let s be Real_Sequence; :: thesis: ( ( for n being Nat holds s . n = (n * ((n + 1) |^ 2)) * (n + 2) ) implies for n being Nat holds (Partial_Sums s) . n = ((((n * (n + 1)) * (n + 2)) * (n + 3)) * ((2 * n) + 3)) / 10 )
defpred S₁[ Nat] means (Partial_Sums s) . $1 = (((($1 * ($1 + 1)) * ($1 + 2)) * ($1 + 3)) * ((2 * $1) + 3)) / 10;
assume A1: for n being Nat holds s . n = (n * ((n + 1) |^ 2)) * (n + 2) ; :: thesis: for n being Nat holds (Partial_Sums s) . n = ((((n * (n + 1)) * (n + 2)) * (n + 3)) * ((2 * n) + 3)) / 10
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 * (n + 1)) * (n + 2)) * (n + 3)) * ((2 * n) + 3)) / 10 ; :: thesis: S₁[n + 1]
then (Partial_Sums s) . (n + 1) = (((((n * (n + 1)) * (n + 2)) * (n + 3)) * ((2 * n) + 3)) / 10) + (s . (n + 1)) by SERIES_1:def 1
.= (((((n * (n + 1)) * (n + 2)) * (n + 3)) * ((2 * n) + 3)) / 10) + (((n + 1) * (((n + 1) + 1) |^ 2)) * ((n + 1) + 2)) by A1
.= (((((n * (n + 1)) * (n + 2)) * (n + 3)) * ((2 * n) + 3)) / 10) + (((((n + 1) * ((n + 2) * (n + 2))) * (n + 3)) * 10) / 10) by WSIERP_1:1
.= (((((n + 1) * (n + 2)) * (n + 3)) * (n + 4)) * ((2 * (n + 1)) + 3)) / 10 ;
hence S₁[n + 1] ; :: thesis: verum

end;

(Partial_Sums s) . 0 = s . 0 by SERIES_1:def 1
.= (0 * ((0 + 1) |^ 2)) * (0 + 2) by A1
.= ((((0 * (0 + 1)) * (0 + 2)) * (0 + 3)) * ((2 * 0) + 3)) / 10 ;
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 * (n + 1)) * (n + 2)) * (n + 3)) * ((2 * n) + 3)) / 10 ; :: thesis: verum