
theorem
  1693 is prime
proof
  now
    1693 = 2*846 + 1; hence not 2 divides 1693 by NAT_4:9;
    1693 = 3*564 + 1; hence not 3 divides 1693 by NAT_4:9;
    1693 = 5*338 + 3; hence not 5 divides 1693 by NAT_4:9;
    1693 = 7*241 + 6; hence not 7 divides 1693 by NAT_4:9;
    1693 = 11*153 + 10; hence not 11 divides 1693 by NAT_4:9;
    1693 = 13*130 + 3; hence not 13 divides 1693 by NAT_4:9;
    1693 = 17*99 + 10; hence not 17 divides 1693 by NAT_4:9;
    1693 = 19*89 + 2; hence not 19 divides 1693 by NAT_4:9;
    1693 = 23*73 + 14; hence not 23 divides 1693 by NAT_4:9;
    1693 = 29*58 + 11; hence not 29 divides 1693 by NAT_4:9;
    1693 = 31*54 + 19; hence not 31 divides 1693 by NAT_4:9;
    1693 = 37*45 + 28; hence not 37 divides 1693 by NAT_4:9;
    1693 = 41*41 + 12; hence not 41 divides 1693 by NAT_4:9;
  end;
  then for n being Element of NAT st 1 < n & n*n <= 1693 & n is prime
  holds not n divides 1693 by XPRIMET1:26;
  hence thesis by NAT_4:14;
end;
