Its been awhile since I've done any PHP, but shouldn't:
$query = "INSERT INTO proofreading_sought (firstname, lastname, email) " .
"VALUES ($first, $last, $theMail)";
Be:
$query = "INSERT INTO proofreading_sought (firstname, lastname, email) ";
$query.= "VALUES ($first, $last, $theMail)";
other thing that looked odd was:
$dbc = @mysqli_connect($mysql_host,$mysql_user,$mysql_password,$mysql_database)
or die("Failed to connect to the database...");
or should be || as in:
$dbc = @mysqli_connect($mysql_host,$mysql_user,$mysql_password,$mysql_database)
|| die("Failed to connect to the database...");