<?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="waybel_0.html#NM7">sup-Semilattice</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be  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">c<sub>1</sub></font></span>)</span>;<br/>



<a NAME="E1:55"/><i><font color="Green">E27</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">c<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:55_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<a NAME="E2:55_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K7">Ids</a> <font color="Maroon">c<sub>1</sub></font>
 
;<br/>
<a NAME="E3:55_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">c<sub>1</sub></font> : verum </span>}</span> 
 
;<br/>
<a NAME="E4:55_1"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 
;<br/>
<b>hence </b><a NAME="E5:55_1"/>
<font color="Maroon">c<sub>3</sub></font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">c<sub>1</sub></font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E2:55"/>
<font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a>  <a href="zfmisc_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></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="E1:55_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<a NAME="E2:55_2"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="yellow_2.html#E1:55"><i><font color="Green">E27</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:55_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">c<sub>2</sub></font> as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E4:55"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> is <a href="waybel_0.html#V12">lower</a>
 
<b>by </b><i><a class="txt" href="yellow_2.html#E1:55"><i><font color="Green">E27</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>4</sub></font> =  <a href="setfam_1.html#K1">meet</a> <font color="Maroon">c<sub>2</sub></font> as  <a href="waybel_0.html#V12">lower</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="yellow_2.html#T38" target="_self">Th38</a></i>;<br/>
<a NAME="E6:55"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> holds <br/> <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:55_3"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> as   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="yellow_2.html#E1:55"><i><font color="Green">E27</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>7</sub></font> being    <a href="struct_0.html#M1">Element</a> of <font color="Maroon">c<sub>6</sub></font>;<br/>
<a NAME="E4:55_3"/>
 <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">c<sub>1</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T44">YELLOW_0:44</a></i>;<br/>
<b>hence </b><a NAME="E5:55_3"/>
 <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="waybel_0.html#D19">WAYBEL_0:def 19</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> as  non <a href="xboole_0.html#V1">empty</a> <a href="waybel_0.html#V12">lower</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/>
<a NAME="E8:55"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> holds <br/>( <font color="Olive">b<sub>1</sub></font> is <a href="waybel_0.html#V12">lower</a> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="waybel_0.html#V1">directed</a> )
 
<b>by </b><i><a class="txt" href="yellow_2.html#E1:55"><i><font color="Green">E27</font></i></a></i>;<br/>
<a NAME="E9:55"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="yellow_2.html#T40" target="_self">Th40</a></i>;<br/>
<b>hence </b><a NAME="E10:55"/>
 <a href="setfam_1.html#K1">meet</a> <font color="Maroon">c<sub>2</sub></font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">c<sub>1</sub></font>
 ;<br/>


</div>
