let s be Real_Sequence; :: thesis: ( ( for n being Nat holds s . n = ((n !) * n) + (n / ((n + 1) !)) ) implies for n being Nat st n >= 1 holds
(Partial_Sums s) . n = ((n + 1) !) - (1 / ((n + 1) !)) )
defpred S₁[ Nat] means (Partial_Sums s) . $1 = (($1 + 1) !) - (1 / (($1 + 1) !));
assume A1: for n being Nat holds s . n = ((n !) * n) + (n / ((n + 1) !)) ; :: thesis: for n being Nat st n >= 1 holds
(Partial_Sums s) . n = ((n + 1) !) - (1 / ((n + 1) !))
then A2: s . 0 = ((0 !) * 0) + (0 / ((0 + 1) !))
.= 0 ;
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 = ((n + 1) !) - (1 / ((n + 1) !)) ; :: thesis: S₁[n + 1]
(Partial_Sums s) . (n + 1) = (((n + 1) !) - (1 / ((n + 1) !))) + (s . (n + 1)) by A4, SERIES_1:def 1
.= (((n + 1) !) - (1 / ((n + 1) !))) + ((((n + 1) !) * (n + 1)) + ((n + 1) / (((n + 1) + 1) !))) by A1
.= ((((n + 1) !) + (((n + 1) !) * (n + 1))) - (1 / ((n + 1) !))) + ((n + 1) / (((n + 1) + 1) !))
.= ((((n + 1) !) * ((n + 1) + 1)) - ((1 * (n + 2)) / (((n + 1) !) * ((n + 1) + 1)))) + ((n + 1) / ((n + 2) !)) by XCMPLX_1:91
.= ((((n + 1) !) * ((n + 1) + 1)) - ((1 * (n + 2)) / (((n + 1) + 1) !))) + ((n + 1) / ((n + 2) !)) by NEWTON:15
.= (((n + 1) !) * ((n + 1) + 1)) - (((n + 2) / (((n + 1) + 1) !)) - ((n + 1) / ((n + 2) !)))
.= (((n + 1) !) * ((n + 1) + 1)) - (((n + 2) - (n + 1)) / ((n + 2) !)) by XCMPLX_1:120
.= (((n + 1) + 1) !) - (1 / (((n + 1) + 1) !)) by NEWTON:15 ;
hence S₁[n + 1] ; :: thesis: verum

end;

(Partial_Sums s) . (1 + 0) = ((Partial_Sums s) . 0) + (s . (1 + 0)) by SERIES_1:def 1
.= 0 + (s . 1) by A2, SERIES_1:def 1
.= ((1 !) * 1) + (1 / ((1 + 1) !)) by A1
.= ((((1 + 1) !) - 1) + 1) - (1 / ((1 + 1) !)) by NEWTON:13, NEWTON:14 ;
then A5: S₁[1] ;
for n being Nat st n >= 1 holds
S₁[n] from NAT_1:sch 8(A5, A3);
hence for n being Nat st n >= 1 holds
(Partial_Sums s) . n = ((n + 1) !) - (1 / ((n + 1) !)) ; :: thesis: verum