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

<b>let </b><font color="Maroon">c<sub>1</sub></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="E1:20"/>
not <font color="Maroon">c<sub>1</sub></font> is <a href="finset_1.html#V1">finite</a>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>2</sub></font> being    <a href="trees_2.html#M1">Chain</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:20"/><i><font color="Green">E19</font></i>: 
not <font color="Maroon">c<sub>2</sub></font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i><a class="ref" href="qc_lang4.html#T21" target="_self">Th21</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>3</sub></font> being   <a href="trees_2.html#NM3">Branch</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E5:20"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="trees_2.html#T30">TREES_2:30</a></i>;<br/>
<a NAME="E6:20"/>
not <font color="Maroon">c<sub>3</sub></font> is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i><a class="txt" href="qc_lang4.html#E3:20"><i><font color="Green">E19</font></i></a>, <a class="txt" href="qc_lang4.html#E5:20"><i><font color="Green">E20</font></i></a>, <a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/>
<b>hence </b><a NAME="E7:20"/>
not for <font color="Olive">b<sub>1</sub></font> being  <a href="trees_2.html#NM3">Branch</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Olive">b<sub>1</sub></font> is <a href="finset_1.html#V1">finite</a>
 ;<br/>


</div>
