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

<b>let </b><font color="Maroon" title="c1">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:90"/><i><font color="Green" title="E52">A1</font></i>: 
<font color="Maroon" title="c1">X</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>and </b><br/><a NAME="E2:90"/><i><font color="Green" title="E53">A2</font></i>: 
for <font color="Olive" title="b1">Z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">Z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Olive" title="b1">Z</font> is <a href="ordinal1.html#V6" title="ORDINAL1:attr.6">c=-linear</a> holds <br/> ex <font color="Olive" title="b2">Y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp; ( for <font color="Olive" title="b3">X1</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b3">X1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Z</font> holds <br/><font color="Olive" title="b2">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b3">X1</font> ) )
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c2">D</font> = <font color="Maroon" title="c1">X</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  <b>by </b><i><a class="txt" href="orders_1.html#E1:90"><i><font color="Green" title="E52">A1</font></i></a></i>;<br/>
<b>set </b><font color="Maroon" title="c3">R</font> = <span class="p1">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c2">D</font></span>)</span> <a href="relat_1.html#K4" title="RELAT_1:func.4">~</a> ;<br/>
<i><font color="Green" title="E54">A3</font></i>:  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c2">D</font></span>)</span> <a href="relat_1.html#K4" title="RELAT_1:func.4">~</a> </span>)</span> = 
 <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c2">D</font></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T38" title="RELAT_1:th.38">RELAT_1:38</a></i>
<br/>.= 
<font color="Maroon" title="c2">D</font>
<b>by </b><i><a class="ref" href="wellord2.html#D1" title="WELLORD2:def.1">WELLORD2:def 1</a></i>
;<br/>
<a NAME="E5:90"/><i><font color="Green" title="E55">A4</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">D</font> holds <br/><span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b1">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c2">D</font></span>)</span> <a href="relat_1.html#K4" title="RELAT_1:func.4">~</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="90_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<a NAME="E1:90_2"/>
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">D</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c4">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c2">D</font> )
 
<b>by </b><i><a class="ref" href="wellord2.html#D1" title="WELLORD2:def.1">WELLORD2:def 1</a></i>;<br/>
<b>hence </b><a NAME="E2:90_2"/>
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">D</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c4">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c2">D</font></span>)</span> <a href="relat_1.html#K4" title="RELAT_1:func.4">~</a>  )
 <b>by </b><i><a class="ref" href="relat_1.html#D7" title="RELAT_1:def.7">RELAT_1:def 7</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:90"/><i><font color="Green" title="E56">A5</font></i>: 
<span class="p1">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c2">D</font></span>)</span> <a href="relat_1.html#K4" title="RELAT_1:func.4">~</a>  <a href="orders_1.html#R2" title="ORDERS_1:pred.2">partially_orders</a> <font color="Maroon" title="c2">D</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="90_3"><b>proof </b></a><div class="add">

<a NAME="E1:90_3"/>
(  <a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c2">D</font> is <a href="relat_2.html#V1" title="RELAT_2:attr.1">reflexive</a> &amp;  <a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c2">D</font> is <a href="relat_2.html#V8" title="RELAT_2:attr.8">transitive</a> &amp;  <a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c2">D</font> is <a href="relat_2.html#V4" title="RELAT_2:attr.4">antisymmetric</a> &amp;  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c2">D</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">D</font> )
 
<b>by </b><i><a class="ref" href="wellord2.html#T2" title="WELLORD2:th.2">WELLORD2:2</a>, <a class="ref" href="wellord2.html#T3" title="WELLORD2:th.3">WELLORD2:3</a>, <a class="ref" href="wellord2.html#T5" title="WELLORD2:th.5">WELLORD2:5</a>, <a class="ref" href="wellord2.html#D1" title="WELLORD2:def.1">WELLORD2:def 1</a></i>;<br/>
<a NAME="E2:90_3"/><b>then </b>
(  <a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c2">D</font> <a href="relat_2.html#R1" title="RELAT_2:pred.1">is_reflexive_in</a> <font color="Maroon" title="c2">D</font> &amp;  <a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c2">D</font> <a href="relat_2.html#R8" title="RELAT_2:pred.8">is_transitive_in</a> <font color="Maroon" title="c2">D</font> &amp;  <a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c2">D</font> <a href="relat_2.html#R4" title="RELAT_2:pred.4">is_antisymmetric_in</a> <font color="Maroon" title="c2">D</font> )
 
<b>by </b><i><a class="ref" href="relat_2.html#D9" title="RELAT_2:def.9">RELAT_2:def 9</a>, <a class="ref" href="relat_2.html#D12" title="RELAT_2:def.12">RELAT_2:def 12</a>, <a class="ref" href="relat_2.html#D16" title="RELAT_2:def.16">RELAT_2:def 16</a></i>;<br/>
<b>hence </b><a NAME="E3:90_3"/>
( <span class="p1">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c2">D</font></span>)</span> <a href="relat_1.html#K4" title="RELAT_1:func.4">~</a>  <a href="relat_2.html#R1" title="RELAT_2:pred.1">is_reflexive_in</a> <font color="Maroon" title="c2">D</font> &amp; <span class="p1">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c2">D</font></span>)</span> <a href="relat_1.html#K4" title="RELAT_1:func.4">~</a>  <a href="relat_2.html#R8" title="RELAT_2:pred.8">is_transitive_in</a> <font color="Maroon" title="c2">D</font> &amp; <span class="p1">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c2">D</font></span>)</span> <a href="relat_1.html#K4" title="RELAT_1:func.4">~</a>  <a href="relat_2.html#R4" title="RELAT_2:pred.4">is_antisymmetric_in</a> <font color="Maroon" title="c2">D</font> )
 <b>by </b><i><a class="txt" href="orders_1.html#E154"><i><font color="Green" title="E31">Lm11</font></i></a>, <a class="txt" href="orders_1.html#E155"><i><font color="Green" title="E32">Lm12</font></i></a>, <a class="txt" href="orders_1.html#E156"><i><font color="Green" title="E33">Lm13</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="orders_1.html#D7" target="_self" title="ORDERS_1:def.7">ORDERS_1:def 7</a><br/>


</div><b>end;</b></div>
<a NAME="E7:90"/>
<font color="Maroon" title="c2">D</font> <a href="orders_1.html#R4" title="ORDERS_1:pred.4">has_upper_Zorn_property_wrt</a> <span class="p1">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c2">D</font></span>)</span> <a href="relat_1.html#K4" title="RELAT_1:func.4">~</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="90_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">Z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="orders_1.html#D9" target="_self" title="ORDERS_1:def.9">ORDERS_1:def 9</a><br/>

<b>assume </b><a NAME="E1:90_4"/><i><font color="Green" title="E57">A6</font></i>: 
( <font color="Maroon" title="c4">Z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">D</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c2">D</font></span>)</span> <a href="relat_1.html#K4" title="RELAT_1:func.4">~</a> </span>)</span> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <font color="Maroon" title="c4">Z</font> <a href="orders_1.html#NR3" title="ORDERS_1:NR.3">is_linear-order</a>  )
 ;<br/>

<a NAME="E2:90_4"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c2">D</font></span>)</span> <a href="relat_1.html#K4" title="RELAT_1:func.4">~</a> </span>)</span> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <font color="Maroon" title="c4">Z</font> is <a href="relat_2.html#V6" title="RELAT_2:attr.6">connected</a>
 
<b>by </b><i><a class="ref" href="orders_1.html#D5" target="_self" title="ORDERS_1:def.5">Def5</a></i>;<br/>
<a NAME="E3:90_4"/><b>then </b><i><font color="Green" title="E58">A7</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c2">D</font></span>)</span> <a href="relat_1.html#K4" title="RELAT_1:func.4">~</a> </span>)</span> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <font color="Maroon" title="c4">Z</font> <a href="relat_2.html#R6" title="RELAT_2:pred.6">is_connected_in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c2">D</font></span>)</span> <a href="relat_1.html#K4" title="RELAT_1:func.4">~</a> </span>)</span> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <font color="Maroon" title="c4">Z</font></span>)</span>
 
<b>by </b><i><a class="ref" href="relat_2.html#D14" title="RELAT_2:def.14">RELAT_2:def 14</a></i>;<br/>
<a NAME="E4:90_4"/><i><font color="Green" title="E59">A8</font></i>: 
<font color="Maroon" title="c4">Z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c2">D</font></span>)</span> <a href="relat_1.html#K4" title="RELAT_1:func.4">~</a> </span>)</span> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <font color="Maroon" title="c4">Z</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="90_4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:90_4_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Z</font>
 ;<br/>

<a NAME="E2:90_4_1"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c5">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c4">Z</font>,<font color="Maroon" title="c4">Z</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c5">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c2">D</font></span>)</span> <a href="relat_1.html#K4" title="RELAT_1:func.4">~</a>  )
 
<b>by </b><i><a class="txt" href="orders_1.html#E5:90"><i><font color="Green" title="E55">A4</font></i></a>, <a class="txt" href="orders_1.html#E1:90_4"><i><font color="Green" title="E57">A6</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E3:90_4_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c5">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c2">D</font></span>)</span> <a href="relat_1.html#K4" title="RELAT_1:func.4">~</a> </span>)</span> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <font color="Maroon" title="c4">Z</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E4:90_4_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c2">D</font></span>)</span> <a href="relat_1.html#K4" title="RELAT_1:func.4">~</a> </span>)</span> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <font color="Maroon" title="c4">Z</font></span>)</span>
 <b>by </b><i><a class="ref" href="relat_1.html#T30" title="RELAT_1:th.30">RELAT_1:30</a></i>;<br/>


</div><b>end;</b></div>
<b>set </b><font color="Maroon" title="c5">Q</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c2">D</font></span>)</span> <a href="relat_1.html#K4" title="RELAT_1:func.4">~</a> </span>)</span> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <font color="Maroon" title="c4">Z</font>;<br/>
<a NAME="E5:90_4"/>
<font color="Maroon" title="c4">Z</font> is <a href="ordinal1.html#V6" title="ORDINAL1:attr.6">c=-linear</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="90_4_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">X1</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="ordinal1.html#D9" title="ORDINAL1:def.9">ORDINAL1:def 9</a><br/><b>let </b><font color="Maroon" title="c7">X2</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>



<b>assume </b><a NAME="E1:90_4_2"/><i><font color="Green" title="E60">A9</font></i>: 
( <font color="Maroon" title="c6">X1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Z</font> &amp; <font color="Maroon" title="c7">X2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Z</font> )
 ;<br/>

<a NAME="E2:90_4_2"/><b>then </b>
(  not <font color="Maroon" title="c6">X1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">X2</font> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">X1</font>,<font color="Maroon" title="c7">X2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c2">D</font></span>)</span> <a href="relat_1.html#K4" title="RELAT_1:func.4">~</a> </span>)</span> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <font color="Maroon" title="c4">Z</font> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">X2</font>,<font color="Maroon" title="c6">X1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c2">D</font></span>)</span> <a href="relat_1.html#K4" title="RELAT_1:func.4">~</a> </span>)</span> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <font color="Maroon" title="c4">Z</font> )
 
<b>by </b><i><a class="txt" href="orders_1.html#E3:90_4"><i><font color="Green" title="E58">A7</font></i></a>, <a class="txt" href="orders_1.html#E4:90_4"><i><font color="Green" title="E59">A8</font></i></a>, <a class="ref" href="relat_2.html#D6" title="RELAT_2:def.6">RELAT_2:def 6</a></i>;<br/>
<a NAME="E3:90_4_2"/><b>then </b>
( (  not <font color="Maroon" title="c6">X1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">X2</font> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">X1</font>,<font color="Maroon" title="c7">X2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c2">D</font></span>)</span> <a href="relat_1.html#K4" title="RELAT_1:func.4">~</a>  or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">X2</font>,<font color="Maroon" title="c6">X1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c2">D</font></span>)</span> <a href="relat_1.html#K4" title="RELAT_1:func.4">~</a>  ) &amp; <font color="Maroon" title="c6">X1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">D</font> &amp; <font color="Maroon" title="c7">X2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">D</font> )
 
<b>by </b><i><a class="txt" href="orders_1.html#E1:90_4"><i><font color="Green" title="E57">A6</font></i></a>, <a class="txt" href="orders_1.html#E1:90_4_2"><i><font color="Green" title="E60">A9</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E4:90_4_2"/><b>then </b>
( (  not <font color="Maroon" title="c6">X1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">X2</font> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">X1</font>,<font color="Maroon" title="c7">X2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c2">D</font> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">X2</font>,<font color="Maroon" title="c6">X1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c2">D</font> ) &amp; <font color="Maroon" title="c6">X1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">D</font> &amp; <font color="Maroon" title="c7">X2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">D</font> &amp; <font color="Maroon" title="c6">X1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">X1</font> )
 
<b>by </b><i><a class="ref" href="relat_1.html#D7" title="RELAT_1:def.7">RELAT_1:def 7</a></i>;<br/>
<b>hence </b><a NAME="E5:90_4_2"/>
( <font color="Maroon" title="c6">X1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">X2</font> or <font color="Maroon" title="c7">X2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">X1</font> )
 <b>by </b><i><a class="ref" href="wellord2.html#D1" title="WELLORD2:def.1">WELLORD2:def 1</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D9" title="XBOOLE_0:def.9">XBOOLE_0:def 9</a><br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon" title="c6">Y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E7:90_4"/><i><font color="Green" title="E60">A10</font></i>: 
( <font color="Maroon" title="c6">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp; ( for <font color="Olive" title="b1">X1</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">X1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Z</font> holds <br/><font color="Maroon" title="c6">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">X1</font> ) )
 <b>by </b><i><a class="txt" href="orders_1.html#E2:90"><i><font color="Green" title="E53">A2</font></i></a>, <a class="txt" href="orders_1.html#E1:90_4"><i><font color="Green" title="E57">A6</font></i></a></i>;<br/>
<b>take </b><font color="Maroon" title="c7">x</font> = <font color="Maroon" title="c6">Y</font>;<br/>

<b>thus </b><a NAME="E8:90_4"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">D</font>
 <b>by </b><i><a class="txt" href="orders_1.html#E7:90_4"><i><font color="Green" title="E60">A10</font></i></a></i>;<br/>

<b>let </b><font color="Maroon" title="c8">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E9:90_4"/>
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Z</font>
 ;<br/>

<a NAME="E10:90_4"/><b>then </b>
( <font color="Maroon" title="c6">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">y</font> &amp; <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">D</font> )
 
<b>by </b><i><a class="txt" href="orders_1.html#E1:90_4"><i><font color="Green" title="E57">A6</font></i></a>, <a class="txt" href="orders_1.html#E7:90_4"><i><font color="Green" title="E60">A10</font></i></a></i>;<br/>
<a NAME="E11:90_4"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">Y</font>,<font color="Maroon" title="c8">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c2">D</font>
 
<b>by </b><i><a class="txt" href="orders_1.html#E7:90_4"><i><font color="Green" title="E60">A10</font></i></a>, <a class="ref" href="wellord2.html#D1" title="WELLORD2:def.1">WELLORD2:def 1</a></i>;<br/>
<b>hence </b><a NAME="E12:90_4"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">y</font>,<font color="Maroon" title="c7">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c2">D</font></span>)</span> <a href="relat_1.html#K4" title="RELAT_1:func.4">~</a> 
 <b>by </b><i><a class="ref" href="relat_1.html#D7" title="RELAT_1:def.7">RELAT_1:def 7</a></i>;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon" title="c4">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E9:90"/><i><font color="Green" title="E57">A11</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="orders_1.html#R6" title="ORDERS_1:pred.6">is_maximal_in</a> <span class="p1">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c2">D</font></span>)</span> <a href="relat_1.html#K4" title="RELAT_1:func.4">~</a> 
 <b>by </b><i><a class="txt" href="orders_1.html#E4:90"><i><font color="Green" title="E54">A3</font></i></a>, <a class="txt" href="orders_1.html#E6:90"><i><font color="Green" title="E56">A5</font></i></a>, <a class="ref" href="orders_1.html#T173" target="_self" title="ORDERS_1:th.173">Th173</a></i>;<br/>
<b>take </b><font color="Maroon" title="c5">Y</font> = <font color="Maroon" title="c4">x</font>;<br/>

<a NAME="E10:90"/><i><font color="Green" title="E58">A12</font></i>: 
( <font color="Maroon" title="c5">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c2">D</font></span>)</span> <a href="relat_1.html#K4" title="RELAT_1:func.4">~</a> </span>)</span> &amp; ( for <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c2">D</font></span>)</span> <a href="relat_1.html#K4" title="RELAT_1:func.4">~</a> </span>)</span> or  not <font color="Olive" title="b1">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">Y</font> or  not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">Y</font>,<font color="Olive" title="b1">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c2">D</font></span>)</span> <a href="relat_1.html#K4" title="RELAT_1:func.4">~</a>  ) ) )
 
<b>by </b><i><a class="txt" href="orders_1.html#E9:90"><i><font color="Green" title="E57">A11</font></i></a>, <a class="ref" href="orders_1.html#D11" target="_self" title="ORDERS_1:def.11">Def11</a></i>;<br/>
<b>thus </b><a NAME="E11:90"/>
<font color="Maroon" title="c5">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="txt" href="orders_1.html#E4:90"><i><font color="Green" title="E54">A3</font></i></a>, <a class="txt" href="orders_1.html#E9:90"><i><font color="Green" title="E57">A11</font></i></a>, <a class="ref" href="orders_1.html#D11" target="_self" title="ORDERS_1:def.11">Def11</a></i>;<br/>

<b>let </b><font color="Maroon" title="c6">Z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E12:90"/><i><font color="Green" title="E59">A13</font></i>: 
( <font color="Maroon" title="c6">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c6">Z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">Y</font> )
 ;<br/>

<a NAME="E13:90"/><b>then </b>
not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">Y</font>,<font color="Maroon" title="c6">Z</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c2">D</font></span>)</span> <a href="relat_1.html#K4" title="RELAT_1:func.4">~</a> 
 
<b>by </b><i><a class="txt" href="orders_1.html#E4:90"><i><font color="Green" title="E54">A3</font></i></a>, <a class="txt" href="orders_1.html#E9:90"><i><font color="Green" title="E57">A11</font></i></a>, <a class="ref" href="orders_1.html#D11" target="_self" title="ORDERS_1:def.11">Def11</a></i>;<br/>
<a NAME="E14:90"/><b>then </b>
not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">Z</font>,<font color="Maroon" title="c5">Y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c2">D</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#D7" title="RELAT_1:def.7">RELAT_1:def 7</a></i>;<br/>
<b>hence </b><a NAME="E15:90"/>
not <font color="Maroon" title="c6">Z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">Y</font>
 <b>by </b><i><a class="txt" href="orders_1.html#E4:90"><i><font color="Green" title="E54">A3</font></i></a>, <a class="txt" href="orders_1.html#E10:90"><i><font color="Green" title="E58">A12</font></i></a>, <a class="txt" href="orders_1.html#E12:90"><i><font color="Green" title="E59">A13</font></i></a>, <a class="ref" href="wellord2.html#D1" title="WELLORD2:def.1">WELLORD2:def 1</a></i>;<br/>


</div>
