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

<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>set </b><font color="Maroon">CR</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>;<br/>
<a NAME="E1:12_1_1"/>
<font color="Maroon">R</font> is <a href="orders_2.html#V2">reflexive</a>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E2:12_1_1"/><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#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="E3:12_1_1"/>
<font color="Maroon">R</font> is <a href="orders_2.html#V3">transitive</a>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:12_1_1"/><b>then </b><i><font color="Green">E30</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/>
<a NAME="E5:12_1_1"/><b>then </b><i><font color="Green">E31</font></i>: 
the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> <a href="orders_1.html#R1">quasi_orders</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_1.html#D6">ORDERS_1:def 6</a></i>;<br/>
<a NAME="E6:12_1_1"/>
( 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> is <a href="partfun1.html#V1">total</a> &amp; 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> is <a href="relat_2.html#V3">symmetric</a> &amp; 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> is <a href="relat_2.html#V8">transitive</a> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:12_1_1_1"/>
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> <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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1_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="relat_2.html#D1">RELAT_2:def 1</a><br/>

<b>assume </b><a NAME="E1:12_1_1_1_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>
 ;<br/>

<a NAME="E2:12_1_1_1_1"/><i><font color="Green">E33</font></i>: 
<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> 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/>
<a NAME="E3:12_1_1_1_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> 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/>
<b>hence </b><a NAME="E4:12_1_1_1_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> 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="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:12_1_1_1"/><b>then </b><i><font color="Green">E34</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> <a href="eqrel_1.html#K2">/\</a> <span class="p2">(<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></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> &amp;  <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> <a href="eqrel_1.html#K2">/\</a> <span class="p2">(<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></span>)</span> <a href="hidden.html#R1">=</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_1.html#T98">ORDERS_1:98</a></i>;<br/>
<b>hence </b><a NAME="E3:12_1_1_1"/>
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> is <a href="partfun1.html#V1">total</a>
 <b>by </b><i><a class="ref" href="partfun1.html#D4">PARTFUN1:def 4</a></i>;<br/>

<a NAME="E4:12_1_1_1"/>
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> <a href="relat_2.html#R3">is_symmetric_in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1_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> , <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#D3">RELAT_2:def 3</a><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:12_1_1_1_2"/>
( <font color="Maroon">x</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">y</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="E2:12_1_1_1_2"/><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> 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>
 ;<br/>

<a NAME="E3:12_1_1_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> 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">x</font>,<font color="Maroon">y</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="E4:12_1_1_1_2"/><b>then </b>
( <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> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> <a href="relset_1.html#K6">~</a>  &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> 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/>
<b>hence </b><a NAME="E5:12_1_1_1_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> 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="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:12_1_1_1"/>
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> is <a href="relat_2.html#V3">symmetric</a>
 <b>by </b><i>, <a class="ref" href="relat_2.html#D11">RELAT_2:def 11</a></i>;<br/>

<a NAME="E6:12_1_1_1"/>
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> <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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1_1_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> , <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="E1:12_1_1_1_3"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">x</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="E2:12_1_1_1_3"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">y</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="E3:12_1_1_1_3"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">z</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="E4:12_1_1_1_3"/><i><font color="Green">E71</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> 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>and </b><br/><a NAME="E5:12_1_1_1_3"/><i><font color="Green">E72</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> 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>
 ;<br/>

<a NAME="E6:12_1_1_1_3"/><i><font color="Green">E73</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> 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">x</font>,<font color="Maroon">y</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="E7:12_1_1_1_3"/><i><font color="Green">E74</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> 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">y</font>,<font color="Maroon">z</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="E8:12_1_1_1_3"/><b>then </b><i><font color="Green">E75</font></i>: 
<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> 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#D8">RELAT_2:def 8</a></i>;<br/>
<a NAME="E9:12_1_1_1_3"/>
the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> <a href="relset_1.html#K6">~</a>  <a href="orders_1.html#R1">quasi_orders</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_1.html#T136">ORDERS_1:136</a></i>;<br/>
<a NAME="E10:12_1_1_1_3"/><b>then </b>
the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> <a href="relset_1.html#K6">~</a>  <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_1.html#D6">ORDERS_1:def 6</a></i>;<br/>
<a NAME="E11:12_1_1_1_3"/><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> 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_2.html#D8">RELAT_2:def 8</a></i>;<br/>
<b>hence </b><a NAME="E12:12_1_1_1_3"/>
<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> 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="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:12_1_1_1"/>
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> is <a href="relat_2.html#V8">transitive</a>
 <b>by </b><i>, <a class="ref" href="relat_2.html#D16">RELAT_2:def 16</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:12_1_1"/>
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> is   <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>
 ;<br/>


</div>
