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<b>let </b><font color="Maroon" title="c1">L</font> be  <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a> <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</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="waybel_1.html#V7" title="WAYBEL_1:attr.7">closure</a> <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> 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;  <a href="waybel_1.html#K3" title="WAYBEL_1:func.3">inclusion</a> <font color="Olive" title="b1">c</font> is <a href="waybel_0.html#V17" title="WAYBEL_0:attr.17">infs-preserving</a> &amp;  <a href="waybel34.html#K2" title="WAYBEL34:func.2">UpperAdj</a> <span class="p1">(<span class="default"><a href="waybel_1.html#K2" title="WAYBEL_1:func.2">corestr</a> <font color="Olive" title="b1">c</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="waybel_1.html#K3" title="WAYBEL_1:func.3">inclusion</a> <font color="Olive" title="b1">c</font> &amp;  <a href="waybel34.html#K1" title="WAYBEL34:func.1">LowerAdj</a> <span class="p1">(<span class="default"><a href="waybel_1.html#K3" title="WAYBEL_1:func.3">inclusion</a> <font color="Olive" title="b1">c</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="waybel_1.html#K2" title="WAYBEL_1:func.2">corestr</a> <font color="Olive" title="b1">c</font> )</span><br/><b>let </b><font color="Maroon" title="c2">c</font> be  <a href="waybel_1.html#V7" title="WAYBEL_1:attr.7">closure</a> <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"> (  <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;  <a href="waybel_1.html#K3" title="WAYBEL_1:func.3">inclusion</a> <font color="Maroon" title="c2">c</font> is <a href="waybel_0.html#V17" title="WAYBEL_0:attr.17">infs-preserving</a> &amp;  <a href="waybel34.html#K2" title="WAYBEL34:func.2">UpperAdj</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">c</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="waybel_1.html#K3" title="WAYBEL_1:func.3">inclusion</a> <font color="Maroon" title="c2">c</font> &amp;  <a href="waybel34.html#K1" title="WAYBEL34:func.1">LowerAdj</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">c</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="waybel_1.html#K2" title="WAYBEL_1:func.2">corestr</a> <font color="Maroon" title="c2">c</font> )</span><br/>



<a NAME="E1:56"/><i><font color="Green" title="E37">A1</font></i>: 
<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" title="WAYBEL_1:th.39">WAYBEL_1:39</a></i>;<br/>
<a NAME="E2:56"/><b>then </b>
(  <a href="waybel_1.html#K3" title="WAYBEL_1:func.3">inclusion</a> <font color="Maroon" title="c2">c</font> is <a href="waybel_1.html#V4" title="WAYBEL_1:attr.4">upper_adjoint</a> &amp;  <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#D11" title="WAYBEL_1:def.11">WAYBEL_1:def 11</a>, <a class="ref" href="waybel_1.html#D12" title="WAYBEL_1:def.12">WAYBEL_1:def 12</a></i>;<br/>
<b>hence </b><a NAME="E3:56"/>
(  <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;  <a href="waybel_1.html#K3" title="WAYBEL_1:func.3">inclusion</a> <font color="Maroon" title="c2">c</font> is <a href="waybel_0.html#V17" title="WAYBEL_0:attr.17">infs-preserving</a> )
 <b>by </b><i><a class="ref" href="waybel_1.html#T13" title="WAYBEL_1:th.13">WAYBEL_1:13</a>, <a class="ref" href="waybel_1.html#T14" title="WAYBEL_1:th.14">WAYBEL_1:14</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="waybel34.html#K2" title="WAYBEL34:func.2">UpperAdj</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">c</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="waybel_1.html#K3" title="WAYBEL_1:func.3">inclusion</a> <font color="Maroon" title="c2">c</font> &amp;  <a href="waybel34.html#K1" title="WAYBEL34:func.1">LowerAdj</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">c</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="waybel_1.html#K2" title="WAYBEL_1:func.2">corestr</a> <font color="Maroon" title="c2">c</font> )</span><br/>

<b>hence </b><a NAME="E4:56"/>
(  <a href="waybel34.html#K2" title="WAYBEL34:func.2">UpperAdj</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">c</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="waybel_1.html#K3" title="WAYBEL_1:func.3">inclusion</a> <font color="Maroon" title="c2">c</font> &amp;  <a href="waybel34.html#K1" title="WAYBEL34:func.1">LowerAdj</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">c</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="waybel_1.html#K2" title="WAYBEL_1:func.2">corestr</a> <font color="Maroon" title="c2">c</font> )
 <b>by </b><i><a class="txt" href="waybel34.html#E1:56"><i><font color="Green" title="E37">A1</font></i></a>, <a class="ref" href="waybel34.html#D1" target="_self" title="WAYBEL34:def.1">Def1</a>, <a class="ref" href="waybel34.html#D2" target="_self" title="WAYBEL34:def.2">Def2</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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