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

<b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">P</font> be  <a href="relat_1.html#V2">non-empty</a> <a href="funct_1.html#NM1">Function</a>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:31"/><i><font color="Green">E29</font></i>: 
 <a href="card_3.html#K3">Union</a> <font color="Maroon">P</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font>
 <b>and </b><br/><a NAME="E2:31"/><i><font color="Green">E36</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">P</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">P</font> &amp; <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">y</font> holds <br/><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font>
 ;<br/>

<a NAME="E3:31"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">P</font> <a href="tarski.html#R1">c=</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:31_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">P</font>
 ;<br/>

<a NAME="E2:31_1"/><b>then </b>
<font color="Maroon">y</font> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">P</font></span>)</span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/>
<a NAME="E3:31_1"/><b>then </b>
<font color="Maroon">y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font>
 
<b>by </b><i>, <a class="ref" href="card_3.html#D4">CARD_3:def 4</a></i>;<br/>
<b>hence </b><a NAME="E4:31_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">X</font>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">Y</font> =  <a href="relat_1.html#K2">rng</a> <font color="Maroon">P</font> as   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">X</font> ;<br/>
<a NAME="E5:31"/>
<font color="Maroon">Y</font> is    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:31_2_1"/>
( <font color="Maroon">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_2_1_1"><b>case </b></a><a NAME="E1:31_2_1_1"/>
<font color="Maroon">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>thus </b><a NAME="E2:31_2_1_1"/>
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font>
 <b>by </b><i>, <a class="ref" href="card_3.html#D4">CARD_3:def 4</a></i>;<br/><b>let </b><font color="Maroon">A</font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>;<br/><b>assume </b><a NAME="E3:31_2_1_1"/>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 ;<br/><a NAME="E4:31_2_1_1"/><b>then </b><i><font color="Green">E37</font></i>: 
 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>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">P</font> &amp; <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>hence </b><a NAME="E5:31_2_1_1"/>
<font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="funct_1.html#D15">FUNCT_1:def 15</a></i>;<br/><b>let </b><font color="Maroon">B</font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>;<br/><b>assume </b><a NAME="E6:31_2_1_1"/>
<font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 ;<br/><a NAME="E7:31_2_1_1"/><b>then </b>
 ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">P</font> &amp; <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>hence </b><a NAME="E8:31_2_1_1"/>
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> or <font color="Maroon">A</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">B</font> )
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_2_1_2"><b>case </b></a><a NAME="E1:31_2_1_2"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>assume </b><a NAME="E2:31_2_1_2"/>
<font color="Maroon">Y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><b>then reconsider </b><font color="Maroon">Y</font> = <font color="Maroon">Y</font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ;<br/><b>consider </b><font color="Maroon">y</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font>;<br/><a NAME="E5:31_2_1_2"/>
 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>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">P</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>then reconsider </b><font color="Maroon">y</font> = <font color="Maroon">y</font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="ref" href="funct_1.html#D15">FUNCT_1:def 15</a></i>;<br/><b>consider </b><font color="Maroon">x</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">y</font>;<br/><a NAME="E8:31_2_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">P</font></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>hence </b><a NAME="E9:31_2_1_2"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="card_3.html#D4">CARD_3:def 4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E6:31"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">P</font> is    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">X</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="pua2mss1.html#D3" target="_self">PUA2MSS1:def 3</a><br/>

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a><br/>

<b>assume </b><a NAME="E7:31"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">P</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">P</font> &amp; <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y</font> )
 ;<br/>

<b>then reconsider </b><font color="Maroon">Px</font> = <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>, <font color="Maroon">Py</font> = <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="ref" href="funct_1.html#D15">FUNCT_1:def 15</a></i>;<br/>
<b>consider </b><font color="Maroon">a</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Px</font>;<br/>
<a NAME="E10:31"/>
<font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E11:31"/><b>then </b>
not <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Py</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<b>hence </b><a NAME="E12:31"/>
contradiction
 <b>by </b><i/>;<br/>


</div>
