
theorem
  9689 is prime
proof
  now
    9689 = 2*4844 + 1; hence not 2 divides 9689 by NAT_4:9;
    9689 = 3*3229 + 2; hence not 3 divides 9689 by NAT_4:9;
    9689 = 5*1937 + 4; hence not 5 divides 9689 by NAT_4:9;
    9689 = 7*1384 + 1; hence not 7 divides 9689 by NAT_4:9;
    9689 = 11*880 + 9; hence not 11 divides 9689 by NAT_4:9;
    9689 = 13*745 + 4; hence not 13 divides 9689 by NAT_4:9;
    9689 = 17*569 + 16; hence not 17 divides 9689 by NAT_4:9;
    9689 = 19*509 + 18; hence not 19 divides 9689 by NAT_4:9;
    9689 = 23*421 + 6; hence not 23 divides 9689 by NAT_4:9;
    9689 = 29*334 + 3; hence not 29 divides 9689 by NAT_4:9;
    9689 = 31*312 + 17; hence not 31 divides 9689 by NAT_4:9;
    9689 = 37*261 + 32; hence not 37 divides 9689 by NAT_4:9;
    9689 = 41*236 + 13; hence not 41 divides 9689 by NAT_4:9;
    9689 = 43*225 + 14; hence not 43 divides 9689 by NAT_4:9;
    9689 = 47*206 + 7; hence not 47 divides 9689 by NAT_4:9;
    9689 = 53*182 + 43; hence not 53 divides 9689 by NAT_4:9;
    9689 = 59*164 + 13; hence not 59 divides 9689 by NAT_4:9;
    9689 = 61*158 + 51; hence not 61 divides 9689 by NAT_4:9;
    9689 = 67*144 + 41; hence not 67 divides 9689 by NAT_4:9;
    9689 = 71*136 + 33; hence not 71 divides 9689 by NAT_4:9;
    9689 = 73*132 + 53; hence not 73 divides 9689 by NAT_4:9;
    9689 = 79*122 + 51; hence not 79 divides 9689 by NAT_4:9;
    9689 = 83*116 + 61; hence not 83 divides 9689 by NAT_4:9;
    9689 = 89*108 + 77; hence not 89 divides 9689 by NAT_4:9;
    9689 = 97*99 + 86; hence not 97 divides 9689 by NAT_4:9;
  end;
  then for n being Element of NAT st 1 < n & n*n <= 9689 & n is prime
  holds not n divides 9689 by XPRIMET1:50;
  hence thesis by NAT_4:14;
end;
