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

<b>let </b><font color="Maroon">L</font> be  <a href="yellow_0.html#V1">lower-bounded</a> <a href="waybel_0.html#NM7">sup-Semilattice</a>;<br/><b>let </b><font color="Maroon">A</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</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_0.html#K7">Ids</a> <font color="Maroon">L</font></span>)</span></span>)</span>;<br/>



<b>set </b><font color="Maroon">P</font> =  <a href="yellow_1.html#K2">InclPoset</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K7">Ids</a> <font color="Maroon">L</font></span>)</span>;<br/>
<a NAME="E1:56"/>
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</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_0.html#K7">Ids</a> <font color="Maroon">L</font></span>)</span></span>)</span>
 
;<br/>
<a NAME="E2:56"/><b>then </b>
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a>  <a href="waybel_0.html#K7">Ids</a> <font color="Maroon">L</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">A'</font> = <font color="Maroon">A</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="waybel_0.html#K7">Ids</a> <font color="Maroon">L</font></span>)</span> ;<br/>
<a NAME="E4:56"/>
 <a href="setfam_1.html#K1">meet</a> <font color="Maroon">A'</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font>
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">I</font> =  <a href="setfam_1.html#K1">meet</a> <font color="Maroon">A</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_0.html#K7">Ids</a> <font color="Maroon">L</font></span>)</span></span>)</span> <b>by </b><i/>;<br/>
<a NAME="E6:56"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">I</font> <a href="lattice3.html#R1">is_&lt;=_than</a> <font color="Maroon">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</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_0.html#K7">Ids</a> <font color="Maroon">L</font></span>)</span></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D8">LATTICE3:def 8</a><br/>

<b>assume </b><a NAME="E1:56_1"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/>

<a NAME="E2:56_1"/>
<font color="Maroon">I</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:56_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 ;<br/>

<b>hence </b><a NAME="E2:56_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y</font>
 <b>by </b><i>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:56_1"/>
<font color="Maroon">I</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="yellow_1.html#T3">YELLOW_1:3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:56"/>
for <font color="Olive">b</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_0.html#K7">Ids</a> <font color="Maroon">L</font></span>)</span></span>)</span>  st <font color="Olive">b</font> <a href="lattice3.html#R1">is_&lt;=_than</a> <font color="Maroon">A</font> holds <br/><font color="Maroon">I</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Olive">b</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">b</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_0.html#K7">Ids</a> <font color="Maroon">L</font></span>)</span></span>)</span>;<br/>

<b>assume </b><a NAME="E1:56_2"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">A</font> <a href="lattice3.html#NR3" target="_self">is_&gt;=_than</a> <font color="Maroon">b</font>
 ;<br/>

<a NAME="E2:56_2"/><i><font color="Green">E28</font></i>: 
for <font color="Olive">J</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">J</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> holds <br/><font color="Maroon">b</font> <a href="tarski.html#R1">c=</a> <font color="Olive">J</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">J</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:56_2_1"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">J</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">y</font> = <font color="Maroon">J</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_0.html#K7">Ids</a> <font color="Maroon">L</font></span>)</span></span>)</span> ;<br/>
<a NAME="E3:56_2_1"/>
<font color="Maroon">b</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">y</font>
 
<b>by </b><i><a class="txt" href="yellow_2.html#E1:56_2_1"><i><font color="Green">E29</font></i></a>, , <a class="ref" href="lattice3.html#D8">LATTICE3:def 8</a></i>;<br/>
<b>hence </b><a NAME="E4:56_2_1"/>
<font color="Maroon">b</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">J</font>
 <b>by </b><i><a class="ref" href="yellow_1.html#T3">YELLOW_1:3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:56_2"/>
<font color="Maroon">b</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">I</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:56_2_2"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">b</font>
 ;<br/>

<a NAME="E2:56_2_2"/>
for <font color="Olive">J</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">J</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> holds <br/><font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Olive">J</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_2_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">J</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:56_2_2_1"/>
<font color="Maroon">J</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/>

<a NAME="E2:56_2_2_1"/><b>then </b>
<font color="Maroon">b</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">J</font>
 
<b>by </b><i><a class="txt" href="yellow_2.html#E1:56_2_2"><i><font color="Green">E30</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:56_2_2_1"/>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:56_2_2"/>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 <b>by </b><i><a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:56_2"/>
<font color="Maroon">I</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">b</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="E8:56"/>
(  <a href="yellow_0.html#R2">ex_inf_of</a> <font color="Maroon">A</font>, <a href="yellow_1.html#K2">InclPoset</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K7">Ids</a> <font color="Maroon">L</font></span>)</span> &amp;  <a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K1">meet</a> <font color="Maroon">A</font> )
 <b>by </b><i>, <a class="ref" href="yellow_0.html#T31">YELLOW_0:31</a></i>;<br/>


</div>
