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

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

<b>assume </b><b>that </b><br/><a NAME="E1:89"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>and </b><br/><a NAME="E2:89"/><i><font color="Green">E19</font></i>: 
for <font color="Olive">Z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">Z</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> &amp; <font color="Olive">Z</font> is <a href="ordinal1.html#V6">c=-linear</a> holds <br/> ex <font color="Olive">Y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; ( for <font color="Olive">X1</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">X1</font> <a href="hidden.html#R2">in</a> <font color="Olive">Z</font> holds <br/><font color="Olive">X1</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Y</font> ) )
 ;<br/>

<b>reconsider </b><font color="Maroon">D</font> = <font color="Maroon">X</font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">R</font> =  <a href="wellord2.html#K1">RelIncl</a> <font color="Maroon">D</font>;<br/>
<a NAME="E4:89"/><i><font color="Green">E20</font></i>: 
 <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><a href="wellord2.html#K1">RelIncl</a> <font color="Maroon">D</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font>
 
<b>by </b><i><a class="ref" href="wellord2.html#D1">WELLORD2:def 1</a></i>;<br/>
<a NAME="E5:89"/><i><font color="Green">E24</font></i>: 
 <a href="wellord2.html#K1">RelIncl</a> <font color="Maroon">D</font> <a href="orders_1.html#R2" target="_self">partially_orders</a> <font color="Maroon">D</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="89_1"><b>proof </b></a><div class="add">

<a NAME="E1:89_1"/>
(  <a href="wellord2.html#K1">RelIncl</a> <font color="Maroon">D</font> is <a href="relat_2.html#V1">reflexive</a> &amp;  <a href="wellord2.html#K1">RelIncl</a> <font color="Maroon">D</font> is <a href="relat_2.html#V8">transitive</a> &amp;  <a href="wellord2.html#K1">RelIncl</a> <font color="Maroon">D</font> is <a href="relat_2.html#V4">antisymmetric</a> )
 
<b>by </b><i><a class="ref" href="wellord2.html#T2">WELLORD2:2</a>, <a class="ref" href="wellord2.html#T3">WELLORD2:3</a>, <a class="ref" href="wellord2.html#T5">WELLORD2:5</a></i>;<br/>
<b>hence </b><a NAME="E2:89_1"/>
(  <a href="wellord2.html#K1">RelIncl</a> <font color="Maroon">D</font> <a href="relat_2.html#R1">is_reflexive_in</a> <font color="Maroon">D</font> &amp;  <a href="wellord2.html#K1">RelIncl</a> <font color="Maroon">D</font> <a href="relat_2.html#R8">is_transitive_in</a> <font color="Maroon">D</font> &amp;  <a href="wellord2.html#K1">RelIncl</a> <font color="Maroon">D</font> <a href="relat_2.html#R4">is_antisymmetric_in</a> <font color="Maroon">D</font> )
 <b>by </b><i>, <a class="ref" href="relat_2.html#D9">RELAT_2:def 9</a>, <a class="ref" href="relat_2.html#D12">RELAT_2:def 12</a>, <a class="ref" href="relat_2.html#D16">RELAT_2:def 16</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="orders_1.html#D7" target="_self">ORDERS_1:def 7</a><br/>


</div><b>end;</b></div>
<a NAME="E6:89"/>
<font color="Maroon">D</font> <a href="orders_1.html#R4" target="_self">has_upper_Zorn_property_wrt</a>  <a href="wellord2.html#K1">RelIncl</a> <font color="Maroon">D</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="89_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">Z</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="orders_1.html#D9" target="_self">ORDERS_1:def 9</a><br/>

<b>assume </b><a NAME="E1:89_2"/><i><font color="Green">E25</font></i>: 
( <font color="Maroon">Z</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">D</font> &amp; <span class="p1">(<span class="default"><a href="wellord2.html#K1">RelIncl</a> <font color="Maroon">D</font></span>)</span> <a href="wellord1.html#K2">|_2</a> <font color="Maroon">Z</font> <a href="orders_1.html#NR3" target="_self">is_linear-order</a>  )
 ;<br/>

<a NAME="E2:89_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="wellord2.html#K1">RelIncl</a> <font color="Maroon">D</font></span>)</span> <a href="wellord1.html#K2">|_2</a> <font color="Maroon">Z</font> is <a href="relat_2.html#V6">connected</a>
 
<b>by </b><i/>;<br/>
<a NAME="E3:89_2"/><b>then </b><i><font color="Green">E27</font></i>: 
<span class="p1">(<span class="default"><a href="wellord2.html#K1">RelIncl</a> <font color="Maroon">D</font></span>)</span> <a href="wellord1.html#K2">|_2</a> <font color="Maroon">Z</font> <a href="relat_2.html#R6">is_connected_in</a>  <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="wellord2.html#K1">RelIncl</a> <font color="Maroon">D</font></span>)</span> <a href="wellord1.html#K2">|_2</a> <font color="Maroon">Z</font></span>)</span>
 
<b>by </b><i><a class="ref" href="relat_2.html#D14">RELAT_2:def 14</a></i>;<br/>
<a NAME="E4:89_2"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">Z</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="wellord2.html#K1">RelIncl</a> <font color="Maroon">D</font></span>)</span> <a href="wellord1.html#K2">|_2</a> <font color="Maroon">Z</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="89_2_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="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:89_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 ;<br/>

<a NAME="E2:89_2_1"/><b>then </b>
( <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> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">Z</font>,<font color="Maroon">Z</font></span><a href="zfmisc_1.html#K2">:]</a></span> &amp; <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>  <a href="wellord2.html#K1">RelIncl</a> <font color="Maroon">D</font> )
 
<b>by </b><i>, <a class="ref" href="wellord2.html#D1">WELLORD2:def 1</a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E3:89_2_1"/><b>then </b>
<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> <span class="p1">(<span class="default"><a href="wellord2.html#K1">RelIncl</a> <font color="Maroon">D</font></span>)</span> <a href="wellord1.html#K2">|_2</a> <font color="Maroon">Z</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E4:89_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="wellord2.html#K1">RelIncl</a> <font color="Maroon">D</font></span>)</span> <a href="wellord1.html#K2">|_2</a> <font color="Maroon">Z</font></span>)</span>
 <b>by </b><i><a class="ref" href="relat_1.html#T30">RELAT_1:30</a></i>;<br/>


</div><b>end;</b></div>
<b>set </b><font color="Maroon">Q</font> = <span class="p1">(<span class="default"><a href="wellord2.html#K1">RelIncl</a> <font color="Maroon">D</font></span>)</span> <a href="wellord1.html#K2">|_2</a> <font color="Maroon">Z</font>;<br/>
<a NAME="E5:89_2"/>
<font color="Maroon">Z</font> is <a href="ordinal1.html#V6">c=-linear</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="89_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">X1</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="ordinal1.html#D9">ORDINAL1:def 9</a><br/><b>let </b><font color="Maroon">X2</font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><a NAME="E1:89_2_2"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">X1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> &amp; <font color="Maroon">X2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> )
 ;<br/>

<a NAME="E2:89_2_2"/><b>then </b>
(  not <font color="Maroon">X1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">X2</font> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">X1</font>,<font color="Maroon">X2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="wellord2.html#K1">RelIncl</a> <font color="Maroon">D</font></span>)</span> <a href="wellord1.html#K2">|_2</a> <font color="Maroon">Z</font> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">X2</font>,<font color="Maroon">X1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="wellord2.html#K1">RelIncl</a> <font color="Maroon">D</font></span>)</span> <a href="wellord1.html#K2">|_2</a> <font color="Maroon">Z</font> )
 
<b>by </b><i>, , <a class="ref" href="relat_2.html#D6">RELAT_2:def 6</a></i>;<br/>
<a NAME="E3:89_2_2"/><b>then </b>
( (  not <font color="Maroon">X1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">X2</font> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">X1</font>,<font color="Maroon">X2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="wellord2.html#K1">RelIncl</a> <font color="Maroon">D</font> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">X2</font>,<font color="Maroon">X1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="wellord2.html#K1">RelIncl</a> <font color="Maroon">D</font> ) &amp; <font color="Maroon">X1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> &amp; <font color="Maroon">X2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> &amp; <font color="Maroon">X1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X1</font> )
 
<b>by </b><i>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E4:89_2_2"/>
( <font color="Maroon">X1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X2</font> or <font color="Maroon">X2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X1</font> )
 <b>by </b><i><a class="ref" href="wellord2.html#D1">WELLORD2:def 1</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D9">XBOOLE_0:def 9</a><br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon">Y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:89_2"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; ( for <font color="Olive">X1</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">X1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> holds <br/><font color="Olive">X1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> ) )
 <b>by </b><i>, </i>;<br/>
<b>take </b><font color="Maroon">x</font> = <font color="Maroon">Y</font>;<br/>

<b>thus </b><a NAME="E8:89_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 <b>by </b><i/>;<br/>

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

<b>assume </b><a NAME="E9:89_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 ;<br/>

<a NAME="E10:89_2"/><b>then </b>
( <font color="Maroon">y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> )
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E11:89_2"/>
<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="wellord2.html#K1">RelIncl</a> <font color="Maroon">D</font>
 <b>by </b><i>, <a class="ref" href="wellord2.html#D1">WELLORD2:def 1</a></i>;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E8:89"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">x</font> <a href="orders_1.html#R6" target="_self">is_maximal_in</a>  <a href="wellord2.html#K1">RelIncl</a> <font color="Maroon">D</font>
 <b>by </b><i>, , </i>;<br/>
<b>take </b><font color="Maroon">Y</font> = <font color="Maroon">x</font>;<br/>

<a NAME="E9:89"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><a href="wellord2.html#K1">RelIncl</a> <font color="Maroon">D</font></span>)</span> &amp; ( for <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>   holds <br/>(  not <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><a href="wellord2.html#K1">RelIncl</a> <font color="Maroon">D</font></span>)</span> or  not <font color="Olive">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">Y</font> or  not <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">Y</font>,<font color="Olive">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="wellord2.html#K1">RelIncl</a> <font color="Maroon">D</font> ) ) )
 
<b>by </b><i>, </i>;<br/>
<b>thus </b><a NAME="E10:89"/>
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i>, , </i>;<br/>

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

<b>assume </b><a NAME="E11:89"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">Z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">Z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">Y</font> )
 ;<br/>

<a NAME="E12:89"/><b>then </b>
not <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">Y</font>,<font color="Maroon">Z</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="wellord2.html#K1">RelIncl</a> <font color="Maroon">D</font>
 
<b>by </b><i>, , </i>;<br/>
<b>hence </b><a NAME="E13:89"/>
not <font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Z</font>
 <b>by </b><i>, , , <a class="ref" href="wellord2.html#D1">WELLORD2:def 1</a></i>;<br/>


</div>
