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

theorem
  (for n holds s.n = 1/(n*(n+1))) implies for n holds Partial_Sums(s).n
  = 1-1/(n+1)
proof
  defpred X[Nat] means Partial_Sums(s).$1 = 1-1/($1+1);
  assume
A1: for n holds s.n = 1/(n*(n+1));
A2: for n st X[n] holds X[n+1]
  proof
    let n;
A3: n+1<>0 by NAT_1:5;
    assume Partial_Sums(s).n =1-1/(n+1);
    then Partial_Sums(s).(n+1) = 1-1/(n+1) + s.(n+1) by SERIES_1:def 1
      .=1-1/(n+1) + 1/((n+1)*(n+1+1)) by A1
      .=1-(1/(n+1) - 1/((n+1)*(n+2)))
      .=1-(1*(1/(n+1)) - (1/(n+1))*(1/(n+2))) by XCMPLX_1:102
      .=1-1/(n+1)*(1 - 1/(n+2))
      .=1-1/(n+1)*((1*(n+2)-1)/(n+2)) by Lm7
      .=1-(1/(n+1)*(n+1))/(n+2) by XCMPLX_1:74
      .=1-1/(n+2) by A3,XCMPLX_1:87;
    hence thesis;
  end;
  Partial_Sums(s).0 = s.0 by SERIES_1:def 1
    .=1/(0*(0+1)) by A1
    .=1-1/1 by XCMPLX_1:49;
  then
A4: X[0];
  for n holds X[n] from NAT_1:sch 2(A4,A2);
  hence thesis;
end;
