
theorem
  5689 is prime
proof
  now
    5689 = 2*2844 + 1; hence not 2 divides 5689 by NAT_4:9;
    5689 = 3*1896 + 1; hence not 3 divides 5689 by NAT_4:9;
    5689 = 5*1137 + 4; hence not 5 divides 5689 by NAT_4:9;
    5689 = 7*812 + 5; hence not 7 divides 5689 by NAT_4:9;
    5689 = 11*517 + 2; hence not 11 divides 5689 by NAT_4:9;
    5689 = 13*437 + 8; hence not 13 divides 5689 by NAT_4:9;
    5689 = 17*334 + 11; hence not 17 divides 5689 by NAT_4:9;
    5689 = 19*299 + 8; hence not 19 divides 5689 by NAT_4:9;
    5689 = 23*247 + 8; hence not 23 divides 5689 by NAT_4:9;
    5689 = 29*196 + 5; hence not 29 divides 5689 by NAT_4:9;
    5689 = 31*183 + 16; hence not 31 divides 5689 by NAT_4:9;
    5689 = 37*153 + 28; hence not 37 divides 5689 by NAT_4:9;
    5689 = 41*138 + 31; hence not 41 divides 5689 by NAT_4:9;
    5689 = 43*132 + 13; hence not 43 divides 5689 by NAT_4:9;
    5689 = 47*121 + 2; hence not 47 divides 5689 by NAT_4:9;
    5689 = 53*107 + 18; hence not 53 divides 5689 by NAT_4:9;
    5689 = 59*96 + 25; hence not 59 divides 5689 by NAT_4:9;
    5689 = 61*93 + 16; hence not 61 divides 5689 by NAT_4:9;
    5689 = 67*84 + 61; hence not 67 divides 5689 by NAT_4:9;
    5689 = 71*80 + 9; hence not 71 divides 5689 by NAT_4:9;
    5689 = 73*77 + 68; hence not 73 divides 5689 by NAT_4:9;
  end;
  then for n being Element of NAT st 1 < n & n*n <= 5689 & n is prime
  holds not n divides 5689 by XPRIMET1:42;
  hence thesis by NAT_4:14;
end;
