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<b>let </b><font color="Maroon" title="c1">L</font> be  <a href="struct_0.html#V8" title="STRUCT_0:attr.8">finite</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_1.html#V2" title="WAYBEL_1:attr.2">distributive</a> iff  ex <font color="Olive" title="b1">X</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="lattice7.html#M2" title="LATTICE7:mode.2">Ring_of_sets</a>  st <font color="Maroon" title="c1">L</font>, <a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <font color="Olive" title="b1">X</font> <a href="waybel_1.html#R5" title="WAYBEL_1:pred.5">are_isomorphic</a>  )</span><br/>

<b>thus </b><a NAME="E1:32"/>
( <font color="Maroon" title="c1">L</font> is <a href="waybel_1.html#V2" title="WAYBEL_1:attr.2">distributive</a> implies  ex <font color="Olive" title="b1">X</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="lattice7.html#M2" title="LATTICE7:mode.2">Ring_of_sets</a>  st <font color="Maroon" title="c1">L</font>, <a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <font color="Olive" title="b1">X</font> <a href="waybel_1.html#R5" title="WAYBEL_1:pred.5">are_isomorphic</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 non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="lattice7.html#M2" title="LATTICE7:mode.2">Ring_of_sets</a>  st <font color="Maroon" title="c1">L</font>, <a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</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_1.html#V2" title="WAYBEL_1:attr.2">distributive</a> )</span><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" title="E23">A1</font></i>: 
<font color="Maroon" title="c1">L</font> is <a href="waybel_1.html#V2" title="WAYBEL_1:attr.2">distributive</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 non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="lattice7.html#M2" title="LATTICE7:mode.2">Ring_of_sets</a>  st <font color="Maroon" title="c1">L</font>, <a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <font color="Olive" title="b1">X</font> <a href="waybel_1.html#R5" title="WAYBEL_1:pred.5">are_isomorphic</a> </span><br/>

<b>consider </b><font color="Maroon" title="c2">X</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:32_1"/><i><font color="Green" title="E24">A2</font></i>: 
<font color="Maroon" title="c2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice7.html#K4" title="LATTICE7:func.4">LOWER</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2">(<span class="default"><a href="lattice7.html#K3" title="LATTICE7:func.3">Join-IRR</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span>
 ;<br/>
<a NAME="E4:32_1"/><i><font color="Green" title="E25">A3</font></i>: 
<font color="Maroon" title="c2">X</font> is    <a href="lattice7.html#M2" title="LATTICE7:mode.2">Ring_of_sets</a> 
 
<b>by </b><i><a class="txt" href="#E3:32_1"><i><font color="Green" title="E24">A2</font></i></a>, <a class="ref" href="lattice7.html#T15" target="_self" title="LATTICE7:th.15">Th15</a></i>;<br/>
<a NAME="E5:32_1"/>
<font color="Maroon" title="c1">L</font>, <a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <font color="Maroon" title="c2">X</font> <a href="waybel_1.html#R5" title="WAYBEL_1:pred.5">are_isomorphic</a> 
 
<b>by </b><i><a class="txt" href="#E1:32_1"><i><font color="Green" title="E23">A1</font></i></a>, <a class="txt" href="#E3:32_1"><i><font color="Green" title="E24">A2</font></i></a>, <a class="ref" href="lattice7.html#T14" target="_self" title="LATTICE7:th.14">Th14</a></i>;<br/>
<b>hence </b><a NAME="E6:32_1"/>
 ex <font color="Olive" title="b1">X</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="lattice7.html#M2" title="LATTICE7:mode.2">Ring_of_sets</a>  st <font color="Maroon" title="c1">L</font>, <a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</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="#E3:32_1"><i><font color="Green" title="E24">A2</font></i></a>, <a class="txt" href="#E4:32_1"><i><font color="Green" title="E25">A3</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>given </b><font color="Maroon" title="c2">X</font> being  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="lattice7.html#M2" title="LATTICE7:mode.2">Ring_of_sets</a> <b> such that </b><a NAME="E3:32"/><i><font color="Green" title="E23">A4</font></i>: 
<font color="Maroon" title="c1">L</font>, <a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <font color="Maroon" title="c2">X</font> <a href="waybel_1.html#R5" title="WAYBEL_1:pred.5">are_isomorphic</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_1.html#V2" title="WAYBEL_1:attr.2">distributive</a></span><br/>

<b>consider </b><font color="Maroon" title="c3">f</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">L</font>,<span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <font color="Maroon" title="c2">X</font></span>)</span><b> such that </b><br/><a NAME="E5:32"/><i><font color="Green" title="E24">A5</font></i>: 
<font color="Maroon" title="c3">f</font> is <a href="waybel_0.html#V23" title="WAYBEL_0:attr.23">isomorphic</a>
 <b>by </b><i><a class="txt" href="#E3:32"><i><font color="Green" title="E23">A4</font></i></a>, <a class="ref" href="waybel_1.html#D8" title="WAYBEL_1:def.8">WAYBEL_1:def 8</a></i>;<br/>
<a NAME="E6:32"/><i><font color="Green" title="E25">A6</font></i>: 
<font color="Maroon" title="c3">f</font> is <a href="waybel_0.html#V17" title="WAYBEL_0:attr.17">infs-preserving</a>
 
<b>by </b><i><a class="txt" href="#E5:32"><i><font color="Green" title="E24">A5</font></i></a>, <a class="ref" href="waybel13.html#T20" title="WAYBEL13:th.20">WAYBEL13:20</a></i>;<br/>
<a NAME="E7:32"/>
<font color="Maroon" title="c3">f</font> is <a href="waybel_0.html#V18" title="WAYBEL_0:attr.18">sups-preserving</a>
 
<b>by </b><i><a class="txt" href="#E5:32"><i><font color="Green" title="E24">A5</font></i></a>, <a class="ref" href="waybel13.html#T20" title="WAYBEL13:th.20">WAYBEL13:20</a></i>;<br/>
<a NAME="E8:32"/><b>then </b><i><font color="Green" title="E26">A7</font></i>: 
<font color="Maroon" title="c3">f</font> is <a href="waybel_0.html#V20" title="WAYBEL_0:attr.20">join-preserving</a>
 
<b>by </b><i><a class="txt" href="lattice7.html#E25"><i><font color="Green" title="E21">Lm3</font></i></a></i>;<br/>
<a NAME="E9:32"/>
<font color="Maroon" title="c3">f</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 
<b>by </b><i><a class="txt" href="#E5:32"><i><font color="Green" title="E24">A5</font></i></a>, <a class="ref" href="waybel_0.html#T66" title="WAYBEL_0:th.66">WAYBEL_0:66</a></i>;<br/>
<b>hence </b><a NAME="E10:32"/>
<font color="Maroon" title="c1">L</font> is <a href="waybel_1.html#V2" title="WAYBEL_1:attr.2">distributive</a>
 <b>by </b><i><a class="txt" href="#E6:32"><i><font color="Green" title="E25">A6</font></i></a>, <a class="txt" href="#E8:32"><i><font color="Green" title="E26">A7</font></i></a>, <a class="txt" href="lattice7.html#E24"><i><font color="Green" title="E20">Lm2</font></i></a>, <a class="ref" href="waybel_6.html#T3" title="WAYBEL_6:th.3">WAYBEL_6:3</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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