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Sort XML nodes by multiple criteria

I'm trying to sort the following XML by date as main criteria and then price as second criteria.

I thought it's sufficient to sort it based on price first and then sort it by date. I expected book bk108 to end up first due to being the newest with highest price, but instead the first book is bkXXX even though its price is lower than bk108 's.

$xml = [xml]@'
<?xml version="1.0"?>
<catalog>
   <book id="bk101">
      <author>Gambardella, Matthew</author>
      <title>XML Developer's Guide</title>
      <genre>Computer</genre>
      <price>44.95</price>
      <publish_date>2000-10-01</publish_date>
      <description>An in-depth look at creating applications 
      with XML.</description>
   </book>
   <book id="bk102">
      <author>Ralls, Kim</author>
      <title>Midnight Rain</title>
      <genre>Fantasy</genre>
      <price>5.95</price>
      <publish_date>2000-12-16</publish_date>
      <description>A former architect battles corporate zombies, 
      an evil sorceress, and her own childhood to become queen 
      of the world.</description>
   </book>
   <book id="bk103">
      <author>Corets, Eva</author>
      <title>Maeve Ascendant</title>
      <genre>Fantasy</genre>
      <price>5.95</price>
      <publish_date>2000-11-17</publish_date>
      <description>After the collapse of a nanotechnology 
      society in England, the young survivors lay the 
      foundation for a new society.</description>
   </book>
   <book id="bk104">
      <author>Corets, Eva</author>
      <title>Oberon's Legacy</title>
      <genre>Fantasy</genre>
      <price>5.95</price>
      <publish_date>2001-03-10</publish_date>
      <description>In post-apocalypse England, the mysterious 
      agent known only as Oberon helps to create a new life 
      for the inhabitants of London. Sequel to Maeve 
      Ascendant.</description>
   </book>
   <book id="bk105">
      <author>Corets, Eva</author>
      <title>The Sundered Grail</title>
      <genre>Fantasy</genre>
      <price>5.95</price>
      <publish_date>2001-09-10</publish_date>
      <description>The two daughters of Maeve, half-sisters, 
      battle one another for control of England. Sequel to 
      Oberon's Legacy.</description>
   </book>
   <book id="bk106">
      <author>Randall, Cynthia</author>
      <title>Lover Birds</title>
      <genre>Romance</genre>
      <price>1000</price>
      <publish_date>2000-09-02</publish_date>
      <description>When Carla meets Paul at an ornithology 
      conference, tempers fly as feathers get ruffled.</description>
   </book>
   <book id="bk107">
      <author>Thurman, Paula</author>
      <title>Splish Splash</title>
      <genre>Romance</genre>
      <price>1000</price>
      <publish_date>2000-10-02</publish_date>
      <description>A deep sea diver finds true love twenty 
      thousand leagues beneath the sea.</description>
   </book>
   <book id="bk108">
      <author>Knorr, Stefan</author>
      <title>Creepy Crawlies</title>
      <genre>Horror</genre>
      <price>999</price>
      <publish_date>2001-12-06</publish_date>
      <description>An anthology of horror stories about roaches,
      centipedes, scorpions  and other insects.</description>
   </book>
    <book id="bkXXX">
      <author>new auth</author>
      <title>some title</title>
      <genre>Horror</genre>
      <price>998</price>
      <publish_date>2001-12-06</publish_date>
      <description>would go here</description>
   </book>
     <book id="bk113">
      <author>Knorr, Stefan</author>
      <title>Creepy Crawlies</title>
      <genre>Horror</genre>
      <price>999</price>
      <publish_date>2000-12-06</publish_date>
      <description>An anthology of horror stories about roaches,
      centipedes, scorpions  and other insects.</description>
   </book>
   <book id="bk109">
      <author>Kress, Peter</author>
      <title>Paradox Lost</title>
      <genre>Science Fiction</genre>
      <price>6.95</price>
      <publish_date>2000-11-02</publish_date>
      <description>After an inadvertant trip through a Heisenberg
      Uncertainty Device, James Salway discovers the problems 
      of being quantum.</description>
   </book>
</catalog>
'@
[System.Xml.XmlNode]$orig = $xml.catalog
$orig.book | sort { [double] $_.price}  |
  foreach { [void]$xml.catalog.PrependChild($_) }
$xml.catalog.book

[System.Xml.XmlNode]$orig = $xml.catalog
$orig.book | sort { [DateTime] $_.publish_date}  |
  foreach { [void]$xml.catalog.PrependChild($_) }
$xml.catalog.book

$xml.Save('c:\work\foo.xml')

This was my starting point powershell xml sort nodes and replacechild

Replace your code after [System.Xml.XmlNode]$orig = $xml.catalog with this:

[xml]$sortedXml = @"
    <catalog>
        $(($orig.book | sort-object -Property {[double] $_.price},{[datetime]$_.publish_date} -descending).OuterXml)
    </catalog>
"@

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