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

<b>let </b><font color="Maroon">R</font> be    <a href="orders_2.html#L1">RelStr</a> ;<br/>

<b>set </b><font color="Maroon">CR</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>;<br/>
<b>set </b><font color="Maroon">IR</font> = the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font>;<br/>
<b>assume </b><a NAME="E1:15"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">R</font> is <a href="dickson.html#V3" target="_self">quasi_ordered</a>
 ;<br/>

<a NAME="E2:15"/><b>then </b>
<font color="Maroon">R</font> is <a href="orders_2.html#V3">transitive</a>
 
<b>by </b><i/>;<br/>
<a NAME="E3:15"/><b>then </b><i><font color="Green">E29</font></i>: 
the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> <a href="relat_2.html#R8">is_transitive_in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>
 
<b>by </b><i><a class="ref" href="orders_2.html#D5">ORDERS_2:def 5</a></i>;<br/>
<b>thus </b><a NAME="E4:15"/>
 <a href="dickson.html#K2" target="_self">&lt;=E</a> <font color="Maroon">R</font> <a href="relat_2.html#R1">is_reflexive_in</a>  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="dickson.html#K1" target="_self">EqRel</a> <font color="Maroon">R</font></span>)</span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="orders_1.html#D7">ORDERS_1:def 7</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_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="E1:15_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="dickson.html#K1" target="_self">EqRel</a> <font color="Maroon">R</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">a</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:15_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>
 <b>and </b><br/><a NAME="E4:15_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="dickson.html#K1" target="_self">EqRel</a> <font color="Maroon">R</font></span>)</span>,<font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="eqrel_1.html#D5">EQREL_1:def 5</a></i>;<br/>
<a NAME="E5:15_1"/>
<font color="Maroon">R</font> is <a href="orders_2.html#V2">reflexive</a>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E6:15_1"/><b>then </b>
the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> <a href="relat_2.html#R1">is_reflexive_in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>
 
<b>by </b><i><a class="ref" href="orders_2.html#D4">ORDERS_2:def 4</a></i>;<br/>
<a NAME="E7:15_1"/><b>then </b><i><font color="Green">E32</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">a</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font>
 
<b>by </b><i>, <a class="ref" href="relat_2.html#D1">RELAT_2:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">a'</font> = <font color="Maroon">a</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> <b>by </b><i/>;<br/>
<a NAME="E9:15_1"/>
<font color="Maroon">a'</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">a'</font>
 
<b>by </b><i>, <a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/>
<b>hence </b><a NAME="E10:15_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="dickson.html#K2" target="_self">&lt;=E</a> <font color="Maroon">R</font>
 <b>by </b><i>, </i>;<br/>


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

<b>thus </b><a NAME="E5:15"/>
 <a href="dickson.html#K2" target="_self">&lt;=E</a> <font color="Maroon">R</font> <a href="relat_2.html#R8">is_transitive_in</a>  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="dickson.html#K1" target="_self">EqRel</a> <font color="Maroon">R</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> , <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#D8">RELAT_2:def 8</a><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:15_2"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="dickson.html#K1" target="_self">EqRel</a> <font color="Maroon">R</font></span>)</span> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="dickson.html#K1" target="_self">EqRel</a> <font color="Maroon">R</font></span>)</span> &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="dickson.html#K1" target="_self">EqRel</a> <font color="Maroon">R</font></span>)</span> )
 <b>and </b><br/><a NAME="E2:15_2"/><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>  <a href="dickson.html#K2" target="_self">&lt;=E</a> <font color="Maroon">R</font>
 <b>and </b><br/><a NAME="E3:15_2"/><i><font color="Green">E35</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>  <a href="dickson.html#K2" target="_self">&lt;=E</a> <font color="Maroon">R</font>
 ;<br/>

<b>consider </b><font color="Maroon">a</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>, <font color="Maroon">b</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font><b> such that </b><br/><a NAME="E5:15_2"/><i><font color="Green">E68</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="dickson.html#K1" target="_self">EqRel</a> <font color="Maroon">R</font></span>)</span>,<font color="Maroon">a</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="dickson.html#K1" target="_self">EqRel</a> <font color="Maroon">R</font></span>)</span>,<font color="Maroon">b</font> &amp; <font color="Maroon">a</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">b</font> )
 <b>by </b><i>, </i>;<br/>
<b>consider </b><font color="Maroon">c</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>, <font color="Maroon">d</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font><b> such that </b><br/><a NAME="E7:15_2"/><i><font color="Green">E69</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="dickson.html#K1" target="_self">EqRel</a> <font color="Maroon">R</font></span>)</span>,<font color="Maroon">c</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="dickson.html#K1" target="_self">EqRel</a> <font color="Maroon">R</font></span>)</span>,<font color="Maroon">d</font> &amp; <font color="Maroon">c</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">d</font> )
 <b>by </b><i>, </i>;<br/>
<a NAME="E8:15_2"/><i><font color="Green">E70</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font>
 
<b>by </b><i>, <a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/>
<a NAME="E9:15_2"/><i><font color="Green">E71</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">d</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font>
 
<b>by </b><i>, <a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/>
<b>consider </b><font color="Maroon">x1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E11:15_2"/><i><font color="Green">E72</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="dickson.html#K1" target="_self">EqRel</a> <font color="Maroon">R</font></span>)</span>,<font color="Maroon">x1</font> )
 <b>by </b><i>, <a class="ref" href="eqrel_1.html#D5">EQREL_1:def 5</a></i>;<br/>
<a NAME="E12:15_2"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="dickson.html#K1" target="_self">EqRel</a> <font color="Maroon">R</font></span>)</span>,<font color="Maroon">c</font>
 
<b>by </b><i>, , <a class="ref" href="dickson.html#T1" target="_self">Th1</a>, <a class="ref" href="eqrel_1.html#T31">EQREL_1:31</a></i>;<br/>
<a NAME="E13:15_2"/><b>then </b>
<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>  <a href="dickson.html#K1" target="_self">EqRel</a> <font color="Maroon">R</font>
 
<b>by </b><i><a class="ref" href="eqrel_1.html#T27">EQREL_1:27</a></i>;<br/>
<a NAME="E14:15_2"/><b>then </b>
<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> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> <a href="eqrel_1.html#K2">/\</a> <span class="p1">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> <a href="relset_1.html#K6">~</a> </span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E15:15_2"/><b>then </b>
<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> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E16:15_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> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font>
 
<b>by </b><i>, , <a class="ref" href="dickson.html#T1" target="_self">Th1</a>, <a class="ref" href="relat_2.html#D8">RELAT_2:def 8</a></i>;<br/>
<a NAME="E17:15_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font>
 
<b>by </b><i>, , <a class="ref" href="dickson.html#T1" target="_self">Th1</a>, <a class="ref" href="relat_2.html#D8">RELAT_2:def 8</a></i>;<br/>
<a NAME="E18:15_2"/><b>then </b>
<font color="Maroon">a</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">d</font>
 
<b>by </b><i><a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/>
<b>hence </b><a NAME="E19:15_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>  <a href="dickson.html#K2" target="_self">&lt;=E</a> <font color="Maroon">R</font>
 <b>by </b><i>, , </i>;<br/>


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

<b>thus </b><a NAME="E6:15"/>
 <a href="dickson.html#K2" target="_self">&lt;=E</a> <font color="Maroon">R</font> <a href="relat_2.html#R4">is_antisymmetric_in</a>  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="dickson.html#K1" target="_self">EqRel</a> <font color="Maroon">R</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> , <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="relat_2.html#D4">RELAT_2:def 4</a><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:15_3"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="dickson.html#K1" target="_self">EqRel</a> <font color="Maroon">R</font></span>)</span>
 <b>and </b><br/><a NAME="E2:15_3"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="dickson.html#K1" target="_self">EqRel</a> <font color="Maroon">R</font></span>)</span>
 <b>and </b><br/><a NAME="E3:15_3"/><i><font color="Green">E74</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>  <a href="dickson.html#K2" target="_self">&lt;=E</a> <font color="Maroon">R</font>
 <b>and </b><br/><a NAME="E4:15_3"/><i><font color="Green">E75</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>  <a href="dickson.html#K2" target="_self">&lt;=E</a> <font color="Maroon">R</font>
 ;<br/>

<b>consider </b><font color="Maroon">a</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>, <font color="Maroon">b</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font><b> such that </b><br/><a NAME="E6:15_3"/><i><font color="Green">E89</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="dickson.html#K1" target="_self">EqRel</a> <font color="Maroon">R</font></span>)</span>,<font color="Maroon">a</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="dickson.html#K1" target="_self">EqRel</a> <font color="Maroon">R</font></span>)</span>,<font color="Maroon">b</font> &amp; <font color="Maroon">a</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">b</font> )
 <b>by </b><i>, </i>;<br/>
<b>consider </b><font color="Maroon">c</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>, <font color="Maroon">d</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font><b> such that </b><br/><a NAME="E8:15_3"/><i><font color="Green">E90</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="dickson.html#K1" target="_self">EqRel</a> <font color="Maroon">R</font></span>)</span>,<font color="Maroon">c</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="dickson.html#K1" target="_self">EqRel</a> <font color="Maroon">R</font></span>)</span>,<font color="Maroon">d</font> &amp; <font color="Maroon">c</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">d</font> )
 <b>by </b><i>, </i>;<br/>
<a NAME="E9:15_3"/><i><font color="Green">E91</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font>
 
<b>by </b><i>, <a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/>
<a NAME="E10:15_3"/><i><font color="Green">E92</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">d</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font>
 
<b>by </b><i>, <a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/>
<b>consider </b><font color="Maroon">x1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E12:15_3"/><i><font color="Green">E93</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="dickson.html#K1" target="_self">EqRel</a> <font color="Maroon">R</font></span>)</span>,<font color="Maroon">x1</font> )
 <b>by </b><i>, <a class="ref" href="eqrel_1.html#D5">EQREL_1:def 5</a></i>;<br/>
<a NAME="E13:15_3"/><i><font color="Green">E94</font></i>: 
<font color="Maroon">d</font> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="dickson.html#K1" target="_self">EqRel</a> <font color="Maroon">R</font></span>)</span>,<font color="Maroon">a</font>
 
<b>by </b><i>, , <a class="ref" href="dickson.html#T1" target="_self">Th1</a>, <a class="ref" href="eqrel_1.html#T31">EQREL_1:31</a></i>;<br/>
<a NAME="E14:15_3"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="dickson.html#K1" target="_self">EqRel</a> <font color="Maroon">R</font></span>)</span>,<font color="Maroon">a</font>
 
<b>by </b><i><a class="ref" href="dickson.html#T1" target="_self">Th1</a>, <a class="ref" href="eqrel_1.html#T28">EQREL_1:28</a></i>;<br/>
<a NAME="E15:15_3"/><b>then </b><i><font color="Green">E95</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="dickson.html#K1" target="_self">EqRel</a> <font color="Maroon">R</font>
 
<b>by </b><i>, <a class="ref" href="eqrel_1.html#T30">EQREL_1:30</a></i>;<br/>
<a NAME="E16:15_3"/><i><font color="Green">E96</font></i>: 
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="dickson.html#K1" target="_self">EqRel</a> <font color="Maroon">R</font></span>)</span>,<font color="Maroon">b</font>
 
<b>by </b><i>, , <a class="ref" href="dickson.html#T1" target="_self">Th1</a>, <a class="ref" href="eqrel_1.html#T31">EQREL_1:31</a></i>;<br/>
<a NAME="E17:15_3"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="dickson.html#K1" target="_self">EqRel</a> <font color="Maroon">R</font></span>)</span>,<font color="Maroon">b</font>
 
<b>by </b><i><a class="ref" href="dickson.html#T1" target="_self">Th1</a>, <a class="ref" href="eqrel_1.html#T28">EQREL_1:28</a></i>;<br/>
<a NAME="E18:15_3"/><b>then </b><i><font color="Green">E97</font></i>: 
<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>  <a href="dickson.html#K1" target="_self">EqRel</a> <font color="Maroon">R</font>
 
<b>by </b><i><a class="txt" href="dickson.html#E1:15"><i><font color="Green">E26</font></i></a>, <a class="ref" href="eqrel_1.html#T30">EQREL_1:30</a></i>;<br/>
<a NAME="E19:15_3"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> <a href="eqrel_1.html#K2">/\</a> <span class="p1">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> <a href="relset_1.html#K6">~</a> </span>)</span>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E20:15_3"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> <a href="relset_1.html#K6">~</a>  )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E21:15_3"/><b>then </b><i><font color="Green">E98</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">d</font>,<font color="Maroon">a</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#D7">RELAT_1:def 7</a></i>;<br/>
<a NAME="E22:15_3"/>
<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> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> <a href="eqrel_1.html#K2">/\</a> <span class="p1">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> <a href="relset_1.html#K6">~</a> </span>)</span>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E23:15_3"/><b>then </b>
( <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> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</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> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> <a href="relset_1.html#K6">~</a>  )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E24:15_3"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">d</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font>
 
<b>by </b><i>, , <a class="ref" href="dickson.html#T1" target="_self">Th1</a>, <a class="ref" href="relat_2.html#D8">RELAT_2:def 8</a></i>;<br/>
<a NAME="E25:15_3"/><b>then </b><i><font color="Green">E99</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">a</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font>
 
<b>by </b><i>, <a class="ref" href="dickson.html#T1" target="_self">Th1</a>, , <a class="ref" href="relat_2.html#D8">RELAT_2:def 8</a></i>;<br/>
<a NAME="E26:15_3"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">a</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> <a href="relset_1.html#K6">~</a> 
 
<b>by </b><i>, <a class="ref" href="relat_1.html#D7">RELAT_1:def 7</a></i>;<br/>
<a NAME="E27:15_3"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">a</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> <a href="eqrel_1.html#K2">/\</a> <span class="p1">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> <a href="relset_1.html#K6">~</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="dickson.html#E3:15"><i><font color="Green">E29</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E28:15_3"/><b>then </b>
<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>  <a href="dickson.html#K1" target="_self">EqRel</a> <font color="Maroon">R</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E29:15_3"/><b>then </b>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="dickson.html#K1" target="_self">EqRel</a> <font color="Maroon">R</font></span>)</span>,<font color="Maroon">a</font>
 
<b>by </b><i><a class="ref" href="eqrel_1.html#T27">EQREL_1:27</a></i>;<br/>
<b>hence </b><a NAME="E30:15_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="eqrel_1.html#T31">EQREL_1:31</a></i>;<br/>


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


</div>
