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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="trees_1.html#NM1">Tree</a>;<br/>

<a NAME="E1:63"/>
 <a href="xboole_0.html#K1">{}</a>  is <a href="trees_1.html#V2" target="_self">AntiChain_of_Prefixes-like</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:63_1"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="xboole_0.html#K1">{}</a>  holds <br/><font color="Olive">b<sub>1</sub></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" target="_self">TREES_1:def 13</a><br/>

<b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be   <a href="finseq_1.html#NM1">FinSequence</a>;<br/>



<b>thus </b><a NAME="E2:63_1"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font> implies  not <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#R3">are_c=-comparable</a>  )
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>2</sub></font> =  <a href="xboole_0.html#K1">{}</a>  as   <a href="trees_1.html#NM5">AntiChain_of_Prefixes</a> ;<br/>
<a NAME="E3:63"/>
<font color="Maroon">c<sub>2</sub></font> is    <a href="trees_1.html#M4" target="_self">AntiChain_of_Prefixes</a> of <font color="Maroon">c<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:63_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T2">XBOOLE_1:2</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="trees_1.html#D14" target="_self">TREES_1:def 14</a><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:63"/>
 <a href="xboole_0.html#K1">{}</a>  is    <a href="trees_1.html#M4" target="_self">AntiChain_of_Prefixes</a> of <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>3</sub></font> = <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span> as   <a href="trees_1.html#NM5">AntiChain_of_Prefixes</a> <b>by </b><i><a class="ref" href="trees_1.html#T70" target="_self">Th70</a></i>;<br/>
<a NAME="E6:63"/>
<font color="Maroon">c<sub>3</sub></font> is    <a href="trees_1.html#M4" target="_self">AntiChain_of_Prefixes</a> of <font color="Maroon">c<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a>,<a class="ref" href="trees_1.html#D14" target="_self">TREES_1:def 14</a><br/>

<b>assume </b><a NAME="E1:63_3"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<a NAME="E2:63_3"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:63_3"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="trees_1.html#T47" target="_self">Th47</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:63"/>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span> is    <a href="trees_1.html#M4" target="_self">AntiChain_of_Prefixes</a> of <font color="Maroon">c<sub>1</sub></font>
 ;<br/>


</div>
