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

theorem
  (for n holds s.n = (-1)|^(n+1)*n|^2) implies for n holds Partial_Sums(
  s).n = (-1)|^(n+1)*n*(n+1)/2
proof
  defpred X[Nat] means Partial_Sums(s).$1 = (-1)|^($1+1)*$1*($1+1)/
  2;
  assume
A1: for n holds s.n = (-1)|^(n+1)*n|^2;
A2: for n st X[n] holds X[n+1]
  proof
    let n;
    assume Partial_Sums(s).n = (-1)|^(n+1)*n*(n+1)/2;
    then Partial_Sums(s).(n+1) = (-1)|^(n+1)*n*(n+1)/2 + s.(n+1) by
SERIES_1:def 1
      .=(-1)|^(n+1)*n*(n+1)/2 + (-1)|^(n+1+1)*(n+1)|^2 by A1
      .=((-1)|^(n+1)*n*(n+1) + (-1)|^(n+1+1)*(n+1)|^2*2)/2
      .=((-1)|^(n+1)*n*(n+1) + (-1)|^(n+1)*(-1)*(n+1)|^2*2)/2 by NEWTON:6
      .=((-1)|^(n+1)*(-1)*((-1)*(n*(n+1))+(n+1)|^2*2))/2
      .=((-1)|^(n+1+1)*((-1)*(n*(n+1))+(n+1)|^2*2))/2 by NEWTON:6
      .=((-1)|^(n+2)*((-1)*n*n+(-1)*n*1+(n|^2+2*n*1+1|^2 )*2))/2 by Lm3
      .=((-1)|^(n+2)*((-1)*n*n+(-1)*n*1+(n|^2+2*n+1)*2))/2
      .=((-1)|^(n+2)*((-1)*(n*n)+(-1)*n+(2*n|^2+2*n*2+1*2)))/2
      .=((-1)|^(n+2)*((-1)*(n|^1*n)+(-1)*n+(2*n|^2+2*n*2+1*2)))/2
      .=((-1)|^(n+2)*((-1)*n|^(1+1)+(-1)*n+(2*n|^2+2*n*2+1*2)))/2 by NEWTON:6
      .=((-1)|^(n+2)*(1*n|^2+3*n+2))/2
      .=(-1)|^(n+2)*((n+1)*(n+2))/2 by Lm10
      .=(-1)|^(n+2)*(n+1)*(n+2)/2;
    hence thesis;
  end;
  Partial_Sums(s).0 = s.0 by SERIES_1:def 1
    .=(-1)|^(0+1)*0|^2 by A1
    .=(-1)|^(0+1)*0*(0+1)/2 by NEWTON:11;
  then
A3: X[0];
  for n holds X[n] from NAT_1:sch 2(A3,A2);
  hence thesis;
end;
