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

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

<a NAME="E1:15"/>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> is    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="t_1topsp.html#T7">T_1TOPSP:7</a></i>;<br/>
<a NAME="E2:15"/><b>then </b>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R2">in</a>  <a href="partit1.html#K1">PARTITIONS</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="partit1.html#D3">PARTIT1:def 3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">S</font> = <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> as   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="partit1.html#K1">PARTITIONS</a> <font color="Maroon">A</font></span>)</span> <b>by </b><i><a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E4:15"/>
<font color="Maroon">S</font> is    <a href="taxonom1.html#M1" target="_self">Classification</a> of <font color="Maroon">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">X</font> be    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">A</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="taxonom1.html#D1" target="_self">TAXONOM1:def 1</a><br/>

<b>thus </b><a NAME="E1:15_1"/>
for <font color="Olive">Y</font> being   <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">A</font>  st <font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> &amp; <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> &amp; not <font color="Maroon">X</font> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Olive">Y</font> holds <br/><font color="Olive">Y</font> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Maroon">X</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:15"/>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> is    <a href="taxonom1.html#M1" target="_self">Classification</a> of <font color="Maroon">A</font>
 ;<br/>


</div>
