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

<b>let </b><font color="Maroon">A</font> be  non <a href="struct_0.html#V3">empty</a> <a href="orders_2.html#V2" target="_self">reflexive</a>  <a href="orders_2.html#L1" target="_self">RelStr</a> ;<br/><b>let </b><font color="Maroon">a1</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font>, <font color="Maroon">a2</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font>;<br/>



<b>thus </b><a NAME="E1:25"/>
(  not <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">a2</font></span><a href="domain_1.html#K7">}</a></span> is   <a href="orders_2.html#NM2">Chain</a> of <font color="Maroon">A</font> or <font color="Maroon">a1</font> <a href="orders_2.html#R3" target="_self">&lt;=</a> <font color="Maroon">a2</font> or <font color="Maroon">a2</font> <a href="orders_2.html#R3" target="_self">&lt;=</a> <font color="Maroon">a1</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1"><b>proof </b></a><div class="add">

<a NAME="E1:25_1"/><i><font color="Green">E18</font></i>: 
( <font color="Maroon">a1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">a2</font></span><a href="domain_1.html#K7">}</a></span> &amp; <font color="Maroon">a2</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">a2</font></span><a href="domain_1.html#K7">}</a></span> )
 
<b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<b>assume </b><a NAME="E2:25_1"/>
<span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">a2</font></span><a href="domain_1.html#K7">}</a></span> is   <a href="orders_2.html#NM2">Chain</a> of <font color="Maroon">A</font>
 ;<br/>

<a NAME="E3:25_1"/><b>then </b>
the <a href="orders_2.html#U1" target="_self">InternalRel</a> of <font color="Maroon">A</font> <a href="relat_2.html#R7">is_strongly_connected_in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">a2</font></span><a href="domain_1.html#K7">}</a></span>
 
<b>by </b><i><a class="ref" href="orders_2.html#D4" target="_self">Def4</a></i>;<br/>
<a NAME="E4:25_1"/><b>then </b>
( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">a2</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1" target="_self">InternalRel</a> of <font color="Maroon">A</font> or <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">a1</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1" target="_self">InternalRel</a> of <font color="Maroon">A</font> )
 
<b>by </b><i>, <a class="ref" href="relat_2.html#D7">RELAT_2:def 7</a></i>;<br/>
<b>hence </b><a NAME="E5:25_1"/>
( <font color="Maroon">a1</font> <a href="orders_2.html#R3" target="_self">&lt;=</a> <font color="Maroon">a2</font> or <font color="Maroon">a2</font> <a href="orders_2.html#R3" target="_self">&lt;=</a> <font color="Maroon">a1</font> )
 <b>by </b><i/>;<br/>


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

<b>assume </b><a NAME="E2:25"/><i><font color="Green">E19</font></i>: 
( <font color="Maroon">a1</font> <a href="orders_2.html#R3" target="_self">&lt;=</a> <font color="Maroon">a2</font> or <font color="Maroon">a2</font> <a href="orders_2.html#R3" target="_self">&lt;=</a> <font color="Maroon">a1</font> )
 ;<br/>

<a NAME="E3:25"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">a1</font> <a href="orders_2.html#R3" target="_self">&lt;=</a> <font color="Maroon">a1</font> &amp; <font color="Maroon">a2</font> <a href="orders_2.html#R3" target="_self">&lt;=</a> <font color="Maroon">a2</font> )
 
;<br/>
<a NAME="E4:25"/>
<span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">a2</font></span><a href="domain_1.html#K7">}</a></span> is <a href="orders_2.html#V5" target="_self">strongly_connected</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_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#D7">RELAT_2:def 7</a>,<a class="ref" href="orders_2.html#D11" target="_self">ORDERS_2:def 11</a><br/><b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><a NAME="E1:25_2"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">a2</font></span><a href="domain_1.html#K7">}</a></span> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">a2</font></span><a href="domain_1.html#K7">}</a></span> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:25_2_1_1"/>
( ( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a1</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a1</font> ) or ( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a1</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a2</font> ) or ( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a2</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a1</font> ) or ( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a2</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a2</font> ) )
 </span><b>by </b><i><a class="txt" href="orders_2.html#E2:25"><i><font color="Green">E19</font></i></a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_2_1_1_1"><b>suppose </b></a><a NAME="E1:25_2_1_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a1</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a1</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:25_2_1_1_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> the <a href="orders_2.html#U1" target="_self">InternalRel</a> of <font color="Maroon">A</font> or <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> the <a href="orders_2.html#U1" target="_self">InternalRel</a> of <font color="Maroon">A</font> )
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_2_1_1_2"><b>suppose </b></a><a NAME="E1:25_2_1_1_2"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a1</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a2</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:25_2_1_1_2"/>
( <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> the <a href="orders_2.html#U1" target="_self">InternalRel</a> of <font color="Maroon">A</font> or <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> the <a href="orders_2.html#U1" target="_self">InternalRel</a> of <font color="Maroon">A</font> )
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_2_1_1_3"><b>suppose </b></a><a NAME="E1:25_2_1_1_3"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a2</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a1</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:25_2_1_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> the <a href="orders_2.html#U1" target="_self">InternalRel</a> of <font color="Maroon">A</font> or <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> the <a href="orders_2.html#U1" target="_self">InternalRel</a> of <font color="Maroon">A</font> )
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_2_1_1_4"><b>suppose </b></a><a NAME="E1:25_2_1_1_4"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a2</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a2</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:25_2_1_1_4"/>
( <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> the <a href="orders_2.html#U1" target="_self">InternalRel</a> of <font color="Maroon">A</font> or <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> the <a href="orders_2.html#U1" target="_self">InternalRel</a> of <font color="Maroon">A</font> )
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:25_2"/>
( <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> the <a href="orders_2.html#U1" target="_self">InternalRel</a> of <font color="Maroon">A</font> or <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> the <a href="orders_2.html#U1" target="_self">InternalRel</a> of <font color="Maroon">A</font> )
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:25"/>
<span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">a2</font></span><a href="domain_1.html#K7">}</a></span> is   <a href="orders_2.html#NM2">Chain</a> of <font color="Maroon">A</font>
 ;<br/>


</div>
