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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="waybel_0.html#NM8">LATTICE</a>;<br/>

<b>thus </b><a NAME="E1:13"/>
( <font color="Maroon">c<sub>1</sub></font> is <a href="waybel_8.html#V2" target="_self">algebraic</a> implies ( <font color="Maroon">c<sub>1</sub></font> is <a href="waybel_3.html#V3">continuous</a> &amp; ( for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Olive">b<sub>2</sub></font> holds <br/>ex <font color="Olive">b<sub>3</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>3</sub></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">c<sub>1</sub></font></span>)</span> &amp; <font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">b<sub>2</sub></font> ) ) ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:13_1"/><i><font color="Green">E8</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="waybel_8.html#V2" target="_self">algebraic</a>
 ;<br/>

<a NAME="E2:13_1"/><b>then </b><i><font color="Green">E9</font></i>: 
( ( for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <br/>( not  <a href="waybel_8.html#K2" target="_self">compactbelow</a> <font color="Olive">b<sub>1</sub></font> is <a href="xboole_0.html#V1">empty</a> &amp;  <a href="waybel_8.html#K2" target="_self">compactbelow</a> <font color="Olive">b<sub>1</sub></font> is <a href="waybel_0.html#V1">directed</a> ) ) &amp; <font color="Maroon">c<sub>1</sub></font> is <a href="waybel_0.html#V24">up-complete</a> &amp; <font color="Maroon">c<sub>1</sub></font> is <a href="waybel_8.html#V1" target="_self">satisfying_axiom_K</a> )
 
<b>by </b><i><a class="ref" href="waybel_8.html#D4" target="_self">Def4</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><a NAME="E1:13_1_1"/><i><font color="Green">E10</font></i>: 
( not  <a href="waybel_8.html#K2" target="_self">compactbelow</a> <font color="Maroon">c<sub>2</sub></font> is <a href="xboole_0.html#V1">empty</a> &amp;  <a href="waybel_8.html#K2" target="_self">compactbelow</a> <font color="Maroon">c<sub>2</sub></font> is <a href="waybel_0.html#V1">directed</a> )
 <b>by </b><i><a class="txt" href="waybel_8.html#E1:13_1"><i><font color="Green">E8</font></i></a>, <a class="ref" href="waybel_8.html#D4" target="_self">Def4</a></i>;<br/><a NAME="E2:13_1_1"/><b>then </b><i><font color="Green">E11</font></i>: 
 <a href="yellow_0.html#R1">ex_sup_of</a>  <a href="waybel_8.html#K2" target="_self">compactbelow</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="waybel_8.html#E2:13_1"><i><font color="Green">E9</font></i></a>, <a class="ref" href="waybel_0.html#T75">WAYBEL_0:75</a></i>;<br/><a NAME="E3:13_1_1"/><i><font color="Green">E12</font></i>: 
 <a href="waybel_8.html#K2" target="_self">compactbelow</a> <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a>  <a href="waybel_3.html#K1">waybelow</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="waybel_8.html#T6" target="_self">Th6</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>3</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:13_1_1"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="waybel_8.html#K2" target="_self">compactbelow</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="waybel_8.html#E1:13_1_1"><i><font color="Green">E10</font></i></a>, <a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/><a NAME="E6:13_1_1"/><i><font color="Green">E14</font></i>: 
 <a href="yellow_0.html#R1">ex_sup_of</a>  <a href="waybel_3.html#K1">waybelow</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="waybel_8.html#E2:13_1"><i><font color="Green">E9</font></i></a>, <a class="txt" href="waybel_8.html#E3:13_1_1"><i><font color="Green">E12</font></i></a>, <a class="txt" href="waybel_8.html#E5:13_1_1"><i><font color="Green">E13</font></i></a>, <a class="ref" href="waybel_0.html#T75">WAYBEL_0:75</a></i>;<br/><a NAME="E7:13_1_1"/><b>then </b>
 <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> <font color="Maroon">c<sub>2</sub></font></span>)</span> <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_3.html#K1">waybelow</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel_8.html#E2:13_1_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="waybel_8.html#E3:13_1_1"><i><font color="Green">E12</font></i></a>, <a class="ref" href="yellow_0.html#T34">YELLOW_0:34</a></i>;<br/><a NAME="E8:13_1_1"/><b>then </b><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>2</sub></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_3.html#K1">waybelow</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel_8.html#E2:13_1"><i><font color="Green">E9</font></i></a>, <a class="ref" href="waybel_8.html#D3" target="_self">Def3</a></i>;<br/><a NAME="E9:13_1_1"/>
 <a href="waybel_3.html#K1">waybelow</a> <font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="waybel_3.html#T9">WAYBEL_3:9</a></i>;<br/><a NAME="E10:13_1_1"/><b>then </b>
 <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><a href="waybel_3.html#K1">waybelow</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="waybel_8.html#E6:13_1_1"><i><font color="Green">E14</font></i></a>, <a class="ref" href="yellow_0.html#D9">YELLOW_0:def 9</a></i>;<br/><b>hence </b><a NAME="E11:13_1_1"/>
<font color="Maroon">c<sub>2</sub></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_3.html#K1">waybelow</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel_8.html#E8:13_1_1"><i><font color="Green">E15</font></i></a>, <a class="ref" href="orders_2.html#T25">ORDERS_2:25</a></i>;<br/></div><b>end;</b></div>
<a NAME="E4:13_1"/><b>then </b><i><font color="Green">E10</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="waybel_3.html#V2">satisfying_axiom_of_approximation</a>
 
<b>by </b><i><a class="ref" href="waybel_3.html#D5">WAYBEL_3:def 5</a></i>;<br/>
<a NAME="E5:13_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <br/>( not  <a href="waybel_3.html#K1">waybelow</a> <font color="Olive">b<sub>1</sub></font> is <a href="xboole_0.html#V1">empty</a> &amp;  <a href="waybel_3.html#K1">waybelow</a> <font color="Olive">b<sub>1</sub></font> is <a href="waybel_0.html#V1">directed</a> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<a NAME="E1:13_1_2"/>
not  <a href="waybel_8.html#K2" target="_self">compactbelow</a> <font color="Maroon">c<sub>2</sub></font> is <a href="xboole_0.html#V1">empty</a>
 
<b>by </b><i><a class="txt" href="waybel_8.html#E1:13_1"><i><font color="Green">E8</font></i></a>, <a class="ref" href="waybel_8.html#D4" target="_self">Def4</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>3</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:13_1_2"/><i><font color="Green">E11</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="waybel_8.html#K2" target="_self">compactbelow</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<a NAME="E4:13_1_2"/>
 <a href="waybel_8.html#K2" target="_self">compactbelow</a> <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a>  <a href="waybel_3.html#K1">waybelow</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="waybel_8.html#T6" target="_self">Th6</a></i>;<br/>
<b>hence </b><a NAME="E5:13_1_2"/>
( not  <a href="waybel_3.html#K1">waybelow</a> <font color="Maroon">c<sub>2</sub></font> is <a href="xboole_0.html#V1">empty</a> &amp;  <a href="waybel_3.html#K1">waybelow</a> <font color="Maroon">c<sub>2</sub></font> is <a href="waybel_0.html#V1">directed</a> )
 <b>by </b><i><a class="txt" href="waybel_8.html#E3:13_1_2"><i><font color="Green">E11</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:13_1"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="waybel_3.html#V3">continuous</a>
 <b>by </b><i><a class="txt" href="waybel_8.html#E2:13_1"><i><font color="Green">E9</font></i></a>, <a class="txt" href="waybel_8.html#E4:13_1"><i><font color="Green">E10</font></i></a>, <a class="ref" href="waybel_3.html#D6">WAYBEL_3:def 6</a></i>;<br/>

<b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><a NAME="E7:13_1"/><i><font color="Green">E11</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> =  <a href="waybel_8.html#K2" target="_self">compactbelow</a> <font color="Maroon">c<sub>3</sub></font> as  non <a href="xboole_0.html#V1">empty</a> <a href="waybel_0.html#V1">directed</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="waybel_8.html#E1:13_1"><i><font color="Green">E8</font></i></a>, <a class="ref" href="waybel_8.html#D4" target="_self">Def4</a></i>;<br/>
<a NAME="E9:13_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="waybel_8.html#E2:13_1"><i><font color="Green">E9</font></i></a>, <a class="ref" href="waybel_8.html#D3" target="_self">Def3</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E11:13_1"/><i><font color="Green">E12</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="waybel_8.html#E7:13_1"><i><font color="Green">E11</font></i></a>, <a class="ref" href="waybel_3.html#D1">WAYBEL_3:def 1</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>5</sub></font>
;<br/>

<a NAME="E12:13_1"/>
<font color="Maroon">c<sub>5</sub></font> is <a href="waybel_3.html#V1">compact</a>
 
<b>by </b><i><a class="txt" href="waybel_8.html#E11:13_1"><i><font color="Green">E12</font></i></a>, <a class="ref" href="waybel_8.html#T4" target="_self">Th4</a></i>;<br/>
<b>hence </b><a NAME="E13:13_1"/>
<font color="Maroon">c<sub>5</sub></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">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="waybel_8.html#D1" target="_self">Def1</a></i>;<br/>

<b>thus </b><a NAME="E14:13_1"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="waybel_8.html#E11:13_1"><i><font color="Green">E12</font></i></a>, <a class="ref" href="waybel_8.html#T4" target="_self">Th4</a></i>;<br/>


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

<b>assume </b><b>that </b><br/><a NAME="E2:13"/><i><font color="Green">E8</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="waybel_3.html#V3">continuous</a>
 <b>and </b><br/><a NAME="E3:13"/><i><font color="Green">E9</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Olive">b<sub>2</sub></font> holds <br/>ex <font color="Olive">b<sub>3</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>3</sub></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">c<sub>1</sub></font></span>)</span> &amp; <font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">b<sub>2</sub></font> )
 ;<br/>

<a NAME="E4:13"/><i><font color="Green">E10</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="waybel_0.html#V24">up-complete</a>
 
<b>by </b><i><a class="txt" href="waybel_8.html#E2:13"><i><font color="Green">E8</font></i></a>, <a class="ref" href="waybel_3.html#D6">WAYBEL_3:def 6</a></i>;<br/>
<a NAME="E5:13"/><i><font color="Green">E11</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <br/>( not  <a href="waybel_8.html#K2" target="_self">compactbelow</a> <font color="Olive">b<sub>1</sub></font> is <a href="xboole_0.html#V1">empty</a> &amp;  <a href="waybel_8.html#K2" target="_self">compactbelow</a> <font color="Olive">b<sub>1</sub></font> is <a href="waybel_0.html#V1">directed</a> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<a NAME="E1:13_2"/><i><font color="Green">E12</font></i>: 
( not  <a href="waybel_3.html#K1">waybelow</a> <font color="Maroon">c<sub>2</sub></font> is <a href="xboole_0.html#V1">empty</a> &amp;  <a href="waybel_3.html#K1">waybelow</a> <font color="Maroon">c<sub>2</sub></font> is <a href="waybel_0.html#V1">directed</a> )
 
<b>by </b><i><a class="txt" href="waybel_8.html#E2:13"><i><font color="Green">E8</font></i></a>, <a class="ref" href="waybel_3.html#D6">WAYBEL_3:def 6</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>3</sub></font> be  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="waybel_8.html#K2" target="_self">compactbelow</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>;<br/><a NAME="E1:13_2_1"/>
 <a href="waybel_8.html#K2" target="_self">compactbelow</a> <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a>  <a href="waybel_3.html#K1">waybelow</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="waybel_8.html#T6" target="_self">Th6</a></i>;<br/><a NAME="E2:13_2_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> is  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="waybel_3.html#K1">waybelow</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E4:13_2_1"/><i><font color="Green">E13</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="waybel_3.html#K1">waybelow</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="waybel_8.html#E1:13_2"><i><font color="Green">E12</font></i></a>, <a class="ref" href="waybel_0.html#T1">WAYBEL_0:1</a></i>;<br/><a NAME="E5:13_2_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="waybel_8.html#E4:13_2_1"><i><font color="Green">E13</font></i></a>, <a class="ref" href="waybel_3.html#T7">WAYBEL_3:7</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E7:13_2_1"/><i><font color="Green">E14</font></i>: 
( <font color="Maroon">c<sub>5</sub></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">c<sub>1</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="waybel_8.html#E3:13"><i><font color="Green">E9</font></i></a></i>;<br/><b>take </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font>;<br/><a NAME="E8:13_2_1"/>
<font color="Maroon">c<sub>6</sub></font> is <a href="waybel_3.html#V1">compact</a>
 <b>by </b><i><a class="txt" href="waybel_8.html#E7:13_2_1"><i><font color="Green">E14</font></i></a>, <a class="ref" href="waybel_8.html#D1" target="_self">Def1</a></i>;<br/><b>hence </b><a NAME="E9:13_2_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="waybel_8.html#K2" target="_self">compactbelow</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="waybel_8.html#E7:13_2_1"><i><font color="Green">E14</font></i></a></i>;<br/><b>thus </b><a NAME="E10:13_2_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="waybel_8.html#E4:13_2_1"><i><font color="Green">E13</font></i></a>, <a class="txt" href="waybel_8.html#E7:13_2_1"><i><font color="Green">E14</font></i></a>, <a class="ref" href="yellow_0.html#T4">YELLOW_0:4</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:13_2"/>
( not  <a href="waybel_8.html#K2" target="_self">compactbelow</a> <font color="Maroon">c<sub>2</sub></font> is <a href="xboole_0.html#V1">empty</a> &amp;  <a href="waybel_8.html#K2" target="_self">compactbelow</a> <font color="Maroon">c<sub>2</sub></font> is <a href="waybel_0.html#V1">directed</a> )
 <b>by </b><i><a class="ref" href="waybel_0.html#T1">WAYBEL_0:1</a></i>;<br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><a NAME="E1:13_3"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="waybel_3.html#V2">satisfying_axiom_of_approximation</a>
 <b>by </b><i><a class="txt" href="waybel_8.html#E2:13"><i><font color="Green">E8</font></i></a>, <a class="ref" href="waybel_3.html#D6">WAYBEL_3:def 6</a></i>;<br/><a NAME="E2:13_3"/><b>then </b><i><font color="Green">E12</font></i>: 
<font color="Maroon">c<sub>2</sub></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_3.html#K1">waybelow</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="waybel_3.html#D5">WAYBEL_3:def 5</a></i>;<br/><a NAME="E3:13_3"/>
not  <a href="waybel_8.html#K2" target="_self">compactbelow</a> <font color="Maroon">c<sub>2</sub></font> is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i><a class="txt" href="waybel_8.html#E5:13"><i><font color="Green">E11</font></i></a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>3</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:13_3"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="waybel_8.html#K2" target="_self">compactbelow</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/><a NAME="E6:13_3"/>
 <a href="waybel_8.html#K2" target="_self">compactbelow</a> <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a>  <a href="waybel_3.html#K1">waybelow</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="waybel_8.html#T6" target="_self">Th6</a></i>;<br/><a NAME="E7:13_3"/><b>then </b><i><font color="Green">E14</font></i>: 
 <a href="yellow_0.html#R1">ex_sup_of</a>  <a href="waybel_3.html#K1">waybelow</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="waybel_8.html#E4:13"><i><font color="Green">E10</font></i></a>, <a class="txt" href="waybel_8.html#E5:13_3"><i><font color="Green">E13</font></i></a>, <a class="ref" href="waybel_0.html#T75">WAYBEL_0:75</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>thus </b><a NAME="E1:13_3_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a>  <a href="waybel_3.html#K1">waybelow</a> <font color="Maroon">c<sub>2</sub></font> implies <font color="Maroon">c<sub>4</sub></font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a>  <a href="waybel_8.html#K2" target="_self">compactbelow</a> <font color="Maroon">c<sub>2</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_3_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:13_3_1_1"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a>  <a href="waybel_3.html#K1">waybelow</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_3_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>assume </b><a NAME="E1:13_3_1_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="waybel_8.html#K2" target="_self">compactbelow</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><a NAME="E2:13_3_1_1_1"/><b>then </b><i><font color="Green">E16</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> is <a href="waybel_3.html#V1">compact</a> )
 <b>by </b><i><a class="ref" href="waybel_8.html#T4" target="_self">Th4</a></i>;<br/><a NAME="E3:13_3_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="waybel_3.html#D2">WAYBEL_3:def 2</a></i>;<br/><a NAME="E4:13_3_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="waybel_8.html#E2:13_3_1_1_1"><i><font color="Green">E16</font></i></a>, <a class="ref" href="waybel_3.html#T2">WAYBEL_3:2</a></i>;<br/><a NAME="E5:13_3_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="waybel_3.html#K1">waybelow</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="waybel_3.html#T7">WAYBEL_3:7</a></i>;<br/><b>hence </b><a NAME="E6:13_3_1_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="waybel_8.html#E1:13_3_1_1"><i><font color="Green">E15</font></i></a>, <a class="ref" href="lattice3.html#D9">LATTICE3:def 9</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:13_3_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a>  <a href="waybel_8.html#K2" target="_self">compactbelow</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="lattice3.html#D9">LATTICE3:def 9</a></i>;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E2:13_3_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a>  <a href="waybel_8.html#K2" target="_self">compactbelow</a> <font color="Maroon">c<sub>2</sub></font> implies <font color="Maroon">c<sub>4</sub></font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a>  <a href="waybel_3.html#K1">waybelow</a> <font color="Maroon">c<sub>2</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_3_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:13_3_1_2"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a>  <a href="waybel_8.html#K2" target="_self">compactbelow</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_3_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>assume </b><a NAME="E1:13_3_1_2_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="waybel_3.html#K1">waybelow</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><a NAME="E2:13_3_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="waybel_3.html#T7">WAYBEL_3:7</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E4:13_3_1_2_1"/><i><font color="Green">E16</font></i>: 
( <font color="Maroon">c<sub>6</sub></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">c<sub>1</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="waybel_8.html#E3:13"><i><font color="Green">E9</font></i></a></i>;<br/><a NAME="E5:13_3_1_2_1"/>
<font color="Maroon">c<sub>6</sub></font> is <a href="waybel_3.html#V1">compact</a>
 <b>by </b><i><a class="txt" href="waybel_8.html#E4:13_3_1_2_1"><i><font color="Green">E16</font></i></a>, <a class="ref" href="waybel_8.html#D1" target="_self">Def1</a></i>;<br/><a NAME="E6:13_3_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="waybel_8.html#K2" target="_self">compactbelow</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="waybel_8.html#E4:13_3_1_2_1"><i><font color="Green">E16</font></i></a></i>;<br/><a NAME="E7:13_3_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="waybel_8.html#E1:13_3_1_2"><i><font color="Green">E15</font></i></a>, <a class="ref" href="lattice3.html#D9">LATTICE3:def 9</a></i>;<br/><b>hence </b><a NAME="E8:13_3_1_2_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="waybel_8.html#E4:13_3_1_2_1"><i><font color="Green">E16</font></i></a>, <a class="ref" href="orders_2.html#T26">ORDERS_2:26</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:13_3_1_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a>  <a href="waybel_3.html#K1">waybelow</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="lattice3.html#D9">LATTICE3:def 9</a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E9:13_3"/>
<font color="Maroon">c<sub>2</sub></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> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel_8.html#E2:13_3"><i><font color="Green">E12</font></i></a>, <a class="txt" href="waybel_8.html#E7:13_3"><i><font color="Green">E14</font></i></a>, <a class="ref" href="yellow_0.html#T47">YELLOW_0:47</a></i>;<br/></div><b>end;</b></div>
<a NAME="E7:13"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> is <a href="waybel_8.html#V1" target="_self">satisfying_axiom_K</a>
 
<b>by </b><i><a class="ref" href="waybel_8.html#D3" target="_self">Def3</a></i>;<br/>
<b>hence </b><a NAME="E8:13"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="waybel_8.html#V2" target="_self">algebraic</a>
 <b>by </b><i><a class="txt" href="waybel_8.html#E4:13"><i><font color="Green">E10</font></i></a>, <a class="txt" href="waybel_8.html#E5:13"><i><font color="Green">E11</font></i></a>, <a class="ref" href="waybel_8.html#D4" target="_self">Def4</a></i>;<br/>


</div>
