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

<span class="kw">let </span><font color="Maroon" title="c1">T</font> be   <a href="trees_9.html#V1" title="TREES_9:attr.1">finite-branching</a>  <a href="trees_1.html#NM1" title="TREES_1:NM.1">Tree</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( not <font color="Maroon" title="c1">T</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  implies  ex <font color="Olive" title="b1">C</font> being   <a href="trees_2.html#M1" title="TREES_2:mode.1">Chain</a> of <font color="Maroon" title="c1">T</font> st not <font color="Olive" title="b1">C</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:22"/><span class="lab"><font color="Green" title="E18">A1</font></span>: 
not <font color="Maroon" title="c1">T</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> not for <font color="Olive" title="b1">C</font> being   <a href="trees_2.html#M1" title="TREES_2:mode.1">Chain</a> of <font color="Maroon" title="c1">T</font> holds  <font color="Olive" title="b1">C</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> </span><br/>

<a NAME="E2:22"/><span class="lab"><font color="Green" title="E19">A2</font></span>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   ex <font color="Olive" title="b2">C</font> being  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="trees_2.html#M1" title="TREES_2:mode.1">Chain</a> of <font color="Maroon" title="c1">T</font> st  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b2">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c2">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">C</font> being  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="trees_2.html#M1" title="TREES_2:mode.1">Chain</a> of <font color="Maroon" title="c1">T</font> st  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b1">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font></span><br/>

<a NAME="E1:22_1"/>
<font color="Maroon" title="c1">T</font> <a href="trees_2.html#K2" title="TREES_2:func.2">-level</a> <font color="Maroon" title="c2">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22"><span class="lab"><font color="Green" title="E18">A1</font></span></a>, <a class="ref" href="qc_lang4.html#T20" target="_self" title="QC_LANG4:th.20">Th20</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c3">t</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:22_1"/><span class="lab"><font color="Green" title="E20">A3</font></span>: 
<font color="Maroon" title="c3">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">T</font> <a href="trees_2.html#K2" title="TREES_2:func.2">-level</a> <font color="Maroon" title="c2">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></span>;<br/>
<a NAME="E4:22_1"/><span class="lab"><font color="Green" title="E21">A4</font></span>: 
 ex <font color="Olive" title="b1">w</font> being   <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of <font color="Maroon" title="c1">T</font> st <br/>( <font color="Maroon" title="c3">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">w</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:22_1"><span class="lab"><font color="Green" title="E20">A3</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c4">t</font> = <font color="Maroon" title="c3">t</font> as    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of <font color="Maroon" title="c1">T</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:22_1"><span class="lab"><font color="Green" title="E20">A3</font></span></a></span>;<br/>
<a NAME="E6:22_1"/>
 <a href="trees_1.html#K1" title="TREES_1:func.1">ProperPrefixes</a> <font color="Maroon" title="c4">t</font> is   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="trees_2.html#M1" title="TREES_2:mode.1">Chain</a> of <font color="Maroon" title="c1">T</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="qc_lang4.html#T16" target="_self" title="QC_LANG4:th.16">Th16</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">C</font> being   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="trees_2.html#M1" title="TREES_2:mode.1">Chain</a> of <font color="Maroon" title="c1">T</font><span class="kw"> such that </span><br/><a NAME="E8:22_1"/><span class="lab"><font color="Green" title="E22">A5</font></span>: 
<font color="Maroon" title="c5">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="trees_1.html#K1" title="TREES_1:func.1">ProperPrefixes</a> <font color="Maroon" title="c4">t</font>
 ;<br/>
<a NAME="E9:22_1"/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c5">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:22_1"><span class="lab"><font color="Green" title="E21">A4</font></span></a>, <a class="txt" href="#E8:22_1"><span class="lab"><font color="Green" title="E22">A5</font></span></a>, <a class="ref" href="trees_1.html#T68" title="TREES_1:th.68">TREES_1:68</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:22_1"/>
 ex <font color="Olive" title="b1">C</font> being  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="trees_2.html#M1" title="TREES_2:mode.1">Chain</a> of <font color="Maroon" title="c1">T</font> st  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b1">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E3:22"/>
for <font color="Olive" title="b1">t</font> being   <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of <font color="Maroon" title="c1">T</font> holds   <a href="trees_2.html#K1" title="TREES_2:func.1">succ</a> <font color="Olive" title="b1">t</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> 
 
;<br/>
<span class="kw">hence </span><a NAME="E4:22"/>
not for <font color="Olive" title="b1">C</font> being   <a href="trees_2.html#M1" title="TREES_2:mode.1">Chain</a> of <font color="Maroon" title="c1">T</font> holds  <font color="Olive" title="b1">C</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:22"><span class="lab"><font color="Green" title="E19">A2</font></span></a>, <a class="ref" href="trees_2.html#T31" title="TREES_2:th.31">TREES_2:31</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
