
theorem
  6343 is prime
proof
  now
    6343 = 2*3171 + 1; hence not 2 divides 6343 by NAT_4:9;
    6343 = 3*2114 + 1; hence not 3 divides 6343 by NAT_4:9;
    6343 = 5*1268 + 3; hence not 5 divides 6343 by NAT_4:9;
    6343 = 7*906 + 1; hence not 7 divides 6343 by NAT_4:9;
    6343 = 11*576 + 7; hence not 11 divides 6343 by NAT_4:9;
    6343 = 13*487 + 12; hence not 13 divides 6343 by NAT_4:9;
    6343 = 17*373 + 2; hence not 17 divides 6343 by NAT_4:9;
    6343 = 19*333 + 16; hence not 19 divides 6343 by NAT_4:9;
    6343 = 23*275 + 18; hence not 23 divides 6343 by NAT_4:9;
    6343 = 29*218 + 21; hence not 29 divides 6343 by NAT_4:9;
    6343 = 31*204 + 19; hence not 31 divides 6343 by NAT_4:9;
    6343 = 37*171 + 16; hence not 37 divides 6343 by NAT_4:9;
    6343 = 41*154 + 29; hence not 41 divides 6343 by NAT_4:9;
    6343 = 43*147 + 22; hence not 43 divides 6343 by NAT_4:9;
    6343 = 47*134 + 45; hence not 47 divides 6343 by NAT_4:9;
    6343 = 53*119 + 36; hence not 53 divides 6343 by NAT_4:9;
    6343 = 59*107 + 30; hence not 59 divides 6343 by NAT_4:9;
    6343 = 61*103 + 60; hence not 61 divides 6343 by NAT_4:9;
    6343 = 67*94 + 45; hence not 67 divides 6343 by NAT_4:9;
    6343 = 71*89 + 24; hence not 71 divides 6343 by NAT_4:9;
    6343 = 73*86 + 65; hence not 73 divides 6343 by NAT_4:9;
    6343 = 79*80 + 23; hence not 79 divides 6343 by NAT_4:9;
  end;
  then for n being Element of NAT st 1 < n & n*n <= 6343 & n is prime
  holds not n divides 6343 by XPRIMET1:44;
  hence thesis by NAT_4:14;
end;
