
theorem
  2689 is prime
proof
  now
    2689 = 2*1344 + 1; hence not 2 divides 2689 by NAT_4:9;
    2689 = 3*896 + 1; hence not 3 divides 2689 by NAT_4:9;
    2689 = 5*537 + 4; hence not 5 divides 2689 by NAT_4:9;
    2689 = 7*384 + 1; hence not 7 divides 2689 by NAT_4:9;
    2689 = 11*244 + 5; hence not 11 divides 2689 by NAT_4:9;
    2689 = 13*206 + 11; hence not 13 divides 2689 by NAT_4:9;
    2689 = 17*158 + 3; hence not 17 divides 2689 by NAT_4:9;
    2689 = 19*141 + 10; hence not 19 divides 2689 by NAT_4:9;
    2689 = 23*116 + 21; hence not 23 divides 2689 by NAT_4:9;
    2689 = 29*92 + 21; hence not 29 divides 2689 by NAT_4:9;
    2689 = 31*86 + 23; hence not 31 divides 2689 by NAT_4:9;
    2689 = 37*72 + 25; hence not 37 divides 2689 by NAT_4:9;
    2689 = 41*65 + 24; hence not 41 divides 2689 by NAT_4:9;
    2689 = 43*62 + 23; hence not 43 divides 2689 by NAT_4:9;
    2689 = 47*57 + 10; hence not 47 divides 2689 by NAT_4:9;
  end;
  then for n being Element of NAT st 1 < n & n*n <= 2689 & n is prime
  holds not n divides 2689 by XPRIMET1:30;
  hence thesis by NAT_4:14;
end;
