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

<b>let </b><font color="Maroon">R</font> be   <a href="relat_1.html#NM1">Relation</a>;<br/><b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><a NAME="E1:95"/><i><font color="Green">E18</font></i>: 
( <font color="Maroon">R</font> <a href="orders_1.html#R2" target="_self">partially_orders</a> <font color="Maroon">X</font> &amp;  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> )
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">P</font> being  <a href="relat_1.html#NM1">Relation</a> st <br/>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">P</font> &amp; <font color="Maroon">R</font> <a href="tarski.html#R1">c=</a> <font color="Olive">P</font> &amp; <font color="Olive">P</font> <a href="orders_1.html#R2" target="_self">partially_orders</a> <font color="Maroon">X</font> &amp;  <a href="relat_1.html#K3">field</a> <font color="Olive">P</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> );<br/>
<b>consider </b><font color="Maroon">Rel</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:95"/><i><font color="Green">E19</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Rel</font> iff ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>] ) )
 <b>from </b><i><a class="ref" href="xboole_0.html#S1">XBOOLE_0:sch 1</a>();<br/></i>
<a NAME="E4:95"/>
<font color="Maroon">R</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="txt" href="orders_1.html#E10:95"><i><font color="Green">E54</font></i></a>, <a class="txt" href="orders_1.html#E1"><i><font color="Green">Lemma17</font></i></a></i>;<br/>
<a NAME="E5:95"/><b>then </b><i><font color="Green">E20</font></i>: 
<font color="Maroon">Rel</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="txt" href="orders_1.html#E10:95"><i><font color="Green">E54</font></i></a>, <a class="txt" href="orders_1.html#E12:95"><i><font color="Green">E55</font></i></a></i>;<br/>
<a NAME="E6:95"/>
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> <font color="Maroon">Rel</font> &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> <font color="Maroon">Rel</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_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="E1:95_1"/><i><font color="Green">E24</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> <font color="Maroon">Rel</font> )
 <b>and </b><br/><a NAME="E2:95_1"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">Z</font> is <a href="ordinal1.html#V6">c=-linear</a>
 ;<br/>

<b>reconsider </b><font color="Maroon">T</font> = <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span> as   <a href="relat_1.html#NM1">Relation</a> <b>by </b><i><a class="ref" href="relat_1.html#T6">RELAT_1:6</a></i>;<br/>
<a NAME="E4:95_1"/>
<font color="Maroon">Z</font> <a href="tarski.html#R1">c=</a>  <a href="zfmisc_1.html#K1">bool</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1_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="E1:95_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 ;<br/>

<b>hence </b><a NAME="E2:95_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="txt" href="orders_1.html#E12:95"><i><font color="Green">E55</font></i></a>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:95_1"/><b>then </b><i><font color="Green">E27</font></i>: 
(  <a href="tarski.html#K3">union</a> <font color="Maroon">Z</font> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> &amp;  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">T</font> )
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T95">ZFMISC_1:95</a>, <a class="ref" href="zfmisc_1.html#T99">ZFMISC_1:99</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">S</font> =  <a href="tarski.html#K3">union</a> <font color="Maroon">Z</font> as   <a href="relat_1.html#NM1">Relation</a> <b>by </b><i><a class="ref" href="relat_1.html#T3">RELAT_1:3</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="E8:95_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Rel</font>
 
<b>by </b><i>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">P</font> being   <a href="relat_1.html#NM1">Relation</a><b> such that </b><br/><a NAME="E10:95_1"/><i><font color="Green">E28</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> &amp; <font color="Maroon">R</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font> &amp; <font color="Maroon">P</font> <a href="orders_1.html#R2" target="_self">partially_orders</a> <font color="Maroon">X</font> &amp;  <a href="relat_1.html#K3">field</a> <font color="Maroon">P</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> )
 <b>by </b><i><a class="txt" href="orders_1.html#E12:95"><i><font color="Green">E55</font></i></a></i>;<br/>
<a NAME="E11:95_1"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">P</font> <a href="relat_2.html#R1">is_reflexive_in</a> <font color="Maroon">X</font>
 
<b>by </b><i>, <a class="ref" href="orders_1.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E12:95_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">R</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">S</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1_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_1.html#D3">RELAT_1:def 3</a><br/><b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><a NAME="E1:95_1_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 ;<br/>

<b>hence </b><a NAME="E2:95_1_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 <b>by </b><i>, , <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E13:95_1"/><i><font color="Green">E31</font></i>: 
 <a href="relat_1.html#K3">field</a> <font color="Maroon">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1_3"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:95_1_3"/>
 <a href="relat_1.html#K3">field</a> <font color="Maroon">S</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1_3_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:95_1_3_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">S</font>
 ;<br/>

<a NAME="E2:95_1_3_1"/><b>then </b><i><font color="Green">E32</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">S</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">S</font></span>)</span>
 
;<br/>
<div><i><font color="Green">E33</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1_3_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:95_1_3_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">S</font>
 ;<br/><b>then consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:95_1_3_1_1"/><i><font color="Green">E34</font></i>: 
<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> <font color="Maroon">S</font>
 <b>by </b><i><a class="ref" href="relat_1.html#D4">RELAT_1:def 4</a></i>;<br/><b>consider </b><font color="Maroon">Y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:95_1_3_1_1"/><i><font color="Green">E35</font></i>: 
( <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> <font color="Maroon">Y</font> &amp; <font color="Maroon">Y</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="E6:95_1_3_1_1"/>
 ex <font color="Olive">P</font> being  <a href="relat_1.html#NM1">Relation</a> st <br/>( <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <font color="Olive">P</font> &amp; <font color="Maroon">R</font> <a href="tarski.html#R1">c=</a> <font color="Olive">P</font> &amp; <font color="Olive">P</font> <a href="orders_1.html#R2" target="_self">partially_orders</a> <font color="Maroon">X</font> &amp;  <a href="relat_1.html#K3">field</a> <font color="Olive">P</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> )
 <b>by </b><i><a class="txt" href="orders_1.html#E12:95"><i><font color="Green">E55</font></i></a>, , </i>;<br/><b>hence </b><a NAME="E7:95_1_3_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i>, <a class="ref" href="relat_1.html#T30">RELAT_1:30</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1_3_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:95_1_3_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">S</font>
 ;<br/><b>then consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:95_1_3_1_2"/><i><font color="Green">E36</font></i>: 
<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> <font color="Maroon">S</font>
 <b>by </b><i><a class="ref" href="relat_1.html#D5">RELAT_1:def 5</a></i>;<br/><b>consider </b><font color="Maroon">Y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:95_1_3_1_2"/><i><font color="Green">E37</font></i>: 
( <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> <font color="Maroon">Y</font> &amp; <font color="Maroon">Y</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="E6:95_1_3_1_2"/>
 ex <font color="Olive">P</font> being  <a href="relat_1.html#NM1">Relation</a> st <br/>( <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <font color="Olive">P</font> &amp; <font color="Maroon">R</font> <a href="tarski.html#R1">c=</a> <font color="Olive">P</font> &amp; <font color="Olive">P</font> <a href="orders_1.html#R2" target="_self">partially_orders</a> <font color="Maroon">X</font> &amp;  <a href="relat_1.html#K3">field</a> <font color="Olive">P</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> )
 <b>by </b><i><a class="txt" href="orders_1.html#E12:95"><i><font color="Green">E55</font></i></a>, , </i>;<br/><b>hence </b><a NAME="E7:95_1_3_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i>, <a class="ref" href="relat_1.html#T30">RELAT_1:30</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:95_1_3_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


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

<b>thus </b><a NAME="E2:95_1_3"/>
<font color="Maroon">X</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">S</font>
 <b>by </b><i><a class="txt" href="orders_1.html#E10:95"><i><font color="Green">E54</font></i></a>, , <a class="ref" href="relat_1.html#T31">RELAT_1:31</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E14:95_1"/>
<font color="Maroon">S</font> <a href="orders_1.html#R2" target="_self">partially_orders</a> <font color="Maroon">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1_4"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:95_1_4"/>
<font color="Maroon">S</font> <a href="relat_2.html#R1">is_reflexive_in</a> <font color="Maroon">X</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="orders_1.html#D7" target="_self">ORDERS_1:def 7</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1_4_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><br/>

<b>assume </b><a NAME="E1:95_1_4_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<a NAME="E2:95_1_4_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> <font color="Maroon">P</font>
 
<b>by </b><i>, <a class="ref" href="relat_2.html#D1">RELAT_2:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:95_1_4_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> <font color="Maroon">S</font>
 <b>by </b><i>, , <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>


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

<b>thus </b><a NAME="E2:95_1_4"/>
<font color="Maroon">S</font> <a href="relat_2.html#R8">is_transitive_in</a> <font color="Maroon">X</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1_4_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><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="E1:95_1_4_2"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <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> <font color="Maroon">S</font> &amp; <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> <font color="Maroon">S</font> )
 ;<br/>

<b>then consider </b><font color="Maroon">X1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:95_1_4_2"/><i><font color="Green">E41</font></i>: 
( <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> <font color="Maroon">X1</font> &amp; <font color="Maroon">X1</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">X2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:95_1_4_2"/><i><font color="Green">E42</font></i>: 
( <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> <font color="Maroon">X2</font> &amp; <font color="Maroon">X2</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="E6:95_1_4_2"/>
<font color="Maroon">X1</font>,<font color="Maroon">X2</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="E7:95_1_4_2"/><b>then </b><i><font color="Green">E43</font></i>: 
( <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="xboole_0.html#D9">XBOOLE_0:def 9</a></i>;<br/>
<b>consider </b><font color="Maroon">P1</font> being   <a href="relat_1.html#NM1">Relation</a><b> such that </b><br/><a NAME="E9:95_1_4_2"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">X1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P1</font> &amp; <font color="Maroon">R</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P1</font> &amp; <font color="Maroon">P1</font> <a href="orders_1.html#R2" target="_self">partially_orders</a> <font color="Maroon">X</font> &amp;  <a href="relat_1.html#K3">field</a> <font color="Maroon">P1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> )
 <b>by </b><i><a class="txt" href="orders_1.html#E12:95"><i><font color="Green">E55</font></i></a>, , </i>;<br/>
<b>consider </b><font color="Maroon">P2</font> being   <a href="relat_1.html#NM1">Relation</a><b> such that </b><br/><a NAME="E11:95_1_4_2"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">X2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P2</font> &amp; <font color="Maroon">R</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P2</font> &amp; <font color="Maroon">P2</font> <a href="orders_1.html#R2" target="_self">partially_orders</a> <font color="Maroon">X</font> &amp;  <a href="relat_1.html#K3">field</a> <font color="Maroon">P2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> )
 <b>by </b><i><a class="txt" href="orders_1.html#E12:95"><i><font color="Green">E55</font></i></a>, , </i>;<br/>
<a NAME="E12:95_1_4_2"/>
( <font color="Maroon">P1</font> <a href="relat_2.html#R8">is_transitive_in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">P2</font> <a href="relat_2.html#R8">is_transitive_in</a> <font color="Maroon">X</font> )
 
<b>by </b><i>, , <a class="ref" href="orders_1.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E13:95_1_4_2"/><b>then </b>
( <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> <font color="Maroon">P1</font> 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> <font color="Maroon">P2</font> )
 
<b>by </b><i>, , , , , , <a class="ref" href="relat_2.html#D8">RELAT_2:def 8</a></i>;<br/>
<b>hence </b><a NAME="E14:95_1_4_2"/>
<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> <font color="Maroon">S</font>
 <b>by </b><i>, , , , <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>


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

<b>thus </b><a NAME="E3:95_1_4"/>
<font color="Maroon">S</font> <a href="relat_2.html#R4">is_antisymmetric_in</a> <font color="Maroon">X</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1_4_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><br/><b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><a NAME="E1:95_1_4_3"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <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> <font color="Maroon">S</font> &amp; <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> <font color="Maroon">S</font> )
 ;<br/>

<b>then consider </b><font color="Maroon">X1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:95_1_4_3"/><i><font color="Green">E47</font></i>: 
( <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> <font color="Maroon">X1</font> &amp; <font color="Maroon">X1</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">X2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:95_1_4_3"/><i><font color="Green">E48</font></i>: 
( <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> <font color="Maroon">X2</font> &amp; <font color="Maroon">X2</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="E6:95_1_4_3"/>
<font color="Maroon">X1</font>,<font color="Maroon">X2</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="E7:95_1_4_3"/><b>then </b><i><font color="Green">E50</font></i>: 
( <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="xboole_0.html#D9">XBOOLE_0:def 9</a></i>;<br/>
<b>consider </b><font color="Maroon">P1</font> being   <a href="relat_1.html#NM1">Relation</a><b> such that </b><br/><a NAME="E9:95_1_4_3"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">X1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P1</font> &amp; <font color="Maroon">R</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P1</font> &amp; <font color="Maroon">P1</font> <a href="orders_1.html#R2" target="_self">partially_orders</a> <font color="Maroon">X</font> &amp;  <a href="relat_1.html#K3">field</a> <font color="Maroon">P1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> )
 <b>by </b><i><a class="txt" href="orders_1.html#E12:95"><i><font color="Green">E55</font></i></a>, , </i>;<br/>
<b>consider </b><font color="Maroon">P2</font> being   <a href="relat_1.html#NM1">Relation</a><b> such that </b><br/><a NAME="E11:95_1_4_3"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">X2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P2</font> &amp; <font color="Maroon">R</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P2</font> &amp; <font color="Maroon">P2</font> <a href="orders_1.html#R2" target="_self">partially_orders</a> <font color="Maroon">X</font> &amp;  <a href="relat_1.html#K3">field</a> <font color="Maroon">P2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> )
 <b>by </b><i><a class="txt" href="orders_1.html#E12:95"><i><font color="Green">E55</font></i></a>, , </i>;<br/>
<a NAME="E12:95_1_4_3"/>
( <font color="Maroon">P1</font> <a href="relat_2.html#R4">is_antisymmetric_in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">P2</font> <a href="relat_2.html#R4">is_antisymmetric_in</a> <font color="Maroon">X</font> )
 
<b>by </b><i>, , <a class="ref" href="orders_1.html#D1" target="_self">Def1</a></i>;<br/>
<b>hence </b><a NAME="E13:95_1_4_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 <b>by </b><i>, , , , , , <a class="ref" href="relat_2.html#D4">RELAT_2:def 4</a></i>;<br/>


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


</div><b>end;</b></div>
<b>hence </b><a NAME="E15:95_1"/>
 <a href="tarski.html#K3">union</a> <font color="Maroon">Z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Rel</font>
 <b>by </b><i><a class="txt" href="orders_1.html#E12:95"><i><font color="Green">E55</font></i></a>, , , </i>;<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="E8:95"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Rel</font> &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> <font color="Maroon">Rel</font> &amp; <font color="Olive">Z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">Y</font> holds <br/> not <font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Z</font> ) )
 <b>by </b><i>, <a class="txt" href="orders_1.html#E8:95"><i><font color="Green">E53</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">P</font> being   <a href="relat_1.html#NM1">Relation</a><b> such that </b><br/><a NAME="E10:95"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> &amp; <font color="Maroon">R</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font> &amp; <font color="Maroon">P</font> <a href="orders_1.html#R2" target="_self">partially_orders</a> <font color="Maroon">X</font> &amp;  <a href="relat_1.html#K3">field</a> <font color="Maroon">P</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> )
 <b>by </b><i><a class="txt" href="orders_1.html#E12:95"><i><font color="Green">E55</font></i></a>, </i>;<br/>
<b>take </b>
<font color="Maroon">P</font>
;<br/>

<b>thus </b><a NAME="E11:95"/>
<font color="Maroon">R</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font>
 <b>by </b><i/>;<br/>

<b>thus </b><a NAME="E12:95"/><i><font color="Green">E55</font></i>: 
( <font color="Maroon">P</font> <a href="relat_2.html#R1">is_reflexive_in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">P</font> <a href="relat_2.html#R8">is_transitive_in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">P</font> <a href="relat_2.html#R4">is_antisymmetric_in</a> <font color="Maroon">X</font> )
 <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#D8" target="_self">ORDERS_1:def 8</a><br/>

<b>thus </b><a NAME="E13:95"/>
<font color="Maroon">P</font> <a href="relat_2.html#R6">is_connected_in</a> <font color="Maroon">X</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_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#D6">RELAT_2:def 6</a><br/><b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><a NAME="E1:95_2"/><i><font color="Green">E56</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y</font> &amp; 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> <font color="Maroon">P</font> &amp; 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> <font color="Maroon">P</font> )
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">a<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> or ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">a<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> ) );<br/>
<b>consider </b><font color="Maroon">Q</font> being   <a href="relat_1.html#NM1">Relation</a><b> such that </b><br/><a NAME="E3:95_2"/><i><font color="Green">E57</font></i>: 
for <font color="Olive">z</font>, <font color="Olive">u</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">z</font>,<font color="Olive">u</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> iff ( <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">z</font>,<font color="Olive">u</font>] ) )
 <b>from </b><i><a class="ref" href="relat_1.html#S1">RELAT_1:sch 1</a>();<br/></i>
<a NAME="E4:95_2"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">P</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Q</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_2_1"><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="relat_1.html#D3">RELAT_1:def 3</a><br/><b>let </b><font color="Maroon">u</font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><a NAME="E1:95_2_1"/><i><font color="Green">E59</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">z</font>,<font color="Maroon">u</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 ;<br/>

<a NAME="E2:95_2_1"/><b>then </b>
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">P</font> &amp; <font color="Maroon">u</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">P</font> )
 
<b>by </b><i><a class="ref" href="relat_1.html#T30">RELAT_1:30</a></i>;<br/>
<b>hence </b><a NAME="E3:95_2_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">z</font>,<font color="Maroon">u</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font>
 <b>by </b><i><a class="txt" href="orders_1.html#E4:95_2"><i><font color="Green">E58</font></i></a>, <a class="txt" href="orders_1.html#E6:95_2"><i><font color="Green">E61</font></i></a>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:95_2"/><b>then </b><i><font color="Green">E60</font></i>: 
<font color="Maroon">R</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Q</font>
 
<b>by </b><i><a class="txt" href="orders_1.html#E4:95_2"><i><font color="Green">E58</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E6:95_2"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">Q</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_2_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="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:95_2_2"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font>
 ;<br/>

<b>then consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:95_2_2"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <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>
 <b>by </b><i><a class="ref" href="relat_1.html#D1">RELAT_1:def 1</a></i>;<br/>
<a NAME="E4:95_2_2"/>
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 
<b>by </b><i><a class="txt" href="orders_1.html#E6:95_2"><i><font color="Green">E61</font></i></a>, , </i>;<br/>
<b>hence </b><a NAME="E5:95_2_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:95_2"/><i><font color="Green">E64</font></i>: 
 <a href="relat_1.html#K3">field</a> <font color="Maroon">Q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_2_3"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:95_2_3"/>
 <a href="relat_1.html#K3">field</a> <font color="Maroon">Q</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_2_3_1"><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="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:95_2_3_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">Q</font>
 ;<br/>

<a NAME="E2:95_2_3_1"/><b>then </b><i><font color="Green">E65</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">Q</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">Q</font></span>)</span>
 
;<br/>
<div><i><font color="Green">E67</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="95_2_3_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:95_2_3_1_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">Q</font>
 ;<br/><a NAME="E2:95_2_3_1_1"/><b>then </b>
 ex <font color="Olive">u</font> being   <a href="hidden.html#M1">set</a>  st <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">z</font>,<font color="Olive">u</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font>
 <b>by </b><i><a class="ref" href="relat_1.html#D4">RELAT_1:def 4</a></i>;<br/><b>hence </b><a NAME="E3:95_2_3_1_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="txt" href="orders_1.html#E6:95_2"><i><font color="Green">E61</font></i></a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_2_3_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:95_2_3_1_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">Q</font>
 ;<br/><a NAME="E2:95_2_3_1_2"/><b>then </b>
 ex <font color="Olive">u</font> being   <a href="hidden.html#M1">set</a>  st <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">u</font>,<font color="Maroon">z</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font>
 <b>by </b><i><a class="ref" href="relat_1.html#D5">RELAT_1:def 5</a></i>;<br/><b>hence </b><a NAME="E3:95_2_3_1_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="txt" href="orders_1.html#E6:95_2"><i><font color="Green">E61</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:95_2_3_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i>, <a class="txt" href="orders_1.html#E3:95_2_3_1"><i><font color="Green">E67</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


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

<b>thus </b><a NAME="E2:95_2_3"/>
<font color="Maroon">X</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">Q</font>
 <b>by </b><i><a class="txt" href="orders_1.html#E4:95_2"><i><font color="Green">E58</font></i></a>, , <a class="ref" href="relat_1.html#T31">RELAT_1:31</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:95_2"/>
<font color="Maroon">Q</font> <a href="orders_1.html#R2" target="_self">partially_orders</a> <font color="Maroon">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_2_4"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:95_2_4"/>
<font color="Maroon">Q</font> <a href="relat_2.html#R1">is_reflexive_in</a> <font color="Maroon">X</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="orders_1.html#D7" target="_self">ORDERS_1:def 7</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_2_4_1"><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="relat_2.html#D1">RELAT_2:def 1</a><br/>

<b>assume </b><a NAME="E1:95_2_4_1"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<a NAME="E2:95_2_4_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">z</font>,<font color="Maroon">z</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 
<b>by </b><i>, <a class="ref" href="relat_2.html#D1">RELAT_2:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:95_2_4_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">z</font>,<font color="Maroon">z</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font>
 <b>by </b><i><a class="txt" href="orders_1.html#E6:95_2"><i><font color="Green">E61</font></i></a>, </i>;<br/>


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

<b>thus </b><a NAME="E2:95_2_4"/>
<font color="Maroon">Q</font> <a href="relat_2.html#R8">is_transitive_in</a> <font color="Maroon">X</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_2_4_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> , <font color="Maroon">c</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><br/>

<b>assume </b><a NAME="E1:95_2_4_2"/><i><font color="Green">E69</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <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> <font color="Maroon">Q</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">c</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> )
 ;<br/>

<a NAME="E2:95_2_4_2"/><b>then </b>
( ( <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> <font color="Maroon">P</font> or ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> ) ) &amp; ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">c</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> or ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">c</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> ) ) )
 
<b>by </b><i><a class="txt" href="orders_1.html#E6:95_2"><i><font color="Green">E61</font></i></a></i>;<br/>
<a NAME="E3:95_2_4_2"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> or ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">c</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> ) or ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">c</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> ) or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> )
 
<b>by </b><i>, <a class="txt" href="orders_1.html#E5:95_2"><i><font color="Green">E60</font></i></a>, , <a class="ref" href="relat_2.html#D8">RELAT_2:def 8</a></i>;<br/>
<b>hence </b><a NAME="E4:95_2_4_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font>
 <b>by </b><i><a class="txt" href="orders_1.html#E5:95_2"><i><font color="Green">E60</font></i></a>, <a class="txt" href="orders_1.html#E6:95_2"><i><font color="Green">E61</font></i></a>, </i>;<br/>


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

<b>thus </b><a NAME="E3:95_2_4"/>
<font color="Maroon">Q</font> <a href="relat_2.html#R4">is_antisymmetric_in</a> <font color="Maroon">X</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_2_4_3"><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#D4">RELAT_2:def 4</a><br/>

<b>assume </b><a NAME="E1:95_2_4_3"/><i><font color="Green">E70</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <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> <font color="Maroon">Q</font> &amp; <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> <font color="Maroon">Q</font> )
 ;<br/>

<a NAME="E2:95_2_4_3"/><b>then </b>
( ( <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> <font color="Maroon">P</font> or ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> ) ) &amp; ( <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> <font color="Maroon">P</font> or ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">a</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> ) ) )
 
<b>by </b><i><a class="txt" href="orders_1.html#E6:95_2"><i><font color="Green">E61</font></i></a></i>;<br/>
<a NAME="E3:95_2_4_3"/><b>then </b>
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> or ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">a</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> ) or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> )
 
<b>by </b><i>, <a class="txt" href="orders_1.html#E5:95_2"><i><font color="Green">E60</font></i></a>, , <a class="ref" href="relat_2.html#D4">RELAT_2:def 4</a>, <a class="ref" href="relat_2.html#D8">RELAT_2:def 8</a></i>;<br/>
<b>hence </b><a NAME="E4:95_2_4_3"/>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 <b>by </b><i>, <a class="txt" href="orders_1.html#E5:95_2"><i><font color="Green">E60</font></i></a>, , <a class="ref" href="relat_2.html#D8">RELAT_2:def 8</a></i>;<br/>


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


</div><b>end;</b></div>
<a NAME="E9:95_2"/><b>then </b>
<font color="Maroon">Q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Rel</font>
 
<b>by </b><i><a class="txt" href="orders_1.html#E12:95"><i><font color="Green">E55</font></i></a>, , <a class="txt" href="orders_1.html#E7:95_2"><i><font color="Green">E64</font></i></a>, </i>;<br/>
<a NAME="E10:95_2"/><b>then </b><i><font color="Green">E71</font></i>: 
<font color="Maroon">Q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font>
 
<b>by </b><i><a class="txt" href="orders_1.html#E3:95_2"><i><font color="Green">E57</font></i></a>, <a class="txt" href="orders_1.html#E4:95_2"><i><font color="Green">E58</font></i></a>, </i>;<br/>
<a NAME="E11:95_2"/>
( <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> <font color="Maroon">P</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 
<b>by </b><i>, <a class="txt" href="orders_1.html#E5:95_2"><i><font color="Green">E60</font></i></a>, <a class="ref" href="relat_2.html#D1">RELAT_2:def 1</a></i>;<br/>
<b>hence </b><a NAME="E12:95_2"/>
contradiction
 <b>by </b><i><a class="txt" href="orders_1.html#E5:95_2"><i><font color="Green">E60</font></i></a>, <a class="txt" href="orders_1.html#E6:95_2"><i><font color="Green">E61</font></i></a>, </i>;<br/>


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

<b>thus </b><a NAME="E14:95"/>
 <a href="relat_1.html#K3">field</a> <font color="Maroon">P</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font>
 <b>by </b><i/>;<br/>


</div>
