theorem
  x > 0 implies x|^k > 0
proof
  defpred P[Nat] means for x st x > 0 holds x|^$1 > 0;
A1: for k holds P[k] implies P[k+1]
  proof
    let k such that
A2: for x st x > 0 holds x|^k > 0;
    let x;
A3: x|^(k+1) = x * x|^k by Th6;
    assume
A4: x > 0;
    then x|^k > 0 by A2;
    hence thesis by A4,A3;
  end;
A5: P[0] by RVSUM_1:94;
  for k holds P[k] from NAT_1:sch 2(A5,A1);
  hence thesis;
end;
