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<b>let </b><font color="Maroon">T</font> be   <a href="trees_1.html#NM1">Tree</a>;<br/>

<b>assume </b><a NAME="E1:10_1_1"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">T</font> is <a href="bintree1.html#V1" target="_self">binary</a>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">t</font> be    <a href="trees_1.html#M1">Element</a> of <font color="Maroon">T</font>;<br/><b>assume </b><a NAME="E1:10_1_1_1"/><i><font color="Green">E30</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">2</span><a href="lang1.html#K3">*&gt;</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">T</font>
 ;<br/><a NAME="E2:10_1_1_1"/><b>then </b><i><font color="Green">E32</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">0</span><a href="lang1.html#K3">*&gt;</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">T</font>
 <b>by </b><i><a class="ref" href="trees_1.html#D5">TREES_1:def 5</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:10_1_1_1_1"/>
( <font color="Maroon">t</font> <a href="hidden.html#R2">in</a>  <a href="trees_1.html#K3">Leaves</a> <font color="Maroon">T</font> or  not <font color="Maroon">t</font> <a href="hidden.html#R2">in</a>  <a href="trees_1.html#K3">Leaves</a> <font color="Maroon">T</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:10_1_1_1_1_1"/>
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a>  <a href="trees_1.html#K3">Leaves</a> <font color="Maroon">T</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:10_1_1_1_1_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="modal_1.html#T53">MODAL_1:53</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:10_1_1_1_1_2"/>
not <font color="Maroon">t</font> <a href="hidden.html#R2">in</a>  <a href="trees_1.html#K3">Leaves</a> <font color="Maroon">T</font>
 ;<br/><div class="add"><a NAME="E2:10_1_1_1_1_2"/><b>then </b><i><font color="Green">E35</font></i>: 
 <a href="trees_2.html#K1">succ</a> <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="lang1.html#K3">&lt;*</a><span class="default">0</span><a href="lang1.html#K3">*&gt;</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="lang1.html#K3">&lt;*</a><span class="default">1</span><a href="lang1.html#K3">*&gt;</a></span></span>)</span></span><a href="tarski.html#K2">}</a></span>
 <b>by </b><i>, </i>;<br/><a NAME="E3:10_1_1_1_1_2"/>
<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">2</span><a href="lang1.html#K3">*&gt;</a></span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="lang1.html#K3">&lt;*</a><span class="default"><font color="Olive">n</font></span><a href="lang1.html#K3">*&gt;</a></span></span>)</span> where <font color="Olive">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <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="Olive">n</font></span><a href="lang1.html#K3">*&gt;</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">T</font> </span>}</span> 
 <b>by </b><i/>;<br/><a NAME="E4:10_1_1_1_1_2"/><b>then </b>
<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">2</span><a href="lang1.html#K3">*&gt;</a></span> <a href="hidden.html#R2">in</a>  <a href="trees_2.html#K1">succ</a> <font color="Maroon">t</font>
 <b>by </b><i><a class="ref" href="trees_2.html#D5">TREES_2:def 5</a></i>;<br/><a NAME="E5:10_1_1_1_1_2"/><b>then </b><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">2</span><a href="lang1.html#K3">*&gt;</a></span> <a href="hidden.html#R1">=</a> <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">0</span><a href="lang1.html#K3">*&gt;</a></span> or <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">2</span><a href="lang1.html#K3">*&gt;</a></span> <a href="hidden.html#R1">=</a> <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">1</span><a href="lang1.html#K3">*&gt;</a></span> )
 <b>by </b><i>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><div><i><font color="Green">E38</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1_1_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:10_1_1_1_1_2_1"/>
<span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default">2</span><a href="lang1.html#K3">*&gt;</a></span> <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>
 ;<br/><a NAME="E2:10_1_1_1_1_2_1"/><b>then </b>
( <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default">2</span><a href="lang1.html#K3">*&gt;</a></span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> 2 &amp; <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default">2</span><a href="lang1.html#K3">*&gt;</a></span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i><a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>;<br/><b>hence </b><a NAME="E3:10_1_1_1_1_2_1"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1_1_1_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:10_1_1_1_1_2_2"/>
<span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default">2</span><a href="lang1.html#K3">*&gt;</a></span> <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>
 ;<br/><a NAME="E2:10_1_1_1_1_2_2"/><b>then </b>
( <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default">2</span><a href="lang1.html#K3">*&gt;</a></span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> 2 &amp; <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default">2</span><a href="lang1.html#K3">*&gt;</a></span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i><a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>;<br/><b>hence </b><a NAME="E3:10_1_1_1_1_2_2"/>
contradiction
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E8:10_1_1_1_1_2"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#T46">FINSEQ_1:46</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:10_1_1"/>
<font color="Maroon">T</font> is <a href="trees_2.html#V1">finite-order</a>
 <b>by </b><i><a class="ref" href="trees_2.html#D2">TREES_2:def 2</a></i>;<br/>


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