
theorem
  691 is prime
proof
  now
    691 = 2*345 + 1; hence not 2 divides 691 by NAT_4:9;
    691 = 3*230 + 1; hence not 3 divides 691 by NAT_4:9;
    691 = 5*138 + 1; hence not 5 divides 691 by NAT_4:9;
    691 = 7*98 + 5; hence not 7 divides 691 by NAT_4:9;
    691 = 11*62 + 9; hence not 11 divides 691 by NAT_4:9;
    691 = 13*53 + 2; hence not 13 divides 691 by NAT_4:9;
    691 = 17*40 + 11; hence not 17 divides 691 by NAT_4:9;
    691 = 19*36 + 7; hence not 19 divides 691 by NAT_4:9;
    691 = 23*30 + 1; hence not 23 divides 691 by NAT_4:9;
  end;
  then for n being Element of NAT st 1 < n & n*n <= 691 & n is prime
  holds not n divides 691 by XPRIMET1:18;
  hence thesis by NAT_4:14;
