
theorem
  1627 is prime
proof
  now
    1627 = 2*813 + 1; hence not 2 divides 1627 by NAT_4:9;
    1627 = 3*542 + 1; hence not 3 divides 1627 by NAT_4:9;
    1627 = 5*325 + 2; hence not 5 divides 1627 by NAT_4:9;
    1627 = 7*232 + 3; hence not 7 divides 1627 by NAT_4:9;
    1627 = 11*147 + 10; hence not 11 divides 1627 by NAT_4:9;
    1627 = 13*125 + 2; hence not 13 divides 1627 by NAT_4:9;
    1627 = 17*95 + 12; hence not 17 divides 1627 by NAT_4:9;
    1627 = 19*85 + 12; hence not 19 divides 1627 by NAT_4:9;
    1627 = 23*70 + 17; hence not 23 divides 1627 by NAT_4:9;
    1627 = 29*56 + 3; hence not 29 divides 1627 by NAT_4:9;
    1627 = 31*52 + 15; hence not 31 divides 1627 by NAT_4:9;
    1627 = 37*43 + 36; hence not 37 divides 1627 by NAT_4:9;
  end;
  then for n being Element of NAT st 1 < n & n*n <= 1627 & n is prime
  holds not n divides 1627 by XPRIMET1:24;
  hence thesis by NAT_4:14;
end;
