<?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>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of <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/>









<b>thus </b><a NAME="E1:8"/>
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="mcart_1.html#K5">`1</a>  implies for <font color="Olive">x1</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X1</font><br/> for <font color="Olive">x2</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X2</font><br/> for <font color="Olive">x3</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X3</font>  st <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K4" target="_self">[</a><span class="default"><font color="Olive">x1</font>,<font color="Olive">x2</font>,<font color="Olive">x3</font></span><a href="domain_1.html#K4" target="_self">]</a></span> holds <br/><font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">x1</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:8_1"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="mcart_1.html#K5">`1</a> 
 ;<br/>

<b>let </b><font color="Maroon">x1</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X1</font>;<br/><b>let </b><font color="Maroon">x2</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X2</font>;<br/><b>let </b><font color="Maroon">x3</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X3</font>;<br/>





<b>assume </b><a NAME="E2:8_1"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K4" target="_self">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font>,<font color="Maroon">x3</font></span><a href="domain_1.html#K4" target="_self">]</a></span>
 ;<br/>

<a NAME="E3:8_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K4" target="_self">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="mcart_1.html#K5">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="mcart_1.html#K6">`2</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="mcart_1.html#K7">`3</a> </span>)</span></span><a href="domain_1.html#K4" target="_self">]</a></span>
 
<b>by </b><i><a class="ref" href="mcart_1.html#T48">MCART_1:48</a></i>;<br/>
<b>hence </b><a NAME="E4:8_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font>
 <b>by </b><i>, , <a class="ref" href="mcart_1.html#T28">MCART_1:28</a></i>;<br/>


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

<b>thus </b><a NAME="E2:8"/>
( ( for <font color="Olive">x1</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X1</font><br/> for <font color="Olive">x2</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X2</font><br/> for <font color="Olive">x3</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X3</font>  st <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K4" target="_self">[</a><span class="default"><font color="Olive">x1</font>,<font color="Olive">x2</font>,<font color="Olive">x3</font></span><a href="domain_1.html#K4" target="_self">]</a></span> holds <br/><font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">x1</font> ) implies <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="mcart_1.html#K5">`1</a>  )
 <b>by </b><i><a class="ref" href="mcart_1.html#T69">MCART_1:69</a></i>;<br/>


</div>
