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

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s1</font> &amp; <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="osalg_4.html#K5" target="_self">CompClass</a> <font color="Maroon">E</font>,<span class="p2">(<span class="default"><a href="osalg_4.html#K3" target="_self">CComp</a> <font color="Maroon">s1</font></span>)</span></span>)</span>,<font color="Olive">x</font> );<br/>
<b>let </b><font color="Maroon">X</font> be   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="eqrel_1.html#K9">Class</a> <span class="p2">(<span class="default"><a href="osalg_4.html#K5" target="_self">CompClass</a> <font color="Maroon">E</font>,<span class="p3">(<span class="default"><a href="osalg_4.html#K3" target="_self">CComp</a> <font color="Maroon">s1</font></span>)</span></span>)</span></span>)</span>, <font color="Maroon">Y</font> be   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="eqrel_1.html#K9">Class</a> <span class="p2">(<span class="default"><a href="osalg_4.html#K5" target="_self">CompClass</a> <font color="Maroon">E</font>,<span class="p3">(<span class="default"><a href="osalg_4.html#K3" target="_self">CComp</a> <font color="Maroon">s1</font></span>)</span></span>)</span></span>)</span>;<br/>

<b>assume </b><a NAME="E1:33_1_2"/><i><font color="Green">E67</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> iff <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>] )
 ;<br/>

<b>assume </b><a NAME="E2:33_1_2"/><i><font color="Green">E68</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> iff <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>] )
 ;<br/>

<a NAME="E3:33_1_2"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> iff <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:33_1_2_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/><a NAME="E2:33_1_2_1_1"/><b>then </b>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x</font>]
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:33_1_2_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>by </b><i/>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:33_1_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 ;<br/>

<a NAME="E3:33_1_2_1"/><b>then </b>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x</font>]
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E4:33_1_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:33_1_2"/>
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>


</div>
