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

<b>let </b><font color="Maroon">x</font> be   <a href="qc_lang1.html#NM3">bound_QC-variable</a>;<br/><b>let </b><font color="Maroon">y</font> be   <a href="qc_lang1.html#NM3">bound_QC-variable</a>;<br/><b>let </b><font color="Maroon">s'</font> be   <a href="qc_lang1.html#NM8">QC-formula</a>;<br/>





<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K8">QC-WFF</a> ] <b>means </b>( <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y</font> &amp; not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">a<sub>1</sub></font> implies  not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="cqc_lang.html#K17">.</a> <font color="Maroon">y</font></span>)</span> );<br/>
<div><i><font color="Green">E25</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">s</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K8">QC-WFF</a> ;<br/><b>thus </b><a NAME="E1:47_1"/>
( <font color="Maroon">s</font> is <a href="qc_lang1.html#V2">atomic</a> implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">s</font>] )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_1"><b>proof </b></a><div class="add">

<b>assume </b><b>that </b><br/><a NAME="E1:47_1_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">s</font> is <a href="qc_lang1.html#V2">atomic</a>
 <b>and </b><br/><a NAME="E2:47_1_1"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y</font> &amp; not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">s</font> )
 ;<br/>

<b>thus </b><a NAME="E3:47_1_1"/>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="cqc_lang.html#K17">.</a> <font color="Maroon">y</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_1_1"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">l</font> =  <a href="qc_lang1.html#K16">the_arguments_of</a> <font color="Maroon">s</font>;<br/>
<i><font color="Green">E33</font></i>:  <a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">s</font> = 
 <a href="qc_lang1.html#K22">still_not-bound_in</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K16">the_arguments_of</a> <font color="Maroon">s</font></span>)</span>
<b>by </b><i>, <a class="ref" href="qc_lang3.html#T8">QC_LANG3:8</a></i>
<br/>.= 
 <a href="qc_lang3.html#K1">variables_in</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K16">the_arguments_of</a> <font color="Maroon">s</font></span>)</span>,<a href="qc_lang1.html#K2">bound_QC-variables</a> 
<b>by </b><i><a class="ref" href="qc_lang3.html#T6">QC_LANG3:6</a></i>
<br/>.= 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qc_lang1.html#K16">the_arguments_of</a> <font color="Maroon">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font></span>)</span> where <font color="Olive">k</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K16">the_arguments_of</a> <font color="Maroon">s</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="qc_lang1.html#K16">the_arguments_of</a> <font color="Maroon">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K2">bound_QC-variables</a>  ) </span>}</span> 
<b>by </b><i><a class="ref" href="qc_lang3.html#D2">QC_LANG3:def 2</a></i>
;<br/>
<a NAME="E2:47_1_1_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">s</font> <a href="cqc_lang.html#K17">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K15">the_pred_symbol_of</a> <font color="Maroon">s</font></span>)</span> <a href="qc_lang1.html#K9">!</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K4">Subst</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K16">the_arguments_of</a> <font color="Maroon">s</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qc_lang3.html#K3">a.</a> 0</span>)</span> <a href="cqc_lang.html#K6">.--&gt;</a> <font color="Maroon">y</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="cqc_lang.html#T29">CQC_LANG:29</a></i>;<br/>
<b>set </b><font color="Maroon">ll</font> =  <a href="cqc_lang.html#K4">Subst</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K16">the_arguments_of</a> <font color="Maroon">s</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="qc_lang3.html#K3">a.</a> 0</span>)</span> <a href="cqc_lang.html#K6">.--&gt;</a> <font color="Maroon">y</font></span>)</span>;<br/>
<a NAME="E3:47_1_1_1"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">s</font> <a href="cqc_lang.html#K17">.</a> <font color="Maroon">y</font> is <a href="qc_lang1.html#V2">atomic</a> &amp;  <a href="qc_lang1.html#K16">the_arguments_of</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="cqc_lang.html#K17">.</a> <font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cqc_lang.html#K4">Subst</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K16">the_arguments_of</a> <font color="Maroon">s</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="qc_lang3.html#K3">a.</a> 0</span>)</span> <a href="cqc_lang.html#K6">.--&gt;</a> <font color="Maroon">y</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_1_1_2"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">p</font> being   <a href="qc_lang1.html#NM6">QC-pred_symbol</a> of <font color="Maroon">k</font>, <font color="Maroon">l2</font> being    <a href="qc_lang1.html#M1">QC-variable_list</a> of <font color="Maroon">k</font><b> such that </b><br/><a NAME="E2:47_1_1_1_2"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="qc_lang1.html#K9">!</a> <font color="Maroon">l2</font>
 <b>by </b><i>, <a class="ref" href="qc_lang1.html#D17">QC_LANG1:def 17</a></i>;<br/>
<a NAME="E3:47_1_1_1_2"/>
<font color="Maroon">l2</font> <a href="hidden.html#R1">=</a>  <a href="qc_lang1.html#K16">the_arguments_of</a> <font color="Maroon">s</font>
 
<b>by </b><i>, <a class="txt" href="valuat_1.html#E4:47_1_1_1"><i><font color="Green">E40</font></i></a>, <a class="ref" href="qc_lang1.html#D22">QC_LANG1:def 22</a></i>;<br/>
<a NAME="E4:47_1_1_1_2"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K16">the_arguments_of</a> <font color="Maroon">s</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 
<b>by </b><i><a class="ref" href="qc_lang1.html#D8">QC_LANG1:def 8</a></i>;<br/>
<a NAME="E5:47_1_1_1_2"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K4">Subst</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K16">the_arguments_of</a> <font color="Maroon">s</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qc_lang3.html#K3">a.</a> 0</span>)</span> <a href="cqc_lang.html#K6">.--&gt;</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 
<b>by </b><i><a class="ref" href="cqc_lang.html#D3">CQC_LANG:def 3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">l3</font> =  <a href="cqc_lang.html#K4">Subst</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K16">the_arguments_of</a> <font color="Maroon">s</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="qc_lang3.html#K3">a.</a> 0</span>)</span> <a href="cqc_lang.html#K6">.--&gt;</a> <font color="Maroon">y</font></span>)</span> as    <a href="qc_lang1.html#M1">QC-variable_list</a> of <font color="Maroon">k</font> <b>by </b><i><a class="ref" href="qc_lang1.html#D8">QC_LANG1:def 8</a></i>;<br/>
<a NAME="E7:47_1_1_1_2"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">s</font> <a href="cqc_lang.html#K17">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="qc_lang1.html#K9">!</a> <font color="Maroon">l3</font>
 
<b>by </b><i>, <a class="txt" href="valuat_1.html#E2:47_1_1_1_2"><i><font color="Green">E38</font></i></a>, <a class="txt" href="valuat_1.html#E4:47_1_1_1"><i><font color="Green">E40</font></i></a>, <a class="ref" href="qc_lang1.html#D21">QC_LANG1:def 21</a></i>;<br/>
<b>hence </b><a NAME="E8:47_1_1_1_2"/>
<font color="Maroon">s</font> <a href="cqc_lang.html#K17">.</a> <font color="Maroon">y</font> is <a href="qc_lang1.html#V2">atomic</a>
 <b>by </b><i><a class="ref" href="qc_lang1.html#D17">QC_LANG1:def 17</a></i>;<br/>

<b>hence </b><a NAME="E9:47_1_1_1_2"/>
 <a href="cqc_lang.html#K4">Subst</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K16">the_arguments_of</a> <font color="Maroon">s</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="qc_lang3.html#K3">a.</a> 0</span>)</span> <a href="cqc_lang.html#K6">.--&gt;</a> <font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="qc_lang1.html#K16">the_arguments_of</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="cqc_lang.html#K17">.</a> <font color="Maroon">y</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="qc_lang1.html#D22">QC_LANG1:def 22</a></i>;<br/>


</div><b>end;</b></div>
<b>then </b><i><font color="Green">E40</font></i>:  <a href="qc_lang1.html#K22">still_not-bound_in</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K16">the_arguments_of</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="cqc_lang.html#K17">.</a> <font color="Maroon">y</font></span>)</span></span>)</span> = 
 <a href="qc_lang3.html#K1">variables_in</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K4">Subst</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K16">the_arguments_of</a> <font color="Maroon">s</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qc_lang3.html#K3">a.</a> 0</span>)</span> <a href="cqc_lang.html#K6">.--&gt;</a> <font color="Maroon">y</font></span>)</span></span>)</span>,<a href="qc_lang1.html#K2">bound_QC-variables</a> 
<b>by </b><i><a class="ref" href="qc_lang3.html#T6">QC_LANG3:6</a></i>
<br/>.= 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="cqc_lang.html#K4">Subst</a> <span class="p4">(<span class="default"><a href="qc_lang1.html#K16">the_arguments_of</a> <font color="Maroon">s</font></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="qc_lang3.html#K3">a.</a> 0</span>)</span> <a href="cqc_lang.html#K6">.--&gt;</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font></span>)</span> where <font color="Olive">k</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K4">Subst</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K16">the_arguments_of</a> <font color="Maroon">s</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qc_lang3.html#K3">a.</a> 0</span>)</span> <a href="cqc_lang.html#K6">.--&gt;</a> <font color="Maroon">y</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="cqc_lang.html#K4">Subst</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K16">the_arguments_of</a> <font color="Maroon">s</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qc_lang3.html#K3">a.</a> 0</span>)</span> <a href="cqc_lang.html#K6">.--&gt;</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K2">bound_QC-variables</a>  ) </span>}</span> 
<b>by </b><i><a class="ref" href="qc_lang3.html#D2">QC_LANG3:def 2</a></i>
;<br/>
<a NAME="E5:47_1_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="cqc_lang.html#K4">Subst</a> <span class="p4">(<span class="default"><a href="qc_lang1.html#K16">the_arguments_of</a> <font color="Maroon">s</font></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="qc_lang3.html#K3">a.</a> 0</span>)</span> <a href="cqc_lang.html#K6">.--&gt;</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font></span>)</span> where <font color="Olive">k</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K4">Subst</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K16">the_arguments_of</a> <font color="Maroon">s</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qc_lang3.html#K3">a.</a> 0</span>)</span> <a href="cqc_lang.html#K6">.--&gt;</a> <font color="Maroon">y</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="cqc_lang.html#K4">Subst</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K16">the_arguments_of</a> <font color="Maroon">s</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qc_lang3.html#K3">a.</a> 0</span>)</span> <a href="cqc_lang.html#K6">.--&gt;</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K2">bound_QC-variables</a>  ) </span>}</span>  implies <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qc_lang1.html#K16">the_arguments_of</a> <font color="Maroon">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K16">the_arguments_of</a> <font color="Maroon">s</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="qc_lang1.html#K16">the_arguments_of</a> <font color="Maroon">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K2">bound_QC-variables</a>  ) </span>}</span>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_1_1_4"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:47_1_1_1_4"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="cqc_lang.html#K4">Subst</a> <span class="p4">(<span class="default"><a href="qc_lang1.html#K16">the_arguments_of</a> <font color="Maroon">s</font></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="qc_lang3.html#K3">a.</a> 0</span>)</span> <a href="cqc_lang.html#K6">.--&gt;</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font></span>)</span> where <font color="Olive">k</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K4">Subst</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K16">the_arguments_of</a> <font color="Maroon">s</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qc_lang3.html#K3">a.</a> 0</span>)</span> <a href="cqc_lang.html#K6">.--&gt;</a> <font color="Maroon">y</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="cqc_lang.html#K4">Subst</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K16">the_arguments_of</a> <font color="Maroon">s</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qc_lang3.html#K3">a.</a> 0</span>)</span> <a href="cqc_lang.html#K6">.--&gt;</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K2">bound_QC-variables</a>  ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:47_1_1_1_4"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K4">Subst</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K16">the_arguments_of</a> <font color="Maroon">s</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qc_lang3.html#K3">a.</a> 0</span>)</span> <a href="cqc_lang.html#K6">.--&gt;</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K4">Subst</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K16">the_arguments_of</a> <font color="Maroon">s</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qc_lang3.html#K3">a.</a> 0</span>)</span> <a href="cqc_lang.html#K6">.--&gt;</a> <font color="Maroon">y</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="cqc_lang.html#K4">Subst</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K16">the_arguments_of</a> <font color="Maroon">s</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qc_lang3.html#K3">a.</a> 0</span>)</span> <a href="cqc_lang.html#K6">.--&gt;</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K2">bound_QC-variables</a>  )
 ;<br/>
<a NAME="E4:47_1_1_1_4"/><i><font color="Green">E42</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K16">the_arguments_of</a> <font color="Maroon">s</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="valuat_1.html#E3:47_1_1_1_4"><i><font color="Green">E41</font></i></a>, <a class="ref" href="cqc_lang.html#D3">CQC_LANG:def 3</a></i>;<br/>
<a NAME="E5:47_1_1_1_4"/><b>then </b>
<span class="p1">(<span class="default"><a href="qc_lang1.html#K16">the_arguments_of</a> <font color="Maroon">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="qc_lang3.html#K3">a.</a> 0
 
<b>by </b><i><a class="txt" href="valuat_1.html#E2:47_1_1_1"><i><font color="Green">E36</font></i></a>, <a class="txt" href="valuat_1.html#E3:47_1_1_1_4"><i><font color="Green">E41</font></i></a>, <a class="ref" href="cqc_lang.html#T11">CQC_LANG:11</a></i>;<br/>
<a NAME="E6:47_1_1_1_4"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K16">the_arguments_of</a> <font color="Maroon">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> &amp; <span class="p1">(<span class="default"><a href="qc_lang1.html#K16">the_arguments_of</a> <font color="Maroon">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K2">bound_QC-variables</a>  )
 
<b>by </b><i><a class="txt" href="valuat_1.html#E3:47_1_1_1_4"><i><font color="Green">E41</font></i></a>, <a class="txt" href="valuat_1.html#E4:47_1_1_1_4"><i><font color="Green">E42</font></i></a>, <a class="ref" href="cqc_lang.html#T11">CQC_LANG:11</a></i>;<br/>
<b>hence </b><a NAME="E7:47_1_1_1_4"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qc_lang1.html#K16">the_arguments_of</a> <font color="Maroon">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K16">the_arguments_of</a> <font color="Maroon">s</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="qc_lang1.html#K16">the_arguments_of</a> <font color="Maroon">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K2">bound_QC-variables</a>  ) </span>}</span> 
 <b>by </b><i><a class="txt" href="valuat_1.html#E4:47_1_1_1_4"><i><font color="Green">E42</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:47_1_1_1"/>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="cqc_lang.html#K17">.</a> <font color="Maroon">y</font></span>)</span>
 <b>by </b><i><a class="txt" href="valuat_1.html#E2:47_1_1_1"><i><font color="Green">E36</font></i></a>, <a class="txt" href="valuat_1.html#E3:47_1_1_1"><i><font color="Green">E37</font></i></a>, , <a class="txt" href="valuat_1.html#E4:47_1_1_1"><i><font color="Green">E40</font></i></a>, <a class="ref" href="qc_lang3.html#T8">QC_LANG3:8</a></i>;<br/>


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


</div><b>end;</b></div><b>thus </b><a NAME="E2:47_1"/>
<font color="Maroon">S<sub>1</sub></font>[ <a href="cqc_lang.html#K9">VERUM</a> ]
 <b>by </b><i><a class="ref" href="cqc_lang.html#T28">CQC_LANG:28</a></i>;<br/><b>thus </b><a NAME="E3:47_1"/>
( <font color="Maroon">s</font> is <a href="qc_lang1.html#V3">negative</a> &amp; <font color="Maroon">S<sub>1</sub></font>[ <a href="qc_lang1.html#K17">the_argument_of</a> <font color="Maroon">s</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">s</font>] )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_2"><b>proof </b></a><div class="add">

<b>assume </b><b>that </b><br/><a NAME="E1:47_1_2"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">s</font> is <a href="qc_lang1.html#V3">negative</a>
 <b>and </b><br/><a NAME="E2:47_1_2"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y</font> &amp; not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K17">the_argument_of</a> <font color="Maroon">s</font></span>)</span> implies  not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="qc_lang1.html#K17">the_argument_of</a> <font color="Maroon">s</font></span>)</span> <a href="cqc_lang.html#K17">.</a> <font color="Maroon">y</font></span>)</span> )
 <b>and </b><br/><a NAME="E3:47_1_2"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y</font> &amp; not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">s</font> )
 ;<br/>

<a NAME="E4:47_1_2"/>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K12">'not'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qc_lang1.html#K17">the_argument_of</a> <font color="Maroon">s</font></span>)</span> <a href="cqc_lang.html#K17">.</a> <font color="Maroon">y</font></span>)</span></span>)</span>
 
<b>by </b><i>, , , <a class="ref" href="qc_lang3.html#T10">QC_LANG3:10</a>, <a class="ref" href="qc_lang3.html#T11">QC_LANG3:11</a></i>;<br/>
<b>hence </b><a NAME="E5:47_1_2"/>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="cqc_lang.html#K17">.</a> <font color="Maroon">y</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="cqc_lang.html#T31">CQC_LANG:31</a></i>;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E4:47_1"/>
( <font color="Maroon">s</font> is <a href="qc_lang1.html#V4">conjunctive</a> &amp; <font color="Maroon">S<sub>1</sub></font>[ <a href="qc_lang1.html#K18">the_left_argument_of</a> <font color="Maroon">s</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[ <a href="qc_lang1.html#K19">the_right_argument_of</a> <font color="Maroon">s</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">s</font>] )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_3"><b>proof </b></a><div class="add">

<b>assume </b><b>that </b><br/><a NAME="E1:47_1_3"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">s</font> is <a href="qc_lang1.html#V4">conjunctive</a>
 <b>and </b><br/><a NAME="E2:47_1_3"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y</font> &amp; not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K18">the_left_argument_of</a> <font color="Maroon">s</font></span>)</span> implies  not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="qc_lang1.html#K18">the_left_argument_of</a> <font color="Maroon">s</font></span>)</span> <a href="cqc_lang.html#K17">.</a> <font color="Maroon">y</font></span>)</span> )
 <b>and </b><br/><a NAME="E3:47_1_3"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y</font> &amp; not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K19">the_right_argument_of</a> <font color="Maroon">s</font></span>)</span> implies  not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="qc_lang1.html#K19">the_right_argument_of</a> <font color="Maroon">s</font></span>)</span> <a href="cqc_lang.html#K17">.</a> <font color="Maroon">y</font></span>)</span> )
 <b>and </b><br/><a NAME="E4:47_1_3"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y</font> &amp; not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">s</font> )
 ;<br/>

<a NAME="E5:47_1_3"/>
 <a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K23">still_not-bound_in</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K18">the_left_argument_of</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K23">still_not-bound_in</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K19">the_right_argument_of</a> <font color="Maroon">s</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="qc_lang3.html#T13">QC_LANG3:13</a></i>;<br/>
<a NAME="E6:47_1_3"/><b>then </b>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K23">still_not-bound_in</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qc_lang1.html#K18">the_left_argument_of</a> <font color="Maroon">s</font></span>)</span> <a href="cqc_lang.html#K17">.</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K23">still_not-bound_in</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qc_lang1.html#K19">the_right_argument_of</a> <font color="Maroon">s</font></span>)</span> <a href="cqc_lang.html#K17">.</a> <font color="Maroon">y</font></span>)</span></span>)</span>
 
<b>by </b><i>, , , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E7:47_1_3"/><b>then </b>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qc_lang1.html#K18">the_left_argument_of</a> <font color="Maroon">s</font></span>)</span> <a href="cqc_lang.html#K17">.</a> <font color="Maroon">y</font></span>)</span> <a href="qc_lang1.html#K13">'&amp;'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qc_lang1.html#K19">the_right_argument_of</a> <font color="Maroon">s</font></span>)</span> <a href="cqc_lang.html#K17">.</a> <font color="Maroon">y</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="qc_lang3.html#T14">QC_LANG3:14</a></i>;<br/>
<b>hence </b><a NAME="E8:47_1_3"/>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="cqc_lang.html#K17">.</a> <font color="Maroon">y</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="cqc_lang.html#T33">CQC_LANG:33</a></i>;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E5:47_1"/>
( <font color="Maroon">s</font> is <a href="qc_lang1.html#V5">universal</a> &amp; <font color="Maroon">S<sub>1</sub></font>[ <a href="qc_lang1.html#K21">the_scope_of</a> <font color="Maroon">s</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">s</font>] )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_4"><b>proof </b></a><div class="add">

<b>assume </b><b>that </b><br/><a NAME="E1:47_1_4"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">s</font> is <a href="qc_lang1.html#V5">universal</a>
 <b>and </b><br/><a NAME="E2:47_1_4"/><i><font color="Green">E55</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y</font> &amp; not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K21">the_scope_of</a> <font color="Maroon">s</font></span>)</span> implies  not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="qc_lang1.html#K21">the_scope_of</a> <font color="Maroon">s</font></span>)</span> <a href="cqc_lang.html#K17">.</a> <font color="Maroon">y</font></span>)</span> )
 <b>and </b><br/><a NAME="E3:47_1_4"/><i><font color="Green">E56</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y</font> &amp; not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">s</font> )
 ;<br/>

<a NAME="E4:47_1_4"/><i><font color="Green">E119</font></i>: 
 <a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K23">still_not-bound_in</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K21">the_scope_of</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="qc_lang1.html#K20">bound_in</a> <font color="Maroon">s</font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i>, <a class="ref" href="qc_lang3.html#T15">QC_LANG3:15</a></i>;<br/>
<b>thus </b><a NAME="E5:47_1_4"/>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="cqc_lang.html#K17">.</a> <font color="Maroon">y</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_4_1"><b>proof </b></a><div class="add">

<div><i><font color="Green">E120</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_4_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:47_1_4_1_1"/>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K21">the_scope_of</a> <font color="Maroon">s</font></span>)</span>
 ;<br/><a NAME="E2:47_1_4_1_1"/><b>then </b>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K23">still_not-bound_in</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qc_lang1.html#K21">the_scope_of</a> <font color="Maroon">s</font></span>)</span> <a href="cqc_lang.html#K17">.</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="qc_lang1.html#K20">bound_in</a> <font color="Maroon">s</font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E3:47_1_4_1_1"/><b>then </b>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K14">All</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K20">bound_in</a> <font color="Maroon">s</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qc_lang1.html#K21">the_scope_of</a> <font color="Maroon">s</font></span>)</span> <a href="cqc_lang.html#K17">.</a> <font color="Maroon">y</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="qc_lang3.html#T16">QC_LANG3:16</a></i>;<br/><a NAME="E4:47_1_4_1_1"/><b>then </b>
( <font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="qc_lang1.html#K20">bound_in</a> <font color="Maroon">s</font> implies  not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="cqc_lang.html#K17">.</a> <font color="Maroon">y</font></span>)</span> )
 <b>by </b><i>, <a class="ref" href="cqc_lang.html#T36">CQC_LANG:36</a></i>;<br/><b>hence </b><a NAME="E5:47_1_4_1_1"/>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="cqc_lang.html#K17">.</a> <font color="Maroon">y</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="cqc_lang.html#T35">CQC_LANG:35</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_4_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:47_1_4_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="qc_lang1.html#K20">bound_in</a> <font color="Maroon">s</font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 ;<br/><a NAME="E2:47_1_4_1_2"/><b>then </b><i><font color="Green">E121</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="qc_lang1.html#K20">bound_in</a> <font color="Maroon">s</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E3:47_1_4_1_2"/>
 <a href="qc_lang1.html#K23">still_not-bound_in</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K14">All</a> <font color="Maroon">x</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qc_lang1.html#K21">the_scope_of</a> <font color="Maroon">s</font></span>)</span> <a href="cqc_lang.html#K17">.</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K23">still_not-bound_in</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qc_lang1.html#K21">the_scope_of</a> <font color="Maroon">s</font></span>)</span> <a href="cqc_lang.html#K17">.</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="qc_lang3.html#T16">QC_LANG3:16</a></i>;<br/><a NAME="E4:47_1_4_1_2"/><b>then </b>
( (  not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="qc_lang1.html#K21">the_scope_of</a> <font color="Maroon">s</font></span>)</span> <a href="cqc_lang.html#K17">.</a> <font color="Maroon">y</font></span>)</span> or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> ) iff  not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K14">All</a> <font color="Maroon">x</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qc_lang1.html#K21">the_scope_of</a> <font color="Maroon">s</font></span>)</span> <a href="cqc_lang.html#K17">.</a> <font color="Maroon">y</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><b>hence </b><a NAME="E5:47_1_4_1_2"/>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="cqc_lang.html#K17">.</a> <font color="Maroon">y</font></span>)</span>
 <b>by </b><i>, , , <a class="ref" href="cqc_lang.html#T36">CQC_LANG:36</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:47_1_4_1"/>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="cqc_lang.html#K17">.</a> <font color="Maroon">y</font></span>)</span>
 <b>by </b><i>, , , <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>


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


</div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E2:47"/>
for <font color="Olive">s</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K8">QC-WFF</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">s</font>]
 
<b>from </b><i><a class="ref" href="qc_lang1.html#S2">QC_LANG1:sch 2</a>();<br/></i>
<b>hence </b><a NAME="E3:47"/>
( <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y</font> &amp; not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">s'</font> implies  not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <span class="p1">(<span class="default"><font color="Maroon">s'</font> <a href="cqc_lang.html#K17">.</a> <font color="Maroon">y</font></span>)</span> )
 ;<br/>


</div>
