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

<b>set </b><font color="Maroon">A</font> = <font color="Maroon">W</font> <a href="waybel34.html#K4" target="_self">-INF_category</a> ;<br/>
<b>set </b><font color="Maroon">B</font> = <font color="Maroon">W</font> <a href="waybel34.html#K5" target="_self">-SUP_category</a> ;<br/>
<b>set </b><font color="Maroon">F</font> = <font color="Maroon">W</font> <a href="waybel34.html#K6" target="_self">LowerAdj</a> ;<br/>
<a NAME="E1:34_1_1"/><i><font color="Green">E44</font></i>: 
 ex <font color="Olive">x</font> being non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font>
 
<b>by </b><i><a class="ref" href="setfam_1.html#D11">SETFAM_1:def 11</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="altcat_1.html#NM1">object</a> of <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="waybel34.html#K4" target="_self">-INF_category</a> </span>)</span>) <b>-&gt; </b>   <a href="orders_2.html#L1">RelStr</a>  =  <a href="yellow21.html#K4">latt</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(  <a href="altcat_1.html#NM1">object</a> of <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="waybel34.html#K4" target="_self">-INF_category</a> </span>)</span>,  <a href="altcat_1.html#NM1">object</a> of <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="waybel34.html#K4" target="_self">-INF_category</a> </span>)</span>,  <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">a<sub>2</sub></font>) <b>-&gt; </b>  <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="yellow21.html#K3">latt</a> <font color="Maroon">a<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="yellow21.html#K3">latt</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> =  <a href="waybel34.html#K1" target="_self">LowerAdj</a> <span class="p1">(<span class="default"><a href="yellow21.html#K5">@</a> <font color="Maroon">a<sub>3</sub></font></span>)</span>;<br/>
<a NAME="E2:34_1_1"/><i><font color="Green">E45</font></i>: 
for <font color="Olive">a</font> being  <a href="altcat_1.html#NM1">object</a> of <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="waybel34.html#K4" target="_self">-INF_category</a> </span>)</span> holds  <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="waybel34.html#K6" target="_self">LowerAdj</a> </span>)</span> <a href="functor0.html#K3">.</a> <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">a</font>)
 
<b>by </b><i/>;<br/>
<a NAME="E3:34_1_1"/><i><font color="Green">E46</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="altcat_1.html#NM1">object</a> of <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="waybel34.html#K4" target="_self">-INF_category</a> </span>)</span>  st <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Olive">a</font>,<font color="Olive">b</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>  holds <br/>for <font color="Olive">f</font> being  <a href="altcat_1.html#NM2">Morphism</a> of <font color="Olive">a</font>,<font color="Olive">b</font> holds  <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="waybel34.html#K6" target="_self">LowerAdj</a> </span>)</span> <a href="functor0.html#K8">.</a> <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">a</font>,<font color="Olive">b</font>,<font color="Olive">f</font>)
 
<b>by </b><i/>;<br/>
<a NAME="E4:34_1_1"/><i><font color="Green">E47</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="altcat_1.html#NM1">object</a> of <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="waybel34.html#K4" target="_self">-INF_category</a> </span>)</span>  st <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">a</font>) <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b</font>) holds <br/><font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font>
 
;<br/>
<div><i><font color="Green">E48</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be   <a href="altcat_1.html#NM1">object</a> of <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="waybel34.html#K4" target="_self">-INF_category</a> </span>)</span>, <font color="Maroon">b</font> be   <a href="altcat_1.html#NM1">object</a> of <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="waybel34.html#K4" target="_self">-INF_category</a> </span>)</span>;<br/><b>assume </b><a NAME="E1:34_1_1_1"/><i><font color="Green">E88</font></i>: 
<span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</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">f</font> be   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">a</font>,<font color="Maroon">b</font>, <font color="Maroon">g</font> be   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">a</font>,<font color="Maroon">b</font>;<br/><a NAME="E2:34_1_1_1"/><i><font color="Green">E89</font></i>: 
(  <a href="yellow21.html#K5">@</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> &amp;  <a href="yellow21.html#K5">@</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> )
 <b>by </b><i>, <a class="ref" href="yellow21.html#D7">YELLOW21:def 7</a></i>;<br/><a NAME="E3:34_1_1_1"/><b>then </b><i><font color="Green">E90</font></i>: 
(  <a href="yellow21.html#K4">latt</a> <font color="Maroon">a</font> is <a href="lattice3.html#V3">complete</a> &amp;  <a href="yellow21.html#K4">latt</a> <font color="Maroon">b</font> is <a href="lattice3.html#V3">complete</a> &amp;  <a href="yellow21.html#K5">@</a> <font color="Maroon">f</font> is <a href="waybel_0.html#V17">infs-preserving</a> &amp;  <a href="yellow21.html#K5">@</a> <font color="Maroon">g</font> is <a href="waybel_0.html#V17">infs-preserving</a> )
 <b>by </b><i>, , </i>;<br/><b>assume </b><a NAME="E4:34_1_1_1"/>
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">f</font>) <a href="funct_2.html#R2">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">g</font>)
 ;<br/><b>hence </b><font color="Maroon">f</font> = 
 <a href="waybel34.html#K2" target="_self">UpperAdj</a> <span class="p1">(<span class="default"><a href="waybel34.html#K1" target="_self">LowerAdj</a> <span class="p2">(<span class="default"><a href="yellow21.html#K5">@</a> <font color="Maroon">g</font></span>)</span></span>)</span>
<b>by </b><i>, , </i>
<br/>.= 
<font color="Maroon">g</font>
<b>by </b><i>, , </i>
;<br/><br/></div><b>end;</b></div>
<div><i><font color="Green">E92</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be   <a href="altcat_1.html#NM1">object</a> of <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="waybel34.html#K5" target="_self">-SUP_category</a> </span>)</span>, <font color="Maroon">b</font> be   <a href="altcat_1.html#NM1">object</a> of <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="waybel34.html#K5" target="_self">-SUP_category</a> </span>)</span>;<br/><b>assume </b><a NAME="E1:34_1_1_2"/><i><font color="Green">E93</font></i>: 
<span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</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">f</font> be   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">a</font>,<font color="Maroon">b</font>;<br/><a NAME="E2:34_1_1_2"/>
(  <a href="yellow21.html#K4">latt</a> <font color="Maroon">a</font> is <a href="orders_2.html#V1">strict</a> &amp;  <a href="yellow21.html#K4">latt</a> <font color="Maroon">a</font> is <a href="lattice3.html#V3">complete</a> &amp; 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">a</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> &amp;  <a href="yellow21.html#K4">latt</a> <font color="Maroon">b</font> is <a href="orders_2.html#V1">strict</a> &amp;  <a href="yellow21.html#K4">latt</a> <font color="Maroon">b</font> is <a href="lattice3.html#V3">complete</a> &amp; 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">b</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> )
 <b>by </b><i>, </i>;<br/><b>then reconsider </b><font color="Maroon">c</font> =  <a href="yellow21.html#K4">latt</a> <font color="Maroon">b</font>, <font color="Maroon">d</font> =  <a href="yellow21.html#K4">latt</a> <font color="Maroon">a</font> as   <a href="altcat_1.html#NM1">object</a> of <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="waybel34.html#K4" target="_self">-INF_category</a> </span>)</span> <b>by </b><i/>;<br/><b>take </b><font color="Maroon">c</font> = <font color="Maroon">c</font>;<br/><b>take </b><font color="Maroon">d</font> = <font color="Maroon">d</font>;<br/><a NAME="E4:34_1_1_2"/><i><font color="Green">E109</font></i>: 
(  <a href="yellow21.html#K4">latt</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> &amp;  <a href="yellow21.html#K4">latt</a> <font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font> )
 ;<br/><a NAME="E5:34_1_1_2"/><i><font color="Green">E110</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="yellow21.html#K5">@</a> <font color="Maroon">f</font>
 <b>by </b><i>, <a class="ref" href="yellow21.html#D7">YELLOW21:def 7</a></i>;<br/><a NAME="E6:34_1_1_2"/><b>then </b><i><font color="Green">E111</font></i>: 
 <a href="yellow21.html#K5">@</a> <font color="Maroon">f</font> is <a href="waybel_0.html#V18">sups-preserving</a>
 <b>by </b><i>, , </i>;<br/><a NAME="E7:34_1_1_2"/><b>then </b><i><font color="Green">E112</font></i>: 
(  <a href="waybel34.html#K2" target="_self">UpperAdj</a> <span class="p1">(<span class="default"><a href="yellow21.html#K5">@</a> <font color="Maroon">f</font></span>)</span> is <a href="orders_3.html#V5">monotone</a> &amp;  <a href="waybel34.html#K2" target="_self">UpperAdj</a> <span class="p1">(<span class="default"><a href="yellow21.html#K5">@</a> <font color="Maroon">f</font></span>)</span> is <a href="waybel_0.html#V17">infs-preserving</a> )
 <b>by </b><i>, </i>;<br/><a NAME="E8:34_1_1_2"/><b>then </b><i><font color="Green">E113</font></i>: 
 <a href="waybel34.html#K2" target="_self">UpperAdj</a> <span class="p1">(<span class="default"><a href="yellow21.html#K5">@</a> <font color="Maroon">f</font></span>)</span> <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</font>,<font color="Maroon">d</font></span><a href="altcat_1.html#K1">^&gt;</a></span>
 <b>by </b><i>, </i>;<br/><b>reconsider </b><font color="Maroon">g</font> =  <a href="waybel34.html#K2" target="_self">UpperAdj</a> <span class="p1">(<span class="default"><a href="yellow21.html#K5">@</a> <font color="Maroon">f</font></span>)</span> as   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">c</font>,<font color="Maroon">d</font> <b>by </b><i>, , </i>;<br/><b>take </b><font color="Maroon">g</font> = <font color="Maroon">g</font>;<br/><b>thus </b><a NAME="E10:34_1_1_2"/>
( <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c</font>) &amp; <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">d</font>) &amp; <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">d</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>  )
 <b>by </b><i>, , </i>;<br/><a NAME="E11:34_1_1_2"/>
<font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="yellow21.html#K5">@</a> <font color="Maroon">g</font>
 <b>by </b><i>, <a class="ref" href="yellow21.html#D7">YELLOW21:def 7</a></i>;<br/><b>hence </b><a NAME="E12:34_1_1_2"/>
<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">c</font>,<font color="Maroon">d</font>,<font color="Maroon">g</font>)
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div>
<b>thus </b><a NAME="E7:34_1_1"/>
<font color="Maroon">W</font> <a href="waybel34.html#K6" target="_self">LowerAdj</a>  is <a href="functor0.html#V21">bijective</a>
 <b>from </b><i><a class="ref" href="yellow18.html#S12">YELLOW18:sch 12</a>(, , , , );</i><br/>


</div>
