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<div class="add">

<b>let </b><font color="Maroon">L</font> be  <a href="group_1.html#V6">finite</a> <a href="waybel_0.html#NM8">LATTICE</a>;<br/>

<b>thus </b><a NAME="E1:32"/>
( <font color="Maroon">L</font> is <a href="waybel_1.html#V2">distributive</a> implies  ex <font color="Olive">X</font> being non <a href="xboole_0.html#V1">empty</a>  <a href="lattice7.html#M2" target="_self">Ring_of_sets</a>  st <font color="Maroon">L</font>, <a href="yellow_1.html#K2">InclPoset</a> <font color="Olive">X</font> <a href="waybel_1.html#R5">are_isomorphic</a>  )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:32_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">L</font> is <a href="waybel_1.html#V2">distributive</a>
 ;<br/>

<b>consider </b><font color="Maroon">X</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:32_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="lattice7.html#K4" target="_self">LOWER</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <span class="p2">(<span class="default"><a href="lattice7.html#K3" target="_self">Join-IRR</a> <font color="Maroon">L</font></span>)</span></span>)</span>
 ;<br/>
<a NAME="E4:32_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">X</font> is    <a href="lattice7.html#M2" target="_self">Ring_of_sets</a> 
 
<b>by </b><i>, </i>;<br/>
<a NAME="E5:32_1"/>
<font color="Maroon">L</font>, <a href="yellow_1.html#K2">InclPoset</a> <font color="Maroon">X</font> <a href="waybel_1.html#R5">are_isomorphic</a> 
 
<b>by </b><i>, , </i>;<br/>
<b>hence </b><a NAME="E6:32_1"/>
 ex <font color="Olive">X</font> being non <a href="xboole_0.html#V1">empty</a>  <a href="lattice7.html#M2" target="_self">Ring_of_sets</a>  st <font color="Maroon">L</font>, <a href="yellow_1.html#K2">InclPoset</a> <font color="Olive">X</font> <a href="waybel_1.html#R5">are_isomorphic</a> 
 <b>by </b><i>, </i>;<br/>


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

<b>given </b><font color="Maroon">X</font> being  non <a href="xboole_0.html#V1">empty</a>  <a href="lattice7.html#M2" target="_self">Ring_of_sets</a> <b> such that </b><a NAME="E2:32"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">L</font>, <a href="yellow_1.html#K2">InclPoset</a> <font color="Maroon">X</font> <a href="waybel_1.html#R5">are_isomorphic</a> 
 ;<br/>

<b>consider </b><font color="Maroon">f</font> being   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">L</font>,<span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <font color="Maroon">X</font></span>)</span><b> such that </b><br/><a NAME="E4:32"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">f</font> is <a href="waybel_0.html#V23">isomorphic</a>
 <b>by </b><i>, <a class="ref" href="waybel_1.html#D8">WAYBEL_1:def 8</a></i>;<br/>
<a NAME="E5:32"/>
<font color="Maroon">f</font> is <a href="waybel_0.html#V17">infs-preserving</a>
 
<b>by </b><i>, <a class="ref" href="waybel13.html#T20">WAYBEL13:20</a></i>;<br/>
<a NAME="E6:32"/><b>then </b><i><font color="Green">E48</font></i>: 
<font color="Maroon">f</font> is <a href="waybel_0.html#V19">meet-preserving</a>
 
<b>by </b><i/>;<br/>
<a NAME="E7:32"/>
<font color="Maroon">f</font> is <a href="waybel_0.html#V18">sups-preserving</a>
 
<b>by </b><i>, <a class="ref" href="waybel13.html#T20">WAYBEL13:20</a></i>;<br/>
<a NAME="E8:32"/><b>then </b><i><font color="Green">E49</font></i>: 
<font color="Maroon">f</font> is <a href="waybel_0.html#V20">join-preserving</a>
 
<b>by </b><i/>;<br/>
<a NAME="E9:32"/>
<font color="Maroon">f</font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i>, <a class="ref" href="waybel_0.html#T66">WAYBEL_0:66</a></i>;<br/>
<b>hence </b><a NAME="E10:32"/>
<font color="Maroon">L</font> is <a href="waybel_1.html#V2">distributive</a>
 <b>by </b><i>, , <a class="ref" href="waybel_6.html#T3">WAYBEL_6:3</a></i>;<br/>


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