let s be Real_Sequence; :: thesis: ( ( for n being Nat holds s . n = 1 / ((n * (n + 1)) * (n + 2)) ) implies for n being Nat holds (Partial_Sums s) . n = (1 / 4) - (1 / ((2 * (n + 1)) * (n + 2))) )
defpred S₁[ Nat] means (Partial_Sums s) . $1 = (1 / 4) - (1 / ((2 * ($1 + 1)) * ($1 + 2)));
assume A1: for n being Nat holds s . n = 1 / ((n * (n + 1)) * (n + 2)) ; :: thesis: for n being Nat holds (Partial_Sums s) . n = (1 / 4) - (1 / ((2 * (n + 1)) * (n + 2)))
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 = (1 / 4) - (1 / ((2 * (n + 1)) * (n + 2))) ; :: thesis: S₁[n + 1]
then (Partial_Sums s) . (n + 1) = ((1 / 4) - (1 / ((2 * (n + 1)) * (n + 2)))) + (s . (n + 1)) by SERIES_1:def 1
.= ((1 / 4) - (1 / ((2 * (n + 1)) * (n + 2)))) + (1 / (((n + 1) * ((n + 1) + 1)) * ((n + 1) + 2))) by A1
.= (1 / 4) - ((1 / ((2 * (n + 1)) * (n + 2))) - (1 / (((n + 1) * (n + 2)) * (n + 3))))
.= (1 / 4) - ((1 / ((2 * (n + 2)) * (n + 1))) - ((1 * 2) / ((((n + 2) * (n + 1)) * (n + 3)) * 2))) by XCMPLX_1:91
.= (1 / 4) - ((1 / ((2 * (n + 2)) * (n + 1))) - ((1 * 2) / (((n + 2) * 2) * ((n + 1) * (n + 3)))))
.= (1 / 4) - ((1 / ((2 * (n + 2)) * (n + 1))) - ((1 / ((n + 2) * 2)) * (2 / ((n + 1) * (n + 3))))) by XCMPLX_1:76
.= (1 / 4) - (((1 / (2 * (n + 2))) * (1 / (n + 1))) - ((1 / (2 * (n + 2))) * (2 / ((n + 1) * (n + 3))))) by XCMPLX_1:102
.= (1 / 4) - ((1 / (2 * (n + 2))) * ((1 / (n + 1)) - (2 / ((n + 1) * (n + 3)))))
.= (1 / 4) - ((1 / (2 * (n + 2))) * (1 / (n + 3))) by Lm8
.= (1 / 4) - (1 / ((2 * ((n + 1) + 1)) * ((n + 1) + 2))) by XCMPLX_1:102 ;
hence S₁[n + 1] ; :: thesis: verum

end;

(Partial_Sums s) . 0 = s . 0 by SERIES_1:def 1
.= 1 / ((0 * (0 + 1)) * (0 + 2)) by A1
.= (1 / 4) - (1 / ((2 * (0 + 1)) * (0 + 2))) by XCMPLX_1:49 ;
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 = (1 / 4) - (1 / ((2 * (n + 1)) * (n + 2))) ; :: thesis: verum