let s be Real_Sequence; :: thesis: ( ( for n being Nat holds s . n = (((1 / 3) |^ n) + (3 |^ n)) |^ 2 ) implies for n being Nat holds (Partial_Sums s) . n = (((- (((1 / 9) |^ n) / 8)) + ((9 |^ (n + 1)) / 8)) + (2 * n)) + 3 )
defpred S₁[ Nat] means (Partial_Sums s) . $1 = (((- (((1 / 9) |^ $1) / 8)) + ((9 |^ ($1 + 1)) / 8)) + (2 * $1)) + 3;
assume A1: for n being Nat holds s . n = (((1 / 3) |^ n) + (3 |^ n)) |^ 2 ; :: thesis: for n being Nat holds (Partial_Sums s) . n = (((- (((1 / 9) |^ n) / 8)) + ((9 |^ (n + 1)) / 8)) + (2 * n)) + 3
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 = (((- (((1 / 9) |^ n) / 8)) + ((9 |^ (n + 1)) / 8)) + (2 * n)) + 3 ; :: thesis: S₁[n + 1]
then (Partial_Sums s) . (n + 1) = ((((- (((1 / 9) |^ n) / 8)) + ((9 |^ (n + 1)) / 8)) + (2 * n)) + 3) + (s . (n + 1)) by SERIES_1:def 1
.= ((((- (((1 / 9) |^ n) / 8)) + ((9 |^ (n + 1)) / 8)) + (2 * n)) + 3) + ((((1 / 3) |^ (n + 1)) + (3 |^ (n + 1))) |^ 2) by A1
.= ((((- (((1 / 9) |^ n) / 8)) + ((9 |^ (n + 1)) / 8)) + (2 * n)) + 3) + ((((1 / 9) |^ (n + 1)) + (9 |^ (n + 1))) + 2) by Lm3
.= (((((- (((1 / 9) |^ n) / 8)) + ((1 / 9) |^ (n + 1))) + ((9 |^ (n + 1)) / 8)) + (9 |^ (n + 1))) + (2 * (n + 1))) + 3
.= (((((- (((1 / 9) |^ n) / 8)) + (((1 / 9) |^ n) * (1 / 9))) + ((9 |^ (n + 1)) / 8)) + (9 |^ (n + 1))) + (2 * (n + 1))) + 3 by NEWTON:6
.= ((((- ((((1 / 9) |^ n) * (1 / 9)) / 8)) + ((9 |^ (n + 1)) / 8)) + (9 |^ (n + 1))) + (2 * (n + 1))) + 3
.= ((((- (((1 / 9) |^ (n + 1)) / 8)) + ((9 |^ (n + 1)) / 8)) + (9 |^ (n + 1))) + (2 * (n + 1))) + 3 by NEWTON:6
.= (((- (((1 / 9) |^ (n + 1)) / 8)) + (((9 |^ (n + 1)) * 9) / 8)) + (2 * (n + 1))) + 3
.= (((- (((1 / 9) |^ (n + 1)) / 8)) + ((9 |^ ((n + 1) + 1)) / 8)) + (2 * (n + 1))) + 3 by NEWTON:6 ;
hence S₁[n + 1] ; :: thesis: verum

end;

(Partial_Sums s) . 0 = s . 0 by SERIES_1:def 1
.= (((1 / 3) |^ 0) + (3 |^ 0)) |^ 2 by A1
.= (1 + (3 |^ 0)) |^ 2 by NEWTON:4
.= (1 + 1) |^ 2 by NEWTON:4
.= (((- (1 / 8)) + (9 / 8)) + (2 * 0)) + 3 by Lm1
.= (((- (((1 / 9) |^ 0) / 8)) + (9 / 8)) + (2 * 0)) + 3 by NEWTON:4
.= (((- (((1 / 9) |^ 0) / 8)) + ((9 |^ (0 + 1)) / 8)) + (2 * 0)) + 3 ;
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 = (((- (((1 / 9) |^ n) / 8)) + ((9 |^ (n + 1)) / 8)) + (2 * n)) + 3 ; :: thesis: verum