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

<a NAME="E1:19_1_1"/>
 <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c1">r</font> = <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> ,<a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> </span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> as   <a href="relset_1.html#NM3" title="RELSET_1:NM.3">Relation</a> of <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <b>by </b><i><a class="ref" href="relset_1.html#T8" title="RELSET_1:th.8">RELSET_1:8</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c2">T</font> =  <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<font color="Maroon" title="c1">r</font> #) as  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a>  ;<br/>
<b>take </b>
<font color="Maroon" title="c2">T</font>
;<br/>

<b>thus </b><a NAME="E4:19_1_1"/>
<font color="Maroon" title="c2">T</font> is <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a>
 ;<br/>

<b>thus </b><a NAME="E5:19_1_1"/>
<font color="Maroon" title="c2">T</font> is <a href="orders_2.html#V3" title="ORDERS_2:attr.3">reflexive</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">T</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_0.html#D1" title="YELLOW_0:def.1">YELLOW_0:def 1</a><br/>

<a NAME="E1:19_1_1_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E2:19_1_1_1"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c3">x</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> ,<a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> </span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:19_1_1_1"/>
<font color="Maroon" title="c3">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c3">x</font>
 <b>by </b><i><a class="ref" href="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</a></i>;<br/>


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

<b>then reconsider </b><font color="Maroon" title="c3">T'</font> = <font color="Maroon" title="c2">T</font> as  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a> <a href="orders_2.html#V3" title="ORDERS_2:attr.3">reflexive</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a>  ;<br/>
<a NAME="E7:19_1_1"/>
<font color="Maroon" title="c3">T'</font> is <a href="orders_2.html#V4" title="ORDERS_2:attr.4">transitive</a>
 
;<br/>
<b>hence </b><a NAME="E8:19_1_1"/>
<font color="Maroon" title="c2">T</font> is <a href="orders_2.html#V4" title="ORDERS_2:attr.4">transitive</a>
 ;<br/>

<a NAME="E9:19_1_1"/>
<font color="Maroon" title="c3">T'</font> is <a href="orders_2.html#V5" title="ORDERS_2:attr.5">antisymmetric</a>
 
;<br/>
<b>hence </b><a NAME="E10:19_1_1"/>
<font color="Maroon" title="c2">T</font> is <a href="orders_2.html#V5" title="ORDERS_2:attr.5">antisymmetric</a>
 ;<br/>

<a NAME="E11:19_1_1"/>
<font color="Maroon" title="c3">T'</font> is <a href="lattice3.html#V1" title="LATTICE3:attr.1">with_suprema</a>
 
;<br/>
<b>hence </b><a NAME="E12:19_1_1"/>
<font color="Maroon" title="c2">T</font> is <a href="lattice3.html#V1" title="LATTICE3:attr.1">with_suprema</a>
 ;<br/>

<a NAME="E13:19_1_1"/>
<font color="Maroon" title="c3">T'</font> is <a href="lattice3.html#V2" title="LATTICE3:attr.2">with_infima</a>
 
;<br/>
<b>hence </b><a NAME="E14:19_1_1"/>
<font color="Maroon" title="c2">T</font> is <a href="lattice3.html#V2" title="LATTICE3:attr.2">with_infima</a>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c4">T''</font> = <font color="Maroon" title="c3">T'</font> as   <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a> ;<br/>
<a NAME="E16:19_1_1"/>
<font color="Maroon" title="c4">T''</font> is <a href="yellow_0.html#V1" title="YELLOW_0:attr.1">lower-bounded</a>
 
;<br/>
<b>hence </b><a NAME="E17:19_1_1"/>
<font color="Maroon" title="c2">T</font> is <a href="yellow_0.html#V1" title="YELLOW_0:attr.1">lower-bounded</a>
 ;<br/>

<b>thus </b><a NAME="E18:19_1_1"/>
not <font color="Maroon" title="c2">T</font> is <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>
 ;<br/>

<b>thus </b><a NAME="E19:19_1_1"/>
<font color="Maroon" title="c2">T</font> is <a href="struct_0.html#V8" title="STRUCT_0:attr.8">finite</a>
 ;<br/>

<b>thus </b><a NAME="E20:19_1_1"/>
<font color="Maroon" title="c2">T</font> is <a href="orders_2.html#V1" title="ORDERS_2:attr.1">strict</a>
 ;<br/>


</div>
