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

<b>let </b><font color="Maroon">A</font> be  non <a href="struct_0.html#V3">empty</a> <a href="orders_2.html#NM1">Poset</a>;<br/><b>let </b><font color="Maroon">c</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font>;<br/><b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font>;<br/>





<b>thus </b><a NAME="E1:40"/>
( <font color="Maroon">c</font> <a href="orders_2.html#R2" target="_self">&lt;</a> <font color="Maroon">a</font> implies <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="orders_2.html#K1" target="_self">UpperCone</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c</font></span><a href="domain_1.html#K6">}</a></span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:40_1"/>
<font color="Maroon">c</font> <a href="orders_2.html#R2" target="_self">&lt;</a> <font color="Maroon">a</font>
 ;<br/>

<a NAME="E2:40_1"/><b>then </b>
for <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font>  st <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c</font></span><a href="domain_1.html#K6">}</a></span> holds <br/><font color="Olive">b</font> <a href="orders_2.html#R2" target="_self">&lt;</a> <font color="Maroon">a</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E3:40_1"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">a1</font> where <font color="Olive">a1</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font> : for <font color="Olive">a2</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font>  st <font color="Olive">a2</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c</font></span><a href="domain_1.html#K6">}</a></span> holds <br/><font color="Olive">a2</font> <a href="orders_2.html#R2" target="_self">&lt;</a> <font color="Olive">a1</font> </span>}</span> 
 
;<br/>
<b>hence </b><a NAME="E4:40_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="orders_2.html#K1" target="_self">UpperCone</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c</font></span><a href="domain_1.html#K6">}</a></span>
 ;<br/>


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

<b>assume </b><a NAME="E2:40"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="orders_2.html#K1" target="_self">UpperCone</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c</font></span><a href="domain_1.html#K6">}</a></span>
 ;<br/>

<a NAME="E3:40"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">a1</font> where <font color="Olive">a1</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font> : for <font color="Olive">a2</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font>  st <font color="Olive">a2</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c</font></span><a href="domain_1.html#K6">}</a></span> holds <br/><font color="Olive">a2</font> <a href="orders_2.html#R2" target="_self">&lt;</a> <font color="Olive">a1</font> </span>}</span> 
 
;<br/>
<a NAME="E4:40"/><b>then </b><i><font color="Green">E18</font></i>: 
 ex <font color="Olive">a1</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font> st <br/>( <font color="Olive">a1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; ( for <font color="Olive">a2</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font>  st <font color="Olive">a2</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c</font></span><a href="domain_1.html#K6">}</a></span> holds <br/><font color="Olive">a2</font> <a href="orders_2.html#R2" target="_self">&lt;</a> <font color="Olive">a1</font> ) )
 
;<br/>
<a NAME="E5:40"/>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c</font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E6:40"/>
<font color="Maroon">c</font> <a href="orders_2.html#R2" target="_self">&lt;</a> <font color="Maroon">a</font>
 <b>by </b><i/>;<br/>


</div>
