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

theorem
  (for n holds s.n = 2*n+1) implies for n holds Partial_Sums(s).n = (n+1 )|^2
proof
  defpred X[Nat] means Partial_Sums(s).$1 = ($1+1)|^2;
  assume
A1: for n holds s.n = 2*n+1;
A2: for n st X[n] holds X[n+1]
  proof
    let n;
    assume Partial_Sums(s).n = (n+1)|^2;
    then Partial_Sums(s).(n+1) = (n+1)|^2 + s.(n+1) by SERIES_1:def 1
      .=(n+1)|^2 + (2*(n+1)+1) by A1
      .=(n+1)|^2 +2*n+3
      .= n|^2+2*n*1+1|^2+2*n+3 by Lm3
      .= n|^2+2*n+1+2*n+3
      .= n|^2+2*n*2+2|^2 by Lm3
      .= (n+2)|^2 by Lm3;
    hence thesis;
  end;
  Partial_Sums(s).0 = s.0 by SERIES_1:def 1
    .=2*0+1 by A1
    .= (0+1)|^2;
  then
A3: X[0];
  for n holds X[n] from NAT_1:sch 2(A3,A2);
  hence thesis;
end;
