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

<b>assume </b><a NAME="E1:6"/><i><font color="Green">E24</font></i>: 
( <font color="Maroon">A</font> is <a href="orders_2.html#V2">reflexive</a> &amp; <font color="Maroon">A</font> is <a href="realset2.html#V3">trivial</a> )
 ;<br/>

<a NAME="E2:6"/><b>then </b><i><font color="Green">E27</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font> is <a href="realset1.html#V1">trivial</a>
 
<b>by </b><i><a class="ref" href="realset2.html#D5">REALSET2:def 5</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:6_1"/>
( the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font> is <a href="xboole_0.html#V1">empty</a> or  ex <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  st the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">x</font></span><a href="tarski.html#K1">}</a></span> )
 </span><b>by </b><i>, <a class="ref" href="realset1.html#D4">REALSET1:def 4</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_1_1"><b>suppose </b></a><a NAME="E1:6_1_1"/><i><font color="Green">E33</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><a NAME="E2:6_1_1"/><b>then </b>
the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="relset_1.html#T26">RELSET_1:26</a></i>;<br/><b>hence </b><a NAME="E3:6_1_1"/>
<font color="Maroon">A</font> is <a href="orders_3.html#V1">discrete</a>
 <b>by </b><i>, <a class="ref" href="orders_3.html#D1">ORDERS_3:def 1</a>, <a class="ref" href="relat_1.html#T81">RELAT_1:81</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_1_2"><b>suppose </b></a><a NAME="E1:6_1_2"/>
 ex <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  st the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">x</font></span><a href="tarski.html#K1">}</a></span>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:6_1_2"/><i><font color="Green">E34</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 ;<br/><a NAME="E4:6_1_2"/>
the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i/>;<br/><a NAME="E5:6_1_2"/><b>then </b>
the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><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></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T35">ZFMISC_1:35</a></i>;<br/><a NAME="E6:6_1_2"/><b>then </b><i><font color="Green">E35</font></i>: 
( the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><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></span><a href="tarski.html#K1">}</a></span> or the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T39">ZFMISC_1:39</a></i>;<br/><a NAME="E7:6_1_2"/><i><font color="Green">E36</font></i>: 
the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">A</font> <a href="relat_2.html#R1">is_reflexive_in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font>
 <b>by </b><i>, <a class="ref" href="orders_2.html#D4">ORDERS_2:def 4</a></i>;<br/><a NAME="E8:6_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">A</font>
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E9:6_1_2"/><b>then </b>
the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="partfun1.html#K6">id</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, , <a class="ref" href="relat_2.html#D1">RELAT_2:def 1</a>, <a class="ref" href="sysrel.html#T30">SYSREL:30</a></i>;<br/><b>hence </b><a NAME="E10:6_1_2"/>
<font color="Maroon">A</font> is <a href="orders_3.html#V1">discrete</a>
 <b>by </b><i>, <a class="ref" href="orders_3.html#D1">ORDERS_3:def 1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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