<?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#NM1">Poset</a>;<br/>

<b>assume </b><a NAME="E1:82"/><i><font color="Green">E18</font></i>: 
for <font color="Olive">C</font> being  <a href="orders_2.html#NM2">Chain</a> of <font color="Maroon">A</font>  ex <font color="Olive">a</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font> st <br/>for <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font>  st <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> holds <br/><font color="Olive">a</font> <a href="orders_2.html#R3" target="_self">&lt;=</a> <font color="Olive">b</font>
 ;<br/>

<b>set </b><font color="Maroon">X</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font>;<br/>
<b>set </b><font color="Maroon">R</font> = the <a href="orders_2.html#U1" target="_self">InternalRel</a> of <font color="Maroon">A</font> <a href="relset_1.html#K6">~</a> ;<br/>
<a NAME="E2:82"/><i><font color="Green">E19</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font> holds <br/> ( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">a</font>,<font color="Olive">b</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> <a href="relset_1.html#K6">~</a>  iff <font color="Olive">b</font> <a href="orders_2.html#R3" target="_self">&lt;=</a> <font color="Olive">a</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="82_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font>, <font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font>;<br/>

<a NAME="E1:82_1"/>
( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</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> <a href="relset_1.html#K6">~</a>  iff <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">a</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_1.html#D7">RELAT_1:def 7</a></i>;<br/>
<b>hence </b><a NAME="E2:82_1"/>
( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</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> <a href="relset_1.html#K6">~</a>  iff <font color="Maroon">b</font> <a href="orders_2.html#R3" target="_self">&lt;=</a> <font color="Maroon">a</font> )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default">the <a href="orders_2.html#U1" target="_self">InternalRel</a> of <font color="Maroon">A</font> <a href="relset_1.html#K6">~</a> </span>)</span> = 
 <a href="relset_1.html#K4">dom</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#T29">RELAT_2:29</a></i>
<br/>.= 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font>
<b>by </b><i><a class="ref" href="partfun1.html#D4">PARTFUN1:def 4</a></i>
;<br/>
<b>then reconsider </b><font color="Maroon">R</font> = the <a href="orders_2.html#U1" target="_self">InternalRel</a> of <font color="Maroon">A</font> <a href="relset_1.html#K6">~</a>  as   <a href="orders_1.html#NM2">Order</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font> <b>by </b><i><a class="ref" href="partfun1.html#D4">PARTFUN1:def 4</a>, <a class="ref" href="relat_2.html#T27">RELAT_2:27</a>, <a class="ref" href="relat_2.html#T40">RELAT_2:40</a>, <a class="ref" href="relat_2.html#T42">RELAT_2:42</a></i>;<br/>
<b>set </b><font color="Maroon">B</font> =  <a href="orders_2.html#G1" target="_self">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font>,<font color="Maroon">R</font> #);<br/>
<a NAME="E5:82"/>
for <font color="Olive">E</font> being  <a href="orders_2.html#NM2">Chain</a> of <a href="orders_2.html#G1" target="_self">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font>,<font color="Maroon">R</font> #)  ex <font color="Olive">e</font> being  <a href="struct_0.html#NM1">Element</a> of <a href="orders_2.html#G1" target="_self">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font>,<font color="Maroon">R</font> #) st <br/>for <font color="Olive">f</font> being  <a href="struct_0.html#NM1">Element</a> of <a href="orders_2.html#G1" target="_self">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font>,<font color="Maroon">R</font> #)  st <font color="Olive">f</font> <a href="hidden.html#R2">in</a> <font color="Olive">E</font> holds <br/><font color="Olive">f</font> <a href="orders_2.html#R3" target="_self">&lt;=</a> <font color="Olive">e</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="82_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">E</font> be   <a href="orders_2.html#NM2">Chain</a> of <a href="orders_2.html#G1" target="_self">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font>,<font color="Maroon">R</font> #);<br/>

<b>reconsider </b><font color="Maroon">C</font> = <font color="Maroon">E</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">A</font> ;<br/>
<a NAME="E2:82_3"/>
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> <font color="Maroon">C</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="82_3_1"><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><br/><b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><a NAME="E1:82_3_1"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> )
 ;<br/>

<b>then reconsider </b><font color="Maroon">e</font> = <font color="Maroon">x</font>, <font color="Maroon">f</font> = <font color="Maroon">y</font> as   <a href="struct_0.html#NM1">Element</a> of <a href="orders_2.html#G1" target="_self">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font>,<font color="Maroon">R</font> #) ;<br/>
<a NAME="E3:82_3_1"/><i><font color="Green">E40</font></i>: 
( ( <font color="Maroon">e</font> <a href="orders_2.html#R3" target="_self">&lt;=</a> <font color="Maroon">f</font> or <font color="Maroon">f</font> <a href="orders_2.html#R3" target="_self">&lt;=</a> <font color="Maroon">e</font> ) &amp; <font color="Maroon">R</font> <a href="hidden.html#R1">=</a> the <a href="orders_2.html#U1" target="_self">InternalRel</a> of <a href="orders_2.html#G1" target="_self">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font>,<font color="Maroon">R</font> #) )
 
<b>by </b><i>, </i>;<br/>
<b>reconsider </b><font color="Maroon">e1</font> = <font color="Maroon">e</font>, <font color="Maroon">f1</font> = <font color="Maroon">f</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="82_3_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="82_3_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:82_3_1_1_1"/>
( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">e</font>,<font color="Maroon">f</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> or <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">f</font>,<font color="Maroon">e</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> )
 </span><b>by </b><i><a class="ref" href="orders_2.html#T39" target="_self">Th39</a>, </i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="82_3_1_1_1_1"><b>suppose </b></a><a NAME="E1:82_3_1_1_1_1"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">e</font>,<font color="Maroon">f</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 ;<br/><div class="add"><a NAME="E2:82_3_1_1_1_1"/><b>then </b>
<font color="Maroon">f1</font> <a href="orders_2.html#R3" target="_self">&lt;=</a> <font color="Maroon">e1</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:82_3_1_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/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="82_3_1_1_1_2"><b>suppose </b></a><a NAME="E1:82_3_1_1_1_2"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">f</font>,<font color="Maroon">e</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 ;<br/><div class="add"><a NAME="E2:82_3_1_1_1_2"/><b>then </b>
<font color="Maroon">e1</font> <a href="orders_2.html#R3" target="_self">&lt;=</a> <font color="Maroon">f1</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:82_3_1_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/>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E6:82_3_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> )
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">C</font> = <font color="Maroon">C</font> as   <a href="orders_2.html#NM2">Chain</a> of <font color="Maroon">A</font> <b>by </b><i><a class="ref" href="orders_2.html#D4" target="_self">Def4</a></i>;<br/>
<b>consider </b><font color="Maroon">a</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font><b> such that </b><br/><a NAME="E5:82_3"/><i><font color="Green">E41</font></i>: 
for <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font>  st <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> holds <br/><font color="Maroon">a</font> <a href="orders_2.html#R3" target="_self">&lt;=</a> <font color="Olive">b</font>
 <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">e</font> = <font color="Maroon">a</font> as   <a href="struct_0.html#NM1">Element</a> of <a href="orders_2.html#G1" target="_self">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font>,<font color="Maroon">R</font> #) ;<br/>
<b>take </b>
<font color="Maroon">e</font>
;<br/>

<b>let </b><font color="Maroon">f</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="orders_2.html#G1" target="_self">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font>,<font color="Maroon">R</font> #);<br/>

<b>reconsider </b><font color="Maroon">b</font> = <font color="Maroon">f</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font> ;<br/>
<b>assume </b><a NAME="E8:82_3"/>
<font color="Maroon">f</font> <a href="hidden.html#R2">in</a> <font color="Maroon">E</font>
 ;<br/>

<a NAME="E9:82_3"/><b>then </b>
<font color="Maroon">a</font> <a href="orders_2.html#R3" target="_self">&lt;=</a> <font color="Maroon">b</font>
 
<b>by </b><i/>;<br/>
<a NAME="E10:82_3"/><b>then </b>
( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">f</font>,<font color="Maroon">e</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> &amp; <font color="Maroon">R</font> <a href="hidden.html#R1">=</a> the <a href="orders_2.html#U1" target="_self">InternalRel</a> of <a href="orders_2.html#G1" target="_self">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font>,<font color="Maroon">R</font> #) )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E11:82_3"/>
<font color="Maroon">f</font> <a href="orders_2.html#R3" target="_self">&lt;=</a> <font color="Maroon">e</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon">e</font> being   <a href="struct_0.html#NM1">Element</a> of <a href="orders_2.html#G1" target="_self">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font>,<font color="Maroon">R</font> #)<b> such that </b><br/><a NAME="E7:82"/><i><font color="Green">E42</font></i>: 
for <font color="Olive">f</font> being  <a href="struct_0.html#NM1">Element</a> of <a href="orders_2.html#G1" target="_self">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font>,<font color="Maroon">R</font> #)  holds not <font color="Maroon">e</font> <a href="orders_2.html#R2" target="_self">&lt;</a> <font color="Olive">f</font>
 <b>by </b><i><a class="ref" href="orders_2.html#T38" target="_self">Th38</a></i>;<br/>
<b>reconsider </b><font color="Maroon">d</font> = <font color="Maroon">e</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font> ;<br/>
<b>take </b>
<font color="Maroon">d</font>
;<br/>

<b>let </b><font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font>;<br/>

<b>reconsider </b><font color="Maroon">f</font> = <font color="Maroon">b</font> as   <a href="struct_0.html#NM1">Element</a> of <a href="orders_2.html#G1" target="_self">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font>,<font color="Maroon">R</font> #) ;<br/>
<b>assume </b><a NAME="E10:82"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">b</font> <a href="orders_2.html#R2" target="_self">&lt;</a> <font color="Maroon">d</font>
 ;<br/>

<a NAME="E11:82"/><b>then </b>
<font color="Maroon">b</font> <a href="orders_2.html#R3" target="_self">&lt;=</a> <font color="Maroon">d</font>
 
<b>by </b><i/>;<br/>
<a NAME="E12:82"/><b>then </b>
( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">e</font>,<font color="Maroon">f</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> &amp; <font color="Maroon">R</font> <a href="hidden.html#R1">=</a> the <a href="orders_2.html#U1" target="_self">InternalRel</a> of <a href="orders_2.html#G1" target="_self">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font>,<font color="Maroon">R</font> #) )
 
<b>by </b><i/>;<br/>
<a NAME="E13:82"/><b>then </b>
( <font color="Maroon">e</font> <a href="orders_2.html#R3" target="_self">&lt;=</a> <font color="Maroon">f</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">d</font> )
 
<b>by </b><i><a class="ref" href="orders_2.html#T39" target="_self">Th39</a>, </i>;<br/>
<a NAME="E14:82"/><b>then </b>
<font color="Maroon">e</font> <a href="orders_2.html#R2" target="_self">&lt;</a> <font color="Maroon">f</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E15:82"/>
contradiction
 <b>by </b><i/>;<br/>


</div>
