let s be Real_Sequence; :: thesis: ( ( for n being Nat holds s . n = n * (2 |^ n) ) implies for n being Nat holds (Partial_Sums s) . n = ((n * (2 |^ (n + 1))) - (2 |^ (n + 1))) + 2 )
defpred S₁[ Nat] means (Partial_Sums s) . $1 = (($1 * (2 |^ ($1 + 1))) - (2 |^ ($1 + 1))) + 2;
assume A1: for n being Nat holds s . n = n * (2 |^ n) ; :: thesis: for n being Nat holds (Partial_Sums s) . n = ((n * (2 |^ (n + 1))) - (2 |^ (n + 1))) + 2
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 * (2 |^ (n + 1))) - (2 |^ (n + 1))) + 2 ; :: thesis: S₁[n + 1]
then (Partial_Sums s) . (n + 1) = (((n * (2 |^ (n + 1))) - (2 |^ (n + 1))) + 2) + (s . (n + 1)) by SERIES_1:def 1
.= (((n * (2 |^ (n + 1))) - (2 |^ (n + 1))) + 2) + ((n + 1) * (2 |^ (n + 1))) by A1
.= (n * ((2 |^ (n + 1)) * 2)) + 2
.= (n * (2 |^ ((n + 1) + 1))) + 2 by NEWTON:6 ;
hence S₁[n + 1] ; :: thesis: verum

end;

(Partial_Sums s) . 0 = s . 0 by SERIES_1:def 1
.= 0 * (2 |^ 0) by A1
.= ((0 * (2 |^ (0 + 1))) - 2) + 2
.= ((0 * (2 |^ (0 + 1))) - (2 |^ (0 + 1))) + 2 ;
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 * (2 |^ (n + 1))) - (2 |^ (n + 1))) + 2 ; :: thesis: verum