
theorem
  9629 is prime
proof
  now
    9629 = 2*4814 + 1; hence not 2 divides 9629 by NAT_4:9;
    9629 = 3*3209 + 2; hence not 3 divides 9629 by NAT_4:9;
    9629 = 5*1925 + 4; hence not 5 divides 9629 by NAT_4:9;
    9629 = 7*1375 + 4; hence not 7 divides 9629 by NAT_4:9;
    9629 = 11*875 + 4; hence not 11 divides 9629 by NAT_4:9;
    9629 = 13*740 + 9; hence not 13 divides 9629 by NAT_4:9;
    9629 = 17*566 + 7; hence not 17 divides 9629 by NAT_4:9;
    9629 = 19*506 + 15; hence not 19 divides 9629 by NAT_4:9;
    9629 = 23*418 + 15; hence not 23 divides 9629 by NAT_4:9;
    9629 = 29*332 + 1; hence not 29 divides 9629 by NAT_4:9;
    9629 = 31*310 + 19; hence not 31 divides 9629 by NAT_4:9;
    9629 = 37*260 + 9; hence not 37 divides 9629 by NAT_4:9;
    9629 = 41*234 + 35; hence not 41 divides 9629 by NAT_4:9;
    9629 = 43*223 + 40; hence not 43 divides 9629 by NAT_4:9;
    9629 = 47*204 + 41; hence not 47 divides 9629 by NAT_4:9;
    9629 = 53*181 + 36; hence not 53 divides 9629 by NAT_4:9;
    9629 = 59*163 + 12; hence not 59 divides 9629 by NAT_4:9;
    9629 = 61*157 + 52; hence not 61 divides 9629 by NAT_4:9;
    9629 = 67*143 + 48; hence not 67 divides 9629 by NAT_4:9;
    9629 = 71*135 + 44; hence not 71 divides 9629 by NAT_4:9;
    9629 = 73*131 + 66; hence not 73 divides 9629 by NAT_4:9;
    9629 = 79*121 + 70; hence not 79 divides 9629 by NAT_4:9;
    9629 = 83*116 + 1; hence not 83 divides 9629 by NAT_4:9;
    9629 = 89*108 + 17; hence not 89 divides 9629 by NAT_4:9;
    9629 = 97*99 + 26; hence not 97 divides 9629 by NAT_4:9;
  end;
  then for n being Element of NAT st 1 < n & n*n <= 9629 & n is prime
  holds not n divides 9629 by XPRIMET1:50;
  hence thesis by NAT_4:14;
end;
