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

<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K8">QC-WFF</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() = <font color="Maroon">F<sub>3</sub></font>(<span class="p1">(<span class="default"><a href="qc_lang1.html#K6">the_arity_of</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K15">the_pred_symbol_of</a> <font color="Maroon">a<sub>1</sub></font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="qc_lang1.html#K15">the_pred_symbol_of</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="qc_lang1.html#K16">the_arguments_of</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>);<br/>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K8">QC-WFF</a> ,   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() = <font color="Maroon">F<sub>6</sub></font>(<span class="p1">(<span class="default"><a href="qc_lang1.html#K20">bound_in</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>,<font color="Maroon">a<sub>2</sub></font>);<br/>
<b>consider </b><font color="Maroon">F</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="qc_lang1.html#K8">QC-WFF</a> ,<font color="Maroon">F<sub>1</sub></font>()<b> such that </b><br/><a NAME="E2:35_1"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <a href="cqc_lang.html#K9" target="_self">VERUM</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>() &amp; ( 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>  holds <br/> ( ( <font color="Olive">p</font> is <a href="qc_lang1.html#V2">atomic</a> implies <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">p</font>) ) &amp; ( <font color="Olive">p</font> is <a href="qc_lang1.html#V3">negative</a> implies <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K17">the_argument_of</a> <font color="Olive">p</font></span>)</span></span>)</span>) ) &amp; ( <font color="Olive">p</font> is <a href="qc_lang1.html#V4">conjunctive</a> implies <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K18">the_left_argument_of</a> <font color="Olive">p</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K19">the_right_argument_of</a> <font color="Olive">p</font></span>)</span></span>)</span>) ) &amp; ( <font color="Olive">p</font> is <a href="qc_lang1.html#V5">universal</a> implies <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">p</font>,<font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K21">the_scope_of</a> <font color="Olive">p</font></span>)</span>) ) ) ) )
 <b>from </b><i><a class="ref" href="qc_lang1.html#S3">QC_LANG1:sch 3</a>();<br/></i>
<b>reconsider </b><font color="Maroon">G</font> = <font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <a href="cqc_lang.html#K7" target="_self">CQC-WFF</a>  as   <a href="funct_2.html#NM1">Function</a> of  <a href="cqc_lang.html#K7" target="_self">CQC-WFF</a> ,<font color="Maroon">F<sub>1</sub></font>() <b>by </b><i><a class="ref" href="funct_2.html#T38">FUNCT_2:38</a></i>;<br/>
<b>take </b>
<font color="Maroon">G</font>
;<br/>

<b>thus </b><a NAME="E4:35_1"/>
<font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <a href="cqc_lang.html#K9" target="_self">VERUM</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>()
 <b>by </b><i>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/>

<b>let </b><font color="Maroon">r</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7" target="_self">CQC-WFF</a> ;<br/><b>let </b><font color="Maroon">s</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7" target="_self">CQC-WFF</a> ;<br/><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">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">l</font> be   <a href="cqc_lang.html#NM2">CQC-variable_list</a> of <font color="Maroon">k</font>;<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>set </b><font color="Maroon">r'</font> = <font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">r</font>;<br/>
<b>set </b><font color="Maroon">s'</font> = <font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font>;<br/>
<a NAME="E5:35_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">P</font> <a href="cqc_lang.html#K8" target="_self">!</a> <font color="Maroon">l</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/>
<a NAME="E6:35_1"/><b>then </b><i><font color="Green">E51</font></i>: 
(  <a href="qc_lang1.html#K16">the_arguments_of</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="cqc_lang.html#K8" target="_self">!</a> <font color="Maroon">l</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">l</font> &amp;  <a href="qc_lang1.html#K15">the_pred_symbol_of</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="cqc_lang.html#K8" target="_self">!</a> <font color="Maroon">l</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> &amp;  <a href="qc_lang1.html#K6">the_arity_of</a> <font color="Maroon">P</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> )
 
<b>by </b><i><a class="ref" href="qc_lang1.html#T35">QC_LANG1:35</a>, <a class="ref" href="qc_lang1.html#D21">QC_LANG1:def 21</a>, <a class="ref" href="qc_lang1.html#D22">QC_LANG1:def 22</a></i>;<br/>
<b>thus </b><font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="cqc_lang.html#K8" target="_self">!</a> <font color="Maroon">l</font></span>)</span> = 
<font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="cqc_lang.html#K8" target="_self">!</a> <font color="Maroon">l</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">k</font>,<font color="Maroon">P</font>,<font color="Maroon">l</font>)
<b>by </b><i>, , </i>
;<br/>

<a NAME="E8:35_1"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">r</font> &amp; <font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font> )
 
<b>by </b><i><a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/>
<a NAME="E9:35_1"/><i><font color="Green">E53</font></i>: 
 <a href="cqc_lang.html#K10" target="_self">'not'</a> <font color="Maroon">r</font> is <a href="qc_lang1.html#V3">negative</a>
 
<b>by </b><i><a class="ref" href="qc_lang1.html#D18">QC_LANG1:def 18</a></i>;<br/>
<a NAME="E10:35_1"/><b>then </b><i><font color="Green">E54</font></i>: 
 <a href="qc_lang1.html#K17">the_argument_of</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K10" target="_self">'not'</a> <font color="Maroon">r</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font>
 
<b>by </b><i><a class="ref" href="qc_lang1.html#D23">QC_LANG1:def 23</a></i>;<br/>
<b>thus </b><font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K10" target="_self">'not'</a> <font color="Maroon">r</font></span>)</span> = 
<font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K10" target="_self">'not'</a> <font color="Maroon">r</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon">F<sub>4</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">r</font></span>)</span>)
<b>by </b><i>, , , </i>
;<br/>

<a NAME="E12:35_1"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">r</font> <a href="cqc_lang.html#K11" target="_self">'&amp;'</a> <font color="Maroon">s</font> is <a href="qc_lang1.html#V4">conjunctive</a>
 
<b>by </b><i><a class="ref" href="qc_lang1.html#D19">QC_LANG1:def 19</a></i>;<br/>
<a NAME="E13:35_1"/><b>then </b><i><font color="Green">E56</font></i>: 
(  <a href="qc_lang1.html#K18">the_left_argument_of</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="cqc_lang.html#K11" target="_self">'&amp;'</a> <font color="Maroon">s</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> &amp;  <a href="qc_lang1.html#K19">the_right_argument_of</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="cqc_lang.html#K11" target="_self">'&amp;'</a> <font color="Maroon">s</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> )
 
<b>by </b><i><a class="ref" href="qc_lang1.html#D24">QC_LANG1:def 24</a>, <a class="ref" href="qc_lang1.html#D25">QC_LANG1:def 25</a></i>;<br/>
<b>thus </b><font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="cqc_lang.html#K11" target="_self">'&amp;'</a> <font color="Maroon">s</font></span>)</span> = 
<font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="cqc_lang.html#K11" target="_self">'&amp;'</a> <font color="Maroon">s</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">r</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font></span>)</span>)
<b>by </b><i>, , , <a class="ref" href="cqc_lang.html#D1" target="_self">Def1</a></i>
;<br/>

<a NAME="E15:35_1"/><i><font color="Green">E57</font></i>: 
 <a href="cqc_lang.html#K15" target="_self">All</a> <font color="Maroon">x</font>,<font color="Maroon">r</font> is <a href="qc_lang1.html#V5">universal</a>
 
<b>by </b><i><a class="ref" href="qc_lang1.html#D20">QC_LANG1:def 20</a></i>;<br/>
<a NAME="E16:35_1"/><b>then </b><i><font color="Green">E58</font></i>: 
(  <a href="qc_lang1.html#K20">bound_in</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K15" target="_self">All</a> <font color="Maroon">x</font>,<font color="Maroon">r</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp;  <a href="qc_lang1.html#K21">the_scope_of</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K15" target="_self">All</a> <font color="Maroon">x</font>,<font color="Maroon">r</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> )
 
<b>by </b><i><a class="ref" href="qc_lang1.html#D26">QC_LANG1:def 26</a>, <a class="ref" href="qc_lang1.html#D27">QC_LANG1:def 27</a></i>;<br/>
<b>thus </b><font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K15" target="_self">All</a> <font color="Maroon">x</font>,<font color="Maroon">r</font></span>)</span> = 
<font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K15" target="_self">All</a> <font color="Maroon">x</font>,<font color="Maroon">r</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon">F<sub>6</sub></font>(<font color="Maroon">x</font>,<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">r</font></span>)</span>)
<b>by </b><i>, , , <a class="ref" href="cqc_lang.html#T5" target="_self">Th5</a></i>
;<br/>


</div>
