<?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>consider </b><font color="Maroon" title="c2">w</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> <b> such that </b><br/><a NAME="E2:74"/><i><font color="Green" title="E46">A1</font></i>: 
<font color="Maroon" title="c2">w</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/>
<b>consider </b><font color="Maroon" title="c3">r</font> being  <a href="wellord1.html#V2" title="WELLORD1:attr.2">well-ordering</a> <a href="yellow21.html#V5" title="YELLOW21:attr.5">upper-bounded</a> <a href="orders_1.html#NM2" title="ORDERS_1:NM.2">Order</a> of <font color="Maroon" title="c2">w</font>;<br/>
<div><i><font color="Green" title="E47">A2</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" title="c4">a</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>let </b><font color="Maroon" title="c5">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="yellow21.html#K6" title="YELLOW21:func.6">-UPS_category</a> </span>)</span>;<br/><a NAME="E1:74_1"/>
(  <a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <span class="p1">(<span class="default"><a href="yellow21.html#K4" title="YELLOW21:func.4">latt</a> <font color="Maroon" title="c4">a</font></span>)</span> &amp;  <a href="yellow21.html#K4" title="YELLOW21:func.4">latt</a> <font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font> &amp;  <a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Maroon" title="c5">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <span class="p1">(<span class="default"><a href="yellow21.html#K4" title="YELLOW21:func.4">latt</a> <font color="Maroon" title="c5">b</font></span>)</span> &amp;  <a href="yellow21.html#K4" title="YELLOW21:func.4">latt</a> <font color="Maroon" title="c5">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b</font> )
 <b>by </b><i><a class="ref" href="yellow21.html#T2" title="YELLOW21:th.2">YELLOW21:2</a></i>;<br/><b>hence </b><a NAME="E2:74_1"/>
( <font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b</font> implies  <a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Maroon" title="c5">b</font> )
 ;<br/></div><b>end;</b></div>
<b>set </b><font color="Maroon" title="c4">B</font> =  <a href="yellow20.html#K2" title="YELLOW20:func.2">Intersect</a> <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>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">W</font> <a href="yellow21.html#K6" title="YELLOW21:func.6">-UPS_category</a> </span>)</span>;<br/>
<a NAME="E5:74"/><i><font color="Green" title="E48">A3</font></i>: 
<font color="Maroon" title="c1">W</font> <a href="waybel34.html#K4" title="WAYBEL34:func.4">-INF_category</a> ,<font color="Maroon" title="c1">W</font> <a href="yellow21.html#K6" title="YELLOW21:func.6">-UPS_category</a>  <a href="yellow20.html#R1" title="YELLOW20:pred.1">have_the_same_composition</a> 
 
<b>by </b><i><a class="ref" href="yellow20.html#T12" title="YELLOW20:th.12">YELLOW20:12</a></i>;<br/>
<a NAME="E6:74"/><b>then </b><i><font color="Green" title="E49">A4</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow20.html#K2" title="YELLOW20:func.2">Intersect</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">W</font> <a href="waybel34.html#K4" title="WAYBEL34:func.4">-INF_category</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">W</font> <a href="yellow21.html#K6" title="YELLOW21:func.6">-UPS_category</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</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> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">W</font> <a href="yellow21.html#K6" title="YELLOW21:func.6">-UPS_category</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="yellow20.html#D3" title="YELLOW20:def.3">YELLOW20:def 3</a></i>;<br/>
<a NAME="E7:74"/>
(  <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# <font color="Maroon" title="c2">w</font>,<font color="Maroon" title="c3">r</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#K4" title="WAYBEL34:func.4">-INF_category</a> </span>)</span> &amp;  <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# <font color="Maroon" title="c2">w</font>,<font color="Maroon" title="c3">r</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="yellow21.html#K6" title="YELLOW21:func.6">-UPS_category</a> </span>)</span> )
 
<b>by </b><i><a class="txt" href="waybel34.html#E2:74"><i><font color="Green" title="E46">A1</font></i></a>, <a class="ref" href="waybel34.html#T13" target="_self" title="WAYBEL34:th.13">Th13</a>, <a class="ref" href="yellow21.html#T14" title="YELLOW21:th.14">YELLOW21:14</a></i>;<br/>
<a NAME="E8:74"/><b>then </b>
 <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# <font color="Maroon" title="c2">w</font>,<font color="Maroon" title="c3">r</font> #) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow20.html#K2" title="YELLOW20:func.2">Intersect</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">W</font> <a href="waybel34.html#K4" title="WAYBEL34:func.4">-INF_category</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">W</font> <a href="yellow21.html#K6" title="YELLOW21:func.6">-UPS_category</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel34.html#E6:74"><i><font color="Green" title="E49">A4</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E9:74"/><b>then </b>
not  <a href="yellow20.html#K2" title="YELLOW20:func.2">Intersect</a> <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>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">W</font> <a href="yellow21.html#K6" title="YELLOW21:func.6">-UPS_category</a> </span>)</span> is <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>
 
<b>by </b><i><a class="ref" href="struct_0.html#D1" title="STRUCT_0:def.1">STRUCT_0:def 1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c5">I</font> =  <a href="yellow20.html#K2" title="YELLOW20:func.2">Intersect</a> <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>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">W</font> <a href="yellow21.html#K6" title="YELLOW21:func.6">-UPS_category</a> </span>)</span> as  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="altcat_2.html#NM2" title="ALTCAT_2:NM.2">subcategory</a> of <font color="Maroon" title="c1">W</font> <a href="waybel34.html#K4" title="WAYBEL34:func.4">-INF_category</a>  <b>by </b><i><a class="txt" href="waybel34.html#E4:74"><i><font color="Green" title="E47">A2</font></i></a>, <a class="txt" href="waybel34.html#E5:74"><i><font color="Green" title="E48">A3</font></i></a>, <a class="ref" href="yellow20.html#T26" title="YELLOW20:th.26">YELLOW20:26</a></i>;<br/>
<b>set </b><font color="Maroon" title="c6">A</font> = <font color="Maroon" title="c1">W</font> <a href="waybel34.html#K8" title="WAYBEL34:func.8">-INF(SC)_category</a> ;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</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> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> $1,$2;<br/>
<a NAME="E11:74"/><i><font color="Green" title="E50">A5</font></i>: 
for <font color="Olive" title="b1">C1</font>, <font color="Olive" title="b2">C2</font> being <a href="altcat_1.html#V9" title="ALTCAT_1:attr.9">semi-functional</a> <a href="yellow18.html#V2" title="YELLOW18:attr.2">para-functional</a> <a href="altcat_1.html#NM4" title="ALTCAT_1:NM.4">category</a>  st the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">C1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</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> &amp; ( for <font color="Olive" title="b3">a</font>, <font color="Olive" title="b4">b</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Olive" title="b1">C1</font> holds  <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Olive" title="b3">a</font>,<font color="Olive" title="b4">b</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b3">a</font>,<font color="Olive" title="b4">b</font>) ) &amp; the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b2">C2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</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> &amp; ( for <font color="Olive" title="b3">a</font>, <font color="Olive" title="b4">b</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Olive" title="b2">C2</font> holds  <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Olive" title="b3">a</font>,<font color="Olive" title="b4">b</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b3">a</font>,<font color="Olive" title="b4">b</font>) ) holds <br/> <a href="altcat_1.html#G2" title="ALTCAT_1:aggr.2">AltCatStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">C1</font>,the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Olive" title="b1">C1</font>,the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Olive" title="b1">C1</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="altcat_1.html#G2" title="ALTCAT_1:aggr.2">AltCatStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b2">C2</font>,the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Olive" title="b2">C2</font>,the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Olive" title="b2">C2</font> #)
 
<b>from </b><i><a class="ref" href="yellow18.html#S19" title="YELLOW18:sch.19">YELLOW18:sch 19</a>();<br/></i>
<a NAME="E12:74"/><i><font color="Green" title="E51">A6</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</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>
 
<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" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">I</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</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>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">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" title="c7">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:74_2_1"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">I</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c8">x</font> = <font color="Maroon" title="c7">x</font> as   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c5">I</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c9">L</font> = <font color="Maroon" title="c8">x</font> as   <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a> <b>by </b><i><a class="ref" href="yellow21.html#D4" title="YELLOW21:def.4">YELLOW21:def 4</a></i>;<br/>
<a NAME="E4:74_2_1"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">W</font> <a href="yellow21.html#K6" title="YELLOW21:func.6">-UPS_category</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="waybel34.html#E6:74"><i><font color="Green" title="E49">A4</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E5:74_2_1"/><b>then </b>
( <font color="Maroon" title="c9">L</font> is <a href="orders_2.html#V1" title="ORDERS_2:attr.1">strict</a> &amp; <font color="Maroon" title="c9">L</font> is <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a> &amp; the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c9">L</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="txt" href="waybel34.html#E2:74"><i><font color="Green" title="E46">A1</font></i></a>, <a class="ref" href="yellow21.html#D10" title="YELLOW21:def.10">YELLOW21:def 10</a></i>;<br/>
<a NAME="E6:74_2_1"/><b>then </b>
<font color="Maroon" title="c9">L</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>
 
<b>by </b><i><a class="ref" href="waybel34.html#T47" target="_self" title="WAYBEL34:th.47">Th47</a></i>;<br/>
<b>hence </b><a NAME="E7:74_2_1"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</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/>


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

<b>let </b><font color="Maroon" title="c7">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:74_2"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</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>then reconsider </b><font color="Maroon" title="c8">x</font> = <font color="Maroon" title="c7">x</font> as   <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>reconsider </b><font color="Maroon" title="c9">L</font> = <font color="Maroon" title="c8">x</font> as   <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a> <b>by </b><i><a class="ref" href="yellow21.html#D4" title="YELLOW21:def.4">YELLOW21:def 4</a></i>;<br/>
<a NAME="E5:74_2"/>
( <font color="Maroon" title="c9">L</font> is <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a> &amp; <font color="Maroon" title="c9">L</font> is <a href="orders_2.html#V1" title="ORDERS_2:attr.1">strict</a> &amp; the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c9">L</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="waybel34.html#T47" target="_self" title="WAYBEL34:th.47">Th47</a></i>;<br/>
<a NAME="E6:74_2"/><b>then </b>
( <font color="Maroon" title="c8">x</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#K4" title="WAYBEL34:func.4">-INF_category</a> </span>)</span> &amp; <font color="Maroon" title="c8">x</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="yellow21.html#K6" title="YELLOW21:func.6">-UPS_category</a> </span>)</span> )
 
<b>by </b><i><a class="ref" href="waybel34.html#T13" target="_self" title="WAYBEL34:th.13">Th13</a>, <a class="ref" href="yellow21.html#D10" title="YELLOW21:def.10">YELLOW21:def 10</a></i>;<br/>
<b>hence </b><a NAME="E7:74_2"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">I</font>
 <b>by </b><i><a class="txt" href="waybel34.html#E6:74"><i><font color="Green" title="E49">A4</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E13:74"/><i><font color="Green" title="E52">A7</font></i>: 
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</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#K8" title="WAYBEL34:func.8">-INF(SC)_category</a> </span>)</span> holds  <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>)
 
<b>by </b><i><a class="ref" href="altcat_1.html#D2" title="ALTCAT_1:def.2">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" title="c7">a</font>, <font color="Maroon" title="c8">b</font> be   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c5">I</font>;<br/><b>reconsider </b><font color="Maroon" title="c9">a'</font> = <font color="Maroon" title="c7">a</font>, <font color="Maroon" title="c10">b'</font> = <font color="Maroon" title="c8">b</font> as   <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> <b>by </b><i><a class="txt" href="waybel34.html#E12:74"><i><font color="Green" title="E51">A6</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c11">a1</font> = <font color="Maroon" title="c7">a</font>, <font color="Maroon" title="c12">b1</font> = <font color="Maroon" title="c8">b</font> as   <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> <b>by </b><i><a class="txt" href="waybel34.html#E6:74"><i><font color="Green" title="E49">A4</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c13">a2</font> = <font color="Maroon" title="c7">a</font>, <font color="Maroon" title="c14">b2</font> = <font color="Maroon" title="c8">b</font> as   <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="yellow21.html#K6" title="YELLOW21:func.6">-UPS_category</a> </span>)</span> <b>by </b><i><a class="txt" href="waybel34.html#E6:74"><i><font color="Green" title="E49">A4</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E4:74_3"/>
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c1">W</font> <a href="waybel34.html#K4" title="WAYBEL34:func.4">-INF_category</a> </span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c1">W</font> <a href="waybel34.html#K4" title="WAYBEL34:func.4">-INF_category</a> </span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">W</font> <a href="yellow21.html#K6" title="YELLOW21:func.6">-UPS_category</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c1">W</font> <a href="yellow21.html#K6" title="YELLOW21:func.6">-UPS_category</a> </span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c1">W</font> <a href="yellow21.html#K6" title="YELLOW21:func.6">-UPS_category</a> </span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> )
 <b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/><a NAME="E5:74_3"/><b>then </b><i><font color="Green" title="E53">A8</font></i>: 
( <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c1">W</font> <a href="waybel34.html#K4" title="WAYBEL34:func.4">-INF_category</a> </span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c1">W</font> <a href="yellow21.html#K6" title="YELLOW21:func.6">-UPS_category</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p3">(<span class="default"><font color="Maroon" title="c1">W</font> <a href="waybel34.html#K4" title="WAYBEL34:func.4">-INF_category</a> </span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p3">(<span class="default"><font color="Maroon" title="c1">W</font> <a href="yellow21.html#K6" title="YELLOW21:func.6">-UPS_category</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p3">(<span class="default"><font color="Maroon" title="c1">W</font> <a href="waybel34.html#K4" title="WAYBEL34:func.4">-INF_category</a> </span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p3">(<span class="default"><font color="Maroon" title="c1">W</font> <a href="yellow21.html#K6" title="YELLOW21:func.6">-UPS_category</a> </span>)</span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp; <font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">I</font> &amp; <font color="Maroon" title="c8">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">I</font> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T123" title="ZFMISC_1:th.123">ZFMISC_1:123</a></i>;<br/><i><font color="Green" title="E54">A9</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="c7">a</font>,<font color="Maroon" title="c8">b</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> = 
the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon" title="c5">I</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font>
<b>by </b><i><a class="ref" href="altcat_1.html#D2" title="ALTCAT_1:def.2">ALTCAT_1:def 2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="yellow20.html#K1" title="YELLOW20:func.1">Intersect</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c1">W</font> <a href="waybel34.html#K4" title="WAYBEL34:func.4">-INF_category</a> </span>)</span>,the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c1">W</font> <a href="yellow21.html#K6" title="YELLOW21:func.6">-UPS_category</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<b>by </b><i><a class="txt" href="waybel34.html#E5:74"><i><font color="Green" title="E48">A3</font></i></a>, <a class="ref" href="yellow20.html#D3" title="YELLOW20:def.3">YELLOW20:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default">the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c1">W</font> <a href="waybel34.html#K4" title="WAYBEL34:func.4">-INF_category</a> </span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default">the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c1">W</font> <a href="yellow21.html#K6" title="YELLOW21:func.6">-UPS_category</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span>
<b>by </b><i><a class="txt" href="waybel34.html#E6:74"><i><font color="Green" title="E49">A4</font></i></a>, <a class="txt" href="waybel34.html#E5:74_3"><i><font color="Green" title="E53">A8</font></i></a>, <a class="ref" href="yellow20.html#D2" title="YELLOW20:def.2">YELLOW20:def 2</a></i>
<br/>.= 
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c11">a1</font>,<font color="Maroon" title="c12">b1</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default">the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c1">W</font> <a href="yellow21.html#K6" title="YELLOW21:func.6">-UPS_category</a> </span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font></span>)</span>
<b>by </b><i><a class="ref" href="altcat_1.html#D2" title="ALTCAT_1:def.2">ALTCAT_1:def 2</a></i>
<br/>.= 
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c11">a1</font>,<font color="Maroon" title="c12">b1</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c13">a2</font>,<font color="Maroon" title="c14">b2</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span>
<b>by </b><i><a class="ref" href="altcat_1.html#D2" title="ALTCAT_1:def.2">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" title="c15">f</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><a NAME="E1:74_3_2"/>
( <font color="Maroon" title="c15">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="c7">a</font>,<font color="Maroon" title="c8">b</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> iff ( <font color="Maroon" title="c15">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">a1</font>,<font color="Maroon" title="c12">b1</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> &amp; <font color="Maroon" title="c15">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="c13">a2</font>,<font color="Maroon" title="c14">b2</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#E6:74_3"><i><font color="Green" title="E54">A9</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E2:74_3_2"/><b>then </b>
( <font color="Maroon" title="c15">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="c7">a</font>,<font color="Maroon" title="c8">b</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> iff ( <font color="Maroon" title="c15">f</font> is  <a href="waybel_0.html#V22" title="WAYBEL_0:attr.22">directed-sups-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="c13">a2</font></span>)</span>,<span class="p1">(<span class="default"><a href="yellow21.html#K4" title="YELLOW21:func.4">latt</a> <font color="Maroon" title="c14">b2</font></span>)</span> &amp; <font color="Maroon" title="c15">f</font> is  <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="c11">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="c12">b1</font></span>)</span> ) )
 <b>by </b><i><a class="ref" href="waybel34.html#T14" target="_self" title="WAYBEL34:th.14">Th14</a>, <a class="ref" href="yellow21.html#T15" title="YELLOW21:th.15">YELLOW21:15</a></i>;<br/><a NAME="E3:74_3_2"/><b>then </b>
( <font color="Maroon" title="c15">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="c7">a</font>,<font color="Maroon" title="c8">b</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> iff <font color="Maroon" title="c15">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="c9">a'</font>,<font color="Maroon" title="c10">b'</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> )
 <b>by </b><i><a class="ref" href="waybel34.html#T48" target="_self" title="WAYBEL34:th.48">Th48</a></i>;<br/><b>hence </b><a NAME="E4:74_3_2"/>
( <font color="Maroon" title="c15">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="c7">a</font>,<font color="Maroon" title="c8">b</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> iff <font color="Maroon" title="c15">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font>) )
 <b>by </b><i><a class="ref" href="altcat_1.html#D2" title="ALTCAT_1:def.2">ALTCAT_1:def 2</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E8:74_3"/>
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font>)
 <b>by </b><i><a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E15:74"/>
<font color="Maroon" title="c1">W</font> <a href="waybel34.html#K8" title="WAYBEL34:func.8">-INF(SC)_category</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow20.html#K2" title="YELLOW20:func.2">Intersect</a> <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>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">W</font> <a href="yellow21.html#K6" title="YELLOW21:func.6">-UPS_category</a> </span>)</span>
 <b>by </b><i><a class="txt" href="waybel34.html#E11:74"><i><font color="Green" title="E50">A5</font></i></a>, <a class="txt" href="waybel34.html#E12:74"><i><font color="Green" title="E51">A6</font></i></a>, <a class="txt" href="waybel34.html#E13:74"><i><font color="Green" title="E52">A7</font></i></a></i>;<br/>


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