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

<b>let </b><font color="Maroon">R</font> be   <a href="relat_1.html#NM1">Relation</a>;<br/>

<b>assume </b><a NAME="E1:30"/>
<font color="Maroon">R</font> is <a href="relat_2.html#V7" target="_self">strongly_connected</a>
 ;<br/>

<a NAME="E2:30"/><b>then </b><i><font color="Green">E30</font></i>: 
<font color="Maroon">R</font> <a href="relat_2.html#R7" target="_self">is_strongly_connected_in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font>
 
<b>by </b><i><a class="ref" href="relat_2.html#D2" target="_self">Def2</a></i>;<br/>
<b>thus </b><a NAME="E3:30"/>
<font color="Maroon">R</font> is <a href="relat_2.html#V6" target="_self">connected</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1"><b>proof </b></a><div class="add">

<a NAME="E1:30_1"/>
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#K3">field</a> <font color="Maroon">R</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font> &amp; <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">y</font> &amp; not <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> holds <br/><span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">y</font>,<font color="Olive">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 
<b>by </b><i><a class="ref" href="relat_2.html#D8" target="_self">Def8</a>, </i>;<br/>
<a NAME="E2:30_1"/><b>then </b>
<font color="Maroon">R</font> <a href="relat_2.html#R6" target="_self">is_connected_in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:30_1"/>
<font color="Maroon">R</font> is <a href="relat_2.html#V6" target="_self">connected</a>
 <b>by </b><i><a class="ref" href="relat_2.html#D1" target="_self">Def1</a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E4:30"/>
<font color="Maroon">R</font> is <a href="relat_2.html#V1" target="_self">reflexive</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="relat_2.html#D1" target="_self">RELAT_2:def 1</a>,<a class="ref" href="relat_2.html#D9" target="_self">RELAT_2:def 9</a><br/>

<b>assume </b><a NAME="E1:30_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font>
 ;<br/>

<b>hence </b><a NAME="E2:30_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 <b>by </b><i><a class="ref" href="relat_2.html#D8" target="_self">Def8</a>, </i>;<br/>


</div><b>end;</b></div>


</div>
