<?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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <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#K15" title="FUNCTOR0:func.15">"</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">W</font> <a href="waybel34.html#K7" title="WAYBEL34:func.7">UpperAdj</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c1">W</font> <a href="waybel34.html#K7" title="WAYBEL34:func.7">UpperAdj</a> </span>)</span> <a href="functor0.html#K15" title="FUNCTOR0:func.15">"</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">W</font> <a href="waybel34.html#K6" title="WAYBEL34:func.6">LowerAdj</a>  )</span><br/>

<a NAME="E1:35"/><i><font color="Green" title="E25">A1</font></i>: 
 ex <font color="Olive" title="b1">x</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">x</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>set </b><font color="Maroon" title="c2">B</font> = <font color="Maroon" title="c1">W</font> <a href="waybel34.html#K5" title="WAYBEL34:func.5">-SUP_category</a> ;<br/>
<b>set </b><font color="Maroon" title="c3">F</font> = <font color="Maroon" title="c1">W</font> <a href="waybel34.html#K6" title="WAYBEL34:func.6">LowerAdj</a> ;<br/>
<b>set </b><font color="Maroon" title="c4">G</font> = <font color="Maroon" title="c1">W</font> <a href="waybel34.html#K7" title="WAYBEL34:func.7">UpperAdj</a> ;<br/>
<div><i><font color="Green" title="E26">A2</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">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#K5" title="WAYBEL34:func.5">-SUP_category</a> </span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">W</font> <a href="waybel34.html#K7" title="WAYBEL34:func.7">UpperAdj</a> </span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c5">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font></span><br/><b>thus </b><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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">W</font> <a href="waybel34.html#K7" title="WAYBEL34:func.7">UpperAdj</a> </span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c5">a</font></span>)</span> = 
 <a href="yellow21.html#K4" title="YELLOW21:func.4">latt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">W</font> <a href="waybel34.html#K7" title="WAYBEL34:func.7">UpperAdj</a> </span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c5">a</font></span>)</span>
<b>by </b><i><a class="ref" href="waybel34.html#D6" target="_self" title="WAYBEL34:def.6">Def6</a></i>
<br/>.= 
 <a href="yellow21.html#K4" title="YELLOW21:func.4">latt</a> <font color="Maroon" title="c5">a</font>
<b>by </b><i><a class="ref" href="waybel34.html#D7" target="_self" title="WAYBEL34:def.7">Def7</a></i>
<br/>.= 
<font color="Maroon" title="c5">a</font>

; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c6">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#K5" title="WAYBEL34:func.5">-SUP_category</a> </span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">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>  implies for <font color="Olive" title="b1">f</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> holds  <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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">W</font> <a href="waybel34.html#K7" title="WAYBEL34:func.7">UpperAdj</a> </span>)</span> <a href="functor0.html#K8" title="FUNCTOR0:func.8">.</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> )</span><br/><b>assume </b><a NAME="E1:35_2"/><i><font color="Green" title="E27">A3</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="c5">a</font>,<font color="Maroon" title="c6">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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">f</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> holds  <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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">W</font> <a href="waybel34.html#K7" title="WAYBEL34:func.7">UpperAdj</a> </span>)</span> <a href="functor0.html#K8" title="FUNCTOR0:func.8">.</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font></span><br/><a NAME="E2:35_2"/><b>then </b><i><font color="Green" title="E28">A4</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#K7" title="WAYBEL34:func.7">UpperAdj</a> </span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c6">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#K7" title="WAYBEL34:func.7">UpperAdj</a> </span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c5">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="ref" href="functor0.html#D20" title="FUNCTOR0:def.20">FUNCTOR0:def 20</a></i>;<br/><b>let </b><font color="Maroon" title="c7">f</font> be   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">W</font> <a href="waybel34.html#K7" title="WAYBEL34:func.7">UpperAdj</a> </span>)</span> <a href="functor0.html#K8" title="FUNCTOR0:func.8">.</a> <font color="Maroon" title="c7">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">f</font></span><br/><a NAME="E3:35_2"/><i><font color="Green" title="E29">A5</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">W</font> <a href="waybel34.html#K7" title="WAYBEL34:func.7">UpperAdj</a> </span>)</span> <a href="functor0.html#K8" title="FUNCTOR0:func.8">.</a> <font color="Maroon" title="c7">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel34.html#K2" title="WAYBEL34:func.2">UpperAdj</a> <span class="p1">(<span class="default"><a href="yellow21.html#K5" title="YELLOW21:func.5">@</a> <font color="Maroon" title="c7">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel34.html#E1:35_2"><i><font color="Green" title="E27">A3</font></i></a>, <a class="ref" href="waybel34.html#D7" target="_self" title="WAYBEL34:def.7">Def7</a></i>;<br/><a NAME="E4:35_2"/><i><font color="Green" title="E30">A6</font></i>: 
(  <a href="yellow21.html#K5" title="YELLOW21:func.5">@</a> <font color="Maroon" title="c7">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">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#K7" title="WAYBEL34:func.7">UpperAdj</a> </span>)</span> <a href="functor0.html#K8" title="FUNCTOR0:func.8">.</a> <font color="Maroon" title="c7">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#K7" title="WAYBEL34:func.7">UpperAdj</a> </span>)</span> <a href="functor0.html#K8" title="FUNCTOR0:func.8">.</a> <font color="Maroon" title="c7">f</font> )
 <b>by </b><i><a class="txt" href="waybel34.html#E1:35_2"><i><font color="Green" title="E27">A3</font></i></a>, <a class="txt" href="waybel34.html#E2:35_2"><i><font color="Green" title="E28">A4</font></i></a>, <a class="ref" href="yellow21.html#D7" title="YELLOW21:def.7">YELLOW21:def 7</a></i>;<br/><a NAME="E5:35_2"/><i><font color="Green" title="E31">A7</font></i>: 
(  <a href="yellow21.html#K4" title="YELLOW21:func.4">latt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">W</font> <a href="waybel34.html#K7" title="WAYBEL34:func.7">UpperAdj</a> </span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c5">a</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#K7" title="WAYBEL34:func.7">UpperAdj</a> </span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c5">a</font> &amp;  <a href="yellow21.html#K4" title="YELLOW21:func.4">latt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">W</font> <a href="waybel34.html#K7" title="WAYBEL34:func.7">UpperAdj</a> </span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c6">b</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#K7" title="WAYBEL34:func.7">UpperAdj</a> </span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c6">b</font> &amp;  <a href="yellow21.html#K4" title="YELLOW21:func.4">latt</a> <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> &amp;  <a href="yellow21.html#K4" title="YELLOW21:func.4">latt</a> <font color="Maroon" title="c6">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c1">W</font> <a href="waybel34.html#K7" title="WAYBEL34:func.7">UpperAdj</a> </span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c5">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="c5">a</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c1">W</font> <a href="waybel34.html#K7" title="WAYBEL34:func.7">UpperAdj</a> </span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c6">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="c6">b</font> )
 <b>by </b><i><a class="ref" href="waybel34.html#D7" target="_self" title="WAYBEL34:def.7">Def7</a></i>;<br/><a NAME="E6:35_2"/><i><font color="Green" title="E32">A8</font></i>: 
(  <a href="yellow21.html#K4" title="YELLOW21:func.4">latt</a> <font color="Maroon" title="c5">a</font> is <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a> &amp;  <a href="yellow21.html#K4" title="YELLOW21:func.4">latt</a> <font color="Maroon" title="c6">b</font> is <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a> &amp;  <a href="yellow21.html#K5" title="YELLOW21:func.5">@</a> <font color="Maroon" title="c7">f</font> is <a href="waybel_0.html#V18" title="WAYBEL_0:attr.18">sups-preserving</a> )
 <b>by </b><i><a class="txt" href="waybel34.html#E1:35"><i><font color="Green" title="E25">A1</font></i></a>, <a class="txt" href="waybel34.html#E1:35_2"><i><font color="Green" title="E27">A3</font></i></a>, <a class="txt" href="waybel34.html#E4:35_2"><i><font color="Green" title="E30">A6</font></i></a>, <a class="ref" href="waybel34.html#D5" target="_self" title="WAYBEL34:def.5">Def5</a></i>;<br/><b>thus </b><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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">W</font> <a href="waybel34.html#K7" title="WAYBEL34:func.7">UpperAdj</a> </span>)</span> <a href="functor0.html#K8" title="FUNCTOR0:func.8">.</a> <font color="Maroon" title="c7">f</font></span>)</span> = 
 <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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">W</font> <a href="waybel34.html#K7" title="WAYBEL34:func.7">UpperAdj</a> </span>)</span> <a href="functor0.html#K8" title="FUNCTOR0:func.8">.</a> <font color="Maroon" title="c7">f</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="waybel34.html#E2:35_2"><i><font color="Green" title="E28">A4</font></i></a>, <a class="ref" href="waybel34.html#D6" target="_self" title="WAYBEL34:def.6">Def6</a></i>
<br/>.= 
<font color="Maroon" title="c7">f</font>
<b>by </b><i><a class="txt" href="waybel34.html#E3:35_2"><i><font color="Green" title="E29">A5</font></i></a>, <a class="txt" href="waybel34.html#E4:35_2"><i><font color="Green" title="E30">A6</font></i></a>, <a class="txt" href="waybel34.html#E5:35_2"><i><font color="Green" title="E31">A7</font></i></a>, <a class="txt" href="waybel34.html#E6:35_2"><i><font color="Green" title="E32">A8</font></i></a>, <a class="ref" href="waybel34.html#T11" target="_self" title="WAYBEL34:th.11">Th11</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:35"/>
<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#K15" title="FUNCTOR0:func.15">"</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">W</font> <a href="waybel34.html#K7" title="WAYBEL34:func.7">UpperAdj</a> 
 <b>by </b><i><a class="txt" href="waybel34.html#E2:35"><i><font color="Green" title="E26">A2</font></i></a>, <a class="ref" href="yellow20.html#T7" title="YELLOW20:th.7">YELLOW20:7</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c1">W</font> <a href="waybel34.html#K7" title="WAYBEL34:func.7">UpperAdj</a> </span>)</span> <a href="functor0.html#K15" title="FUNCTOR0:func.15">"</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">W</font> <a href="waybel34.html#K6" title="WAYBEL34:func.6">LowerAdj</a> </span><br/>

<b>set </b><font color="Maroon" title="c5">B</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="c6">G</font> = <font color="Maroon" title="c1">W</font> <a href="waybel34.html#K6" title="WAYBEL34:func.6">LowerAdj</a> ;<br/>
<b>set </b><font color="Maroon" title="c7">F</font> = <font color="Maroon" title="c1">W</font> <a href="waybel34.html#K7" title="WAYBEL34:func.7">UpperAdj</a> ;<br/>
<div><i><font color="Green" title="E27">A9</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="35_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c1">W</font> <a href="waybel34.html#K7" title="WAYBEL34:func.7">UpperAdj</a> </span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</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#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c8">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">a</font></span><br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c1">W</font> <a href="waybel34.html#K7" title="WAYBEL34:func.7">UpperAdj</a> </span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</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#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c8">a</font></span>)</span> = 
 <a href="yellow21.html#K4" title="YELLOW21:func.4">latt</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#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c8">a</font></span>)</span>
<b>by </b><i><a class="ref" href="waybel34.html#D7" target="_self" title="WAYBEL34:def.7">Def7</a></i>
<br/>.= 
 <a href="yellow21.html#K4" title="YELLOW21:func.4">latt</a> <font color="Maroon" title="c8">a</font>
<b>by </b><i><a class="ref" href="waybel34.html#D6" target="_self" title="WAYBEL34:def.6">Def6</a></i>
<br/>.= 
<font color="Maroon" title="c8">a</font>

; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">a</font>, <font color="Maroon" title="c9">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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c9">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>  implies for <font color="Olive" title="b1">f</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c9">b</font> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c1">W</font> <a href="waybel34.html#K7" title="WAYBEL34:func.7">UpperAdj</a> </span>)</span> <a href="functor0.html#K8" title="FUNCTOR0:func.8">.</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="Olive" title="b1">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> )</span><br/><b>assume </b><a NAME="E1:35_4"/><i><font color="Green" title="E28">A10</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="c8">a</font>,<font color="Maroon" title="c9">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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">f</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c9">b</font> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c1">W</font> <a href="waybel34.html#K7" title="WAYBEL34:func.7">UpperAdj</a> </span>)</span> <a href="functor0.html#K8" title="FUNCTOR0:func.8">.</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="Olive" title="b1">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font></span><br/><a NAME="E2:35_4"/><b>then </b><i><font color="Green" title="E29">A11</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="c9">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="c8">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="ref" href="functor0.html#D20" title="FUNCTOR0:def.20">FUNCTOR0:def 20</a></i>;<br/><b>let </b><font color="Maroon" title="c10">f</font> be   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c9">b</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c1">W</font> <a href="waybel34.html#K7" title="WAYBEL34:func.7">UpperAdj</a> </span>)</span> <a href="functor0.html#K8" title="FUNCTOR0:func.8">.</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="c10">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">f</font></span><br/><a NAME="E3:35_4"/><i><font color="Green" title="E30">A12</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#K8" title="FUNCTOR0:func.8">.</a> <font color="Maroon" title="c10">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="c10">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel34.html#E1:35_4"><i><font color="Green" title="E28">A10</font></i></a>, <a class="ref" href="waybel34.html#D6" target="_self" title="WAYBEL34:def.6">Def6</a></i>;<br/><a NAME="E4:35_4"/><i><font color="Green" title="E31">A13</font></i>: 
(  <a href="yellow21.html#K5" title="YELLOW21:func.5">@</a> <font color="Maroon" title="c10">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">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="c10">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="c10">f</font> )
 <b>by </b><i><a class="txt" href="waybel34.html#E1:35_4"><i><font color="Green" title="E28">A10</font></i></a>, <a class="txt" href="waybel34.html#E2:35_4"><i><font color="Green" title="E29">A11</font></i></a>, <a class="ref" href="yellow21.html#D7" title="YELLOW21:def.7">YELLOW21:def 7</a></i>;<br/><a NAME="E5:35_4"/><i><font color="Green" title="E32">A14</font></i>: 
(  <a href="yellow21.html#K4" title="YELLOW21:func.4">latt</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#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c8">a</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#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c8">a</font> &amp;  <a href="yellow21.html#K4" title="YELLOW21:func.4">latt</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#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c9">b</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#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c9">b</font> &amp;  <a href="yellow21.html#K4" title="YELLOW21:func.4">latt</a> <font color="Maroon" title="c8">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">a</font> &amp;  <a href="yellow21.html#K4" title="YELLOW21:func.4">latt</a> <font color="Maroon" title="c9">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">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#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c8">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="c8">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="c9">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="c9">b</font> )
 <b>by </b><i><a class="ref" href="waybel34.html#D6" target="_self" title="WAYBEL34:def.6">Def6</a></i>;<br/><a NAME="E6:35_4"/><i><font color="Green" title="E33">A15</font></i>: 
(  <a href="yellow21.html#K4" title="YELLOW21:func.4">latt</a> <font color="Maroon" title="c8">a</font> is <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a> &amp;  <a href="yellow21.html#K4" title="YELLOW21:func.4">latt</a> <font color="Maroon" title="c9">b</font> is <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a> &amp;  <a href="yellow21.html#K5" title="YELLOW21:func.5">@</a> <font color="Maroon" title="c10">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#E1:35"><i><font color="Green" title="E25">A1</font></i></a>, <a class="txt" href="waybel34.html#E1:35_4"><i><font color="Green" title="E28">A10</font></i></a>, <a class="txt" href="waybel34.html#E4:35_4"><i><font color="Green" title="E31">A13</font></i></a>, <a class="ref" href="waybel34.html#D4" target="_self" title="WAYBEL34:def.4">Def4</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c1">W</font> <a href="waybel34.html#K7" title="WAYBEL34:func.7">UpperAdj</a> </span>)</span> <a href="functor0.html#K8" title="FUNCTOR0:func.8">.</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="c10">f</font></span>)</span> = 
 <a href="waybel34.html#K2" title="WAYBEL34:func.2">UpperAdj</a> <span class="p1">(<span class="default"><a href="yellow21.html#K5" title="YELLOW21:func.5">@</a> <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#K8" title="FUNCTOR0:func.8">.</a> <font color="Maroon" title="c10">f</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="waybel34.html#E2:35_4"><i><font color="Green" title="E29">A11</font></i></a>, <a class="ref" href="waybel34.html#D7" target="_self" title="WAYBEL34:def.7">Def7</a></i>
<br/>.= 
<font color="Maroon" title="c10">f</font>
<b>by </b><i><a class="txt" href="waybel34.html#E3:35_4"><i><font color="Green" title="E30">A12</font></i></a>, <a class="txt" href="waybel34.html#E4:35_4"><i><font color="Green" title="E31">A13</font></i></a>, <a class="txt" href="waybel34.html#E5:35_4"><i><font color="Green" title="E32">A14</font></i></a>, <a class="txt" href="waybel34.html#E6:35_4"><i><font color="Green" title="E33">A15</font></i></a>, <a class="ref" href="waybel34.html#T10" target="_self" title="WAYBEL34:th.10">Th10</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:35"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">W</font> <a href="waybel34.html#K7" title="WAYBEL34:func.7">UpperAdj</a> </span>)</span> <a href="functor0.html#K15" title="FUNCTOR0:func.15">"</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">W</font> <a href="waybel34.html#K6" title="WAYBEL34:func.6">LowerAdj</a> 
 <b>by </b><i><a class="txt" href="waybel34.html#E5:35"><i><font color="Green" title="E27">A9</font></i></a>, <a class="ref" href="yellow20.html#T7" title="YELLOW20:th.7">YELLOW20:7</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
