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<b>let </b><font color="Maroon">W</font> be   <a href="trees_1.html#NM1">Tree</a>;<br/><b>let </b><font color="Maroon">L</font> be    <a href="trees_2.html#M2" target="_self">Level</a> of <font color="Maroon">W</font>;<br/>



<b>consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E2:15"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">L</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">w</font> where <font color="Olive">w</font> is    <a href="trees_1.html#M1">Element</a> of <font color="Maroon">W</font> :  <a href="finseq_1.html#K3">len</a> <font color="Olive">w</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> </span>}</span> 
 <b>by </b><i/>;<br/>
<a NAME="E3:15"/>
<font color="Maroon">L</font> is <a href="trees_1.html#V2">AntiChain_of_Prefixes-like</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:15_1"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">L</font> holds <br/><font color="Olive">x</font> is   <a href="finseq_1.html#NM1">FinSequence</a>
  <i><font color="Red">:: according to </font></i><a class="ref" href="trees_1.html#D13">TREES_1:def 13</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:15_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">L</font>
 ;<br/>

<a NAME="E2:15_1_1"/><b>then </b>
<font color="Maroon">x</font> is    <a href="trees_1.html#M1">Element</a> of <font color="Maroon">W</font>
 
;<br/>
<b>hence </b><a NAME="E3:15_1_1"/>
<font color="Maroon">x</font> is   <a href="finseq_1.html#NM1">FinSequence</a>
 ;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon">v1</font> be   <a href="finseq_1.html#NM1">FinSequence</a>;<br/><b>let </b><font color="Maroon">v2</font> be   <a href="finseq_1.html#NM1">FinSequence</a>;<br/>



<b>assume </b><a NAME="E2:15_1"/>
<font color="Maroon">v1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">L</font>
 ;<br/>

<a NAME="E3:15_1"/><b>then </b><i><font color="Green">E52</font></i>: 
 ex <font color="Olive">w1</font> being   <a href="trees_1.html#M1">Element</a> of <font color="Maroon">W</font> st <br/>( <font color="Maroon">v1</font> <a href="hidden.html#R1">=</a> <font color="Olive">w1</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Olive">w1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> )
 
<b>by </b><i/>;<br/>
<b>assume </b><a NAME="E4:15_1"/>
<font color="Maroon">v2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">L</font>
 ;<br/>

<a NAME="E5:15_1"/><b>then </b>
 ex <font color="Olive">w2</font> being   <a href="trees_1.html#M1">Element</a> of <font color="Maroon">W</font> st <br/>( <font color="Maroon">v2</font> <a href="hidden.html#R1">=</a> <font color="Olive">w2</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Olive">w2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E6:15_1"/>
( <font color="Maroon">v1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v2</font> or  not <font color="Maroon">v1</font>,<font color="Maroon">v2</font> <a href="xboole_0.html#R3">are_c=-comparable</a>  )
 <b>by </b><i>, <a class="ref" href="trees_1.html#T19">TREES_1:19</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:15"/>
<font color="Maroon">L</font> is    <a href="trees_1.html#M4">AntiChain_of_Prefixes</a> of <font color="Maroon">W</font>
 <b>by </b><i><a class="ref" href="trees_1.html#D14">TREES_1:def 14</a></i>;<br/>


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