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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="yellow_0.html#V1">lower-bounded</a> <a href="lattice3.html#V1">with_suprema</a> <a href="orders_2.html#NM1">Poset</a>;<br/>

<a NAME="E1:24"/>
 <a href="waybel_4.html#K4" target="_self">AuxBottom</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="waybel_4.html#K3" target="_self">Aux</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="waybel_4.html#D9" target="_self">Def9</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>2</sub></font> =  <a href="waybel_4.html#K4" target="_self">AuxBottom</a> <font color="Maroon">c<sub>1</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_4.html#K3" target="_self">Aux</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <b>by </b><i><a class="ref" href="yellow_1.html#T1">YELLOW_1:1</a></i>;<br/>
<a NAME="E3:24"/>
 <a href="waybel_4.html#K4" target="_self">AuxBottom</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K1">"\/"</a> <a href="xboole_0.html#K1">{}</a> ,<span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_4.html#K3" target="_self">Aux</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1"><b>proof </b></a><div class="add">

<a NAME="E1:24_1"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="yellow_0.html#T6">YELLOW_0:6</a></i>;<br/>
<a NAME="E2:24_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_4.html#K3" target="_self">Aux</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> st <font color="Olive">b<sub>1</sub></font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a>  <a href="xboole_0.html#K1">{}</a>  holds <br/><font color="Maroon">c<sub>2</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_4.html#K3" target="_self">Aux</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>;<br/>

<b>assume </b><a NAME="E1:24_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<a NAME="E2:24_1_1"/>
<font color="Maroon">c<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="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_4.html#K3" target="_self">Aux</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 
;<br/>
<a NAME="E3:24_1_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="waybel_4.html#K3" target="_self">Aux</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="yellow_1.html#T1">YELLOW_1:1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> as  <a href="waybel_4.html#V5" target="_self">auxiliary</a> <a href="alg_1.html#NM1">Relation</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="waybel_4.html#D9" target="_self">Def9</a></i>;<br/>
<a NAME="E5:24_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="waybel_4.html#T4" target="_self">Th4</a></i>;<br/>
<b>hence </b><a NAME="E6:24_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="yellow_1.html#T3">YELLOW_1:3</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:24_1"/>
 <a href="waybel_4.html#K4" target="_self">AuxBottom</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K1">"\/"</a> <a href="xboole_0.html#K1">{}</a> ,<span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_4.html#K3" target="_self">Aux</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="waybel_4.html#E1:24_1"><i><font color="Green">E15</font></i></a>, <a class="ref" href="yellow_0.html#T30">YELLOW_0:30</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:24"/>
 <a href="yellow_0.html#K3">Bottom</a> <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_4.html#K3" target="_self">Aux</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="waybel_4.html#K4" target="_self">AuxBottom</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>


</div>
