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

let s be Real_Sequence; :: thesis: ( ( for n being Nat holds s . n = (n + 1) / (n + 2) ) implies () . n = 1 / (n + 2) )
defpred S1[ Nat] means () . \$1 = 1 / (\$1 + 2);
assume A1: for n being Nat holds s . n = (n + 1) / (n + 2) ; :: thesis: () . n = 1 / (n + 2)
A2: for n being Nat st S1[n] holds
S1[n + 1]
proof
let n be Nat; :: thesis: ( S1[n] implies S1[n + 1] )
assume (Partial_Product s) . n = 1 / (n + 2) ; :: thesis: S1[n + 1]
then () . (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 S1[n + 1] ; :: thesis: verum
end;
() . 0 = s . 0 by SERIES_3:def 1
.= (0 + 1) / (0 + 2) by A1
.= 1 / (0 + 2) ;
then A3: S1[ 0 ] ;
for n being Nat holds S1[n] from NAT_1:sch 2(A3, A2);
hence (Partial_Product s) . n = 1 / (n + 2) ; :: thesis: verum