<?xml version="1.0"?>
<div class="add">

<b>set </b><font color="Maroon">S</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">t</font> <a href="trees_2.html#K5">|</a> <font color="Olive">p</font></span>)</span> where <font color="Olive">t</font> is    <a href="trees_3.html#M3">Element</a> of <font color="Maroon">X</font>, <font color="Olive">p</font> is   <a href="trees_4.html#NM1">Node</a> of <font color="Olive">t</font> : verum </span>}</span> ;<br/>
<b>consider </b><font color="Maroon">t</font> being    <a href="trees_3.html#M3">Element</a> of <font color="Maroon">X</font>, <font color="Maroon">p0</font> being   <a href="trees_4.html#NM1">Node</a> of <font color="Maroon">t</font>;<br/>
<a NAME="E2:51_1_1"/>
<font color="Maroon">t</font> <a href="trees_2.html#K5">|</a> <font color="Maroon">p0</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">t</font> <a href="trees_2.html#K5">|</a> <font color="Olive">p</font></span>)</span> where <font color="Olive">t</font> is    <a href="trees_3.html#M3">Element</a> of <font color="Maroon">X</font>, <font color="Olive">p</font> is   <a href="trees_4.html#NM1">Node</a> of <font color="Olive">t</font> : verum </span>}</span> 
 
;<br/>
<b>then reconsider </b><font color="Maroon">S</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">t</font> <a href="trees_2.html#K5">|</a> <font color="Olive">p</font></span>)</span> where <font color="Olive">t</font> is    <a href="trees_3.html#M3">Element</a> of <font color="Maroon">X</font>, <font color="Olive">p</font> is   <a href="trees_4.html#NM1">Node</a> of <font color="Olive">t</font> : verum </span>}</span>  as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ;<br/>
<a NAME="E4:51_1_1"/>
<font color="Maroon">S</font> is <a href="trees_3.html#V3">constituted-DTrees</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="trees_3.html#D5">TREES_3:def 5</a><br/>

<b>assume </b><a NAME="E1:51_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 ;<br/>

<a NAME="E2:51_1_1_1"/><b>then </b>
 ex <font color="Olive">t</font> being   <a href="trees_3.html#M3">Element</a> of <font color="Maroon">X</font> ex <font color="Olive">p</font> being  <a href="trees_4.html#NM1">Node</a> of <font color="Olive">t</font> st <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">t</font> <a href="trees_2.html#K5">|</a> <font color="Olive">p</font>
 
;<br/>
<b>hence </b><a NAME="E3:51_1_1_1"/>
<font color="Maroon">x</font> is   <a href="trees_2.html#NM4">DecoratedTree</a>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:51_1_1"/>
(  <a href="trees_9.html#K9" target="_self">Subtrees</a> <font color="Maroon">X</font> is <a href="trees_3.html#V3">constituted-DTrees</a> &amp; not  <a href="trees_9.html#K9" target="_self">Subtrees</a> <font color="Maroon">X</font> is <a href="xboole_0.html#V1">empty</a> )
 ;<br/>


</div>
