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

<a NAME="E1:41_1_1"/><i><font color="Green" title="E28">A1</font></i>: 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">Y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> is   <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <font color="Maroon" title="c1">Y</font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T80" title="ZFMISC_1:th.80">ZFMISC_1:80</a></i>;<br/>
<a NAME="E2:41_1_1"/><i><font color="Green" title="E29">A2</font></i>: 
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">Y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">Y</font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T31" title="ZFMISC_1:th.31">ZFMISC_1:31</a></i>;<br/>
<a NAME="E3:41_1_1"/>
for <font color="Olive" title="b1">A</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">Y</font>  st <font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">Y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> holds <br/>( <font color="Olive" title="b1">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; ( for <font color="Olive" title="b2">B</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">Y</font>  holds <br/>(  not <font color="Olive" title="b2">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">Y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> or <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">B</font> or <font color="Olive" title="b1">A</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b2">B</font> ) ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">A</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">Y</font>;<br/>

<b>assume </b><a NAME="E1:41_1_1_1"/><i><font color="Green" title="E30">A3</font></i>: 
<font color="Maroon" title="c2">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">Y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ;<br/>

<a NAME="E2:41_1_1_1"/><b>then </b><i><font color="Green" title="E31">A4</font></i>: 
<font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">Y</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>thus </b><a NAME="E3:41_1_1_1"/>
<font color="Maroon" title="c2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="partit1.html#E1:41_1_1_1"><i><font color="Green" title="E30">A3</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>

<b>let </b><font color="Maroon" title="c3">B</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">Y</font>;<br/>

<b>assume </b><a NAME="E4:41_1_1_1"/>
<font color="Maroon" title="c3">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">Y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ;<br/>

<b>hence </b><a NAME="E5:41_1_1_1"/>
( <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">B</font> or <font color="Maroon" title="c2">A</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">B</font> )
 <b>by </b><i><a class="txt" href="partit1.html#E2:41_1_1_1"><i><font color="Green" title="E31">A4</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:41_1_1"/>
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">Y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> is    <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of <font color="Maroon" title="c1">Y</font>
 <b>by </b><i><a class="txt" href="partit1.html#E1:41_1_1"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="partit1.html#E2:41_1_1"><i><font color="Green" title="E29">A2</font></i></a>, <a class="ref" href="eqrel_1.html#D6" title="EQREL_1:def.6">EQREL_1:def 6</a></i>;<br/>


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