
theorem
  4643 is prime
proof
  now
    4643 = 2*2321 + 1; hence not 2 divides 4643 by NAT_4:9;
    4643 = 3*1547 + 2; hence not 3 divides 4643 by NAT_4:9;
    4643 = 5*928 + 3; hence not 5 divides 4643 by NAT_4:9;
    4643 = 7*663 + 2; hence not 7 divides 4643 by NAT_4:9;
    4643 = 11*422 + 1; hence not 11 divides 4643 by NAT_4:9;
    4643 = 13*357 + 2; hence not 13 divides 4643 by NAT_4:9;
    4643 = 17*273 + 2; hence not 17 divides 4643 by NAT_4:9;
    4643 = 19*244 + 7; hence not 19 divides 4643 by NAT_4:9;
    4643 = 23*201 + 20; hence not 23 divides 4643 by NAT_4:9;
    4643 = 29*160 + 3; hence not 29 divides 4643 by NAT_4:9;
    4643 = 31*149 + 24; hence not 31 divides 4643 by NAT_4:9;
    4643 = 37*125 + 18; hence not 37 divides 4643 by NAT_4:9;
    4643 = 41*113 + 10; hence not 41 divides 4643 by NAT_4:9;
    4643 = 43*107 + 42; hence not 43 divides 4643 by NAT_4:9;
    4643 = 47*98 + 37; hence not 47 divides 4643 by NAT_4:9;
    4643 = 53*87 + 32; hence not 53 divides 4643 by NAT_4:9;
    4643 = 59*78 + 41; hence not 59 divides 4643 by NAT_4:9;
    4643 = 61*76 + 7; hence not 61 divides 4643 by NAT_4:9;
    4643 = 67*69 + 20; hence not 67 divides 4643 by NAT_4:9;
  end;
  then for n being Element of NAT st 1 < n & n*n <= 4643 & n is prime
  holds not n divides 4643 by XPRIMET1:38;
  hence thesis by NAT_4:14;
end;
