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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">c</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">c</font> is <a href="waybel_1.html#V7" title="WAYBEL_1:attr.7">closure</a> holds <br/>(  <a href="waybel_1.html#K2" title="WAYBEL_1:func.2">corestr</a> <font color="Olive" title="b1">c</font> is <a href="waybel_0.html#V18" title="WAYBEL_0:attr.18">sups-preserving</a> &amp; ( for <font color="Olive" title="b2">X</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font>  st <font color="Olive" title="b2">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <font color="Olive" title="b1">c</font></span>)</span> &amp;  <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Olive" title="b2">X</font>,<font color="Maroon" title="c1">L</font> holds <br/>(  <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Olive" title="b2">X</font>, <a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <font color="Olive" title="b1">c</font> &amp;  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Olive" title="b2">X</font>,<span class="p1">(<span class="default"><a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <font color="Olive" title="b1">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">c</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Olive" title="b2">X</font>,<font color="Maroon" title="c1">L</font></span>)</span> ) ) )</span><br/><b>let </b><font color="Maroon" title="c2">c</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">c</font> is <a href="waybel_1.html#V7" title="WAYBEL_1:attr.7">closure</a> implies (  <a href="waybel_1.html#K2" title="WAYBEL_1:func.2">corestr</a> <font color="Maroon" title="c2">c</font> is <a href="waybel_0.html#V18" title="WAYBEL_0:attr.18">sups-preserving</a> &amp; ( for <font color="Olive" title="b1">X</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font>  st <font color="Olive" title="b1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <font color="Maroon" title="c2">c</font></span>)</span> &amp;  <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Olive" title="b1">X</font>,<font color="Maroon" title="c1">L</font> holds <br/>(  <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Olive" title="b1">X</font>, <a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <font color="Maroon" title="c2">c</font> &amp;  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Olive" title="b1">X</font>,<span class="p1">(<span class="default"><a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <font color="Maroon" title="c2">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">c</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Olive" title="b1">X</font>,<font color="Maroon" title="c1">L</font></span>)</span> ) ) ) )</span><br/>



<b>assume </b><a NAME="E1:91"/><i><font color="Green" title="E58">A1</font></i>: 
<font color="Maroon" title="c2">c</font> is <a href="waybel_1.html#V7" title="WAYBEL_1:attr.7">closure</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="waybel_1.html#K2" title="WAYBEL_1:func.2">corestr</a> <font color="Maroon" title="c2">c</font> is <a href="waybel_0.html#V18" title="WAYBEL_0:attr.18">sups-preserving</a> &amp; ( for <font color="Olive" title="b1">X</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font>  st <font color="Olive" title="b1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <font color="Maroon" title="c2">c</font></span>)</span> &amp;  <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Olive" title="b1">X</font>,<font color="Maroon" title="c1">L</font> holds <br/>(  <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Olive" title="b1">X</font>, <a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <font color="Maroon" title="c2">c</font> &amp;  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Olive" title="b1">X</font>,<span class="p1">(<span class="default"><a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <font color="Maroon" title="c2">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">c</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Olive" title="b1">X</font>,<font color="Maroon" title="c1">L</font></span>)</span> ) ) )</span><br/>

<a NAME="E2:91"/><b>then </b>
<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">c</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">c</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="ref" href="waybel_1.html#T39" target="_self" title="WAYBEL_1:th.39">Th39</a></i>;<br/>
<a NAME="E3:91"/><b>then </b>
 <a href="waybel_1.html#K2" title="WAYBEL_1:func.2">corestr</a> <font color="Maroon" title="c2">c</font> is <a href="waybel_1.html#V5" title="WAYBEL_1:attr.5">lower_adjoint</a>
 
<b>by </b><i><a class="ref" href="waybel_1.html#D12" target="_self" title="WAYBEL_1:def.12">Def12</a></i>;<br/>
<b>hence </b><a NAME="E4:91"/><i><font color="Green" title="E59">A2</font></i>: 
 <a href="waybel_1.html#K2" title="WAYBEL_1:func.2">corestr</a> <font color="Maroon" title="c2">c</font> is <a href="waybel_0.html#V18" title="WAYBEL_0:attr.18">sups-preserving</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">X</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font>  st <font color="Olive" title="b1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <font color="Maroon" title="c2">c</font></span>)</span> &amp;  <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Olive" title="b1">X</font>,<font color="Maroon" title="c1">L</font> holds <br/>(  <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Olive" title="b1">X</font>, <a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <font color="Maroon" title="c2">c</font> &amp;  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Olive" title="b1">X</font>,<span class="p1">(<span class="default"><a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <font color="Maroon" title="c2">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">c</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Olive" title="b1">X</font>,<font color="Maroon" title="c1">L</font></span>)</span> )</span><br/>

<b>let </b><font color="Maroon" title="c3">X</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <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="c3">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <font color="Maroon" title="c2">c</font></span>)</span> &amp;  <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c1">L</font> implies (  <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Maroon" title="c3">X</font>, <a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <font color="Maroon" title="c2">c</font> &amp;  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c3">X</font>,<span class="p1">(<span class="default"><a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <font color="Maroon" title="c2">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">c</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c1">L</font></span>)</span> ) )</span><br/>

<b>assume </b><b>that </b><br/><a NAME="E5:91"/><i><font color="Green" title="E60">A3</font></i>: 
<font color="Maroon" title="c3">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <font color="Maroon" title="c2">c</font></span>)</span>
 <b>and </b><br/><a NAME="E6:91"/><i><font color="Green" title="E61">A4</font></i>: 
 <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Maroon" title="c3">X</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"> (  <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Maroon" title="c3">X</font>, <a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <font color="Maroon" title="c2">c</font> &amp;  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c3">X</font>,<span class="p1">(<span class="default"><a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <font color="Maroon" title="c2">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">c</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c1">L</font></span>)</span> )</span><br/>

<a NAME="E7:91"/><i><font color="Green" title="E62">A5</font></i>: 
 <a href="waybel_1.html#K2" title="WAYBEL_1:func.2">corestr</a> <font color="Maroon" title="c2">c</font> <a href="waybel_0.html#R4" title="WAYBEL_0:pred.4">preserves_sup_of</a> <font color="Maroon" title="c3">X</font>
 
<b>by </b><i><a class="txt" href="#E4:91"><i><font color="Green" title="E59">A2</font></i></a>, <a class="ref" href="waybel_0.html#D33" title="WAYBEL_0:def.33">WAYBEL_0:def 33</a></i>;<br/>
<a NAME="E8:91"/><i><font color="Green" title="E63">A6</font></i>: 
 <a href="waybel_1.html#K2" title="WAYBEL_1:func.2">corestr</a> <font color="Maroon" title="c2">c</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c2">c</font>
 
<b>by </b><i><a class="ref" href="waybel_1.html#T32" target="_self" title="WAYBEL_1:th.32">Th32</a></i>;<br/>
<a NAME="E9:91"/><i><font color="Green" title="E64">A7</font></i>: 
<font color="Maroon" title="c3">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">c</font>
 
<b>by </b><i><a class="txt" href="#E5:91"><i><font color="Green" title="E60">A3</font></i></a>, <a class="ref" href="yellow_0.html#D15" title="YELLOW_0:def.15">YELLOW_0:def 15</a></i>;<br/>
<a NAME="E10:91"/>
<font color="Maroon" title="c2">c</font> is <a href="waybel_1.html#V6" title="WAYBEL_1:attr.6">projection</a>
 
<b>by </b><i><a class="txt" href="#E1:91"><i><font color="Green" title="E58">A1</font></i></a></i>;<br/>
<a NAME="E11:91"/><b>then </b>
<font color="Maroon" title="c2">c</font> is <a href="quantal1.html#V11" title="QUANTAL1:attr.11">idempotent</a>
 
<b>by </b><i><a class="ref" href="waybel_1.html#D13" target="_self" title="WAYBEL_1:def.13">Def13</a></i>;<br/>
<a NAME="E12:91"/><b>then </b>
<font color="Maroon" title="c2">c</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c3">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">X</font>
 
<b>by </b><i><a class="txt" href="#E9:91"><i><font color="Green" title="E64">A7</font></i></a>, <a class="ref" href="yellow_2.html#T22" title="YELLOW_2:th.22">YELLOW_2:22</a></i>;<br/>
<b>hence </b><a NAME="E13:91"/>
(  <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Maroon" title="c3">X</font>, <a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <font color="Maroon" title="c2">c</font> &amp;  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c3">X</font>,<span class="p1">(<span class="default"><a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <font color="Maroon" title="c2">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">c</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c1">L</font></span>)</span> )
 <b>by </b><i><a class="txt" href="#E6:91"><i><font color="Green" title="E61">A4</font></i></a>, <a class="txt" href="#E7:91"><i><font color="Green" title="E62">A5</font></i></a>, <a class="txt" href="#E8:91"><i><font color="Green" title="E63">A6</font></i></a>, <a class="ref" href="waybel_0.html#D31" title="WAYBEL_0:def.31">WAYBEL_0:def 31</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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