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<b>let </b><font color="Maroon">L</font> be  <a href="lattice3.html#V3">complete</a> <a href="waybel_0.html#NM8">LATTICE</a>;<br/><b>let </b><font color="Maroon">X</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font>, <font color="Maroon">Y</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font>;<br/>



<b>assume </b><a NAME="E1:3"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">Y</font> <a href="yellow_4.html#R4">is_coarser_than</a> <font color="Maroon">X</font>
 ;<br/>

<a NAME="E2:3"/>
 <a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">X</font>,<font color="Maroon">L</font> <a href="lattice3.html#R1">is_&lt;=_than</a> <font color="Maroon">Y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D8">LATTICE3:def 8</a><br/>

<b>assume </b><a NAME="E1:3_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E3:3_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>and </b><br/><a NAME="E4:3_1"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">y</font>
 <b>by </b><i>, <a class="ref" href="yellow_4.html#D2">YELLOW_4:def 2</a></i>;<br/>
<a NAME="E5:3_1"/>
 <a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">X</font>,<font color="Maroon">L</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">x</font>
 
<b>by </b><i>, <a class="ref" href="yellow_2.html#T24">YELLOW_2:24</a></i>;<br/>
<b>hence </b><a NAME="E6:3_1"/>
 <a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">X</font>,<font color="Maroon">L</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">y</font>
 <b>by </b><i>, <a class="ref" href="yellow_0.html#D2">YELLOW_0:def 2</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:3"/>
 <a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">X</font>,<font color="Maroon">L</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">Y</font>,<font color="Maroon">L</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T33">YELLOW_0:33</a></i>;<br/>


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