
theorem
  6823 is prime
proof
  now
    6823 = 2*3411 + 1; hence not 2 divides 6823 by NAT_4:9;
    6823 = 3*2274 + 1; hence not 3 divides 6823 by NAT_4:9;
    6823 = 5*1364 + 3; hence not 5 divides 6823 by NAT_4:9;
    6823 = 7*974 + 5; hence not 7 divides 6823 by NAT_4:9;
    6823 = 11*620 + 3; hence not 11 divides 6823 by NAT_4:9;
    6823 = 13*524 + 11; hence not 13 divides 6823 by NAT_4:9;
    6823 = 17*401 + 6; hence not 17 divides 6823 by NAT_4:9;
    6823 = 19*359 + 2; hence not 19 divides 6823 by NAT_4:9;
    6823 = 23*296 + 15; hence not 23 divides 6823 by NAT_4:9;
    6823 = 29*235 + 8; hence not 29 divides 6823 by NAT_4:9;
    6823 = 31*220 + 3; hence not 31 divides 6823 by NAT_4:9;
    6823 = 37*184 + 15; hence not 37 divides 6823 by NAT_4:9;
    6823 = 41*166 + 17; hence not 41 divides 6823 by NAT_4:9;
    6823 = 43*158 + 29; hence not 43 divides 6823 by NAT_4:9;
    6823 = 47*145 + 8; hence not 47 divides 6823 by NAT_4:9;
    6823 = 53*128 + 39; hence not 53 divides 6823 by NAT_4:9;
    6823 = 59*115 + 38; hence not 59 divides 6823 by NAT_4:9;
    6823 = 61*111 + 52; hence not 61 divides 6823 by NAT_4:9;
    6823 = 67*101 + 56; hence not 67 divides 6823 by NAT_4:9;
    6823 = 71*96 + 7; hence not 71 divides 6823 by NAT_4:9;
    6823 = 73*93 + 34; hence not 73 divides 6823 by NAT_4:9;
    6823 = 79*86 + 29; hence not 79 divides 6823 by NAT_4:9;
  end;
  then for n being Element of NAT st 1 < n & n*n <= 6823 & n is prime
  holds not n divides 6823 by XPRIMET1:44;
  hence thesis by NAT_4:14;
end;
