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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="yellow_0.html#V1">lower-bounded</a> <a href="waybel_3.html#V3">continuous</a> <a href="waybel_0.html#NM8">LATTICE</a>;<br/>

<b>thus </b><a NAME="E1:54"/>
( <font color="Maroon">c<sub>1</sub></font> is <a href="waybel_1.html#V2">distributive</a> implies  <a href="waybel_6.html#K4" target="_self">PRIME</a> <font color="Maroon">c<sub>1</sub></font> is <a href="waybel_6.html#V4" target="_self">order-generating</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:54_1"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="waybel_1.html#V2">distributive</a>
 ;<br/>

<a NAME="E2:54_1"/><b>then </b><i><font color="Green">E35</font></i>: 
 <a href="waybel_6.html#K4" target="_self">PRIME</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="waybel_6.html#K3" target="_self">IRR</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="waybel_6.html#T28" target="_self">Th28</a></i>;<br/>
<a NAME="E3:54_1"/><i><font color="Green">E36</font></i>: 
<span class="p1">(<span class="default"><a href="waybel_6.html#K3" target="_self">IRR</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K4">Top</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="waybel_6.html#K3" target="_self">IRR</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T36">XBOOLE_1:36</a></i>;<br/>
<a NAME="E4:54_1"/>
<span class="p1">(<span class="default"><a href="waybel_6.html#K3" target="_self">IRR</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K4">Top</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span> is <a href="waybel_6.html#V4" target="_self">order-generating</a>
 
<b>by </b><i><a class="ref" href="waybel_6.html#T18" target="_self">Th18</a></i>;<br/>
<b>hence </b><a NAME="E5:54_1"/>
 <a href="waybel_6.html#K4" target="_self">PRIME</a> <font color="Maroon">c<sub>1</sub></font> is <a href="waybel_6.html#V4" target="_self">order-generating</a>
 <b>by </b><i><a class="txt" href="waybel_6.html#E2:54_1"><i><font color="Green">E35</font></i></a>, <a class="txt" href="waybel_6.html#E3:54_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="waybel_6.html#T19" target="_self">Th19</a></i>;<br/>


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

<b>thus </b><a NAME="E2:54"/>
(  <a href="waybel_6.html#K4" target="_self">PRIME</a> <font color="Maroon">c<sub>1</sub></font> is <a href="waybel_6.html#V4" target="_self">order-generating</a> implies <font color="Maroon">c<sub>1</sub></font> is <a href="waybel_1.html#V2">distributive</a> )
 <b>by </b><i><a class="ref" href="waybel_6.html#T34" target="_self">Th34</a></i>;<br/>


</div>
