) . z
then consider n, m being object such that
A1: ( n in NAT & m in NAT & z = [n,m] ) by ZFMISC_1:def 2;
reconsider n = n, m = m as Element of NAT by A1;
defpred S₁[ Nat] means (Partial_Sums_in_cod2 f) . (n,f) <= 0 ;
(Partial_Sums_in_cod2 f) . (n,0) = f . (n,0) by DefCSM;
then A2: S₁[ 0 ] by MESFUNC5:8;
A3: for k being Nat st S₁[k] holds
S₁[k + 1]

proof

let k be Nat; :: thesis: ( S₁[k] implies S₁[k + 1] )
assume A4: S₁[k] ; :: thesis: S₁[k + 1]
A5: f . (n,(k + 1)) <= 0 by MESFUNC5:8;
(Partial_Sums_in_cod2 f) . (n,(k + 1)) = ((Partial_Sums_in_cod2 f) . (n,k)) + (f . (n,(k + 1))) by DefCSM;
hence (Partial_Sums_in_cod2 f) . (n,(k + 1)) <= 0 by A5, A4; :: thesis: verum

end;

for k being Nat holds S₁[k] from NAT_1:sch 2(A2, A3);
then (Partial_Sums_in_cod2 f) . (n,m) <= 0 ;
hence 0. >= (Partial_Sums_in_cod2 f) . z by A1; :: thesis: verum

end;

hence for b₁ being Function of [:NAT,NAT:],ExtREAL st b₁ = Partial_Sums_in_cod2 f holds
b₁ is nonpositive by MESFUNC5:9; :: thesis: verum