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

<b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">X1</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">X2</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">X3</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/>







<b>thus </b><a NAME="E1:5"/>
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K3">[:</a><span class="default"><font color="Maroon">X1</font>,<font color="Maroon">X2</font>,<font color="Maroon">X3</font></span><a href="zfmisc_1.html#K3">:]</a></span> implies  ex <font color="Olive">x1</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X1</font> ex <font color="Olive">x2</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X2</font> ex <font color="Olive">x3</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X3</font> st <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="mcart_1.html#K3">[</a><span class="default"><font color="Olive">x1</font>,<font color="Olive">x2</font>,<font color="Olive">x3</font></span><a href="mcart_1.html#K3">]</a></span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:5_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K3">[:</a><span class="default"><font color="Maroon">X1</font>,<font color="Maroon">X2</font>,<font color="Maroon">X3</font></span><a href="zfmisc_1.html#K3">:]</a></span>
 ;<br/>

<a NAME="E2:5_1"/><b>then </b>
<font color="Maroon">a</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="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">X1</font>,<font color="Maroon">X2</font></span><a href="zfmisc_1.html#K2">:]</a></span>,<font color="Maroon">X3</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#D3">ZFMISC_1:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">x12</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">x3</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:5_1"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">x12</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">X1</font>,<font color="Maroon">X2</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>and </b><br/><a NAME="E5:5_1"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">x3</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X3</font>
 <b>and </b><br/><a NAME="E6:5_1"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x12</font>,<font color="Maroon">x3</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> <b> such that </b><br/><a NAME="E8:5_1"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X1</font>
 <b>and </b><br/><a NAME="E9:5_1"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X2</font>
 <b>and </b><br/><a NAME="E10:5_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">x12</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</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>reconsider </b><font color="Maroon">x1</font> = <font color="Maroon">x1</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X1</font> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">x2</font> = <font color="Maroon">x2</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X2</font> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">x3</font> = <font color="Maroon">x3</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X3</font> <b>by </b><i/>;<br/>
<a NAME="E14:5_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="mcart_1.html#K3">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font>,<font color="Maroon">x3</font></span><a href="mcart_1.html#K3">]</a></span>
 
<b>by </b><i>, , <a class="ref" href="mcart_1.html#D3">MCART_1:def 3</a></i>;<br/>
<b>hence </b><a NAME="E15:5_1"/>
 ex <font color="Olive">x1</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X1</font> ex <font color="Olive">x2</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X2</font> ex <font color="Olive">x3</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X3</font> st <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="mcart_1.html#K3">[</a><span class="default"><font color="Olive">x1</font>,<font color="Olive">x2</font>,<font color="Olive">x3</font></span><a href="mcart_1.html#K3">]</a></span>
 ;<br/>


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

<b>given </b><font color="Maroon">x1</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X1</font>, <font color="Maroon">x2</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X2</font>, <font color="Maroon">x3</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X3</font><b> such that </b><a NAME="E2:5"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="mcart_1.html#K3">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font>,<font color="Maroon">x3</font></span><a href="mcart_1.html#K3">]</a></span>
 ;<br/>

<a NAME="E3:5"/>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1" target="_self">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1" target="_self">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="domain_1.html#K1" target="_self">]</a></span>,<font color="Maroon">x3</font></span><a href="domain_1.html#K1" target="_self">]</a></span>
 
<b>by </b><i>, <a class="ref" href="mcart_1.html#D3">MCART_1:def 3</a></i>;<br/>
<a NAME="E4:5"/><b>then </b>
<font color="Maroon">a</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="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">X1</font>,<font color="Maroon">X2</font></span><a href="zfmisc_1.html#K2">:]</a></span>,<font color="Maroon">X3</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
;<br/>
<b>hence </b><a NAME="E5:5"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K3">[:</a><span class="default"><font color="Maroon">X1</font>,<font color="Maroon">X2</font>,<font color="Maroon">X3</font></span><a href="zfmisc_1.html#K3">:]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D3">ZFMISC_1:def 3</a></i>;<br/>


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