
theorem
  821 is prime
proof
  now
    821 = 2*410 + 1; hence not 2 divides 821 by NAT_4:9;
    821 = 3*273 + 2; hence not 3 divides 821 by NAT_4:9;
    821 = 5*164 + 1; hence not 5 divides 821 by NAT_4:9;
    821 = 7*117 + 2; hence not 7 divides 821 by NAT_4:9;
    821 = 11*74 + 7; hence not 11 divides 821 by NAT_4:9;
    821 = 13*63 + 2; hence not 13 divides 821 by NAT_4:9;
    821 = 17*48 + 5; hence not 17 divides 821 by NAT_4:9;
    821 = 19*43 + 4; hence not 19 divides 821 by NAT_4:9;
    821 = 23*35 + 16; hence not 23 divides 821 by NAT_4:9;
  end;
  then for n being Element of NAT st 1 < n & n*n <= 821 & n is prime
  holds not n divides 821 by XPRIMET1:18;
  hence thesis by NAT_4:14;
end;
