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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="qc_lang1.html#NM8">QC-formula</a>;<br/>

<b>set </b><font color="Maroon">c<sub>2</sub></font> =  <a href="qc_lang1.html#K2">bound_QC-variables</a> ;<br/>
<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#NM2">Subset</a> of <a href="qc_lang1.html#K2">bound_QC-variables</a>  =  <a href="substut1.html#K9" target="_self">Bound_Vars</a> <font color="Maroon">a<sub>1</sub></font>;<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> ) <b>-&gt; </b>  <a href="subset_1.html#NM2">Subset</a> of <a href="qc_lang1.html#K2">bound_QC-variables</a>  =  <a href="substut1.html#K8" target="_self">Bound_Vars</a> <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>3</sub></font>(  <a href="subset_1.html#NM2">Subset</a> of <a href="qc_lang1.html#K2">bound_QC-variables</a> ) <b>-&gt; </b>  <a href="subset_1.html#NM2">Subset</a> of <a href="qc_lang1.html#K2">bound_QC-variables</a>  = <font color="Maroon">a<sub>1</sub></font>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>4</sub></font>(  <a href="subset_1.html#NM2">Subset</a> of <a href="qc_lang1.html#K2">bound_QC-variables</a> ,  <a href="subset_1.html#NM2">Subset</a> of <a href="qc_lang1.html#K2">bound_QC-variables</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of  <a href="zfmisc_1.html#K1">bool</a> <a href="qc_lang1.html#K2">bound_QC-variables</a>  = <font color="Maroon">a<sub>1</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">a<sub>2</sub></font>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>5</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#NM2">Subset</a> of <a href="qc_lang1.html#K2">bound_QC-variables</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of  <a href="zfmisc_1.html#K1">bool</a> <a href="qc_lang1.html#K2">bound_QC-variables</a>  = <font color="Maroon">a<sub>2</sub></font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="qc_lang1.html#K20">bound_in</a> <font color="Maroon">a<sub>1</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span>;<br/>
<a NAME="E1:15"/><i><font color="Green">E5</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="qc_lang1.html#NM8">QC-formula</a><br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="subset_1.html#NM2">Subset</a> of <a href="qc_lang1.html#K2">bound_QC-variables</a>  holds <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>) iff ex <font color="Olive">b<sub>3</sub></font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="qc_lang1.html#K8">QC-WFF</a> , <a href="zfmisc_1.html#K1">bool</a> <a href="qc_lang1.html#K2">bound_QC-variables</a>  st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>4</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K8">QC-WFF</a> <br/>for <font color="Olive">b<sub>5</sub></font>, <font color="Olive">b<sub>6</sub></font> being  <a href="subset_1.html#NM2">Subset</a> of <a href="qc_lang1.html#K2">bound_QC-variables</a>  holds <br/>( ( <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="cqc_lang.html#K9">VERUM</a>  implies <font color="Olive">b<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="subset_1.html#K1">{}</a> <a href="qc_lang1.html#K2">bound_QC-variables</a>  ) &amp; ( <font color="Olive">b<sub>4</sub></font> is <a href="qc_lang1.html#V2">atomic</a> implies <font color="Olive">b<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b<sub>4</sub></font>) ) &amp; ( <font color="Olive">b<sub>4</sub></font> is <a href="qc_lang1.html#V3">negative</a> &amp; <font color="Olive">b<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K17">the_argument_of</a> <font color="Olive">b<sub>4</sub></font></span>)</span> implies <font color="Olive">b<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive">b<sub>5</sub></font>) ) &amp; ( <font color="Olive">b<sub>4</sub></font> is <a href="qc_lang1.html#V4">conjunctive</a> &amp; <font color="Olive">b<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K18">the_left_argument_of</a> <font color="Olive">b<sub>4</sub></font></span>)</span> &amp; <font color="Olive">b<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K19">the_right_argument_of</a> <font color="Olive">b<sub>4</sub></font></span>)</span> implies <font color="Olive">b<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive">b<sub>5</sub></font>,<font color="Olive">b<sub>6</sub></font>) ) &amp; ( <font color="Olive">b<sub>4</sub></font> is <a href="qc_lang1.html#V5">universal</a> &amp; <font color="Olive">b<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></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">b<sub>4</sub></font></span>)</span> implies <font color="Olive">b<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>5</sub></font>(<font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>5</sub></font>) ) ) ) ) )
 
<b>by </b><i><a class="ref" href="substut1.html#D8" target="_self">Def8</a></i>;<br/>
<font color="Maroon">H<sub>1</sub></font>( <a href="cqc_lang.html#K9">VERUM</a> ) = 
 <a href="subset_1.html#K1">{}</a> <a href="qc_lang1.html#K2">bound_QC-variables</a> 
<b>from </b><i><a class="ref" href="qc_lang3.html#S3">QC_LANG3:sch 3</a>(<a class="txt" href="substut1.html#E1:15"><i><font color="Green">E5</font></i></a>)</i>
<br/>.= 
 <a href="xboole_0.html#K1">{}</a> 

;<br/>
<b>hence </b><a NAME="E3:15"/>
 <a href="substut1.html#K9" target="_self">Bound_Vars</a> <a href="cqc_lang.html#K9">VERUM</a>  <a href="hidden.html#R1">=</a>  <a href="subset_1.html#K1">{}</a> <a href="qc_lang1.html#K2">bound_QC-variables</a> 
 ;<br/>

<b>thus </b><a NAME="E4:15"/>
( <font color="Maroon">c<sub>1</sub></font> is <a href="qc_lang1.html#V2">atomic</a> implies  <a href="substut1.html#K9" target="_self">Bound_Vars</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="substut1.html#K8" target="_self">Bound_Vars</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K16">the_arguments_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:15_2"/><i><font color="Green">E6</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="qc_lang1.html#V2">atomic</a>
 ;<br/>

<b>thus </b><a NAME="E2:15_2"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>1</sub></font>) <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">c<sub>1</sub></font>)
 <b>from </b><i><a class="ref" href="qc_lang3.html#S4">QC_LANG3:sch 4</a>(<a class="txt" href="substut1.html#E1:15"><i><font color="Green">E5</font></i></a>, <a class="txt" href="substut1.html#E1:15_2"><i><font color="Green">E6</font></i></a>);</i><br/>


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

<b>thus </b><a NAME="E5:15"/>
( <font color="Maroon">c<sub>1</sub></font> is <a href="qc_lang1.html#V3">negative</a> implies  <a href="substut1.html#K9" target="_self">Bound_Vars</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="substut1.html#K9" target="_self">Bound_Vars</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K17">the_argument_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:15_3"/><i><font color="Green">E6</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="qc_lang1.html#V3">negative</a>
 ;<br/>

<b>thus </b><a NAME="E2:15_3"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>1</sub></font>) <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">H<sub>1</sub></font>( <a href="qc_lang1.html#K17">the_argument_of</a> <font color="Maroon">c<sub>1</sub></font>))
 <b>from </b><i><a class="ref" href="qc_lang3.html#S5">QC_LANG3:sch 5</a>(<a class="txt" href="substut1.html#E1:15"><i><font color="Green">E5</font></i></a>, <a class="txt" href="substut1.html#E1:15_3"><i><font color="Green">E6</font></i></a>);</i><br/>


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

<b>thus </b><a NAME="E6:15"/>
( <font color="Maroon">c<sub>1</sub></font> is <a href="qc_lang1.html#V4">conjunctive</a> implies  <a href="substut1.html#K9" target="_self">Bound_Vars</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="substut1.html#K9" target="_self">Bound_Vars</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K18">the_left_argument_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="substut1.html#K9" target="_self">Bound_Vars</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K19">the_right_argument_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_4"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:15_4"/><i><font color="Green">E6</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="qc_lang1.html#V4">conjunctive</a>
 ;<br/>

<a NAME="E2:15_4"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="subset_1.html#NM2">Subset</a> of <a href="qc_lang1.html#K2">bound_QC-variables</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>( <a href="qc_lang1.html#K18">the_left_argument_of</a> <font color="Maroon">c<sub>1</sub></font>) &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>( <a href="qc_lang1.html#K19">the_right_argument_of</a> <font color="Maroon">c<sub>1</sub></font>) holds <br/><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>1</sub></font>) <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>)
 
<b>from </b><i><a class="ref" href="qc_lang3.html#S6">QC_LANG3:sch 6</a>(<a class="txt" href="substut1.html#E1:15"><i><font color="Green">E5</font></i></a>, <a class="txt" href="substut1.html#E1:15_4"><i><font color="Green">E6</font></i></a>);<br/></i>
<b>hence </b><a NAME="E3:15_4"/>
 <a href="substut1.html#K9" target="_self">Bound_Vars</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="substut1.html#K9" target="_self">Bound_Vars</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K18">the_left_argument_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="substut1.html#K9" target="_self">Bound_Vars</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K19">the_right_argument_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 ;<br/>


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

<b>thus </b><a NAME="E7:15"/>
( <font color="Maroon">c<sub>1</sub></font> is <a href="qc_lang1.html#V5">universal</a> implies  <a href="substut1.html#K9" target="_self">Bound_Vars</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="substut1.html#K9" target="_self">Bound_Vars</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K21">the_scope_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="qc_lang1.html#K20">bound_in</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_5"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:15_5"/><i><font color="Green">E6</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="qc_lang1.html#V5">universal</a>
 ;<br/>

<b>thus </b><a NAME="E2:15_5"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>1</sub></font>) <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>5</sub></font>(<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">H<sub>1</sub></font>( <a href="qc_lang1.html#K21">the_scope_of</a> <font color="Maroon">c<sub>1</sub></font>))
 <b>from </b><i><a class="ref" href="qc_lang3.html#S7">QC_LANG3:sch 7</a>(<a class="txt" href="substut1.html#E1:15"><i><font color="Green">E5</font></i></a>, <a class="txt" href="substut1.html#E1:15_5"><i><font color="Green">E6</font></i></a>);</i><br/>


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


</div>
