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

<b>let </b><font color="Maroon">P1</font> be    <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>, <font color="Maroon">P2</font> be    <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>;<br/>

<b>assume </b><a NAME="E1:47_1_2"/><i><font color="Green">E43</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">P1</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> )
 ;<br/>

<b>assume </b><a NAME="E2:47_1_2"/><i><font color="Green">E44</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">P2</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> )
 ;<br/>

<b>thus </b><a NAME="E3:47_1_2"/>
<font color="Maroon">P1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P2</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_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> ; <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_2_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P1</font>
 ;<br/>

<a NAME="E2:47_1_2_1"/>
<font color="Maroon">P1</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>
 
<b>by </b><i><a class="ref" href="trees_1.html#D14">TREES_1:def 14</a></i>;<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  <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/>
<a NAME="E4:47_1_2_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>, </i>;<br/>
<b>hence </b><a NAME="E5:47_1_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P2</font>
 <b>by </b><i/>;<br/>


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

<b>thus </b><a NAME="E4:47_1_2"/>
<font color="Maroon">P2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P1</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_2_2"><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_2_2"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P2</font>
 ;<br/>

<a NAME="E2:47_1_2_2"/>
<font color="Maroon">P2</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>
 
<b>by </b><i><a class="ref" href="trees_1.html#D14">TREES_1:def 14</a></i>;<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  <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/>
<a NAME="E4:47_1_2_2"/>
<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>, </i>;<br/>
<b>hence </b><a NAME="E5:47_1_2_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P1</font>
 <b>by </b><i/>;<br/>


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


</div>
