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

<b>let </b><font color="Maroon">I</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">A</font> be    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">I</font>;<br/><b>let </b><font color="Maroon">B</font> be    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">I</font>;<br/>





<b>assume </b><a NAME="E1:32"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">A</font> is <a href="relat_1.html#V2">non-empty</a> &amp; <span class="p1"><a href="pboole.html#K10">[|</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font></span><a href="pboole.html#K10">|]</a></span> is <a href="finset_1.html#V2">locally-finite</a> )
 ;<br/>

<b>let </b><font color="Maroon">i</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="pre_circ.html#D3">PRE_CIRC:def 3</a><br/>

<b>assume </b><a NAME="E2:32"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 ;<br/>

<a NAME="E3:32"/><b>then </b>
<span class="p1"><a href="pboole.html#K10">[|</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font></span><a href="pboole.html#K10">|]</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i>, <a class="ref" href="pre_circ.html#D3">PRE_CIRC:def 3</a></i>;<br/>
<a NAME="E4:32"/><b>then </b>
( <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> is <a href="finset_1.html#V1">finite</a> )
 
<b>by </b><i>, , <a class="ref" href="pboole.html#D16">PBOOLE:def 16</a>, <a class="ref" href="pboole.html#D21">PBOOLE:def 21</a></i>;<br/>
<b>hence </b><a NAME="E5:32"/>
<font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i><a class="ref" href="finset_1.html#T23">FINSET_1:23</a></i>;<br/>


</div>
