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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="lattice3.html#V3">complete</a> <a href="waybel_0.html#NM8">LATTICE</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be  <a href="waybel_1.html#V8">kernel</a> <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/>



<a NAME="E1:52"/><i><font color="Green">E36</font></i>: 
<span class="p1"><a href="waybel_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><a href="waybel_1.html#K2">corestr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="waybel_1.html#K3">inclusion</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="waybel_1.html#K1">]</a></span> is <a href="waybel_1.html#V3">Galois</a>
 
<b>by </b><i><a class="ref" href="waybel_1.html#T42">WAYBEL_1:42</a></i>;<br/>
<a NAME="E2:52"/><b>then </b>
(  <a href="waybel_1.html#K3">inclusion</a> <font color="Maroon">c<sub>2</sub></font> is <a href="waybel_1.html#V5">lower_adjoint</a> &amp;  <a href="waybel_1.html#K2">corestr</a> <font color="Maroon">c<sub>2</sub></font> is <a href="waybel_1.html#V4">upper_adjoint</a> )
 
<b>by </b><i><a class="ref" href="waybel_1.html#D11">WAYBEL_1:def 11</a>, <a class="ref" href="waybel_1.html#D12">WAYBEL_1:def 12</a></i>;<br/>
<a NAME="E3:52"/><b>then </b><i><font color="Green">E37</font></i>: 
(  <a href="waybel_1.html#K3">inclusion</a> <font color="Maroon">c<sub>2</sub></font> is  <a href="waybel_0.html#V18">sups-preserving</a> <a href="struct_0.html#NM7">Function</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>,<font color="Maroon">c<sub>1</sub></font> &amp;  <a href="waybel_1.html#K2">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">WAYBEL_1:13</a>, <a class="ref" href="waybel_1.html#T14">WAYBEL_1:14</a></i>;<br/>
<a NAME="E4:52"/><b>then </b><i><font color="Green">E38</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><a href="waybel_1.html#K3">inclusion</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><a href="waybel_1.html#K2">corestr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> &amp;  <a href="waybel_1.html#K3">inclusion</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#R2">=</a>  <a href="waybel34.html#K1" target="_self">LowerAdj</a> <span class="p1">(<span class="default"><a href="waybel_1.html#K2">corestr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="waybel34.html#E1:52"><i><font color="Green">E36</font></i></a>, <a class="ref" href="waybel34.html#D1" target="_self">Def1</a>, <a class="ref" href="waybel_1.html#T35">WAYBEL_1:35</a></i>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:52_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="waybel_0.html#V22">directed-sups-preserving</a>
 ;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be  <a href="waybel11.html#V4">Scott</a>  <a href="yellow_9.html#M1">TopAugmentation</a> of  <a href="yellow_2.html#K1">Image</a> <font color="Maroon">c<sub>2</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be  <a href="waybel11.html#V4">Scott</a>  <a href="yellow_9.html#M1">TopAugmentation</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><a NAME="E2:52_1"/><i><font color="Green">E40</font></i>: 
(  <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>3</sub></font> #) <a href="hidden.html#R1">=</a>  <a href="yellow_2.html#K1">Image</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>4</sub></font> #) <a href="hidden.html#R1">=</a>  <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font> #) )
 <b>by </b><i><a class="ref" href="yellow_9.html#D4">YELLOW_9:def 4</a></i>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>assume </b><a NAME="E3:52_1"/>
<font color="Maroon">c<sub>5</sub></font> is  <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>3</sub></font>
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> as  <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>3</sub></font> ;<br/><b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font> as   <a href="struct_0.html#NM2">Subset</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>by </b><i><a class="txt" href="waybel34.html#E2:52_1"><i><font color="Green">E40</font></i></a></i>;<br/><a NAME="E6:52_1"/><i><font color="Green">E41</font></i>: 
 <a href="waybel_1.html#K2">corestr</a> <font color="Maroon">c<sub>2</sub></font> is <a href="waybel_0.html#V22">directed-sups-preserving</a>
 <b>by </b><i><a class="txt" href="waybel34.html#E1:52_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="waybel34.html#T34" target="_self">Th34</a></i>;<br/><a NAME="E7:52_1"/>
<span class="p1">(<span class="default"><a href="waybel_1.html#K3">inclusion</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="waybel15.html#T14">WAYBEL15:14</a></i>;<br/><b>hence </b><a NAME="E8:52_1"/>
 <a href="waybel_0.html#K4">uparrow</a> <font color="Maroon">c<sub>5</sub></font> is  <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="waybel34.html#E3:52"><i><font color="Green">E37</font></i></a>, <a class="txt" href="waybel34.html#E4:52"><i><font color="Green">E38</font></i></a>, <a class="txt" href="waybel34.html#E6:52_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="waybel34.html#T23" target="_self">Th23</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E6:52"/><i><font color="Green">E39</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being <a href="waybel11.html#V4">Scott</a>  <a href="yellow_9.html#M1">TopAugmentation</a> of  <a href="yellow_2.html#K1">Image</a> <font color="Maroon">c<sub>2</sub></font><br/>for <font color="Olive">b<sub>2</sub></font> being <a href="waybel11.html#V4">Scott</a>  <a href="yellow_9.html#M1">TopAugmentation</a> of <font color="Maroon">c<sub>1</sub></font><br/>for <font color="Olive">b<sub>3</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>3</sub></font> is  <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Olive">b<sub>1</sub></font> holds <br/> <a href="waybel_0.html#K4">uparrow</a> <font color="Olive">b<sub>3</sub></font> is  <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Olive">b<sub>2</sub></font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_2"><b>now </b></a><div class="add"><b>set </b><font color="Maroon">c<sub>3</sub></font> =  <a href="waybel_1.html#K2">corestr</a> <font color="Maroon">c<sub>2</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be  <a href="waybel11.html#V4">Scott</a>  <a href="yellow_9.html#M1">TopAugmentation</a> of  <a href="yellow_2.html#K1">Image</a> <font color="Maroon">c<sub>2</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be  <a href="waybel11.html#V4">Scott</a>  <a href="yellow_9.html#M1">TopAugmentation</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>6</sub></font> be  <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>4</sub></font>;<br/><a NAME="E1:52_2"/>
(  <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>4</sub></font> #) <a href="hidden.html#R1">=</a>  <a href="yellow_2.html#K1">Image</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>5</sub></font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>5</sub></font> #) <a href="hidden.html#R1">=</a>  <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font> #) )
 <b>by </b><i><a class="ref" href="yellow_9.html#D4">YELLOW_9:def 4</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font> as   <a href="struct_0.html#NM2">Subset</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> ;<br/><a NAME="E3:52_2"/>
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> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="yellow_0.html#D13">YELLOW_0:def 13</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>7</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E5:52_2"/>
 <a href="waybel_0.html#K4">uparrow</a> <font color="Maroon">c<sub>8</sub></font> is  <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="waybel34.html#E6:52"><i><font color="Green">E39</font></i></a></i>;<br/><b>hence </b><a NAME="E6:52_2"/>
 <a href="waybel_0.html#K4">uparrow</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel34.html#K1" target="_self">LowerAdj</a> <span class="p3">(<span class="default"><a href="waybel_1.html#K2">corestr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> is  <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="waybel34.html#E4:52"><i><font color="Green">E38</font></i></a>, <a class="ref" href="waybel15.html#T14">WAYBEL15:14</a></i>;<br/></div><b>end;</b></div>
<a NAME="E8:52"/><b>then </b>
 <a href="waybel_1.html#K2">corestr</a> <font color="Maroon">c<sub>2</sub></font> is <a href="waybel_0.html#V22">directed-sups-preserving</a>
 
<b>by </b><i><a class="txt" href="waybel34.html#E3:52"><i><font color="Green">E37</font></i></a>, <a class="ref" href="waybel34.html#T23" target="_self">Th23</a></i>;<br/>
<b>hence </b><a NAME="E9:52"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="waybel_0.html#V22">directed-sups-preserving</a>
 <b>by </b><i><a class="ref" href="waybel34.html#T34" target="_self">Th34</a></i>;<br/>


</div>
