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<b>let </b><font color="Maroon">tau</font> be   <a href="setfam_1.html#NM1">Subset-Family</a> of <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>;<br/><b>let </b><font color="Maroon">r</font> be   <a href="relset_1.html#NM3">Relation</a> of <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:69"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">tau</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> ,<span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K2">}</a></span>
 <b>and </b><br/><a NAME="E2:69"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">r</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">0,0</span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span>
 ;<br/>

<b>set </b><font color="Maroon">T</font> =  <a href="waybel_9.html#G1" target="_self">TopRelStr</a>(# <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<font color="Maroon">r</font>,<font color="Maroon">tau</font> #);<br/>
<b>thus </b><a NAME="E3:69"/>
 <a href="waybel_9.html#G1" target="_self">TopRelStr</a>(# <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<font color="Maroon">r</font>,<font color="Maroon">tau</font> #) is <a href="realset2.html#V3">trivial</a>
 ;<br/>

<b>thus </b><a NAME="E4:69"/>
 <a href="waybel_9.html#G1" target="_self">TopRelStr</a>(# <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<font color="Maroon">r</font>,<font color="Maroon">tau</font> #) is <a href="orders_2.html#V2">reflexive</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="waybel_9.html#G1" target="_self">TopRelStr</a>(# <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<font color="Maroon">r</font>,<font color="Maroon">tau</font> #); <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_0.html#D1">YELLOW_0:def 1</a><br/>

<a NAME="E1:69_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E2:69_1"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">x</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</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">0,0</span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:69_1"/>
<font color="Maroon">x</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">x</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>thus </b><a NAME="E5:69"/>
not  <a href="waybel_9.html#G1" target="_self">TopRelStr</a>(# <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<font color="Maroon">r</font>,<font color="Maroon">tau</font> #) is <a href="struct_0.html#V3">empty</a>
 ;<br/>

<a NAME="E6:69"/>
the <a href="pre_topc.html#U1">topology</a> of <a href="waybel_9.html#G1" target="_self">TopRelStr</a>(# <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<font color="Maroon">r</font>,<font color="Maroon">tau</font> #) <a href="hidden.html#R1">=</a>  <a href="pcomps_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <a href="waybel_9.html#G1" target="_self">TopRelStr</a>(# <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<font color="Maroon">r</font>,<font color="Maroon">tau</font> #)
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T30">ZFMISC_1:30</a></i>;<br/>
<b>hence </b><a NAME="E7:69"/>
 <a href="waybel_9.html#G1" target="_self">TopRelStr</a>(# <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<font color="Maroon">r</font>,<font color="Maroon">tau</font> #) is <a href="tdlat_3.html#V1">discrete</a>
 <b>by </b><i><a class="ref" href="tdlat_3.html#D1">TDLAT_3:def 1</a></i>;<br/>

<b>thus </b><a NAME="E8:69"/>
the <a href="struct_0.html#U1">carrier</a> of <a href="waybel_9.html#G1" target="_self">TopRelStr</a>(# <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<font color="Maroon">r</font>,<font color="Maroon">tau</font> #) is <a href="finset_1.html#V1">finite</a>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="group_1.html#D14">GROUP_1:def 14</a><br/>


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