
theorem
  829 is prime
proof
  now
    829 = 2*414 + 1; hence not 2 divides 829 by NAT_4:9;
    829 = 3*276 + 1; hence not 3 divides 829 by NAT_4:9;
    829 = 5*165 + 4; hence not 5 divides 829 by NAT_4:9;
    829 = 7*118 + 3; hence not 7 divides 829 by NAT_4:9;
    829 = 11*75 + 4; hence not 11 divides 829 by NAT_4:9;
    829 = 13*63 + 10; hence not 13 divides 829 by NAT_4:9;
    829 = 17*48 + 13; hence not 17 divides 829 by NAT_4:9;
    829 = 19*43 + 12; hence not 19 divides 829 by NAT_4:9;
    829 = 23*36 + 1; hence not 23 divides 829 by NAT_4:9;
  end;
  then for n being Element of NAT st 1 < n & n*n <= 829 & n is prime
  holds not n divides 829 by XPRIMET1:18;
  hence thesis by NAT_4:14;
