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

<b>let </b><font color="Maroon">Z</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">X1</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">X2</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">X3</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">X4</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">X5</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">X6</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">X7</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">X8</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">X9</font> be    <a href="hidden.html#M1">set</a> ;<br/>



















<b>assume </b><a NAME="E1:41"/><i><font color="Green">E73</font></i>: 
for <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> iff  ex <font color="Olive">x1</font>, <font color="Olive">x2</font>, <font color="Olive">x3</font>, <font color="Olive">x4</font>, <font color="Olive">x5</font>, <font color="Olive">x6</font>, <font color="Olive">x7</font>, <font color="Olive">x8</font>, <font color="Olive">x9</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X1</font> &amp; <font color="Olive">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X2</font> &amp; <font color="Olive">x3</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X3</font> &amp; <font color="Olive">x4</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X4</font> &amp; <font color="Olive">x5</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X5</font> &amp; <font color="Olive">x6</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X6</font> &amp; <font color="Olive">x7</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X7</font> &amp; <font color="Olive">x8</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X8</font> &amp; <font color="Olive">x9</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X9</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="mcart_6.html#K1" target="_self">[</a><span class="default"><font color="Olive">x1</font>,<font color="Olive">x2</font>,<font color="Olive">x3</font>,<font color="Olive">x4</font>,<font color="Olive">x5</font>,<font color="Olive">x6</font>,<font color="Olive">x7</font>,<font color="Olive">x8</font>,<font color="Olive">x9</font></span><a href="mcart_6.html#K1" target="_self">]</a></span> ) )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>thus </b><a NAME="E1:41_1"/>
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> implies <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="mcart_5.html#K2">[:</a><span class="default"><font color="Maroon">X1</font>,<font color="Maroon">X2</font>,<font color="Maroon">X3</font>,<font color="Maroon">X4</font>,<font color="Maroon">X5</font>,<font color="Maroon">X6</font>,<font color="Maroon">X7</font>,<font color="Maroon">X8</font></span><a href="mcart_5.html#K2">:]</a></span>,<font color="Maroon">X9</font></span><a href="zfmisc_1.html#K2">:]</a></span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:41_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 ;<br/>

<b>then consider </b><font color="Maroon">x1</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">x2</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">x3</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">x4</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">x5</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">x6</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">x7</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">x8</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">x9</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:41_1_1"/><i><font color="Green">E75</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X1</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X2</font> &amp; <font color="Maroon">x3</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X3</font> &amp; <font color="Maroon">x4</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X4</font> &amp; <font color="Maroon">x5</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X5</font> &amp; <font color="Maroon">x6</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X6</font> &amp; <font color="Maroon">x7</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X7</font> &amp; <font color="Maroon">x8</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X8</font> &amp; <font color="Maroon">x9</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X9</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="mcart_6.html#K1" target="_self">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font>,<font color="Maroon">x3</font>,<font color="Maroon">x4</font>,<font color="Maroon">x5</font>,<font color="Maroon">x6</font>,<font color="Maroon">x7</font>,<font color="Maroon">x8</font>,<font color="Maroon">x9</font></span><a href="mcart_6.html#K1" target="_self">]</a></span> )
 <b>by </b><i/>;<br/>
<a NAME="E4:41_1_1"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="mcart_5.html#K1">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font>,<font color="Maroon">x3</font>,<font color="Maroon">x4</font>,<font color="Maroon">x5</font>,<font color="Maroon">x6</font>,<font color="Maroon">x7</font>,<font color="Maroon">x8</font></span><a href="mcart_5.html#K1">]</a></span>,<font color="Maroon">x9</font></span><a href="tarski.html#K4">]</a></span>
 
<b>by </b><i/>;<br/>
<a NAME="E5:41_1_1"/>
( <span class="p1"><a href="mcart_5.html#K1">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font>,<font color="Maroon">x3</font>,<font color="Maroon">x4</font>,<font color="Maroon">x5</font>,<font color="Maroon">x6</font>,<font color="Maroon">x7</font>,<font color="Maroon">x8</font></span><a href="mcart_5.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="mcart_5.html#K2">[:</a><span class="default"><font color="Maroon">X1</font>,<font color="Maroon">X2</font>,<font color="Maroon">X3</font>,<font color="Maroon">X4</font>,<font color="Maroon">X5</font>,<font color="Maroon">X6</font>,<font color="Maroon">X7</font>,<font color="Maroon">X8</font></span><a href="mcart_5.html#K2">:]</a></span> &amp; <font color="Maroon">x9</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X9</font> )
 
<b>by </b><i>, <a class="ref" href="mcart_5.html#T40">MCART_5:40</a></i>;<br/>
<b>hence </b><a NAME="E6:41_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="mcart_5.html#K2">[:</a><span class="default"><font color="Maroon">X1</font>,<font color="Maroon">X2</font>,<font color="Maroon">X3</font>,<font color="Maroon">X4</font>,<font color="Maroon">X5</font>,<font color="Maroon">X6</font>,<font color="Maroon">X7</font>,<font color="Maroon">X8</font></span><a href="mcart_5.html#K2">:]</a></span>,<font color="Maroon">X9</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>


</div><b>end;</b></div><b>assume </b><a NAME="E2:41_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="mcart_5.html#K2">[:</a><span class="default"><font color="Maroon">X1</font>,<font color="Maroon">X2</font>,<font color="Maroon">X3</font>,<font color="Maroon">X4</font>,<font color="Maroon">X5</font>,<font color="Maroon">X6</font>,<font color="Maroon">X7</font>,<font color="Maroon">X8</font></span><a href="mcart_5.html#K2">:]</a></span>,<font color="Maroon">X9</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/><b>then consider </b><font color="Maroon">x12345678</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">x9</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:41_1"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">x12345678</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="mcart_5.html#K2">[:</a><span class="default"><font color="Maroon">X1</font>,<font color="Maroon">X2</font>,<font color="Maroon">X3</font>,<font color="Maroon">X4</font>,<font color="Maroon">X5</font>,<font color="Maroon">X6</font>,<font color="Maroon">X7</font>,<font color="Maroon">X8</font></span><a href="mcart_5.html#K2">:]</a></span>
 <b>and </b><br/><a NAME="E5:41_1"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">x9</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X9</font>
 <b>and </b><br/><a NAME="E6:41_1"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x12345678</font>,<font color="Maroon">x9</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/><b>consider </b><font color="Maroon">x1</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">x2</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">x3</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">x4</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">x5</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">x6</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">x7</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">x8</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E8:41_1"/><i><font color="Green">E85</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X1</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X2</font> &amp; <font color="Maroon">x3</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X3</font> &amp; <font color="Maroon">x4</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X4</font> &amp; <font color="Maroon">x5</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X5</font> &amp; <font color="Maroon">x6</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X6</font> &amp; <font color="Maroon">x7</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X7</font> &amp; <font color="Maroon">x8</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X8</font> &amp; <font color="Maroon">x12345678</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="mcart_5.html#K1">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font>,<font color="Maroon">x3</font>,<font color="Maroon">x4</font>,<font color="Maroon">x5</font>,<font color="Maroon">x6</font>,<font color="Maroon">x7</font>,<font color="Maroon">x8</font></span><a href="mcart_5.html#K1">]</a></span> )
 <b>by </b><i>, <a class="ref" href="mcart_5.html#T39">MCART_5:39</a></i>;<br/><a NAME="E9:41_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="mcart_6.html#K1" target="_self">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font>,<font color="Maroon">x3</font>,<font color="Maroon">x4</font>,<font color="Maroon">x5</font>,<font color="Maroon">x6</font>,<font color="Maroon">x7</font>,<font color="Maroon">x8</font>,<font color="Maroon">x9</font></span><a href="mcart_6.html#K1" target="_self">]</a></span>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E10:41_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div>
<a NAME="E3:41"/><b>then </b>
<font color="Maroon">Z</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="mcart_5.html#K2">[:</a><span class="default"><font color="Maroon">X1</font>,<font color="Maroon">X2</font>,<font color="Maroon">X3</font>,<font color="Maroon">X4</font>,<font color="Maroon">X5</font>,<font color="Maroon">X6</font>,<font color="Maroon">X7</font>,<font color="Maroon">X8</font></span><a href="mcart_5.html#K2">:]</a></span>,<font color="Maroon">X9</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>
<b>hence </b><a NAME="E4:41"/>
<font color="Maroon">Z</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="mcart_6.html#K2" target="_self">[:</a><span class="default"><font color="Maroon">X1</font>,<font color="Maroon">X2</font>,<font color="Maroon">X3</font>,<font color="Maroon">X4</font>,<font color="Maroon">X5</font>,<font color="Maroon">X6</font>,<font color="Maroon">X7</font>,<font color="Maroon">X8</font>,<font color="Maroon">X9</font></span><a href="mcart_6.html#K2" target="_self">:]</a></span>
 ;<br/>


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