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

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

<b>consider </b><font color="Maroon">w</font> being  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E2:74"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">w</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>consider </b><font color="Maroon">r</font> being  <a href="wellord1.html#V2">well-ordering</a> <a href="yellow21.html#V5">upper-bounded</a> <a href="orders_1.html#NM2">Order</a> of <font color="Maroon">w</font>;<br/>
<div><i><font color="Green">E45</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="74_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>;<br/><b>let </b><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="yellow21.html#K6">-UPS_category</a> </span>)</span>;<br/><a NAME="E1:74_1"/>
(  <a href="altcat_1.html#K8">idm</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="grcat_1.html#K7">id</a> <span class="p1">(<span class="default"><a href="yellow21.html#K4">latt</a> <font color="Maroon">a</font></span>)</span> &amp;  <a href="yellow21.html#K4">latt</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp;  <a href="altcat_1.html#K8">idm</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="grcat_1.html#K7">id</a> <span class="p1">(<span class="default"><a href="yellow21.html#K4">latt</a> <font color="Maroon">b</font></span>)</span> &amp;  <a href="yellow21.html#K4">latt</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> )
 <b>by </b><i><a class="ref" href="yellow21.html#T2">YELLOW21:2</a></i>;<br/><b>hence </b><a NAME="E2:74_1"/>
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> implies  <a href="altcat_1.html#K8">idm</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="altcat_1.html#K8">idm</a> <font color="Maroon">b</font> )
 ;<br/></div><b>end;</b></div>
<b>set </b><font color="Maroon">B</font> =  <a href="yellow20.html#K2">Intersect</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="waybel34.html#K4" target="_self">-INF_category</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="yellow21.html#K6">-UPS_category</a> </span>)</span>;<br/>
<a NAME="E5:74"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">W</font> <a href="waybel34.html#K4" target="_self">-INF_category</a> ,<font color="Maroon">W</font> <a href="yellow21.html#K6">-UPS_category</a>  <a href="yellow20.html#R1">have_the_same_composition</a> 
 
<b>by </b><i><a class="ref" href="yellow20.html#T12">YELLOW20:12</a></i>;<br/>
<a NAME="E6:74"/><b>then </b><i><font color="Green">E47</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow20.html#K2">Intersect</a> <span class="p2">(<span class="default"><font color="Maroon">W</font> <a href="waybel34.html#K4" target="_self">-INF_category</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">W</font> <a href="yellow21.html#K6">-UPS_category</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</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="xboole_0.html#K3">/\</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="yellow21.html#K6">-UPS_category</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="yellow20.html#D3">YELLOW20:def 3</a></i>;<br/>
<a NAME="E7:74"/>
(  <a href="orders_2.html#G1">RelStr</a>(# <font color="Maroon">w</font>,<font color="Maroon">r</font> #) is   <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> &amp;  <a href="orders_2.html#G1">RelStr</a>(# <font color="Maroon">w</font>,<font color="Maroon">r</font> #) is   <a href="altcat_1.html#NM1">object</a> of <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="yellow21.html#K6">-UPS_category</a> </span>)</span> )
 
<b>by </b><i><a class="txt" href="waybel34.html#E6:74"><i><font color="Green">E47</font></i></a>, , <a class="ref" href="yellow21.html#T14">YELLOW21:14</a></i>;<br/>
<a NAME="E8:74"/><b>then </b>
 <a href="orders_2.html#G1">RelStr</a>(# <font color="Maroon">w</font>,<font color="Maroon">r</font> #) <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow20.html#K2">Intersect</a> <span class="p2">(<span class="default"><font color="Maroon">W</font> <a href="waybel34.html#K4" target="_self">-INF_category</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">W</font> <a href="yellow21.html#K6">-UPS_category</a> </span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E9:74"/><b>then </b>
not  <a href="yellow20.html#K2">Intersect</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="waybel34.html#K4" target="_self">-INF_category</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="yellow21.html#K6">-UPS_category</a> </span>)</span> is <a href="struct_0.html#V3">empty</a>
 
<b>by </b><i><a class="ref" href="struct_0.html#D1">STRUCT_0:def 1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">I</font> =  <a href="yellow20.html#K2">Intersect</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="waybel34.html#K4" target="_self">-INF_category</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="yellow21.html#K6">-UPS_category</a> </span>)</span> as  non <a href="struct_0.html#V3">empty</a> <a href="altcat_2.html#NM2">subcategory</a> of <font color="Maroon">W</font> <a href="waybel34.html#K4" target="_self">-INF_category</a>  <b>by </b><i><a class="txt" href="waybel34.html#E11:74"><i><font color="Green">E48</font></i></a>, , <a class="ref" href="yellow20.html#T26">YELLOW20:26</a></i>;<br/>
<b>set </b><font color="Maroon">A</font> = <font color="Maroon">W</font> <a href="waybel34.html#K8" target="_self">-INF(SC)_category</a> ;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = the <a href="altcat_1.html#U1">Arrows</a> of <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="waybel34.html#K8" target="_self">-INF(SC)_category</a> </span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">a<sub>2</sub></font>;<br/>
<a NAME="E11:74"/><i><font color="Green">E48</font></i>: 
for <font color="Olive">C1</font>, <font color="Olive">C2</font> being <a href="altcat_1.html#V9">semi-functional</a> <a href="yellow18.html#V2">para-functional</a> <a href="altcat_1.html#NM4">category</a>  st the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">C1</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="waybel34.html#K8" target="_self">-INF(SC)_category</a> </span>)</span> &amp; ( for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="altcat_1.html#NM1">object</a> of <font color="Olive">C1</font> holds  <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#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">a</font>,<font color="Olive">b</font>) ) &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">C2</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="waybel34.html#K8" target="_self">-INF(SC)_category</a> </span>)</span> &amp; ( for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="altcat_1.html#NM1">object</a> of <font color="Olive">C2</font> holds  <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#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">a</font>,<font color="Olive">b</font>) ) holds <br/> <a href="altcat_1.html#G2">AltCatStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">C1</font>,the <a href="altcat_1.html#U1">Arrows</a> of <font color="Olive">C1</font>,the <a href="altcat_1.html#U2">Comp</a> of <font color="Olive">C1</font> #) <a href="hidden.html#R1">=</a>  <a href="altcat_1.html#G2">AltCatStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">C2</font>,the <a href="altcat_1.html#U1">Arrows</a> of <font color="Olive">C2</font>,the <a href="altcat_1.html#U2">Comp</a> of <font color="Olive">C2</font> #)
 
<b>from </b><i><a class="ref" href="yellow18.html#S19">YELLOW18:sch 19</a>();<br/></i>
<a NAME="E12:74"/><i><font color="Green">E88</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">I</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="waybel34.html#K8" target="_self">-INF(SC)_category</a> </span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:74_2"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">I</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="waybel34.html#K8" target="_self">-INF(SC)_category</a> </span>)</span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:74_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">I</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">I</font> ;<br/>
<b>reconsider </b><font color="Maroon">L</font> = <font color="Maroon">x</font> as   <a href="waybel_0.html#NM8">LATTICE</a> <b>by </b><i><a class="ref" href="yellow21.html#D4">YELLOW21:def 4</a></i>;<br/>
<a NAME="E4:74_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="yellow21.html#K6">-UPS_category</a> </span>)</span>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E5:74_2_1"/><b>then </b>
( <font color="Maroon">L</font> is <a href="orders_2.html#V1">strict</a> &amp; <font color="Maroon">L</font> is <a href="lattice3.html#V3">complete</a> &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> )
 
<b>by </b><i><a class="txt" href="waybel34.html#E6:74"><i><font color="Green">E47</font></i></a>, <a class="ref" href="yellow21.html#D10">YELLOW21:def 10</a></i>;<br/>
<a NAME="E6:74_2_1"/><b>then </b>
<font color="Maroon">L</font> is   <a href="altcat_1.html#NM1">object</a> of <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="waybel34.html#K8" target="_self">-INF(SC)_category</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="waybel34.html#E2:74"><i><font color="Green">E44</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:74_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="waybel34.html#K8" target="_self">-INF(SC)_category</a> </span>)</span>
 ;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:74_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="waybel34.html#K8" target="_self">-INF(SC)_category</a> </span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</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#K8" target="_self">-INF(SC)_category</a> </span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon">L</font> = <font color="Maroon">x</font> as   <a href="waybel_0.html#NM8">LATTICE</a> <b>by </b><i><a class="ref" href="yellow21.html#D4">YELLOW21:def 4</a></i>;<br/>
<a NAME="E5:74_2"/>
( <font color="Maroon">L</font> is <a href="lattice3.html#V3">complete</a> &amp; <font color="Maroon">L</font> is <a href="orders_2.html#V1">strict</a> &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> )
 
<b>by </b><i><a class="txt" href="waybel34.html#E2:74"><i><font color="Green">E44</font></i></a></i>;<br/>
<a NAME="E6:74_2"/><b>then </b>
( <font color="Maroon">x</font> is   <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> &amp; <font color="Maroon">x</font> is   <a href="altcat_1.html#NM1">object</a> of <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="yellow21.html#K6">-UPS_category</a> </span>)</span> )
 
<b>by </b><i>, <a class="ref" href="yellow21.html#D10">YELLOW21:def 10</a></i>;<br/>
<b>hence </b><a NAME="E7:74_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">I</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E13:74"/><i><font color="Green">E89</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#K8" target="_self">-INF(SC)_category</a> </span>)</span> holds  <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#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">a</font>,<font color="Olive">b</font>)
 
<b>by </b><i><a class="ref" href="altcat_1.html#D2">ALTCAT_1:def 2</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_3"><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 <font color="Maroon">I</font>, <font color="Maroon">b</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">I</font>;<br/><b>reconsider </b><font color="Maroon">a'</font> = <font color="Maroon">a</font>, <font color="Maroon">b'</font> = <font color="Maroon">b</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#K8" target="_self">-INF(SC)_category</a> </span>)</span> <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">a1</font> = <font color="Maroon">a</font>, <font color="Maroon">b1</font> = <font color="Maroon">b</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>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon">a2</font> = <font color="Maroon">a</font>, <font color="Maroon">b2</font> = <font color="Maroon">b</font> as   <a href="altcat_1.html#NM1">object</a> of <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="yellow21.html#K6">-UPS_category</a> </span>)</span> <b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E4:74_3"/>
(  <a href="relat_1.html#K1">dom</a> the <a href="altcat_1.html#U1">Arrows</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="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><font color="Maroon">W</font> <a href="waybel34.html#K4" target="_self">-INF_category</a> </span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><font color="Maroon">W</font> <a href="waybel34.html#K4" target="_self">-INF_category</a> </span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> &amp;  <a href="relat_1.html#K1">dom</a> the <a href="altcat_1.html#U1">Arrows</a> of <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="yellow21.html#K6">-UPS_category</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><font color="Maroon">W</font> <a href="yellow21.html#K6">-UPS_category</a> </span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><font color="Maroon">W</font> <a href="yellow21.html#K6">-UPS_category</a> </span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> )
 <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><a NAME="E5:74_3"/><b>then </b>
( <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> the <a href="altcat_1.html#U1">Arrows</a> of <span class="p2">(<span class="default"><font color="Maroon">W</font> <a href="waybel34.html#K4" target="_self">-INF_category</a> </span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> the <a href="altcat_1.html#U1">Arrows</a> of <span class="p2">(<span class="default"><font color="Maroon">W</font> <a href="yellow21.html#K6">-UPS_category</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p3">(<span class="default"><font color="Maroon">W</font> <a href="waybel34.html#K4" target="_self">-INF_category</a> </span>)</span> <a href="xboole_0.html#K3">/\</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p3">(<span class="default"><font color="Maroon">W</font> <a href="yellow21.html#K6">-UPS_category</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p3">(<span class="default"><font color="Maroon">W</font> <a href="waybel34.html#K4" target="_self">-INF_category</a> </span>)</span> <a href="xboole_0.html#K3">/\</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p3">(<span class="default"><font color="Maroon">W</font> <a href="yellow21.html#K6">-UPS_category</a> </span>)</span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">I</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">I</font> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T123">ZFMISC_1:123</a></i>;<br/><a NAME="E6:74_3"/><b>then </b><i><font color="Green">E90</font></i>: 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> the <a href="altcat_1.html#U1">Arrows</a> of <span class="p2">(<span class="default"><font color="Maroon">W</font> <a href="waybel34.html#K4" target="_self">-INF_category</a> </span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> the <a href="altcat_1.html#U1">Arrows</a> of <span class="p2">(<span class="default"><font color="Maroon">W</font> <a href="yellow21.html#K6">-UPS_category</a> </span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/><i><font color="Green">E92</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> = 
the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">I</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>
<b>by </b><i><a class="ref" href="altcat_1.html#D2">ALTCAT_1:def 2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="yellow20.html#K1">Intersect</a> the <a href="altcat_1.html#U1">Arrows</a> of <span class="p2">(<span class="default"><font color="Maroon">W</font> <a href="waybel34.html#K4" target="_self">-INF_category</a> </span>)</span>,the <a href="altcat_1.html#U1">Arrows</a> of <span class="p2">(<span class="default"><font color="Maroon">W</font> <a href="yellow21.html#K6">-UPS_category</a> </span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i>, <a class="ref" href="yellow20.html#D3">YELLOW20:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default">the <a href="altcat_1.html#U1">Arrows</a> of <span class="p2">(<span class="default"><font color="Maroon">W</font> <a href="waybel34.html#K4" target="_self">-INF_category</a> </span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default">the <a href="altcat_1.html#U1">Arrows</a> of <span class="p2">(<span class="default"><font color="Maroon">W</font> <a href="yellow21.html#K6">-UPS_category</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K1">]</a></span></span>)</span>
<b>by </b><i>, <a class="ref" href="yellow20.html#D2">YELLOW20:def 2</a></i>
<br/>.= 
<span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default">the <a href="altcat_1.html#U1">Arrows</a> of <span class="p2">(<span class="default"><font color="Maroon">W</font> <a href="yellow21.html#K6">-UPS_category</a> </span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span>
<b>by </b><i><a class="ref" href="altcat_1.html#D2">ALTCAT_1:def 2</a></i>
<br/>.= 
<span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="xboole_0.html#K3">/\</a> <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="altcat_1.html#K1">^&gt;</a></span>
<b>by </b><i><a class="ref" href="altcat_1.html#D2">ALTCAT_1:def 2</a></i>
;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_3_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">f</font> be    <a href="hidden.html#M1">set</a> ;<br/><a NAME="E1:74_3_2"/>
( <font color="Maroon">f</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">a</font>,<font color="Maroon">b</font></span><a href="altcat_1.html#K1">^&gt;</a></span> iff ( <font color="Maroon">f</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">a1</font>,<font color="Maroon">b1</font></span><a href="altcat_1.html#K1">^&gt;</a></span> &amp; <font color="Maroon">f</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">a2</font>,<font color="Maroon">b2</font></span><a href="altcat_1.html#K1">^&gt;</a></span> ) )
 <b>by </b><i><a class="ref" href="waybel34.html#D1" target="_self">Def1</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E2:74_3_2"/><b>then </b>
( <font color="Maroon">f</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">a</font>,<font color="Maroon">b</font></span><a href="altcat_1.html#K1">^&gt;</a></span> iff ( <font color="Maroon">f</font> is  <a href="waybel_0.html#V22">directed-sups-preserving</a> <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="yellow21.html#K4">latt</a> <font color="Maroon">a2</font></span>)</span>,<span class="p1">(<span class="default"><a href="yellow21.html#K4">latt</a> <font color="Maroon">b2</font></span>)</span> &amp; <font color="Maroon">f</font> is  <a href="waybel_0.html#V17">infs-preserving</a> <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="yellow21.html#K4">latt</a> <font color="Maroon">a1</font></span>)</span>,<span class="p1">(<span class="default"><a href="yellow21.html#K4">latt</a> <font color="Maroon">b1</font></span>)</span> ) )
 <b>by </b><i>, <a class="ref" href="yellow21.html#T15">YELLOW21:15</a></i>;<br/><a NAME="E3:74_3_2"/><b>then </b>
( <font color="Maroon">f</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">a</font>,<font color="Maroon">b</font></span><a href="altcat_1.html#K1">^&gt;</a></span> iff <font color="Maroon">f</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">a'</font>,<font color="Maroon">b'</font></span><a href="altcat_1.html#K1">^&gt;</a></span> )
 <b>by </b><i><a class="txt" href="waybel34.html#E4:74"><i><font color="Green">E45</font></i></a></i>;<br/><b>hence </b><a NAME="E4:74_3_2"/>
( <font color="Maroon">f</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">a</font>,<font color="Maroon">b</font></span><a href="altcat_1.html#K1">^&gt;</a></span> iff <font color="Maroon">f</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">a</font>,<font color="Maroon">b</font>) )
 <b>by </b><i><a class="ref" href="altcat_1.html#D2">ALTCAT_1:def 2</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E9:74_3"/>
<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#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">a</font>,<font color="Maroon">b</font>)
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E15:74"/>
<font color="Maroon">W</font> <a href="waybel34.html#K8" target="_self">-INF(SC)_category</a>  <a href="hidden.html#R1">=</a>  <a href="yellow20.html#K2">Intersect</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="waybel34.html#K4" target="_self">-INF_category</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="yellow21.html#K6">-UPS_category</a> </span>)</span>
 <b>by </b><i>, , </i>;<br/>


</div>
