
theorem
  4349 is prime
proof
  now
    4349 = 2*2174 + 1; hence not 2 divides 4349 by NAT_4:9;
    4349 = 3*1449 + 2; hence not 3 divides 4349 by NAT_4:9;
    4349 = 5*869 + 4; hence not 5 divides 4349 by NAT_4:9;
    4349 = 7*621 + 2; hence not 7 divides 4349 by NAT_4:9;
    4349 = 11*395 + 4; hence not 11 divides 4349 by NAT_4:9;
    4349 = 13*334 + 7; hence not 13 divides 4349 by NAT_4:9;
    4349 = 17*255 + 14; hence not 17 divides 4349 by NAT_4:9;
    4349 = 19*228 + 17; hence not 19 divides 4349 by NAT_4:9;
    4349 = 23*189 + 2; hence not 23 divides 4349 by NAT_4:9;
    4349 = 29*149 + 28; hence not 29 divides 4349 by NAT_4:9;
    4349 = 31*140 + 9; hence not 31 divides 4349 by NAT_4:9;
    4349 = 37*117 + 20; hence not 37 divides 4349 by NAT_4:9;
    4349 = 41*106 + 3; hence not 41 divides 4349 by NAT_4:9;
    4349 = 43*101 + 6; hence not 43 divides 4349 by NAT_4:9;
    4349 = 47*92 + 25; hence not 47 divides 4349 by NAT_4:9;
    4349 = 53*82 + 3; hence not 53 divides 4349 by NAT_4:9;
    4349 = 59*73 + 42; hence not 59 divides 4349 by NAT_4:9;
    4349 = 61*71 + 18; hence not 61 divides 4349 by NAT_4:9;
  end;
  then for n being Element of NAT st 1 < n & n*n <= 4349 & n is prime
  holds not n divides 4349 by XPRIMET1:36;
  hence thesis by NAT_4:14;
end;
