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

<b>consider </b><font color="Maroon">X</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E2:47_1_1"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">t</font> where <font color="Olive">t</font> is    <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K2" target="_self">tree_of_subformulae</a> <font color="Maroon">F</font></span>)</span> : <span class="p1">(<span class="default"><a href="qc_lang4.html#K2" target="_self">tree_of_subformulae</a> <font color="Maroon">F</font></span>)</span> <a href="trees_2.html#K3">.</a> <font color="Olive">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> </span>}</span> 
 ;<br/>
<a NAME="E3:47_1_1"/><i><font color="Green">E29</font></i>: 
for <font color="Olive">t</font> being   <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K2" target="_self">tree_of_subformulae</a> <font color="Maroon">F</font></span>)</span> holds <br/> ( <font color="Olive">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> iff <span class="p1">(<span class="default"><a href="qc_lang4.html#K2" target="_self">tree_of_subformulae</a> <font color="Maroon">F</font></span>)</span> <a href="trees_2.html#K3">.</a> <font color="Olive">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">t</font> be    <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K2" target="_self">tree_of_subformulae</a> <font color="Maroon">F</font></span>)</span>;<br/>

<b>thus </b><a NAME="E1:47_1_1_1"/>
( <font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> iff <span class="p1">(<span class="default"><a href="qc_lang4.html#K2" target="_self">tree_of_subformulae</a> <font color="Maroon">F</font></span>)</span> <a href="trees_2.html#K3">.</a> <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_1_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:47_1_1_1_1_1"/>
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/><b>then consider </b><font color="Maroon">t'</font> being    <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K2" target="_self">tree_of_subformulae</a> <font color="Maroon">F</font></span>)</span><b> such that </b><br/><a NAME="E3:47_1_1_1_1_1"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">t'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> &amp; <span class="p1">(<span class="default"><a href="qc_lang4.html#K2" target="_self">tree_of_subformulae</a> <font color="Maroon">F</font></span>)</span> <a href="trees_2.html#K3">.</a> <font color="Maroon">t'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> )
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E4:47_1_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="qc_lang4.html#K2" target="_self">tree_of_subformulae</a> <font color="Maroon">F</font></span>)</span> <a href="trees_2.html#K3">.</a> <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:47_1_1_1_1"/>
( <font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> implies <span class="p1">(<span class="default"><a href="qc_lang4.html#K2" target="_self">tree_of_subformulae</a> <font color="Maroon">F</font></span>)</span> <a href="trees_2.html#K3">.</a> <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> )
 ;<br/>

<b>assume </b><a NAME="E3:47_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="qc_lang4.html#K2" target="_self">tree_of_subformulae</a> <font color="Maroon">F</font></span>)</span> <a href="trees_2.html#K3">.</a> <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font>
 ;<br/>

<b>hence </b><a NAME="E4:47_1_1_1_1"/>
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i/>;<br/>


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


</div><b>end;</b></div>
<a NAME="E4:47_1_1"/>
<font color="Maroon">X</font> is    <a href="trees_1.html#M4">AntiChain_of_Prefixes</a> of  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K2" target="_self">tree_of_subformulae</a> <font color="Maroon">F</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:47_1_1_2"/><i><font color="Green">E32</font></i>: 
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">X</font> holds <br/><font color="Olive">x</font> is   <a href="finseq_1.html#NM1">FinSequence</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_1_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> ;<br/>

<b>assume </b><a NAME="E1:47_1_1_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<b>then consider </b><font color="Maroon">t</font> being    <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K2" target="_self">tree_of_subformulae</a> <font color="Maroon">F</font></span>)</span><b> such that </b><br/><a NAME="E3:47_1_1_2_1"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <span class="p1">(<span class="default"><a href="qc_lang4.html#K2" target="_self">tree_of_subformulae</a> <font color="Maroon">F</font></span>)</span> <a href="trees_2.html#K3">.</a> <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> )
 <b>by </b><i/>;<br/>
<b>thus </b><a NAME="E4:47_1_1_2_1"/>
<font color="Maroon">x</font> is   <a href="finseq_1.html#NM1">FinSequence</a>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E2:47_1_1_2"/>
for <font color="Olive">p1</font>, <font color="Olive">p2</font> being  <a href="finseq_1.html#NM1">FinSequence</a>  st <font color="Olive">p1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Olive">p2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Olive">p1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">p2</font> holds <br/> not <font color="Olive">p1</font>,<font color="Olive">p2</font> <a href="xboole_0.html#R3">are_c=-comparable</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p1</font> be   <a href="finseq_1.html#NM1">FinSequence</a>, <font color="Maroon">p2</font> be   <a href="finseq_1.html#NM1">FinSequence</a>;<br/>

<b>assume </b><a NAME="E1:47_1_1_2_2"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">p1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">p2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">p1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">p2</font> )
 ;<br/>

<b>then consider </b><font color="Maroon">t1</font> being    <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K2" target="_self">tree_of_subformulae</a> <font color="Maroon">F</font></span>)</span><b> such that </b><br/><a NAME="E3:47_1_1_2_2"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">t1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font> &amp; <span class="p1">(<span class="default"><a href="qc_lang4.html#K2" target="_self">tree_of_subformulae</a> <font color="Maroon">F</font></span>)</span> <a href="trees_2.html#K3">.</a> <font color="Maroon">t1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> )
 <b>by </b><i><a class="txt" href="qc_lang4.html#E5:47_1_1_2_2"><i><font color="Green">E40</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">t2</font> being    <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K2" target="_self">tree_of_subformulae</a> <font color="Maroon">F</font></span>)</span><b> such that </b><br/><a NAME="E5:47_1_1_2_2"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">t2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font> &amp; <span class="p1">(<span class="default"><a href="qc_lang4.html#K2" target="_self">tree_of_subformulae</a> <font color="Maroon">F</font></span>)</span> <a href="trees_2.html#K3">.</a> <font color="Maroon">t2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> )
 <b>by </b><i><a class="txt" href="qc_lang4.html#E5:47_1_1_2_2"><i><font color="Green">E40</font></i></a>, </i>;<br/>
<b>thus </b><a NAME="E6:47_1_1_2_2"/>
not <font color="Maroon">p1</font>,<font color="Maroon">p2</font> <a href="xboole_0.html#R3">are_c=-comparable</a> 
 <b>by </b><i>, , , <a class="txt" href="qc_lang4.html#E3:47_1_1_2_2"><i><font color="Green">E39</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">X</font> = <font color="Maroon">X</font> as   <a href="trees_1.html#NM5">AntiChain_of_Prefixes</a> <b>by </b><i>, <a class="ref" href="trees_1.html#D13">TREES_1:def 13</a></i>;<br/>
<a NAME="E4:47_1_1_2"/>
<font color="Maroon">X</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K2" target="_self">tree_of_subformulae</a> <font color="Maroon">F</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_1_2_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="E1:47_1_1_2_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<b>then consider </b><font color="Maroon">t</font> being    <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K2" target="_self">tree_of_subformulae</a> <font color="Maroon">F</font></span>)</span><b> such that </b><br/><a NAME="E3:47_1_1_2_3"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <span class="p1">(<span class="default"><a href="qc_lang4.html#K2" target="_self">tree_of_subformulae</a> <font color="Maroon">F</font></span>)</span> <a href="trees_2.html#K3">.</a> <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> )
 <b>by </b><i/>;<br/>
<b>thus </b><a NAME="E4:47_1_1_2_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K2" target="_self">tree_of_subformulae</a> <font color="Maroon">F</font></span>)</span>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:47_1_1_2"/>
<font color="Maroon">X</font> is    <a href="trees_1.html#M4">AntiChain_of_Prefixes</a> of  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K2" target="_self">tree_of_subformulae</a> <font color="Maroon">F</font></span>)</span>
 <b>by </b><i><a class="ref" href="trees_1.html#D14">TREES_1:def 14</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:47_1_1"/>
 ex <font color="Olive">b<sub>1</sub></font> being   <a href="trees_1.html#M4">AntiChain_of_Prefixes</a> of  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K2" target="_self">tree_of_subformulae</a> <font color="Maroon">F</font></span>)</span> st <br/>for <font color="Olive">t</font> being   <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K2" target="_self">tree_of_subformulae</a> <font color="Maroon">F</font></span>)</span> holds <br/> ( <font color="Olive">t</font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> iff <span class="p1">(<span class="default"><a href="qc_lang4.html#K2" target="_self">tree_of_subformulae</a> <font color="Maroon">F</font></span>)</span> <a href="trees_2.html#K3">.</a> <font color="Olive">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> )
 <b>by </b><i/>;<br/>


</div>
