
theorem
  5167 is prime
proof
  now
    5167 = 2*2583 + 1; hence not 2 divides 5167 by NAT_4:9;
    5167 = 3*1722 + 1; hence not 3 divides 5167 by NAT_4:9;
    5167 = 5*1033 + 2; hence not 5 divides 5167 by NAT_4:9;
    5167 = 7*738 + 1; hence not 7 divides 5167 by NAT_4:9;
    5167 = 11*469 + 8; hence not 11 divides 5167 by NAT_4:9;
    5167 = 13*397 + 6; hence not 13 divides 5167 by NAT_4:9;
    5167 = 17*303 + 16; hence not 17 divides 5167 by NAT_4:9;
    5167 = 19*271 + 18; hence not 19 divides 5167 by NAT_4:9;
    5167 = 23*224 + 15; hence not 23 divides 5167 by NAT_4:9;
    5167 = 29*178 + 5; hence not 29 divides 5167 by NAT_4:9;
    5167 = 31*166 + 21; hence not 31 divides 5167 by NAT_4:9;
    5167 = 37*139 + 24; hence not 37 divides 5167 by NAT_4:9;
    5167 = 41*126 + 1; hence not 41 divides 5167 by NAT_4:9;
    5167 = 43*120 + 7; hence not 43 divides 5167 by NAT_4:9;
    5167 = 47*109 + 44; hence not 47 divides 5167 by NAT_4:9;
    5167 = 53*97 + 26; hence not 53 divides 5167 by NAT_4:9;
    5167 = 59*87 + 34; hence not 59 divides 5167 by NAT_4:9;
    5167 = 61*84 + 43; hence not 61 divides 5167 by NAT_4:9;
    5167 = 67*77 + 8; hence not 67 divides 5167 by NAT_4:9;
    5167 = 71*72 + 55; hence not 71 divides 5167 by NAT_4:9;
  end;
  then for n being Element of NAT st 1 < n & n*n <= 5167 & n is prime
  holds not n divides 5167 by XPRIMET1:40;
  hence thesis by NAT_4:14;
end;
