
theorem
  8111 is prime
proof
  now
    8111 = 2*4055 + 1; hence not 2 divides 8111 by NAT_4:9;
    8111 = 3*2703 + 2; hence not 3 divides 8111 by NAT_4:9;
    8111 = 5*1622 + 1; hence not 5 divides 8111 by NAT_4:9;
    8111 = 7*1158 + 5; hence not 7 divides 8111 by NAT_4:9;
    8111 = 11*737 + 4; hence not 11 divides 8111 by NAT_4:9;
    8111 = 13*623 + 12; hence not 13 divides 8111 by NAT_4:9;
    8111 = 17*477 + 2; hence not 17 divides 8111 by NAT_4:9;
    8111 = 19*426 + 17; hence not 19 divides 8111 by NAT_4:9;
    8111 = 23*352 + 15; hence not 23 divides 8111 by NAT_4:9;
    8111 = 29*279 + 20; hence not 29 divides 8111 by NAT_4:9;
    8111 = 31*261 + 20; hence not 31 divides 8111 by NAT_4:9;
    8111 = 37*219 + 8; hence not 37 divides 8111 by NAT_4:9;
    8111 = 41*197 + 34; hence not 41 divides 8111 by NAT_4:9;
    8111 = 43*188 + 27; hence not 43 divides 8111 by NAT_4:9;
    8111 = 47*172 + 27; hence not 47 divides 8111 by NAT_4:9;
    8111 = 53*153 + 2; hence not 53 divides 8111 by NAT_4:9;
    8111 = 59*137 + 28; hence not 59 divides 8111 by NAT_4:9;
    8111 = 61*132 + 59; hence not 61 divides 8111 by NAT_4:9;
    8111 = 67*121 + 4; hence not 67 divides 8111 by NAT_4:9;
    8111 = 71*114 + 17; hence not 71 divides 8111 by NAT_4:9;
    8111 = 73*111 + 8; hence not 73 divides 8111 by NAT_4:9;
    8111 = 79*102 + 53; hence not 79 divides 8111 by NAT_4:9;
    8111 = 83*97 + 60; hence not 83 divides 8111 by NAT_4:9;
    8111 = 89*91 + 12; hence not 89 divides 8111 by NAT_4:9;
  end;
  then for n being Element of NAT st 1 < n & n*n <= 8111 & n is prime
  holds not n divides 8111 by XPRIMET1:48;
  hence thesis by NAT_4:14;
end;
