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

<b>let </b><font color="Maroon">W</font> be   <a href="trees_1.html#NM1">Tree</a>;<br/><b>let </b><font color="Maroon">C</font> be    <a href="trees_2.html#M1" target="_self">Chain</a> of <font color="Maroon">W</font>;<br/>



<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b>( <font color="Maroon">a<sub>1</sub></font> is    <a href="trees_2.html#M1" target="_self">Chain</a> of <font color="Maroon">W</font> &amp; <font color="Maroon">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">a<sub>1</sub></font> &amp; ( for <font color="Olive">p</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a<sub>1</sub></font> holds <br/> <a href="trees_1.html#K1">ProperPrefixes</a> <font color="Olive">p</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">a<sub>1</sub></font> ) );<br/>
<b>consider </b><font color="Maroon">X</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E2:43"/><i><font color="Green">E50</font></i>: 
for <font color="Olive">Y</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> iff ( <font color="Olive">Y</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">W</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">Y</font>] ) )
 <b>from </b><i><a class="ref" href="xboole_0.html#S1">XBOOLE_0:sch 1</a>();<br/></i>
<b>set </b><font color="Maroon">Z</font> = <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> :  ex <font color="Olive">p</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> &amp; <font color="Olive">w</font> <a href="tarski.html#R1">is_a_prefix_of</a> <font color="Olive">p</font> ) </span>}</span> ;<br/>
<a NAME="E3:43"/><i><font color="Green">E52</font></i>: 
<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> :  ex <font color="Olive">p</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> &amp; <font color="Olive">w</font> <a href="tarski.html#R1">is_a_prefix_of</a> <font color="Olive">p</font> ) </span>}</span>  is    <a href="trees_2.html#M1" target="_self">Chain</a> of <font color="Maroon">W</font>
 
<b>by </b><i/>;<br/>
<a NAME="E4:43"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">C</font> <a href="tarski.html#R1">c=</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> :  ex <font color="Olive">p</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> &amp; <font color="Olive">w</font> <a href="tarski.html#R1">is_a_prefix_of</a> <font color="Olive">p</font> ) </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</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="E1:43_1"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">w</font> = <font color="Maroon">x</font> as    <a href="trees_1.html#M1">Element</a> of <font color="Maroon">W</font> ;<br/>
<a NAME="E3:43_1"/>
<font color="Maroon">w</font> <a href="tarski.html#R1">is_a_prefix_of</a> <font color="Maroon">w</font>
 
;<br/>
<b>hence </b><a NAME="E4:43_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</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> :  ex <font color="Olive">p</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> &amp; <font color="Olive">w</font> <a href="tarski.html#R1">is_a_prefix_of</a> <font color="Olive">p</font> ) </span>}</span> 
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">p</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:43_2"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</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> :  ex <font color="Olive">p</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> &amp; <font color="Olive">w</font> <a href="tarski.html#R1">is_a_prefix_of</a> <font color="Olive">p</font> ) </span>}</span> 
 ;<br/><a NAME="E2:43_2"/><b>then </b><i><font color="Green">E64</font></i>: 
 ex <font color="Olive">w</font> being   <a href="trees_1.html#M1">Element</a> of <font color="Maroon">W</font> st <br/>( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Olive">w</font> &amp;  ex <font color="Olive">p</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> &amp; <font color="Olive">w</font> <a href="tarski.html#R1">is_a_prefix_of</a> <font color="Olive">p</font> ) )
 ;<br/><b>then consider </b><font color="Maroon">q</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E4:43_2"/><i><font color="Green">E65</font></i>: 
( <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> &amp; <font color="Maroon">p</font> <a href="tarski.html#R1">is_a_prefix_of</a> <font color="Maroon">q</font> )
 ;<br/><b>thus </b><a NAME="E5:43_2"/>
 <a href="trees_1.html#K1">ProperPrefixes</a> <font color="Maroon">p</font> <a href="tarski.html#R1">c=</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> :  ex <font color="Olive">p</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> &amp; <font color="Olive">w</font> <a href="tarski.html#R1">is_a_prefix_of</a> <font color="Olive">p</font> ) </span>}</span> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</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="E1:43_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="trees_1.html#K1">ProperPrefixes</a> <font color="Maroon">p</font>
 ;<br/>

<b>then consider </b><font color="Maroon">r</font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E3:43_2_1"/><i><font color="Green">E66</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> &amp; <font color="Maroon">r</font> <a href="trees_1.html#NR3" target="_self">is_a_proper_prefix_of</a> <font color="Maroon">p</font> )
 <b>by </b><i><a class="ref" href="trees_1.html#D4">TREES_1:def 4</a></i>;<br/>
<a NAME="E4:43_2_1"/>
( <font color="Maroon">r</font> <a href="tarski.html#R1">is_a_prefix_of</a> <font color="Maroon">p</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> )
 
<b>by </b><i>, , <a class="ref" href="xboole_0.html#D8">XBOOLE_0:def 8</a></i>;<br/>
<a NAME="E5:43_2_1"/><b>then </b>
( <font color="Maroon">r</font> <a href="tarski.html#R1">is_a_prefix_of</a> <font color="Maroon">q</font> &amp; <font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> )
 
<b>by </b><i>, <a class="ref" href="trees_1.html#T45">TREES_1:45</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>hence </b><a NAME="E6:43_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</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> :  ex <font color="Olive">p</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> &amp; <font color="Olive">w</font> <a href="tarski.html#R1">is_a_prefix_of</a> <font color="Olive">p</font> ) </span>}</span> 
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E6:43"/><b>then </b><i><font color="Green">E90</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i>, , </i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">Z</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><b>that </b><br/><a NAME="E1:43_3"/><i><font color="Green">E91</font></i>: 
( <font color="Maroon">Z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">Z</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> )
 <b>and </b><br/><a NAME="E2:43_3"/><i><font color="Green">E92</font></i>: 
<font color="Maroon">Z</font> is <a href="ordinal1.html#V6">c=-linear</a>
 ;<br/><a NAME="E3:43_3"/>
 <a href="tarski.html#K3">union</a> <font color="Maroon">Z</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">W</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</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="E1:43_3_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">Z</font>
 ;<br/>

<b>then consider </b><font color="Maroon">Y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:43_3_1"/><i><font color="Green">E93</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<a NAME="E4:43_3_1"/>
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">W</font>
 
<b>by </b><i>, , </i>;<br/>
<b>hence </b><a NAME="E5:43_3_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">Z'</font> =  <a href="tarski.html#K3">union</a> <font color="Maroon">Z</font> as   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">W</font> ;<br/><a NAME="E5:43_3"/><i><font color="Green">E94</font></i>: 
<font color="Maroon">Z'</font> is    <a href="trees_2.html#M1" target="_self">Chain</a> of <font color="Maroon">W</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_3_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="trees_2.html#D3" target="_self">TREES_2:def 3</a><br/><b>let </b><font color="Maroon">q</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>



<b>assume </b><a NAME="E1:43_3_2"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z'</font>
 ;<br/>

<b>then consider </b><font color="Maroon">X1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:43_3_2"/><i><font color="Green">E95</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X1</font> &amp; <font color="Maroon">X1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>assume </b><a NAME="E4:43_3_2"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z'</font>
 ;<br/>

<b>then consider </b><font color="Maroon">X2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:43_3_2"/><i><font color="Green">E96</font></i>: 
( <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X2</font> &amp; <font color="Maroon">X2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<a NAME="E7:43_3_2"/>
( <font color="Maroon">X1</font>,<font color="Maroon">X2</font> <a href="xboole_0.html#R3">are_c=-comparable</a>  &amp; <font color="Maroon">X1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">X2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 
<b>by </b><i>, , , , <a class="ref" href="ordinal1.html#D9">ORDINAL1:def 9</a></i>;<br/>
<a NAME="E8:43_3_2"/><b>then </b>
( ( <font color="Maroon">X1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X2</font> or <font color="Maroon">X2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X1</font> ) &amp; <font color="Maroon">X1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">X2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D9">XBOOLE_0:def 9</a></i>;<br/>
<a NAME="E9:43_3_2"/><b>then </b>
( ( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X2</font> or <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X1</font> ) &amp; <font color="Maroon">X1</font> is    <a href="trees_2.html#M1" target="_self">Chain</a> of <font color="Maroon">W</font> &amp; <font color="Maroon">X2</font> is    <a href="trees_2.html#M1" target="_self">Chain</a> of <font color="Maroon">W</font> )
 
<b>by </b><i>, , </i>;<br/>
<b>hence </b><a NAME="E10:43_3_2"/>
<font color="Maroon">p</font>,<font color="Maroon">q</font> <a href="xboole_0.html#R3">are_c=-comparable</a> 
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div><div><i><font color="Green">E97</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="43_3_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">p</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:43_3_3"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">Z</font>
 ;<br/><b>then consider </b><font color="Maroon">X1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:43_3_3"/><i><font color="Green">E98</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X1</font> &amp; <font color="Maroon">X1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><a NAME="E4:43_3_3"/>
(  <a href="trees_1.html#K1">ProperPrefixes</a> <font color="Maroon">p</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X1</font> &amp; <font color="Maroon">X1</font> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">Z</font> )
 <b>by </b><i>, , , <a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/><b>hence </b><a NAME="E5:43_3_3"/>
 <a href="trees_1.html#K1">ProperPrefixes</a> <font color="Maroon">p</font> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">Z</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/></div><b>end;</b></div><b>consider </b><font color="Maroon">x</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Z</font>;<br/><a NAME="E8:43_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><a NAME="E9:43_3"/><b>then </b>
( <font color="Maroon">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x</font> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">Z</font> )
 <b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/><a NAME="E10:43_3"/><b>then </b>
<font color="Maroon">C</font> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">Z</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><b>hence </b><a NAME="E11:43_3"/>
 <a href="tarski.html#K3">union</a> <font color="Maroon">Z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div>
<b>then consider </b><font color="Maroon">Y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E9:43"/><i><font color="Green">E99</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>and </b><br/><a NAME="E10:43"/><i><font color="Green">E108</font></i>: 
for <font color="Olive">Z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">Z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Olive">Z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">Y</font> holds <br/> not <font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Z</font>
 <b>by </b><i>, <a class="ref" href="orders_1.html#T177">ORDERS_1:177</a></i>;<br/>
<b>reconsider </b><font color="Maroon">Y</font> = <font color="Maroon">Y</font> as    <a href="trees_2.html#M1" target="_self">Chain</a> of <font color="Maroon">W</font> <b>by </b><i>, </i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_4"><b>now </b></a><div class="add"><b>thus </b><a NAME="E1:43_4"/>
for <font color="Olive">p</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> holds <br/> <a href="trees_1.html#K1">ProperPrefixes</a> <font color="Olive">p</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font>
 <b>by </b><i>, </i>;<br/><b>given </b><font color="Maroon">p</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><a NAME="E2:43_4"/><i><font color="Green">E109</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font>
 <b>and </b><br/><a NAME="E3:43_4"/><i><font color="Green">E110</font></i>: 
for <font color="Olive">q</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> holds <br/><font color="Olive">q</font> <a href="trees_1.html#NR3" target="_self">is_a_proper_prefix_of</a> <font color="Maroon">p</font>
 ;<br/><b>set </b><font color="Maroon">Z</font> = <span class="p1">(<span class="default"><a href="trees_1.html#K1">ProperPrefixes</a> <font color="Maroon">p</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">p</font></span><a href="tarski.html#K1">}</a></span>;<br/><a NAME="E4:43_4"/><i><font color="Green">E111</font></i>: 
(  <a href="trees_1.html#K1">ProperPrefixes</a> <font color="Maroon">p</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">W</font> &amp; <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">p</font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">W</font> )
 <b>by </b><i>, <a class="ref" href="trees_1.html#D5">TREES_1:def 5</a>, <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/><a NAME="E5:43_4"/><b>then </b><i><font color="Green">E112</font></i>: 
<span class="p1">(<span class="default"><a href="trees_1.html#K1">ProperPrefixes</a> <font color="Maroon">p</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">p</font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">W</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/><b>reconsider </b><font color="Maroon">Z'</font> = <span class="p1">(<span class="default"><a href="trees_1.html#K1">ProperPrefixes</a> <font color="Maroon">p</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">p</font></span><a href="tarski.html#K1">}</a></span> as   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">W</font> <b>by </b><i>, <a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/><a NAME="E7:43_4"/><i><font color="Green">E113</font></i>: 
<font color="Maroon">Z'</font> is    <a href="trees_2.html#M1" target="_self">Chain</a> of <font color="Maroon">W</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">q</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="trees_2.html#D3" target="_self">TREES_2:def 3</a><br/><b>let </b><font color="Maroon">r</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>



<b>assume </b><a NAME="E1:43_4_1"/>
( <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z'</font> &amp; <font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z'</font> )
 ;<br/>

<a NAME="E2:43_4_1"/><b>then </b>
( ( <font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="trees_1.html#K1">ProperPrefixes</a> <font color="Maroon">p</font> or <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">p</font></span><a href="tarski.html#K1">}</a></span> ) &amp; ( <font color="Maroon">r</font> <a href="hidden.html#R2">in</a>  <a href="trees_1.html#K1">ProperPrefixes</a> <font color="Maroon">p</font> or <font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">p</font></span><a href="tarski.html#K1">}</a></span> ) )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E3:43_4_1"/><b>then </b>
( ( <font color="Maroon">q</font> <a href="trees_1.html#NR3" target="_self">is_a_proper_prefix_of</a> <font color="Maroon">p</font> or <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> ) &amp; ( <font color="Maroon">r</font> <a href="trees_1.html#NR3" target="_self">is_a_proper_prefix_of</a> <font color="Maroon">p</font> or <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> ) )
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a>, <a class="ref" href="trees_1.html#T36">TREES_1:36</a></i>;<br/>
<a NAME="E4:43_4_1"/><b>then </b>
( <font color="Maroon">q</font> <a href="tarski.html#R1">is_a_prefix_of</a> <font color="Maroon">p</font> &amp; <font color="Maroon">r</font> <a href="tarski.html#R1">is_a_prefix_of</a> <font color="Maroon">p</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D8">XBOOLE_0:def 8</a></i>;<br/>
<b>hence </b><a NAME="E5:43_4_1"/>
<font color="Maroon">q</font>,<font color="Maroon">r</font> <a href="xboole_0.html#R3">are_c=-comparable</a> 
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><div><i><font color="Green">E114</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="43_4_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">q</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:43_4_2"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="trees_1.html#K1">ProperPrefixes</a> <font color="Maroon">p</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">p</font></span><a href="tarski.html#K1">}</a></span>
 ;<br/><a NAME="E2:43_4_2"/><b>then </b>
( <font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="trees_1.html#K1">ProperPrefixes</a> <font color="Maroon">p</font> or <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">p</font></span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E3:43_4_2"/><b>then </b>
( <font color="Maroon">q</font> <a href="trees_1.html#NR3" target="_self">is_a_proper_prefix_of</a> <font color="Maroon">p</font> or <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a>, <a class="ref" href="trees_1.html#T36">TREES_1:36</a></i>;<br/><a NAME="E4:43_4_2"/><b>then </b>
<font color="Maroon">q</font> <a href="tarski.html#R1">is_a_prefix_of</a> <font color="Maroon">p</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D8">XBOOLE_0:def 8</a></i>;<br/><a NAME="E5:43_4_2"/><b>then </b>
(  <a href="trees_1.html#K1">ProperPrefixes</a> <font color="Maroon">q</font> <a href="tarski.html#R1">c=</a>  <a href="trees_1.html#K1">ProperPrefixes</a> <font color="Maroon">p</font> &amp;  <a href="trees_1.html#K1">ProperPrefixes</a> <font color="Maroon">p</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="trees_1.html#K1">ProperPrefixes</a> <font color="Maroon">p</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">p</font></span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i><a class="ref" href="trees_1.html#T41">TREES_1:41</a>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><b>hence </b><a NAME="E6:43_4_2"/>
 <a href="trees_1.html#K1">ProperPrefixes</a> <font color="Maroon">q</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="trees_1.html#K1">ProperPrefixes</a> <font color="Maroon">p</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">p</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/></div><b>end;</b></div><a NAME="E9:43_4"/><i><font color="Green">E115</font></i>: 
<font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="trees_1.html#K1">ProperPrefixes</a> <font color="Maroon">p</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">p</font></span><a href="tarski.html#K1">}</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_4_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</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="E1:43_4_3"/><i><font color="Green">E116</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">t</font> = <font color="Maroon">x</font> as    <a href="trees_1.html#M1">Element</a> of <font color="Maroon">W</font> ;<br/>
<a NAME="E3:43_4_3"/>
<font color="Maroon">t</font> <a href="trees_1.html#NR3" target="_self">is_a_proper_prefix_of</a> <font color="Maroon">p</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:43_4_3"/><b>then </b>
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a>  <a href="trees_1.html#K1">ProperPrefixes</a> <font color="Maroon">p</font>
 
<b>by </b><i><a class="ref" href="trees_1.html#T36">TREES_1:36</a></i>;<br/>
<b>hence </b><a NAME="E5:43_4_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="trees_1.html#K1">ProperPrefixes</a> <font color="Maroon">p</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">p</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


</div><b>end;</b></div><a NAME="E10:43_4"/>
<font color="Maroon">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font>
 <b>by </b><i>, </i>;<br/><a NAME="E11:43_4"/><b>then </b>
( <span class="p1">(<span class="default"><a href="trees_1.html#K1">ProperPrefixes</a> <font color="Maroon">p</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">p</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">W</font> &amp; <font color="Maroon">C</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="trees_1.html#K1">ProperPrefixes</a> <font color="Maroon">p</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">p</font></span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i>, , <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E12:43_4"/><b>then </b><i><font color="Green">E117</font></i>: 
<span class="p1">(<span class="default"><a href="trees_1.html#K1">ProperPrefixes</a> <font color="Maroon">p</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">p</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i>, , </i>;<br/><a NAME="E13:43_4"/>
( not <font color="Maroon">p</font> <a href="trees_1.html#NR3" target="_self">is_a_proper_prefix_of</a> <font color="Maroon">p</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">p</font></span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E14:43_4"/><b>then </b>
( not <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="trees_1.html#K1">ProperPrefixes</a> <font color="Maroon">p</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">p</font></span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>hence </b><a NAME="E15:43_4"/>
contradiction
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">Y</font> = <font color="Maroon">Y</font> as   <a href="trees_2.html#NM3">Branch</a> of <font color="Maroon">W</font> <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">Y</font>
;<br/>

<b>thus </b><a NAME="E14:43"/>
<font color="Maroon">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font>
 <b>by </b><i>, </i>;<br/>


</div>
