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<b>let </b><font color="Maroon">T</font> be  <a href="trees_9.html#V2">finite-branching</a> <a href="trees_1.html#NM1">Tree</a>;<br/>

<b>assume </b><a NAME="E1:19"/><i><font color="Green">E22</font></i>: 
not <font color="Maroon">T</font> is <a href="finset_1.html#V1">finite</a>
 ;<br/>

<a NAME="E2:19"/><i><font color="Green">E29</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   ex <font color="Olive">C</font> being <a href="finset_1.html#V1">finite</a>  <a href="trees_2.html#M1">Chain</a> of <font color="Maroon">T</font> st  <a href="card_1.html#K4">card</a> <font color="Olive">C</font> <a href="hidden.html#R1">=</a> <font color="Olive">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<a NAME="E1:19_1"/>
<font color="Maroon">T</font> <a href="trees_2.html#K2">-level</a> <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="qc_lang4.html#T4" target="_self">Th4</a>, </i>;<br/>
<b>then consider </b><font color="Maroon">t</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:19_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">T</font> <a href="trees_2.html#K2">-level</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon">w</font> being    <a href="trees_1.html#M1">Element</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E5:19_1"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> )
 <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">t</font> = <font color="Maroon">t</font> as    <a href="trees_1.html#M1">Element</a> of <font color="Maroon">T</font> <b>by </b><i/>;<br/>
<a NAME="E7:19_1"/>
 <a href="trees_1.html#K1">ProperPrefixes</a> <font color="Maroon">t</font> is  <a href="finset_1.html#V1">finite</a>  <a href="trees_2.html#M1">Chain</a> of <font color="Maroon">T</font>
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">C</font> being  <a href="finset_1.html#V1">finite</a>  <a href="trees_2.html#M1">Chain</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E9:19_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">C</font> <a href="hidden.html#R1">=</a>  <a href="trees_1.html#K1">ProperPrefixes</a> <font color="Maroon">t</font>
 ;<br/>
<a NAME="E10:19_1"/>
 <a href="card_1.html#K4">card</a> <font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 
<b>by </b><i>, <a class="txt" href="qc_lang4.html#E1:19"><i><font color="Green">E22</font></i></a>, <a class="ref" href="trees_1.html#T68">TREES_1:68</a></i>;<br/>
<b>hence </b><a NAME="E11:19_1"/>
 ex <font color="Olive">C</font> being <a href="finset_1.html#V1">finite</a>  <a href="trees_2.html#M1">Chain</a> of <font color="Maroon">T</font> st  <a href="card_1.html#K4">card</a> <font color="Olive">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E3:19"/>
for <font color="Olive">t</font> being   <a href="trees_1.html#M1">Element</a> of <font color="Maroon">T</font> holds   <a href="trees_2.html#K1">succ</a> <font color="Olive">t</font> is <a href="finset_1.html#V1">finite</a>
 
;<br/>
<b>hence </b><a NAME="E4:19"/>
not for <font color="Olive">C</font> being   <a href="trees_2.html#M1">Chain</a> of <font color="Maroon">T</font> holds  <font color="Olive">C</font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i>, <a class="ref" href="trees_2.html#T31">TREES_2:31</a></i>;<br/>


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