
theorem
  6869 is prime
proof
  now
    6869 = 2*3434 + 1; hence not 2 divides 6869 by NAT_4:9;
    6869 = 3*2289 + 2; hence not 3 divides 6869 by NAT_4:9;
    6869 = 5*1373 + 4; hence not 5 divides 6869 by NAT_4:9;
    6869 = 7*981 + 2; hence not 7 divides 6869 by NAT_4:9;
    6869 = 11*624 + 5; hence not 11 divides 6869 by NAT_4:9;
    6869 = 13*528 + 5; hence not 13 divides 6869 by NAT_4:9;
    6869 = 17*404 + 1; hence not 17 divides 6869 by NAT_4:9;
    6869 = 19*361 + 10; hence not 19 divides 6869 by NAT_4:9;
    6869 = 23*298 + 15; hence not 23 divides 6869 by NAT_4:9;
    6869 = 29*236 + 25; hence not 29 divides 6869 by NAT_4:9;
    6869 = 31*221 + 18; hence not 31 divides 6869 by NAT_4:9;
    6869 = 37*185 + 24; hence not 37 divides 6869 by NAT_4:9;
    6869 = 41*167 + 22; hence not 41 divides 6869 by NAT_4:9;
    6869 = 43*159 + 32; hence not 43 divides 6869 by NAT_4:9;
    6869 = 47*146 + 7; hence not 47 divides 6869 by NAT_4:9;
    6869 = 53*129 + 32; hence not 53 divides 6869 by NAT_4:9;
    6869 = 59*116 + 25; hence not 59 divides 6869 by NAT_4:9;
    6869 = 61*112 + 37; hence not 61 divides 6869 by NAT_4:9;
    6869 = 67*102 + 35; hence not 67 divides 6869 by NAT_4:9;
    6869 = 71*96 + 53; hence not 71 divides 6869 by NAT_4:9;
    6869 = 73*94 + 7; hence not 73 divides 6869 by NAT_4:9;
    6869 = 79*86 + 75; hence not 79 divides 6869 by NAT_4:9;
  end;
  then for n being Element of NAT st 1 < n & n*n <= 6869 & n is prime
  holds not n divides 6869 by XPRIMET1:44;
  hence thesis by NAT_4:14;
end;
