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

<b>let </b><font color="Maroon">T</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="orders_2.html#L1">RelStr</a> ;<br/><b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font>;<br/>



<b>assume </b><a NAME="E1:13"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">T</font> is <a href="orders_2.html#V2">reflexive</a>
 ;<br/>

<b>assume </b><a NAME="E2:13"/>
 <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span>
 ;<br/>

<a NAME="E3:13"/><b>then </b><i><font color="Green">E21</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T12">PRE_TOPC:12</a></i>;<br/>
<b>set </b><font color="Maroon">R</font> = the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">T</font>;<br/>
<a NAME="E4:13"/>
the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">T</font> <a href="hidden.html#R1">=</a>  <a href="partfun1.html#K6">id</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1"><b>proof </b></a><div class="add">

<a NAME="E1:13_1"/><i><font color="Green">E30</font></i>: 
( the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">T</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span></span><a href="mcart_1.html#K12">:]</a></span> &amp;  <a href="partfun1.html#K6">id</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">a</font></span><a href="domain_1.html#K1">]</a></span></span><a href="domain_1.html#K6">}</a></span> )
 
<b>by </b><i>, <a class="ref" href="sysrel.html#T30">SYSREL:30</a></i>;<br/>
<b>hence </b><a NAME="E2:13_1"/>
the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">T</font> <a href="tarski.html#R1">c=</a>  <a href="partfun1.html#K6">id</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T35">ZFMISC_1:35</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><br/>

<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="E3:13_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="partfun1.html#K6">id</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span>
 ;<br/>

<a NAME="E4:13_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">a</font></span><a href="domain_1.html#K1">]</a></span> &amp; <font color="Maroon">a</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">a</font> )
 
<b>by </b><i>, , <a class="ref" href="orders_2.html#T24">ORDERS_2:24</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E5:13_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">T</font>
 <b>by </b><i><a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:13"/>
<font color="Maroon">T</font> is <a href="orders_3.html#V1" target="_self">discrete</a>
 <b>by </b><i>, </i>;<br/>


</div>
