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

<b>let </b><font color="Maroon">F</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K8">QC-WFF</a> ;<br/><b>let </b><font color="Maroon">G1</font> be    <a href="qc_lang4.html#M1" target="_self">Subformula</a> of <font color="Maroon">F</font>;<br/><b>let </b><font color="Maroon">G2</font> be    <a href="qc_lang4.html#M1" target="_self">Subformula</a> of <font color="Maroon">F</font>;<br/>





<b>thus </b><a NAME="E1:72"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">t</font> <a href="finseq_1.html#K8">^</a> <font color="Olive">s</font></span>)</span> where <font color="Olive">t</font> is    <a href="qc_lang4.html#M2" target="_self">Entry_Point_in_Subformula_Tree</a> of <font color="Maroon">G1</font>, <font color="Olive">s</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">G1</font></span>)</span> : <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G1</font> <a href="qc_lang4.html#K3" target="_self">-entry_points_in_subformula_tree_of</a> <font color="Maroon">G2</font> </span>}</span>  <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">t</font> <a href="finseq_1.html#K8">^</a> <font color="Olive">s</font></span>)</span> 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>, <font color="Olive">s</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">G1</font></span>)</span> : ( <font color="Olive">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> <a href="qc_lang4.html#K3" target="_self">-entry_points_in_subformula_tree_of</a> <font color="Maroon">G1</font> &amp; <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G1</font> <a href="qc_lang4.html#K3" target="_self">-entry_points_in_subformula_tree_of</a> <font color="Maroon">G2</font> ) </span>}</span> 
  <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="72_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="E1:72_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">t</font> <a href="finseq_1.html#K8">^</a> <font color="Olive">s</font></span>)</span> where <font color="Olive">t</font> is    <a href="qc_lang4.html#M2" target="_self">Entry_Point_in_Subformula_Tree</a> of <font color="Maroon">G1</font>, <font color="Olive">s</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">G1</font></span>)</span> : <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G1</font> <a href="qc_lang4.html#K3" target="_self">-entry_points_in_subformula_tree_of</a> <font color="Maroon">G2</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">t1</font> being    <a href="qc_lang4.html#M2" target="_self">Entry_Point_in_Subformula_Tree</a> of <font color="Maroon">G1</font>, <font color="Maroon">s1</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">G1</font></span>)</span><b> such that </b><br/><a NAME="E3:72_1"/><i><font color="Green">E22</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">s1</font> &amp; <font color="Maroon">s1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G1</font> <a href="qc_lang4.html#K3" target="_self">-entry_points_in_subformula_tree_of</a> <font color="Maroon">G2</font> )
 ;<br/>
<a NAME="E4:72_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">t1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G1</font>
 
<b>by </b><i><a class="ref" href="qc_lang4.html#T13" target="_self">Th13</a></i>;<br/>
<a NAME="E5:72_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">s1</font> &amp; <font color="Maroon">t1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> <a href="qc_lang4.html#K3" target="_self">-entry_points_in_subformula_tree_of</a> <font color="Maroon">G1</font> &amp; <font color="Maroon">s1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G1</font> <a href="qc_lang4.html#K3" target="_self">-entry_points_in_subformula_tree_of</a> <font color="Maroon">G2</font> )
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E6:72_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">t</font> <a href="finseq_1.html#K8">^</a> <font color="Olive">s</font></span>)</span> 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>, <font color="Olive">s</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">G1</font></span>)</span> : ( <font color="Olive">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> <a href="qc_lang4.html#K3" target="_self">-entry_points_in_subformula_tree_of</a> <font color="Maroon">G1</font> &amp; <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G1</font> <a href="qc_lang4.html#K3" target="_self">-entry_points_in_subformula_tree_of</a> <font color="Maroon">G2</font> ) </span>}</span> 
 ;<br/>


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

<b>thus </b><a NAME="E2:72"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">t</font> <a href="finseq_1.html#K8">^</a> <font color="Olive">s</font></span>)</span> 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>, <font color="Olive">s</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">G1</font></span>)</span> : ( <font color="Olive">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> <a href="qc_lang4.html#K3" target="_self">-entry_points_in_subformula_tree_of</a> <font color="Maroon">G1</font> &amp; <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G1</font> <a href="qc_lang4.html#K3" target="_self">-entry_points_in_subformula_tree_of</a> <font color="Maroon">G2</font> ) </span>}</span>  <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">t</font> <a href="finseq_1.html#K8">^</a> <font color="Olive">s</font></span>)</span> where <font color="Olive">t</font> is    <a href="qc_lang4.html#M2" target="_self">Entry_Point_in_Subformula_Tree</a> of <font color="Maroon">G1</font>, <font color="Olive">s</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">G1</font></span>)</span> : <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G1</font> <a href="qc_lang4.html#K3" target="_self">-entry_points_in_subformula_tree_of</a> <font color="Maroon">G2</font> </span>}</span> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_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="E1:72_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">t</font> <a href="finseq_1.html#K8">^</a> <font color="Olive">s</font></span>)</span> 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>, <font color="Olive">s</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">G1</font></span>)</span> : ( <font color="Olive">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> <a href="qc_lang4.html#K3" target="_self">-entry_points_in_subformula_tree_of</a> <font color="Maroon">G1</font> &amp; <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G1</font> <a href="qc_lang4.html#K3" target="_self">-entry_points_in_subformula_tree_of</a> <font color="Maroon">G2</font> ) </span>}</span> 
 ;<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>, <font color="Maroon">s1</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">G1</font></span>)</span><b> such that </b><br/><a NAME="E3:72_2"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">s1</font> &amp; <font color="Maroon">t1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> <a href="qc_lang4.html#K3" target="_self">-entry_points_in_subformula_tree_of</a> <font color="Maroon">G1</font> &amp; <font color="Maroon">s1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G1</font> <a href="qc_lang4.html#K3" target="_self">-entry_points_in_subformula_tree_of</a> <font color="Maroon">G2</font> )
 ;<br/>
<a NAME="E4:72_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">t1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G1</font>
 
<b>by </b><i>, </i>;<br/>
<b>then reconsider </b><font color="Maroon">t1</font> = <font color="Maroon">t1</font> as    <a href="qc_lang4.html#M2" target="_self">Entry_Point_in_Subformula_Tree</a> of <font color="Maroon">G1</font> <b>by </b><i><a class="ref" href="qc_lang4.html#T13" target="_self">Th13</a></i>;<br/>
<a NAME="E6:72_2"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">s1</font> &amp; <font color="Maroon">s1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G1</font> <a href="qc_lang4.html#K3" target="_self">-entry_points_in_subformula_tree_of</a> <font color="Maroon">G2</font> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E7:72_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">t</font> <a href="finseq_1.html#K8">^</a> <font color="Olive">s</font></span>)</span> where <font color="Olive">t</font> is    <a href="qc_lang4.html#M2" target="_self">Entry_Point_in_Subformula_Tree</a> of <font color="Maroon">G1</font>, <font color="Olive">s</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">G1</font></span>)</span> : <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G1</font> <a href="qc_lang4.html#K3" target="_self">-entry_points_in_subformula_tree_of</a> <font color="Maroon">G2</font> </span>}</span> 
 ;<br/>


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


</div>
