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<b>let </b><font color="Maroon">X</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="trees_3.html#V3">constituted-DTrees</a>  <a href="hidden.html#M1">set</a> ;<br/>

<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a>,<a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a>
<div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:58_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="trees_9.html#K9" target="_self">Subtrees</a> <font color="Maroon">X</font>
 ;<br/><b>then consider </b><font color="Maroon">t</font> being    <a href="trees_3.html#M3">Element</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E3:58_1"/><i><font color="Green">E22</font></i>: 
 ex <font color="Olive">p</font> being  <a href="trees_4.html#NM1">Node</a> of <font color="Maroon">t</font> st <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> <a href="trees_2.html#K5">|</a> <font color="Olive">p</font>
 ;<br/><a NAME="E4:58_1"/>
(  <a href="trees_9.html#K3" target="_self">Subtrees</a> <font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="trees_9.html#K3" target="_self">Subtrees</a> <font color="Olive">s</font></span>)</span> where <font color="Olive">s</font> is    <a href="trees_3.html#M3">Element</a> of <font color="Maroon">X</font> : verum </span>}</span>  &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="trees_9.html#K3" target="_self">Subtrees</a> <font color="Maroon">t</font> )
 <b>by </b><i><a class="txt" href="trees_9.html#E4:58"><i><font color="Green">E25</font></i></a></i>;<br/><b>hence </b><a NAME="E5:58_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="trees_9.html#K3" target="_self">Subtrees</a> <font color="Olive">s</font></span>)</span> where <font color="Olive">s</font> is    <a href="trees_3.html#M3">Element</a> of <font color="Maroon">X</font> : verum </span>}</span> 
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/></div>
<b>end;</b></div>

<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="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:58"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="trees_9.html#K3" target="_self">Subtrees</a> <font color="Olive">s</font></span>)</span> where <font color="Olive">s</font> is    <a href="trees_3.html#M3">Element</a> of <font color="Maroon">X</font> : verum </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">Y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:58"/><i><font color="Green">E25</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="trees_9.html#K3" target="_self">Subtrees</a> <font color="Olive">s</font></span>)</span> where <font color="Olive">s</font> is    <a href="trees_3.html#M3">Element</a> of <font color="Maroon">X</font> : verum </span>}</span>  )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">t</font> being    <a href="trees_3.html#M3">Element</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E6:58"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R1">=</a>  <a href="trees_9.html#K3" target="_self">Subtrees</a> <font color="Maroon">t</font>
 <b>by </b><i><a class="txt" href="trees_9.html#E4:58"><i><font color="Green">E25</font></i></a></i>;<br/>
<a NAME="E7:58"/>
 ex <font color="Olive">p</font> being  <a href="trees_4.html#NM1">Node</a> of <font color="Maroon">t</font> st <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> <a href="trees_2.html#K5">|</a> <font color="Olive">p</font>
 
<b>by </b><i><a class="txt" href="trees_9.html#E4:58"><i><font color="Green">E25</font></i></a>, </i>;<br/>
<b>hence </b><a NAME="E8:58"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="trees_9.html#K9" target="_self">Subtrees</a> <font color="Maroon">X</font>
 ;<br/>


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