diff --git a/strip-frames.xsl b/strip-frames.xsl
index 8e39c3f4689cfe4fc5d22bdc9c8d37cdd0fe395c..c849e7ec46bff006d1efaa2086984fc1952d829f 100644
--- a/strip-frames.xsl
+++ b/strip-frames.xsl
@@ -24,7 +24,8 @@
     <!-- TODO: add styles for new sections based on the frames? -->
     <!-- create a derived style for each paragraph we strip from around a frame,
          which should be identical but for having a 0 line height -->
-    <xsl:for-each select="//text:p[draw:frame]">
+    <xsl:for-each select="//text:p[draw:frame]
+                         |//text:h[draw:frame]">
       <xsl:element name="style:style">
         <xsl:variable name="style-name" select="@text:style-name"/>
 
@@ -56,8 +57,9 @@
 <!-- moves children of a draw:p containing a draw:frame outside of itself,
      because we want to replace the frame with sections, which do not belong
      inside a text:p -->
-<xsl:template match="text:p[draw:frame]">
-  <xsl:comment>Removed outer text:p</xsl:comment>
+<xsl:template match="text:p[draw:frame] |
+                     text:h[draw:frame]">
+  <xsl:comment>Removed outer <xsl:value-of select="name()"/></xsl:comment>
   <xsl:copy>
     <!-- FIXME: there's probably a better way to do that -->
     <xsl:for-each select="@*">
@@ -84,17 +86,29 @@
 </xsl:template>
 
 <!-- this is kind of a hack, as we stripped the text:p, we add it back -->
-<xsl:template match="text:p/draw:frame[not(draw:text-box)] |
-                     text:p[draw:frame]/*[not(self::draw:frame)][not(self::text:p)]">
-  <xsl:comment>Re-added text:p wrapping element</xsl:comment>
-  <xsl:element name="text:p">
-    <!-- TODO: add the stripped p's attributes here -->
-    <xsl:call-template name="copy-and-recurse"/>
-  </xsl:element>
+<xsl:template match="text:p/draw:frame[not(draw:text-box)]
+                    |text:h/draw:frame[not(draw:text-box)]
+                    |text:p[draw:frame]/*[not(self::draw:frame)][not(self::text:p)]
+                    |text:h[draw:frame]/*[not(self::draw:frame)][not(self::text:h)]">
+  <xsl:comment>Re-added <xsl:value-of select="name(..)" /> wrapping element</xsl:comment>
+  <!-- FIXME: there's probably a better way to do that.
+       Here in order to change the context (and because XSLT 1.0 doesn't have
+       the "select" attribute on xsl:copy), we use a xsl:for-each that matches
+       a single element in order to switch context -->
+  <xsl:variable name="self" select="."/>
+  <xsl:for-each select="..">
+    <xsl:copy>
+      <!-- TODO: add the stripped p's attributes here (?) -->
+      <xsl:for-each select="$self">
+        <xsl:call-template name="copy-and-recurse"/>
+      </xsl:for-each>
+    </xsl:copy>
+  </xsl:for-each>
 </xsl:template>
 
 <!-- Replaces draw:frame/draw:text-box with a section -->
-<xsl:template match="text:p/draw:frame[count(draw:text-box) = count(*)]">
+<xsl:template match="text:p/draw:frame[count(draw:text-box) = count(*)]
+                    |text:h/draw:frame[count(draw:text-box) = count(*)]">
   <xsl:comment>draw:frame</xsl:comment>
   <xsl:comment>draw:text-box</xsl:comment>
   <xsl:element name="text:section">
@@ -107,9 +121,9 @@
 <!-- companion for the above, catches frames not handled because outside of
      supported elements.  Should never happen, but helps debugging of unexpected
      cases. -->
-<xsl:template match="*[not(self::text:p)][draw:frame[count(draw:text-box) = count(*)]]">
-  <xsl:message>Unsupported input with a draw:frame not inside a text:p (but a <xsl:value-of select="name()"/>) but with draw:text-box children</xsl:message>
-  <xsl:comment>Unsupported input with a draw:frame not inside a text:p (but a <xsl:value-of select="name()"/>) but with draw:text-box children</xsl:comment>
+<xsl:template match="*[not(self::text:p) and not(self::text:h)][draw:frame[count(draw:text-box) = count(*)]]">
+  <xsl:message>Unsupported input with a draw:frame not inside a text:p or text:h (but a <xsl:value-of select="name()"/>) but with draw:text-box children</xsl:message>
+  <xsl:comment>Unsupported input with a draw:frame not inside a text:p or text:h (but a <xsl:value-of select="name()"/>) but with draw:text-box children</xsl:comment>
   <xsl:call-template name="copy-and-recurse"/>
 </xsl:template>