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

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K8">QC-WFF</a> ;<br/>



<b>thus </b><a NAME="E1:48"/>
<font color="Maroon">c<sub>1</sub></font> <a href="qc_lang4.html#K3" target="_self">-entry_points_in_subformula_tree_of</a> <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  <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">c<sub>1</sub></font></span>)</span> : <span class="p1">(<span class="default"><a href="qc_lang4.html#K2" target="_self">tree_of_subformulae</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="trees_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></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="48_1"><b>proof </b></a><div class="add">

<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="E1:48_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="qc_lang4.html#K3" target="_self">-entry_points_in_subformula_tree_of</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<a NAME="E2:48_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="qc_lang4.html#K3" target="_self">-entry_points_in_subformula_tree_of</a> <font color="Maroon">c<sub>2</sub></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">c<sub>1</sub></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">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></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">c<sub>1</sub></font></span>)</span> <b>by </b><i><a class="txt" href="qc_lang4.html#E1:48_1"><i><font color="Green">E42</font></i></a></i>;<br/>
<a NAME="E4:48_1"/>
<span class="p1">(<span class="default"><a href="qc_lang4.html#K2" target="_self">tree_of_subformulae</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="trees_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="qc_lang4.html#E1:48_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="qc_lang4.html#D3" target="_self">Def3</a></i>;<br/>
<b>hence </b><a NAME="E5:48_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  <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">c<sub>1</sub></font></span>)</span> : <span class="p1">(<span class="default"><a href="qc_lang4.html#K2" target="_self">tree_of_subformulae</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="trees_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> </span>}</span> 
 ;<br/>


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

<b>thus </b><a NAME="E2:48"/>
<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  <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">c<sub>1</sub></font></span>)</span> : <span class="p1">(<span class="default"><a href="qc_lang4.html#K2" target="_self">tree_of_subformulae</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="trees_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> </span>}</span>  <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></font> <a href="qc_lang4.html#K3" target="_self">-entry_points_in_subformula_tree_of</a> <font color="Maroon">c<sub>2</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_2"><b>proof </b></a><div class="add">

<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="E1:48_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  <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">c<sub>1</sub></font></span>)</span> : <span class="p1">(<span class="default"><a href="qc_lang4.html#K2" target="_self">tree_of_subformulae</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="trees_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</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  <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">c<sub>1</sub></font></span>)</span><b> such that </b><br/><a NAME="E3:48_2"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <span class="p1">(<span class="default"><a href="qc_lang4.html#K2" target="_self">tree_of_subformulae</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="trees_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> )
 ;<br/>
<b>thus </b><a NAME="E4:48_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="qc_lang4.html#K3" target="_self">-entry_points_in_subformula_tree_of</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="qc_lang4.html#E3:48_2"><i><font color="Green">E42</font></i></a>, <a class="ref" href="qc_lang4.html#D3" target="_self">Def3</a></i>;<br/>


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


</div>
