
theorem
  827 is prime
proof
  now
    827 = 2*413 + 1; hence not 2 divides 827 by NAT_4:9;
    827 = 3*275 + 2; hence not 3 divides 827 by NAT_4:9;
    827 = 5*165 + 2; hence not 5 divides 827 by NAT_4:9;
    827 = 7*118 + 1; hence not 7 divides 827 by NAT_4:9;
    827 = 11*75 + 2; hence not 11 divides 827 by NAT_4:9;
    827 = 13*63 + 8; hence not 13 divides 827 by NAT_4:9;
    827 = 17*48 + 11; hence not 17 divides 827 by NAT_4:9;
    827 = 19*43 + 10; hence not 19 divides 827 by NAT_4:9;
    827 = 23*35 + 22; hence not 23 divides 827 by NAT_4:9;
  end;
  then for n being Element of NAT st 1 < n & n*n <= 827 & n is prime
  holds not n divides 827 by XPRIMET1:18;
  hence thesis by NAT_4:14;
