
theorem
  1307 is prime
proof
  now
    1307 = 2*653 + 1; hence not 2 divides 1307 by NAT_4:9;
    1307 = 3*435 + 2; hence not 3 divides 1307 by NAT_4:9;
    1307 = 5*261 + 2; hence not 5 divides 1307 by NAT_4:9;
    1307 = 7*186 + 5; hence not 7 divides 1307 by NAT_4:9;
    1307 = 11*118 + 9; hence not 11 divides 1307 by NAT_4:9;
    1307 = 13*100 + 7; hence not 13 divides 1307 by NAT_4:9;
    1307 = 17*76 + 15; hence not 17 divides 1307 by NAT_4:9;
    1307 = 19*68 + 15; hence not 19 divides 1307 by NAT_4:9;
    1307 = 23*56 + 19; hence not 23 divides 1307 by NAT_4:9;
    1307 = 29*45 + 2; hence not 29 divides 1307 by NAT_4:9;
    1307 = 31*42 + 5; hence not 31 divides 1307 by NAT_4:9;
  end;
  then for n being Element of NAT st 1 < n & n*n <= 1307 & n is prime
  holds not n divides 1307 by XPRIMET1:22;
  hence thesis by NAT_4:14;
