<?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="E1:55"/>
( <font color="Maroon">c<sub>1</sub></font> is <a href="waybel_1.html#V2">distributive</a> implies <font color="Maroon">c<sub>1</sub></font> is <a href="waybel_1.html#V9">Heyting</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:55_1"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="waybel_1.html#V2">distributive</a>
 ;<br/>

<a NAME="E2:55_1"/><b>then </b>
 <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="ref" href="waybel_6.html#T35" target="_self">Th35</a></i>;<br/>
<a NAME="E3:55_1"/><b>then </b>
( <font color="Maroon">c<sub>1</sub></font> is <a href="waybel_1.html#V2">distributive</a> &amp; <font color="Maroon">c<sub>1</sub></font> is <a href="waybel_2.html#V2">meet-continuous</a> )
 
<b>by </b><i><a class="ref" href="waybel_6.html#T34" target="_self">Th34</a></i>;<br/>
<b>hence </b><a NAME="E4:55_1"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="waybel_1.html#V9">Heyting</a>
 <b>by </b><i><a class="ref" href="waybel_2.html#T56">WAYBEL_2:56</a></i>;<br/>


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

<b>thus </b><a NAME="E2:55"/>
( <font color="Maroon">c<sub>1</sub></font> is <a href="waybel_1.html#V9">Heyting</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_1.html#T69">WAYBEL_1:69</a></i>;<br/>


</div>
