
theorem
  7669 is prime
proof
  now
    7669 = 2*3834 + 1; hence not 2 divides 7669 by NAT_4:9;
    7669 = 3*2556 + 1; hence not 3 divides 7669 by NAT_4:9;
    7669 = 5*1533 + 4; hence not 5 divides 7669 by NAT_4:9;
    7669 = 7*1095 + 4; hence not 7 divides 7669 by NAT_4:9;
    7669 = 11*697 + 2; hence not 11 divides 7669 by NAT_4:9;
    7669 = 13*589 + 12; hence not 13 divides 7669 by NAT_4:9;
    7669 = 17*451 + 2; hence not 17 divides 7669 by NAT_4:9;
    7669 = 19*403 + 12; hence not 19 divides 7669 by NAT_4:9;
    7669 = 23*333 + 10; hence not 23 divides 7669 by NAT_4:9;
    7669 = 29*264 + 13; hence not 29 divides 7669 by NAT_4:9;
    7669 = 31*247 + 12; hence not 31 divides 7669 by NAT_4:9;
    7669 = 37*207 + 10; hence not 37 divides 7669 by NAT_4:9;
    7669 = 41*187 + 2; hence not 41 divides 7669 by NAT_4:9;
    7669 = 43*178 + 15; hence not 43 divides 7669 by NAT_4:9;
    7669 = 47*163 + 8; hence not 47 divides 7669 by NAT_4:9;
    7669 = 53*144 + 37; hence not 53 divides 7669 by NAT_4:9;
    7669 = 59*129 + 58; hence not 59 divides 7669 by NAT_4:9;
    7669 = 61*125 + 44; hence not 61 divides 7669 by NAT_4:9;
    7669 = 67*114 + 31; hence not 67 divides 7669 by NAT_4:9;
    7669 = 71*108 + 1; hence not 71 divides 7669 by NAT_4:9;
    7669 = 73*105 + 4; hence not 73 divides 7669 by NAT_4:9;
    7669 = 79*97 + 6; hence not 79 divides 7669 by NAT_4:9;
    7669 = 83*92 + 33; hence not 83 divides 7669 by NAT_4:9;
  end;
  then for n being Element of NAT st 1 < n & n*n <= 7669 & n is prime
  holds not n divides 7669 by XPRIMET1:46;
  hence thesis by NAT_4:14;
end;
