let n be Nat; :: thesis: for s being Real_Sequence st ( for n being Nat holds s . n = 1 / (n + 1) ) holds

(Partial_Product s) . n = 1 / ((n + 1) !)

let s be Real_Sequence; :: thesis: ( ( for n being Nat holds s . n = 1 / (n + 1) ) implies (Partial_Product s) . n = 1 / ((n + 1) !) )

defpred S_{1}[ Nat] means (Partial_Product s) . $1 = 1 / (($1 + 1) !);

assume A1: for n being Nat holds s . n = 1 / (n + 1) ; :: thesis: (Partial_Product s) . n = 1 / ((n + 1) !)

A2: for n being Nat st S_{1}[n] holds

S_{1}[n + 1]

.= 1 / ((0 + 1) !) by A1, NEWTON:13 ;

then A3: S_{1}[ 0 ]
;

for n being Nat holds S_{1}[n]
from NAT_1:sch 2(A3, A2);

hence (Partial_Product s) . n = 1 / ((n + 1) !) ; :: thesis: verum

(Partial_Product s) . n = 1 / ((n + 1) !)

let s be Real_Sequence; :: thesis: ( ( for n being Nat holds s . n = 1 / (n + 1) ) implies (Partial_Product s) . n = 1 / ((n + 1) !) )

defpred S

assume A1: for n being Nat holds s . n = 1 / (n + 1) ; :: thesis: (Partial_Product s) . n = 1 / ((n + 1) !)

A2: for n being Nat st S

S

proof

(Partial_Product s) . 0 =
s . 0
by SERIES_3:def 1
let n be Nat; :: thesis: ( S_{1}[n] implies S_{1}[n + 1] )

assume (Partial_Product s) . n = 1 / ((n + 1) !) ; :: thesis: S_{1}[n + 1]

then (Partial_Product s) . (n + 1) = (1 / ((n + 1) !)) * (s . (n + 1)) by SERIES_3:def 1

.= (1 / ((n + 1) !)) * (1 / ((n + 1) + 1)) by A1

.= ((1 / ((n + 1) !)) * 1) / ((n + 1) + 1) by XCMPLX_1:74

.= 1 / (((n + 1) !) * ((n + 1) + 1)) by XCMPLX_1:78

.= 1 / ((n + 2) !) by NEWTON:15 ;

hence S_{1}[n + 1]
; :: thesis: verum

end;assume (Partial_Product s) . n = 1 / ((n + 1) !) ; :: thesis: S

then (Partial_Product s) . (n + 1) = (1 / ((n + 1) !)) * (s . (n + 1)) by SERIES_3:def 1

.= (1 / ((n + 1) !)) * (1 / ((n + 1) + 1)) by A1

.= ((1 / ((n + 1) !)) * 1) / ((n + 1) + 1) by XCMPLX_1:74

.= 1 / (((n + 1) !) * ((n + 1) + 1)) by XCMPLX_1:78

.= 1 / ((n + 2) !) by NEWTON:15 ;

hence S

.= 1 / ((0 + 1) !) by A1, NEWTON:13 ;

then A3: S

for n being Nat holds S

hence (Partial_Product s) . n = 1 / ((n + 1) !) ; :: thesis: verum