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<b>let </b><font color="Maroon">C</font> be    <a href="hidden.html#M1">set</a> ;<br/><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>
<div class="add"><b>assume </b><a NAME="E1:60_1"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">C</font> <a href="trees_9.html#K12" target="_self">-Subtrees</a> <font color="Maroon">X</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><b>let </b><font color="Maroon">t</font> be    <a href="trees_3.html#M3">Element</a> of <font color="Maroon">X</font>;<br/><b>reconsider </b><font color="Maroon">e</font> =  <a href="xboole_0.html#K1">{}</a>  as   <a href="trees_4.html#NM1">Node</a> of <font color="Maroon">t</font> <b>by </b><i><a class="ref" href="trees_1.html#T47">TREES_1:47</a></i>;<br/><a NAME="E3:60_1"/>
not <font color="Maroon">t</font> <a href="trees_2.html#K5">|</a> <font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> <a href="trees_9.html#K12" target="_self">-Subtrees</a> <font color="Maroon">X</font>
 <b>by </b><i/>;<br/><a NAME="E4:60_1"/><b>then </b>
( <font color="Maroon">e</font> <a href="hidden.html#R2">in</a>  <a href="trees_1.html#K3">Leaves</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">t</font></span>)</span> &amp; not <font color="Maroon">t</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> )
 ;<br/><b>hence </b><a NAME="E5:60_1"/>
( <font color="Maroon">t</font> is <a href="trees_9.html#V1" target="_self">root</a> &amp; not <font color="Maroon">t</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> )
 <b>by </b><i/>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:60"/><i><font color="Green">E25</font></i>: 
for <font color="Olive">t</font> being   <a href="trees_3.html#M3">Element</a> of <font color="Maroon">X</font> holds <br/> ( <font color="Olive">t</font> is <a href="trees_9.html#V1" target="_self">root</a> &amp; not <font color="Olive">t</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> )
 ;<br/>

<b>assume </b><a NAME="E3:60"/>
not <font color="Maroon">C</font> <a href="trees_9.html#K12" target="_self">-Subtrees</a> <font color="Maroon">X</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/>

<b>then reconsider </b><font color="Maroon">S</font> = <font color="Maroon">C</font> <a href="trees_9.html#K12" target="_self">-Subtrees</a> <font color="Maroon">X</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="trees_9.html#K9" target="_self">Subtrees</a> <font color="Maroon">X</font></span>)</span> ;<br/>
<b>consider </b><font color="Maroon">s</font> being    <a href="trees_3.html#M3">Element</a> of <font color="Maroon">S</font>;<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">n</font> being   <a href="trees_4.html#NM1">Node</a> of <font color="Maroon">t</font><b> such that </b><br/><a NAME="E7:60"/><i><font color="Green">E27</font></i>: 
( <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> <a href="trees_2.html#K5">|</a> <font color="Maroon">n</font> &amp; (  not <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="trees_1.html#K3">Leaves</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">t</font></span>)</span> or <font color="Maroon">t</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> ) )
 <b>by </b><i><a class="txt" href="trees_9.html#E2:60"><i><font color="Green">E25</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">e</font> =  <a href="xboole_0.html#K1">{}</a>  as   <a href="trees_4.html#NM1">Node</a> of <font color="Maroon">t</font> <b>by </b><i><a class="ref" href="trees_1.html#T47">TREES_1:47</a></i>;<br/>
<a NAME="E9:60"/>
<font color="Maroon">t</font> is <a href="trees_9.html#V1" target="_self">root</a>
 
<b>by </b><i/>;<br/>
<a NAME="E10:60"/><b>then </b><i><font color="Green">E28</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i>, <a class="ref" href="trees_1.html#T56">TREES_1:56</a></i>;<br/>
<a NAME="E11:60"/><b>then </b>
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E12:60"/><b>then </b>
( <font color="Maroon">e</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">t</font> &amp; <font color="Maroon">e</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12">*&gt;</a></span> )
 
<b>by </b><i>, <a class="txt" href="trees_9.html#E7:60"><i><font color="Green">E27</font></i></a>, <a class="ref" href="finseq_1.html#T47">FINSEQ_1:47</a>, <a class="ref" href="modal_1.html#T53">MODAL_1:53</a></i>;<br/>
<a NAME="E13:60"/><b>then </b>
 <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12">*&gt;</a></span>
 
<b>by </b><i><a class="txt" href="trees_9.html#E10:60"><i><font color="Green">E28</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E14:60"/>
contradiction
 ;<br/>


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