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







<b>assume </b><a NAME="E1:30"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 ;<br/>

<a NAME="E2:30"/><i><font color="Green">E39</font></i>: 
( <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">EqR</font> implies  <a href="eqrel_1.html#K7" target="_self">Class</a> <font color="Maroon">EqR</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">EqR</font>,<font color="Maroon">y</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:30_1"/>
<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">EqR</font>
 ;<br/>

<a NAME="E2:30_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K7" target="_self">Class</a> <font color="Maroon">EqR</font>,<font color="Maroon">y</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:30_1"/>
 <a href="eqrel_1.html#K7" target="_self">Class</a> <font color="Maroon">EqR</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">EqR</font>,<font color="Maroon">y</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:30"/>
( not <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">EqR</font> implies  <a href="eqrel_1.html#K7" target="_self">Class</a> <font color="Maroon">EqR</font>,<font color="Maroon">x</font> <a href="xboole_0.html#R1">misses</a>  <a href="eqrel_1.html#K7" target="_self">Class</a> <font color="Maroon">EqR</font>,<font color="Maroon">y</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:30_2"/><i><font color="Green">E48</font></i>: 
not <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">EqR</font>
 ;<br/>

<b>assume </b><a NAME="E2:30_2"/>
 <a href="eqrel_1.html#K7" target="_self">Class</a> <font color="Maroon">EqR</font>,<font color="Maroon">x</font> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="eqrel_1.html#K7" target="_self">Class</a> <font color="Maroon">EqR</font>,<font color="Maroon">y</font>
 ;<br/>

<b>then consider </b><font color="Maroon">z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:30_2"/><i><font color="Green">E49</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">EqR</font>,<font color="Maroon">x</font> &amp; <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">EqR</font>,<font color="Maroon">y</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<a NAME="E5:30_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">EqR</font> &amp; <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">EqR</font> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E6:30_2"/><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">EqR</font> &amp; <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">EqR</font> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E7:30_2"/>
contradiction
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:30"/>
(  <a href="eqrel_1.html#K7" target="_self">Class</a> <font color="Maroon">EqR</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">EqR</font>,<font color="Maroon">y</font> or  <a href="eqrel_1.html#K7" target="_self">Class</a> <font color="Maroon">EqR</font>,<font color="Maroon">x</font> <a href="xboole_0.html#R1">misses</a>  <a href="eqrel_1.html#K7" target="_self">Class</a> <font color="Maroon">EqR</font>,<font color="Maroon">y</font> )
 <b>by </b><i/>;<br/>


</div>
