
theorem
  8123 is prime
proof
  now
    8123 = 2*4061 + 1; hence not 2 divides 8123 by NAT_4:9;
    8123 = 3*2707 + 2; hence not 3 divides 8123 by NAT_4:9;
    8123 = 5*1624 + 3; hence not 5 divides 8123 by NAT_4:9;
    8123 = 7*1160 + 3; hence not 7 divides 8123 by NAT_4:9;
    8123 = 11*738 + 5; hence not 11 divides 8123 by NAT_4:9;
    8123 = 13*624 + 11; hence not 13 divides 8123 by NAT_4:9;
    8123 = 17*477 + 14; hence not 17 divides 8123 by NAT_4:9;
    8123 = 19*427 + 10; hence not 19 divides 8123 by NAT_4:9;
    8123 = 23*353 + 4; hence not 23 divides 8123 by NAT_4:9;
    8123 = 29*280 + 3; hence not 29 divides 8123 by NAT_4:9;
    8123 = 31*262 + 1; hence not 31 divides 8123 by NAT_4:9;
    8123 = 37*219 + 20; hence not 37 divides 8123 by NAT_4:9;
    8123 = 41*198 + 5; hence not 41 divides 8123 by NAT_4:9;
    8123 = 43*188 + 39; hence not 43 divides 8123 by NAT_4:9;
    8123 = 47*172 + 39; hence not 47 divides 8123 by NAT_4:9;
    8123 = 53*153 + 14; hence not 53 divides 8123 by NAT_4:9;
    8123 = 59*137 + 40; hence not 59 divides 8123 by NAT_4:9;
    8123 = 61*133 + 10; hence not 61 divides 8123 by NAT_4:9;
    8123 = 67*121 + 16; hence not 67 divides 8123 by NAT_4:9;
    8123 = 71*114 + 29; hence not 71 divides 8123 by NAT_4:9;
    8123 = 73*111 + 20; hence not 73 divides 8123 by NAT_4:9;
    8123 = 79*102 + 65; hence not 79 divides 8123 by NAT_4:9;
    8123 = 83*97 + 72; hence not 83 divides 8123 by NAT_4:9;
    8123 = 89*91 + 24; hence not 89 divides 8123 by NAT_4:9;
  end;
  then for n being Element of NAT st 1 < n & n*n <= 8123 & n is prime
  holds not n divides 8123 by XPRIMET1:48;
  hence thesis by NAT_4:14;
end;
