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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="E1:97"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="waybel_1.html#NR12" target="_self">is_a_kernel_operator</a> 
 ;<br/>

<a NAME="E2:97"/><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#K2" target="_self">corestr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<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><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#T42" target="_self">Th42</a></i>;<br/>
<a NAME="E3:97"/><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#V4" target="_self">upper_adjoint</a>
 
<b>by </b><i><a class="ref" href="waybel_1.html#D11" target="_self">Def11</a></i>;<br/>
<b>hence </b><a NAME="E4:97"/><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#V17">infs-preserving</a>
 <b>by </b><i><a class="ref" href="waybel_1.html#T13" target="_self">Th13</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="E5:97"/><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="E6:97"/><i><font color="Green">E62</font></i>: 
 <a href="yellow_0.html#R2">ex_inf_of</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<a NAME="E7:97"/><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#R3">preserves_inf_of</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="waybel_1.html#E4:97"><i><font color="Green">E60</font></i></a>, <a class="ref" href="waybel_0.html#D32">WAYBEL_0:def 32</a></i>;<br/>
<a NAME="E8:97"/><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="E9:97"/><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#E5:97"><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="E10:97"/>
<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#E1:97"><i><font color="Green">E59</font></i></a>, <a class="ref" href="waybel_1.html#D15" target="_self">Def15</a></i>;<br/>
<a NAME="E11:97"/><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="E12:97"/><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#E9:97"><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="E13:97"/>
(  <a href="yellow_0.html#R2">ex_inf_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#K2">"/\"</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#K2">"/\"</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#E6:97"><i><font color="Green">E62</font></i></a>, <a class="txt" href="waybel_1.html#E7:97"><i><font color="Green">E63</font></i></a>, <a class="txt" href="waybel_1.html#E8:97"><i><font color="Green">E64</font></i></a>, <a class="ref" href="waybel_0.html#D30">WAYBEL_0:def 30</a></i>;<br/>


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