<?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"> <font color="Maroon" title="c1">W</font> <a href="waybel34.html#K10" title="WAYBEL34:func.10">-CL_category</a> ,<font color="Maroon" title="c1">W</font> <a href="waybel34.html#K11" title="WAYBEL34:func.11">-CL-opp_category</a>  <a href="yellow20.html#R3" title="YELLOW20:pred.3">are_anti-isomorphic_under</a> <font color="Maroon" title="c1">W</font> <a href="waybel34.html#K6" title="WAYBEL34:func.6">LowerAdj</a> </span><br/>

<b>set </b><font color="Maroon" title="c2">A1</font> = <font color="Maroon" title="c1">W</font> <a href="waybel34.html#K4" title="WAYBEL34:func.4">-INF_category</a> ;<br/>
<b>set </b><font color="Maroon" title="c3">A2</font> = <font color="Maroon" title="c1">W</font> <a href="waybel34.html#K5" title="WAYBEL34:func.5">-SUP_category</a> ;<br/>
<b>reconsider </b><font color="Maroon" title="c4">B1</font> = <font color="Maroon" title="c1">W</font> <a href="waybel34.html#K10" title="WAYBEL34:func.10">-CL_category</a>  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="ref" href="altcat_4.html#T36" title="ALTCAT_4:th.36">ALTCAT_4:36</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">B2</font> = <font color="Maroon" title="c1">W</font> <a href="waybel34.html#K11" title="WAYBEL34:func.11">-CL-opp_category</a>  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#K5" title="WAYBEL34:func.5">-SUP_category</a>  <b>by </b><i><a class="ref" href="altcat_4.html#T36" title="ALTCAT_4:th.36">ALTCAT_4:36</a></i>;<br/>
<b>set </b><font color="Maroon" title="c6">F</font> = <font color="Maroon" title="c1">W</font> <a href="waybel34.html#K6" title="WAYBEL34:func.6">LowerAdj</a> ;<br/>
<a NAME="E3:84"/><i><font color="Green" title="E54">A1</font></i>: 
 ex <font color="Olive" title="b1">a</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">W</font>
 
<b>by </b><i><a class="ref" href="setfam_1.html#D11" title="SETFAM_1:def.11">SETFAM_1:def 11</a></i>;<br/>
<a NAME="E4:84"/><i><font color="Green" title="E55">A2</font></i>: 
<font color="Maroon" title="c1">W</font> <a href="waybel34.html#K6" title="WAYBEL34:func.6">LowerAdj</a>  is <a href="functor0.html#V21" title="FUNCTOR0:attr.21">bijective</a>
 
;<br/>
<a NAME="E5:84"/><i><font color="Green" title="E56">A3</font></i>: 
for <font color="Olive" title="b1">a</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">W</font> <a href="waybel34.html#K4" title="WAYBEL34:func.4">-INF_category</a> </span>)</span> holds <br/> ( <font color="Olive" title="b1">a</font> is   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c4">B1</font> iff <span class="p1">(<span class="default"><font color="Maroon" title="c1">W</font> <a href="waybel34.html#K6" title="WAYBEL34:func.6">LowerAdj</a> </span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Olive" title="b1">a</font> is   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c5">B2</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">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"> ( <font color="Maroon" title="c7">a</font> is   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c4">B1</font> iff <span class="p1">(<span class="default"><font color="Maroon" title="c1">W</font> <a href="waybel34.html#K6" title="WAYBEL34:func.6">LowerAdj</a> </span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c7">a</font> is   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c5">B2</font> )</span><br/>

<a NAME="E1:84_1"/><i><font color="Green" title="E57">A4</font></i>: 
(  <a href="yellow21.html#K4" title="YELLOW21:func.4">latt</a> <font color="Maroon" title="c7">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">a</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c1">W</font> <a href="waybel34.html#K6" title="WAYBEL34:func.6">LowerAdj</a> </span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c7">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="c7">a</font> )
 
<b>by </b><i><a class="ref" href="waybel34.html#D6" target="_self" title="WAYBEL34:def.6">Def6</a></i>;<br/>
<a NAME="E2:84_1"/><i><font color="Green" title="E58">A5</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="yellow21.html#K4" title="YELLOW21:func.4">latt</a> <font color="Maroon" title="c7">a</font></span>)</span> <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="#E3:84"><i><font color="Green" title="E54">A1</font></i></a>, <a class="ref" href="waybel34.html#D4" target="_self" title="WAYBEL34:def.4">Def4</a></i>;<br/>
<a NAME="E3:84_1"/><b>then </b>
( <font color="Maroon" title="c7">a</font> is   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c4">B1</font> iff (  <a href="yellow21.html#K4" title="YELLOW21:func.4">latt</a> <font color="Maroon" title="c7">a</font> is <a href="orders_2.html#V1" title="ORDERS_2:attr.1">strict</a> &amp;  <a href="yellow21.html#K4" title="YELLOW21:func.4">latt</a> <font color="Maroon" title="c7">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="c7">a</font> is <a href="waybel_3.html#V3" title="WAYBEL_3:attr.3">continuous</a> ) )
 
<b>by </b><i><a class="ref" href="waybel34.html#T52" target="_self" title="WAYBEL34:th.52">Th52</a></i>;<br/>
<b>hence </b><a NAME="E4:84_1"/>
( <font color="Maroon" title="c7">a</font> is   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c4">B1</font> iff <span class="p1">(<span class="default"><font color="Maroon" title="c1">W</font> <a href="waybel34.html#K6" title="WAYBEL34:func.6">LowerAdj</a> </span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c7">a</font> is   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c5">B2</font> )
 <b>by </b><i><a class="txt" href="#E1:84_1"><i><font color="Green" title="E57">A4</font></i></a>, <a class="txt" href="#E2:84_1"><i><font color="Green" title="E58">A5</font></i></a>, <a class="ref" href="waybel34.html#T54" target="_self" title="WAYBEL34:th.54">Th54</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>
<div><i><font color="Green" title="E57">A6</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="84_2"><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 <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="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#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">a1</font>, <font color="Olive" title="b2">b1</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c4">B1</font>  st <font color="Olive" title="b1">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">a</font> &amp; <font color="Olive" title="b2">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b</font> holds <br/>for <font color="Olive" title="b3">a2</font>, <font color="Olive" title="b4">b2</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c5">B2</font>  st <font color="Olive" title="b3">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">W</font> <a href="waybel34.html#K6" title="WAYBEL34:func.6">LowerAdj</a> </span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c7">a</font> &amp; <font color="Olive" title="b4">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">W</font> <a href="waybel34.html#K6" title="WAYBEL34:func.6">LowerAdj</a> </span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c8">b</font> holds <br/>for <font color="Olive" title="b5">f</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font> holds <br/> ( ( <font color="Olive" title="b5">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="Olive" title="b1">a1</font>,<font color="Olive" title="b2">b1</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> implies <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="Olive" title="b5">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="Olive" title="b4">b2</font>,<font color="Olive" title="b3">a2</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> ) &amp; ( <span class="p1">(<span class="default"><font color="Maroon" title="c1">W</font> <a href="waybel34.html#K6" title="WAYBEL34:func.6">LowerAdj</a> </span>)</span> <a href="functor0.html#K8" title="FUNCTOR0:func.8">.</a> <font color="Olive" title="b5">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="Olive" title="b4">b2</font>,<font color="Olive" title="b3">a2</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> implies <font color="Olive" title="b5">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="Olive" title="b1">a1</font>,<font color="Olive" title="b2">b1</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> ) ) )</span><br/><b>assume </b><a NAME="E1:84_2"/><i><font color="Green" title="E58">A7</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> <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">a1</font>, <font color="Olive" title="b2">b1</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c4">B1</font>  st <font color="Olive" title="b1">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">a</font> &amp; <font color="Olive" title="b2">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b</font> holds <br/>for <font color="Olive" title="b3">a2</font>, <font color="Olive" title="b4">b2</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c5">B2</font>  st <font color="Olive" title="b3">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">W</font> <a href="waybel34.html#K6" title="WAYBEL34:func.6">LowerAdj</a> </span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c7">a</font> &amp; <font color="Olive" title="b4">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">W</font> <a href="waybel34.html#K6" title="WAYBEL34:func.6">LowerAdj</a> </span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c8">b</font> holds <br/>for <font color="Olive" title="b5">f</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font> holds <br/> ( ( <font color="Olive" title="b5">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="Olive" title="b1">a1</font>,<font color="Olive" title="b2">b1</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> implies <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="Olive" title="b5">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="Olive" title="b4">b2</font>,<font color="Olive" title="b3">a2</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> ) &amp; ( <span class="p1">(<span class="default"><font color="Maroon" title="c1">W</font> <a href="waybel34.html#K6" title="WAYBEL34:func.6">LowerAdj</a> </span>)</span> <a href="functor0.html#K8" title="FUNCTOR0:func.8">.</a> <font color="Olive" title="b5">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="Olive" title="b4">b2</font>,<font color="Olive" title="b3">a2</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> implies <font color="Olive" title="b5">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="Olive" title="b1">a1</font>,<font color="Olive" title="b2">b1</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> ) )</span><br/><b>let </b><font color="Maroon" title="c9">a1</font>, <font color="Maroon" title="c10">b1</font> be   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c4">B1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c9">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">a</font> &amp; <font color="Maroon" title="c10">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b</font> implies for <font color="Olive" title="b1">a2</font>, <font color="Olive" title="b2">b2</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c5">B2</font>  st <font color="Olive" title="b1">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">W</font> <a href="waybel34.html#K6" title="WAYBEL34:func.6">LowerAdj</a> </span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c7">a</font> &amp; <font color="Olive" title="b2">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">W</font> <a href="waybel34.html#K6" title="WAYBEL34:func.6">LowerAdj</a> </span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c8">b</font> holds <br/>for <font color="Olive" title="b3">f</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font> holds <br/> ( ( <font color="Olive" title="b3">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">a1</font>,<font color="Maroon" title="c10">b1</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> implies <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="Olive" title="b3">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="Olive" title="b2">b2</font>,<font color="Olive" title="b1">a2</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> ) &amp; ( <span class="p1">(<span class="default"><font color="Maroon" title="c1">W</font> <a href="waybel34.html#K6" title="WAYBEL34:func.6">LowerAdj</a> </span>)</span> <a href="functor0.html#K8" title="FUNCTOR0:func.8">.</a> <font color="Olive" title="b3">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="Olive" title="b2">b2</font>,<font color="Olive" title="b1">a2</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> implies <font color="Olive" title="b3">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">a1</font>,<font color="Maroon" title="c10">b1</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> ) ) )</span><br/><b>assume </b><a NAME="E2:84_2"/><i><font color="Green" title="E59">A8</font></i>: 
( <font color="Maroon" title="c9">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">a</font> &amp; <font color="Maroon" title="c10">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b</font> )
 ; <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">a2</font>, <font color="Olive" title="b2">b2</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c5">B2</font>  st <font color="Olive" title="b1">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">W</font> <a href="waybel34.html#K6" title="WAYBEL34:func.6">LowerAdj</a> </span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c7">a</font> &amp; <font color="Olive" title="b2">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">W</font> <a href="waybel34.html#K6" title="WAYBEL34:func.6">LowerAdj</a> </span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c8">b</font> holds <br/>for <font color="Olive" title="b3">f</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font> holds <br/> ( ( <font color="Olive" title="b3">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">a1</font>,<font color="Maroon" title="c10">b1</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> implies <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="Olive" title="b3">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="Olive" title="b2">b2</font>,<font color="Olive" title="b1">a2</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> ) &amp; ( <span class="p1">(<span class="default"><font color="Maroon" title="c1">W</font> <a href="waybel34.html#K6" title="WAYBEL34:func.6">LowerAdj</a> </span>)</span> <a href="functor0.html#K8" title="FUNCTOR0:func.8">.</a> <font color="Olive" title="b3">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="Olive" title="b2">b2</font>,<font color="Olive" title="b1">a2</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> implies <font color="Olive" title="b3">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">a1</font>,<font color="Maroon" title="c10">b1</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> ) )</span><br/><b>let </b><font color="Maroon" title="c11">a2</font>, <font color="Maroon" title="c12">b2</font> be   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c5">B2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c11">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">W</font> <a href="waybel34.html#K6" title="WAYBEL34:func.6">LowerAdj</a> </span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c7">a</font> &amp; <font color="Maroon" title="c12">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">W</font> <a href="waybel34.html#K6" title="WAYBEL34:func.6">LowerAdj</a> </span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c8">b</font> 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="c7">a</font>,<font color="Maroon" title="c8">b</font> holds <br/> ( ( <font color="Olive" title="b1">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">a1</font>,<font color="Maroon" title="c10">b1</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> implies <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="Olive" title="b1">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="c12">b2</font>,<font color="Maroon" title="c11">a2</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> ) &amp; ( <span class="p1">(<span class="default"><font color="Maroon" title="c1">W</font> <a href="waybel34.html#K6" title="WAYBEL34:func.6">LowerAdj</a> </span>)</span> <a href="functor0.html#K8" title="FUNCTOR0:func.8">.</a> <font color="Olive" title="b1">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="c12">b2</font>,<font color="Maroon" title="c11">a2</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> implies <font color="Olive" title="b1">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">a1</font>,<font color="Maroon" title="c10">b1</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> ) ) )</span><br/><b>assume </b><a NAME="E3:84_2"/><i><font color="Green" title="E60">A9</font></i>: 
( <font color="Maroon" title="c11">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">W</font> <a href="waybel34.html#K6" title="WAYBEL34:func.6">LowerAdj</a> </span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c7">a</font> &amp; <font color="Maroon" title="c12">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">W</font> <a href="waybel34.html#K6" title="WAYBEL34:func.6">LowerAdj</a> </span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c8">b</font> )
 ; <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="c7">a</font>,<font color="Maroon" title="c8">b</font> holds <br/> ( ( <font color="Olive" title="b1">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">a1</font>,<font color="Maroon" title="c10">b1</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> implies <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="Olive" title="b1">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="c12">b2</font>,<font color="Maroon" title="c11">a2</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> ) &amp; ( <span class="p1">(<span class="default"><font color="Maroon" title="c1">W</font> <a href="waybel34.html#K6" title="WAYBEL34:func.6">LowerAdj</a> </span>)</span> <a href="functor0.html#K8" title="FUNCTOR0:func.8">.</a> <font color="Olive" title="b1">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="c12">b2</font>,<font color="Maroon" title="c11">a2</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> implies <font color="Olive" title="b1">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">a1</font>,<font color="Maroon" title="c10">b1</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> ) )</span><br/><b>let </b><font color="Maroon" title="c13">f</font> be   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( <font color="Maroon" title="c13">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">a1</font>,<font color="Maroon" title="c10">b1</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> implies <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="c13">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="c12">b2</font>,<font color="Maroon" title="c11">a2</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> ) &amp; ( <span class="p1">(<span class="default"><font color="Maroon" title="c1">W</font> <a href="waybel34.html#K6" title="WAYBEL34:func.6">LowerAdj</a> </span>)</span> <a href="functor0.html#K8" title="FUNCTOR0:func.8">.</a> <font color="Maroon" title="c13">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="c12">b2</font>,<font color="Maroon" title="c11">a2</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> implies <font color="Maroon" title="c13">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">a1</font>,<font color="Maroon" title="c10">b1</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> ) )</span><br/><a NAME="E4:84_2"/>
<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="c8">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="c7">a</font></span>)</span></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="#E1:84_2"><i><font color="Green" title="E58">A7</font></i></a>, <a class="ref" href="functor0.html#D20" title="FUNCTOR0:def.20">FUNCTOR0:def 20</a></i>;<br/><a NAME="E5:84_2"/><b>then </b><i><font color="Green" title="E61">A10</font></i>: 
(  <a href="yellow21.html#K5" title="YELLOW21:func.5">@</a> <font color="Maroon" title="c13">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">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="c13">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="c13">f</font> &amp;  <a href="yellow21.html#K4" title="YELLOW21:func.4">latt</a> <font color="Maroon" title="c9">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">a1</font> &amp;  <a href="yellow21.html#K4" title="YELLOW21:func.4">latt</a> <font color="Maroon" title="c10">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">b1</font> &amp;  <a href="yellow21.html#K4" title="YELLOW21:func.4">latt</a> <font color="Maroon" title="c7">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">a</font> &amp;  <a href="yellow21.html#K4" title="YELLOW21:func.4">latt</a> <font color="Maroon" title="c8">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b</font> &amp;  <a href="yellow21.html#K4" title="YELLOW21:func.4">latt</a> <font color="Maroon" title="c11">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">a2</font> &amp;  <a href="yellow21.html#K4" title="YELLOW21:func.4">latt</a> <font color="Maroon" title="c12">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">b2</font> )
 <b>by </b><i><a class="txt" href="#E1:84_2"><i><font color="Green" title="E58">A7</font></i></a>, <a class="ref" href="yellow21.html#D7" title="YELLOW21:def.7">YELLOW21:def 7</a></i>;<br/><a NAME="E6:84_2"/><b>then </b><i><font color="Green" title="E62">A11</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c1">W</font> <a href="waybel34.html#K6" title="WAYBEL34:func.6">LowerAdj</a> </span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c7">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="c7">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="c8">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="c8">b</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c1">W</font> <a href="waybel34.html#K6" title="WAYBEL34:func.6">LowerAdj</a> </span>)</span> <a href="functor0.html#K8" title="FUNCTOR0:func.8">.</a> <font color="Maroon" title="c13">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="c13">f</font></span>)</span> &amp;  <a href="yellow21.html#K5" title="YELLOW21:func.5">@</a> <font color="Maroon" title="c13">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="#E3:84"><i><font color="Green" title="E54">A1</font></i></a>, <a class="txt" href="#E1:84_2"><i><font color="Green" title="E58">A7</font></i></a>, <a class="ref" href="waybel34.html#D4" target="_self" title="WAYBEL34:def.4">Def4</a>, <a class="ref" href="waybel34.html#D6" target="_self" title="WAYBEL34:def.6">Def6</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c14">g</font> = <font color="Maroon" title="c13">f</font> as  <a href="waybel_0.html#V17" title="WAYBEL_0:attr.17">infs-preserving</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="yellow21.html#K4" title="YELLOW21:func.4">latt</a> <font color="Maroon" title="c9">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="c10">b1</font></span>)</span> <b>by </b><i><a class="txt" href="#E3:84"><i><font color="Green" title="E54">A1</font></i></a>, <a class="txt" href="#E1:84_2"><i><font color="Green" title="E58">A7</font></i></a>, <a class="txt" href="#E2:84_2"><i><font color="Green" title="E59">A8</font></i></a>, <a class="txt" href="#E5:84_2"><i><font color="Green" title="E61">A10</font></i></a>, <a class="ref" href="waybel34.html#D4" target="_self" title="WAYBEL34:def.4">Def4</a></i>;<br/><a NAME="E8:84_2"/><i><font color="Green" title="E63">A12</font></i>: 
 <a href="waybel34.html#K2" title="WAYBEL34:func.2">UpperAdj</a> <span class="p1">(<span class="default"><a href="waybel34.html#K1" title="WAYBEL34:func.1">LowerAdj</a> <font color="Maroon" title="c14">g</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c14">g</font>
 <b>by </b><i><a class="ref" href="waybel34.html#T10" target="_self" title="WAYBEL34:th.10">Th10</a></i>;<br/><a NAME="E9:84_2"/><b>then </b>
( <font color="Maroon" title="c13">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">a1</font>,<font color="Maroon" title="c10">b1</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> iff  <a href="waybel34.html#K2" title="WAYBEL34:func.2">UpperAdj</a> <span class="p1">(<span class="default"><a href="waybel34.html#K1" title="WAYBEL34:func.1">LowerAdj</a> <font color="Maroon" title="c14">g</font></span>)</span> is <a href="waybel_0.html#V22" title="WAYBEL_0:attr.22">directed-sups-preserving</a> )
 <b>by </b><i><a class="ref" href="waybel34.html#T53" target="_self" title="WAYBEL34:th.53">Th53</a></i>;<br/><b>hence </b><a NAME="E10:84_2"/>
( <font color="Maroon" title="c13">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">a1</font>,<font color="Maroon" title="c10">b1</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> implies <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="c13">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="c12">b2</font>,<font color="Maroon" title="c11">a2</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> )
 <b>by </b><i><a class="txt" href="#E2:84_2"><i><font color="Green" title="E59">A8</font></i></a>, <a class="txt" href="#E3:84_2"><i><font color="Green" title="E60">A9</font></i></a>, <a class="txt" href="#E5:84_2"><i><font color="Green" title="E61">A10</font></i></a>, <a class="txt" href="#E6:84_2"><i><font color="Green" title="E62">A11</font></i></a>, <a class="ref" href="waybel34.html#T55" target="_self" title="WAYBEL34:th.55">Th55</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#K6" title="WAYBEL34:func.6">LowerAdj</a> </span>)</span> <a href="functor0.html#K8" title="FUNCTOR0:func.8">.</a> <font color="Maroon" title="c13">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="c12">b2</font>,<font color="Maroon" title="c11">a2</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> implies <font color="Maroon" title="c13">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">a1</font>,<font color="Maroon" title="c10">b1</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> )</span><br/><b>assume </b><a NAME="E11:84_2"/>
<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="c13">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="c12">b2</font>,<font color="Maroon" title="c11">a2</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c13">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">a1</font>,<font color="Maroon" title="c10">b1</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span></span><br/><a NAME="E12:84_2"/><b>then </b>
 ex <font color="Olive" title="b1">g</font> being <a href="waybel_0.html#V18" title="WAYBEL_0:attr.18">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="c12">b2</font></span>)</span>,<span class="p1">(<span class="default"><a href="yellow21.html#K4" title="YELLOW21:func.4">latt</a> <font color="Maroon" title="c11">a2</font></span>)</span> st <br/>( <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="c13">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g</font> &amp;  <a href="waybel34.html#K2" title="WAYBEL34:func.2">UpperAdj</a> <font color="Olive" title="b1">g</font> is <a href="waybel_0.html#V22" title="WAYBEL_0:attr.22">directed-sups-preserving</a> )
 <b>by </b><i><a class="ref" href="waybel34.html#T55" target="_self" title="WAYBEL34:th.55">Th55</a></i>;<br/><b>hence </b><a NAME="E13:84_2"/>
<font color="Maroon" title="c13">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">a1</font>,<font color="Maroon" title="c10">b1</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span>
 <b>by </b><i><a class="txt" href="#E2:84_2"><i><font color="Green" title="E59">A8</font></i></a>, <a class="txt" href="#E3:84_2"><i><font color="Green" title="E60">A9</font></i></a>, <a class="txt" href="#E5:84_2"><i><font color="Green" title="E61">A10</font></i></a>, <a class="txt" href="#E6:84_2"><i><font color="Green" title="E62">A11</font></i></a>, <a class="txt" href="#E8:84_2"><i><font color="Green" title="E63">A12</font></i></a>, <a class="ref" href="waybel34.html#T53" target="_self" title="WAYBEL34:th.53">Th53</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>
<a NAME="E7:84"/>
<font color="Maroon" title="c4">B1</font>,<font color="Maroon" title="c5">B2</font> <a href="yellow20.html#R3" title="YELLOW20:pred.3">are_anti-isomorphic_under</a> <font color="Maroon" title="c1">W</font> <a href="waybel34.html#K6" title="WAYBEL34:func.6">LowerAdj</a> 
 
<b>from </b><i><a class="ref" href="yellow20.html#S5" title="YELLOW20:sch.5">YELLOW20:sch 5</a>(<a class="txt" href="#E4:84"><i><font color="Green" title="E55">A2</font></i></a>, <a class="txt" href="#E5:84"><i><font color="Green" title="E56">A3</font></i></a>, <a class="txt" href="#E6:84"><i><font color="Green" title="E57">A6</font></i></a>);<br/></i>
<b>hence </b><a NAME="E8:84"/>
<font color="Maroon" title="c1">W</font> <a href="waybel34.html#K10" title="WAYBEL34:func.10">-CL_category</a> ,<font color="Maroon" title="c1">W</font> <a href="waybel34.html#K11" title="WAYBEL34:func.11">-CL-opp_category</a>  <a href="yellow20.html#R3" title="YELLOW20:pred.3">are_anti-isomorphic_under</a> <font color="Maroon" title="c1">W</font> <a href="waybel34.html#K6" title="WAYBEL34:func.6">LowerAdj</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
