
theorem
  8167 is prime
proof
  now
    8167 = 2*4083 + 1; hence not 2 divides 8167 by NAT_4:9;
    8167 = 3*2722 + 1; hence not 3 divides 8167 by NAT_4:9;
    8167 = 5*1633 + 2; hence not 5 divides 8167 by NAT_4:9;
    8167 = 7*1166 + 5; hence not 7 divides 8167 by NAT_4:9;
    8167 = 11*742 + 5; hence not 11 divides 8167 by NAT_4:9;
    8167 = 13*628 + 3; hence not 13 divides 8167 by NAT_4:9;
    8167 = 17*480 + 7; hence not 17 divides 8167 by NAT_4:9;
    8167 = 19*429 + 16; hence not 19 divides 8167 by NAT_4:9;
    8167 = 23*355 + 2; hence not 23 divides 8167 by NAT_4:9;
    8167 = 29*281 + 18; hence not 29 divides 8167 by NAT_4:9;
    8167 = 31*263 + 14; hence not 31 divides 8167 by NAT_4:9;
    8167 = 37*220 + 27; hence not 37 divides 8167 by NAT_4:9;
    8167 = 41*199 + 8; hence not 41 divides 8167 by NAT_4:9;
    8167 = 43*189 + 40; hence not 43 divides 8167 by NAT_4:9;
    8167 = 47*173 + 36; hence not 47 divides 8167 by NAT_4:9;
    8167 = 53*154 + 5; hence not 53 divides 8167 by NAT_4:9;
    8167 = 59*138 + 25; hence not 59 divides 8167 by NAT_4:9;
    8167 = 61*133 + 54; hence not 61 divides 8167 by NAT_4:9;
    8167 = 67*121 + 60; hence not 67 divides 8167 by NAT_4:9;
    8167 = 71*115 + 2; hence not 71 divides 8167 by NAT_4:9;
    8167 = 73*111 + 64; hence not 73 divides 8167 by NAT_4:9;
    8167 = 79*103 + 30; hence not 79 divides 8167 by NAT_4:9;
    8167 = 83*98 + 33; hence not 83 divides 8167 by NAT_4:9;
    8167 = 89*91 + 68; hence not 89 divides 8167 by NAT_4:9;
  end;
  then for n being Element of NAT st 1 < n & n*n <= 8167 & n is prime
  holds not n divides 8167 by XPRIMET1:48;
  hence thesis by NAT_4:14;
end;
