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<b>let </b><font color="Maroon">c<sub>1</sub></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">c<sub>2</sub></font> be   <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font>;<br/>



<b>assume </b><a NAME="E1:96"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="waybel_1.html#NR11" target="_self">is_a_closure_operator</a> 
 ;<br/>

<a NAME="E2:96"/><b>then </b>
<span class="p1"><a href="waybel_1.html#K1" target="_self">[</a><span class="default"><span class="p2">(<span class="default"><a href="waybel_1.html#K3" target="_self">inclusion</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="waybel_1.html#K2" target="_self">corestr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="waybel_1.html#K1" target="_self">]</a></span> is <a href="waybel_1.html#V3" target="_self">Galois</a>
 
<b>by </b><i><a class="ref" href="waybel_1.html#T39" target="_self">Th39</a></i>;<br/>
<a NAME="E3:96"/><b>then </b>
 <a href="waybel_1.html#K2" target="_self">corestr</a> <font color="Maroon">c<sub>2</sub></font> is <a href="waybel_1.html#V5" target="_self">lower_adjoint</a>
 
<b>by </b><i><a class="ref" href="waybel_1.html#D12" target="_self">Def12</a></i>;<br/>
<b>hence </b><a NAME="E4:96"/><i><font color="Green">E60</font></i>: 
 <a href="waybel_1.html#K2" target="_self">corestr</a> <font color="Maroon">c<sub>2</sub></font> is <a href="waybel_0.html#V18">sups-preserving</a>
 <b>by </b><i><a class="ref" href="waybel_1.html#T14" target="_self">Th14</a></i>;<br/>

<b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E5:96"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> 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">c<sub>2</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E6:96"/><i><font color="Green">E62</font></i>: 
 <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<a NAME="E7:96"/><i><font color="Green">E63</font></i>: 
 <a href="waybel_1.html#K2" target="_self">corestr</a> <font color="Maroon">c<sub>2</sub></font> <a href="waybel_0.html#R4">preserves_sup_of</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="waybel_1.html#E4:96"><i><font color="Green">E60</font></i></a>, <a class="ref" href="waybel_0.html#D33">WAYBEL_0:def 33</a></i>;<br/>
<a NAME="E8:96"/><i><font color="Green">E64</font></i>: 
 <a href="waybel_1.html#K2" target="_self">corestr</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="waybel_1.html#T32" target="_self">Th32</a></i>;<br/>
<a NAME="E9:96"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="waybel_1.html#E5:96"><i><font color="Green">E61</font></i></a>, <a class="ref" href="yellow_0.html#D15">YELLOW_0:def 15</a></i>;<br/>
<a NAME="E10:96"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="waybel_1.html#V6" target="_self">projection</a>
 
<b>by </b><i><a class="txt" href="waybel_1.html#E1:96"><i><font color="Green">E59</font></i></a>, <a class="ref" href="waybel_1.html#D14" target="_self">Def14</a></i>;<br/>
<a NAME="E11:96"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> is <a href="quantal1.html#V11">idempotent</a>
 
<b>by </b><i><a class="ref" href="waybel_1.html#D13" target="_self">Def13</a></i>;<br/>
<a NAME="E12:96"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="waybel_1.html#E9:96"><i><font color="Green">E65</font></i></a>, <a class="ref" href="yellow_2.html#T22">YELLOW_2:22</a></i>;<br/>
<b>hence </b><a NAME="E13:96"/>
(  <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">c<sub>3</sub></font>, <a href="yellow_2.html#K1">Image</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="yellow_0.html#K1">"\/"</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><a href="yellow_2.html#K1">Image</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K1">"\/"</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="waybel_1.html#E6:96"><i><font color="Green">E62</font></i></a>, <a class="txt" href="waybel_1.html#E7:96"><i><font color="Green">E63</font></i></a>, <a class="txt" href="waybel_1.html#E8:96"><i><font color="Green">E64</font></i></a>, <a class="ref" href="waybel_0.html#D31">WAYBEL_0:def 31</a></i>;<br/>


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