let s be Real_Sequence; :: thesis: ( ( for n being Nat holds s . n = ((- 1) |^ (n + 1)) * (n |^ 4) ) implies for n being Nat holds (Partial_Sums s) . n = (((((- 1) |^ (n + 1)) * n) * (n + 1)) * (((n |^ 2) + n) - 1)) / 2 )
defpred S₁[ Nat] means (Partial_Sums s) . $1 = (((((- 1) |^ ($1 + 1)) * $1) * ($1 + 1)) * ((($1 |^ 2) + $1) - 1)) / 2;
assume A1: for n being Nat holds s . n = ((- 1) |^ (n + 1)) * (n |^ 4) ; :: thesis: for n being Nat holds (Partial_Sums s) . n = (((((- 1) |^ (n + 1)) * n) * (n + 1)) * (((n |^ 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 = (((((- 1) |^ (n + 1)) * n) * (n + 1)) * (((n |^ 2) + n) - 1)) / 2 ; :: thesis: S₁[n + 1]
then (Partial_Sums s) . (n + 1) = ((((((- 1) |^ (n + 1)) * n) * (n + 1)) * (((n |^ 2) + n) - 1)) / 2) + (s . (n + 1)) by SERIES_1:def 1
.= ((((((- 1) |^ (n + 1)) * n) * (n + 1)) * (((n |^ 2) + n) - 1)) / 2) + (((- 1) |^ ((n + 1) + 1)) * ((n + 1) |^ 4)) by A1
.= ((((((((- 1) |^ (n + 1)) * (- 1)) * (- 1)) * n) * (n + 1)) * (((n |^ 2) + n) - 1)) + ((((- 1) |^ (n + 2)) * ((n + 1) |^ 4)) * 2)) / 2
.= (((((((- 1) |^ ((n + 1) + 1)) * (- 1)) * n) * (n + 1)) * (((n |^ 2) + n) - 1)) + ((((- 1) |^ (n + 2)) * ((n + 1) |^ 4)) * 2)) / 2 by NEWTON:6
.= (((- 1) |^ (n + 2)) * (((((- 1) * n) * (n + 1)) * (((n |^ 2) + n) - 1)) + (((n + 1) |^ (3 + 1)) * 2))) / 2
.= (((- 1) |^ (n + 2)) * (((((- 1) * n) * (n + 1)) * (((n |^ 2) + n) - 1)) + ((((n + 1) |^ 3) * (n + 1)) * 2))) / 2 by NEWTON:6
.= ((((- 1) |^ (n + 2)) * (n + 1)) * (((- 1) * (((n * (n |^ 2)) + (n * n)) - (n * 1))) + (((n + 1) |^ 3) * 2))) / 2
.= ((((- 1) |^ (n + 2)) * (n + 1)) * (((- 1) * (((n |^ (2 + 1)) + (n * n)) - (n * 1))) + (((n + 1) |^ 3) * 2))) / 2 by NEWTON:6
.= ((((- 1) |^ (n + 2)) * (n + 1)) * (((- 1) * (((n |^ 3) + (n |^ 2)) - n)) + (((n + 1) |^ 3) * 2))) / 2 by WSIERP_1:1
.= ((((- 1) |^ (n + 2)) * (n + 1)) * (((- 1) * (((n |^ 3) + (n |^ 2)) - n)) + (((((n |^ 3) + (3 * (n |^ 2))) + (3 * n)) + 1) * 2))) / 2 by Lm4
.= ((((- 1) |^ (n + 2)) * (n + 1)) * ((((n |^ 3) + (5 * (n |^ 2))) + (7 * n)) + 2)) / 2
.= ((((- 1) |^ (n + 2)) * (n + 1)) * ((n + 2) * ((((n + 1) |^ 2) + (n + 1)) - 1))) / 2 by Lm13
.= (((((- 1) |^ (n + 2)) * (n + 1)) * (n + 2)) * ((((n + 1) |^ 2) + (n + 1)) - 1)) / 2 ;
hence S₁[n + 1] ; :: thesis: verum

end;

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