[OOP]problema con variabili di sessione

[Raulken]

<?php require_once('conn.php');?>
<?php
class login{
    private $email;
    private $password;
    
    public $nome;
    public $cognome;
    
    public function __construct($email,$password){
        $this->email=$email;
        $this->password=md5($password);    
        $this->query();
    }

    private function query(){
        $query=mysql_query("SELECT * FROM info_utente WHERE email='".$this->email."' AND password='".$this->password."'");
        $row=mysql_fetch_array($query);
        $lista=mysql_num_rows($query);
        if($lista!=0){
            echo "ok esisti
";
            $this->nome=$row['nome'];
            $this->cognome=$row['cognome'];
            echo "<a href=\"prova.php\">prova</a>";
        }else{
            echo "non esisti";
        }
        
    }
    
    
}


if((isset($_POST['email']))and(isset($_POST['password']))){
$utente= new login($_POST['email'],$_POST['password']);

echo $utente->nome;
echo $_SESSION['nome']=$utente->nome;

}

?>

prova.php

<?php
session_start();
echo $_SESSION['nome'];
?>

perchè in prova.php non mi stampa la variabile di sessione?