1. Enable module rewrite

fulmanp@ubuntu:~$ sudo a2enmod rewrite 
[sudo] password for fulmanp: 
Enabling module rewrite.
To activate the new configuration, you need to run:
  service apache2 restart
fulmanp@ubuntu:~$ sudo service apache2 restart 
 * Restarting web server apache2
 
 
2. In file

sudo gedit /etc/apache2/apache2.conf 

add

<Directory /var/www/html/fulmanp/pai/rest/>
  Options Indexes FollowSymLinks
  AllowOverride All
  Require all granted
</Directory>

where

Options (description from: http://httpd.apache.org/docs/2.2/mod/core.html#options)

+FollowSymLinks makes Apache follow system symbolic links (shortcuts, if you would) in your file system.
+Indexes allows access to open folders within your file system, e.g. if you forgot to put an index.{ext_here} file down within a directory, +Indexes would allow list that directory's contents to the user.
AllowOverride All: This is telling apache, that all configurations can be overridden in the /var/www/html/fulmanp/pai/rest and all its sub-directories (recursively).
Require all granted: this is new equivalent of Allow from all (no IP addresses blocked from accessing the service)

3. Restart Apache

sudo service apache2 restart

4. Create file

/var/www/html/fulmanp/pai/rest.htaccess

with content

# use mod_rewrite for pretty URL support
RewriteEngine on
# If a directory or a file exists, use the request directly
RewriteCond %{REQUEST_FILENAME} !-f
RewriteCond %{REQUEST_FILENAME} !-d
# Otherwise forward the request to rest.php
RewriteRule . rest.php