<?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">G</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">H</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">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>let </b><font color="Maroon">s</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">G</font></span>)</span>;<br/>









<b>assume </b><a NAME="E1:57"/><i><font color="Green">E22</font></i>: 
( <font color="Maroon">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">G</font> &amp; <font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="qc_lang4.html#K3" target="_self">-entry_points_in_subformula_tree_of</a> <font color="Maroon">H</font> )
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>for <font color="Olive">F</font>, <font color="Olive">G</font>, <font color="Olive">H</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K8">QC-WFF</a> <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="Olive">F</font></span>)</span><br/> for <font color="Olive">s</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="Olive">G</font></span>)</span>  st <font color="Olive">G</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K2" target="_self">tree_of_subformulae</a> <font color="Olive">F</font></span>)</span> <a href="trees_2.html#K3">.</a> <font color="Olive">t</font> &amp; <font color="Olive">H</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K2" target="_self">tree_of_subformulae</a> <font color="Olive">G</font></span>)</span> <a href="trees_2.html#K3">.</a> <font color="Olive">s</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Olive">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> holds <br/><font color="Olive">t</font> <a href="finseq_1.html#K8">^</a> <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Olive">F</font> <a href="qc_lang4.html#K3" target="_self">-entry_points_in_subformula_tree_of</a> <font color="Olive">H</font>;<br/>
<a NAME="E2:57"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1"><b>proof </b></a><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">G</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">H</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">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>let </b><font color="Maroon">s</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">G</font></span>)</span>;<br/>









<b>assume </b><a NAME="E1:57_1"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">G</font> <a href="hidden.html#R1">=</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> <a href="trees_2.html#K3">.</a> <font color="Maroon">t</font> &amp; <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K2" target="_self">tree_of_subformulae</a> <font color="Maroon">G</font></span>)</span> <a href="trees_2.html#K3">.</a> <font color="Maroon">s</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> 0 )
 ;<br/>

<a NAME="E2:57_1"/><b>then </b><i><font color="Green">E32</font></i>: 
<font color="Maroon">s</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/>
<a NAME="E3:57_1"/><b>then </b><i><font color="Green">E35</font></i>: 
<font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font>
 
<b>by </b><i><a class="ref" href="qc_lang4.html#T10" target="_self">Th10</a>, </i>;<br/>
<a NAME="E4:57_1"/>
<font color="Maroon">t</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font>
 
<b>by </b><i><a class="ref" href="qc_lang4.html#T11" target="_self">Th11</a>, <a class="ref" href="finseq_1.html#T47">FINSEQ_1:47</a></i>;<br/>
<b>hence </b><a NAME="E5:57_1"/>
<font color="Maroon">t</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">s</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">H</font>
 <b>by </b><i><a class="ref" href="qc_lang4.html#T10" target="_self">Th10</a>, <a class="ref" href="qc_lang4.html#T12" target="_self">Th12</a>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:57"/><i><font color="Green">E37</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:57_2"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>]
 ;<br/>

<b>thus </b><a NAME="E2:57_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_2_1"><b>proof </b></a><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">G</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">H</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">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>let </b><font color="Maroon">s</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">G</font></span>)</span>;<br/>









<b>assume </b><a NAME="E1:57_2_1"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">G</font> <a href="hidden.html#R1">=</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> <a href="trees_2.html#K3">.</a> <font color="Maroon">t</font> &amp; <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K2" target="_self">tree_of_subformulae</a> <font color="Maroon">G</font></span>)</span> <a href="trees_2.html#K3">.</a> <font color="Maroon">s</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/>

<b>then consider </b><font color="Maroon">v</font> being   <a href="finseq_1.html#NM1">FinSequence</a>, <font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:57_2_1"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9">*&gt;</a></span> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> )
 <b>by </b><i><a class="ref" href="trees_2.html#T4">TREES_2:4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">v</font> = <font color="Maroon">v</font> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="qc_lang4.html#T13" target="_self">Th13</a>, <a class="ref" href="finseq_1.html#T50">FINSEQ_1:50</a></i>;<br/>
<b>reconsider </b><font color="Maroon">s'</font> = <font color="Maroon">v</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">G</font></span>)</span> <b>by </b><i><a class="ref" href="qc_lang4.html#T13" target="_self">Th13</a>, <a class="ref" href="trees_1.html#T46">TREES_1:46</a></i>;<br/>
<b>reconsider </b><font color="Maroon">u</font> = <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9">*&gt;</a></span> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="qc_lang4.html#T13" target="_self">Th13</a>, <a class="ref" href="finseq_1.html#T50">FINSEQ_1:50</a></i>;<br/>
<a NAME="E7:57_2_1"/><i><font color="Green">E43</font></i>: 
(  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">u</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> 1 &amp; <font color="Maroon">u</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#D8">FINSEQ_1:def 8</a></i>;<br/>
<a NAME="E8:57_2_1"/>
1 <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">1</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="E9:57_2_1"/><b>then </b><i><font color="Green">E44</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">u</font>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E10:57_2_1"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">u</font> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">m</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/>
<a NAME="E12:57_2_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s'</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">m</font></span><a href="finseq_1.html#K12">*&gt;</a></span>
 
<b>by </b><i><a class="ref" href="qc_lang4.html#T13" target="_self">Th13</a></i>;<br/>
<b>consider </b><font color="Maroon">H'</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K8">QC-WFF</a> <b> such that </b><br/><a NAME="E14:57_2_1"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">H'</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K2" target="_self">tree_of_subformulae</a> <font color="Maroon">G</font></span>)</span> <a href="trees_2.html#K3">.</a> <font color="Maroon">s'</font>
 ;<br/>
<a NAME="E15:57_2_1"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">t</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">s'</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">H'</font>
 
<b>by </b><i><a class="ref" href="qc_lang4.html#T11" target="_self">Th11</a>, <a class="ref" href="qc_lang4.html#T12" target="_self">Th12</a>, <a class="ref" href="qc_lang4.html#T13" target="_self">Th13</a>, <a class="ref" href="qc_lang4.html#T15" target="_self">Th15</a></i>;<br/>
<a NAME="E16:57_2_1"/>
<font color="Maroon">F</font> <a href="qc_lang4.html#K3" target="_self">-entry_points_in_subformula_tree_of</a> <font color="Maroon">H'</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 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="E18:57_2_1"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">t'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">s'</font>
 <b>by </b><i/>;<br/>
<a NAME="E19:57_2_1"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">H'</font> <a href="hidden.html#R1">=</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> <a href="trees_2.html#K3">.</a> <font color="Maroon">t'</font>
 
<b>by </b><i>, , <a class="txt" href="qc_lang4.html#E1:57_2_1"><i><font color="Green">E40</font></i></a></i>;<br/>
<a NAME="E20:57_2_1"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">H</font> <a href="qc_lang2.html#R1">is_immediate_constituent_of</a> <font color="Maroon">H'</font>
 
<b>by </b><i><a class="ref" href="qc_lang4.html#T12" target="_self">Th12</a>, <a class="ref" href="qc_lang4.html#T14" target="_self">Th14</a>, <a class="ref" href="qc_lang4.html#T15" target="_self">Th15</a>, </i>;<br/>
<a NAME="E21:57_2_1"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">t</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t'</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">m</font></span><a href="finseq_1.html#K12">*&gt;</a></span>
 
<b>by </b><i><a class="ref" href="qc_lang4.html#T13" target="_self">Th13</a>, , <a class="ref" href="finseq_1.html#T45">FINSEQ_1:45</a></i>;<br/>
<a NAME="E22:57_2_1"/>
( <font color="Maroon">H</font> <a href="hidden.html#R1">=</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> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">t'</font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">m</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> &amp; <font color="Maroon">t'</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">m</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <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> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_2_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:57_2_1_1"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">H'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="qc_lang1.html#K11">VERUM</a> 
 
<b>by </b><i>, <a class="ref" href="qc_lang2.html#T58">QC_LANG2:58</a></i>;<br/>
<a NAME="E2:57_2_1_1"/><i><font color="Green">E82</font></i>: 
not <font color="Maroon">H'</font> is <a href="qc_lang1.html#V2">atomic</a>
 
<b>by </b><i>, <a class="ref" href="qc_lang2.html#T64">QC_LANG2:64</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_2_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_2_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:57_2_1_1_1_1"/>
( <font color="Maroon">H'</font> is <a href="qc_lang1.html#V3">negative</a> or <font color="Maroon">H'</font> is <a href="qc_lang1.html#V4">conjunctive</a> or <font color="Maroon">H'</font> is <a href="qc_lang1.html#V5">universal</a> )
 </span><b>by </b><i>, , <a class="ref" href="qc_lang1.html#T33">QC_LANG1:33</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_2_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:57_2_1_1_1_1_1"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">H'</font> is <a href="qc_lang1.html#V3">negative</a>
 ;<br/><div class="add"><a NAME="E2:57_2_1_1_1_1_1"/><b>then </b>
( <font color="Maroon">H</font> <a href="hidden.html#R1">=</a>  <a href="qc_lang1.html#K17">the_argument_of</a> <font color="Maroon">H'</font> &amp; <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i><a class="ref" href="qc_lang4.html#T12" target="_self">Th12</a>, <a class="ref" href="qc_lang4.html#T14" target="_self">Th14</a>, <a class="ref" href="qc_lang4.html#T15" target="_self">Th15</a>, <a class="txt" href="qc_lang4.html#E7:57_2_1"><i><font color="Green">E43</font></i></a></i>;<br/><b>hence </b><a NAME="E3:57_2_1_1_1_1_1"/>
( <font color="Maroon">H</font> <a href="hidden.html#R1">=</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> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">t'</font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">m</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> &amp; <font color="Maroon">t'</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">m</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <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>, , <a class="ref" href="qc_lang4.html#T7" target="_self">Th7</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_2_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:57_2_1_1_1_1_2"/><i><font color="Green">E85</font></i>: 
<font color="Maroon">H'</font> is <a href="qc_lang1.html#V4">conjunctive</a>
 ;<br/><div class="add"><a NAME="E2:57_2_1_1_1_1_2"/><b>then </b>
( ( <font color="Maroon">H</font> <a href="hidden.html#R1">=</a>  <a href="qc_lang1.html#K18">the_left_argument_of</a> <font color="Maroon">H'</font> &amp; <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 0 ) or ( <font color="Maroon">H</font> <a href="hidden.html#R1">=</a>  <a href="qc_lang1.html#K19">the_right_argument_of</a> <font color="Maroon">H'</font> &amp; <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 1 ) )
 <b>by </b><i><a class="ref" href="qc_lang4.html#T12" target="_self">Th12</a>, <a class="ref" href="qc_lang4.html#T14" target="_self">Th14</a>, <a class="ref" href="qc_lang4.html#T15" target="_self">Th15</a>, <a class="txt" href="qc_lang4.html#E9:57_2_1"><i><font color="Green">E44</font></i></a></i>;<br/><b>hence </b><a NAME="E3:57_2_1_1_1_1_2"/>
( <font color="Maroon">H</font> <a href="hidden.html#R1">=</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> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">t'</font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">m</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> &amp; <font color="Maroon">t'</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">m</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <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>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_2_1_1_1_1_3"><b>suppose </b></a><a NAME="E1:57_2_1_1_1_1_3"/><i><font color="Green">E86</font></i>: 
<font color="Maroon">H'</font> is <a href="qc_lang1.html#V5">universal</a>
 ;<br/><div class="add"><a NAME="E2:57_2_1_1_1_1_3"/><b>then </b>
( <font color="Maroon">H</font> <a href="hidden.html#R1">=</a>  <a href="qc_lang1.html#K21">the_scope_of</a> <font color="Maroon">H'</font> &amp; <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i><a class="ref" href="qc_lang4.html#T12" target="_self">Th12</a>, <a class="ref" href="qc_lang4.html#T14" target="_self">Th14</a>, <a class="ref" href="qc_lang4.html#T15" target="_self">Th15</a>, <a class="txt" href="qc_lang4.html#E12:57_2_1"><i><font color="Green">E45</font></i></a></i>;<br/><b>hence </b><a NAME="E3:57_2_1_1_1_1_3"/>
( <font color="Maroon">H</font> <a href="hidden.html#R1">=</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> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">t'</font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">m</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> &amp; <font color="Maroon">t'</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">m</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <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>, , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:57_2_1_1"/>
( <font color="Maroon">H</font> <a href="hidden.html#R1">=</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> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">t'</font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">m</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> &amp; <font color="Maroon">t'</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">m</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <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> )
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E23:57_2_1"/>
<font color="Maroon">t</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">s</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">H</font>
 <b>by </b><i>, <a class="txt" href="qc_lang4.html#E1:57_2_1"><i><font color="Green">E40</font></i></a></i>;<br/>


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


</div><b>end;</b></div>
<a NAME="E4:57"/>
for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="ref" href="qc_lang4.html#T9" target="_self">Th9</a>, <a class="ref" href="qc_lang4.html#T10" target="_self">Th10</a>);<br/></i>
<a NAME="E5:57"/><b>then </b>
( <font color="Maroon">G</font> <a href="hidden.html#R1">=</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> <a href="trees_2.html#K3">.</a> <font color="Maroon">t</font> &amp; <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K2" target="_self">tree_of_subformulae</a> <font color="Maroon">G</font></span>)</span> <a href="trees_2.html#K3">.</a> <font color="Maroon">s</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">s</font> implies <font color="Maroon">t</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">s</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">H</font> )
 
;<br/>
<b>hence </b><a NAME="E6:57"/>
<font color="Maroon">t</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">s</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">H</font>
 <b>by </b><i>, </i>;<br/>


</div>
