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<b>let </b><font color="Maroon" title="c1">L</font> be  <a href="yellow_0.html#V1" title="YELLOW_0:attr.1">lower-bounded</a> <a href="waybel_0.html#NM7" title="WAYBEL_0:NM.7">sup-Semilattice</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="yellow_2.html#K3" title="YELLOW_2:func.3">IdsMap</a> <font color="Maroon" title="c1">L</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></span><br/>

<b>set </b><font color="Maroon" title="c2">s</font> =  <a href="yellow_2.html#K3" title="YELLOW_2:func.3">IdsMap</a> <font color="Maroon" title="c1">L</font>;<br/>
<a NAME="E1:50"/>
 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/>(  <a href="yellow_2.html#K3" title="YELLOW_2:func.3">IdsMap</a> <font color="Maroon" title="c1">L</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="50_1"><b>proof </b></a><div class="add">

<b>take </b>
 <a href="yellow_2.html#K3" title="YELLOW_2:func.3">IdsMap</a> <font color="Maroon" title="c1">L</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="yellow_2.html#K3" title="YELLOW_2:func.3">IdsMap</a> <font color="Maroon" title="c1">L</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="yellow_2.html#K3" title="YELLOW_2:func.3">IdsMap</a> <font color="Maroon" title="c1">L</font> &amp;  <a href="yellow_2.html#K3" title="YELLOW_2:func.3">IdsMap</a> <font color="Maroon" title="c1">L</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  <span class="p1">(<span class="default"><a href="yellow_2.html#K3" title="YELLOW_2:func.3">IdsMap</a> <font color="Maroon" title="c1">L</font></span>)</span> <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> ) )</span><br/>

<b>thus </b><a NAME="E1:50_1"/>
(  <a href="yellow_2.html#K3" title="YELLOW_2:func.3">IdsMap</a> <font color="Maroon" title="c1">L</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="yellow_2.html#K3" title="YELLOW_2:func.3">IdsMap</a> <font color="Maroon" title="c1">L</font> &amp;  <a href="yellow_2.html#K3" title="YELLOW_2:func.3">IdsMap</a> <font color="Maroon" title="c1">L</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  <span class="p1">(<span class="default"><a href="yellow_2.html#K3" title="YELLOW_2:func.3">IdsMap</a> <font color="Maroon" title="c1">L</font></span>)</span> <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="ref" href="yellow_2.html#D4" title="YELLOW_2:def.4">YELLOW_2:def 4</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:50"/>
 <a href="yellow_2.html#K3" title="YELLOW_2:func.3">IdsMap</a> <font color="Maroon" title="c1">L</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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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