let s be Real_Sequence; :: thesis: ( ( for n being Nat holds s . n = 2 * n ) implies for n being Nat holds (Partial_Sums s) . n = n * (n + 1) )
defpred S₁[ Nat] means (Partial_Sums s) . $1 = $1 * ($1 + 1);
assume A1: for n being Nat holds s . n = 2 * n ; :: thesis: for n being Nat holds (Partial_Sums s) . n = n * (n + 1)
A2: for n being Nat st S₁[n] holds
S₁[n + 1]

proof

let n be Nat; :: thesis: ( S₁[n] implies S₁[n + 1] )
assume (Partial_Sums s) . n = n * (n + 1) ; :: thesis: S₁[n + 1]
then (Partial_Sums s) . (n + 1) = (n * (n + 1)) + (s . (n + 1)) by SERIES_1:def 1
.= (n * (n + 1)) + (2 * (n + 1)) by A1
.= (n + 2) * (n + 1) ;
hence S₁[n + 1] ; :: thesis: verum

end;

(Partial_Sums s) . 0 = s . 0 by SERIES_1:def 1
.= 2 * 0 by A1
.= 0 * (0 + 1) ;
then A3: S₁[ 0 ] ;
for n being Nat holds S₁[n] from NAT_1:sch 2(A3, A2);
hence for n being Nat holds (Partial_Sums s) . n = n * (n + 1) ; :: thesis: verum