<?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>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="qc_lang1.html#NM8">QC-formula</a>] <b>means </b> <a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">a<sub>1</sub></font> <a href="tarski.html#R1">c=</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">x</font></span>)</span>;<br/>
<a NAME="E1:70"/><i><font color="Green">E39</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <br/> for <font color="Olive">P</font> being  <a href="qc_lang1.html#NM6">QC-pred_symbol</a> of <font color="Olive">k</font><br/> for <font color="Olive">ll</font> being   <a href="qc_lang1.html#M1">QC-variable_list</a> of <font color="Olive">k</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">P</font> <a href="qc_lang1.html#K9">!</a> <font color="Olive">ll</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">P</font> be   <a href="qc_lang1.html#NM6">QC-pred_symbol</a> of <font color="Maroon">k</font>;<br/><b>let </b><font color="Maroon">ll</font> be    <a href="qc_lang1.html#M1">QC-variable_list</a> of <font color="Maroon">k</font>;<br/>





<a NAME="E1:70_1"/><i><font color="Green">E42</font></i>: 
 <a href="qc_lang1.html#K22">still_not-bound_in</a> <font color="Maroon">ll</font> <a href="tarski.html#R1">c=</a>  <a href="qc_lang1.html#K22">still_not-bound_in</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K5">Subst</a> <font color="Maroon">ll</font>,<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">x</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="cqc_the3.html#T9" target="_self">Th9</a></i>;<br/>
 <a href="qc_lang1.html#K23">still_not-bound_in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="qc_lang1.html#K9">!</a> <font color="Maroon">ll</font></span>)</span> <a href="cqc_lang.html#K17">.</a> <font color="Maroon">x</font></span>)</span> = 
 <a href="qc_lang1.html#K23">still_not-bound_in</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="qc_lang1.html#K9">!</a> <span class="p2">(<span class="default"><a href="cqc_lang.html#K5">Subst</a> <font color="Maroon">ll</font>,<span class="p3">(<span class="default"><span class="p4">(<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">x</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="cqc_lang.html#T30">CQC_LANG:30</a></i>
<br/>.= 
 <a href="qc_lang1.html#K22">still_not-bound_in</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K5">Subst</a> <font color="Maroon">ll</font>,<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">x</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="qc_lang3.html#T9">QC_LANG3:9</a></i>
;<br/>
<b>hence </b><a NAME="E3:70_1"/>
 <a href="qc_lang1.html#K23">still_not-bound_in</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="qc_lang1.html#K9">!</a> <font color="Maroon">ll</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="qc_lang1.html#K9">!</a> <font color="Maroon">ll</font></span>)</span> <a href="cqc_lang.html#K17">.</a> <font color="Maroon">x</font></span>)</span>
 <b>by </b><i><a class="ref" href="cqc_the3.html#T14" target="_self">Th14</a>, <a class="ref" href="qc_lang3.html#T9">QC_LANG3:9</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:70"/><i><font color="Green">E43</font></i>: 
<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/>
<a NAME="E3:70"/><i><font color="Green">E60</font></i>: 
for <font color="Olive">p</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K8">QC-WFF</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">p</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[ <a href="qc_lang1.html#K12">'not'</a> <font color="Olive">p</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_2"><b>proof </b></a><div class="add">

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

 <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#K12">'not'</a> <font color="Maroon">p</font></span>)</span> <a href="cqc_lang.html#K17">.</a> <font color="Maroon">x</font></span>)</span> = 
 <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"><font color="Maroon">p</font> <a href="cqc_lang.html#K17">.</a> <font color="Maroon">x</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="cqc_lang.html#T32">CQC_LANG:32</a></i>
<br/>.= 
 <a href="qc_lang1.html#K23">still_not-bound_in</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="cqc_lang.html#K17">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="ref" href="qc_lang3.html#T11">QC_LANG3:11</a></i>
;<br/>
<b>hence </b><a NAME="E2:70_2"/>
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">p</font>] implies <font color="Maroon">S<sub>1</sub></font>[ <a href="qc_lang1.html#K12">'not'</a> <font color="Maroon">p</font>] )
 <b>by </b><i><a class="ref" href="qc_lang3.html#T11">QC_LANG3:11</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:70"/><i><font color="Green">E61</font></i>: 
for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K8">QC-WFF</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">p</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">q</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">p</font> <a href="qc_lang1.html#K13">'&amp;'</a> <font color="Olive">q</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_3"><b>proof </b></a><div class="add">

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

<b>assume </b><b>that </b><br/><a NAME="E1:70_3"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">p</font>]
 <b>and </b><br/><a NAME="E2:70_3"/><i><font color="Green">E85</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">q</font>]
 ;<br/>

<a NAME="E3:70_3"/><i><font color="Green">E86</font></i>: 
 <a href="qc_lang1.html#K23">still_not-bound_in</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="qc_lang1.html#K13">'&amp;'</a> <font color="Maroon">q</font></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> <font color="Maroon">p</font></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> <font color="Maroon">q</font></span>)</span>
 
<b>by </b><i><a class="ref" href="qc_lang3.html#T14">QC_LANG3:14</a></i>;<br/>
 <a href="qc_lang1.html#K23">still_not-bound_in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="qc_lang1.html#K13">'&amp;'</a> <font color="Maroon">q</font></span>)</span> <a href="cqc_lang.html#K17">.</a> <font color="Maroon">x</font></span>)</span> = 
 <a href="qc_lang1.html#K23">still_not-bound_in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="cqc_lang.html#K17">.</a> <font color="Maroon">x</font></span>)</span> <a href="qc_lang1.html#K13">'&amp;'</a> <span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="cqc_lang.html#K17">.</a> <font color="Maroon">x</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="cqc_lang.html#T34">CQC_LANG:34</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="qc_lang1.html#K23">still_not-bound_in</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="cqc_lang.html#K17">.</a> <font color="Maroon">x</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"><font color="Maroon">q</font> <a href="cqc_lang.html#K17">.</a> <font color="Maroon">x</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="E5:70_3"/>
 <a href="qc_lang1.html#K23">still_not-bound_in</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="qc_lang1.html#K13">'&amp;'</a> <font color="Maroon">q</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="qc_lang1.html#K13">'&amp;'</a> <font color="Maroon">q</font></span>)</span> <a href="cqc_lang.html#K17">.</a> <font color="Maroon">x</font></span>)</span>
 <b>by </b><i><a class="ref" href="cqc_the3.html#D4" target="_self">Def4</a>, <a class="ref" href="cqc_the3.html#T18" target="_self">Th18</a>, <a class="txt" href="cqc_the3.html#E3:70"><i><font color="Green">E60</font></i></a>, <a class="ref" href="xboole_1.html#T13">XBOOLE_1:13</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:70"/><i><font color="Green">E87</font></i>: 
for <font color="Olive">x</font> being  <a href="qc_lang1.html#NM3">bound_QC-variable</a><br/> for <font color="Olive">p</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K8">QC-WFF</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">p</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[ <a href="qc_lang1.html#K14">All</a> <font color="Olive">x</font>,<font color="Olive">p</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_4"><b>proof </b></a><div class="add">

<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">p</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K8">QC-WFF</a> ;<br/>



<b>assume </b><a NAME="E1:70_4"/><i><font color="Green">E88</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">p</font>]
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="70_4_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:70_4_1"/>
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> or <font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="70_4_1_1"><b>suppose </b></a><a NAME="E1:70_4_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:70_4_1_1"/>
 <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">y</font>,<font color="Maroon">p</font></span>)</span> <a href="tarski.html#R1">c=</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#K14">All</a> <font color="Maroon">y</font>,<font color="Maroon">p</font></span>)</span> <a href="cqc_lang.html#K17">.</a> <font color="Maroon">x</font></span>)</span>
 <b>by </b><i><a class="ref" href="cqc_lang.html#T37">CQC_LANG:37</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="70_4_1_2"><b>suppose </b></a><a NAME="E1:70_4_1_2"/>
<font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font>
 ;<br/><div class="add"><b>then </b><i><font color="Green">E89</font></i>:  <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#K14">All</a> <font color="Maroon">y</font>,<font color="Maroon">p</font></span>)</span> <a href="cqc_lang.html#K17">.</a> <font color="Maroon">x</font></span>)</span> = 
 <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">y</font>,<span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="cqc_lang.html#K17">.</a> <font color="Maroon">x</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="cqc_lang.html#T38">CQC_LANG:38</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="qc_lang1.html#K23">still_not-bound_in</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="cqc_lang.html#K17">.</a> <font color="Maroon">x</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">y</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="E3:70_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">y</font>,<font color="Maroon">p</font></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> <font color="Maroon">p</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</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/><b>hence </b><a NAME="E4:70_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">y</font>,<font color="Maroon">p</font></span>)</span> <a href="tarski.html#R1">c=</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#K14">All</a> <font color="Maroon">y</font>,<font color="Maroon">p</font></span>)</span> <a href="cqc_lang.html#K17">.</a> <font color="Maroon">x</font></span>)</span>
 <b>by </b><i><a class="ref" href="cqc_the3.html#T18" target="_self">Th18</a>, <a class="txt" href="cqc_the3.html#E3:70"><i><font color="Green">E60</font></i></a>, <a class="ref" href="xboole_1.html#T33">XBOOLE_1:33</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>thus </b><a NAME="E6:70"/>
for <font color="Olive">h</font> being  <a href="qc_lang1.html#NM8">QC-formula</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">h</font>]
 <b>from </b><i><a class="ref" href="qc_lang1.html#S1">QC_LANG1:sch 1</a>(<a class="ref" href="cqc_the3.html#T13" target="_self">Th13</a>, <a class="ref" href="cqc_the3.html#T14" target="_self">Th14</a>, <a class="ref" href="cqc_the3.html#D3" target="_self">Def3</a>, <a class="ref" href="cqc_the3.html#T15" target="_self">Th15</a>, <a class="ref" href="cqc_the3.html#D4" target="_self">Def4</a>);</i><br/>


</div>
