
theorem
  8563 is prime
proof
  now
    8563 = 2*4281 + 1; hence not 2 divides 8563 by NAT_4:9;
    8563 = 3*2854 + 1; hence not 3 divides 8563 by NAT_4:9;
    8563 = 5*1712 + 3; hence not 5 divides 8563 by NAT_4:9;
    8563 = 7*1223 + 2; hence not 7 divides 8563 by NAT_4:9;
    8563 = 11*778 + 5; hence not 11 divides 8563 by NAT_4:9;
    8563 = 13*658 + 9; hence not 13 divides 8563 by NAT_4:9;
    8563 = 17*503 + 12; hence not 17 divides 8563 by NAT_4:9;
    8563 = 19*450 + 13; hence not 19 divides 8563 by NAT_4:9;
    8563 = 23*372 + 7; hence not 23 divides 8563 by NAT_4:9;
    8563 = 29*295 + 8; hence not 29 divides 8563 by NAT_4:9;
    8563 = 31*276 + 7; hence not 31 divides 8563 by NAT_4:9;
    8563 = 37*231 + 16; hence not 37 divides 8563 by NAT_4:9;
    8563 = 41*208 + 35; hence not 41 divides 8563 by NAT_4:9;
    8563 = 43*199 + 6; hence not 43 divides 8563 by NAT_4:9;
    8563 = 47*182 + 9; hence not 47 divides 8563 by NAT_4:9;
    8563 = 53*161 + 30; hence not 53 divides 8563 by NAT_4:9;
    8563 = 59*145 + 8; hence not 59 divides 8563 by NAT_4:9;
    8563 = 61*140 + 23; hence not 61 divides 8563 by NAT_4:9;
    8563 = 67*127 + 54; hence not 67 divides 8563 by NAT_4:9;
    8563 = 71*120 + 43; hence not 71 divides 8563 by NAT_4:9;
    8563 = 73*117 + 22; hence not 73 divides 8563 by NAT_4:9;
    8563 = 79*108 + 31; hence not 79 divides 8563 by NAT_4:9;
    8563 = 83*103 + 14; hence not 83 divides 8563 by NAT_4:9;
    8563 = 89*96 + 19; hence not 89 divides 8563 by NAT_4:9;
  end;
  then for n being Element of NAT st 1 < n & n*n <= 8563 & n is prime
  holds not n divides 8563 by XPRIMET1:48;
  hence thesis by NAT_4:14;
end;
