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

<b>let </b><font color="Maroon" title="c1">W</font> be  <a href="setfam_1.html#NV3" title="SETFAM_1:NV.3">with_non-empty_element</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>set </b><font color="Maroon" title="c2">A1</font> = <font color="Maroon" title="c1">W</font> <a href="waybel34.html#K4" title="WAYBEL34:func.4">-INF_category</a> ;<br/>
<b>set </b><font color="Maroon" title="c3">B1</font> = <font color="Maroon" title="c1">W</font> <a href="waybel34.html#K8" title="WAYBEL34:func.8">-INF(SC)_category</a> ;<br/>
<b>set </b><font color="Maroon" title="c4">B2</font> = <font color="Maroon" title="c1">W</font> <a href="waybel34.html#K9" title="WAYBEL34:func.9">-SUP(SO)_category</a> ;<br/>
<b>set </b><font color="Maroon" title="c5">F</font> = <font color="Maroon" title="c1">W</font> <a href="waybel34.html#K6" title="WAYBEL34:func.6">LowerAdj</a> ;<br/>
<a NAME="E1:82"/><i><font color="Green" title="E54">A1</font></i>: 
 ex <font color="Olive" title="b1">a</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">W</font>
 
<b>by </b><i><a class="ref" href="setfam_1.html#D11" title="SETFAM_1:def.11">SETFAM_1:def 11</a></i>;<br/>
<a NAME="E2:82"/><i><font color="Green" title="E55">A2</font></i>: 
<font color="Maroon" title="c1">W</font> <a href="waybel34.html#K6" title="WAYBEL34:func.6">LowerAdj</a>  is <a href="functor0.html#V21" title="FUNCTOR0:attr.21">bijective</a>
 
;<br/>
<a NAME="E3:82"/><i><font color="Green" title="E56">A3</font></i>: 
for <font color="Olive" title="b1">a</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">W</font> <a href="waybel34.html#K4" title="WAYBEL34:func.4">-INF_category</a> </span>)</span> holds <br/> ( <font color="Olive" title="b1">a</font> is   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">W</font> <a href="waybel34.html#K8" title="WAYBEL34:func.8">-INF(SC)_category</a> </span>)</span> iff <span class="p1">(<span class="default"><font color="Maroon" title="c1">W</font> <a href="waybel34.html#K6" title="WAYBEL34:func.6">LowerAdj</a> </span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Olive" title="b1">a</font> is   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">W</font> <a href="waybel34.html#K9" title="WAYBEL34:func.9">-SUP(SO)_category</a> </span>)</span> )
 
<b>by </b><i><a class="ref" href="waybel34.html#D10" target="_self" title="WAYBEL34:def.10">Def10</a>, <a class="ref" href="waybel34.html#D11" target="_self" title="WAYBEL34:def.11">Def11</a></i>;<br/>
<div><i><font color="Green" title="E57">A4</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="82_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">a</font>, <font color="Maroon" title="c7">b</font> be   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">W</font> <a href="waybel34.html#K4" title="WAYBEL34:func.4">-INF_category</a> </span>)</span>;<br/><b>assume </b><a NAME="E1:82_1"/><i><font color="Green" title="E58">A5</font></i>: 
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><b>let </b><font color="Maroon" title="c8">a1</font>, <font color="Maroon" title="c9">b1</font> be   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">W</font> <a href="waybel34.html#K8" title="WAYBEL34:func.8">-INF(SC)_category</a> </span>)</span>;<br/><b>assume </b><a NAME="E2:82_1"/><i><font color="Green" title="E59">A6</font></i>: 
( <font color="Maroon" title="c8">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a</font> &amp; <font color="Maroon" title="c9">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">b</font> )
 ;<br/><b>let </b><font color="Maroon" title="c10">a2</font>, <font color="Maroon" title="c11">b2</font> be   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">W</font> <a href="waybel34.html#K9" title="WAYBEL34:func.9">-SUP(SO)_category</a> </span>)</span>;<br/><b>assume </b><a NAME="E3:82_1"/><i><font color="Green" title="E60">A7</font></i>: 
( <font color="Maroon" title="c10">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">W</font> <a href="waybel34.html#K6" title="WAYBEL34:func.6">LowerAdj</a> </span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c6">a</font> &amp; <font color="Maroon" title="c11">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">W</font> <a href="waybel34.html#K6" title="WAYBEL34:func.6">LowerAdj</a> </span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c7">b</font> )
 ;<br/><b>let </b><font color="Maroon" title="c12">f</font> be   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font>;<br/><a NAME="E4:82_1"/><i><font color="Green" title="E61">A8</font></i>: 
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">W</font> <a href="waybel34.html#K6" title="WAYBEL34:func.6">LowerAdj</a> </span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c7">b</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">W</font> <a href="waybel34.html#K6" title="WAYBEL34:func.6">LowerAdj</a> </span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c6">a</font></span>)</span></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="waybel34.html#E1:82_1"><i><font color="Green" title="E58">A5</font></i></a>, <a class="ref" href="functor0.html#D20" title="FUNCTOR0:def.20">FUNCTOR0:def 20</a></i>;<br/><a NAME="E5:82_1"/><b>then </b><i><font color="Green" title="E62">A9</font></i>: 
(  <a href="yellow21.html#K5" title="YELLOW21:func.5">@</a> <font color="Maroon" title="c12">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">f</font> &amp;  <a href="yellow21.html#K5" title="YELLOW21:func.5">@</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">W</font> <a href="waybel34.html#K6" title="WAYBEL34:func.6">LowerAdj</a> </span>)</span> <a href="functor0.html#K8" title="FUNCTOR0:func.8">.</a> <font color="Maroon" title="c12">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">W</font> <a href="waybel34.html#K6" title="WAYBEL34:func.6">LowerAdj</a> </span>)</span> <a href="functor0.html#K8" title="FUNCTOR0:func.8">.</a> <font color="Maroon" title="c12">f</font> &amp;  <a href="yellow21.html#K4" title="YELLOW21:func.4">latt</a> <font color="Maroon" title="c8">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">a1</font> &amp;  <a href="yellow21.html#K4" title="YELLOW21:func.4">latt</a> <font color="Maroon" title="c9">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">b1</font> &amp;  <a href="yellow21.html#K4" title="YELLOW21:func.4">latt</a> <font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a</font> &amp;  <a href="yellow21.html#K4" title="YELLOW21:func.4">latt</a> <font color="Maroon" title="c7">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">b</font> &amp;  <a href="yellow21.html#K4" title="YELLOW21:func.4">latt</a> <font color="Maroon" title="c10">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">a2</font> &amp;  <a href="yellow21.html#K4" title="YELLOW21:func.4">latt</a> <font color="Maroon" title="c11">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">b2</font> )
 <b>by </b><i><a class="txt" href="waybel34.html#E1:82_1"><i><font color="Green" title="E58">A5</font></i></a>, <a class="ref" href="yellow21.html#D7" title="YELLOW21:def.7">YELLOW21:def 7</a></i>;<br/><a NAME="E6:82_1"/><b>then </b><i><font color="Green" title="E63">A10</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c1">W</font> <a href="waybel34.html#K6" title="WAYBEL34:func.6">LowerAdj</a> </span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow21.html#K4" title="YELLOW21:func.4">latt</a> <font color="Maroon" title="c6">a</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c1">W</font> <a href="waybel34.html#K6" title="WAYBEL34:func.6">LowerAdj</a> </span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c7">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow21.html#K4" title="YELLOW21:func.4">latt</a> <font color="Maroon" title="c7">b</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c1">W</font> <a href="waybel34.html#K6" title="WAYBEL34:func.6">LowerAdj</a> </span>)</span> <a href="functor0.html#K8" title="FUNCTOR0:func.8">.</a> <font color="Maroon" title="c12">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel34.html#K1" title="WAYBEL34:func.1">LowerAdj</a> <span class="p1">(<span class="default"><a href="yellow21.html#K5" title="YELLOW21:func.5">@</a> <font color="Maroon" title="c12">f</font></span>)</span> &amp;  <a href="yellow21.html#K5" title="YELLOW21:func.5">@</a> <font color="Maroon" title="c12">f</font> is <a href="waybel_0.html#V17" title="WAYBEL_0:attr.17">infs-preserving</a> )
 <b>by </b><i><a class="txt" href="waybel34.html#E1:82"><i><font color="Green" title="E54">A1</font></i></a>, <a class="txt" href="waybel34.html#E1:82_1"><i><font color="Green" title="E58">A5</font></i></a>, <a class="ref" href="waybel34.html#D4" target="_self" title="WAYBEL34:def.4">Def4</a>, <a class="ref" href="waybel34.html#D6" target="_self" title="WAYBEL34:def.6">Def6</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c13">g</font> = <font color="Maroon" title="c12">f</font> as  <a href="waybel_0.html#V17" title="WAYBEL_0:attr.17">infs-preserving</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="yellow21.html#K4" title="YELLOW21:func.4">latt</a> <font color="Maroon" title="c8">a1</font></span>)</span>,<span class="p1">(<span class="default"><a href="yellow21.html#K4" title="YELLOW21:func.4">latt</a> <font color="Maroon" title="c9">b1</font></span>)</span> <b>by </b><i><a class="txt" href="waybel34.html#E1:82"><i><font color="Green" title="E54">A1</font></i></a>, <a class="txt" href="waybel34.html#E1:82_1"><i><font color="Green" title="E58">A5</font></i></a>, <a class="txt" href="waybel34.html#E2:82_1"><i><font color="Green" title="E59">A6</font></i></a>, <a class="txt" href="waybel34.html#E5:82_1"><i><font color="Green" title="E62">A9</font></i></a>, <a class="ref" href="waybel34.html#D4" target="_self" title="WAYBEL34:def.4">Def4</a></i>;<br/><a NAME="E8:82_1"/>
 <a href="waybel34.html#K2" title="WAYBEL34:func.2">UpperAdj</a> <span class="p1">(<span class="default"><a href="waybel34.html#K1" title="WAYBEL34:func.1">LowerAdj</a> <font color="Maroon" title="c13">g</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c13">g</font>
 <b>by </b><i><a class="ref" href="waybel34.html#T10" target="_self" title="WAYBEL34:th.10">Th10</a></i>;<br/><a NAME="E9:82_1"/><b>then </b>
( <font color="Maroon" title="c12">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c8">a1</font>,<font color="Maroon" title="c9">b1</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> iff  <a href="waybel34.html#K2" title="WAYBEL34:func.2">UpperAdj</a> <span class="p1">(<span class="default"><a href="waybel34.html#K1" title="WAYBEL34:func.1">LowerAdj</a> <font color="Maroon" title="c13">g</font></span>)</span> is <a href="waybel_0.html#V22" title="WAYBEL_0:attr.22">directed-sups-preserving</a> )
 <b>by </b><i><a class="txt" href="waybel34.html#E1:82_1"><i><font color="Green" title="E58">A5</font></i></a>, <a class="txt" href="waybel34.html#E2:82_1"><i><font color="Green" title="E59">A6</font></i></a>, <a class="txt" href="waybel34.html#E5:82_1"><i><font color="Green" title="E62">A9</font></i></a>, <a class="ref" href="waybel34.html#D10" target="_self" title="WAYBEL34:def.10">Def10</a></i>;<br/><b>hence </b><a NAME="E10:82_1"/>
( <font color="Maroon" title="c12">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c8">a1</font>,<font color="Maroon" title="c9">b1</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> iff <span class="p1">(<span class="default"><font color="Maroon" title="c1">W</font> <a href="waybel34.html#K6" title="WAYBEL34:func.6">LowerAdj</a> </span>)</span> <a href="functor0.html#K8" title="FUNCTOR0:func.8">.</a> <font color="Maroon" title="c12">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c11">b2</font>,<font color="Maroon" title="c10">a2</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> )
 <b>by </b><i><a class="txt" href="waybel34.html#E2:82_1"><i><font color="Green" title="E59">A6</font></i></a>, <a class="txt" href="waybel34.html#E3:82_1"><i><font color="Green" title="E60">A7</font></i></a>, <a class="txt" href="waybel34.html#E4:82_1"><i><font color="Green" title="E61">A8</font></i></a>, <a class="txt" href="waybel34.html#E5:82_1"><i><font color="Green" title="E62">A9</font></i></a>, <a class="txt" href="waybel34.html#E6:82_1"><i><font color="Green" title="E63">A10</font></i></a>, <a class="ref" href="waybel34.html#D11" target="_self" title="WAYBEL34:def.11">Def11</a></i>;<br/></div><b>end;</b></div>
<b>thus </b><a NAME="E5:82"/>
<font color="Maroon" title="c1">W</font> <a href="waybel34.html#K8" title="WAYBEL34:func.8">-INF(SC)_category</a> ,<font color="Maroon" title="c1">W</font> <a href="waybel34.html#K9" title="WAYBEL34:func.9">-SUP(SO)_category</a>  <a href="yellow20.html#R3" title="YELLOW20:pred.3">are_anti-isomorphic_under</a> <font color="Maroon" title="c1">W</font> <a href="waybel34.html#K6" title="WAYBEL34:func.6">LowerAdj</a> 
 <b>from </b><i><a class="ref" href="yellow20.html#S5" title="YELLOW20:sch.5">YELLOW20:sch 5</a>(<a class="txt" href="waybel34.html#E2:82"><i><font color="Green" title="E55">A2</font></i></a>, <a class="txt" href="waybel34.html#E3:82"><i><font color="Green" title="E56">A3</font></i></a>, <a class="txt" href="waybel34.html#E4:82"><i><font color="Green" title="E57">A4</font></i></a>);</i><br/>


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