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

<a NAME="E1:35"/><i><font color="Green">E18</font></i>: 
 <a href="relat_1.html#K3">field</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="relat_1.html#T60">RELAT_1:60</a></i>;<br/>
<b>thus </b><a NAME="E2:35"/><i><font color="Green">E19</font></i>: 
 <a href="xboole_0.html#K1">{}</a>  is <a href="relat_2.html#V1">reflexive</a>
  <i><font color="Red">:: according to </font></i><a class="ref" href="orders_1.html#D3" target="_self">ORDERS_1:def 3</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_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#D1">RELAT_2:def 1</a>,<a class="ref" href="relat_2.html#D9">RELAT_2:def 9</a><br/>

<b>assume </b><a NAME="E1:35_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>hence </b><a NAME="E2:35_1"/>
<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="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="orders_1.html#E3:35"><i><font color="Green">E20</font></i></a></i>;<br/>


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

<b>thus </b><a NAME="E3:35"/><i><font color="Green">E20</font></i>: 
 <a href="xboole_0.html#K1">{}</a>  is <a href="relat_2.html#V8">transitive</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="relat_2.html#D8">RELAT_2:def 8</a>,<a class="ref" href="relat_2.html#D16">RELAT_2:def 16</a><br/><b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ;<br/>





<b>assume </b><a NAME="E1:35_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>thus </b><a NAME="E2:35_2"/>
(  not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <a href="xboole_0.html#K1">{}</a>  or  not <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <a href="xboole_0.html#K1">{}</a>  or  not <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>  <a href="xboole_0.html#K1">{}</a>  or  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="xboole_0.html#K1">{}</a>  or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">z</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="xboole_0.html#K1">{}</a>  )
 ;<br/>


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

<b>hence </b><a NAME="E4:35"/>
(  <a href="xboole_0.html#K1">{}</a>  is <a href="relat_2.html#V1">reflexive</a> &amp;  <a href="xboole_0.html#K1">{}</a>  is <a href="relat_2.html#V8">transitive</a> )
 <b>by </b><i/>; <i><font color="Red">:: according to </font></i><a class="ref" href="orders_1.html#D4" target="_self">ORDERS_1:def 4</a><br/>

<b>thus </b><a NAME="E5:35"/><i><font color="Green">E24</font></i>: 
 <a href="xboole_0.html#K1">{}</a>  is <a href="relat_2.html#V4">antisymmetric</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_3"><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#D4">RELAT_2:def 4</a>,<a class="ref" href="relat_2.html#D12">RELAT_2:def 12</a><br/><b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><a NAME="E1:35_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>thus </b><a NAME="E2:35_3"/>
(  not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <a href="xboole_0.html#K1">{}</a>  or  not <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>  <a href="xboole_0.html#K1">{}</a>  or  not <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="xboole_0.html#K1">{}</a>  or <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> )
 ;<br/>


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

<b>hence </b><a NAME="E6:35"/>
(  <a href="xboole_0.html#K1">{}</a>  is <a href="relat_2.html#V1">reflexive</a> &amp;  <a href="xboole_0.html#K1">{}</a>  is <a href="relat_2.html#V8">transitive</a> &amp;  <a href="xboole_0.html#K1">{}</a>  is <a href="relat_2.html#V4">antisymmetric</a> )
 <b>by </b><i>, <a class="ref" href="orders_1.html#D1" target="_self">Def1</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="orders_1.html#D5" target="_self">ORDERS_1:def 5</a><br/>

<b>thus </b><a NAME="E7:35"/>
 <a href="xboole_0.html#K1">{}</a>  is <a href="relat_2.html#V6">connected</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_4"><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#D6">RELAT_2:def 6</a>,<a class="ref" href="relat_2.html#D14">RELAT_2:def 14</a><br/><b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><a NAME="E1:35_4"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>thus </b><a NAME="E2:35_4"/>
(  not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> or <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>  <a href="xboole_0.html#K1">{}</a>  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>  <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i><a class="txt" href="orders_1.html#E3:35"><i><font color="Green">E20</font></i></a></i>;<br/>


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

<b>hence </b><a NAME="E8:35"/>
(  <a href="xboole_0.html#K1">{}</a>  is <a href="relat_2.html#V1">reflexive</a> &amp;  <a href="xboole_0.html#K1">{}</a>  is <a href="relat_2.html#V8">transitive</a> &amp;  <a href="xboole_0.html#K1">{}</a>  is <a href="relat_2.html#V4">antisymmetric</a> &amp;  <a href="xboole_0.html#K1">{}</a>  is <a href="relat_2.html#V6">connected</a> )
 <b>by </b><i>, <a class="ref" href="orders_1.html#D1" target="_self">Def1</a>, </i>; <i><font color="Red">:: according to </font></i><a class="ref" href="wellord1.html#D4">WELLORD1:def 4</a><br/>

<b>let </b><font color="Maroon">Y</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="wellord1.html#D2">WELLORD1:def 2</a><br/>

<b>assume </b><a NAME="E9:35"/>
( <font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K3">field</a> <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">Y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 ;<br/>

<b>hence </b><a NAME="E10:35"/>
 ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <a href="xboole_0.html#K1">{}</a>  <a href="wellord1.html#K1">-Seg</a> <font color="Olive">b<sub>1</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">Y</font> )
 <b>by </b><i><a class="txt" href="orders_1.html#E3:35"><i><font color="Green">E20</font></i></a>, <a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/>


</div>
