
theorem PrimesNotPowers:
  for p being Prime holds
    p is not perfect_power
  proof
    let p be Prime;
A0: p > 1 by INT_2:def 4;
    assume p is perfect_power; then
    consider x being Nat, k being Nat such that
A1: k > 1 &
    p = x |^ k;
    x |^ 1 = x; then
    per cases by A1,INT_2:def 4,NEWTON:89;
    suppose x = 1;
      hence thesis by A1,INT_2:def 4;
    end;
    suppose x = p; then
      p |^ 1 = p |^ k by A1;
      hence thesis by A1,A0,PEPIN:30;
    end;
  end;
