A2: Partial_Sums s is total ;

now :: thesis: for y being object st y in rng (Partial_Sums s) holds
y in NAT

let y be object ; :: thesis: ( y in rng (Partial_Sums s) implies y in NAT )
assume y in rng (Partial_Sums s) ; :: thesis: y in NAT
then consider n being object such that
A3: ( n in dom (Partial_Sums s) & y = (Partial_Sums s) . n ) by FUNCT_1:def 3;
reconsider n = n as Nat by A3;
defpred S₁[ Nat] means (Partial_Sums s) . $1 is Nat;
(Partial_Sums s) . 0 = s . 0 by SERIES_1:def 1;
then A4: S₁[ 0 ] ;
A5: for k being Nat st S₁[k] holds
S₁[k + 1]

proof

let k be Nat; :: thesis: ( S₁[k] implies S₁[k + 1] )
assume S₁[k] ; :: thesis: S₁[k + 1]
then reconsider Pk = (Partial_Sums s) . k as Nat ;
(Partial_Sums s) . (k + 1) = Pk + (s . (k + 1)) by SERIES_1:def 1;
hence S₁[k + 1] ; :: thesis: verum

end;

for k being Nat holds S₁[k] from NAT_1:sch 2(A4, A5);
then (Partial_Sums s) . n is Nat ;
hence y in NAT by A3, ORDINAL1:def 12; :: thesis: verum

end;

hence Partial_Sums s is sequence of NAT by A2, TARSKI:def 3, FUNCT_2:2; :: thesis: verum