reserve k,n,m,l,p for Nat;
reserve n0,m0 for non zero Nat;
reserve f for FinSequence;
reserve x,X,Y for set;
reserve f1,f2,f3 for FinSequence of REAL;
reserve n1,n2,m1,m2 for Nat;

theorem Th19:
  p is prime implies NatDivisors(p|^n) = {p|^k where k is Element
  of NAT : k <= n}
proof
  assume
A1: p is prime;
  for x being object holds x in NatDivisors(p|^n) iff x in {p|^k where k is
  Element of NAT : k <= n}
  proof
    let x be object;
    hereby
      assume
A2:   x in NatDivisors(p|^n);
      then reconsider x9=x as Nat;
      x9 divides p|^n by A2,MOEBIUS1:39;
      then ex t be Element of NAT st x9 = p|^t & t<=n by A1,PEPIN:34;
      hence x in {p|^k where k is Element of NAT : k <= n};
    end;
    assume x in {p|^k where k is Element of NAT : k <= n};
    then
A3: ex t be Element of NAT st x=p|^t & t <= n;
    then reconsider x9=x as Nat;
    x9 divides p|^n by A3,NEWTON:89;
    hence x in NatDivisors(p|^n) by A1,A3;
  end;
  hence thesis by TARSKI:2;
end;
