
theorem
  8693 is prime
proof
  now
    8693 = 2*4346 + 1; hence not 2 divides 8693 by NAT_4:9;
    8693 = 3*2897 + 2; hence not 3 divides 8693 by NAT_4:9;
    8693 = 5*1738 + 3; hence not 5 divides 8693 by NAT_4:9;
    8693 = 7*1241 + 6; hence not 7 divides 8693 by NAT_4:9;
    8693 = 11*790 + 3; hence not 11 divides 8693 by NAT_4:9;
    8693 = 13*668 + 9; hence not 13 divides 8693 by NAT_4:9;
    8693 = 17*511 + 6; hence not 17 divides 8693 by NAT_4:9;
    8693 = 19*457 + 10; hence not 19 divides 8693 by NAT_4:9;
    8693 = 23*377 + 22; hence not 23 divides 8693 by NAT_4:9;
    8693 = 29*299 + 22; hence not 29 divides 8693 by NAT_4:9;
    8693 = 31*280 + 13; hence not 31 divides 8693 by NAT_4:9;
    8693 = 37*234 + 35; hence not 37 divides 8693 by NAT_4:9;
    8693 = 41*212 + 1; hence not 41 divides 8693 by NAT_4:9;
    8693 = 43*202 + 7; hence not 43 divides 8693 by NAT_4:9;
    8693 = 47*184 + 45; hence not 47 divides 8693 by NAT_4:9;
    8693 = 53*164 + 1; hence not 53 divides 8693 by NAT_4:9;
    8693 = 59*147 + 20; hence not 59 divides 8693 by NAT_4:9;
    8693 = 61*142 + 31; hence not 61 divides 8693 by NAT_4:9;
    8693 = 67*129 + 50; hence not 67 divides 8693 by NAT_4:9;
    8693 = 71*122 + 31; hence not 71 divides 8693 by NAT_4:9;
    8693 = 73*119 + 6; hence not 73 divides 8693 by NAT_4:9;
    8693 = 79*110 + 3; hence not 79 divides 8693 by NAT_4:9;
    8693 = 83*104 + 61; hence not 83 divides 8693 by NAT_4:9;
    8693 = 89*97 + 60; hence not 89 divides 8693 by NAT_4:9;
  end;
  then for n being Element of NAT st 1 < n & n*n <= 8693 & n is prime
  holds not n divides 8693 by XPRIMET1:48;
  hence thesis by NAT_4:14;
end;
