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<b>let </b><font color="Maroon">L</font> be  non <a href="struct_0.html#V3">empty</a> <a href="orders_2.html#NM1">Poset</a>;<br/><b>let </b><font color="Maroon">f</font> be   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">L</font>,<font color="Maroon">L</font>;<br/>



<b>assume </b><a NAME="E1:74"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">f</font> is <a href="quantal1.html#V11">idempotent</a>
 ;<br/>

<a NAME="E2:74"/>
for <font color="Olive">s</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_2.html#K1">Image</a> <font color="Maroon">f</font></span>)</span> holds  <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_1.html#K2" target="_self">corestr</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="waybel_1.html#K3" target="_self">inclusion</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">s</font> <a href="hidden.html#R1">=</a> <font color="Olive">s</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">s</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_2.html#K1">Image</a> <font color="Maroon">f</font></span>)</span>;<br/>

<a NAME="E1:74_1"/>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_2.html#K1">Image</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="waybel_1.html#K2" target="_self">corestr</a> <font color="Maroon">f</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D3">FUNCT_2:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">l</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:74_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">l</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>
 <b>and </b><br/><a NAME="E4:74_1"/><i><font color="Green">E45</font></i>: 
<span class="p1">(<span class="default"><a href="waybel_1.html#K2" target="_self">corestr</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">l</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T17">FUNCT_2:17</a></i>;<br/>
<b>reconsider </b><font color="Maroon">l</font> = <font color="Maroon">l</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> <b>by </b><i/>;<br/>
<a NAME="E6:74_1"/><i><font color="Green">E48</font></i>: 
<span class="p1">(<span class="default"><a href="waybel_1.html#K2" target="_self">corestr</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">l</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">l</font>
 
<b>by </b><i><a class="ref" href="waybel_1.html#D5" target="_self">Def5</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_1.html#K2" target="_self">corestr</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="waybel_1.html#K3" target="_self">inclusion</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font> = 
<span class="p1">(<span class="default"><a href="waybel_1.html#K2" target="_self">corestr</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_1.html#K3" target="_self">inclusion</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="waybel_1.html#K2" target="_self">corestr</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>
<b>by </b><i><a class="ref" href="waybel_1.html#D7" target="_self">Def7</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">l</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="waybel_1.html#D5" target="_self">Def5</a></i>
<br/>.= 
<font color="Maroon">s</font>
<b>by </b><i><a class="txt" href="waybel_1.html#E6:74_1"><i><font color="Green">E48</font></i></a>, , , <a class="ref" href="yellow_2.html#T20">YELLOW_2:20</a></i>
;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:74"/>
<span class="p1">(<span class="default"><a href="waybel_1.html#K2" target="_self">corestr</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><a href="waybel_1.html#K3" target="_self">inclusion</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#R2">=</a>  <a href="grcat_1.html#K7">id</a> <span class="p1">(<span class="default"><a href="yellow_2.html#K1">Image</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i><a class="ref" href="tmap_1.html#T92">TMAP_1:92</a></i>;<br/>


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