
theorem
  for n being Nat, a being Nat st n >= 1 holds
    PrimeDivisors (a |^ n) = PrimeDivisors a
  proof
    let n be Nat, a be Nat;
    assume
A0: n >= 1;
    thus PrimeDivisors (a |^ n) c= PrimeDivisors a
    proof
      let x be object;
      assume x in PrimeDivisors (a |^ n); then
      consider k being Prime such that
S1:   k = x & k divides a |^ n;
      k divides a by S1,NAT_3:5;
      hence thesis by S1;
    end;
    let x be object;
    assume x in PrimeDivisors a; then
    consider k being Prime such that
S1: k = x & k divides a;
    a = a |^ 1; then
    a divides a |^ n by A0,NEWTON:89; then
    k divides a |^ n by S1,NAT_D:4;
    hence thesis by S1;
  end;
