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

<b>let </b><font color="Maroon">L</font> be   <a href="waybel_0.html#NM8">LATTICE</a>;<br/>

<b>assume </b><a NAME="E1:19_1_1"/>
<font color="Maroon">L</font> is <a href="realset2.html#V3">trivial</a>
 ;<br/>

<b>then reconsider </b><font color="Maroon">L'</font> = <font color="Maroon">L</font> as  <a href="realset2.html#V3">trivial</a> <a href="waybel_0.html#NM8">LATTICE</a> ;<br/>
<a NAME="E3:19_1_1"/>
for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L'</font>  st <font color="Olive">x</font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Olive">y</font> holds <br/> ex <font color="Olive">k</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L'</font> st <br/>( <font color="Olive">k</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel_8.html#K1" target="_self">CompactSublatt</a> <font color="Maroon">L'</font></span>)</span> &amp; <font color="Olive">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">y</font> )
 
<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">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L'</font>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L'</font>;<br/>

<a NAME="E1:19_1_1_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 
<b>by </b><i><a class="ref" href="realset2.html#D7">REALSET2:def 7</a></i>;<br/>
<b>assume </b><a NAME="E2:19_1_1_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">x</font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">y</font>
 ;<br/>

<b>take </b><font color="Maroon">k</font> = <font color="Maroon">x</font>;<br/>

<a NAME="E3:19_1_1_1"/>
<font color="Maroon">k</font> is <a href="waybel_3.html#V1">compact</a>
 
<b>by </b><i>, , <a class="ref" href="waybel_3.html#D2">WAYBEL_3:def 2</a></i>;<br/>
<b>hence </b><a NAME="E4:19_1_1_1"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel_8.html#K1" target="_self">CompactSublatt</a> <font color="Maroon">L'</font></span>)</span>
 <b>by </b><i/>;<br/>

<b>thus </b><a NAME="E5:19_1_1_1"/>
( <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">y</font> )
 <b>by </b><i><a class="ref" href="realset2.html#D7">REALSET2:def 7</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:19_1_1"/><b>then </b><i><font color="Green">E36</font></i>: 
<font color="Maroon">L'</font> is <a href="waybel_8.html#V2" target="_self">algebraic</a>
 
<b>by </b><i/>;<br/>
<a NAME="E5:19_1_1"/>
for <font color="Olive">z</font>, <font color="Olive">v</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L'</font>  st <font color="Olive">z</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel_8.html#K1" target="_self">CompactSublatt</a> <font color="Maroon">L'</font></span>)</span> &amp; <font color="Olive">v</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel_8.html#K1" target="_self">CompactSublatt</a> <font color="Maroon">L'</font></span>)</span> &amp;  <a href="yellow_0.html#R2">ex_inf_of</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Olive">z</font>,<font color="Olive">v</font></span><a href="domain_1.html#K7">}</a></span>,<font color="Maroon">L'</font> holds <br/> <a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Olive">z</font>,<font color="Olive">v</font></span><a href="domain_1.html#K7">}</a></span> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel_8.html#K1" target="_self">CompactSublatt</a> <font color="Maroon">L'</font></span>)</span>
 
<b>by </b><i><a class="ref" href="realset2.html#D7">REALSET2:def 7</a></i>;<br/>
<a NAME="E6:19_1_1"/><b>then </b>
 <a href="waybel_8.html#K1" target="_self">CompactSublatt</a> <font color="Maroon">L'</font> is <a href="yellow_0.html#V5">meet-inheriting</a>
 
<b>by </b><i><a class="ref" href="yellow_0.html#D16">YELLOW_0:def 16</a></i>;<br/>
<b>hence </b><a NAME="E7:19_1_1"/>
<font color="Maroon">L</font> is <a href="waybel_8.html#V3" target="_self">arithmetic</a>
 <b>by </b><i>, </i>;<br/>


</div>
