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<b>let </b><font color="Maroon" title="c1">L</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="orders_2.html#NM1" title="ORDERS_2:NM.1">Poset</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">f</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c1">L</font>  st <font color="Olive" title="b1">f</font> <a href="waybel_1.html#NR11" title="WAYBEL_1:NR.11">is_a_closure_operator</a>  holds <br/><span class="p1"><a href="waybel_1.html#K1" title="WAYBEL_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><a href="waybel_1.html#K3" title="WAYBEL_1:func.3">inclusion</a> <font color="Olive" title="b1">f</font></span>)</span>,<span class="p2">(<span class="default"><a href="waybel_1.html#K2" title="WAYBEL_1:func.2">corestr</a> <font color="Olive" title="b1">f</font></span>)</span></span><a href="waybel_1.html#K1" title="WAYBEL_1:func.1">]</a></span> is <a href="waybel_1.html#V3" title="WAYBEL_1:attr.3">Galois</a></span><br/><b>let </b><font color="Maroon" title="c2">f</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">L</font>,<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"> ( <font color="Maroon" title="c2">f</font> <a href="waybel_1.html#NR11" title="WAYBEL_1:NR.11">is_a_closure_operator</a>  implies <span class="p1"><a href="waybel_1.html#K1" title="WAYBEL_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><a href="waybel_1.html#K3" title="WAYBEL_1:func.3">inclusion</a> <font color="Maroon" title="c2">f</font></span>)</span>,<span class="p2">(<span class="default"><a href="waybel_1.html#K2" title="WAYBEL_1:func.2">corestr</a> <font color="Maroon" title="c2">f</font></span>)</span></span><a href="waybel_1.html#K1" title="WAYBEL_1:func.1">]</a></span> is <a href="waybel_1.html#V3" title="WAYBEL_1:attr.3">Galois</a> )</span><br/>



<b>assume </b><b>that </b><br/><a NAME="E1:77"/><i><font color="Green" title="E48">A1</font></i>: 
( <font color="Maroon" title="c2">f</font> is <a href="quantal1.html#V11" title="QUANTAL1:attr.11">idempotent</a> &amp; <font color="Maroon" title="c2">f</font> is <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a> )
 <b>and </b><br/><a NAME="E2:77"/><i><font color="Green" title="E49">A2</font></i>: 
 <a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c1">L</font> <a href="yellow_2.html#R1" title="YELLOW_2:pred.1">&lt;=</a> <font color="Maroon" title="c2">f</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_1.html#D13" target="_self" title="WAYBEL_1:def.13">WAYBEL_1:def 13</a>,<a class="ref" href="waybel_1.html#D14" target="_self" title="WAYBEL_1:def.14">WAYBEL_1:def 14</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1"><a href="waybel_1.html#K1" title="WAYBEL_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><a href="waybel_1.html#K3" title="WAYBEL_1:func.3">inclusion</a> <font color="Maroon" title="c2">f</font></span>)</span>,<span class="p2">(<span class="default"><a href="waybel_1.html#K2" title="WAYBEL_1:func.2">corestr</a> <font color="Maroon" title="c2">f</font></span>)</span></span><a href="waybel_1.html#K1" title="WAYBEL_1:func.1">]</a></span> is <a href="waybel_1.html#V3" title="WAYBEL_1:attr.3">Galois</a></span><br/>

<b>set </b><font color="Maroon" title="c3">g</font> =  <a href="waybel_1.html#K2" title="WAYBEL_1:func.2">corestr</a> <font color="Maroon" title="c2">f</font>;<br/>
<b>set </b><font color="Maroon" title="c4">d</font> =  <a href="waybel_1.html#K3" title="WAYBEL_1:func.3">inclusion</a> <font color="Maroon" title="c2">f</font>;<br/>
<a NAME="E3:77"/><i><font color="Green" title="E50">A3</font></i>: 
 <a href="waybel_1.html#K2" title="WAYBEL_1:func.2">corestr</a> <font color="Maroon" title="c2">f</font> is <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a>
 
<b>by </b><i><a class="txt" href="waybel_1.html#E1:77"><i><font color="Green" title="E48">A1</font></i></a>, <a class="ref" href="waybel_1.html#T33" target="_self" title="WAYBEL_1:th.33">Th33</a></i>;<br/>
<a NAME="E4:77"/>
<span class="p1">(<span class="default"><a href="waybel_1.html#K2" title="WAYBEL_1:func.2">corestr</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="waybel_1.html#K3" title="WAYBEL_1:func.3">inclusion</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <span class="p1">(<span class="default"><a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <font color="Maroon" title="c2">f</font></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel_1.html#E1:77"><i><font color="Green" title="E48">A1</font></i></a>, <a class="ref" href="waybel_1.html#T36" target="_self" title="WAYBEL_1:th.36">Th36</a></i>;<br/>
<a NAME="E5:77"/><b>then </b><i><font color="Green" title="E51">A4</font></i>: 
<span class="p1">(<span class="default"><a href="waybel_1.html#K2" title="WAYBEL_1:func.2">corestr</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="waybel_1.html#K3" title="WAYBEL_1:func.3">inclusion</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="yellow_2.html#R1" title="YELLOW_2:pred.1">&lt;=</a>  <a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <span class="p1">(<span class="default"><a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <font color="Maroon" title="c2">f</font></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel_1.html#E44"><i><font color="Green" title="E36">Lm1</font></i></a></i>;<br/>
<a NAME="E6:77"/>
 <a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c1">L</font> <a href="yellow_2.html#R1" title="YELLOW_2:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="waybel_1.html#K3" title="WAYBEL_1:func.3">inclusion</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="waybel_1.html#K2" title="WAYBEL_1:func.2">corestr</a> <font color="Maroon" title="c2">f</font></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel_1.html#E2:77"><i><font color="Green" title="E49">A2</font></i></a>, <a class="ref" href="waybel_1.html#T35" target="_self" title="WAYBEL_1:th.35">Th35</a></i>;<br/>
<b>hence </b><a NAME="E7:77"/>
<span class="p1"><a href="waybel_1.html#K1" title="WAYBEL_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><a href="waybel_1.html#K3" title="WAYBEL_1:func.3">inclusion</a> <font color="Maroon" title="c2">f</font></span>)</span>,<span class="p2">(<span class="default"><a href="waybel_1.html#K2" title="WAYBEL_1:func.2">corestr</a> <font color="Maroon" title="c2">f</font></span>)</span></span><a href="waybel_1.html#K1" title="WAYBEL_1:func.1">]</a></span> is <a href="waybel_1.html#V3" title="WAYBEL_1:attr.3">Galois</a>
 <b>by </b><i><a class="txt" href="waybel_1.html#E3:77"><i><font color="Green" title="E50">A3</font></i></a>, <a class="txt" href="waybel_1.html#E5:77"><i><font color="Green" title="E51">A4</font></i></a>, <a class="ref" href="waybel_1.html#T20" target="_self" title="WAYBEL_1:th.20">Th20</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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