
theorem
  5563 is prime
proof
  now
    5563 = 2*2781 + 1; hence not 2 divides 5563 by NAT_4:9;
    5563 = 3*1854 + 1; hence not 3 divides 5563 by NAT_4:9;
    5563 = 5*1112 + 3; hence not 5 divides 5563 by NAT_4:9;
    5563 = 7*794 + 5; hence not 7 divides 5563 by NAT_4:9;
    5563 = 11*505 + 8; hence not 11 divides 5563 by NAT_4:9;
    5563 = 13*427 + 12; hence not 13 divides 5563 by NAT_4:9;
    5563 = 17*327 + 4; hence not 17 divides 5563 by NAT_4:9;
    5563 = 19*292 + 15; hence not 19 divides 5563 by NAT_4:9;
    5563 = 23*241 + 20; hence not 23 divides 5563 by NAT_4:9;
    5563 = 29*191 + 24; hence not 29 divides 5563 by NAT_4:9;
    5563 = 31*179 + 14; hence not 31 divides 5563 by NAT_4:9;
    5563 = 37*150 + 13; hence not 37 divides 5563 by NAT_4:9;
    5563 = 41*135 + 28; hence not 41 divides 5563 by NAT_4:9;
    5563 = 43*129 + 16; hence not 43 divides 5563 by NAT_4:9;
    5563 = 47*118 + 17; hence not 47 divides 5563 by NAT_4:9;
    5563 = 53*104 + 51; hence not 53 divides 5563 by NAT_4:9;
    5563 = 59*94 + 17; hence not 59 divides 5563 by NAT_4:9;
    5563 = 61*91 + 12; hence not 61 divides 5563 by NAT_4:9;
    5563 = 67*83 + 2; hence not 67 divides 5563 by NAT_4:9;
    5563 = 71*78 + 25; hence not 71 divides 5563 by NAT_4:9;
    5563 = 73*76 + 15; hence not 73 divides 5563 by NAT_4:9;
  end;
  then for n being Element of NAT st 1 < n & n*n <= 5563 & n is prime
  holds not n divides 5563 by XPRIMET1:42;
  hence thesis by NAT_4:14;
end;
