reserve i,j,k,n,m,l,s,t for Nat;
reserve a,b for Real;
reserve F for real-valued FinSequence;
reserve z for Complex;
reserve x,y for Complex;
reserve r,s,t for natural Number;
reserve p,q for natural Number;
reserve i0,i,i1,i2,i4 for Integer;
reserve x for set;
reserve p for Prime;
reserve i for Nat;
reserve x for Real;

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;
