set f = Partial_Sums s;
let x be object ; :: according to VALUED_0:def 9 :: thesis: ( not x in dom (Partial_Sums s) or (Partial_Sums s) . x is real )
assume x in dom (Partial_Sums s) ; :: thesis: (Partial_Sums s) . x is real
then reconsider n = x as Element of NAT ;
defpred S₁[ Nat] means (Partial_Sums s) . s is real ;
(Partial_Sums s) . 0 = s . 0 by Def1;
then A1: S₁[ 0 ] ;
A2: for k being Nat st S₁[k] holds
S₁[k + 1]

proof

let k be Nat; :: thesis: ( S₁[k] implies S₁[k + 1] )
assume A3: S₁[k] ; :: thesis: S₁[k + 1]
reconsider k = k as Element of NAT by ORDINAL1:def 12;
(Partial_Sums s) . (k + 1) = ((Partial_Sums s) . k) + (s . (k + 1)) by Def1;
hence S₁[k + 1] by A3; :: thesis: verum

end;

for k being Nat holds S₁[k] from NAT_1:sch 2(A1, A2);
then (Partial_Sums s) . n is real ;
hence (Partial_Sums s) . x is real ; :: thesis: verum