let n be Nat; :: thesis: for s being Real_Sequence st ( for n being Nat holds s . n = (n + 1) / (n + 2) ) holds

(Partial_Product s) . n = 1 / (n + 2)

let s be Real_Sequence; :: thesis: ( ( for n being Nat holds s . n = (n + 1) / (n + 2) ) implies (Partial_Product s) . n = 1 / (n + 2) )

defpred S_{1}[ Nat] means (Partial_Product s) . $1 = 1 / ($1 + 2);

assume A1: for n being Nat holds s . n = (n + 1) / (n + 2) ; :: thesis: (Partial_Product s) . n = 1 / (n + 2)

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

S_{1}[n + 1]

.= (0 + 1) / (0 + 2) by A1

.= 1 / (0 + 2) ;

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 + 2) ; :: thesis: verum

(Partial_Product s) . n = 1 / (n + 2)

let s be Real_Sequence; :: thesis: ( ( for n being Nat holds s . n = (n + 1) / (n + 2) ) implies (Partial_Product s) . n = 1 / (n + 2) )

defpred S

assume A1: for n being Nat holds s . n = (n + 1) / (n + 2) ; :: thesis: (Partial_Product s) . n = 1 / (n + 2)

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 + 2) ; :: thesis: S_{1}[n + 1]

then (Partial_Product s) . (n + 1) = (1 / (n + 2)) * (s . (n + 1)) by SERIES_3:def 1

.= (1 / (n + 2)) * (((n + 1) + 1) / ((n + 1) + 2)) by A1

.= ((1 / (n + 2)) * (n + 2)) / ((n + 1) + 2) by XCMPLX_1:74

.= 1 / ((n + 1) + 2) by XCMPLX_1:106 ;

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

end;assume (Partial_Product s) . n = 1 / (n + 2) ; :: thesis: S

then (Partial_Product s) . (n + 1) = (1 / (n + 2)) * (s . (n + 1)) by SERIES_3:def 1

.= (1 / (n + 2)) * (((n + 1) + 1) / ((n + 1) + 2)) by A1

.= ((1 / (n + 2)) * (n + 2)) / ((n + 1) + 2) by XCMPLX_1:74

.= 1 / ((n + 1) + 2) by XCMPLX_1:106 ;

hence S

.= (0 + 1) / (0 + 2) by A1

.= 1 / (0 + 2) ;

then A3: S

for n being Nat holds S

hence (Partial_Product s) . n = 1 / (n + 2) ; :: thesis: verum