let s be Real_Sequence; :: thesis: ( ( for n being Nat holds s . n = ((2 * n) - 1) / ((n * (n + 1)) * (n + 2)) ) implies for n being Nat st n >= 1 holds
(Partial_Sums s) . n = ((3 / 4) - (2 / (n + 2))) + (1 / ((2 * (n + 1)) * (n + 2))) )
defpred S₁[ Nat] means (Partial_Sums s) . $1 = ((3 / 4) - (2 / ($1 + 2))) + (1 / ((2 * ($1 + 1)) * ($1 + 2)));
assume A1: for n being Nat holds s . n = ((2 * n) - 1) / ((n * (n + 1)) * (n + 2)) ; :: thesis: for n being Nat st n >= 1 holds
(Partial_Sums s) . n = ((3 / 4) - (2 / (n + 2))) + (1 / ((2 * (n + 1)) * (n + 2)))
then A2: s . 0 = ((2 * 0) - 1) / ((0 * (0 + 1)) * (0 + 2))
.= 0 by XCMPLX_1:49 ;
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 that
n >= 1 and
A4: (Partial_Sums s) . n = ((3 / 4) - (2 / (n + 2))) + (1 / ((2 * (n + 1)) * (n + 2))) ; :: thesis: S₁[n + 1]
n + 1 >= 1 + 0 by NAT_1:11;
then A5: n + 1 > 0 by NAT_1:13;
n + 2 >= 2 by NAT_1:11;
then n + 2 > 0 by XXREAL_0:2;
then A6: (n + 1) * (n + 2) > 0 by A5, XREAL_1:129;
n + 3 >= 3 by NAT_1:11;
then A7: n + 3 > 0 by XXREAL_0:2;
(Partial_Sums s) . (n + 1) = (((3 / 4) - (2 / (n + 2))) + (1 / ((2 * (n + 1)) * (n + 2)))) + (s . (n + 1)) by A4, SERIES_1:def 1
.= (((3 / 4) - (2 / (n + 2))) + (1 / ((2 * (n + 1)) * (n + 2)))) + (((2 * (n + 1)) - 1) / (((n + 1) * ((n + 1) + 1)) * ((n + 1) + 2))) by A1
.= ((3 / 4) - ((2 / (n + 2)) - (1 / (2 * ((n + 1) * (n + 2)))))) + (((2 * n) + 1) / (((n + 1) * (n + 2)) * (n + 3)))
.= ((3 / 4) - (((2 * (n + 1)) / ((n + 2) * (n + 1))) - (1 / (2 * ((n + 1) * (n + 2)))))) + (((2 * n) + 1) / (((n + 1) * (n + 2)) * (n + 3))) by A5, XCMPLX_1:91
.= ((3 / 4) - ((((2 * (n + 1)) * 2) / ((2 * (n + 2)) * (n + 1))) - (1 / (2 * ((n + 1) * (n + 2)))))) + (((2 * n) + 1) / (((n + 1) * (n + 2)) * (n + 3))) by XCMPLX_1:91
.= ((3 / 4) - (((4 * (n + 1)) - 1) / (2 * ((n + 2) * (n + 1))))) + (((2 * n) + 1) / (((n + 1) * (n + 2)) * (n + 3))) by XCMPLX_1:120
.= ((3 / 4) - ((((4 * n) + 3) * (n + 3)) / ((2 * ((n + 2) * (n + 1))) * (n + 3)))) + (((2 * n) + 1) / (((n + 1) * (n + 2)) * (n + 3))) by A7, XCMPLX_1:91
.= ((3 / 4) - ((((4 * n) + 3) * (n + 3)) / ((2 * ((n + 2) * (n + 1))) * (n + 3)))) + ((((2 * n) + 1) * 2) / ((((n + 1) * (n + 2)) * (n + 3)) * 2)) by XCMPLX_1:91
.= (3 / 4) - (((((4 * n) + 3) * (n + 3)) / ((2 * ((n + 2) * (n + 1))) * (n + 3))) - ((((2 * n) + 1) * 2) / ((((n + 1) * (n + 2)) * (n + 3)) * 2)))
.= (3 / 4) - (((((4 * n) + 3) * (n + 3)) - (((2 * n) + 1) * 2)) / ((2 * ((n + 2) * (n + 1))) * (n + 3))) by XCMPLX_1:120
.= (3 / 4) - ((((4 * (n + 1)) * (n + 2)) - (n + 1)) / ((2 * ((n + 2) * (n + 1))) * (n + 3)))
.= (3 / 4) - (((4 * ((n + 1) * (n + 2))) / ((2 * (n + 3)) * ((n + 2) * (n + 1)))) - ((n + 1) / ((2 * ((n + 2) * (n + 1))) * (n + 3)))) by XCMPLX_1:120
.= (3 / 4) - ((4 / (2 * (n + 3))) - ((1 * (n + 1)) / (((2 * (n + 2)) * (n + 3)) * (n + 1)))) by A6, XCMPLX_1:91
.= (3 / 4) - (((2 * 2) / ((n + 3) * 2)) - (1 / ((2 * (n + 2)) * (n + 3)))) by A5, XCMPLX_1:91
.= (3 / 4) - ((2 / ((n + 1) + 2)) - (1 / ((2 * ((n + 1) + 1)) * ((n + 1) + 2)))) by XCMPLX_1:91 ;
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
.= ((2 * 1) - 1) / ((1 * (1 + 1)) * (1 + 2)) by A1, A2
.= ((3 / 4) - (2 / (1 + 2))) + (1 / ((2 * (1 + 1)) * (1 + 2))) ;
then A8: S₁[1] ;
for n being Nat st n >= 1 holds
S₁[n] from NAT_1:sch 8(A8, A3);
hence for n being Nat st n >= 1 holds
(Partial_Sums s) . n = ((3 / 4) - (2 / (n + 2))) + (1 / ((2 * (n + 1)) * (n + 2))) ; :: thesis: verum