let s be Real_Sequence; :: thesis: ( ( for n being Element of NAT st n >= 1 holds
( s . n = (((n |^ 2) + n) - 1) / ((n + 2) ! ) & s . 0 = 0 ) ) implies for n being Element of NAT st n >= 1 holds
(Partial_Sums s) . n = (1 / 2) - ((n + 1) / ((n + 2) ! )) )
assume A1: for n being Element of NAT st n >= 1 holds
( s . n = (((n |^ 2) + n) - 1) / ((n + 2) ! ) & s . 0 = 0 ) ; :: thesis: for n being Element of 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) ! ));
(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) ! ) by WSIERP_1:2
.= 1 / (2 * 3) by NEWTON:20, NEWTON:21
.= (1 / 2) - (2 / ((2 ! ) * (2 + 1))) by NEWTON:20
.= (1 / 2) - (2 / ((2 + 1) ! )) by NEWTON:21 ;
then A2: S₁[1] ;
A3: 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 A4: ( n >= 1 & (Partial_Sums s) . n = (1 / 2) - ((n + 1) / ((n + 2) ! )) ) ; :: thesis: S₁[n + 1]
A5: n + 1 >= 1 by NAT_1:11;
n + 3 >= 3 by NAT_1:11;
then A6: n + 3 > 0 by XXREAL_0:2;
n in NAT by ORDINAL1:def 13;
then (Partial_Sums s) . (n + 1) = ((1 / 2) - ((n + 1) / ((n + 2) ! ))) + (s . (n + 1)) by A4, SERIES_1:def 1
.= ((1 / 2) - ((n + 1) / ((n + 2) ! ))) + (((((n + 1) |^ 2) + (n + 1)) - 1) / (((n + 1) + 2) ! )) by A1, A5
.= ((1 / 2) - (((n + 1) * (n + 3)) / (((n + 2) ! ) * ((n + 2) + 1)))) + ((((n + 1) |^ 2) + n) / ((n + 3) ! )) by A6, XCMPLX_1:92
.= ((1 / 2) - (((n + 1) * (n + 3)) / (((n + 2) + 1) ! ))) + ((((n + 1) |^ 2) + n) / ((n + 3) ! )) by NEWTON:21
.= (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_1:121
.= (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_1:2
.= (1 / 2) - (((n + 1) + 1) / (((n + 1) + 2) ! )) ;
hence S₁[n + 1] ; :: thesis: verum

end;

for n being Nat st n >= 1 holds
S₁[n] from NAT_1:sch 8(A2, A3);
hence for n being Element of NAT st n >= 1 holds
(Partial_Sums s) . n = (1 / 2) - ((n + 1) / ((n + 2) ! )) ; :: thesis: verum