
theorem
  5009 is prime
proof
  now
    5009 = 2*2504 + 1; hence not 2 divides 5009 by NAT_4:9;
    5009 = 3*1669 + 2; hence not 3 divides 5009 by NAT_4:9;
    5009 = 5*1001 + 4; hence not 5 divides 5009 by NAT_4:9;
    5009 = 7*715 + 4; hence not 7 divides 5009 by NAT_4:9;
    5009 = 11*455 + 4; hence not 11 divides 5009 by NAT_4:9;
    5009 = 13*385 + 4; hence not 13 divides 5009 by NAT_4:9;
    5009 = 17*294 + 11; hence not 17 divides 5009 by NAT_4:9;
    5009 = 19*263 + 12; hence not 19 divides 5009 by NAT_4:9;
    5009 = 23*217 + 18; hence not 23 divides 5009 by NAT_4:9;
    5009 = 29*172 + 21; hence not 29 divides 5009 by NAT_4:9;
    5009 = 31*161 + 18; hence not 31 divides 5009 by NAT_4:9;
    5009 = 37*135 + 14; hence not 37 divides 5009 by NAT_4:9;
    5009 = 41*122 + 7; hence not 41 divides 5009 by NAT_4:9;
    5009 = 43*116 + 21; hence not 43 divides 5009 by NAT_4:9;
    5009 = 47*106 + 27; hence not 47 divides 5009 by NAT_4:9;
    5009 = 53*94 + 27; hence not 53 divides 5009 by NAT_4:9;
    5009 = 59*84 + 53; hence not 59 divides 5009 by NAT_4:9;
    5009 = 61*82 + 7; hence not 61 divides 5009 by NAT_4:9;
    5009 = 67*74 + 51; hence not 67 divides 5009 by NAT_4:9;
  end;
  then for n being Element of NAT st 1 < n & n*n <= 5009 & n is prime
  holds not n divides 5009 by XPRIMET1:38;
  hence thesis by NAT_4:14;
end;
