let s be Real_Sequence; :: thesis: ( ( for n being Nat holds s . n = ((- 1) |^ (n + 1)) * (n |^ 2) ) implies for n being Nat holds (Partial_Sums s) . n = ((((- 1) |^ (n + 1)) * n) * (n + 1)) / 2 )
defpred S₁[ Nat] means (Partial_Sums s) . $1 = ((((- 1) |^ ($1 + 1)) * $1) * ($1 + 1)) / 2;
assume A1: for n being Nat holds s . n = ((- 1) |^ (n + 1)) * (n |^ 2) ; :: thesis: for n being Nat holds (Partial_Sums s) . n = ((((- 1) |^ (n + 1)) * n) * (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 = ((((- 1) |^ (n + 1)) * n) * (n + 1)) / 2 ; :: thesis: S₁[n + 1]
then (Partial_Sums s) . (n + 1) = (((((- 1) |^ (n + 1)) * n) * (n + 1)) / 2) + (s . (n + 1)) by SERIES_1:def 1
.= (((((- 1) |^ (n + 1)) * n) * (n + 1)) / 2) + (((- 1) |^ ((n + 1) + 1)) * ((n + 1) |^ 2)) by A1
.= (((((- 1) |^ (n + 1)) * n) * (n + 1)) + ((((- 1) |^ ((n + 1) + 1)) * ((n + 1) |^ 2)) * 2)) / 2
.= (((((- 1) |^ (n + 1)) * n) * (n + 1)) + (((((- 1) |^ (n + 1)) * (- 1)) * ((n + 1) |^ 2)) * 2)) / 2 by NEWTON:6
.= ((((- 1) |^ (n + 1)) * (- 1)) * (((- 1) * (n * (n + 1))) + (((n + 1) |^ 2) * 2))) / 2
.= (((- 1) |^ ((n + 1) + 1)) * (((- 1) * (n * (n + 1))) + (((n + 1) |^ 2) * 2))) / 2 by NEWTON:6
.= (((- 1) |^ (n + 2)) * (((((- 1) * n) * n) + (((- 1) * n) * 1)) + ((((n |^ 2) + ((2 * n) * 1)) + (1 |^ 2)) * 2))) / 2 by Lm3
.= (((- 1) |^ (n + 2)) * (((((- 1) * n) * n) + (((- 1) * n) * 1)) + ((((n |^ 2) + (2 * n)) + 1) * 2))) / 2
.= (((- 1) |^ (n + 2)) * ((((- 1) * (n * n)) + ((- 1) * n)) + (((2 * (n |^ 2)) + ((2 * n) * 2)) + (1 * 2)))) / 2
.= (((- 1) |^ (n + 2)) * ((((- 1) * ((n |^ 1) * n)) + ((- 1) * n)) + (((2 * (n |^ 2)) + ((2 * n) * 2)) + (1 * 2)))) / 2
.= (((- 1) |^ (n + 2)) * ((((- 1) * (n |^ (1 + 1))) + ((- 1) * n)) + (((2 * (n |^ 2)) + ((2 * n) * 2)) + (1 * 2)))) / 2 by NEWTON:6
.= (((- 1) |^ (n + 2)) * (((1 * (n |^ 2)) + (3 * n)) + 2)) / 2
.= (((- 1) |^ (n + 2)) * ((n + 1) * (n + 2))) / 2 by Lm10
.= ((((- 1) |^ (n + 2)) * (n + 1)) * (n + 2)) / 2 ;
hence S₁[n + 1] ; :: thesis: verum

end;

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