
theorem
  401 is prime
proof
  now
    401 = 2*200 + 1; hence not 2 divides 401 by NAT_4:9;
    401 = 3*133 + 2; hence not 3 divides 401 by NAT_4:9;
    401 = 5*80 + 1; hence not 5 divides 401 by NAT_4:9;
    401 = 7*57 + 2; hence not 7 divides 401 by NAT_4:9;
    401 = 11*36 + 5; hence not 11 divides 401 by NAT_4:9;
    401 = 13*30 + 11; hence not 13 divides 401 by NAT_4:9;
    401 = 17*23 + 10; hence not 17 divides 401 by NAT_4:9;
    401 = 19*21 + 2; hence not 19 divides 401 by NAT_4:9;
  end;
  then for n being Element of NAT st 1 < n & n*n <= 401 & n is prime
  holds not n divides 401 by XPRIMET1:16;
  hence thesis by NAT_4:14;
