
theorem
  823 is prime
proof
  now
    823 = 2*411 + 1; hence not 2 divides 823 by NAT_4:9;
    823 = 3*274 + 1; hence not 3 divides 823 by NAT_4:9;
    823 = 5*164 + 3; hence not 5 divides 823 by NAT_4:9;
    823 = 7*117 + 4; hence not 7 divides 823 by NAT_4:9;
    823 = 11*74 + 9; hence not 11 divides 823 by NAT_4:9;
    823 = 13*63 + 4; hence not 13 divides 823 by NAT_4:9;
    823 = 17*48 + 7; hence not 17 divides 823 by NAT_4:9;
    823 = 19*43 + 6; hence not 19 divides 823 by NAT_4:9;
    823 = 23*35 + 18; hence not 23 divides 823 by NAT_4:9;
  end;
  then for n being Element of NAT st 1 < n & n*n <= 823 & n is prime
  holds not n divides 823 by XPRIMET1:18;
  hence thesis by NAT_4:14;
end;
