
theorem
  5623 is prime
proof
  now
    5623 = 2*2811 + 1; hence not 2 divides 5623 by NAT_4:9;
    5623 = 3*1874 + 1; hence not 3 divides 5623 by NAT_4:9;
    5623 = 5*1124 + 3; hence not 5 divides 5623 by NAT_4:9;
    5623 = 7*803 + 2; hence not 7 divides 5623 by NAT_4:9;
    5623 = 11*511 + 2; hence not 11 divides 5623 by NAT_4:9;
    5623 = 13*432 + 7; hence not 13 divides 5623 by NAT_4:9;
    5623 = 17*330 + 13; hence not 17 divides 5623 by NAT_4:9;
    5623 = 19*295 + 18; hence not 19 divides 5623 by NAT_4:9;
    5623 = 23*244 + 11; hence not 23 divides 5623 by NAT_4:9;
    5623 = 29*193 + 26; hence not 29 divides 5623 by NAT_4:9;
    5623 = 31*181 + 12; hence not 31 divides 5623 by NAT_4:9;
    5623 = 37*151 + 36; hence not 37 divides 5623 by NAT_4:9;
    5623 = 41*137 + 6; hence not 41 divides 5623 by NAT_4:9;
    5623 = 43*130 + 33; hence not 43 divides 5623 by NAT_4:9;
    5623 = 47*119 + 30; hence not 47 divides 5623 by NAT_4:9;
    5623 = 53*106 + 5; hence not 53 divides 5623 by NAT_4:9;
    5623 = 59*95 + 18; hence not 59 divides 5623 by NAT_4:9;
    5623 = 61*92 + 11; hence not 61 divides 5623 by NAT_4:9;
    5623 = 67*83 + 62; hence not 67 divides 5623 by NAT_4:9;
    5623 = 71*79 + 14; hence not 71 divides 5623 by NAT_4:9;
    5623 = 73*77 + 2; hence not 73 divides 5623 by NAT_4:9;
  end;
  then for n being Element of NAT st 1 < n & n*n <= 5623 & n is prime
  holds not n divides 5623 by XPRIMET1:42;
  hence thesis by NAT_4:14;
end;
