
theorem KeyPerfectPower:
  for n being Nat, p being Prime st n is perfect_power & p divides n
    holds
      p |^ 2 divides n
  proof
    let n be Nat, p be Prime;
    assume
A0: n is perfect_power & p divides n; then
    consider x being Nat, k being Nat such that
A1: k > 1 & n = x |^ k by NUMBER06:def 10;
A2: 1 + 1 <= k by A1,NAT_1:13;
A3: p |^ k divides x |^ k by NEWTON03:15,A1,A0,NAT_3:5;
    p |^ 2 divides p |^ k by A2,NEWTON:89;
    hence thesis by A1,A3,NAT_D:4;
  end;
