Ajax with JQuery

I have used Ajax link and live validation in index.rhtml
Ajax link requires JavaScript "prototype" and live validation requires
"jquery".
i got "element.dispatchEvent is not a function" on the firebug console
when i used "prototype" and "jquery" in same page index.rhtml
after i removed javascript "prototype" library, i got "Ajax is not
defined" on the firebug console when i clicking ajax link.
how can i use both ajax and jquery in the same page?
Can anybody give me a solution for this?

For example:

<%=javascript_include_tag "prototype"%>
<%=javascript_include_tag "jquery"%>
<%= javascript_include_tag 'livevalidation/jquery.validate' %>
<%= javascript_include_tag 'livevalidation/jquery.validation.functions'
%>
<%= stylesheet_link_tag 'livevalidation/jquery.validate' %>
<%= stylesheet_link_tag 'livevalidation/style' %>

<%=link_to_remote "new user",:url=>{:action=>:a},:update=>"div"%>
<div id="div"></div>
<% form_for(@user) do |f| %>
  <p>
    <%= f.label :name %><br />
    <%= f.text_field :name %><br />
  </p>
  <p>
    <%= f.submit "Register"%>
  </p>
<% end %>
<script type="text/javascript">
  jQuery(function(){
  jQuery("#user_name").validate({
    expression: "if (VAL) return true; else return false;",
      message: "Please enter the Password"
    });});
</script>

Try using the noConflict setting

<script type="text/javascript">
//define function to be executed on document ready
var $j = jQuery.noConflict();

$j(function(){
   $j("#user_name").validate({
    expression: "if (VAL) return true; else return false;",
      message: "Please enter the Password"
     });
  });
</script>