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<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">EqR1</font> be   <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">EqR2</font> be   <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">E</font> be   <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of <font color="Maroon">X</font>;<br/>







<b>assume </b><a NAME="E1:35"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">E</font> <a href="hidden.html#R1">=</a> <font color="Maroon">EqR1</font> <a href="eqrel_1.html#K3" target="_self">\/</a> <font color="Maroon">EqR2</font>
 ;<br/>

<a NAME="E2:35"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   holds <br/>(  not <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> or  <a href="eqrel_1.html#K7" target="_self">Class</a> <font color="Maroon">E</font>,<font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7" target="_self">Class</a> <font color="Maroon">EqR1</font>,<font color="Olive">x</font> or  <a href="eqrel_1.html#K7" target="_self">Class</a> <font color="Maroon">E</font>,<font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7" target="_self">Class</a> <font color="Maroon">EqR2</font>,<font color="Olive">x</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:35_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E2:35_1"/><i><font color="Green">E39</font></i>: 
not  <a href="eqrel_1.html#K7" target="_self">Class</a> <font color="Maroon">E</font>,<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7" target="_self">Class</a> <font color="Maroon">EqR1</font>,<font color="Maroon">x</font>
 <b>and </b><br/><a NAME="E3:35_1"/><i><font color="Green">E48</font></i>: 
not  <a href="eqrel_1.html#K7" target="_self">Class</a> <font color="Maroon">E</font>,<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7" target="_self">Class</a> <font color="Maroon">EqR2</font>,<font color="Maroon">x</font>
 ;<br/>

<b>consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:35_1"/><i><font color="Green">E49</font></i>: 
( ( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K7" target="_self">Class</a> <font color="Maroon">E</font>,<font color="Maroon">x</font> &amp; not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K7" target="_self">Class</a> <font color="Maroon">EqR1</font>,<font color="Maroon">x</font> ) or ( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K7" target="_self">Class</a> <font color="Maroon">EqR1</font>,<font color="Maroon">x</font> &amp; not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K7" target="_self">Class</a> <font color="Maroon">E</font>,<font color="Maroon">x</font> ) )
 <b>by </b><i>, <a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>
<div><i><font color="Green">E52</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1"><b>now </b></a><div class="add"><b>assume </b><b>that </b><br/><a NAME="E1:35_1_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K7" target="_self">Class</a> <font color="Maroon">EqR1</font>,<font color="Maroon">x</font>
 <b>and </b><br/><a NAME="E2:35_1_1"/><i><font color="Green">E54</font></i>: 
not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K7" target="_self">Class</a> <font color="Maroon">E</font>,<font color="Maroon">x</font>
 ;<br/><a NAME="E3:35_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">EqR1</font>
 <b>by </b><i>, </i>;<br/><a NAME="E4:35_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">E</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>hence </b><a NAME="E5:35_1_1"/>
contradiction
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<b>consider </b><font color="Maroon">z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E8:35_1"/><i><font color="Green">E55</font></i>: 
( ( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K7" target="_self">Class</a> <font color="Maroon">E</font>,<font color="Maroon">x</font> &amp; not <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K7" target="_self">Class</a> <font color="Maroon">EqR2</font>,<font color="Maroon">x</font> ) or ( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K7" target="_self">Class</a> <font color="Maroon">EqR2</font>,<font color="Maroon">x</font> &amp; not <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K7" target="_self">Class</a> <font color="Maroon">E</font>,<font color="Maroon">x</font> ) )
 <b>by </b><i>, <a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>
<div><i><font color="Green">E56</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_2"><b>now </b></a><div class="add"><b>assume </b><b>that </b><br/><a NAME="E1:35_1_2"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K7" target="_self">Class</a> <font color="Maroon">EqR2</font>,<font color="Maroon">x</font>
 <b>and </b><br/><a NAME="E2:35_1_2"/><i><font color="Green">E58</font></i>: 
not <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K7" target="_self">Class</a> <font color="Maroon">E</font>,<font color="Maroon">x</font>
 ;<br/><a NAME="E3:35_1_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">z</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">EqR2</font>
 <b>by </b><i>, </i>;<br/><a NAME="E4:35_1_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">z</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">E</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>hence </b><a NAME="E5:35_1_2"/>
contradiction
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<a NAME="E10:35_1"/><i><font color="Green">E59</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">E</font>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E11:35_1"/>
not <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">EqR1</font>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E12:35_1"/><b>then </b><i><font color="Green">E60</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">EqR2</font>
 
<b>by </b><i>, , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E13:35_1"/><i><font color="Green">E76</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">z</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">E</font>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E14:35_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">z</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">E</font>
 
<b>by </b><i/>;<br/>
<a NAME="E15:35_1"/><b>then </b><i><font color="Green">E77</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">E</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E16:35_1"/>
not <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">z</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">EqR2</font>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E17:35_1"/><b>then </b><i><font color="Green">E78</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">z</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">EqR1</font>
 
<b>by </b><i>, , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<div><i><font color="Green">E79</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:35_1_3"/><i><font color="Green">E80</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">EqR2</font>
 ;<br/><a NAME="E2:35_1_3"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">EqR2</font>
 <b>by </b><i>, </i>;<br/><a NAME="E3:35_1_3"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">z</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">EqR2</font>
 <b>by </b><i>, </i>;<br/><a NAME="E4:35_1_3"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">z</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">EqR2</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E5:35_1_3"/>
contradiction
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:35_1_4"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">EqR1</font>
 ;<br/><a NAME="E2:35_1_4"/><b>then </b><i><font color="Green">E81</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">z</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">EqR1</font>
 <b>by </b><i/>;<br/><a NAME="E3:35_1_4"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">z</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">EqR1</font>
 <b>by </b><i>, </i>;<br/><a NAME="E4:35_1_4"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">EqR1</font>
 <b>by </b><i>, </i>;<br/><a NAME="E5:35_1_4"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">EqR1</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E6:35_1_4"/>
contradiction
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E20:35_1"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:35"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   holds <br/>(  not <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> or  <a href="eqrel_1.html#K7" target="_self">Class</a> <font color="Maroon">E</font>,<font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7" target="_self">Class</a> <font color="Maroon">EqR1</font>,<font color="Olive">x</font> or  <a href="eqrel_1.html#K7" target="_self">Class</a> <font color="Maroon">E</font>,<font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7" target="_self">Class</a> <font color="Maroon">EqR2</font>,<font color="Olive">x</font> )
 ;<br/>


</div>
