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<b>let </b><font color="Maroon" title="c1">L</font> be  <a href="waybel_1.html#V11" title="WAYBEL_1:attr.11">Boolean</a> <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">L</font> is <a href="waybel_8.html#V3" title="WAYBEL_8:attr.3">arithmetic</a> iff  ex <font color="Olive" title="b1">X</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon" title="c1">L</font>, <a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> <font color="Olive" title="b1">X</font> <a href="waybel_1.html#R5" title="WAYBEL_1:pred.5">are_isomorphic</a>  )</span><br/>

<div><b>hereby </b> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  ex <font color="Olive" title="b1">X</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon" title="c1">L</font>, <a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> <font color="Olive" title="b1">X</font> <a href="waybel_1.html#R5" title="WAYBEL_1:pred.5">are_isomorphic</a>  implies <font color="Maroon" title="c1">L</font> is <a href="waybel_8.html#V3" title="WAYBEL_8:attr.3">arithmetic</a> )</span>
<div class="add"><b>assume </b><a NAME="E1:36_1"/>
<font color="Maroon" title="c1">L</font> is <a href="waybel_8.html#V3" title="WAYBEL_8:attr.3">arithmetic</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">X</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon" title="c1">L</font>, <a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> <font color="Olive" title="b1">X</font> <a href="waybel_1.html#R5" title="WAYBEL_1:pred.5">are_isomorphic</a> </span><br/><a NAME="E2:36_1"/><b>then </b>
( <font color="Maroon" title="c1">L</font> is <a href="waybel_3.html#V3" title="WAYBEL_3:attr.3">continuous</a> &amp; <font color="Maroon" title="c1">L</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a>  is <a href="waybel_3.html#V3" title="WAYBEL_3:attr.3">continuous</a> )
 <b>by </b><i><a class="ref" href="waybel15.html#T11" target="_self" title="WAYBEL15:th.11">Th11</a>, <a class="ref" href="yellow_7.html#T38" title="YELLOW_7:th.38">YELLOW_7:38</a></i>;<br/><a NAME="E3:36_1"/><b>then </b>
<font color="Maroon" title="c1">L</font> is <a href="waybel_5.html#V1" title="WAYBEL_5:attr.1">completely-distributive</a>
 <b>by </b><i><a class="ref" href="waybel_6.html#T39" title="WAYBEL_6:th.39">WAYBEL_6:39</a></i>;<br/><a NAME="E4:36_1"/><b>then </b>
( <font color="Maroon" title="c1">L</font> is <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a> &amp; ( for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>  ex <font color="Olive" title="b2">X</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b2">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="waybel15.html#K1" title="WAYBEL15:func.1">ATOM</a> <font color="Maroon" title="c1">L</font> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Olive" title="b2">X</font> ) ) )
 <b>by </b><i><a class="txt" href="waybel15.html#E35"><i><font color="Green" title="E24">Lm6</font></i></a></i>;<br/><b>hence </b><a NAME="E5:36_1"/>
 ex <font color="Olive" title="b1">X</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon" title="c1">L</font>, <a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> <font color="Olive" title="b1">X</font> <a href="waybel_1.html#R5" title="WAYBEL_1:pred.5">are_isomorphic</a> 
 <b>by </b><i><a class="txt" href="waybel15.html#E36"><i><font color="Green" title="E25">Lm7</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div>
<b>end;</b></div>

<b>thus </b><a NAME="E2:36"/>
(  ex <font color="Olive" title="b1">X</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon" title="c1">L</font>, <a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> <font color="Olive" title="b1">X</font> <a href="waybel_1.html#R5" title="WAYBEL_1:pred.5">are_isomorphic</a>  implies <font color="Maroon" title="c1">L</font> is <a href="waybel_8.html#V3" title="WAYBEL_8:attr.3">arithmetic</a> )
 <b>by </b><i><a class="ref" href="waybel15.html#T12" target="_self" title="WAYBEL15:th.12">Th12</a>, <a class="ref" href="waybel_1.html#T7" title="WAYBEL_1:th.7">WAYBEL_1:7</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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