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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="waybel_1.html#V11">Boolean</a> <a href="waybel_0.html#NM8">LATTICE</a>;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:40_1"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="waybel_8.html#V3">arithmetic</a>
 ;<br/><a NAME="E2:40_1"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> is <a href="waybel_8.html#V2">algebraic</a>
 <b>by </b><i><a class="ref" href="waybel_8.html#D5">WAYBEL_8:def 5</a></i>;<br/><a NAME="E3:40_1"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> is <a href="waybel_3.html#V3">continuous</a>
 <b>by </b><i><a class="ref" href="waybel_8.html#T7">WAYBEL_8:7</a></i>;<br/><a NAME="E4:40_1"/><b>then </b>
( <font color="Maroon">c<sub>1</sub></font> is <a href="waybel_3.html#V3">continuous</a> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="yellow_7.html#NK1" target="_self">opp</a>  is <a href="waybel_3.html#V3">continuous</a> )
 <b>by </b><i><a class="txt" href="waybel15.html#E37"><i><font color="Green">Lemma28</font></i></a></i>;<br/><a NAME="E5:40_1"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> is <a href="waybel_5.html#V1">completely-distributive</a>
 <b>by </b><i><a class="txt" href="waybel15.html#E38"><i><font color="Green">Lemma29</font></i></a></i>;<br/><a NAME="E6:40_1"/><b>then </b>
( <font color="Maroon">c<sub>1</sub></font> is <a href="lattice3.html#V3">complete</a> &amp; ( 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>ex <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="tarski.html#R1">c=</a>  <a href="waybel15.html#K1" target="_self">ATOM</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">b<sub>2</sub></font> ) ) )
 <b>by </b><i><a class="txt" href="waybel15.html#E39"><i><font color="Green">Lemma30</font></i></a></i>;<br/><b>hence </b><a NAME="E7:40_1"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">c<sub>1</sub></font>, <a href="yellow_1.html#K3">BoolePoset</a> <font color="Olive">b<sub>1</sub></font> <a href="waybel_1.html#R5">are_isomorphic</a> 
 <b>by </b><i><a class="txt" href="waybel15.html#E40"><i><font color="Green">Lemma31</font></i></a></i>;<br/></div>
<b>end;</b></div>

<b>thus </b><a NAME="E2:40"/>
( ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">c<sub>1</sub></font>, <a href="yellow_1.html#K3">BoolePoset</a> <font color="Olive">b<sub>1</sub></font> <a href="waybel_1.html#R5">are_isomorphic</a>  implies <font color="Maroon">c<sub>1</sub></font> is <a href="waybel_8.html#V3">arithmetic</a> )
 <b>by </b><i><a class="txt" href="waybel15.html#E36"><i><font color="Green">Lemma27</font></i></a></i>;<br/>


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