
theorem
  569 is prime
proof
  now
    569 = 2*284 + 1; hence not 2 divides 569 by NAT_4:9;
    569 = 3*189 + 2; hence not 3 divides 569 by NAT_4:9;
    569 = 5*113 + 4; hence not 5 divides 569 by NAT_4:9;
    569 = 7*81 + 2; hence not 7 divides 569 by NAT_4:9;
    569 = 11*51 + 8; hence not 11 divides 569 by NAT_4:9;
    569 = 13*43 + 10; hence not 13 divides 569 by NAT_4:9;
    569 = 17*33 + 8; hence not 17 divides 569 by NAT_4:9;
    569 = 19*29 + 18; hence not 19 divides 569 by NAT_4:9;
    569 = 23*24 + 17; hence not 23 divides 569 by NAT_4:9;
  end;
  then for n being Element of NAT st 1 < n & n*n <= 569 & n is prime
  holds not n divides 569 by XPRIMET1:18;
  hence thesis by NAT_4:14;
