<?xml version="1.0"?>
<div class="add">

<a NAME="E1:30_1_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">X</font> <a href="cohsp_1.html#R1">includes_lattice_of</a> <font color="Maroon">X</font>
 
<b>by </b><i/>;<br/>
<a NAME="E2:30_1_1"/><i><font color="Green">E37</font></i>: 
 <a href="yellow_1.html#K2">InclPoset</a> <font color="Maroon">X</font> is   <a href="waybel_0.html#NM8">LATTICE</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:30_1_1_1"/><i><font color="Green">E38</font></i>: 
 <a href="yellow_1.html#K2">InclPoset</a> <font color="Maroon">X</font> is <a href="lattice3.html#V2">with_infima</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:30_1_1_1_1"/>
for <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> holds <br/><font color="Olive">x</font> <a href="xboole_0.html#K3">/\</a> <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 
<b>by </b><i>, <a class="ref" href="cohsp_1.html#D8">COHSP_1:def 8</a></i>;<br/>
<b>hence </b><a NAME="E2:30_1_1_1_1"/>
 <a href="yellow_1.html#K2">InclPoset</a> <font color="Maroon">X</font> is <a href="lattice3.html#V2">with_infima</a>
 <b>by </b><i><a class="ref" href="yellow_1.html#T12">YELLOW_1:12</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:30_1_1_1"/>
 <a href="yellow_1.html#K2">InclPoset</a> <font color="Maroon">X</font> is <a href="lattice3.html#V1">with_suprema</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:30_1_1_1_2"/>
for <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> holds <br/><font color="Olive">x</font> <a href="xboole_0.html#K2">\/</a> <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 
<b>by </b><i>, <a class="ref" href="cohsp_1.html#D8">COHSP_1:def 8</a></i>;<br/>
<b>hence </b><a NAME="E2:30_1_1_1_2"/>
 <a href="yellow_1.html#K2">InclPoset</a> <font color="Maroon">X</font> is <a href="lattice3.html#V1">with_suprema</a>
 <b>by </b><i><a class="ref" href="yellow_1.html#T11">YELLOW_1:11</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:30_1_1_1"/>
 <a href="yellow_1.html#K2">InclPoset</a> <font color="Maroon">X</font> is   <a href="waybel_0.html#NM8">LATTICE</a>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E3:30_1_1"/>
 <a href="yellow_1.html#K2">InclPoset</a> <font color="Maroon">X</font> is <a href="waybel_1.html#V2">distributive</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <font color="Maroon">X</font></span>)</span>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <font color="Maroon">X</font></span>)</span>, <font color="Maroon">z</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <font color="Maroon">X</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_1.html#D3">WAYBEL_1:def 3</a><br/>

<a NAME="E1:30_1_1_2"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">x</font> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">y</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">z</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T23">XBOOLE_1:23</a></i>;<br/>
<a NAME="E2:30_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <font color="Maroon">X</font></span>)</span>
 
;<br/>
<a NAME="E3:30_1_1_2"/><b>then </b><i><font color="Green">E41</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="yellow_1.html#T1">YELLOW_1:1</a></i>;<br/>
<a NAME="E4:30_1_1_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <font color="Maroon">X</font></span>)</span>
 
;<br/>
<a NAME="E5:30_1_1_2"/><b>then </b><i><font color="Green">E48</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="yellow_1.html#T1">YELLOW_1:1</a></i>;<br/>
<a NAME="E6:30_1_1_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <font color="Maroon">X</font></span>)</span>
 
;<br/>
<a NAME="E7:30_1_1_2"/><b>then </b><i><font color="Green">E49</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="yellow_1.html#T1">YELLOW_1:1</a></i>;<br/>
<a NAME="E8:30_1_1_2"/><b>then </b><i><font color="Green">E50</font></i>: 
<font color="Maroon">y</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 
<b>by </b><i>, , <a class="ref" href="cohsp_1.html#D8">COHSP_1:def 8</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">r</font> = <font color="Maroon">y</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">z</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <font color="Maroon">X</font></span>)</span> <b>by </b><i><a class="ref" href="yellow_1.html#T1">YELLOW_1:1</a></i>;<br/>
<a NAME="E10:30_1_1_2"/>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <font color="Maroon">X</font></span>)</span>
 
;<br/>
<a NAME="E11:30_1_1_2"/><b>then </b>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="yellow_1.html#T1">YELLOW_1:1</a></i>;<br/>
<a NAME="E12:30_1_1_2"/><b>then </b>
<font color="Maroon">x</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 
<b>by </b><i>, , <a class="ref" href="cohsp_1.html#D8">COHSP_1:def 8</a></i>;<br/>
<a NAME="E13:30_1_1_2"/><b>then </b>
<font color="Maroon">x</font> <a href="lattice3.html#K11">"/\"</a> <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">r</font>
 
<b>by </b><i><a class="ref" href="yellow_1.html#T9">YELLOW_1:9</a></i>;<br/>
<a NAME="E14:30_1_1_2"/><b>then </b><i><font color="Green">E51</font></i>: 
<font color="Maroon">x</font> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="lattice3.html#K11">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="lattice3.html#K10">"\/"</a> <font color="Maroon">z</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="yellow_1.html#T8">YELLOW_1:8</a></i>;<br/>
<a NAME="E15:30_1_1_2"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">x</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 
<b>by </b><i>, , , <a class="ref" href="cohsp_1.html#D8">COHSP_1:def 8</a></i>;<br/>
<a NAME="E16:30_1_1_2"/><b>then </b><i><font color="Green">E53</font></i>: 
<font color="Maroon">x</font> <a href="lattice3.html#K11">"/\"</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">y</font>
 
<b>by </b><i><a class="ref" href="yellow_1.html#T9">YELLOW_1:9</a></i>;<br/>
<a NAME="E17:30_1_1_2"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">x</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 
<b>by </b><i>, , , <a class="ref" href="cohsp_1.html#D8">COHSP_1:def 8</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">r1</font> = <font color="Maroon">x</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">y</font>, <font color="Maroon">r2</font> = <font color="Maroon">x</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">z</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <font color="Maroon">X</font></span>)</span> <b>by </b><i>, <a class="ref" href="yellow_1.html#T1">YELLOW_1:1</a></i>;<br/>
<a NAME="E19:30_1_1_2"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">x</font> <a href="lattice3.html#K11">"/\"</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">z</font>
 
<b>by </b><i>, <a class="ref" href="yellow_1.html#T9">YELLOW_1:9</a></i>;<br/>
<a NAME="E20:30_1_1_2"/>
<font color="Maroon">r2</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <font color="Maroon">X</font></span>)</span>
 
;<br/>
<a NAME="E21:30_1_1_2"/><b>then </b><i><font color="Green">E59</font></i>: 
<font color="Maroon">r2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="yellow_1.html#T1">YELLOW_1:1</a></i>;<br/>
<a NAME="E22:30_1_1_2"/>
<font color="Maroon">r1</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <font color="Maroon">X</font></span>)</span>
 
;<br/>
<a NAME="E23:30_1_1_2"/><b>then </b>
<font color="Maroon">r1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="yellow_1.html#T1">YELLOW_1:1</a></i>;<br/>
<a NAME="E24:30_1_1_2"/><b>then </b>
<font color="Maroon">r1</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">r2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 
<b>by </b><i>, , <a class="ref" href="cohsp_1.html#D8">COHSP_1:def 8</a></i>;<br/>
<b>hence </b><a NAME="E25:30_1_1_2"/>
<font color="Maroon">x</font> <a href="lattice3.html#K11">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="lattice3.html#K10">"\/"</a> <font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="lattice3.html#K11">"/\"</a> <font color="Maroon">y</font></span>)</span> <a href="lattice3.html#K10">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="lattice3.html#K11">"/\"</a> <font color="Maroon">z</font></span>)</span>
 <b>by </b><i>, , , <a class="txt" href="lattice7.html#E1:30_1_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="yellow_1.html#T8">YELLOW_1:8</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:30_1_1"/>
(  <a href="yellow_1.html#K2">InclPoset</a> <font color="Maroon">X</font> is <a href="lattice3.html#V1">with_suprema</a> &amp;  <a href="yellow_1.html#K2">InclPoset</a> <font color="Maroon">X</font> is <a href="lattice3.html#V2">with_infima</a> &amp;  <a href="yellow_1.html#K2">InclPoset</a> <font color="Maroon">X</font> is <a href="waybel_1.html#V2">distributive</a> )
 <b>by </b><i/>;<br/>


</div>
