for z being object st z in dom (Partial_Sums_in_cod1 f) holds
0. <= (Partial_Sums_in_cod1 f) . z

proof

let z be object ; :: thesis: ( z in dom (Partial_Sums_in_cod1 f) implies 0. <= (Partial_Sums_in_cod1 f) . z )
assume z in dom (Partial_Sums_in_cod1 f) ; :: thesis: 0. <= (Partial_Sums_in_cod1 f) . z
then consider m, n being object such that
A1: ( m in NAT & n in NAT & z = [m,n] ) by ZFMISC_1:def 2;
reconsider n = n, m = m as Element of NAT by A1;
defpred S₁[ Nat] means (Partial_Sums_in_cod1 f) . (f,n) >= 0 ;
(Partial_Sums_in_cod1 f) . (0,n) = f . (0,n) by DefRSM;
then A2: S₁[ 0 ] by SUPINF_2:51;
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 . ((k + 1),n) >= 0 by SUPINF_2:51;
(Partial_Sums_in_cod1 f) . ((k + 1),n) = ((Partial_Sums_in_cod1 f) . (k,n)) + (f . ((k + 1),n)) by DefRSM;
hence (Partial_Sums_in_cod1 f) . ((k + 1),n) >= 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_cod1 f) . (m,n) >= 0 ;
hence 0. <= (Partial_Sums_in_cod1 f) . z by A1; :: thesis: verum

end;

hence for b₁ being Function of [:NAT,NAT:],ExtREAL st b₁ = Partial_Sums_in_cod1 f holds
b₁ is nonnegative by SUPINF_2:52; :: thesis: verum