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

<b>let </b><font color="Maroon">L</font> be  <a href="lattice3.html#V3">complete</a> <a href="waybel_0.html#NM8">LATTICE</a>;<br/><b>let </b><font color="Maroon">X</font> be  <a href="waybel_0.html#V13">upper</a> <a href="waybel_6.html#V1" target="_self">Open</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font>;<br/><b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/>





<b>assume </b><a NAME="E1:18"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="subset_1.html#K3">`</a> 
 ;<br/>

<b>set </b><font color="Maroon">A</font> = <span class="p1">{<span class="default"> <font color="Olive">C</font> where <font color="Olive">C</font> is   <a href="orders_2.html#NM2">Chain</a> of <font color="Maroon">L</font> : ( <font color="Olive">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> <a href="subset_1.html#K3">`</a>  &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> ) </span>}</span> ;<br/>
<b>reconsider </b><font color="Maroon">x1</font> = <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span> as   <a href="orders_2.html#NM2">Chain</a> of <font color="Maroon">L</font> <b>by </b><i><a class="ref" href="orders_2.html#T35">ORDERS_2:35</a></i>;<br/>
<a NAME="E3:18"/>
( <font color="Maroon">x1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> <a href="subset_1.html#K3">`</a>  &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x1</font> )
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E4:18"/><b>then </b><i><font color="Green">E47</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">C</font> where <font color="Olive">C</font> is   <a href="orders_2.html#NM2">Chain</a> of <font color="Maroon">L</font> : ( <font color="Olive">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> <a href="subset_1.html#K3">`</a>  &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> ) </span>}</span> 
 
;<br/>
<a NAME="E5:18"/>
for <font color="Olive">Z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">Z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Olive">Z</font> <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <font color="Olive">C</font> where <font color="Olive">C</font> is   <a href="orders_2.html#NM2">Chain</a> of <font color="Maroon">L</font> : ( <font color="Olive">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> <a href="subset_1.html#K3">`</a>  &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> ) </span>}</span>  &amp; <font color="Olive">Z</font> is <a href="ordinal1.html#V6">c=-linear</a> holds <br/> <a href="tarski.html#K3">union</a> <font color="Olive">Z</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">C</font> where <font color="Olive">C</font> is   <a href="orders_2.html#NM2">Chain</a> of <font color="Maroon">L</font> : ( <font color="Olive">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> <a href="subset_1.html#K3">`</a>  &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> ) </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1"><b>proof </b></a><div class="add">

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

<b>assume </b><b>that </b><br/><a NAME="E1:18_1"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">Z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">Z</font> <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <font color="Olive">C</font> where <font color="Olive">C</font> is   <a href="orders_2.html#NM2">Chain</a> of <font color="Maroon">L</font> : ( <font color="Olive">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> <a href="subset_1.html#K3">`</a>  &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> ) </span>}</span>  )
 <b>and </b><br/><a NAME="E2:18_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">Z</font> is <a href="ordinal1.html#V6">c=-linear</a>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">Y</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:18_1_1"/>
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 ;<br/><a NAME="E2:18_1_1"/><b>then </b>
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">C</font> where <font color="Olive">C</font> is   <a href="orders_2.html#NM2">Chain</a> of <font color="Maroon">L</font> : ( <font color="Olive">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> <a href="subset_1.html#K3">`</a>  &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> ) </span>}</span> 
 <b>by </b><i/>;<br/><a NAME="E3:18_1_1"/><b>then </b>
 ex <font color="Olive">C</font> being  <a href="orders_2.html#NM2">Chain</a> of <font color="Maroon">L</font> st <br/>( <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <font color="Olive">C</font> &amp; <font color="Olive">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> <a href="subset_1.html#K3">`</a>  &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> )
 ;<br/><b>hence </b><a NAME="E4:18_1_1"/>
<font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>
 ;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">UZ</font> =  <a href="tarski.html#K3">union</a> <font color="Maroon">Z</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> <b>by </b><i><a class="ref" href="zfmisc_1.html#T94">ZFMISC_1:94</a></i>;<br/>
<a NAME="E5:18_1"/>
the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">L</font> <a href="relat_2.html#R7">is_strongly_connected_in</a> <font color="Maroon">UZ</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">b</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>assume </b><a NAME="E1:18_1_2"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">UZ</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">UZ</font> )
 ;<br/>

<b>then consider </b><font color="Maroon">az</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:18_1_2"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">az</font> &amp; <font color="Maroon">az</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">bz</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:18_1_2"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">bz</font> &amp; <font color="Maroon">bz</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> )
 <b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<a NAME="E6:18_1_2"/>
<font color="Maroon">az</font>,<font color="Maroon">bz</font> <a href="xboole_0.html#R3">are_c=-comparable</a> 
 
<b>by </b><i>, , , <a class="ref" href="ordinal1.html#D9">ORDINAL1:def 9</a></i>;<br/>
<a NAME="E7:18_1_2"/><b>then </b><i><font color="Green">E55</font></i>: 
( <font color="Maroon">az</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">bz</font> or <font color="Maroon">bz</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">az</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D9">XBOOLE_0:def 9</a></i>;<br/>
<a NAME="E8:18_1_2"/>
<font color="Maroon">az</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">C</font> where <font color="Olive">C</font> is   <a href="orders_2.html#NM2">Chain</a> of <font color="Maroon">L</font> : ( <font color="Olive">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> <a href="subset_1.html#K3">`</a>  &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> ) </span>}</span> 
 
<b>by </b><i>, </i>;<br/>
<a NAME="E9:18_1_2"/><b>then </b>
 ex <font color="Olive">C</font> being  <a href="orders_2.html#NM2">Chain</a> of <font color="Maroon">L</font> st <br/>( <font color="Maroon">az</font> <a href="hidden.html#R1">=</a> <font color="Olive">C</font> &amp; <font color="Olive">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> <a href="subset_1.html#K3">`</a>  &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> )
 
;<br/>
<b>then reconsider </b><font color="Maroon">az</font> = <font color="Maroon">az</font> as   <a href="orders_2.html#NM2">Chain</a> of <font color="Maroon">L</font> ;<br/>
<a NAME="E11:18_1_2"/>
<font color="Maroon">bz</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">C</font> where <font color="Olive">C</font> is   <a href="orders_2.html#NM2">Chain</a> of <font color="Maroon">L</font> : ( <font color="Olive">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> <a href="subset_1.html#K3">`</a>  &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> ) </span>}</span> 
 
<b>by </b><i>, </i>;<br/>
<a NAME="E12:18_1_2"/><b>then </b>
 ex <font color="Olive">C</font> being  <a href="orders_2.html#NM2">Chain</a> of <font color="Maroon">L</font> st <br/>( <font color="Maroon">bz</font> <a href="hidden.html#R1">=</a> <font color="Olive">C</font> &amp; <font color="Olive">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> <a href="subset_1.html#K3">`</a>  &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> )
 
;<br/>
<b>then reconsider </b><font color="Maroon">bz</font> = <font color="Maroon">bz</font> as   <a href="orders_2.html#NM2">Chain</a> of <font color="Maroon">L</font> ;<br/>
<a NAME="E14:18_1_2"/>
( the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">L</font> <a href="relat_2.html#R7">is_strongly_connected_in</a> <font color="Maroon">az</font> &amp; the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">L</font> <a href="relat_2.html#R7">is_strongly_connected_in</a> <font color="Maroon">bz</font> )
 
<b>by </b><i><a class="ref" href="orders_2.html#D11">ORDERS_2:def 11</a></i>;<br/>
<b>hence </b><a NAME="E15:18_1_2"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">L</font> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">a</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">L</font> )
 <b>by </b><i>, , , <a class="ref" href="relat_2.html#D7">RELAT_2:def 7</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">UZ</font> = <font color="Maroon">UZ</font> as   <a href="orders_2.html#NM2">Chain</a> of <font color="Maroon">L</font> <b>by </b><i><a class="ref" href="orders_2.html#D11">ORDERS_2:def 11</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">Y</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:18_1_3"/>
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 ;<br/><a NAME="E2:18_1_3"/><b>then </b>
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">C</font> where <font color="Olive">C</font> is   <a href="orders_2.html#NM2">Chain</a> of <font color="Maroon">L</font> : ( <font color="Olive">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> <a href="subset_1.html#K3">`</a>  &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> ) </span>}</span> 
 <b>by </b><i/>;<br/><a NAME="E3:18_1_3"/><b>then </b>
 ex <font color="Olive">C</font> being  <a href="orders_2.html#NM2">Chain</a> of <font color="Maroon">L</font> st <br/>( <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <font color="Olive">C</font> &amp; <font color="Olive">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> <a href="subset_1.html#K3">`</a>  &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> )
 ;<br/><b>hence </b><a NAME="E4:18_1_3"/>
<font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> <a href="subset_1.html#K3">`</a> 
 ;<br/></div><b>end;</b></div>
<a NAME="E8:18_1"/><b>then </b><i><font color="Green">E79</font></i>: 
<font color="Maroon">UZ</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> <a href="subset_1.html#K3">`</a> 
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T94">ZFMISC_1:94</a></i>;<br/>
<b>consider </b><font color="Maroon">Y</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Z</font>;<br/>
<a NAME="E10:18_1"/>
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 
<b>by </b><i/>;<br/>
<a NAME="E11:18_1"/><b>then </b>
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">C</font> where <font color="Olive">C</font> is   <a href="orders_2.html#NM2">Chain</a> of <font color="Maroon">L</font> : ( <font color="Olive">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> <a href="subset_1.html#K3">`</a>  &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> ) </span>}</span> 
 
<b>by </b><i/>;<br/>
<a NAME="E12:18_1"/><b>then </b>
 ex <font color="Olive">C</font> being  <a href="orders_2.html#NM2">Chain</a> of <font color="Maroon">L</font> st <br/>( <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <font color="Olive">C</font> &amp; <font color="Olive">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> <a href="subset_1.html#K3">`</a>  &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> )
 
;<br/>
<a NAME="E13:18_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">UZ</font>
 
<b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>hence </b><a NAME="E14:18_1"/>
 <a href="tarski.html#K3">union</a> <font color="Maroon">Z</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">C</font> where <font color="Olive">C</font> is   <a href="orders_2.html#NM2">Chain</a> of <font color="Maroon">L</font> : ( <font color="Olive">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> <a href="subset_1.html#K3">`</a>  &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> ) </span>}</span> 
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon">Y1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:18"/><i><font color="Green">E80</font></i>: 
( <font color="Maroon">Y1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">C</font> where <font color="Olive">C</font> is   <a href="orders_2.html#NM2">Chain</a> of <font color="Maroon">L</font> : ( <font color="Olive">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> <a href="subset_1.html#K3">`</a>  &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> ) </span>}</span>  &amp; ( for <font color="Olive">Z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">Z</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">C</font> where <font color="Olive">C</font> is   <a href="orders_2.html#NM2">Chain</a> of <font color="Maroon">L</font> : ( <font color="Olive">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> <a href="subset_1.html#K3">`</a>  &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> ) </span>}</span>  &amp; <font color="Olive">Z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">Y1</font> holds <br/> not <font color="Maroon">Y1</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Z</font> ) )
 <b>by </b><i>, <a class="ref" href="orders_1.html#T177">ORDERS_1:177</a></i>;<br/>
<b>consider </b><font color="Maroon">Y</font> being   <a href="orders_2.html#NM2">Chain</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E9:18"/><i><font color="Green">E82</font></i>: 
( <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y1</font> &amp; <font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> <a href="subset_1.html#K3">`</a>  &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> )
 <b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">m</font> =  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">Y</font>;<br/>
<a NAME="E10:18"/><i><font color="Green">E84</font></i>: 
 <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">Y</font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a> <font color="Maroon">Y</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T32">YELLOW_0:32</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_2"/>
 <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/><b>then consider </b><font color="Maroon">y</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E3:18_2"/><i><font color="Green">E85</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">y</font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">Y</font> )
 <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">d</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E5:18_2"/><i><font color="Green">E86</font></i>: 
( <font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">y</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">d</font> )
 <b>by </b><i>, , <a class="ref" href="waybel_3.html#D1">WAYBEL_3:def 1</a></i>;<br/><a NAME="E6:18_2"/>
<font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i>, , <a class="ref" href="waybel_0.html#D20">WAYBEL_0:def 20</a></i>;<br/><b>hence </b><a NAME="E7:18_2"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/></div><b>end;</b></div>
<a NAME="E12:18"/><b>then </b><i><font color="Green">E88</font></i>: 
 <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="subset_1.html#K3">`</a> 
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E13:18"/>
 <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">Y</font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a> <font color="Maroon">Y</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T32">YELLOW_0:32</a></i>;<br/>
<a NAME="E14:18"/><b>then </b><i><font color="Green">E89</font></i>: 
<font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">Y</font>
 
<b>by </b><i>, <a class="ref" href="lattice3.html#D9">LATTICE3:def 9</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_3"><b>now </b></a><div class="add"><b>given </b><font color="Maroon">y</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><a NAME="E1:18_3"/><i><font color="Green">E91</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="subset_1.html#K3">`</a>  &amp; <font color="Maroon">y</font> <a href="orders_2.html#NR4" target="_self">&gt;</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">Y</font> )
 ;<br/><a NAME="E2:18_3"/><i><font color="Green">E100</font></i>: 
(  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">Y</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">y</font> &amp;  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">Y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y</font> )
 <b>by </b><i>, <a class="ref" href="orders_2.html#D10">ORDERS_2:def 10</a></i>;<br/><b>set </b><font color="Maroon">Y2</font> = <font color="Maroon">Y</font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y</font></span><a href="domain_1.html#K6">}</a></span>;<br/><a NAME="E3:18_3"/>
the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">L</font> <a href="relat_2.html#R7">is_strongly_connected_in</a> <font color="Maroon">Y</font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y</font></span><a href="domain_1.html#K6">}</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">b</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>assume </b><a NAME="E1:18_3_1"/><i><font color="Green">E101</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y</font></span><a href="domain_1.html#K6">}</a></span> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y</font></span><a href="domain_1.html#K6">}</a></span> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:18_3_1_1"/>
( ( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> ) or ( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y</font></span><a href="domain_1.html#K6">}</a></span> ) or ( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y</font></span><a href="domain_1.html#K6">}</a></span> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> ) or ( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y</font></span><a href="domain_1.html#K6">}</a></span> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y</font></span><a href="domain_1.html#K6">}</a></span> ) )
 </span><b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_3_1_1_1"><b>suppose </b></a><a NAME="E1:18_3_1_1_1"/><i><font color="Green">E102</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> )
 ;<br/><div class="add"><a NAME="E2:18_3_1_1_1"/>
the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">L</font> <a href="relat_2.html#R7">is_strongly_connected_in</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="orders_2.html#D11">ORDERS_2:def 11</a></i>;<br/><b>hence </b><a NAME="E3:18_3_1_1_1"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">L</font> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">a</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">L</font> )
 <b>by </b><i>, <a class="ref" href="relat_2.html#D7">RELAT_2:def 7</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_3_1_1_2"><b>suppose </b></a><a NAME="E1:18_3_1_1_2"/><i><font color="Green">E103</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y</font></span><a href="domain_1.html#K6">}</a></span> )
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">b1</font> = <font color="Maroon">b</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> ;<br/><b>reconsider </b><font color="Maroon">a1</font> = <font color="Maroon">a</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> <b>by </b><i/>;<br/><a NAME="E4:18_3_1_1_2"/><i><font color="Green">E105</font></i>: 
<font color="Maroon">b1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E5:18_3_1_1_2"/>
<font color="Maroon">a1</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">Y</font>
 <b>by </b><i>, , <a class="ref" href="lattice3.html#D9">LATTICE3:def 9</a></i>;<br/><a NAME="E6:18_3_1_1_2"/><b>then </b>
<font color="Maroon">a1</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">b1</font>
 <b>by </b><i>, , <a class="ref" href="orders_2.html#T26">ORDERS_2:26</a></i>;<br/><b>hence </b><a NAME="E7:18_3_1_1_2"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">L</font> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">a</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">L</font> )
 <b>by </b><i><a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_3_1_1_3"><b>suppose </b></a><a NAME="E1:18_3_1_1_3"/><i><font color="Green">E107</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y</font></span><a href="domain_1.html#K6">}</a></span> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> )
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">b1</font> = <font color="Maroon">a</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> ;<br/><b>reconsider </b><font color="Maroon">a1</font> = <font color="Maroon">b</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> <b>by </b><i/>;<br/><a NAME="E4:18_3_1_1_3"/><i><font color="Green">E108</font></i>: 
<font color="Maroon">b1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E5:18_3_1_1_3"/>
<font color="Maroon">a1</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">Y</font>
 <b>by </b><i>, , <a class="ref" href="lattice3.html#D9">LATTICE3:def 9</a></i>;<br/><a NAME="E6:18_3_1_1_3"/><b>then </b>
<font color="Maroon">a1</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">b1</font>
 <b>by </b><i>, , <a class="ref" href="orders_2.html#T26">ORDERS_2:26</a></i>;<br/><b>hence </b><a NAME="E7:18_3_1_1_3"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">L</font> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">a</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">L</font> )
 <b>by </b><i><a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_3_1_1_4"><b>suppose </b></a><a NAME="E1:18_3_1_1_4"/><i><font color="Green">E110</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y</font></span><a href="domain_1.html#K6">}</a></span> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y</font></span><a href="domain_1.html#K6">}</a></span> )
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">a1</font> = <font color="Maroon">a</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> ;<br/><a NAME="E3:18_3_1_1_4"/>
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">a1</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">a1</font> )
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E4:18_3_1_1_4"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">L</font> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">a</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">L</font> )
 <b>by </b><i><a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">Y2</font> = <font color="Maroon">Y</font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y</font></span><a href="domain_1.html#K6">}</a></span> as   <a href="orders_2.html#NM2">Chain</a> of <font color="Maroon">L</font> <b>by </b><i><a class="ref" href="orders_2.html#D11">ORDERS_2:def 11</a></i>;<br/><a NAME="E5:18_3"/>
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y</font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> <a href="subset_1.html#K3">`</a> 
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/><a NAME="E6:18_3"/><b>then </b>
( <font color="Maroon">Y2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> <a href="subset_1.html#K3">`</a>  &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y2</font> )
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a>, <a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/><a NAME="E7:18_3"/><b>then </b><i><font color="Green">E111</font></i>: 
<font color="Maroon">Y2</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">C</font> where <font color="Olive">C</font> is   <a href="orders_2.html#NM2">Chain</a> of <font color="Maroon">L</font> : ( <font color="Olive">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> <a href="subset_1.html#K3">`</a>  &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> ) </span>}</span> 
 ;<br/><div><i><font color="Green">E112</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="18_3_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_3_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 ;<br/><a NAME="E2:18_3_2"/><b>then </b>
<font color="Maroon">y</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">Y</font>
 <b>by </b><i>, <a class="ref" href="lattice3.html#D9">LATTICE3:def 9</a></i>;<br/><b>hence </b><a NAME="E3:18_3_2"/>
contradiction
 <b>by </b><i>, <a class="ref" href="orders_2.html#T30">ORDERS_2:30</a></i>;<br/></div><b>end;</b></div><a NAME="E9:18_3"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E10:18_3"/><b>then </b><i><font color="Green">E113</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y2</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E11:18_3"/>
<font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y2</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><b>hence </b><a NAME="E12:18_3"/>
contradiction
 <b>by </b><i>, , , , </i>;<br/></div><b>end;</b></div>
<a NAME="E16:18"/><b>then </b>
 <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">Y</font> <a href="waybel_4.html#R3">is_maximal_in</a> <font color="Maroon">X</font> <a href="subset_1.html#K3">`</a> 
 
<b>by </b><i>, <a class="ref" href="waybel_4.html#T56">WAYBEL_4:56</a></i>;<br/>
<b>hence </b><a NAME="E17:18"/>
 ex <font color="Olive">m</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <br/>( <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">m</font> &amp; <font color="Olive">m</font> <a href="waybel_4.html#R3">is_maximal_in</a> <font color="Maroon">X</font> <a href="subset_1.html#K3">`</a>  )
 <b>by </b><i/>;<br/>


</div>
