let s be Real_Sequence; :: thesis: ( ( for n being Nat st n >= 1 holds
( s . n = 1 / ((((2 * n) - 1) * ((2 * n) + 1)) * ((2 * n) + 3)) & s . 0 = 0 ) ) implies for n being Nat st n >= 1 holds
(Partial_Sums s) . n = (1 / 12) - (1 / ((4 * ((2 * n) + 1)) * ((2 * n) + 3))) )
defpred S₁[ Nat] means (Partial_Sums s) . $1 = (1 / 12) - (1 / ((4 * ((2 * $1) + 1)) * ((2 * $1) + 3)));
assume A1: for n being Nat st n >= 1 holds
( s . n = 1 / ((((2 * n) - 1) * ((2 * n) + 1)) * ((2 * n) + 3)) & s . 0 = 0 ) ; :: thesis: for n being Nat st n >= 1 holds
(Partial_Sums s) . n = (1 / 12) - (1 / ((4 * ((2 * n) + 1)) * ((2 * n) + 3)))
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 / 12) - (1 / ((4 * ((2 * n) + 1)) * ((2 * n) + 3))) ; :: thesis: S₁[n + 1]
A4: n + 1 >= 1 by NAT_1:11;
(2 * n) + 1 >= 1 by NAT_1:11;
then A5: (2 * n) + 1 > 0 by XXREAL_0:2;
(2 * n) + 5 >= 5 by NAT_1:11;
then A6: (2 * n) + 5 > 0 by XXREAL_0:2;
(Partial_Sums s) . (n + 1) = ((1 / 12) - (1 / ((4 * ((2 * n) + 1)) * ((2 * n) + 3)))) + (s . (n + 1)) by A3, SERIES_1:def 1
.= ((1 / 12) - (1 / ((4 * ((2 * n) + 1)) * ((2 * n) + 3)))) + (1 / ((((2 * (n + 1)) - 1) * ((2 * (n + 1)) + 1)) * ((2 * (n + 1)) + 3))) by A1, A4
.= ((1 / 12) - (1 / (4 * (((2 * n) + 1) * ((2 * n) + 3))))) + (1 / ((((2 * n) + 1) * ((2 * n) + 3)) * ((2 * n) + 5)))
.= ((1 / 12) - ((1 / 4) * (1 / (((2 * n) + 1) * ((2 * n) + 3))))) + (1 / ((((2 * n) + 1) * ((2 * n) + 3)) * ((2 * n) + 5))) by XCMPLX_1:102
.= ((1 / 12) - ((1 / 4) * (1 / (((2 * n) + 1) * ((2 * n) + 3))))) + ((1 / (((2 * n) + 1) * ((2 * n) + 3))) * (1 / ((2 * n) + 5))) by XCMPLX_1:102
.= (1 / 12) - ((1 / (((2 * n) + 1) * ((2 * n) + 3))) * ((1 / 4) - (1 / ((2 * n) + 5))))
.= (1 / 12) - ((1 / (((2 * n) + 1) * ((2 * n) + 3))) * (((1 * ((2 * n) + 5)) - (1 * 4)) / (4 * ((2 * n) + 5)))) by A6, XCMPLX_1:130
.= (1 / 12) - (((1 / ((2 * n) + 1)) * (1 / ((2 * n) + 3))) * (((2 * n) + 1) / (4 * ((2 * n) + 5)))) by XCMPLX_1:102
.= (1 / 12) - ((((1 / ((2 * n) + 1)) * (1 / ((2 * n) + 3))) * ((2 * n) + 1)) / (4 * ((2 * n) + 5))) by XCMPLX_1:74
.= (1 / 12) - ((1 / ((2 * n) + 3)) / (4 * ((2 * n) + 5))) by A5, XCMPLX_1:109
.= (1 / 12) - (1 / (((2 * n) + 3) * (4 * ((2 * n) + 5)))) by XCMPLX_1:78
.= (1 / 12) - (1 / ((4 * ((2 * (n + 1)) + 1)) * ((2 * (n + 1)) + 3))) ;
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) * ((2 * 1) + 1)) * ((2 * 1) + 3)) by A1
.= (1 / 12) - (1 / ((4 * ((2 * 1) + 1)) * ((2 * 1) + 3))) ;
then A7: S₁[1] ;
for n being Nat st n >= 1 holds
S₁[n] from NAT_1:sch 8(A7, A2);
hence for n being Nat st n >= 1 holds
(Partial_Sums s) . n = (1 / 12) - (1 / ((4 * ((2 * n) + 1)) * ((2 * n) + 3))) ; :: thesis: verum