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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="lattices.html#NM1">Lattice</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E2:37"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>
<a NAME="E3:37"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="txt" href="lattice6.html#E2:37"><i><font color="Green">E27</font></i></a>, <a class="ref" href="lattices.html#T23">LATTICES:23</a></i>;<br/>
<a NAME="E4:37"/><b>then </b><i><font color="Green">E28</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="lattice3.html#K4">%</a>  <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K4">%</a>  &amp; <font color="Maroon">c<sub>4</sub></font> <a href="lattice3.html#K4">%</a>  <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> <a href="lattice3.html#K4">%</a>  )
 
<b>by </b><i><a class="ref" href="lattice3.html#T7">LATTICE3:7</a></i>;<br/>
<a NAME="E5:37"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> st <font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K4">%</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> <a href="lattice3.html#K4">%</a>  holds <br/><font color="Maroon">c<sub>4</sub></font> <a href="lattice3.html#K4">%</a>  <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/>

<b>assume </b><a NAME="E1:37_1"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K4">%</a>  &amp; <font color="Maroon">c<sub>5</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> <a href="lattice3.html#K4">%</a>  )
 ;<br/>

<i><font color="Green">E30</font></i>: <span class="p1">(<span class="default"><a href="lattice3.html#K5">%</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="lattice3.html#K4">%</a>  = 
 <a href="lattice3.html#K5">%</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="ref" href="lattice3.html#D3">LATTICE3:def 3</a></i>
<br/>.= 
<font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="ref" href="lattice3.html#D4">LATTICE3:def 4</a></i>
;<br/>
<a NAME="E3:37_1"/><b>then </b><i><font color="Green">E31</font></i>: 
(  <a href="lattice3.html#K5">%</a> <font color="Maroon">c<sub>5</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="lattice3.html#K5">%</a> <font color="Maroon">c<sub>5</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="txt" href="lattice6.html#E1:37_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="lattice3.html#T7">LATTICE3:7</a></i>;<br/>
<span class="p1">(<span class="default"><a href="lattice3.html#K5">%</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>4</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice3.html#K5">%</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="lattice6.html#E2:37"><i><font color="Green">E27</font></i></a>, <a class="ref" href="lattices.html#D7">LATTICES:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="lattice3.html#K5">%</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="lattice6.html#E3:37_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="lattices.html#T21">LATTICES:21</a></i>
<br/>.= 
 <a href="lattice3.html#K5">%</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="txt" href="lattice6.html#E3:37_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="lattices.html#T21">LATTICES:21</a></i>
;<br/>
<a NAME="E5:37_1"/><b>then </b>
 <a href="lattice3.html#K5">%</a> <font color="Maroon">c<sub>5</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="lattices.html#T21">LATTICES:21</a></i>;<br/>
<b>hence </b><a NAME="E6:37_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="lattice3.html#K4">%</a>  <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="lattice6.html#E2:37_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="lattice3.html#T7">LATTICE3:7</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:37"/><b>then </b><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="lattice3.html#K4">%</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K4">%</a> </span>)</span> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="lattice3.html#K4">%</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="lattice6.html#E4:37"><i><font color="Green">E28</font></i></a>, <a class="ref" href="yellow_0.html#T23">YELLOW_0:23</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E8:37"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>
<a NAME="E9:37"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>5</sub></font> )
 
<b>by </b><i><a class="txt" href="lattice6.html#E8:37"><i><font color="Green">E30</font></i></a>, <a class="ref" href="lattices.html#T22">LATTICES:22</a></i>;<br/>
<a NAME="E10:37"/><b>then </b><i><font color="Green">E31</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K4">%</a>  <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> <a href="lattice3.html#K4">%</a>  &amp; <font color="Maroon">c<sub>3</sub></font> <a href="lattice3.html#K4">%</a>  <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> <a href="lattice3.html#K4">%</a>  )
 
<b>by </b><i><a class="ref" href="lattice3.html#T7">LATTICE3:7</a></i>;<br/>
<a NAME="E11:37"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> st <font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K4">%</a>  <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="lattice3.html#K4">%</a>  <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>5</sub></font> <a href="lattice3.html#K4">%</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/>

<b>assume </b><a NAME="E1:37_2"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K4">%</a>  <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="lattice3.html#K4">%</a>  <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> )
 ;<br/>

<i><font color="Green">E33</font></i>: <span class="p1">(<span class="default"><a href="lattice3.html#K5">%</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="lattice3.html#K4">%</a>  = 
 <a href="lattice3.html#K5">%</a> <font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="ref" href="lattice3.html#D3">LATTICE3:def 3</a></i>
<br/>.= 
<font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="ref" href="lattice3.html#D4">LATTICE3:def 4</a></i>
;<br/>
<a NAME="E3:37_2"/><b>then </b><i><font color="Green">E34</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="lattices.html#R3">[=</a>  <a href="lattice3.html#K5">%</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="lattices.html#R3">[=</a>  <a href="lattice3.html#K5">%</a> <font color="Maroon">c<sub>6</sub></font> )
 
<b>by </b><i><a class="txt" href="lattice6.html#E1:37_2"><i><font color="Green">E32</font></i></a>, <a class="ref" href="lattice3.html#T7">LATTICE3:7</a></i>;<br/>
<span class="p1">(<span class="default"><a href="lattice3.html#K5">%</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c<sub>5</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice3.html#K5">%</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="lattice6.html#E8:37"><i><font color="Green">E30</font></i></a>, <a class="ref" href="lattices.html#D5">LATTICES:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="lattice3.html#K5">%</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="lattice6.html#E3:37_2"><i><font color="Green">E34</font></i></a>, <a class="ref" href="lattices.html#D3">LATTICES:def 3</a></i>
<br/>.= 
 <a href="lattice3.html#K5">%</a> <font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="txt" href="lattice6.html#E3:37_2"><i><font color="Green">E34</font></i></a>, <a class="ref" href="lattices.html#D3">LATTICES:def 3</a></i>
;<br/>
<a NAME="E5:37_2"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="lattices.html#R3">[=</a>  <a href="lattice3.html#K5">%</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="ref" href="lattices.html#D3">LATTICES:def 3</a></i>;<br/>
<b>hence </b><a NAME="E6:37_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>5</sub></font> <a href="lattice3.html#K4">%</a> 
 <b>by </b><i><a class="txt" href="lattice6.html#E2:37_2"><i><font color="Green">E33</font></i></a>, <a class="ref" href="lattice3.html#T7">LATTICE3:7</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E12:37"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="lattice3.html#K4">%</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K4">%</a> </span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="lattice3.html#K4">%</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="lattice6.html#E10:37"><i><font color="Green">E31</font></i></a>, <a class="ref" href="yellow_0.html#T22">YELLOW_0:22</a></i>;<br/>
<b>hence </b><a NAME="E13:37"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K4">%</a> </span>)</span> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="lattice3.html#K4">%</a> </span>)</span> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K4">%</a> </span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="lattice3.html#K4">%</a> </span>)</span> )
 <b>by </b><i><a class="txt" href="lattice6.html#E2:37"><i><font color="Green">E27</font></i></a>, <a class="txt" href="lattice6.html#E6:37"><i><font color="Green">E29</font></i></a>, <a class="txt" href="lattice6.html#E8:37"><i><font color="Green">E30</font></i></a>, <a class="ref" href="lattice3.html#D3">LATTICE3:def 3</a></i>;<br/>


</div>
