<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="setfam_1.html#V2">with_non-empty_element</a>  <a href="hidden.html#M1">set</a> ;<br/>

<b>set </b><font color="Maroon">c<sub>2</sub></font> = <font color="Maroon">c<sub>1</sub></font> <a href="waybel34.html#K4" target="_self">-INF_category</a> ;<br/>
<b>set </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">c<sub>1</sub></font> <a href="waybel34.html#K5" target="_self">-SUP_category</a> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>1</sub></font> <a href="waybel34.html#K10" target="_self">-CL_category</a>  as  non <a href="struct_0.html#V3">empty</a> <a href="altcat_2.html#NM2">subcategory</a> of <font color="Maroon">c<sub>1</sub></font> <a href="waybel34.html#K4" target="_self">-INF_category</a>  <b>by </b><i><a class="ref" href="altcat_4.html#T36">ALTCAT_4:36</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>1</sub></font> <a href="waybel34.html#K11" target="_self">-CL-opp_category</a>  as  non <a href="struct_0.html#V3">empty</a> <a href="altcat_2.html#NM2">subcategory</a> of <font color="Maroon">c<sub>1</sub></font> <a href="waybel34.html#K5" target="_self">-SUP_category</a>  <b>by </b><i><a class="ref" href="altcat_4.html#T36">ALTCAT_4:36</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>1</sub></font> <a href="waybel34.html#K6" target="_self">LowerAdj</a> ;<br/>
<a NAME="E3:84"/><i><font color="Green">E55</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="setfam_1.html#D11">SETFAM_1:def 11</a></i>;<br/>
<a NAME="E4:84"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="waybel34.html#K6" target="_self">LowerAdj</a>  is <a href="functor0.html#V21">bijective</a>
 
;<br/>
<a NAME="E5:84"/><i><font color="Green">E57</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="altcat_1.html#NM1">object</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="waybel34.html#K4" target="_self">-INF_category</a> </span>)</span> holds <br/>( <font color="Olive">b<sub>1</sub></font> is   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">c<sub>4</sub></font> iff <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="waybel34.html#K6" target="_self">LowerAdj</a> </span>)</span> <a href="functor0.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> is   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">c<sub>5</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>7</sub></font> be   <a href="altcat_1.html#NM1">object</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="waybel34.html#K4" target="_self">-INF_category</a> </span>)</span>;<br/>

<a NAME="E1:84_1"/><i><font color="Green">E58</font></i>: 
(  <a href="yellow21.html#K4">latt</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="waybel34.html#K6" target="_self">LowerAdj</a> </span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="yellow21.html#K4">latt</a> <font color="Maroon">c<sub>7</sub></font> )
 
<b>by </b><i><a class="ref" href="waybel34.html#D6" target="_self">Def6</a></i>;<br/>
<a NAME="E2:84_1"/><i><font color="Green">E59</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow21.html#K4">latt</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="waybel34.html#E3:84"><i><font color="Green">E55</font></i></a>, <a class="ref" href="waybel34.html#D4" target="_self">Def4</a></i>;<br/>
<a NAME="E3:84_1"/><b>then </b>
( <font color="Maroon">c<sub>7</sub></font> is   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">c<sub>4</sub></font> iff (  <a href="yellow21.html#K4">latt</a> <font color="Maroon">c<sub>7</sub></font> is <a href="orders_2.html#V1">strict</a> &amp;  <a href="yellow21.html#K4">latt</a> <font color="Maroon">c<sub>7</sub></font> is <a href="lattice3.html#V3">complete</a> &amp;  <a href="yellow21.html#K4">latt</a> <font color="Maroon">c<sub>7</sub></font> is <a href="waybel_3.html#V3">continuous</a> ) )
 
<b>by </b><i><a class="ref" href="waybel34.html#T52" target="_self">Th52</a></i>;<br/>
<b>hence </b><a NAME="E4:84_1"/>
( <font color="Maroon">c<sub>7</sub></font> is   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">c<sub>4</sub></font> iff <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="waybel34.html#K6" target="_self">LowerAdj</a> </span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font> is   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="waybel34.html#E1:84_1"><i><font color="Green">E58</font></i></a>, <a class="txt" href="waybel34.html#E2:84_1"><i><font color="Green">E59</font></i></a>, <a class="ref" href="waybel34.html#T54" target="_self">Th54</a></i>;<br/>


</div><b>end;</b></div>
<div><i><font color="Green">E58</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="84_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font> be   <a href="altcat_1.html#NM1">object</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="waybel34.html#K4" target="_self">-INF_category</a> </span>)</span>;<br/><b>assume </b><a NAME="E1:84_2"/><i><font color="Green">E59</font></i>: 
<span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><b>let </b><font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>10</sub></font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">c<sub>4</sub></font>;<br/><b>assume </b><a NAME="E2:84_2"/><i><font color="Green">E60</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> )
 ;<br/><b>let </b><font color="Maroon">c<sub>11</sub></font>, <font color="Maroon">c<sub>12</sub></font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">c<sub>5</sub></font>;<br/><b>assume </b><a NAME="E3:84_2"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="waybel34.html#K6" target="_self">LowerAdj</a> </span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="waybel34.html#K6" target="_self">LowerAdj</a> </span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font> )
 ;<br/><b>let </b><font color="Maroon">c<sub>13</sub></font> be   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font>;<br/><a NAME="E4:84_2"/>
<span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="waybel34.html#K6" target="_self">LowerAdj</a> </span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="waybel34.html#K6" target="_self">LowerAdj</a> </span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="waybel34.html#E1:84_2"><i><font color="Green">E59</font></i></a>, <a class="ref" href="functor0.html#D20">FUNCTOR0:def 20</a></i>;<br/><a NAME="E5:84_2"/><b>then </b><i><font color="Green">E62</font></i>: 
(  <a href="yellow21.html#K5">@</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> &amp;  <a href="yellow21.html#K5">@</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="waybel34.html#K6" target="_self">LowerAdj</a> </span>)</span> <a href="functor0.html#K8">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="waybel34.html#K6" target="_self">LowerAdj</a> </span>)</span> <a href="functor0.html#K8">.</a> <font color="Maroon">c<sub>13</sub></font> &amp;  <a href="yellow21.html#K4">latt</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> &amp;  <a href="yellow21.html#K4">latt</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> &amp;  <a href="yellow21.html#K4">latt</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> &amp;  <a href="yellow21.html#K4">latt</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> &amp;  <a href="yellow21.html#K4">latt</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp;  <a href="yellow21.html#K4">latt</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> )
 <b>by </b><i><a class="txt" href="waybel34.html#E1:84_2"><i><font color="Green">E59</font></i></a>, <a class="ref" href="yellow21.html#D7">YELLOW21:def 7</a></i>;<br/><a NAME="E6:84_2"/><b>then </b><i><font color="Green">E63</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="waybel34.html#K6" target="_self">LowerAdj</a> </span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="yellow21.html#K4">latt</a> <font color="Maroon">c<sub>7</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="waybel34.html#K6" target="_self">LowerAdj</a> </span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="yellow21.html#K4">latt</a> <font color="Maroon">c<sub>8</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="waybel34.html#K6" target="_self">LowerAdj</a> </span>)</span> <a href="functor0.html#K8">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a>  <a href="waybel34.html#K1" target="_self">LowerAdj</a> <span class="p1">(<span class="default"><a href="yellow21.html#K5">@</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> &amp;  <a href="yellow21.html#K5">@</a> <font color="Maroon">c<sub>13</sub></font> is <a href="waybel_0.html#V17">infs-preserving</a> )
 <b>by </b><i><a class="txt" href="waybel34.html#E3:84"><i><font color="Green">E55</font></i></a>, <a class="txt" href="waybel34.html#E1:84_2"><i><font color="Green">E59</font></i></a>, <a class="ref" href="waybel34.html#D4" target="_self">Def4</a>, <a class="ref" href="waybel34.html#D6" target="_self">Def6</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>13</sub></font> as  <a href="waybel_0.html#V17">infs-preserving</a> <a href="struct_0.html#NM7">Function</a> of <span class="p1">(<span class="default"><a href="yellow21.html#K4">latt</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="yellow21.html#K4">latt</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <b>by </b><i><a class="txt" href="waybel34.html#E3:84"><i><font color="Green">E55</font></i></a>, <a class="txt" href="waybel34.html#E1:84_2"><i><font color="Green">E59</font></i></a>, <a class="txt" href="waybel34.html#E2:84_2"><i><font color="Green">E60</font></i></a>, <a class="txt" href="waybel34.html#E5:84_2"><i><font color="Green">E62</font></i></a>, <a class="ref" href="waybel34.html#D4" target="_self">Def4</a></i>;<br/><a NAME="E8:84_2"/><i><font color="Green">E64</font></i>: 
 <a href="waybel34.html#K2" target="_self">UpperAdj</a> <span class="p1">(<span class="default"><a href="waybel34.html#K1" target="_self">LowerAdj</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="ref" href="waybel34.html#T10" target="_self">Th10</a></i>;<br/><a NAME="E9:84_2"/><b>then </b>
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="altcat_1.html#K1">^&gt;</a></span> iff  <a href="waybel34.html#K2" target="_self">UpperAdj</a> <span class="p1">(<span class="default"><a href="waybel34.html#K1" target="_self">LowerAdj</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> is <a href="waybel_0.html#V22">directed-sups-preserving</a> )
 <b>by </b><i><a class="ref" href="waybel34.html#T53" target="_self">Th53</a></i>;<br/><b>hence </b><a NAME="E10:84_2"/>
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="altcat_1.html#K1">^&gt;</a></span> implies <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="waybel34.html#K6" target="_self">LowerAdj</a> </span>)</span> <a href="functor0.html#K8">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>11</sub></font></span><a href="altcat_1.html#K1">^&gt;</a></span> )
 <b>by </b><i><a class="txt" href="waybel34.html#E2:84_2"><i><font color="Green">E60</font></i></a>, <a class="txt" href="waybel34.html#E3:84_2"><i><font color="Green">E61</font></i></a>, <a class="txt" href="waybel34.html#E5:84_2"><i><font color="Green">E62</font></i></a>, <a class="txt" href="waybel34.html#E6:84_2"><i><font color="Green">E63</font></i></a>, <a class="ref" href="waybel34.html#T55" target="_self">Th55</a></i>;<br/><b>assume </b><a NAME="E11:84_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="waybel34.html#K6" target="_self">LowerAdj</a> </span>)</span> <a href="functor0.html#K8">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>11</sub></font></span><a href="altcat_1.html#K1">^&gt;</a></span>
 ;<br/><a NAME="E12:84_2"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being <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="yellow21.html#K4">latt</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="yellow21.html#K4">latt</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> st <br/>( <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="waybel34.html#K6" target="_self">LowerAdj</a> </span>)</span> <a href="functor0.html#K8">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp;  <a href="waybel34.html#K2" target="_self">UpperAdj</a> <font color="Olive">b<sub>1</sub></font> is <a href="waybel_0.html#V22">directed-sups-preserving</a> )
 <b>by </b><i><a class="ref" href="waybel34.html#T55" target="_self">Th55</a></i>;<br/><b>hence </b><a NAME="E13:84_2"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="altcat_1.html#K1">^&gt;</a></span>
 <b>by </b><i><a class="txt" href="waybel34.html#E2:84_2"><i><font color="Green">E60</font></i></a>, <a class="txt" href="waybel34.html#E3:84_2"><i><font color="Green">E61</font></i></a>, <a class="txt" href="waybel34.html#E5:84_2"><i><font color="Green">E62</font></i></a>, <a class="txt" href="waybel34.html#E6:84_2"><i><font color="Green">E63</font></i></a>, <a class="txt" href="waybel34.html#E8:84_2"><i><font color="Green">E64</font></i></a>, <a class="ref" href="waybel34.html#T53" target="_self">Th53</a></i>;<br/></div><b>end;</b></div>
<a NAME="E7:84"/>
<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="yellow20.html#R3">are_anti-isomorphic_under</a> <font color="Maroon">c<sub>1</sub></font> <a href="waybel34.html#K6" target="_self">LowerAdj</a> 
 
<b>from </b><i><a class="ref" href="yellow20.html#S5">YELLOW20:sch 5</a>(<a class="txt" href="waybel34.html#E4:84"><i><font color="Green">E56</font></i></a>, <a class="txt" href="waybel34.html#E5:84"><i><font color="Green">E57</font></i></a>, <a class="txt" href="waybel34.html#E6:84"><i><font color="Green">E58</font></i></a>);<br/></i>
<b>hence </b><a NAME="E8:84"/>
<font color="Maroon">c<sub>1</sub></font> <a href="waybel34.html#K10" target="_self">-CL_category</a> ,<font color="Maroon">c<sub>1</sub></font> <a href="waybel34.html#K11" target="_self">-CL-opp_category</a>  <a href="yellow20.html#R3">are_anti-isomorphic_under</a> <font color="Maroon">c<sub>1</sub></font> <a href="waybel34.html#K6" target="_self">LowerAdj</a> 
 ;<br/>


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