
theorem
  5639 is prime
proof
  now
    5639 = 2*2819 + 1; hence not 2 divides 5639 by NAT_4:9;
    5639 = 3*1879 + 2; hence not 3 divides 5639 by NAT_4:9;
    5639 = 5*1127 + 4; hence not 5 divides 5639 by NAT_4:9;
    5639 = 7*805 + 4; hence not 7 divides 5639 by NAT_4:9;
    5639 = 11*512 + 7; hence not 11 divides 5639 by NAT_4:9;
    5639 = 13*433 + 10; hence not 13 divides 5639 by NAT_4:9;
    5639 = 17*331 + 12; hence not 17 divides 5639 by NAT_4:9;
    5639 = 19*296 + 15; hence not 19 divides 5639 by NAT_4:9;
    5639 = 23*245 + 4; hence not 23 divides 5639 by NAT_4:9;
    5639 = 29*194 + 13; hence not 29 divides 5639 by NAT_4:9;
    5639 = 31*181 + 28; hence not 31 divides 5639 by NAT_4:9;
    5639 = 37*152 + 15; hence not 37 divides 5639 by NAT_4:9;
    5639 = 41*137 + 22; hence not 41 divides 5639 by NAT_4:9;
    5639 = 43*131 + 6; hence not 43 divides 5639 by NAT_4:9;
    5639 = 47*119 + 46; hence not 47 divides 5639 by NAT_4:9;
    5639 = 53*106 + 21; hence not 53 divides 5639 by NAT_4:9;
    5639 = 59*95 + 34; hence not 59 divides 5639 by NAT_4:9;
    5639 = 61*92 + 27; hence not 61 divides 5639 by NAT_4:9;
    5639 = 67*84 + 11; hence not 67 divides 5639 by NAT_4:9;
    5639 = 71*79 + 30; hence not 71 divides 5639 by NAT_4:9;
    5639 = 73*77 + 18; hence not 73 divides 5639 by NAT_4:9;
  end;
  then for n being Element of NAT st 1 < n & n*n <= 5639 & n is prime
  holds not n divides 5639 by XPRIMET1:42;
  hence thesis by NAT_4:14;
end;
