
theorem
  5309 is prime
proof
  now
    5309 = 2*2654 + 1; hence not 2 divides 5309 by NAT_4:9;
    5309 = 3*1769 + 2; hence not 3 divides 5309 by NAT_4:9;
    5309 = 5*1061 + 4; hence not 5 divides 5309 by NAT_4:9;
    5309 = 7*758 + 3; hence not 7 divides 5309 by NAT_4:9;
    5309 = 11*482 + 7; hence not 11 divides 5309 by NAT_4:9;
    5309 = 13*408 + 5; hence not 13 divides 5309 by NAT_4:9;
    5309 = 17*312 + 5; hence not 17 divides 5309 by NAT_4:9;
    5309 = 19*279 + 8; hence not 19 divides 5309 by NAT_4:9;
    5309 = 23*230 + 19; hence not 23 divides 5309 by NAT_4:9;
    5309 = 29*183 + 2; hence not 29 divides 5309 by NAT_4:9;
    5309 = 31*171 + 8; hence not 31 divides 5309 by NAT_4:9;
    5309 = 37*143 + 18; hence not 37 divides 5309 by NAT_4:9;
    5309 = 41*129 + 20; hence not 41 divides 5309 by NAT_4:9;
    5309 = 43*123 + 20; hence not 43 divides 5309 by NAT_4:9;
    5309 = 47*112 + 45; hence not 47 divides 5309 by NAT_4:9;
    5309 = 53*100 + 9; hence not 53 divides 5309 by NAT_4:9;
    5309 = 59*89 + 58; hence not 59 divides 5309 by NAT_4:9;
    5309 = 61*87 + 2; hence not 61 divides 5309 by NAT_4:9;
    5309 = 67*79 + 16; hence not 67 divides 5309 by NAT_4:9;
    5309 = 71*74 + 55; hence not 71 divides 5309 by NAT_4:9;
  end;
  then for n being Element of NAT st 1 < n & n*n <= 5309 & n is prime
  holds not n divides 5309 by XPRIMET1:40;
  hence thesis by NAT_4:14;
end;
