
theorem
  3181 is prime
proof
  now
    3181 = 2*1590 + 1; hence not 2 divides 3181 by NAT_4:9;
    3181 = 3*1060 + 1; hence not 3 divides 3181 by NAT_4:9;
    3181 = 5*636 + 1; hence not 5 divides 3181 by NAT_4:9;
    3181 = 7*454 + 3; hence not 7 divides 3181 by NAT_4:9;
    3181 = 11*289 + 2; hence not 11 divides 3181 by NAT_4:9;
    3181 = 13*244 + 9; hence not 13 divides 3181 by NAT_4:9;
    3181 = 17*187 + 2; hence not 17 divides 3181 by NAT_4:9;
    3181 = 19*167 + 8; hence not 19 divides 3181 by NAT_4:9;
    3181 = 23*138 + 7; hence not 23 divides 3181 by NAT_4:9;
    3181 = 29*109 + 20; hence not 29 divides 3181 by NAT_4:9;
    3181 = 31*102 + 19; hence not 31 divides 3181 by NAT_4:9;
    3181 = 37*85 + 36; hence not 37 divides 3181 by NAT_4:9;
    3181 = 41*77 + 24; hence not 41 divides 3181 by NAT_4:9;
    3181 = 43*73 + 42; hence not 43 divides 3181 by NAT_4:9;
    3181 = 47*67 + 32; hence not 47 divides 3181 by NAT_4:9;
    3181 = 53*60 + 1; hence not 53 divides 3181 by NAT_4:9;
  end;
  then for n being Element of NAT st 1 < n & n*n <= 3181 & n is prime
  holds not n divides 3181 by XPRIMET1:32;
  hence thesis by NAT_4:14;
end;
