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

<b>let </b><font color="Maroon">L</font> be  <a href="waybel_8.html#V2" target="_self">algebraic</a> <a href="waybel_0.html#NM8">LATTICE</a>;<br/>

<b>thus </b><a NAME="E1:41"/>
( <font color="Maroon">L</font> is <a href="waybel_8.html#V3" target="_self">arithmetic</a> implies  <a href="waybel_8.html#K1" target="_self">CompactSublatt</a> <font color="Maroon">L</font> is   <a href="waybel_0.html#NM8">LATTICE</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:41_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">L</font> is <a href="waybel_8.html#V3" target="_self">arithmetic</a>
 ;<br/>

<b>consider </b><font color="Maroon">x</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/>
<a NAME="E3:41_1"/>
not  <a href="waybel_8.html#K2" target="_self">compactbelow</a> <font color="Maroon">x</font> is <a href="xboole_0.html#V1">empty</a>
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:41_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="waybel_8.html#K2" target="_self">compactbelow</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<a NAME="E6:41_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">y</font> where <font color="Olive">y</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : ( <font color="Maroon">x</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Olive">y</font> &amp; <font color="Olive">y</font> is <a href="waybel_3.html#V1">compact</a> ) </span>}</span> 
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">z'</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E8:41_1"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">z'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> &amp; <font color="Maroon">x</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">z'</font> &amp; <font color="Maroon">z'</font> is <a href="waybel_3.html#V1">compact</a> )
 ;<br/>
<a NAME="E9:41_1"/>
<font color="Maroon">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>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E10:41_1"/><b>then </b>
 <a href="waybel_8.html#K1" target="_self">CompactSublatt</a> <font color="Maroon">L</font> is  non <a href="struct_0.html#V3">empty</a> <a href="yellow_0.html#V4">full</a> <a href="yellow_0.html#V5">meet-inheriting</a> <a href="yellow_0.html#V6">join-inheriting</a>  <a href="yellow_0.html#M1">SubRelStr</a> of <font color="Maroon">L</font>
 
<b>by </b><i>, , <a class="ref" href="struct_0.html#D1">STRUCT_0:def 1</a></i>;<br/>
<b>hence </b><a NAME="E11:41_1"/>
 <a href="waybel_8.html#K1" target="_self">CompactSublatt</a> <font color="Maroon">L</font> is   <a href="waybel_0.html#NM8">LATTICE</a>
 ;<br/>


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

<b>assume </b><a NAME="E2:41"/><i><font color="Green">E37</font></i>: 
 <a href="waybel_8.html#K1" target="_self">CompactSublatt</a> <font color="Maroon">L</font> is   <a href="waybel_0.html#NM8">LATTICE</a>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_2"><b>now </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/><b>assume </b><b>that </b><br/><a NAME="E1:41_2"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">x</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>and </b><br/><a NAME="E2:41_2"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">y</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>and </b><br/><a NAME="E3:41_2"/>
 <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="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K7">}</a></span>,<font color="Maroon">L</font>
 ;<br/><b>reconsider </b><font color="Maroon">L'</font> =  <a href="waybel_8.html#K1" target="_self">CompactSublatt</a> <font color="Maroon">L</font> as  non <a href="struct_0.html#V3">empty</a> <a href="yellow_0.html#V4">full</a>  <a href="yellow_0.html#M1">SubRelStr</a> of <font color="Maroon">L</font> <b>by </b><i>, <a class="ref" href="struct_0.html#D1">STRUCT_0:def 1</a></i>;<br/><b>reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font>, <font color="Maroon">y'</font> = <font color="Maroon">y</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L'</font> <b>by </b><i>, </i>;<br/><a NAME="E6:41_2"/><i><font color="Green">E41</font></i>: 
 <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="Maroon">x'</font>,<font color="Maroon">y'</font></span><a href="domain_1.html#K7">}</a></span>,<font color="Maroon">L'</font>
 <b>by </b><i>, <a class="ref" href="yellow_0.html#T21">YELLOW_0:21</a></i>;<br/><b>set </b><font color="Maroon">X</font> =  <a href="waybel_8.html#K2" target="_self">compactbelow</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p2"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K7">}</a></span></span>)</span>;<br/><a NAME="E7:41_2"/>
( not  <a href="waybel_8.html#K2" target="_self">compactbelow</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p2"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K7">}</a></span></span>)</span> is <a href="xboole_0.html#V1">empty</a> &amp;  <a href="waybel_8.html#K2" target="_self">compactbelow</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p2"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K7">}</a></span></span>)</span> is <a href="waybel_0.html#V1">directed</a> )
 <b>by </b><i/>;<br/><a NAME="E8:41_2"/><b>then </b><i><font color="Green">E42</font></i>: 
 <a href="yellow_0.html#R1">ex_sup_of</a>  <a href="waybel_8.html#K2" target="_self">compactbelow</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p2"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K7">}</a></span></span>)</span>,<font color="Maroon">L</font>
 <b>by </b><i><a class="ref" href="waybel_0.html#T75">WAYBEL_0:75</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">d</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:41_2_1"/>
<font color="Maroon">d</font> <a href="hidden.html#R2">in</a>  <a href="waybel_8.html#K2" target="_self">compactbelow</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p2"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K7">}</a></span></span>)</span>
 ;<br/><a NAME="E2:41_2_1"/><b>then </b>
<font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">f</font> where <font color="Olive">f</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : (  <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="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K7">}</a></span> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Olive">f</font> &amp; <font color="Olive">f</font> is <a href="waybel_3.html#V1">compact</a> ) </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">d'</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E4:41_2_1"/><i><font color="Green">E59</font></i>: 
( <font color="Maroon">d'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font> &amp;  <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="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K7">}</a></span> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">d'</font> &amp; <font color="Maroon">d'</font> is <a href="waybel_3.html#V1">compact</a> )
 ;<br/><b>thus </b><a NAME="E5:41_2_1"/>
<font color="Maroon">d</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L'</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><a NAME="E10:41_2"/><b>then </b><i><font color="Green">E60</font></i>: 
 <a href="waybel_8.html#K2" target="_self">compactbelow</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p2"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K7">}</a></span></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L'</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><a NAME="E11:41_2"/><i><font color="Green">E61</font></i>: 
 <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="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K7">}</a></span> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><a href="waybel_8.html#K2" target="_self">compactbelow</a> <span class="p2">(<span class="default"><a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p3"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K7">}</a></span></span>)</span></span>)</span>
 <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">c</font> =  <a href="yellow_0.html#K2">"/\"</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K7">}</a></span>,<font color="Maroon">L'</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> <b>by </b><i><a class="ref" href="yellow_0.html#T59">YELLOW_0:59</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:41_2_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="waybel_8.html#K2" target="_self">compactbelow</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p2"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K7">}</a></span></span>)</span>
 ;<br/><a NAME="E2:41_2_2"/><b>then </b>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">v</font> where <font color="Olive">v</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : (  <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="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K7">}</a></span> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Olive">v</font> &amp; <font color="Olive">v</font> is <a href="waybel_3.html#V1">compact</a> ) </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">z1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E4:41_2_2"/><i><font color="Green">E64</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z1</font> &amp;  <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="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K7">}</a></span> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">z1</font> &amp; <font color="Maroon">z1</font> is <a href="waybel_3.html#V1">compact</a> )
 ;<br/><a NAME="E5:41_2_2"/><i><font color="Green">E65</font></i>: 
( <font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">y</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">y</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">y</font> )
 <b>by </b><i><a class="ref" href="yellow_0.html#T23">YELLOW_0:23</a></i>;<br/><a NAME="E6:41_2_2"/>
<font color="Maroon">z1</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">y</font>
 <b>by </b><i>, <a class="ref" href="yellow_0.html#T40">YELLOW_0:40</a></i>;<br/><a NAME="E7:41_2_2"/><b>then </b>
( <font color="Maroon">z1</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">z1</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">y</font> )
 <b>by </b><i>, <a class="ref" href="orders_2.html#T26">ORDERS_2:26</a></i>;<br/><a NAME="E8:41_2_2"/><b>then </b>
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="waybel_8.html#K2" target="_self">compactbelow</a> <font color="Maroon">x</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="waybel_8.html#K2" target="_self">compactbelow</a> <font color="Maroon">y</font> )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E9:41_2_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="waybel_8.html#K2" target="_self">compactbelow</a> <font color="Maroon">x</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="waybel_8.html#K2" target="_self">compactbelow</a> <font color="Maroon">y</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div><a NAME="E14:41_2"/><b>then </b><i><font color="Green">E66</font></i>: 
 <a href="waybel_8.html#K2" target="_self">compactbelow</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p2"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K7">}</a></span></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="waybel_8.html#K2" target="_self">compactbelow</a> <font color="Maroon">x</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="waybel_8.html#K2" target="_self">compactbelow</a> <font color="Maroon">y</font></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_2_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">b'</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L'</font>;<br/><b>reconsider </b><font color="Maroon">b</font> = <font color="Maroon">b'</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> <b>by </b><i><a class="ref" href="yellow_0.html#T59">YELLOW_0:59</a></i>;<br/><b>assume </b><a NAME="E2:41_2_3"/>
<font color="Maroon">b'</font> <a href="hidden.html#R2">in</a>  <a href="waybel_8.html#K2" target="_self">compactbelow</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p2"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K7">}</a></span></span>)</span>
 ;<br/><a NAME="E3:41_2_3"/><b>then </b>
( <font color="Maroon">b'</font> <a href="hidden.html#R2">in</a>  <a href="waybel_8.html#K2" target="_self">compactbelow</a> <font color="Maroon">x</font> &amp; <font color="Maroon">b'</font> <a href="hidden.html#R2">in</a>  <a href="waybel_8.html#K2" target="_self">compactbelow</a> <font color="Maroon">y</font> )
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E4:41_2_3"/><b>then </b>
( <font color="Maroon">b</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">b</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">y</font> )
 <b>by </b><i/>;<br/><a NAME="E5:41_2_3"/><b>then </b>
( <font color="Maroon">b'</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">x'</font> &amp; <font color="Maroon">b'</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">y'</font> )
 <b>by </b><i><a class="ref" href="yellow_0.html#T61">YELLOW_0:61</a></i>;<br/><a NAME="E6:41_2_3"/><b>then </b>
<font color="Maroon">b'</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">x'</font> <a href="lattice3.html#K11">"/\"</a> <font color="Maroon">y'</font>
 <b>by </b><i>, <a class="ref" href="yellow_0.html#T19">YELLOW_0:19</a></i>;<br/><b>hence </b><a NAME="E7:41_2_3"/>
<font color="Maroon">b'</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#K2">"/\"</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K7">}</a></span>,<font color="Maroon">L'</font>
 <b>by </b><i>, <a class="ref" href="yellow_0.html#T40">YELLOW_0:40</a></i>;<br/></div><b>end;</b></div><a NAME="E16:41_2"/><b>then </b>
 <a href="waybel_8.html#K2" target="_self">compactbelow</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p2"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K7">}</a></span></span>)</span> <a href="lattice3.html#R2">is_&lt;=_than</a>  <a href="yellow_0.html#K2">"/\"</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K7">}</a></span>,<font color="Maroon">L'</font>
 <b>by </b><i><a class="ref" href="lattice3.html#D9">LATTICE3:def 9</a></i>;<br/><a NAME="E17:41_2"/><b>then </b>
 <a href="waybel_8.html#K2" target="_self">compactbelow</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p2"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K7">}</a></span></span>)</span> <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Maroon">c</font>
 <b>by </b><i>, <a class="ref" href="yellow_0.html#T63">YELLOW_0:63</a></i>;<br/><a NAME="E18:41_2"/><b>then </b><i><font color="Green">E67</font></i>: 
 <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><a href="waybel_8.html#K2" target="_self">compactbelow</a> <span class="p2">(<span class="default"><a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p3"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K7">}</a></span></span>)</span></span>)</span> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c</font>
 <b>by </b><i>, <a class="ref" href="yellow_0.html#T30">YELLOW_0:30</a></i>;<br/><a NAME="E19:41_2"/><i><font color="Green">E68</font></i>: 
( <font color="Maroon">x'</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K7">}</a></span> &amp; <font color="Maroon">y'</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K7">}</a></span> )
 <b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><a NAME="E20:41_2"/>
 <a href="yellow_0.html#K2">"/\"</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K7">}</a></span>,<font color="Maroon">L'</font> <a href="lattice3.html#R1">is_&lt;=_than</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K7">}</a></span>
 <b>by </b><i>, <a class="ref" href="yellow_0.html#T31">YELLOW_0:31</a></i>;<br/><a NAME="E21:41_2"/><b>then </b>
(  <a href="yellow_0.html#K2">"/\"</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K7">}</a></span>,<font color="Maroon">L'</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">x'</font> &amp;  <a href="yellow_0.html#K2">"/\"</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K7">}</a></span>,<font color="Maroon">L'</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">y'</font> )
 <b>by </b><i>, <a class="ref" href="lattice3.html#D8">LATTICE3:def 8</a></i>;<br/><a NAME="E22:41_2"/><b>then </b>
( <font color="Maroon">c</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">c</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">y</font> )
 <b>by </b><i><a class="ref" href="yellow_0.html#T60">YELLOW_0:60</a></i>;<br/><a NAME="E23:41_2"/><b>then </b>
<font color="Maroon">c</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T23">YELLOW_0:23</a></i>;<br/><a NAME="E24:41_2"/><b>then </b>
<font color="Maroon">c</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><a href="waybel_8.html#K2" target="_self">compactbelow</a> <span class="p2">(<span class="default"><a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p3"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K7">}</a></span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="waybel_8.html#D1" target="_self">Def1</a>, <a class="ref" href="yellow_0.html#T40">YELLOW_0:40</a></i>;<br/><a NAME="E25:41_2"/><b>then </b>
<font color="Maroon">c</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><a href="waybel_8.html#K2" target="_self">compactbelow</a> <span class="p2">(<span class="default"><a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p3"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K7">}</a></span></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="orders_2.html#T25">ORDERS_2:25</a></i>;<br/><b>hence </b><a NAME="E26:41_2"/>
 <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="Maroon">x</font>,<font color="Maroon">y</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="waybel_8.html#D1" target="_self">Def1</a></i>;<br/></div><b>end;</b></div>
<a NAME="E4:41"/><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="E5:41"/>
<font color="Maroon">L</font> is <a href="waybel_8.html#V3" target="_self">arithmetic</a>
 <b>by </b><i/>;<br/>


</div>
