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<b>let </b><font color="Maroon" title="c1">L</font> be   <a href="waybel_0.html#NM6" title="WAYBEL_0:NM.6">Semilattice</a>;<br/><b>let </b><font color="Maroon" title="c2">I</font> be   <a href="waybel_0.html#NM3" title="WAYBEL_0:NM.3">Ideal</a> of <font color="Maroon" title="c1">L</font>;<br/>



<b>reconsider </b><font color="Maroon" title="c3">mI</font> =  <a href="waybel_4.html#K11" title="WAYBEL_4:func.11">DownMap</a> <font color="Maroon" title="c2">I</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">L</font>,<span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K7" title="WAYBEL_0:func.7">Ids</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span> ;<br/>
<a NAME="E2:80"/>
 ex <font color="Olive" title="b1">s</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">L</font>,<span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K7" title="WAYBEL_0:func.7">Ids</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span> st <br/>( <font color="Maroon" title="c3">mI</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b1">s</font> &amp; <font color="Olive" title="b1">s</font> is <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a> &amp; ( for <font color="Olive" title="b2">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> holds  <font color="Olive" title="b1">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Olive" title="b2">x</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_1"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon" title="c3">mI</font>
;<br/>

<b>thus </b><a NAME="E1:80_1"/>
( <font color="Maroon" title="c3">mI</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c3">mI</font> &amp; <font color="Maroon" title="c3">mI</font> is <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a> &amp; ( for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> holds  <font color="Maroon" title="c3">mI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Olive" title="b1">x</font> ) )
 <b>by </b><i><a class="txt" href="waybel_4.html#E60"><i><font color="Green" title="E45">Lm9</font></i></a>, <a class="txt" href="waybel_4.html#E61"><i><font color="Green" title="E46">Lm10</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:80"/>
 <a href="waybel_4.html#K11" title="WAYBEL_4:func.11">DownMap</a> <font color="Maroon" title="c2">I</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel_4.html#K8" title="WAYBEL_4:func.8">MonSet</a> <font color="Maroon" title="c1">L</font></span>)</span>
 <b>by </b><i><a class="ref" href="waybel_4.html#D14" target="_self" title="WAYBEL_4:def.14">Def14</a></i>;<br/>


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