let s be Real_Sequence; :: thesis: ( ( for n being Element of NAT holds s . n = ((n ! ) * n) + (n / ((n + 1) ! )) ) implies for n being Element of NAT st n >= 1 holds
(Partial_Sums s) . n = ((n + 1) ! ) - (1 / ((n + 1) ! )) )
assume A1: for n being Element of NAT holds s . n = ((n ! ) * n) + (n / ((n + 1) ! )) ; :: thesis: for n being Element of NAT st n >= 1 holds
(Partial_Sums s) . n = ((n + 1) ! ) - (1 / ((n + 1) ! ))
then A2: s . 0 = ((0 ! ) * 0 ) + (0 / ((0 + 1) ! ))
.= 0 ;
defpred S₁[ Nat] means (Partial_Sums s) . $1 = (($1 + 1) ! ) - (1 / (($1 + 1) ! ));
(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:19, NEWTON:20 ;
then A3: S₁[1] ;
A4: 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] )
A5: n in NAT by ORDINAL1:def 13;
assume ( n >= 1 & (Partial_Sums s) . n = ((n + 1) ! ) - (1 / ((n + 1) ! )) ) ; :: thesis: S₁[n + 1]
then (Partial_Sums s) . (n + 1) = (((n + 1) ! ) - (1 / ((n + 1) ! ))) + (s . (n + 1)) by A5, 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_1:92
.= ((((n + 1) ! ) * ((n + 1) + 1)) - ((1 * (n + 2)) / (((n + 1) + 1) ! ))) + ((n + 1) / ((n + 2) ! )) by NEWTON:21
.= (((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_1:121
.= (((n + 1) + 1) ! ) - (1 / (((n + 1) + 1) ! )) by NEWTON:21 ;
hence S₁[n + 1] ; :: thesis: verum

end;

for n being Nat st n >= 1 holds
S₁[n] from NAT_1:sch 8(A3, A4);
hence for n being Element of NAT st n >= 1 holds
(Partial_Sums s) . n = ((n + 1) ! ) - (1 / ((n + 1) ! )) ; :: thesis: verum