<?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/><b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="trees_1.html#M4">AntiChain_of_Prefixes</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:6_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="trees_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : for <font color="Olive">b<sub>1</sub></font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> holds <br/> not <font color="Olive">b<sub>2</sub></font> <a href="trees_1.html#NR3" target="_self">is_a_proper_prefix_of</a> <font color="Olive">b<sub>1</sub></font> </span>}</span>  <a href="subset_1.html#K6">\</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="trees_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : for <font color="Olive">b<sub>1</sub></font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> holds <br/> not <font color="Olive">b<sub>2</sub></font> <a href="tarski.html#R1">is_a_prefix_of</a> <font color="Olive">b<sub>1</sub></font> </span>}</span> 
 ;<br/><a NAME="E2:6_1"/><b>then </b><i><font color="Green">E7</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="trees_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : for <font color="Olive">b<sub>1</sub></font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> holds <br/> not <font color="Olive">b<sub>2</sub></font> <a href="trees_1.html#NR3" target="_self">is_a_proper_prefix_of</a> <font color="Olive">b<sub>1</sub></font> </span>}</span>  &amp; not <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="trees_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : for <font color="Olive">b<sub>1</sub></font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> holds <br/> not <font color="Olive">b<sub>2</sub></font> <a href="tarski.html#R1">is_a_prefix_of</a> <font color="Olive">b<sub>1</sub></font> </span>}</span>  )
 <b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><b>assume </b><a NAME="E3:6_1"/><i><font color="Green">E8</font></i>: 
not <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><a NAME="E4:6_1"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="trees_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>2</sub></font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> holds <br/> not <font color="Olive">b<sub>2</sub></font> <a href="tarski.html#R1">is_a_prefix_of</a> <font color="Olive">b<sub>1</sub></font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">c<sub>4</sub></font> being    <a href="trees_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E2:6_1_1"/><i><font color="Green">E9</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> holds <br/> not <font color="Olive">b<sub>1</sub></font> <a href="trees_1.html#NR3" target="_self">is_a_proper_prefix_of</a> <font color="Maroon">c<sub>4</sub></font> ) )
 <b>by </b><i><a class="txt" href="trees_a.html#E2:6_1"><i><font color="Green">E7</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:6_1_1_1"/><i><font color="Green">E10</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><a NAME="E2:6_1_1_1"/><b>then </b><i><font color="Green">E11</font></i>: 
not <font color="Maroon">c<sub>5</sub></font> <a href="trees_1.html#NR3" target="_self">is_a_proper_prefix_of</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="trees_a.html#E2:6_1_1"><i><font color="Green">E9</font></i></a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:6_1_1_1_1"/>
(  not <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">is_a_prefix_of</a> <font color="Maroon">c<sub>4</sub></font> or <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> )
 </span><b>by </b><i><a class="txt" href="trees_a.html#E2:6_1_1_1"><i><font color="Green">E11</font></i></a>, <a class="ref" href="xboole_0.html#D8">XBOOLE_0:def 8</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:6_1_1_1_1_1"/>
not <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">is_a_prefix_of</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:6_1_1_1_1_1"/>
not <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">is_a_prefix_of</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:6_1_1_1_1_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:6_1_1_1_1_2"/>
not <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">is_a_prefix_of</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="trees_a.html#E3:6_1"><i><font color="Green">E8</font></i></a>, <a class="txt" href="trees_a.html#E2:6_1_1"><i><font color="Green">E9</font></i></a>, <a class="txt" href="trees_a.html#E1:6_1_1_1"><i><font color="Green">E10</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:6_1_1"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="trees_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>2</sub></font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> holds <br/> not <font color="Olive">b<sub>2</sub></font> <a href="tarski.html#R1">is_a_prefix_of</a> <font color="Olive">b<sub>1</sub></font> ) )
 <b>by </b><i><a class="txt" href="trees_a.html#E2:6_1_1"><i><font color="Green">E9</font></i></a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E5:6_1"/>
contradiction
 <b>by </b><i><a class="txt" href="trees_a.html#E2:6_1"><i><font color="Green">E7</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:6"/>
<span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="trees_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : for <font color="Olive">b<sub>1</sub></font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> holds <br/> not <font color="Olive">b<sub>2</sub></font> <a href="trees_1.html#NR3" target="_self">is_a_proper_prefix_of</a> <font color="Olive">b<sub>1</sub></font> </span>}</span>  <a href="subset_1.html#K6">\</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="trees_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : for <font color="Olive">b<sub>1</sub></font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> holds <br/> not <font color="Olive">b<sub>2</sub></font> <a href="tarski.html#R1">is_a_prefix_of</a> <font color="Olive">b<sub>1</sub></font> </span>}</span>  <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><br/>

<b>let </b><font color="Maroon">c<sub>3</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><br/>

<b>assume </b><a NAME="E3:6"/><i><font color="Green">E7</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<a NAME="E4:6"/><i><font color="Green">E8</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="trees_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : for <font color="Olive">b<sub>1</sub></font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> holds <br/> not <font color="Olive">b<sub>2</sub></font> <a href="trees_1.html#NR3" target="_self">is_a_proper_prefix_of</a> <font color="Olive">b<sub>1</sub></font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="trees_a.html#T4" target="_self">Th4</a></i>;<br/>
<a NAME="E5:6"/>
not <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="trees_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : for <font color="Olive">b<sub>1</sub></font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> holds <br/> not <font color="Olive">b<sub>2</sub></font> <a href="tarski.html#R1">is_a_prefix_of</a> <font color="Olive">b<sub>1</sub></font> </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:6_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="trees_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : for <font color="Olive">b<sub>1</sub></font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> holds <br/> not <font color="Olive">b<sub>2</sub></font> <a href="tarski.html#R1">is_a_prefix_of</a> <font color="Olive">b<sub>1</sub></font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>4</sub></font> being    <a href="trees_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:6_2"/><i><font color="Green">E9</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> holds <br/> not <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">is_a_prefix_of</a> <font color="Maroon">c<sub>4</sub></font> ) )
 ;<br/>
<b>thus </b><a NAME="E4:6_2"/>
contradiction
 <b>by </b><i><a class="txt" href="trees_a.html#E3:6"><i><font color="Green">E7</font></i></a>, <a class="txt" href="trees_a.html#E3:6_2"><i><font color="Green">E9</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:6"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="trees_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : for <font color="Olive">b<sub>1</sub></font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> holds <br/> not <font color="Olive">b<sub>2</sub></font> <a href="trees_1.html#NR3" target="_self">is_a_proper_prefix_of</a> <font color="Olive">b<sub>1</sub></font> </span>}</span>  <a href="subset_1.html#K6">\</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="trees_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : for <font color="Olive">b<sub>1</sub></font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> holds <br/> not <font color="Olive">b<sub>2</sub></font> <a href="tarski.html#R1">is_a_prefix_of</a> <font color="Olive">b<sub>1</sub></font> </span>}</span> 
 <b>by </b><i><a class="txt" href="trees_a.html#E3:6"><i><font color="Green">E7</font></i></a>, <a class="txt" href="trees_a.html#E4:6"><i><font color="Green">E8</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>


</div>
