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

<b>reconsider </b><font color="Maroon">c<sub>2</sub></font> =  <a href="xboole_0.html#K1">{}</a>  as   <a href="subset_1.html#NM2">Subset</a> of <a href="qc_lang1.html#K2" target="_self">bound_QC-variables</a>  <b>by </b><i><a class="ref" href="xboole_1.html#T2">XBOOLE_1:2</a></i>;<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" target="_self">QC-WFF</a> ) <b>-&gt; </b>  <a href="subset_1.html#NM2">Subset</a> of <a href="qc_lang1.html#K2" target="_self">bound_QC-variables</a>  =  <a href="qc_lang1.html#K22" target="_self">still_not-bound_in</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K16" target="_self">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#NM2">Subset</a> of <a href="qc_lang1.html#K2" target="_self">bound_QC-variables</a> ) <b>-&gt; </b>  <a href="subset_1.html#NM2">Subset</a> of <a href="qc_lang1.html#K2" target="_self">bound_QC-variables</a>  = <font color="Maroon">a<sub>1</sub></font>;<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" target="_self">bound_QC-variables</a> ,  <a href="subset_1.html#NM2">Subset</a> of <a href="qc_lang1.html#K2" target="_self">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" target="_self">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>4</sub></font>(   <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K8" target="_self">QC-WFF</a> ,  <a href="subset_1.html#NM2">Subset</a> of <a href="qc_lang1.html#K2" target="_self">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" target="_self">bound_QC-variables</a>  = <font color="Maroon">a<sub>2</sub></font> <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" target="_self">bound_in</a> <font color="Maroon">a<sub>1</sub></font></span>)</span></span><a href="tarski.html#K1">}</a></span>;<br/>
<b>consider </b><font color="Maroon">c<sub>3</sub></font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="qc_lang1.html#K8" target="_self">QC-WFF</a> , <a href="zfmisc_1.html#K1">bool</a> <a href="qc_lang1.html#K2" target="_self">bound_QC-variables</a> <b> such that </b><br/><a NAME="E3:48_1_1"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <a href="qc_lang1.html#K11" target="_self">VERUM</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K8" target="_self">QC-WFF</a>  holds <br/>( ( <font color="Olive">b<sub>1</sub></font> is <a href="qc_lang1.html#V2" target="_self">atomic</a> implies <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>) ) &amp; ( <font color="Olive">b<sub>1</sub></font> is <a href="qc_lang1.html#V3" target="_self">negative</a> implies <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K17" target="_self">the_argument_of</a> <font color="Olive">b<sub>1</sub></font></span>)</span>) ) &amp; ( <font color="Olive">b<sub>1</sub></font> is <a href="qc_lang1.html#V4" target="_self">conjunctive</a> implies <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K18" target="_self">the_left_argument_of</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K19" target="_self">the_right_argument_of</a> <font color="Olive">b<sub>1</sub></font></span>)</span>) ) &amp; ( <font color="Olive">b<sub>1</sub></font> is <a href="qc_lang1.html#V5" target="_self">universal</a> implies <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<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="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K21" target="_self">the_scope_of</a> <font color="Olive">b<sub>1</sub></font></span>)</span>) ) ) ) )
 <b>from </b><i><a class="ref" href="qc_lang1.html#S3" target="_self">QC_LANG1:sch 3</a>();<br/></i>
<b>take </b>
<font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>3</sub></font>
;<br/>

<b>thus </b><a NAME="E4:48_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K8" target="_self">QC-WFF</a> ;<br/>

<b>thus </b><a NAME="E5:48_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <a href="qc_lang1.html#K11" target="_self">VERUM</a>  <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="qc_lang1.html#E3:48_1_1"><i><font color="Green">E37</font></i></a></i>;<br/>

<b>thus </b><a NAME="E6:48_1_1"/>
( <font color="Maroon">c<sub>4</sub></font> is <a href="qc_lang1.html#V2" target="_self">atomic</a> implies <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</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" target="_self">the_arguments_of</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a> : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></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" target="_self">the_arguments_of</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="qc_lang1.html#K16" target="_self">the_arguments_of</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K2" target="_self">bound_QC-variables</a>  ) </span>}</span>  )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:48_1_1_1"/>
<font color="Maroon">c<sub>4</sub></font> is <a href="qc_lang1.html#V2" target="_self">atomic</a>
 ;<br/>

<a NAME="E2:48_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="qc_lang1.html#K22" target="_self">still_not-bound_in</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K16" target="_self">the_arguments_of</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="qc_lang1.html#E3:48_1_1"><i><font color="Green">E37</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:48_1_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</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" target="_self">the_arguments_of</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a> : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></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" target="_self">the_arguments_of</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="qc_lang1.html#K16" target="_self">the_arguments_of</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K2" target="_self">bound_QC-variables</a>  ) </span>}</span> 
 ;<br/>


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

<b>thus </b><a NAME="E7:48_1_1"/>
( <font color="Maroon">c<sub>4</sub></font> is <a href="qc_lang1.html#V3" target="_self">negative</a> implies <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K17" target="_self">the_argument_of</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="qc_lang1.html#E3:48_1_1"><i><font color="Green">E37</font></i></a></i>;<br/>

<b>thus </b><a NAME="E8:48_1_1"/>
( <font color="Maroon">c<sub>4</sub></font> is <a href="qc_lang1.html#V4" target="_self">conjunctive</a> implies <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K18" target="_self">the_left_argument_of</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K19" target="_self">the_right_argument_of</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="qc_lang1.html#E3:48_1_1"><i><font color="Green">E37</font></i></a></i>;<br/>

<b>assume </b><a NAME="E9:48_1_1"/>
<font color="Maroon">c<sub>4</sub></font> is <a href="qc_lang1.html#V5" target="_self">universal</a>
 ;<br/>

<b>hence </b><a NAME="E10:48_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K21" target="_self">the_scope_of</a> <font color="Maroon">c<sub>4</sub></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" target="_self">bound_in</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="qc_lang1.html#E3:48_1_1"><i><font color="Green">E37</font></i></a></i>;<br/>


</div>
