
theorem
  4013 is prime
proof
  now
    4013 = 2*2006 + 1; hence not 2 divides 4013 by NAT_4:9;
    4013 = 3*1337 + 2; hence not 3 divides 4013 by NAT_4:9;
    4013 = 5*802 + 3; hence not 5 divides 4013 by NAT_4:9;
    4013 = 7*573 + 2; hence not 7 divides 4013 by NAT_4:9;
    4013 = 11*364 + 9; hence not 11 divides 4013 by NAT_4:9;
    4013 = 13*308 + 9; hence not 13 divides 4013 by NAT_4:9;
    4013 = 17*236 + 1; hence not 17 divides 4013 by NAT_4:9;
    4013 = 19*211 + 4; hence not 19 divides 4013 by NAT_4:9;
    4013 = 23*174 + 11; hence not 23 divides 4013 by NAT_4:9;
    4013 = 29*138 + 11; hence not 29 divides 4013 by NAT_4:9;
    4013 = 31*129 + 14; hence not 31 divides 4013 by NAT_4:9;
    4013 = 37*108 + 17; hence not 37 divides 4013 by NAT_4:9;
    4013 = 41*97 + 36; hence not 41 divides 4013 by NAT_4:9;
    4013 = 43*93 + 14; hence not 43 divides 4013 by NAT_4:9;
    4013 = 47*85 + 18; hence not 47 divides 4013 by NAT_4:9;
    4013 = 53*75 + 38; hence not 53 divides 4013 by NAT_4:9;
    4013 = 59*68 + 1; hence not 59 divides 4013 by NAT_4:9;
    4013 = 61*65 + 48; hence not 61 divides 4013 by NAT_4:9;
  end;
  then for n being Element of NAT st 1 < n & n*n <= 4013 & n is prime
  holds not n divides 4013 by XPRIMET1:36;
  hence thesis by NAT_4:14;
end;
