
theorem
  563 is prime
proof
  now
    563 = 2*281 + 1; hence not 2 divides 563 by NAT_4:9;
    563 = 3*187 + 2; hence not 3 divides 563 by NAT_4:9;
    563 = 5*112 + 3; hence not 5 divides 563 by NAT_4:9;
    563 = 7*80 + 3; hence not 7 divides 563 by NAT_4:9;
    563 = 11*51 + 2; hence not 11 divides 563 by NAT_4:9;
    563 = 13*43 + 4; hence not 13 divides 563 by NAT_4:9;
    563 = 17*33 + 2; hence not 17 divides 563 by NAT_4:9;
    563 = 19*29 + 12; hence not 19 divides 563 by NAT_4:9;
    563 = 23*24 + 11; hence not 23 divides 563 by NAT_4:9;
  end;
  then for n being Element of NAT st 1 < n & n*n <= 563 & n is prime
  holds not n divides 563 by XPRIMET1:18;
  hence thesis by NAT_4:14;
