let s be Real_Sequence; :: thesis: ( ( for n being Nat holds s . n = (n * (n + 1)) * (2 |^ n) ) implies for n being Nat holds (Partial_Sums s) . n = ((2 |^ (n + 1)) * (((n |^ 2) - n) + 2)) - 4 )
defpred S₁[ Nat] means (Partial_Sums s) . $1 = ((2 |^ ($1 + 1)) * ((($1 |^ 2) - $1) + 2)) - 4;
assume A1: for n being Nat holds s . n = (n * (n + 1)) * (2 |^ n) ; :: thesis: for n being Nat holds (Partial_Sums s) . n = ((2 |^ (n + 1)) * (((n |^ 2) - n) + 2)) - 4
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 = ((2 |^ (n + 1)) * (((n |^ 2) - n) + 2)) - 4 ; :: thesis: S₁[n + 1]
then (Partial_Sums s) . (n + 1) = (((2 |^ (n + 1)) * (((n |^ 2) - n) + 2)) - 4) + (s . (n + 1)) by SERIES_1:def 1
.= (((2 |^ (n + 1)) * (((n |^ 2) - n) + 2)) - 4) + (((n + 1) * ((n + 1) + 1)) * (2 |^ (n + 1))) by A1
.= ((((2 |^ (n + 1)) * 2) * ((((n |^ 2) - n) + 2) + ((n + 1) * (n + 2)))) / 2) - 4
.= (((2 |^ ((n + 1) + 1)) * ((((n |^ 2) - n) + 2) + ((n + 1) * (n + 2)))) / 2) - 4 by NEWTON:6
.= (((2 |^ (n + 2)) * (((((((n |^ 2) - n) + 2) + (n * n)) + (1 * n)) + (n * 2)) + 2)) / 2) - 4
.= (((2 |^ (n + 2)) * (((((((n |^ 2) - n) + 2) + (n |^ 2)) + (1 * n)) + (n * 2)) + 2)) / 2) - 4 by WSIERP_1:1
.= ((2 |^ (n + 2)) * ((((((n |^ 2) + (2 * n)) + 1) - 1) - n) + 2)) - 4
.= ((2 |^ (n + 2)) * ((((((n |^ 2) + ((2 * n) * 1)) + (1 |^ 2)) - 1) - n) + 2)) - 4
.= ((2 |^ (n + 2)) * (((((n + 1) |^ 2) - 1) - n) + 2)) - 4 by Lm3
.= ((2 |^ (n + 2)) * ((((n + 1) |^ 2) - (1 + n)) + 2)) - 4 ;
hence S₁[n + 1] ; :: thesis: verum

end;

(Partial_Sums s) . 0 = s . 0 by SERIES_1:def 1
.= (0 * (0 + 1)) * (2 |^ 0) by A1
.= (2 * 2) - 4
.= ((2 |^ (0 + 1)) * ((0 - 0) + 2)) - 4
.= ((2 |^ (0 + 1)) * (((0 |^ 2) - 0) + 2)) - 4 by NEWTON:11 ;
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 = ((2 |^ (n + 1)) * (((n |^ 2) - n) + 2)) - 4 ; :: thesis: verum