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<b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">EqR</font> be   <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of <font color="Maroon">X</font>;<br/>



<a NAME="E1:40"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  implies  <a href="eqrel_1.html#K8" target="_self">Class</a> <font color="Maroon">EqR</font> is    <a href="eqrel_1.html#M1" target="_self">a_partition</a> of <font color="Maroon">X</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:40_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:40_1_1_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/><b>consider </b><font color="Maroon">Z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:40_1_1_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">Z</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7" target="_self">Class</a> <font color="Maroon">EqR</font>,<font color="Maroon">x</font>
 ;<br/><a NAME="E4:40_1_1_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 <b>by </b><i>, , </i>;<br/><a NAME="E5:40_1_1_1"/>
<font color="Maroon">Z</font> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K8" target="_self">Class</a> <font color="Maroon">EqR</font>
 <b>by </b><i>, , </i>;<br/><b>hence </b><a NAME="E6:40_1_1_1"/>
 ex <font color="Olive">Z</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">Z</font> &amp; <font color="Olive">Z</font> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K8" target="_self">Class</a> <font color="Maroon">EqR</font> )
 <b>by </b><i/>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E2:40_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> iff  ex <font color="Olive">Z</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">Z</font> &amp; <font color="Olive">Z</font> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K8" target="_self">Class</a> <font color="Maroon">EqR</font> ) )
 ;<br/></div><b>end;</b></div>
<a NAME="E3:40_1"/><b>then </b><i><font color="Green">E53</font></i>: 
 <a href="setfam_1.html#K5">union</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K8" target="_self">Class</a> <font color="Maroon">EqR</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<a NAME="E4:40_1"/>
for <font color="Olive">A</font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>  st <font color="Olive">A</font> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K8" target="_self">Class</a> <font color="Maroon">EqR</font> holds <br/>( <font color="Olive">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; ( for <font color="Olive">B</font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>  holds <br/>(  not <font color="Olive">B</font> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K8" target="_self">Class</a> <font color="Maroon">EqR</font> or <font color="Olive">A</font> <a href="hidden.html#R1">=</a> <font color="Olive">B</font> or <font color="Olive">A</font> <a href="xboole_0.html#R1">misses</a> <font color="Olive">B</font> ) ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_2"><b>proof </b></a><div class="add">

<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="E1:40_1_2"/>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K8" target="_self">Class</a> <font color="Maroon">EqR</font>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:40_1_2"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7" target="_self">Class</a> <font color="Maroon">EqR</font>,<font color="Maroon">x</font> )
 <b>by </b><i/>;<br/>
<b>thus </b><a NAME="E4:40_1_2"/>
<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>, </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="E5:40_1_2"/>
<font color="Maroon">B</font> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K8" target="_self">Class</a> <font color="Maroon">EqR</font>
 ;<br/>

<a NAME="E6:40_1_2"/><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> <font color="Maroon">X</font> &amp; <font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7" target="_self">Class</a> <font color="Maroon">EqR</font>,<font color="Olive">y</font> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E7:40_1_2"/>
( <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>
<b>hence </b><a NAME="E5:40_1"/>
 <a href="eqrel_1.html#K8" target="_self">Class</a> <font color="Maroon">EqR</font> is    <a href="eqrel_1.html#M1" target="_self">a_partition</a> of <font color="Maroon">X</font>
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:40"/>
( <font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  implies  <a href="eqrel_1.html#K8" target="_self">Class</a> <font color="Maroon">EqR</font> is    <a href="eqrel_1.html#M1" target="_self">a_partition</a> of <font color="Maroon">X</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:40_2"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<a NAME="E2:40_2"/><b>then </b>
 <a href="eqrel_1.html#K8" target="_self">Class</a> <font color="Maroon">EqR</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:40_2"/>
 <a href="eqrel_1.html#K8" target="_self">Class</a> <font color="Maroon">EqR</font> is    <a href="eqrel_1.html#M1" target="_self">a_partition</a> of <font color="Maroon">X</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:40"/>
 <a href="eqrel_1.html#K8" target="_self">Class</a> <font color="Maroon">EqR</font> is    <a href="eqrel_1.html#M1" target="_self">a_partition</a> of <font color="Maroon">X</font>
 <b>by </b><i/>;<br/>


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