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<b>let </b><font color="Maroon" title="c1">T</font> be  <a href="trees_9.html#V2" title="TREES_9:attr.2">finite-branching</a> <a href="trees_1.html#NM1" title="TREES_1:NM.1">Tree</a>;<br/>

<b>assume </b><a NAME="E1:19"/><i><font color="Green" title="E18">A1</font></i>: 
not <font color="Maroon" title="c1">T</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 ;<br/>

<a NAME="E2:19"/><i><font color="Green" title="E19">A2</font></i>: 
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="19_1"><b>proof </b></a><div class="add">

<b>let </b><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> ;<br/>

<a NAME="E1:19_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> 
 
<b>by </b><i><a class="txt" href="qc_lang4.html#E1:19"><i><font color="Green" title="E18">A1</font></i></a>, <a class="ref" href="qc_lang4.html#T20" target="_self" title="QC_LANG4:th.20">Th20</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c3">t</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:19_1"/><i><font color="Green" title="E20">A3</font></i>: 
<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>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c4">w</font> being    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of <font color="Maroon" title="c1">T</font><b> such that </b><br/><a NAME="E5:19_1"/><i><font color="Green" title="E21">A4</font></i>: 
( <font color="Maroon" title="c3">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">w</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font> )
 <b>by </b><i><a class="txt" href="qc_lang4.html#E3:19_1"><i><font color="Green" title="E20">A3</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">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> <b>by </b><i><a class="txt" href="qc_lang4.html#E5:19_1"><i><font color="Green" title="E21">A4</font></i></a></i>;<br/>
<a NAME="E7:19_1"/>
 <a href="trees_1.html#K1" title="TREES_1:func.1">ProperPrefixes</a> <font color="Maroon" title="c5">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>
 
<b>by </b><i><a class="ref" href="qc_lang4.html#T16" target="_self" title="QC_LANG4:th.16">Th16</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c6">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><b> such that </b><br/><a NAME="E9:19_1"/><i><font color="Green" title="E22">A5</font></i>: 
<font color="Maroon" title="c6">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="c5">t</font>
 ;<br/>
<a NAME="E10:19_1"/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c6">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font>
 
<b>by </b><i><a class="txt" href="qc_lang4.html#E5:19_1"><i><font color="Green" title="E21">A4</font></i></a>, <a class="txt" href="qc_lang4.html#E9:19_1"><i><font color="Green" title="E22">A5</font></i></a>, <a class="ref" href="trees_1.html#T68" title="TREES_1:th.68">TREES_1:68</a></i>;<br/>
<b>hence </b><a NAME="E11:19_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>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E3:19"/>
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/>
<b>hence </b><a NAME="E4:19"/>
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>
 <b>by </b><i><a class="txt" href="qc_lang4.html#E2:19"><i><font color="Green" title="E19">A2</font></i></a>, <a class="ref" href="trees_2.html#T31" title="TREES_2:th.31">TREES_2:31</a></i>;<br/>


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