<?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#NM8">LATTICE</a>;<br/>

<b>thus </b><a NAME="E1:42"/>
( <font color="Maroon">L</font> is <a href="waybel_8.html#V2">algebraic</a> implies  ex <font color="Olive">X</font> being non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ex <font color="Olive">S</font> being <a href="yellow_0.html#V4">full</a>  <a href="yellow_0.html#M1">SubRelStr</a> of  <a href="yellow_1.html#K3">BoolePoset</a> <font color="Olive">X</font> st <br/>( <font color="Olive">S</font> is <a href="yellow_0.html#V7">infs-inheriting</a> &amp; <font color="Olive">S</font> is <a href="waybel_0.html#V4">directed-sups-inheriting</a> &amp; <font color="Maroon">L</font>,<font color="Olive">S</font> <a href="waybel_1.html#R5">are_isomorphic</a>  ) )
 <b>by </b><i/>;<br/>

<b>thus </b><a NAME="E2:42"/>
(  ex <font color="Olive">X</font> being   <a href="hidden.html#M1">set</a>  ex <font color="Olive">S</font> being <a href="yellow_0.html#V4">full</a>  <a href="yellow_0.html#M1">SubRelStr</a> of  <a href="yellow_1.html#K3">BoolePoset</a> <font color="Olive">X</font> st <br/>( <font color="Olive">S</font> is <a href="yellow_0.html#V7">infs-inheriting</a> &amp; <font color="Olive">S</font> is <a href="waybel_0.html#V4">directed-sups-inheriting</a> &amp; <font color="Maroon">L</font>,<font color="Olive">S</font> <a href="waybel_1.html#R5">are_isomorphic</a>  ) implies <font color="Maroon">L</font> is <a href="waybel_8.html#V2">algebraic</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:42_1"/>
 ex <font color="Olive">X</font> being   <a href="hidden.html#M1">set</a>  ex <font color="Olive">S</font> being <a href="yellow_0.html#V4">full</a>  <a href="yellow_0.html#M1">SubRelStr</a> of  <a href="yellow_1.html#K3">BoolePoset</a> <font color="Olive">X</font> st <br/>( <font color="Olive">S</font> is <a href="yellow_0.html#V7">infs-inheriting</a> &amp; <font color="Olive">S</font> is <a href="waybel_0.html#V4">directed-sups-inheriting</a> &amp; <font color="Maroon">L</font>,<font color="Olive">S</font> <a href="waybel_1.html#R5">are_isomorphic</a>  )
 ;<br/>

<a NAME="E2:42_1"/><b>then </b>
 ex <font color="Olive">X</font> being   <a href="hidden.html#M1">set</a>  ex <font color="Olive">c</font> being <a href="waybel_1.html#V7">closure</a> <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> <font color="Olive">X</font></span>)</span>,<span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> <font color="Olive">X</font></span>)</span> st <br/>( <font color="Olive">c</font> is <a href="waybel_0.html#V22">directed-sups-preserving</a> &amp; <font color="Maroon">L</font>, <a href="yellow_2.html#K1">Image</a> <font color="Olive">c</font> <a href="waybel_1.html#R5">are_isomorphic</a>  )
 
<b>by </b><i><a class="ref" href="waybel13.html#T1" target="_self">Th1</a></i>;<br/>
<b>hence </b><a NAME="E3:42_1"/>
<font color="Maroon">L</font> is <a href="waybel_8.html#V2">algebraic</a>
 <b>by </b><i/>;<br/>


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


</div>
