let y be object ; :: thesis:
assume y in rng (Partial_Sums s) ; :: thesis:
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]

let k be Nat; :: thesis:
assume S₁[k] ; :: thesis:
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:

end;

end;