Session varaibles

/ *  here are the session.php variables...

    private $signed_in = false;
    public  $user_id;
    public $count;
    public $message;
    public $first_name;
    public $username;
    public $password;
*/

//************the login function in session.php*********//

public function login($user) {

// we have a user
    if($user) {

        $this->first_name = $_SESSION['first_name'] = $user->first_name;
        $this->username = $_SESSION['username'] = $user->username;


        $this->user_id = $_SESSION['user_id'] = $user->id;
        $this->signed_in = true;
        }
    }

//********************* the test.php page*******************//


<?php include("includes/init.php"); ?>

<?php if(!$session->is_signed_in()) {redirect("login.php");} ?>

<?php

if(isset($_POST['submit'])){
        echo "first name is ". $_SESSION['first_name']."<br>";
        echo "Username is ". $_SESSION['username']."<br>";
        echo "User id is ".$_SESSION['user_id']."<br>";

}

?>

<!DOCTYPE html>
<html lang="en">
<head>
    <meta charset="UTF-8">
    <title>Document</title>
</head>
<body>

    <form action="" method='post'>
        <input name='user' type="text">
        <input name='submit' type="submit">
    </form>

</body>
</html>

//******************* the return******************//
Notice: Undefined index: first_name in C:\xampp\htdocs\gallery\admin\test.php on line 13
first name is
Username is william
User id is 190