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<b>let </b><font color="Maroon">R</font> be  non <a href="struct_0.html#V3">empty</a> <a href="orders_3.html#V1">discrete</a> <a href="orders_2.html#NM1">Poset</a>;<br/><b>let </b><font color="Maroon">C</font> be   <a href="osalg_4.html#NM5">Component</a> of <font color="Maroon">R</font>;<br/>



<b>consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E2:25"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>
 <b>and </b><br/><a NAME="E3:25"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">C</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="osalg_4.html#K1" target="_self">Path_Rel</a> <font color="Maroon">R</font></span>)</span>,<font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="eqrel_1.html#D5">EQREL_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x1</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">x1</font>
;<br/>

<a NAME="E5:25"/>
for <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> iff <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1"><b>proof </b></a><div class="add">

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

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:25_1_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 ;<br/><a NAME="E2:25_1_1"/><b>then </b><i><font color="Green">E45</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>  <a href="osalg_4.html#K1" target="_self">Path_Rel</a> <font color="Maroon">R</font>
 <b>by </b><i>, <a class="ref" href="eqrel_1.html#T27">EQREL_1:27</a></i>;<br/><b>reconsider </b><font color="Maroon">y1</font> = <font color="Maroon">y</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> <b>by </b><i/>;<br/><a NAME="E4:25_1_1"/>
<font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E5:25_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font>
 ;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:25_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font>
 ;<br/>

<b>hence </b><a NAME="E3:25_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 <b>by </b><i>, <a class="ref" href="eqrel_1.html#T28">EQREL_1:28</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:25"/>
<font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x1</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


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