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

<b>let </b><font color="Maroon" title="c1">L</font> be  <a href="yellow_0.html#V1" title="YELLOW_0:attr.1">lower-bounded</a> <a href="waybel_3.html#V3" title="WAYBEL_3:attr.3">continuous</a> <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">L</font> is <a href="waybel_1.html#V2" title="WAYBEL_1:attr.2">distributive</a> iff  <a href="waybel_6.html#K4" title="WAYBEL_6:func.4">PRIME</a> <font color="Maroon" title="c1">L</font> is <a href="waybel_6.html#V4" title="WAYBEL_6:attr.4">order-generating</a> )</span><br/>

<b>thus </b><a NAME="E1:54"/>
( <font color="Maroon" title="c1">L</font> is <a href="waybel_1.html#V2" title="WAYBEL_1:attr.2">distributive</a> implies  <a href="waybel_6.html#K4" title="WAYBEL_6:func.4">PRIME</a> <font color="Maroon" title="c1">L</font> is <a href="waybel_6.html#V4" title="WAYBEL_6:attr.4">order-generating</a> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="waybel_6.html#K4" title="WAYBEL_6:func.4">PRIME</a> <font color="Maroon" title="c1">L</font> is <a href="waybel_6.html#V4" title="WAYBEL_6:attr.4">order-generating</a> implies <font color="Maroon" title="c1">L</font> is <a href="waybel_1.html#V2" title="WAYBEL_1:attr.2">distributive</a> )</span><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" title="c1">L</font> is <a href="waybel_1.html#V2" title="WAYBEL_1:attr.2">distributive</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="waybel_6.html#K4" title="WAYBEL_6:func.4">PRIME</a> <font color="Maroon" title="c1">L</font> is <a href="waybel_6.html#V4" title="WAYBEL_6:attr.4">order-generating</a></span><br/>

<a NAME="E2:54_1"/><b>then </b><i><font color="Green" title="E35">A1</font></i>: 
 <a href="waybel_6.html#K4" title="WAYBEL_6:func.4">PRIME</a> <font color="Maroon" title="c1">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel_6.html#K3" title="WAYBEL_6:func.3">IRR</a> <font color="Maroon" title="c1">L</font>
 
<b>by </b><i><a class="ref" href="waybel_6.html#T28" target="_self" title="WAYBEL_6:th.28">Th28</a></i>;<br/>
<a NAME="E3:54_1"/>
<span class="p1">(<span class="default"><a href="waybel_6.html#K3" title="WAYBEL_6:func.3">IRR</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K4" title="YELLOW_0:func.4">Top</a> <font color="Maroon" title="c1">L</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="waybel_6.html#K3" title="WAYBEL_6:func.3">IRR</a> <font color="Maroon" title="c1">L</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T36" title="XBOOLE_1:th.36">XBOOLE_1:36</a></i>;<br/>
<b>hence </b><a NAME="E4:54_1"/>
 <a href="waybel_6.html#K4" title="WAYBEL_6:func.4">PRIME</a> <font color="Maroon" title="c1">L</font> is <a href="waybel_6.html#V4" title="WAYBEL_6:attr.4">order-generating</a>
 <b>by </b><i><a class="txt" href="#E2:54_1"><i><font color="Green" title="E35">A1</font></i></a>, <a class="ref" href="waybel_6.html#T18" target="_self" title="WAYBEL_6:th.18">Th18</a>, <a class="ref" href="waybel_6.html#T19" target="_self" title="WAYBEL_6:th.19">Th19</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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

<b>thus </b><a NAME="E2:54"/>
(  <a href="waybel_6.html#K4" title="WAYBEL_6:func.4">PRIME</a> <font color="Maroon" title="c1">L</font> is <a href="waybel_6.html#V4" title="WAYBEL_6:attr.4">order-generating</a> implies <font color="Maroon" title="c1">L</font> is <a href="waybel_1.html#V2" title="WAYBEL_1:attr.2">distributive</a> )
 <b>by </b><i><a class="ref" href="waybel_6.html#T34" target="_self" title="WAYBEL_6:th.34">Th34</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
