XDocument duplicate namespace with different Local Name

Question

I have an XML document that looks like this:

Answer 1

I found a solution all thanks to your ideas. The general Idea is I changed the value of p1 to anything added my new attribute then returned p1 to its original value even before saving the XDocument.

XAttribute p1 = csdlDoc.Root.Attributes()
            .First(a => a.IsNamespaceDeclaration && a.Name.LocalName == "p1");

p1.Value = "http://schemas.microsoft.com/ado/2009/02/edm/annotation/TEMP";

......
// the same update now works as there is only one xmlns 
XNamespace annotation = "http://schemas.microsoft.com/ado/2009/02/edm/annotation";
var attrib = new XAttribute(annotation + "StoreGeneratedPattern", "Computed");

csdlProperty.Add(attrib);

p1.Value = "http://schemas.microsoft.com/ado/2009/02/edm/annotation/";

Another solution that worked is swapping the order or the declarations at the root node (declare p1 before annotation) like below:

In general both look like cheap solutions..