
theorem
  3191 is prime
proof
  now
    3191 = 2*1595 + 1; hence not 2 divides 3191 by NAT_4:9;
    3191 = 3*1063 + 2; hence not 3 divides 3191 by NAT_4:9;
    3191 = 5*638 + 1; hence not 5 divides 3191 by NAT_4:9;
    3191 = 7*455 + 6; hence not 7 divides 3191 by NAT_4:9;
    3191 = 11*290 + 1; hence not 11 divides 3191 by NAT_4:9;
    3191 = 13*245 + 6; hence not 13 divides 3191 by NAT_4:9;
    3191 = 17*187 + 12; hence not 17 divides 3191 by NAT_4:9;
    3191 = 19*167 + 18; hence not 19 divides 3191 by NAT_4:9;
    3191 = 23*138 + 17; hence not 23 divides 3191 by NAT_4:9;
    3191 = 29*110 + 1; hence not 29 divides 3191 by NAT_4:9;
    3191 = 31*102 + 29; hence not 31 divides 3191 by NAT_4:9;
    3191 = 37*86 + 9; hence not 37 divides 3191 by NAT_4:9;
    3191 = 41*77 + 34; hence not 41 divides 3191 by NAT_4:9;
    3191 = 43*74 + 9; hence not 43 divides 3191 by NAT_4:9;
    3191 = 47*67 + 42; hence not 47 divides 3191 by NAT_4:9;
    3191 = 53*60 + 11; hence not 53 divides 3191 by NAT_4:9;
  end;
  then for n being Element of NAT st 1 < n & n*n <= 3191 & n is prime
  holds not n divides 3191 by XPRIMET1:32;
  hence thesis by NAT_4:14;
end;
