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

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:14_1_1_1"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">L</font> is <a href="lattice3.html#V3">complete</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>;<br/><b>set </b><font color="Maroon">Y</font> = <span class="p1">{<span class="default"> <font color="Olive">c</font> where <font color="Olive">c</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : <font color="Olive">c</font> <a href="lattice3.html#R1">is_&lt;=_than</a> <font color="Maroon">X</font> </span>}</span> ;<br/><b>consider </b><font color="Maroon">p</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E3:14_1_1_1"/><i><font color="Green">E37</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive">c</font> where <font color="Olive">c</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : <font color="Olive">c</font> <a href="lattice3.html#R1">is_&lt;=_than</a> <font color="Maroon">X</font> </span>}</span>  <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Maroon">p</font>
 <b>and </b><br/><a NAME="E4:14_1_1_1"/><i><font color="Green">E39</font></i>: 
for <font color="Olive">r</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>  st <span class="p1">{<span class="default"> <font color="Olive">c</font> where <font color="Olive">c</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : <font color="Olive">c</font> <a href="lattice3.html#R1">is_&lt;=_than</a> <font color="Maroon">X</font> </span>}</span>  <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Olive">r</font> holds <br/><font color="Maroon">p</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">r</font>
 <b>by </b><i>, <a class="ref" href="lattice3.html#D12">LATTICE3:def 12</a></i>;<br/><b>take </b><font color="Maroon">p</font> = <font color="Maroon">p</font>;<br/><b>thus </b><a NAME="E5:14_1_1_1"/>
<font color="Maroon">p</font> <a href="lattice3.html#R1">is_&lt;=_than</a> <font color="Maroon">X</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">q</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="E1:14_1_1_1_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<a NAME="E2:14_1_1_1_1"/>
<span class="p1">{<span class="default"> <font color="Olive">c</font> where <font color="Olive">c</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : <font color="Olive">c</font> <a href="lattice3.html#R1">is_&lt;=_than</a> <font color="Maroon">X</font> </span>}</span>  <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Maroon">q</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">s</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#D9">LATTICE3:def 9</a><br/>

<b>assume </b><a NAME="E1:14_1_1_1_1_1"/>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">c</font> where <font color="Olive">c</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : <font color="Olive">c</font> <a href="lattice3.html#R1">is_&lt;=_than</a> <font color="Maroon">X</font> </span>}</span> 
 ;<br/>

<a NAME="E2:14_1_1_1_1_1"/><b>then </b>
 ex <font color="Olive">t</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <br/>( <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Olive">t</font> &amp; <font color="Olive">t</font> <a href="lattice3.html#R1">is_&lt;=_than</a> <font color="Maroon">X</font> )
 
;<br/>
<b>hence </b><a NAME="E3:14_1_1_1_1_1"/>
<font color="Maroon">s</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">q</font>
 <b>by </b><i>, <a class="ref" href="lattice3.html#D8">LATTICE3:def 8</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:14_1_1_1_1"/>
<font color="Maroon">p</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">q</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><b>let </b><font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/><b>assume </b><a NAME="E6:14_1_1_1"/>
<font color="Maroon">b</font> <a href="lattice3.html#R1">is_&lt;=_than</a> <font color="Maroon">X</font>
 ;<br/><a NAME="E7:14_1_1_1"/><b>then </b>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">c</font> where <font color="Olive">c</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : <font color="Olive">c</font> <a href="lattice3.html#R1">is_&lt;=_than</a> <font color="Maroon">X</font> </span>}</span> 
 ;<br/><b>hence </b><a NAME="E8:14_1_1_1"/>
<font color="Maroon">b</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">p</font>
 <b>by </b><i>, <a class="ref" href="lattice3.html#D9">LATTICE3:def 9</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:14_1_1"/><i><font color="Green">E47</font></i>: 
for <font color="Olive">X</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font>  ex <font color="Olive">a</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <br/>( <font color="Olive">a</font> <a href="lattice3.html#R1">is_&lt;=_than</a> <font color="Olive">X</font> &amp; ( for <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>  st <font color="Olive">b</font> <a href="lattice3.html#R1">is_&lt;=_than</a> <font color="Olive">X</font> holds <br/><font color="Olive">b</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">a</font> ) )
 ;<br/>

<b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D12">LATTICE3:def 12</a><br/>

<b>set </b><font color="Maroon">Y</font> = <span class="p1">{<span class="default"> <font color="Olive">c</font> where <font color="Olive">c</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : <font color="Maroon">X</font> <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Olive">c</font> </span>}</span> ;<br/>
<a NAME="E3:14_1_1"/>
<span class="p1">{<span class="default"> <font color="Olive">c</font> where <font color="Olive">c</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : <font color="Maroon">X</font> <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Olive">c</font> </span>}</span>  <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:14_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">c</font> where <font color="Olive">c</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : <font color="Maroon">X</font> <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Olive">c</font> </span>}</span> 
 ;<br/>

<a NAME="E2:14_1_1_2"/><b>then </b>
 ex <font color="Olive">c</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">c</font> &amp; <font color="Maroon">X</font> <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Olive">c</font> )
 
;<br/>
<b>hence </b><a NAME="E3:14_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 <font color="Maroon">L</font>
 ;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon">p</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E5:14_1_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">p</font> <a href="lattice3.html#R1">is_&lt;=_than</a> <span class="p1">{<span class="default"> <font color="Olive">c</font> where <font color="Olive">c</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : <font color="Maroon">X</font> <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Olive">c</font> </span>}</span> 
 <b>and </b><br/><a NAME="E6:14_1_1"/><i><font color="Green">E50</font></i>: 
for <font color="Olive">r</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>  st <font color="Olive">r</font> <a href="lattice3.html#R1">is_&lt;=_than</a> <span class="p1">{<span class="default"> <font color="Olive">c</font> where <font color="Olive">c</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : <font color="Maroon">X</font> <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Olive">c</font> </span>}</span>  holds <br/><font color="Olive">r</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">p</font>
 <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">p</font>
;<br/>

<b>thus </b><a NAME="E7:14_1_1"/>
<font color="Maroon">X</font> <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Maroon">p</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">q</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#D9">LATTICE3:def 9</a><br/>

<b>assume </b><a NAME="E1:14_1_1_3"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<a NAME="E2:14_1_1_3"/>
<font color="Maroon">q</font> <a href="lattice3.html#R1">is_&lt;=_than</a> <span class="p1">{<span class="default"> <font color="Olive">c</font> where <font color="Olive">c</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : <font color="Maroon">X</font> <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Olive">c</font> </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_1_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">s</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="E1:14_1_1_3_1"/>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">c</font> where <font color="Olive">c</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : <font color="Maroon">X</font> <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Olive">c</font> </span>}</span> 
 ;<br/>

<a NAME="E2:14_1_1_3_1"/><b>then </b>
 ex <font color="Olive">t</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <br/>( <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Olive">t</font> &amp; <font color="Maroon">X</font> <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Olive">t</font> )
 
;<br/>
<b>hence </b><a NAME="E3:14_1_1_3_1"/>
<font color="Maroon">q</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">s</font>
 <b>by </b><i>, <a class="ref" href="lattice3.html#D9">LATTICE3:def 9</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:14_1_1_3"/>
<font color="Maroon">q</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">p</font>
 <b>by </b><i/>;<br/>


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

<b>let </b><font color="Maroon">r</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/>

<b>assume </b><a NAME="E8:14_1_1"/>
<font color="Maroon">X</font> <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Maroon">r</font>
 ;<br/>

<a NAME="E9:14_1_1"/><b>then </b>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">c</font> where <font color="Olive">c</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : <font color="Maroon">X</font> <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Olive">c</font> </span>}</span> 
 
;<br/>
<b>hence </b><a NAME="E10:14_1_1"/>
<font color="Maroon">p</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">r</font>
 <b>by </b><i>, <a class="ref" href="lattice3.html#D8">LATTICE3:def 8</a></i>;<br/>


</div>
