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

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be  <a href="yellow_0.html#V1">lower-bounded</a> <a href="waybel_0.html#NM8">LATTICE</a>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be  <a href="waybel34.html#V4" target="_self">finite-sups-preserving</a> <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be  non <a href="struct_0.html#V3">empty</a> <a href="yellow_0.html#V4">full</a> <a href="waybel34.html#V6" target="_self">finite-sups-inheriting</a>  <a href="yellow_0.html#M1">SubRelStr</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be  non <a href="struct_0.html#V3">empty</a> <a href="yellow_0.html#V4">full</a> <a href="waybel34.html#V6" target="_self">finite-sups-inheriting</a>  <a href="yellow_0.html#M1">SubRelStr</a> of <font color="Maroon">c<sub>2</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>;<br/>









<b>assume </b><a NAME="E1:109"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<a NAME="E2:109"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">c<sub>6</sub></font> is <a href="waybel34.html#V5" target="_self">bottom-preserving</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="109_1"><b>proof </b></a><div class="add">

<a NAME="E1:109_1"/>
(  <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">c<sub>1</sub></font> &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 
<b>by </b><i><a class="ref" href="waybel34.html#T64" target="_self">Th64</a></i>;<br/>
<b>hence </b><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> = 
<font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="waybel34.html#E1:109"><i><font color="Green">E68</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
 <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="ref" href="waybel34.html#D17" target="_self">Def17</a></i>
<br/>.= 
 <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="ref" href="waybel34.html#T64" target="_self">Th64</a></i>
;<br/> <i><font color="Red">:: according to </font></i><a class="ref" href="waybel34.html#D17" target="_self">WAYBEL34:def 17</a><br/>


</div><b>end;</b></div>
<a NAME="E3:109"/>
<font color="Maroon">c<sub>6</sub></font> is <a href="waybel_0.html#V20">join-preserving</a>
 
<b>by </b><i><a class="txt" href="waybel34.html#E1:109"><i><font color="Green">E68</font></i></a>, <a class="ref" href="waybel34.html#T68" target="_self">Th68</a></i>;<br/>
<b>hence </b><a NAME="E4:109"/>
<font color="Maroon">c<sub>6</sub></font> is <a href="waybel34.html#V4" target="_self">finite-sups-preserving</a>
 <b>by </b><i><a class="txt" href="waybel34.html#E2:109"><i><font color="Green">E69</font></i></a>, <a class="ref" href="waybel34.html#T66" target="_self">Th66</a></i>;<br/>


</div>
