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

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