
theorem
  6829 is prime
proof
  now
    6829 = 2*3414 + 1; hence not 2 divides 6829 by NAT_4:9;
    6829 = 3*2276 + 1; hence not 3 divides 6829 by NAT_4:9;
    6829 = 5*1365 + 4; hence not 5 divides 6829 by NAT_4:9;
    6829 = 7*975 + 4; hence not 7 divides 6829 by NAT_4:9;
    6829 = 11*620 + 9; hence not 11 divides 6829 by NAT_4:9;
    6829 = 13*525 + 4; hence not 13 divides 6829 by NAT_4:9;
    6829 = 17*401 + 12; hence not 17 divides 6829 by NAT_4:9;
    6829 = 19*359 + 8; hence not 19 divides 6829 by NAT_4:9;
    6829 = 23*296 + 21; hence not 23 divides 6829 by NAT_4:9;
    6829 = 29*235 + 14; hence not 29 divides 6829 by NAT_4:9;
    6829 = 31*220 + 9; hence not 31 divides 6829 by NAT_4:9;
    6829 = 37*184 + 21; hence not 37 divides 6829 by NAT_4:9;
    6829 = 41*166 + 23; hence not 41 divides 6829 by NAT_4:9;
    6829 = 43*158 + 35; hence not 43 divides 6829 by NAT_4:9;
    6829 = 47*145 + 14; hence not 47 divides 6829 by NAT_4:9;
    6829 = 53*128 + 45; hence not 53 divides 6829 by NAT_4:9;
    6829 = 59*115 + 44; hence not 59 divides 6829 by NAT_4:9;
    6829 = 61*111 + 58; hence not 61 divides 6829 by NAT_4:9;
    6829 = 67*101 + 62; hence not 67 divides 6829 by NAT_4:9;
    6829 = 71*96 + 13; hence not 71 divides 6829 by NAT_4:9;
    6829 = 73*93 + 40; hence not 73 divides 6829 by NAT_4:9;
    6829 = 79*86 + 35; hence not 79 divides 6829 by NAT_4:9;
  end;
  then for n being Element of NAT st 1 < n & n*n <= 6829 & n is prime
  holds not n divides 6829 by XPRIMET1:44;
  hence thesis by NAT_4:14;
end;
