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

<b>let </b><font color="Maroon">T0</font> be   <a href="trees_1.html#NM1">Tree</a>, <font color="Maroon">T1</font> be   <a href="trees_1.html#NM1">Tree</a>;<br/><b>let </b><font color="Maroon">t</font> be    <a href="trees_1.html#M1">Element</a> of  <a href="trees_3.html#K15">tree</a> <font color="Maroon">T0</font>,<font color="Maroon">T1</font>;<br/>



<b>set </b><font color="Maroon">RT</font> =  <a href="trees_3.html#K15">tree</a> <font color="Maroon">T0</font>,<font color="Maroon">T1</font>;<br/>
<div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon">p</font> be    <a href="trees_1.html#M1">Element</a> of <font color="Maroon">T0</font>;<br/><b>assume </b><a NAME="E1:11_1"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default">0</span><a href="lang1.html#K3">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">p</font>
 ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:11_1_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a>  <a href="trees_1.html#K3">Leaves</a> <span class="p1">(<span class="default"><a href="trees_3.html#K15">tree</a> <font color="Maroon">T0</font>,<font color="Maroon">T1</font></span>)</span>
 ;<br/><b>assume </b><a NAME="E2:11_1_1"/>
not <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="trees_1.html#K3">Leaves</a> <font color="Maroon">T0</font>
 ;<br/><b>then consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E4:11_1_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">p</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default"><font color="Maroon">n</font></span><a href="lang1.html#K3">*&gt;</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">T0</font>
 <b>by </b><i><a class="ref" href="modal_1.html#T54">MODAL_1:54</a></i>;<br/><a NAME="E5:11_1_1"/>
<span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default">0</span><a href="lang1.html#K3">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="lang1.html#K3">&lt;*</a><span class="default"><font color="Maroon">n</font></span><a href="lang1.html#K3">*&gt;</a></span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="trees_3.html#K15">tree</a> <font color="Maroon">T0</font>,<font color="Maroon">T1</font>
 <b>by </b><i>, <a class="ref" href="trees_3.html#T72">TREES_3:72</a></i>;<br/><a NAME="E6:11_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2"><a href="lang1.html#K3">&lt;*</a><span class="default">0</span><a href="lang1.html#K3">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">p</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default"><font color="Maroon">n</font></span><a href="lang1.html#K3">*&gt;</a></span> <a href="hidden.html#R2">in</a>  <a href="trees_3.html#K15">tree</a> <font color="Maroon">T0</font>,<font color="Maroon">T1</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T45">FINSEQ_1:45</a></i>;<br/><b>hence </b><a NAME="E7:11_1_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="modal_1.html#T54">MODAL_1:54</a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E3:11_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="trees_1.html#K3">Leaves</a> <font color="Maroon">T0</font>
 ;<br/><b>assume </b><a NAME="E4:11_1"/>
not <font color="Maroon">t</font> <a href="hidden.html#R2">in</a>  <a href="trees_1.html#K3">Leaves</a> <span class="p1">(<span class="default"><a href="trees_3.html#K15">tree</a> <font color="Maroon">T0</font>,<font color="Maroon">T1</font></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E6:11_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">t</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default"><font color="Maroon">n</font></span><a href="lang1.html#K3">*&gt;</a></span> <a href="hidden.html#R2">in</a>  <a href="trees_3.html#K15">tree</a> <font color="Maroon">T0</font>,<font color="Maroon">T1</font>
 <b>by </b><i><a class="ref" href="modal_1.html#T54">MODAL_1:54</a></i>;<br/><a NAME="E7:11_1"/>
<span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default">0</span><a href="lang1.html#K3">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="lang1.html#K3">&lt;*</a><span class="default"><font color="Maroon">n</font></span><a href="lang1.html#K3">*&gt;</a></span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="trees_3.html#K15">tree</a> <font color="Maroon">T0</font>,<font color="Maroon">T1</font>
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#T45">FINSEQ_1:45</a></i>;<br/><a NAME="E8:11_1"/><b>then </b>
<font color="Maroon">p</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default"><font color="Maroon">n</font></span><a href="lang1.html#K3">*&gt;</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">T0</font>
 <b>by </b><i><a class="ref" href="trees_3.html#T72">TREES_3:72</a></i>;<br/><b>hence </b><a NAME="E9:11_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="modal_1.html#T54">MODAL_1:54</a></i>;<br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon">p</font> be    <a href="trees_1.html#M1">Element</a> of <font color="Maroon">T1</font>;<br/>

<b>assume </b><a NAME="E2:11"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default">1</span><a href="lang1.html#K3">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">p</font>
 ;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:11_2"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a>  <a href="trees_1.html#K3">Leaves</a> <span class="p1">(<span class="default"><a href="trees_3.html#K15">tree</a> <font color="Maroon">T0</font>,<font color="Maroon">T1</font></span>)</span>
 ;<br/><b>assume </b><a NAME="E2:11_2"/>
not <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="trees_1.html#K3">Leaves</a> <font color="Maroon">T1</font>
 ;<br/><b>then consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E4:11_2"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">p</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default"><font color="Maroon">n</font></span><a href="lang1.html#K3">*&gt;</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">T1</font>
 <b>by </b><i><a class="ref" href="modal_1.html#T54">MODAL_1:54</a></i>;<br/><a NAME="E5:11_2"/>
<span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default">1</span><a href="lang1.html#K3">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="lang1.html#K3">&lt;*</a><span class="default"><font color="Maroon">n</font></span><a href="lang1.html#K3">*&gt;</a></span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="trees_3.html#K15">tree</a> <font color="Maroon">T0</font>,<font color="Maroon">T1</font>
 <b>by </b><i>, <a class="ref" href="trees_3.html#T73">TREES_3:73</a></i>;<br/><a NAME="E6:11_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2"><a href="lang1.html#K3">&lt;*</a><span class="default">1</span><a href="lang1.html#K3">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">p</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default"><font color="Maroon">n</font></span><a href="lang1.html#K3">*&gt;</a></span> <a href="hidden.html#R2">in</a>  <a href="trees_3.html#K15">tree</a> <font color="Maroon">T0</font>,<font color="Maroon">T1</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T45">FINSEQ_1:45</a></i>;<br/><b>hence </b><a NAME="E7:11_2"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="modal_1.html#T54">MODAL_1:54</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E4:11"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="trees_1.html#K3">Leaves</a> <font color="Maroon">T1</font>
 ;<br/>

<b>assume </b><a NAME="E5:11"/>
not <font color="Maroon">t</font> <a href="hidden.html#R2">in</a>  <a href="trees_1.html#K3">Leaves</a> <span class="p1">(<span class="default"><a href="trees_3.html#K15">tree</a> <font color="Maroon">T0</font>,<font color="Maroon">T1</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E7:11"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">t</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default"><font color="Maroon">n</font></span><a href="lang1.html#K3">*&gt;</a></span> <a href="hidden.html#R2">in</a>  <a href="trees_3.html#K15">tree</a> <font color="Maroon">T0</font>,<font color="Maroon">T1</font>
 <b>by </b><i><a class="ref" href="modal_1.html#T54">MODAL_1:54</a></i>;<br/>
<a NAME="E8:11"/>
<span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default">1</span><a href="lang1.html#K3">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="lang1.html#K3">&lt;*</a><span class="default"><font color="Maroon">n</font></span><a href="lang1.html#K3">*&gt;</a></span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="trees_3.html#K15">tree</a> <font color="Maroon">T0</font>,<font color="Maroon">T1</font>
 
<b>by </b><i>, , <a class="ref" href="finseq_1.html#T45">FINSEQ_1:45</a></i>;<br/>
<a NAME="E9:11"/><b>then </b>
<font color="Maroon">p</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default"><font color="Maroon">n</font></span><a href="lang1.html#K3">*&gt;</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">T1</font>
 
<b>by </b><i><a class="ref" href="trees_3.html#T73">TREES_3:73</a></i>;<br/>
<b>hence </b><a NAME="E10:11"/>
contradiction
 <b>by </b><i>, <a class="ref" href="modal_1.html#T54">MODAL_1:54</a></i>;<br/>


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