<?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>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="subset_1.html#NM2">Subset</a> of <a href="qc_lang1.html#K2">bound_QC-variables</a> ;<br/>



<b>thus </b><a NAME="E1:10"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">c<sub>1</sub></font> implies ex <font color="Olive">b<sub>1</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="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>2</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>3</sub></font>, <font color="Olive">b<sub>4</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>  <a href="qc_lang1.html#K11">VERUM</a>  implies <font color="Olive">b<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>2</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>2</sub></font> is <a href="qc_lang1.html#V2">atomic</a> implies <font color="Olive">b<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <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="Olive">b<sub>2</sub></font></span>)</span> ) &amp; ( <font color="Olive">b<sub>2</sub></font> is <a href="qc_lang1.html#V3">negative</a> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</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>2</sub></font></span>)</span> implies <font color="Olive">b<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font> ) &amp; ( <font color="Olive">b<sub>2</sub></font> is <a href="qc_lang1.html#V4">conjunctive</a> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</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>2</sub></font></span>)</span> &amp; <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</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>2</sub></font></span>)</span> implies <font color="Olive">b<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Olive">b<sub>4</sub></font> ) &amp; ( <font color="Olive">b<sub>2</sub></font> is <a href="qc_lang1.html#V5">universal</a> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</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>2</sub></font></span>)</span> implies <font color="Olive">b<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</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">bound_in</a> <font color="Olive">b<sub>2</sub></font></span>)</span></span><a href="tarski.html#K1">}</a></span> ) ) ) ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:10_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<b>then 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">QC-WFF</a> , <a href="zfmisc_1.html#K1">bool</a> <a href="qc_lang1.html#K2">bound_QC-variables</a> <b> such that </b><br/><a NAME="E3:10_1"/><i><font color="Green">E2</font></i>: 
( <font color="Maroon">c<sub>2</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> &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">QC-WFF</a>  holds <br/>( <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <a href="qc_lang1.html#K11">VERUM</a>  <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; ( <font color="Olive">b<sub>1</sub></font> is <a href="qc_lang1.html#V2">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> <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="Olive">b<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>2</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>2</sub></font> &amp; <font color="Olive">b<sub>2</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">the_arguments_of</a> <font color="Olive">b<sub>1</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="qc_lang1.html#K16">the_arguments_of</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K2">bound_QC-variables</a>  ) </span>}</span>  ) &amp; ( <font color="Olive">b<sub>1</sub></font> is <a href="qc_lang1.html#V3">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">c<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>1</sub></font></span>)</span> ) &amp; ( <font color="Olive">b<sub>1</sub></font> is <a href="qc_lang1.html#V4">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> <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">the_left_argument_of</a> <font color="Olive">b<sub>1</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">the_right_argument_of</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span> ) &amp; ( <font color="Olive">b<sub>1</sub></font> is <a href="qc_lang1.html#V5">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> <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">the_scope_of</a> <font color="Olive">b<sub>1</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">bound_in</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span><a href="tarski.html#K1">}</a></span> ) ) ) )
 <b>by </b><i><a class="ref" href="qc_lang1.html#D29">QC_LANG1:def 29</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>3</sub></font>
;<br/>

<b>thus </b><a NAME="E4:10_1"/>
<font color="Maroon">c<sub>2</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>
 <b>by </b><i><a class="txt" href="qc_lang3.html#E3:10_1"><i><font color="Green">E2</font></i></a></i>;<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">QC-WFF</a> ;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> be   <a href="subset_1.html#NM2">Subset</a> of <a href="qc_lang1.html#K2">bound_QC-variables</a> ;<br/>



<b>thus </b><a NAME="E5:10_1"/>
( ( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="qc_lang1.html#K11">VERUM</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>  <a href="subset_1.html#K1">{}</a> <a href="qc_lang1.html#K2">bound_QC-variables</a>  ) &amp; ( <font color="Maroon">c<sub>4</sub></font> is <a href="qc_lang1.html#V2">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>  <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">c<sub>4</sub></font></span>)</span> ) &amp; ( <font color="Maroon">c<sub>4</sub></font> is <a href="qc_lang1.html#V3">negative</a> &amp; <font color="Maroon">c<sub>5</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">the_argument_of</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> 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>5</sub></font> ) )
 <b>by </b><i><a class="txt" href="qc_lang3.html#E3:10_1"><i><font color="Green">E2</font></i></a></i>;<br/>

<b>thus </b><a NAME="E6:10_1"/>
( ( <font color="Maroon">c<sub>4</sub></font> is <a href="qc_lang1.html#V4">conjunctive</a> &amp; <font color="Maroon">c<sub>5</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#K18">the_left_argument_of</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>6</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#K19">the_right_argument_of</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> 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>5</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>6</sub></font> ) &amp; ( <font color="Maroon">c<sub>4</sub></font> is <a href="qc_lang1.html#V5">universal</a> &amp; <font color="Maroon">c<sub>5</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#K21">the_scope_of</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> 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>5</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">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_lang3.html#E3:10_1"><i><font color="Green">E2</font></i></a></i>;<br/>


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

<b>given </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">QC-WFF</a> , <a href="zfmisc_1.html#K1">bool</a> <a href="qc_lang1.html#K2">bound_QC-variables</a> <b> such that </b><a NAME="E2:10"/><i><font color="Green">E2</font></i>: 
<font color="Maroon">c<sub>2</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>
 <b>and </b><br/><a NAME="E3:10"/><i><font color="Green">E3</font></i>: 
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">QC-WFF</a> <br/>for <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</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>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="qc_lang1.html#K11">VERUM</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>  <a href="subset_1.html#K1">{}</a> <a href="qc_lang1.html#K2">bound_QC-variables</a>  ) &amp; ( <font color="Olive">b<sub>1</sub></font> is <a href="qc_lang1.html#V2">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>  <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="Olive">b<sub>1</sub></font></span>)</span> ) &amp; ( <font color="Olive">b<sub>1</sub></font> is <a href="qc_lang1.html#V3">negative</a> &amp; <font color="Olive">b<sub>2</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">the_argument_of</a> <font color="Olive">b<sub>1</sub></font></span>)</span> 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="Olive">b<sub>2</sub></font> ) &amp; ( <font color="Olive">b<sub>1</sub></font> is <a href="qc_lang1.html#V4">conjunctive</a> &amp; <font color="Olive">b<sub>2</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#K18">the_left_argument_of</a> <font color="Olive">b<sub>1</sub></font></span>)</span> &amp; <font color="Olive">b<sub>3</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#K19">the_right_argument_of</a> <font color="Olive">b<sub>1</sub></font></span>)</span> 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="Olive">b<sub>2</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Olive">b<sub>3</sub></font> ) &amp; ( <font color="Olive">b<sub>1</sub></font> is <a href="qc_lang1.html#V5">universal</a> &amp; <font color="Olive">b<sub>2</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#K21">the_scope_of</a> <font color="Olive">b<sub>1</sub></font></span>)</span> 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="Olive">b<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">bound_in</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span><a href="tarski.html#K1">}</a></span> ) )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_2"><b>now </b></a><div class="add"><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">QC-WFF</a> ;<br/><b>thus </b><a NAME="E1:10_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <a href="qc_lang1.html#K11">VERUM</a>  <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="qc_lang3.html#E3:10"><i><font color="Green">E3</font></i></a></i>;<br/><b>thus </b><a NAME="E2:10_2"/>
( <font color="Maroon">c<sub>4</sub></font> is <a href="qc_lang1.html#V2">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">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">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">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">bound_QC-variables</a>  ) </span>}</span>  )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:10_2_1"/>
<font color="Maroon">c<sub>4</sub></font> is <a href="qc_lang1.html#V2">atomic</a>
 ;<br/>

<a NAME="E2:10_2_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">still_not-bound_in</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K16">the_arguments_of</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="qc_lang3.html#E3:10"><i><font color="Green">E3</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:10_2_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">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">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">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">bound_QC-variables</a>  ) </span>}</span> 
 ;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E3:10_2"/>
( <font color="Maroon">c<sub>4</sub></font> is <a href="qc_lang1.html#V3">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">the_argument_of</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="qc_lang3.html#E3:10"><i><font color="Green">E3</font></i></a></i>;<br/><b>thus </b><a NAME="E4:10_2"/>
( <font color="Maroon">c<sub>4</sub></font> is <a href="qc_lang1.html#V4">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">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">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_lang3.html#E3:10"><i><font color="Green">E3</font></i></a></i>;<br/><b>assume </b><a NAME="E5:10_2"/>
<font color="Maroon">c<sub>4</sub></font> is <a href="qc_lang1.html#V5">universal</a>
 ;<br/><b>hence </b><a NAME="E6:10_2"/>
<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">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">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_lang3.html#E3:10"><i><font color="Green">E3</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:10"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>1</sub></font>)
 <b>by </b><i><a class="txt" href="qc_lang3.html#E2:10"><i><font color="Green">E2</font></i></a>, <a class="ref" href="qc_lang1.html#D29">QC_LANG1:def 29</a></i>;<br/>


</div>
