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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="waybel_0.html#NM6">Semilattice</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>reconsider </b><font color="Maroon">c<sub>3</sub></font> =  <a href="waybel_4.html#K11" target="_self">DownMap</a> <font color="Maroon">c<sub>2</sub></font> as   <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K7">Ids</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> ;<br/>
<a NAME="E2:80"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K7">Ids</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> st <br/>( <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#R2">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="orders_3.html#V5">monotone</a> &amp; ( for <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Olive">b<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>2</sub></font> <a href="tarski.html#R1">c=</a>  <a href="waybel_0.html#K5">downarrow</a> <font color="Olive">b<sub>2</sub></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">c<sub>3</sub></font>
;<br/>

<b>thus </b><a NAME="E1:80_1"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> is <a href="orders_3.html#V5">monotone</a> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="waybel_0.html#K5">downarrow</a> <font color="Olive">b<sub>1</sub></font> ) )
 <b>by </b><i><a class="txt" href="waybel_4.html#E61"><i><font color="Green">Lemma46</font></i></a>, <a class="txt" href="waybel_4.html#E62"><i><font color="Green">Lemma47</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:80"/>
 <a href="waybel_4.html#K11" target="_self">DownMap</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel_4.html#K8" target="_self">MonSet</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="waybel_4.html#D14" target="_self">Def14</a></i>;<br/>


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