let s be Real_Sequence; :: thesis: ( ( for n being Nat st n >= 1 holds
( s . n = (((n |^ 2) + n) - 1) / ((n + 2) !) & s . 0 = 0 ) ) implies for n being Nat st n >= 1 holds
(Partial_Sums s) . n = (1 / 2) - ((n + 1) / ((n + 2) !)) )
defpred S₁[ Nat] means (Partial_Sums s) . $1 = (1 / 2) - (($1 + 1) / (($1 + 2) !));
assume A1: for n being Nat st n >= 1 holds
( s . n = (((n |^ 2) + n) - 1) / ((n + 2) !) & s . 0 = 0 ) ; :: thesis: for n being Nat st n >= 1 holds
(Partial_Sums s) . n = (1 / 2) - ((n + 1) / ((n + 2) !))
A2: for n being Nat st n >= 1 & S₁[n] holds
S₁[n + 1]

proof

let n be Nat; :: thesis: ( n >= 1 & S₁[n] implies S₁[n + 1] )
assume that
n >= 1 and
A3: (Partial_Sums s) . n = (1 / 2) - ((n + 1) / ((n + 2) !)) ; :: thesis: S₁[n + 1]
A4: n + 1 >= 1 by NAT_1:11;
n + 3 >= 3 by NAT_1:11;
then A5: n + 3 > 0 by XXREAL_0:2;
(Partial_Sums s) . (n + 1) = ((1 / 2) - ((n + 1) / ((n + 2) !))) + (s . (n + 1)) by A3, SERIES_1:def 1
.= ((1 / 2) - ((n + 1) / ((n + 2) !))) + (((((n + 1) |^ 2) + (n + 1)) - 1) / (((n + 1) + 2) !)) by A1, A4
.= ((1 / 2) - (((n + 1) * (n + 3)) / (((n + 2) !) * ((n + 2) + 1)))) + ((((n + 1) |^ 2) + n) / ((n + 3) !)) by A5, XCMPLX_1:91
.= ((1 / 2) - (((n + 1) * (n + 3)) / (((n + 2) + 1) !))) + ((((n + 1) |^ 2) + n) / ((n + 3) !)) by NEWTON:15
.= (1 / 2) - ((((n + 1) * (n + 3)) / ((n + 3) !)) - ((((n + 1) |^ 2) + n) / ((n + 3) !)))
.= (1 / 2) - ((((n + 1) * (n + 3)) - (((n + 1) |^ 2) + n)) / ((n + 3) !)) by XCMPLX_1:120
.= (1 / 2) - (((((n + 1) * (n + 3)) - ((n + 1) |^ 2)) - n) / ((n + 3) !))
.= (1 / 2) - (((((n + 1) * (n + 3)) - ((n + 1) * (n + 1))) - n) / ((n + 3) !)) by WSIERP_1:1
.= (1 / 2) - (((n + 1) + 1) / (((n + 1) + 2) !)) ;
hence S₁[n + 1] ; :: thesis: verum

end;

(Partial_Sums s) . (1 + 0) = ((Partial_Sums s) . 0) + (s . (1 + 0)) by SERIES_1:def 1
.= (s . 0) + (s . 1) by SERIES_1:def 1
.= 0 + (s . 1) by A1
.= (((1 |^ 2) + 1) - 1) / ((1 + 2) !) by A1
.= (1 * 1) / ((2 + 1) !)
.= 1 / (2 * 3) by NEWTON:14, NEWTON:15
.= (1 / 2) - (2 / ((2 !) * (2 + 1))) by NEWTON:14
.= (1 / 2) - (2 / ((2 + 1) !)) by NEWTON:15 ;
then A6: S₁[1] ;
for n being Nat st n >= 1 holds
S₁[n] from NAT_1:sch 8(A6, A2);
hence for n being Nat st n >= 1 holds
(Partial_Sums s) . n = (1 / 2) - ((n + 1) / ((n + 2) !)) ; :: thesis: verum