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

<b>set </b><font color="Maroon" title="c2">Y</font> = <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default">7 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c1">k</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>;<br/>
<a NAME="E1:11_1_1"/>
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default">7 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c1">k</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="qc_lang1.html#K5" title="QC_LANG1:func.5">QC-pred_symbols</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:11_1_1_1"/><i><font color="Green" title="E5">A1</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default">7 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c1">k</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 ;<br/>

<a NAME="E2:11_1_1_1"/><b>then </b><i><font color="Green" title="E6">A2</font></i>: 
( <font color="Maroon" title="c3">x</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 7 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c1">k</font> &amp; <font color="Maroon" title="c3">x</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  )
 
<b>by </b><i><a class="ref" href="mcart_1.html#T12" title="MCART_1:th.12">MCART_1:12</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c4">x1</font> = <font color="Maroon" title="c3">x</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> , <font color="Maroon" title="c5">x2</font> = <font color="Maroon" title="c3">x</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="qc_lang1.html#E1:11_1_1_1"><i><font color="Green" title="E5">A1</font></i></a>, <a class="ref" href="mcart_1.html#T12" title="MCART_1:th.12">MCART_1:12</a></i>;<br/>
<a NAME="E4:11_1_1_1"/>
7 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">x1</font>
 
<b>by </b><i><a class="txt" href="qc_lang1.html#E2:11_1_1_1"><i><font color="Green" title="E6">A2</font></i></a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></i>;<br/>
<a NAME="E5:11_1_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x1</font>,<font color="Maroon" title="c5">x2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">m</font>,<font color="Olive" title="b2">n</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">m</font>, <font color="Olive" title="b2">n</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : 7 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> </span>}</span> 
 
;<br/>
<b>hence </b><a NAME="E6:11_1_1_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang1.html#K5" title="QC_LANG1:func.5">QC-pred_symbols</a> 
 <b>by </b><i><a class="txt" href="qc_lang1.html#E1:11_1_1_1"><i><font color="Green" title="E5">A1</font></i></a>, <a class="ref" href="mcart_1.html#T23" title="MCART_1:th.23">MCART_1:23</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c3">X</font> = <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default">7 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c1">k</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="qc_lang1.html#K5" title="QC_LANG1:func.5">QC-pred_symbols</a>  ;<br/>
<a NAME="E3:11_1_1"/>
<font color="Maroon" title="c3">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">P</font> where <font color="Olive" title="b1">P</font> is   <a href="qc_lang1.html#NM2" title="QC_LANG1:NM.2">QC-pred_symbol</a> :  <a href="qc_lang1.html#K6" title="QC_LANG1:func.6">the_arity_of</a> <font color="Olive" title="b1">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_1_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:11_1_1_2"/>
<font color="Maroon" title="c3">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">P</font> where <font color="Olive" title="b1">P</font> is   <a href="qc_lang1.html#NM2" title="QC_LANG1:NM.2">QC-pred_symbol</a> :  <a href="qc_lang1.html#K6" title="QC_LANG1:func.6">the_arity_of</a> <font color="Olive" title="b1">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> </span>}</span> 
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:11_1_1_2_1"/><i><font color="Green" title="E5">A3</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c5">Q</font> = <font color="Maroon" title="c4">x</font> as   <a href="qc_lang1.html#NM2" title="QC_LANG1:NM.2">QC-pred_symbol</a> ;<br/>
<a NAME="E3:11_1_1_2_1"/>
<font color="Maroon" title="c4">x</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default">7 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c1">k</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="txt" href="qc_lang1.html#E1:11_1_1_2_1"><i><font color="Green" title="E5">A3</font></i></a>, <a class="ref" href="mcart_1.html#T10" title="MCART_1:th.10">MCART_1:10</a></i>;<br/>
<a NAME="E4:11_1_1_2_1"/><b>then </b>
<font color="Maroon" title="c4">x</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 7 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c1">k</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E5:11_1_1_2_1"/><b>then </b>
 <a href="qc_lang1.html#K6" title="QC_LANG1:func.6">the_arity_of</a> <font color="Maroon" title="c5">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font>
 
<b>by </b><i><a class="ref" href="qc_lang1.html#D6" target="_self" title="QC_LANG1:def.6">Def6</a></i>;<br/>
<b>hence </b><a NAME="E6:11_1_1_2_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">P</font> where <font color="Olive" title="b1">P</font> is   <a href="qc_lang1.html#NM2" title="QC_LANG1:NM.2">QC-pred_symbol</a> :  <a href="qc_lang1.html#K6" title="QC_LANG1:func.6">the_arity_of</a> <font color="Olive" title="b1">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> </span>}</span> 
 ;<br/>


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

<b>let </b><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:11_1_1_2"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">P</font> where <font color="Olive" title="b1">P</font> is   <a href="qc_lang1.html#NM2" title="QC_LANG1:NM.2">QC-pred_symbol</a> :  <a href="qc_lang1.html#K6" title="QC_LANG1:func.6">the_arity_of</a> <font color="Olive" title="b1">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c5">P</font> being   <a href="qc_lang1.html#NM2" title="QC_LANG1:NM.2">QC-pred_symbol</a><b> such that </b><br/><a NAME="E4:11_1_1_2"/><i><font color="Green" title="E5">A4</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P</font>
 <b>and </b><br/><a NAME="E5:11_1_1_2"/><i><font color="Green" title="E6">A5</font></i>: 
 <a href="qc_lang1.html#K6" title="QC_LANG1:func.6">the_arity_of</a> <font color="Maroon" title="c5">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font>
 ;<br/>
<a NAME="E6:11_1_1_2"/><i><font color="Green" title="E7">A6</font></i>: 
<font color="Maroon" title="c5">P</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span>
 
<b>by </b><i><a class="ref" href="qc_lang1.html#T10" target="_self" title="QC_LANG1:th.10">Th10</a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<a NAME="E7:11_1_1_2"/>
<font color="Maroon" title="c5">P</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 7 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c1">k</font>
 
<b>by </b><i><a class="txt" href="qc_lang1.html#E5:11_1_1_2"><i><font color="Green" title="E6">A5</font></i></a>, <a class="ref" href="qc_lang1.html#D6" target="_self" title="QC_LANG1:def.6">Def6</a></i>;<br/>
<a NAME="E8:11_1_1_2"/><b>then </b>
( <font color="Maroon" title="c5">P</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default">7 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c1">k</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c5">P</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  )
 
<b>by </b><i><a class="txt" href="qc_lang1.html#E6:11_1_1_2"><i><font color="Green" title="E7">A6</font></i></a>, <a class="ref" href="mcart_1.html#T10" title="MCART_1:th.10">MCART_1:10</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E9:11_1_1_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<b>hence </b><a NAME="E10:11_1_1_2"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font>
 <b>by </b><i><a class="txt" href="qc_lang1.html#E4:11_1_1_2"><i><font color="Green" title="E5">A4</font></i></a>, <a class="txt" href="qc_lang1.html#E6:11_1_1_2"><i><font color="Green" title="E7">A6</font></i></a>, <a class="ref" href="mcart_1.html#T23" title="MCART_1:th.23">MCART_1:23</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:11_1_1"/>
not <font color="Maroon" title="c1">k</font> <a href="qc_lang1.html#K7" title="QC_LANG1:func.7">-ary_QC-pred_symbols</a>  is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ;<br/>


</div>
