
theorem
  8191 is prime
proof
  now
    8191 = 2*4095 + 1; hence not 2 divides 8191 by NAT_4:9;
    8191 = 3*2730 + 1; hence not 3 divides 8191 by NAT_4:9;
    8191 = 5*1638 + 1; hence not 5 divides 8191 by NAT_4:9;
    8191 = 7*1170 + 1; hence not 7 divides 8191 by NAT_4:9;
    8191 = 11*744 + 7; hence not 11 divides 8191 by NAT_4:9;
    8191 = 13*630 + 1; hence not 13 divides 8191 by NAT_4:9;
    8191 = 17*481 + 14; hence not 17 divides 8191 by NAT_4:9;
    8191 = 19*431 + 2; hence not 19 divides 8191 by NAT_4:9;
    8191 = 23*356 + 3; hence not 23 divides 8191 by NAT_4:9;
    8191 = 29*282 + 13; hence not 29 divides 8191 by NAT_4:9;
    8191 = 31*264 + 7; hence not 31 divides 8191 by NAT_4:9;
    8191 = 37*221 + 14; hence not 37 divides 8191 by NAT_4:9;
    8191 = 41*199 + 32; hence not 41 divides 8191 by NAT_4:9;
    8191 = 43*190 + 21; hence not 43 divides 8191 by NAT_4:9;
    8191 = 47*174 + 13; hence not 47 divides 8191 by NAT_4:9;
    8191 = 53*154 + 29; hence not 53 divides 8191 by NAT_4:9;
    8191 = 59*138 + 49; hence not 59 divides 8191 by NAT_4:9;
    8191 = 61*134 + 17; hence not 61 divides 8191 by NAT_4:9;
    8191 = 67*122 + 17; hence not 67 divides 8191 by NAT_4:9;
    8191 = 71*115 + 26; hence not 71 divides 8191 by NAT_4:9;
    8191 = 73*112 + 15; hence not 73 divides 8191 by NAT_4:9;
    8191 = 79*103 + 54; hence not 79 divides 8191 by NAT_4:9;
    8191 = 83*98 + 57; hence not 83 divides 8191 by NAT_4:9;
    8191 = 89*92 + 3; hence not 89 divides 8191 by NAT_4:9;
  end;
  then for n being Element of NAT st 1 < n & n*n <= 8191 & n is prime
  holds not n divides 8191 by XPRIMET1:48;
  hence thesis by NAT_4:14;
end;
