reserve n for Nat,
  a,b,c,d for Real,
  s for Real_Sequence;

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