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

<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/>





<a NAME="E1:29"/><i><font color="Green">E20</font></i>: 
for <font color="Olive">x</font> being   <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> holds  <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><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> where <font color="Olive">x1</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X1</font>, <font color="Olive">x2</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X2</font>, <font color="Olive">x3</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X3</font> : verum </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1"><b>proof </b></a><div class="add">

<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/>

<a NAME="E1:29_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="E2:29_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><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> where <font color="Olive">x1</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X1</font>, <font color="Olive">x2</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X2</font>, <font color="Olive">x3</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X3</font> : verum </span>}</span> 
 ;<br/>


</div><b>end;</b></div>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b>verum;<br/>
<a NAME="E2:29"/>
for <font color="Olive">X1</font>, <font color="Olive">X2</font>, <font color="Olive">X3</font> being non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  holds  <span class="p1">{<span class="default"> <span class="p2"><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> where <font color="Olive">x1</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Olive">X1</font>, <font color="Olive">x2</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Olive">X2</font>, <font color="Olive">x3</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Olive">X3</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x1</font>,<font color="Olive">x2</font>,<font color="Olive">x3</font>] </span>}</span>  is   <a href="subset_1.html#NM2">Subset</a> of <span class="p1"><a href="zfmisc_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="zfmisc_1.html#K3">:]</a></span>
 
<b>from </b><i><a class="ref" href="domain_1.html#S3" target="_self">DOMAIN_1:sch 3</a>();<br/></i>
<a NAME="E3:29"/><b>then </b>
<span class="p1">{<span class="default"> <span class="p2"><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> where <font color="Olive">x1</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X1</font>, <font color="Olive">x2</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X2</font>, <font color="Olive">x3</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X3</font> : verum </span>}</span>  is   <a href="subset_1.html#NM2">Subset</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>hence </b><a NAME="E4:29"/>
<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> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2"><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> where <font color="Olive">x1</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X1</font>, <font color="Olive">x2</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X2</font>, <font color="Olive">x3</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X3</font> : verum </span>}</span> 
 <b>by </b><i>, <a class="ref" href="subset_1.html#T49">SUBSET_1:49</a></i>;<br/>


</div>
