let s be Real_Sequence; :: thesis: ( ( for n being Nat holds s . n = n * ((n + 1) |^ 2) ) implies for n being Nat holds (Partial_Sums s) . n = (((n * (n + 1)) * (n + 2)) * ((3 * n) + 5)) / 12 )
defpred S₁[ Nat] means (Partial_Sums s) . $1 = ((($1 * ($1 + 1)) * ($1 + 2)) * ((3 * $1) + 5)) / 12;
assume A1: for n being Nat holds s . n = n * ((n + 1) |^ 2) ; :: thesis: for n being Nat holds (Partial_Sums s) . n = (((n * (n + 1)) * (n + 2)) * ((3 * n) + 5)) / 12
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)) * ((3 * n) + 5)) / 12 ; :: thesis: S₁[n + 1]
then (Partial_Sums s) . (n + 1) = ((((n * (n + 1)) * (n + 2)) * ((3 * n) + 5)) / 12) + (s . (n + 1)) by SERIES_1:def 1
.= ((((n * (n + 1)) * (n + 2)) * ((3 * n) + 5)) / 12) + ((n + 1) * (((n + 1) + 1) |^ 2)) by A1
.= ((n + 1) * (((n * (n + 2)) * ((3 * n) + 5)) + (((n + 2) |^ 2) * 12))) / 12
.= ((n + 1) * (((n * (n + 2)) * ((3 * n) + 5)) + (((n + 2) * (n + 2)) * 12))) / 12 by WSIERP_1:1
.= ((((n + 1) * (n + 2)) * (n + 3)) * ((3 * (n + 1)) + 5)) / 12 ;
hence S₁[n + 1] ; :: thesis: verum

end;

(Partial_Sums s) . 0 = s . 0 by SERIES_1:def 1
.= 0 * ((0 + 1) |^ 2) by A1
.= (((0 * (0 + 1)) * (0 + 2)) * ((3 * 0) + 5)) / 12 ;
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)) * ((3 * n) + 5)) / 12 ; :: thesis: verum