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<b>let </b><font color="Maroon">c<sub>1</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>thus </b><a NAME="E1:7"/>
(  not <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K1">QC-variables</a>  or <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K3">fixed_QC-variables</a>  or <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K4">free_QC-variables</a>  or <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K2">bound_QC-variables</a>  )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:7_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K1">QC-variables</a> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:7_1"/><i><font color="Green">E6</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="enumset1.html#K1">{</a><span class="default">4,5,6</span><a href="enumset1.html#K1">}</a></span>
 <b>and </b><br/><a NAME="E4:7_1"/><i><font color="Green">E7</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  &amp; <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i><a class="ref" href="qc_lang1.html#D1">QC_LANG1:def 1</a>, <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<a NAME="E5:7_1"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 4 or <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 5 or <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 6 )
 
<b>by </b><i><a class="txt" href="cqc_lang.html#E3:7_1"><i><font color="Green">E6</font></i></a>, <a class="ref" href="enumset1.html#D1">ENUMSET1:def 1</a></i>;<br/>
<a NAME="E6:7_1"/><b>then </b>
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">4</span><a href="tarski.html#K1">}</a></span> or <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">5</span><a href="tarski.html#K1">}</a></span> or <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">6</span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E7:7_1"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K3">fixed_QC-variables</a>  or <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K4">free_QC-variables</a>  or <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K2">bound_QC-variables</a>  )
 <b>by </b><i><a class="txt" href="cqc_lang.html#E4:7_1"><i><font color="Green">E7</font></i></a>, <a class="ref" href="qc_lang1.html#D2">QC_LANG1:def 2</a>, <a class="ref" href="qc_lang1.html#D3">QC_LANG1:def 3</a>, <a class="ref" href="qc_lang1.html#D4">QC_LANG1:def 4</a>, <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>


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

<b>thus </b><a NAME="E2:7"/>
( ( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K3">fixed_QC-variables</a>  or <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K4">free_QC-variables</a>  or <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K2">bound_QC-variables</a>  ) implies <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K1">QC-variables</a>  )
 ;<br/>


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