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

<b>let </b><font color="Maroon" title="c1">S</font>, <font color="Maroon" title="c2">T</font> be  <a href="yellow_0.html#V1" title="YELLOW_0:attr.1">lower-bounded</a> <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a>;<br/><b>let </b><font color="Maroon" title="c3">f</font> be  <a href="waybel34.html#V4" title="WAYBEL34:attr.4">finite-sups-preserving</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">T</font>;<br/><b>let </b><font color="Maroon" title="c4">S'</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="yellow_0.html#V4" title="YELLOW_0:attr.4">full</a> <a href="waybel34.html#V6" title="WAYBEL34:attr.6">finite-sups-inheriting</a>  <a href="yellow_0.html#M1" title="YELLOW_0:mode.1">SubRelStr</a> of <font color="Maroon" title="c1">S</font>;<br/><b>let </b><font color="Maroon" title="c5">T'</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="yellow_0.html#V4" title="YELLOW_0:attr.4">full</a> <a href="waybel34.html#V6" title="WAYBEL34:attr.6">finite-sups-inheriting</a>  <a href="yellow_0.html#M1" title="YELLOW_0:mode.1">SubRelStr</a> of <font color="Maroon" title="c2">T</font>;<br/><b>let </b><font color="Maroon" title="c6">g</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c4">S'</font>,<font color="Maroon" title="c5">T'</font>;<br/>









<b>assume </b><a NAME="E1:109"/><i><font color="Green" title="E67">A1</font></i>: 
<font color="Maroon" title="c6">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">S'</font>
 ;<br/>

<a NAME="E2:109"/><i><font color="Green" title="E68">A2</font></i>: 
<font color="Maroon" title="c6">g</font> is <a href="waybel34.html#V5" title="WAYBEL34:attr.5">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" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c4">S'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">S</font> &amp; the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">T</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 
<b>by </b><i><a class="ref" href="waybel34.html#T64" target="_self" title="WAYBEL34:th.64">Th64</a></i>;<br/>
<b>hence </b><font color="Maroon" title="c6">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c4">S'</font></span>)</span> = 
<font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">S</font></span>)</span>
<b>by </b><i><a class="txt" href="waybel34.html#E1:109"><i><font color="Green" title="E67">A1</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
 <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c2">T</font>
<b>by </b><i><a class="ref" href="waybel34.html#D17" target="_self" title="WAYBEL34:def.17">Def17</a></i>
<br/>.= 
 <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c5">T'</font>
<b>by </b><i><a class="ref" href="waybel34.html#T64" target="_self" title="WAYBEL34:th.64">Th64</a></i>
;<br/> <i><font color="Red">:: according to </font></i><a class="ref" href="waybel34.html#D17" target="_self" title="WAYBEL34:def.17">WAYBEL34:def 17</a><br/>


</div><b>end;</b></div>
<a NAME="E3:109"/>
<font color="Maroon" title="c6">g</font> is <a href="waybel_0.html#V20" title="WAYBEL_0:attr.20">join-preserving</a>
 
<b>by </b><i><a class="txt" href="waybel34.html#E1:109"><i><font color="Green" title="E67">A1</font></i></a>, <a class="ref" href="waybel34.html#T68" target="_self" title="WAYBEL34:th.68">Th68</a></i>;<br/>
<b>hence </b><a NAME="E4:109"/>
<font color="Maroon" title="c6">g</font> is <a href="waybel34.html#V4" title="WAYBEL34:attr.4">finite-sups-preserving</a>
 <b>by </b><i><a class="txt" href="waybel34.html#E2:109"><i><font color="Green" title="E68">A2</font></i></a>, <a class="ref" href="waybel34.html#T66" target="_self" title="WAYBEL34:th.66">Th66</a></i>;<br/>


</div>
