let s be Real_Sequence; :: thesis: ( ( for n being Nat holds s . n = n |^ 7 ) implies for n being Nat holds (Partial_Sums s) . n = (((n |^ 2) * ((n + 1) |^ 2)) * (((((3 * (n |^ 4)) + (6 * (n |^ 3))) - (n |^ 2)) - (4 * n)) + 2)) / 24 )
defpred S₁[ Nat] means (Partial_Sums s) . $1 = ((($1 |^ 2) * (($1 + 1) |^ 2)) * (((((3 * ($1 |^ 4)) + (6 * ($1 |^ 3))) - ($1 |^ 2)) - (4 * $1)) + 2)) / 24;
assume A1: for n being Nat holds s . n = n |^ 7 ; :: thesis: for n being Nat holds (Partial_Sums s) . n = (((n |^ 2) * ((n + 1) |^ 2)) * (((((3 * (n |^ 4)) + (6 * (n |^ 3))) - (n |^ 2)) - (4 * n)) + 2)) / 24
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)) * (((((3 * (n |^ 4)) + (6 * (n |^ 3))) - (n |^ 2)) - (4 * n)) + 2)) / 24 ; :: thesis: S₁[n + 1]
then (Partial_Sums s) . (n + 1) = ((((n |^ 2) * ((n + 1) |^ 2)) * (((((3 * (n |^ 4)) + (6 * (n |^ 3))) - (n |^ 2)) - (4 * n)) + 2)) / 24) + (s . (n + 1)) by SERIES_1:def 1
.= ((((n |^ 2) * ((n + 1) |^ 2)) * (((((3 * (n |^ 4)) + (6 * (n |^ 3))) - (n |^ 2)) - (4 * n)) + 2)) / 24) + ((n + 1) |^ 7) by A1
.= ((((n |^ 2) * ((n + 1) |^ 2)) * (((((3 * (n |^ 4)) + (6 * (n |^ 3))) - (n |^ 2)) - (4 * n)) + 2)) + (((n + 1) |^ (5 + 2)) * 24)) / 24
.= ((((n |^ 2) * ((n + 1) |^ 2)) * (((((3 * (n |^ 4)) + (6 * (n |^ 3))) - (n |^ 2)) - (4 * n)) + 2)) + ((((n + 1) |^ 5) * ((n + 1) |^ 2)) * 24)) / 24 by NEWTON:8
.= (((n + 1) |^ 2) * (((n |^ 2) * (((((3 * (n |^ 4)) + (6 * (n |^ 3))) - (n |^ 2)) - (4 * n)) + 2)) + (((n + 1) |^ 5) * 24))) / 24
.= (((n + 1) |^ 2) * (((n + 2) |^ 2) * (((((3 * ((n + 1) |^ 4)) + (6 * ((n + 1) |^ 3))) - ((n + 1) |^ 2)) - (4 * (n + 1))) + 2))) / 24 by Lm17
.= ((((n + 1) |^ 2) * (((n + 1) + 1) |^ 2)) * (((((3 * ((n + 1) |^ 4)) + (6 * ((n + 1) |^ 3))) - ((n + 1) |^ 2)) - (4 * (n + 1))) + 2)) / 24 ;
hence S₁[n + 1] ; :: thesis: verum

end;

(Partial_Sums s) . 0 = s . 0 by SERIES_1:def 1
.= 0 |^ 7 by A1
.= ((0 * ((0 + 1) |^ 2)) * (((((3 * (0 |^ 4)) + (6 * (0 |^ 3))) - (0 |^ 2)) - (4 * 0)) + 2)) / 24 by NEWTON:11
.= (((0 |^ 2) * ((0 + 1) |^ 2)) * (((((3 * (0 |^ 4)) + (6 * (0 |^ 3))) - (0 |^ 2)) - (4 * 0)) + 2)) / 24 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 = (((n |^ 2) * ((n + 1) |^ 2)) * (((((3 * (n |^ 4)) + (6 * (n |^ 3))) - (n |^ 2)) - (4 * n)) + 2)) / 24 ; :: thesis: verum