let s be Real_Sequence; :: thesis: ( ( for n being Nat st n >= 1 holds
( s . n = 1 / ((((3 * n) - 2) * ((3 * n) + 1)) * ((3 * n) + 4)) & s . 0 = 0 ) ) implies for n being Nat st n >= 1 holds
(Partial_Sums s) . n = (1 / 24) - (1 / ((6 * ((3 * n) + 1)) * ((3 * n) + 4))) )
defpred S₁[ Nat] means (Partial_Sums s) . $1 = (1 / 24) - (1 / ((6 * ((3 * $1) + 1)) * ((3 * $1) + 4)));
assume A1: for n being Nat st n >= 1 holds
( s . n = 1 / ((((3 * n) - 2) * ((3 * n) + 1)) * ((3 * n) + 4)) & s . 0 = 0 ) ; :: thesis: for n being Nat st n >= 1 holds
(Partial_Sums s) . n = (1 / 24) - (1 / ((6 * ((3 * n) + 1)) * ((3 * n) + 4)))
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 / 24) - (1 / ((6 * ((3 * n) + 1)) * ((3 * n) + 4))) ; :: thesis: S₁[n + 1]
A4: n + 1 >= 1 by NAT_1:11;
(3 * n) + 1 >= 1 by NAT_1:11;
then A5: (3 * n) + 1 > 0 by XXREAL_0:2;
(3 * n) + 7 >= 7 by NAT_1:11;
then A6: (3 * n) + 7 > 0 by XXREAL_0:2;
(Partial_Sums s) . (n + 1) = ((1 / 24) - (1 / ((6 * ((3 * n) + 1)) * ((3 * n) + 4)))) + (s . (n + 1)) by A3, SERIES_1:def 1
.= ((1 / 24) - (1 / ((6 * ((3 * n) + 1)) * ((3 * n) + 4)))) + (1 / ((((3 * (n + 1)) - 2) * ((3 * (n + 1)) + 1)) * ((3 * (n + 1)) + 4))) by A1, A4
.= ((1 / 24) - (1 / (6 * (((3 * n) + 1) * ((3 * n) + 4))))) + (1 / ((((3 * n) + 1) * ((3 * n) + 4)) * ((3 * n) + 7)))
.= ((1 / 24) - ((1 / 6) * (1 / (((3 * n) + 1) * ((3 * n) + 4))))) + (1 / ((((3 * n) + 1) * ((3 * n) + 4)) * ((3 * n) + 7))) by XCMPLX_1:102
.= ((1 / 24) - ((1 / 6) * (1 / (((3 * n) + 1) * ((3 * n) + 4))))) + ((1 / (((3 * n) + 1) * ((3 * n) + 4))) * (1 / ((3 * n) + 7))) by XCMPLX_1:102
.= (1 / 24) - ((1 / (((3 * n) + 1) * ((3 * n) + 4))) * ((1 / 6) - (1 / ((3 * n) + 7))))
.= (1 / 24) - ((1 / (((3 * n) + 1) * ((3 * n) + 4))) * (((1 * ((3 * n) + 7)) - (1 * 6)) / (6 * ((3 * n) + 7)))) by A6, XCMPLX_1:130
.= (1 / 24) - (((1 / ((3 * n) + 4)) * (1 / ((3 * n) + 1))) * (((3 * n) + 1) / (6 * ((3 * n) + 7)))) by XCMPLX_1:102
.= (1 / 24) - ((((1 / ((3 * n) + 4)) * (1 / ((3 * n) + 1))) * ((3 * n) + 1)) / (6 * ((3 * n) + 7))) by XCMPLX_1:74
.= (1 / 24) - ((1 / ((3 * n) + 4)) / (6 * ((3 * n) + 7))) by A5, XCMPLX_1:109
.= (1 / 24) - (1 / (((3 * n) + 4) * (6 * ((3 * n) + 7)))) by XCMPLX_1:78
.= (1 / 24) - (1 / ((6 * ((3 * (n + 1)) + 1)) * ((3 * (n + 1)) + 4))) ;
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 / ((((3 * 1) - 2) * ((3 * 1) + 1)) * ((3 * 1) + 4)) by A1
.= (1 / 24) - (1 / ((6 * ((3 * 1) + 1)) * ((3 * 1) + 4))) ;
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 / 24) - (1 / ((6 * ((3 * n) + 1)) * ((3 * n) + 4))) ; :: thesis: verum