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

<b>let </b><font color="Maroon" title="c1">F</font>, <font color="Maroon" title="c2">G</font>, <font color="Maroon" title="c3">H</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="qc_lang1.html#K8" title="QC_LANG1:func.8">QC-WFF</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">t</font> being   <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K2" title="QC_LANG4:func.2">tree_of_subformulae</a> <font color="Maroon" title="c1">F</font></span>)</span><br/> for <font color="Olive" title="b2">s</font> being  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>  st <font color="Olive" title="b1">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">F</font> <a href="qc_lang4.html#K3" title="QC_LANG4:func.3">-entry_points_in_subformula_tree_of</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Olive" title="b1">t</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Olive" title="b2">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">F</font> <a href="qc_lang4.html#K3" title="QC_LANG4:func.3">-entry_points_in_subformula_tree_of</a> <font color="Maroon" title="c3">H</font> holds <br/><font color="Olive" title="b2">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">G</font> <a href="qc_lang4.html#K3" title="QC_LANG4:func.3">-entry_points_in_subformula_tree_of</a> <font color="Maroon" title="c3">H</font></span><br/><b>let </b><font color="Maroon" title="c4">t</font> be    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K2" title="QC_LANG4:func.2">tree_of_subformulae</a> <font color="Maroon" title="c1">F</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">s</font> being  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>  st <font color="Maroon" title="c4">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">F</font> <a href="qc_lang4.html#K3" title="QC_LANG4:func.3">-entry_points_in_subformula_tree_of</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c4">t</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">F</font> <a href="qc_lang4.html#K3" title="QC_LANG4:func.3">-entry_points_in_subformula_tree_of</a> <font color="Maroon" title="c3">H</font> holds <br/><font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">G</font> <a href="qc_lang4.html#K3" title="QC_LANG4:func.3">-entry_points_in_subformula_tree_of</a> <font color="Maroon" title="c3">H</font></span><br/><b>let </b><font color="Maroon" title="c5">s</font> be   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">F</font> <a href="qc_lang4.html#K3" title="QC_LANG4:func.3">-entry_points_in_subformula_tree_of</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c4">t</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c5">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">F</font> <a href="qc_lang4.html#K3" title="QC_LANG4:func.3">-entry_points_in_subformula_tree_of</a> <font color="Maroon" title="c3">H</font> implies <font color="Maroon" title="c5">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">G</font> <a href="qc_lang4.html#K3" title="QC_LANG4:func.3">-entry_points_in_subformula_tree_of</a> <font color="Maroon" title="c3">H</font> )</span><br/>









<b>assume </b><a NAME="E1:58"/><i><font color="Green" title="E52">A1</font></i>: 
( <font color="Maroon" title="c4">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">F</font> <a href="qc_lang4.html#K3" title="QC_LANG4:func.3">-entry_points_in_subformula_tree_of</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c4">t</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c5">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">F</font> <a href="qc_lang4.html#K3" title="QC_LANG4:func.3">-entry_points_in_subformula_tree_of</a> <font color="Maroon" title="c3">H</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">G</font> <a href="qc_lang4.html#K3" title="QC_LANG4:func.3">-entry_points_in_subformula_tree_of</a> <font color="Maroon" title="c3">H</font></span><br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <b>means </b>for <font color="Olive" title="b1">F</font>, <font color="Olive" title="b2">G</font>, <font color="Olive" title="b3">H</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="qc_lang1.html#K8" title="QC_LANG1:func.8">QC-WFF</a> <br/> for <font color="Olive" title="b4">t</font> being   <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K2" title="QC_LANG4:func.2">tree_of_subformulae</a> <font color="Olive" title="b1">F</font></span>)</span><br/> for <font color="Olive" title="b5">s</font> being  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>  st <font color="Olive" title="b2">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K2" title="QC_LANG4:func.2">tree_of_subformulae</a> <font color="Olive" title="b1">F</font></span>)</span> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Olive" title="b4">t</font> &amp; <font color="Olive" title="b4">t</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Olive" title="b5">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">F</font> <a href="qc_lang4.html#K3" title="QC_LANG4:func.3">-entry_points_in_subformula_tree_of</a> <font color="Olive" title="b3">H</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b5">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 holds <br/><font color="Olive" title="b5">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">G</font> <a href="qc_lang4.html#K3" title="QC_LANG4:func.3">-entry_points_in_subformula_tree_of</a> <font color="Olive" title="b3">H</font>;<br/>
<a NAME="E2:58"/><i><font color="Green" title="E53">A2</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">F</font>, <font color="Maroon" title="c7">G</font>, <font color="Maroon" title="c8">H</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="qc_lang1.html#K8" title="QC_LANG1:func.8">QC-WFF</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">t</font> being   <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K2" title="QC_LANG4:func.2">tree_of_subformulae</a> <font color="Maroon" title="c6">F</font></span>)</span><br/> for <font color="Olive" title="b2">s</font> being  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>  st <font color="Maroon" title="c7">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K2" title="QC_LANG4:func.2">tree_of_subformulae</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Olive" title="b1">t</font> &amp; <font color="Olive" title="b1">t</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Olive" title="b2">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">F</font> <a href="qc_lang4.html#K3" title="QC_LANG4:func.3">-entry_points_in_subformula_tree_of</a> <font color="Maroon" title="c8">H</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/><font color="Olive" title="b2">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">G</font> <a href="qc_lang4.html#K3" title="QC_LANG4:func.3">-entry_points_in_subformula_tree_of</a> <font color="Maroon" title="c8">H</font></span><br/><b>let </b><font color="Maroon" title="c9">t</font> be    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K2" title="QC_LANG4:func.2">tree_of_subformulae</a> <font color="Maroon" title="c6">F</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">s</font> being  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>  st <font color="Maroon" title="c7">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K2" title="QC_LANG4:func.2">tree_of_subformulae</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c9">t</font> &amp; <font color="Maroon" title="c9">t</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">F</font> <a href="qc_lang4.html#K3" title="QC_LANG4:func.3">-entry_points_in_subformula_tree_of</a> <font color="Maroon" title="c8">H</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/><font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">G</font> <a href="qc_lang4.html#K3" title="QC_LANG4:func.3">-entry_points_in_subformula_tree_of</a> <font color="Maroon" title="c8">H</font></span><br/><b>let </b><font color="Maroon" title="c10">s</font> be   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c7">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K2" title="QC_LANG4:func.2">tree_of_subformulae</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c9">t</font> &amp; <font color="Maroon" title="c9">t</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c10">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">F</font> <a href="qc_lang4.html#K3" title="QC_LANG4:func.3">-entry_points_in_subformula_tree_of</a> <font color="Maroon" title="c8">H</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <font color="Maroon" title="c10">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">G</font> <a href="qc_lang4.html#K3" title="QC_LANG4:func.3">-entry_points_in_subformula_tree_of</a> <font color="Maroon" title="c8">H</font> )</span><br/>









<b>assume </b><a NAME="E1:58_1"/><i><font color="Green" title="E54">A3</font></i>: 
( <font color="Maroon" title="c7">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K2" title="QC_LANG4:func.2">tree_of_subformulae</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c9">t</font> &amp; <font color="Maroon" title="c9">t</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c10">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">F</font> <a href="qc_lang4.html#K3" title="QC_LANG4:func.3">-entry_points_in_subformula_tree_of</a> <font color="Maroon" title="c8">H</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c10">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">G</font> <a href="qc_lang4.html#K3" title="QC_LANG4:func.3">-entry_points_in_subformula_tree_of</a> <font color="Maroon" title="c8">H</font></span><br/>

<a NAME="E2:58_1"/>
<font color="Maroon" title="c6">F</font> <a href="qc_lang4.html#K3" title="QC_LANG4:func.3">-entry_points_in_subformula_tree_of</a> <font color="Maroon" title="c8">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">t1</font> where <font color="Olive" title="b1">t1</font> is    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K2" title="QC_LANG4:func.2">tree_of_subformulae</a> <font color="Maroon" title="c6">F</font></span>)</span> : <span class="p1">(<span class="default"><a href="qc_lang4.html#K2" title="QC_LANG4:func.2">tree_of_subformulae</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Olive" title="b1">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">H</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="qc_lang4.html#T49" target="_self" title="QC_LANG4:th.49">Th49</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c11">t'</font> being    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K2" title="QC_LANG4:func.2">tree_of_subformulae</a> <font color="Maroon" title="c6">F</font></span>)</span><b> such that </b><br/><a NAME="E4:58_1"/><i><font color="Green" title="E55">A4</font></i>: 
( <font color="Maroon" title="c11">t'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">t</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c10">s</font> &amp; <span class="p1">(<span class="default"><a href="qc_lang4.html#K2" title="QC_LANG4:func.2">tree_of_subformulae</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c11">t'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">H</font> )
 <b>by </b><i><a class="txt" href="qc_lang4.html#E1:58_1"><i><font color="Green" title="E54">A3</font></i></a></i>;<br/>
<a NAME="E5:58_1"/><i><font color="Green" title="E56">A5</font></i>: 
<font color="Maroon" title="c10">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="txt" href="qc_lang4.html#E1:58_1"><i><font color="Green" title="E54">A3</font></i></a>, <a class="ref" href="card_2.html#T59" title="CARD_2:th.59">CARD_2:59</a></i>;<br/>
<a NAME="E6:58_1"/><b>then </b><i><font color="Green" title="E57">A6</font></i>: 
<font color="Maroon" title="c8">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">G</font>
 
<b>by </b><i><a class="txt" href="qc_lang4.html#E1:58_1"><i><font color="Green" title="E54">A3</font></i></a>, <a class="txt" href="qc_lang4.html#E4:58_1"><i><font color="Green" title="E55">A4</font></i></a>, <a class="ref" href="finseq_1.html#T47" title="FINSEQ_1:th.47">FINSEQ_1:47</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c12">s'</font> = <font color="Maroon" title="c10">s</font> as    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K2" title="QC_LANG4:func.2">tree_of_subformulae</a> <font color="Maroon" title="c7">G</font></span>)</span> <b>by </b><i><a class="txt" href="qc_lang4.html#E5:58_1"><i><font color="Green" title="E56">A5</font></i></a>, <a class="ref" href="trees_1.html#T47" title="TREES_1:th.47">TREES_1:47</a></i>;<br/>
<a NAME="E8:58_1"/>
<font color="Maroon" title="c7">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K2" title="QC_LANG4:func.2">tree_of_subformulae</a> <font color="Maroon" title="c7">G</font></span>)</span> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c12">s'</font>
 
<b>by </b><i><a class="txt" href="qc_lang4.html#E5:58_1"><i><font color="Green" title="E56">A5</font></i></a>, <a class="ref" href="qc_lang4.html#D2" target="_self" title="QC_LANG4:def.2">Def2</a></i>;<br/>
<b>hence </b><a NAME="E9:58_1"/>
<font color="Maroon" title="c10">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">G</font> <a href="qc_lang4.html#K3" title="QC_LANG4:func.3">-entry_points_in_subformula_tree_of</a> <font color="Maroon" title="c8">H</font>
 <b>by </b><i><a class="txt" href="qc_lang4.html#E6:58_1"><i><font color="Green" title="E57">A6</font></i></a>, <a class="ref" href="qc_lang4.html#D3" target="_self" title="QC_LANG4:def.3">Def3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E3:58"/><i><font color="Green" title="E54">A7</font></i>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">k</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1] )</span><br/>

<b>assume </b><a NAME="E1:58_2"/><i><font color="Green" title="E55">A8</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">k</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/>

<b>thus </b><a NAME="E2:58_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">F</font>, <font color="Maroon" title="c8">G</font>, <font color="Maroon" title="c9">H</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="qc_lang1.html#K8" title="QC_LANG1:func.8">QC-WFF</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">t</font> being   <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K2" title="QC_LANG4:func.2">tree_of_subformulae</a> <font color="Maroon" title="c7">F</font></span>)</span><br/> for <font color="Olive" title="b2">s</font> being  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>  st <font color="Maroon" title="c8">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K2" title="QC_LANG4:func.2">tree_of_subformulae</a> <font color="Maroon" title="c7">F</font></span>)</span> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Olive" title="b1">t</font> &amp; <font color="Olive" title="b1">t</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Olive" title="b2">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">F</font> <a href="qc_lang4.html#K3" title="QC_LANG4:func.3">-entry_points_in_subformula_tree_of</a> <font color="Maroon" title="c9">H</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 holds <br/><font color="Olive" title="b2">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">G</font> <a href="qc_lang4.html#K3" title="QC_LANG4:func.3">-entry_points_in_subformula_tree_of</a> <font color="Maroon" title="c9">H</font></span><br/><b>let </b><font color="Maroon" title="c10">t</font> be    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K2" title="QC_LANG4:func.2">tree_of_subformulae</a> <font color="Maroon" title="c7">F</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">s</font> being  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>  st <font color="Maroon" title="c8">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K2" title="QC_LANG4:func.2">tree_of_subformulae</a> <font color="Maroon" title="c7">F</font></span>)</span> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c10">t</font> &amp; <font color="Maroon" title="c10">t</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">F</font> <a href="qc_lang4.html#K3" title="QC_LANG4:func.3">-entry_points_in_subformula_tree_of</a> <font color="Maroon" title="c9">H</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 holds <br/><font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">G</font> <a href="qc_lang4.html#K3" title="QC_LANG4:func.3">-entry_points_in_subformula_tree_of</a> <font color="Maroon" title="c9">H</font></span><br/><b>let </b><font color="Maroon" title="c11">s</font> be   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c8">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K2" title="QC_LANG4:func.2">tree_of_subformulae</a> <font color="Maroon" title="c7">F</font></span>)</span> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c10">t</font> &amp; <font color="Maroon" title="c10">t</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c11">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">F</font> <a href="qc_lang4.html#K3" title="QC_LANG4:func.3">-entry_points_in_subformula_tree_of</a> <font color="Maroon" title="c9">H</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies <font color="Maroon" title="c11">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">G</font> <a href="qc_lang4.html#K3" title="QC_LANG4:func.3">-entry_points_in_subformula_tree_of</a> <font color="Maroon" title="c9">H</font> )</span><br/>









<b>assume </b><a NAME="E1:58_2_1"/><i><font color="Green" title="E56">A9</font></i>: 
( <font color="Maroon" title="c8">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K2" title="QC_LANG4:func.2">tree_of_subformulae</a> <font color="Maroon" title="c7">F</font></span>)</span> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c10">t</font> &amp; <font color="Maroon" title="c10">t</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c11">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">F</font> <a href="qc_lang4.html#K3" title="QC_LANG4:func.3">-entry_points_in_subformula_tree_of</a> <font color="Maroon" title="c9">H</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c11">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">G</font> <a href="qc_lang4.html#K3" title="QC_LANG4:func.3">-entry_points_in_subformula_tree_of</a> <font color="Maroon" title="c9">H</font></span><br/>

<b>then consider </b><font color="Maroon" title="c12">v</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>, <font color="Maroon" title="c13">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:58_2_1"/><i><font color="Green" title="E57">A10</font></i>: 
( <font color="Maroon" title="c11">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">v</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c13">x</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">k</font> )
 <b>by </b><i><a class="ref" href="trees_2.html#T4" title="TREES_2:th.4">TREES_2:4</a></i>;<br/>
<a NAME="E4:58_2_1"/>
<font color="Maroon" title="c7">F</font> <a href="qc_lang4.html#K3" title="QC_LANG4:func.3">-entry_points_in_subformula_tree_of</a> <font color="Maroon" title="c9">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">t1</font> where <font color="Olive" title="b1">t1</font> is    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K2" title="QC_LANG4:func.2">tree_of_subformulae</a> <font color="Maroon" title="c7">F</font></span>)</span> : <span class="p1">(<span class="default"><a href="qc_lang4.html#K2" title="QC_LANG4:func.2">tree_of_subformulae</a> <font color="Maroon" title="c7">F</font></span>)</span> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Olive" title="b1">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">H</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="qc_lang4.html#T49" target="_self" title="QC_LANG4:th.49">Th49</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c14">t''</font> being    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K2" title="QC_LANG4:func.2">tree_of_subformulae</a> <font color="Maroon" title="c7">F</font></span>)</span><b> such that </b><br/><a NAME="E6:58_2_1"/><i><font color="Green" title="E58">A11</font></i>: 
( <font color="Maroon" title="c14">t''</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">t</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c11">s</font> &amp; <span class="p1">(<span class="default"><a href="qc_lang4.html#K2" title="QC_LANG4:func.2">tree_of_subformulae</a> <font color="Maroon" title="c7">F</font></span>)</span> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c14">t''</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">H</font> )
 <b>by </b><i><a class="txt" href="qc_lang4.html#E1:58_2_1"><i><font color="Green" title="E56">A9</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c15">s1</font> = <font color="Maroon" title="c11">s</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="qc_lang4.html#E6:58_2_1"><i><font color="Green" title="E58">A11</font></i></a>, <a class="ref" href="finseq_1.html#T50" title="FINSEQ_1:th.50">FINSEQ_1:50</a></i>;<br/>
<a NAME="E8:58_2_1"/><i><font color="Green" title="E59">A12</font></i>: 
<font color="Maroon" title="c15">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">v</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c13">x</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>
 
<b>by </b><i><a class="txt" href="qc_lang4.html#E3:58_2_1"><i><font color="Green" title="E57">A10</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c16">v</font> = <font color="Maroon" title="c12">v</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="finseq_1.html#T50" title="FINSEQ_1:th.50">FINSEQ_1:50</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c17">u</font> = <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c13">x</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="qc_lang4.html#E8:58_2_1"><i><font color="Green" title="E59">A12</font></i></a>, <a class="ref" href="finseq_1.html#T50" title="FINSEQ_1:th.50">FINSEQ_1:50</a></i>;<br/>
<a NAME="E11:58_2_1"/><i><font color="Green" title="E60">A13</font></i>: 
(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c17">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 1 &amp; <font color="Maroon" title="c17">u</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">x</font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#D8" title="FINSEQ_1:def.8">FINSEQ_1:def 8</a></i>;<br/>
<a NAME="E12:58_2_1"/>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E13:58_2_1"/><b>then </b><i><font color="Green" title="E61">A14</font></i>: 
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c17">u</font>
 
<b>by </b><i><a class="txt" href="qc_lang4.html#E11:58_2_1"><i><font color="Green" title="E60">A13</font></i></a>, <a class="ref" href="finseq_1.html#T4" title="FINSEQ_1:th.4">FINSEQ_1:4</a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E14:58_2_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c17">u</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c18">m</font> = <font color="Maroon" title="c13">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="qc_lang4.html#E13:58_2_1"><i><font color="Green" title="E61">A14</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c19">t'</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E17:58_2_1"/><i><font color="Green" title="E62">A15</font></i>: 
<font color="Maroon" title="c19">t'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">t</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c16">v</font>
 ;<br/>
<a NAME="E18:58_2_1"/><i><font color="Green" title="E63">A16</font></i>: 
<font color="Maroon" title="c14">t''</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">t'</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c18">m</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 
<b>by </b><i><a class="txt" href="qc_lang4.html#E3:58_2_1"><i><font color="Green" title="E57">A10</font></i></a>, <a class="txt" href="qc_lang4.html#E6:58_2_1"><i><font color="Green" title="E58">A11</font></i></a>, <a class="txt" href="qc_lang4.html#E17:58_2_1"><i><font color="Green" title="E62">A15</font></i></a>, <a class="ref" href="finseq_1.html#T45" title="FINSEQ_1:th.45">FINSEQ_1:45</a></i>;<br/>
<a NAME="E19:58_2_1"/><b>then </b>
<font color="Maroon" title="c19">t'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">is_a_prefix_of</a> <font color="Maroon" title="c14">t''</font>
 
<b>by </b><i><a class="ref" href="trees_1.html#T8" title="TREES_1:th.8">TREES_1:8</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c20">t'</font> = <font color="Maroon" title="c19">t'</font> as    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K2" title="QC_LANG4:func.2">tree_of_subformulae</a> <font color="Maroon" title="c7">F</font></span>)</span> <b>by </b><i><a class="ref" href="trees_1.html#T45" title="TREES_1:th.45">TREES_1:45</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c21">H'</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="qc_lang1.html#K8" title="QC_LANG1:func.8">QC-WFF</a> <b> such that </b><br/><a NAME="E22:58_2_1"/><i><font color="Green" title="E64">A17</font></i>: 
<font color="Maroon" title="c21">H'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K2" title="QC_LANG4:func.2">tree_of_subformulae</a> <font color="Maroon" title="c7">F</font></span>)</span> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c20">t'</font>
 ;<br/>
<a NAME="E23:58_2_1"/>
<font color="Maroon" title="c10">t</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c16">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">F</font> <a href="qc_lang4.html#K3" title="QC_LANG4:func.3">-entry_points_in_subformula_tree_of</a> <font color="Maroon" title="c21">H'</font>
 
<b>by </b><i><a class="txt" href="qc_lang4.html#E17:58_2_1"><i><font color="Green" title="E62">A15</font></i></a>, <a class="txt" href="qc_lang4.html#E22:58_2_1"><i><font color="Green" title="E64">A17</font></i></a>, <a class="ref" href="qc_lang4.html#D3" target="_self" title="QC_LANG4:def.3">Def3</a></i>;<br/>
<a NAME="E24:58_2_1"/><b>then </b><i><font color="Green" title="E65">A18</font></i>: 
<font color="Maroon" title="c16">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">G</font> <a href="qc_lang4.html#K3" title="QC_LANG4:func.3">-entry_points_in_subformula_tree_of</a> <font color="Maroon" title="c21">H'</font>
 
<b>by </b><i><a class="txt" href="qc_lang4.html#E1:58_2"><i><font color="Green" title="E55">A8</font></i></a>, <a class="txt" href="qc_lang4.html#E1:58_2_1"><i><font color="Green" title="E56">A9</font></i></a>, <a class="txt" href="qc_lang4.html#E3:58_2_1"><i><font color="Green" title="E57">A10</font></i></a></i>;<br/>
<a NAME="E25:58_2_1"/>
<font color="Maroon" title="c8">G</font> <a href="qc_lang4.html#K3" title="QC_LANG4:func.3">-entry_points_in_subformula_tree_of</a> <font color="Maroon" title="c21">H'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">v1</font> where <font color="Olive" title="b1">v1</font> is    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K2" title="QC_LANG4:func.2">tree_of_subformulae</a> <font color="Maroon" title="c8">G</font></span>)</span> : <span class="p1">(<span class="default"><a href="qc_lang4.html#K2" title="QC_LANG4:func.2">tree_of_subformulae</a> <font color="Maroon" title="c8">G</font></span>)</span> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Olive" title="b1">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">H'</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="qc_lang4.html#T49" target="_self" title="QC_LANG4:th.49">Th49</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c22">v1</font> being    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K2" title="QC_LANG4:func.2">tree_of_subformulae</a> <font color="Maroon" title="c8">G</font></span>)</span><b> such that </b><br/><a NAME="E27:58_2_1"/><i><font color="Green" title="E66">A19</font></i>: 
( <font color="Maroon" title="c16">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">v1</font> &amp; <span class="p1">(<span class="default"><a href="qc_lang4.html#K2" title="QC_LANG4:func.2">tree_of_subformulae</a> <font color="Maroon" title="c8">G</font></span>)</span> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c22">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">H'</font> )
 <b>by </b><i><a class="txt" href="qc_lang4.html#E24:58_2_1"><i><font color="Green" title="E65">A18</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c23">v</font> = <font color="Maroon" title="c16">v</font> as    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K2" title="QC_LANG4:func.2">tree_of_subformulae</a> <font color="Maroon" title="c8">G</font></span>)</span> <b>by </b><i><a class="txt" href="qc_lang4.html#E27:58_2_1"><i><font color="Green" title="E66">A19</font></i></a></i>;<br/>
<a NAME="E29:58_2_1"/><i><font color="Green" title="E67">A20</font></i>: 
<font color="Maroon" title="c9">H</font> <a href="qc_lang2.html#R1" title="QC_LANG2:pred.1">is_immediate_constituent_of</a> <font color="Maroon" title="c21">H'</font>
 
<b>by </b><i><a class="txt" href="qc_lang4.html#E6:58_2_1"><i><font color="Green" title="E58">A11</font></i></a>, <a class="txt" href="qc_lang4.html#E18:58_2_1"><i><font color="Green" title="E63">A16</font></i></a>, <a class="txt" href="qc_lang4.html#E22:58_2_1"><i><font color="Green" title="E64">A17</font></i></a>, <a class="ref" href="qc_lang4.html#T36" target="_self" title="QC_LANG4:th.36">Th36</a></i>;<br/>
<a NAME="E30:58_2_1"/>
( <font color="Maroon" title="c9">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K2" title="QC_LANG4:func.2">tree_of_subformulae</a> <font color="Maroon" title="c8">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">v</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c18">m</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> &amp; <font color="Maroon" title="c23">v</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c18">m</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K2" title="QC_LANG4:func.2">tree_of_subformulae</a> <font color="Maroon" title="c8">G</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_2_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:58_2_1_1"/><i><font color="Green" title="E68">A21</font></i>: 
<font color="Maroon" title="c21">H'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="qc_lang1.html#K11" title="QC_LANG1:func.11">VERUM</a> 
 
<b>by </b><i><a class="txt" href="qc_lang4.html#E29:58_2_1"><i><font color="Green" title="E67">A20</font></i></a>, <a class="ref" href="qc_lang2.html#T58" title="QC_LANG2:th.58">QC_LANG2:58</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_2_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_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="E1:58_2_1_1_1_1"/>
( <font color="Maroon" title="c21">H'</font> is <a href="qc_lang1.html#V3" title="QC_LANG1:attr.3">negative</a> or <font color="Maroon" title="c21">H'</font> is <a href="qc_lang1.html#V4" title="QC_LANG1:attr.4">conjunctive</a> or <font color="Maroon" title="c21">H'</font> is <a href="qc_lang1.html#V5" title="QC_LANG1:attr.5">universal</a> )
 </span><b>by </b><i><a class="txt" href="qc_lang4.html#E29:58_2_1"><i><font color="Green" title="E67">A20</font></i></a>, <a class="txt" href="qc_lang4.html#E1:58_2_1_1"><i><font color="Green" title="E68">A21</font></i></a>, <a class="ref" href="qc_lang1.html#T33" title="QC_LANG1:th.33">QC_LANG1:33</a>, <a class="ref" href="qc_lang2.html#T64" title="QC_LANG2:th.64">QC_LANG2:64</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_2_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:58_2_1_1_1_1_1"/><i><font color="Green" title="E69">A22</font></i>: 
<font color="Maroon" title="c21">H'</font> is <a href="qc_lang1.html#V3" title="QC_LANG1:attr.3">negative</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c9">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K2" title="QC_LANG4:func.2">tree_of_subformulae</a> <font color="Maroon" title="c8">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">v</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c18">m</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> &amp; <font color="Maroon" title="c23">v</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c18">m</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K2" title="QC_LANG4:func.2">tree_of_subformulae</a> <font color="Maroon" title="c8">G</font></span>)</span> )</span><br/><div class="add"><a NAME="E2:58_2_1_1_1_1_1"/><b>then </b>
( <font color="Maroon" title="c9">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="qc_lang1.html#K17" title="QC_LANG1:func.17">the_argument_of</a> <font color="Maroon" title="c21">H'</font> &amp; <font color="Maroon" title="c18">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <b>by </b><i><a class="txt" href="qc_lang4.html#E6:58_2_1"><i><font color="Green" title="E58">A11</font></i></a>, <a class="txt" href="qc_lang4.html#E18:58_2_1"><i><font color="Green" title="E63">A16</font></i></a>, <a class="txt" href="qc_lang4.html#E22:58_2_1"><i><font color="Green" title="E64">A17</font></i></a>, <a class="ref" href="qc_lang4.html#T51" target="_self" title="QC_LANG4:th.51">Th51</a></i>;<br/><b>hence </b><a NAME="E3:58_2_1_1_1_1_1"/>
( <font color="Maroon" title="c9">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K2" title="QC_LANG4:func.2">tree_of_subformulae</a> <font color="Maroon" title="c8">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">v</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c18">m</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> &amp; <font color="Maroon" title="c23">v</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c18">m</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K2" title="QC_LANG4:func.2">tree_of_subformulae</a> <font color="Maroon" title="c8">G</font></span>)</span> )
 <b>by </b><i><a class="txt" href="qc_lang4.html#E27:58_2_1"><i><font color="Green" title="E66">A19</font></i></a>, <a class="txt" href="qc_lang4.html#E1:58_2_1_1_1_1_1"><i><font color="Green" title="E69">A22</font></i></a>, <a class="ref" href="qc_lang4.html#T54" target="_self" title="QC_LANG4:th.54">Th54</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_2_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:58_2_1_1_1_1_2"/><i><font color="Green" title="E69">A23</font></i>: 
<font color="Maroon" title="c21">H'</font> is <a href="qc_lang1.html#V4" title="QC_LANG1:attr.4">conjunctive</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c9">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K2" title="QC_LANG4:func.2">tree_of_subformulae</a> <font color="Maroon" title="c8">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">v</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c18">m</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> &amp; <font color="Maroon" title="c23">v</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c18">m</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K2" title="QC_LANG4:func.2">tree_of_subformulae</a> <font color="Maroon" title="c8">G</font></span>)</span> )</span><br/><div class="add"><a NAME="E2:58_2_1_1_1_1_2"/><b>then </b>
( ( <font color="Maroon" title="c9">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="qc_lang1.html#K18" title="QC_LANG1:func.18">the_left_argument_of</a> <font color="Maroon" title="c21">H'</font> &amp; <font color="Maroon" title="c18">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) or ( <font color="Maroon" title="c9">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="qc_lang1.html#K19" title="QC_LANG1:func.19">the_right_argument_of</a> <font color="Maroon" title="c21">H'</font> &amp; <font color="Maroon" title="c18">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) )
 <b>by </b><i><a class="txt" href="qc_lang4.html#E6:58_2_1"><i><font color="Green" title="E58">A11</font></i></a>, <a class="txt" href="qc_lang4.html#E18:58_2_1"><i><font color="Green" title="E63">A16</font></i></a>, <a class="txt" href="qc_lang4.html#E22:58_2_1"><i><font color="Green" title="E64">A17</font></i></a>, <a class="ref" href="qc_lang4.html#T52" target="_self" title="QC_LANG4:th.52">Th52</a></i>;<br/><b>hence </b><a NAME="E3:58_2_1_1_1_1_2"/>
( <font color="Maroon" title="c9">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K2" title="QC_LANG4:func.2">tree_of_subformulae</a> <font color="Maroon" title="c8">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">v</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c18">m</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> &amp; <font color="Maroon" title="c23">v</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c18">m</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K2" title="QC_LANG4:func.2">tree_of_subformulae</a> <font color="Maroon" title="c8">G</font></span>)</span> )
 <b>by </b><i><a class="txt" href="qc_lang4.html#E27:58_2_1"><i><font color="Green" title="E66">A19</font></i></a>, <a class="txt" href="qc_lang4.html#E1:58_2_1_1_1_1_2"><i><font color="Green" title="E69">A23</font></i></a>, <a class="ref" href="qc_lang4.html#T55" target="_self" title="QC_LANG4:th.55">Th55</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_2_1_1_1_1_3"><b>suppose </b></a><a NAME="E1:58_2_1_1_1_1_3"/><i><font color="Green" title="E69">A24</font></i>: 
<font color="Maroon" title="c21">H'</font> is <a href="qc_lang1.html#V5" title="QC_LANG1:attr.5">universal</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c9">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K2" title="QC_LANG4:func.2">tree_of_subformulae</a> <font color="Maroon" title="c8">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">v</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c18">m</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> &amp; <font color="Maroon" title="c23">v</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c18">m</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K2" title="QC_LANG4:func.2">tree_of_subformulae</a> <font color="Maroon" title="c8">G</font></span>)</span> )</span><br/><div class="add"><a NAME="E2:58_2_1_1_1_1_3"/><b>then </b>
( <font color="Maroon" title="c9">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="qc_lang1.html#K21" title="QC_LANG1:func.21">the_scope_of</a> <font color="Maroon" title="c21">H'</font> &amp; <font color="Maroon" title="c18">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <b>by </b><i><a class="txt" href="qc_lang4.html#E6:58_2_1"><i><font color="Green" title="E58">A11</font></i></a>, <a class="txt" href="qc_lang4.html#E18:58_2_1"><i><font color="Green" title="E63">A16</font></i></a>, <a class="txt" href="qc_lang4.html#E22:58_2_1"><i><font color="Green" title="E64">A17</font></i></a>, <a class="ref" href="qc_lang4.html#T53" target="_self" title="QC_LANG4:th.53">Th53</a></i>;<br/><b>hence </b><a NAME="E3:58_2_1_1_1_1_3"/>
( <font color="Maroon" title="c9">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K2" title="QC_LANG4:func.2">tree_of_subformulae</a> <font color="Maroon" title="c8">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">v</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c18">m</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> &amp; <font color="Maroon" title="c23">v</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c18">m</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K2" title="QC_LANG4:func.2">tree_of_subformulae</a> <font color="Maroon" title="c8">G</font></span>)</span> )
 <b>by </b><i><a class="txt" href="qc_lang4.html#E27:58_2_1"><i><font color="Green" title="E66">A19</font></i></a>, <a class="txt" href="qc_lang4.html#E1:58_2_1_1_1_1_3"><i><font color="Green" title="E69">A24</font></i></a>, <a class="ref" href="qc_lang4.html#T56" target="_self" title="QC_LANG4:th.56">Th56</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:58_2_1_1"/>
( <font color="Maroon" title="c9">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K2" title="QC_LANG4:func.2">tree_of_subformulae</a> <font color="Maroon" title="c8">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">v</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c18">m</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> &amp; <font color="Maroon" title="c23">v</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c18">m</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K2" title="QC_LANG4:func.2">tree_of_subformulae</a> <font color="Maroon" title="c8">G</font></span>)</span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E31:58_2_1"/>
<font color="Maroon" title="c11">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">G</font> <a href="qc_lang4.html#K3" title="QC_LANG4:func.3">-entry_points_in_subformula_tree_of</a> <font color="Maroon" title="c9">H</font>
 <b>by </b><i><a class="txt" href="qc_lang4.html#E3:58_2_1"><i><font color="Green" title="E57">A10</font></i></a>, <a class="ref" href="qc_lang4.html#D3" target="_self" title="QC_LANG4:def.3">Def3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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


</div><b>end;</b></div>
<a NAME="E4:58"/>
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="qc_lang4.html#E2:58"><i><font color="Green" title="E53">A2</font></i></a>, <a class="txt" href="qc_lang4.html#E3:58"><i><font color="Green" title="E54">A7</font></i></a>);<br/></i>
<a NAME="E5:58"/><b>then </b>
( <font color="Maroon" title="c2">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K2" title="QC_LANG4:func.2">tree_of_subformulae</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c4">t</font> &amp; <font color="Maroon" title="c4">t</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c5">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">F</font> <a href="qc_lang4.html#K3" title="QC_LANG4:func.3">-entry_points_in_subformula_tree_of</a> <font color="Maroon" title="c3">H</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">s</font> implies <font color="Maroon" title="c5">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">G</font> <a href="qc_lang4.html#K3" title="QC_LANG4:func.3">-entry_points_in_subformula_tree_of</a> <font color="Maroon" title="c3">H</font> )
 
;<br/>
<b>hence </b><a NAME="E6:58"/>
<font color="Maroon" title="c5">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">G</font> <a href="qc_lang4.html#K3" title="QC_LANG4:func.3">-entry_points_in_subformula_tree_of</a> <font color="Maroon" title="c3">H</font>
 <b>by </b><i><a class="txt" href="qc_lang4.html#E1:58"><i><font color="Green" title="E52">A1</font></i></a>, <a class="ref" href="qc_lang4.html#D3" target="_self" title="QC_LANG4:def.3">Def3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
