
theorem
  6317 is prime
proof
  now
    6317 = 2*3158 + 1; hence not 2 divides 6317 by NAT_4:9;
    6317 = 3*2105 + 2; hence not 3 divides 6317 by NAT_4:9;
    6317 = 5*1263 + 2; hence not 5 divides 6317 by NAT_4:9;
    6317 = 7*902 + 3; hence not 7 divides 6317 by NAT_4:9;
    6317 = 11*574 + 3; hence not 11 divides 6317 by NAT_4:9;
    6317 = 13*485 + 12; hence not 13 divides 6317 by NAT_4:9;
    6317 = 17*371 + 10; hence not 17 divides 6317 by NAT_4:9;
    6317 = 19*332 + 9; hence not 19 divides 6317 by NAT_4:9;
    6317 = 23*274 + 15; hence not 23 divides 6317 by NAT_4:9;
    6317 = 29*217 + 24; hence not 29 divides 6317 by NAT_4:9;
    6317 = 31*203 + 24; hence not 31 divides 6317 by NAT_4:9;
    6317 = 37*170 + 27; hence not 37 divides 6317 by NAT_4:9;
    6317 = 41*154 + 3; hence not 41 divides 6317 by NAT_4:9;
    6317 = 43*146 + 39; hence not 43 divides 6317 by NAT_4:9;
    6317 = 47*134 + 19; hence not 47 divides 6317 by NAT_4:9;
    6317 = 53*119 + 10; hence not 53 divides 6317 by NAT_4:9;
    6317 = 59*107 + 4; hence not 59 divides 6317 by NAT_4:9;
    6317 = 61*103 + 34; hence not 61 divides 6317 by NAT_4:9;
    6317 = 67*94 + 19; hence not 67 divides 6317 by NAT_4:9;
    6317 = 71*88 + 69; hence not 71 divides 6317 by NAT_4:9;
    6317 = 73*86 + 39; hence not 73 divides 6317 by NAT_4:9;
    6317 = 79*79 + 76; hence not 79 divides 6317 by NAT_4:9;
  end;
  then for n being Element of NAT st 1 < n & n*n <= 6317 & n is prime
  holds not n divides 6317 by XPRIMET1:44;
  hence thesis by NAT_4:14;
end;
