
theorem
  6689 is prime
proof
  now
    6689 = 2*3344 + 1; hence not 2 divides 6689 by NAT_4:9;
    6689 = 3*2229 + 2; hence not 3 divides 6689 by NAT_4:9;
    6689 = 5*1337 + 4; hence not 5 divides 6689 by NAT_4:9;
    6689 = 7*955 + 4; hence not 7 divides 6689 by NAT_4:9;
    6689 = 11*608 + 1; hence not 11 divides 6689 by NAT_4:9;
    6689 = 13*514 + 7; hence not 13 divides 6689 by NAT_4:9;
    6689 = 17*393 + 8; hence not 17 divides 6689 by NAT_4:9;
    6689 = 19*352 + 1; hence not 19 divides 6689 by NAT_4:9;
    6689 = 23*290 + 19; hence not 23 divides 6689 by NAT_4:9;
    6689 = 29*230 + 19; hence not 29 divides 6689 by NAT_4:9;
    6689 = 31*215 + 24; hence not 31 divides 6689 by NAT_4:9;
    6689 = 37*180 + 29; hence not 37 divides 6689 by NAT_4:9;
    6689 = 41*163 + 6; hence not 41 divides 6689 by NAT_4:9;
    6689 = 43*155 + 24; hence not 43 divides 6689 by NAT_4:9;
    6689 = 47*142 + 15; hence not 47 divides 6689 by NAT_4:9;
    6689 = 53*126 + 11; hence not 53 divides 6689 by NAT_4:9;
    6689 = 59*113 + 22; hence not 59 divides 6689 by NAT_4:9;
    6689 = 61*109 + 40; hence not 61 divides 6689 by NAT_4:9;
    6689 = 67*99 + 56; hence not 67 divides 6689 by NAT_4:9;
    6689 = 71*94 + 15; hence not 71 divides 6689 by NAT_4:9;
    6689 = 73*91 + 46; hence not 73 divides 6689 by NAT_4:9;
    6689 = 79*84 + 53; hence not 79 divides 6689 by NAT_4:9;
  end;
  then for n being Element of NAT st 1 < n & n*n <= 6689 & n is prime
  holds not n divides 6689 by XPRIMET1:44;
  hence thesis by NAT_4:14;
end;
