
theorem
  5113 is prime
proof
  now
    5113 = 2*2556 + 1; hence not 2 divides 5113 by NAT_4:9;
    5113 = 3*1704 + 1; hence not 3 divides 5113 by NAT_4:9;
    5113 = 5*1022 + 3; hence not 5 divides 5113 by NAT_4:9;
    5113 = 7*730 + 3; hence not 7 divides 5113 by NAT_4:9;
    5113 = 11*464 + 9; hence not 11 divides 5113 by NAT_4:9;
    5113 = 13*393 + 4; hence not 13 divides 5113 by NAT_4:9;
    5113 = 17*300 + 13; hence not 17 divides 5113 by NAT_4:9;
    5113 = 19*269 + 2; hence not 19 divides 5113 by NAT_4:9;
    5113 = 23*222 + 7; hence not 23 divides 5113 by NAT_4:9;
    5113 = 29*176 + 9; hence not 29 divides 5113 by NAT_4:9;
    5113 = 31*164 + 29; hence not 31 divides 5113 by NAT_4:9;
    5113 = 37*138 + 7; hence not 37 divides 5113 by NAT_4:9;
    5113 = 41*124 + 29; hence not 41 divides 5113 by NAT_4:9;
    5113 = 43*118 + 39; hence not 43 divides 5113 by NAT_4:9;
    5113 = 47*108 + 37; hence not 47 divides 5113 by NAT_4:9;
    5113 = 53*96 + 25; hence not 53 divides 5113 by NAT_4:9;
    5113 = 59*86 + 39; hence not 59 divides 5113 by NAT_4:9;
    5113 = 61*83 + 50; hence not 61 divides 5113 by NAT_4:9;
    5113 = 67*76 + 21; hence not 67 divides 5113 by NAT_4:9;
    5113 = 71*72 + 1; hence not 71 divides 5113 by NAT_4:9;
  end;
  then for n being Element of NAT st 1 < n & n*n <= 5113 & n is prime
  holds not n divides 5113 by XPRIMET1:40;
  hence thesis by NAT_4:14;
end;
