let n be Element of NAT ; :: thesis: for s being Real_Sequence st ( for n being Element of 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 Element of NAT holds s . n = (n + 1) / (n + 2) ) implies (Partial_Product s) . n = 1 / (n + 2) )
assume A1: for n being Element of NAT holds s . n = (n + 1) / (n + 2) ; :: thesis: (Partial_Product s) . n = 1 / (n + 2)
defpred S₁[ Element of NAT ] means (Partial_Product s) . $1 = 1 / ($1 + 2);
(Partial_Product s) . 0 = s . 0 by SERIES_3:def 1
.= (0 + 1) / (0 + 2) by A1
.= 1 / (0 + 2) ;
then A2: S₁[ 0 ] ;
A3: for n being Element of NAT st S₁[n] holds
S₁[n + 1]

proof

let n be Element of NAT ; :: thesis: ( S₁[n] implies S₁[n + 1] )
assume (Partial_Product s) . n = 1 / (n + 2) ; :: thesis: S₁[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_1:75
.= 1 / ((n + 1) + 2) by XCMPLX_1:107 ;
hence S₁[n + 1] ; :: thesis: verum

end;

for n being Element of NAT holds S₁[n] from NAT_1:sch 1(A2, A3);
hence (Partial_Product s) . n = 1 / (n + 2) ; :: thesis: verum