<?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"> 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#K9" title="WAYBEL34:func.9">-SUP(SO)_category</a> </span>)</span><br/> for <font color="Olive" title="b3">f</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  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="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> iff  ex <font color="Olive" title="b4">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="Olive" title="b1">a</font></span>)</span>,<span class="p1">(<span class="default"><a href="yellow21.html#K4" title="YELLOW21:func.4">latt</a> <font color="Olive" title="b2">b</font></span>)</span> st <br/>( <font color="Olive" title="b4">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">f</font> &amp;  <a href="waybel34.html#K2" title="WAYBEL34:func.2">UpperAdj</a> <font color="Olive" title="b4">g</font> is <a href="waybel_0.html#V22" title="WAYBEL_0:attr.22">directed-sups-preserving</a> ) )</span><br/><b>let </b><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">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#K9" title="WAYBEL34:func.9">-SUP(SO)_category</a> </span>)</span>; <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="hidden.html#M1" title="HIDDEN:mode.1">set</a>  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="c2">a</font>,<font color="Maroon" title="c3">b</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> iff  ex <font color="Olive" title="b2">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="c2">a</font></span>)</span>,<span class="p1">(<span class="default"><a href="yellow21.html#K4" title="YELLOW21:func.4">latt</a> <font color="Maroon" title="c3">b</font></span>)</span> st <br/>( <font color="Olive" title="b2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> &amp;  <a href="waybel34.html#K2" title="WAYBEL34:func.2">UpperAdj</a> <font color="Olive" title="b2">g</font> is <a href="waybel_0.html#V22" title="WAYBEL_0:attr.22">directed-sups-preserving</a> ) )</span><br/><b>let </b><font color="Maroon" title="c4">f</font> be    <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="c4">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="c2">a</font>,<font color="Maroon" title="c3">b</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> iff  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="c2">a</font></span>)</span>,<span class="p1">(<span class="default"><a href="yellow21.html#K4" title="YELLOW21:func.4">latt</a> <font color="Maroon" title="c3">b</font></span>)</span> st <br/>( <font color="Olive" title="b1">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</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> ) )</span><br/>





<a NAME="E1:73"/>
( <font color="Maroon" title="c2">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 <span class="p1">(<span class="default"><font color="Maroon" title="c1">W</font> <a href="waybel34.html#K9" title="WAYBEL34:func.9">-SUP(SO)_category</a> </span>)</span> &amp; <font color="Maroon" title="c3">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 <span class="p1">(<span class="default"><font color="Maroon" title="c1">W</font> <a href="waybel34.html#K9" title="WAYBEL34:func.9">-SUP(SO)_category</a> </span>)</span> &amp; 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#K9" title="WAYBEL34:func.9">-SUP(SO)_category</a> </span>)</span> <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#K5" title="WAYBEL34:func.5">-SUP_category</a> </span>)</span> )
 
<b>by </b><i><a class="ref" href="altcat_2.html#D11" title="ALTCAT_2:def.11">ALTCAT_2:def 11</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c5">a1</font> = <font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c6">b1</font> = <font color="Maroon" title="c3">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#K5" title="WAYBEL34:func.5">-SUP_category</a> </span>)</span> ;<br/>
<div><b>hereby </b> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  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="c2">a</font></span>)</span>,<span class="p1">(<span class="default"><a href="yellow21.html#K4" title="YELLOW21:func.4">latt</a> <font color="Maroon" title="c3">b</font></span>)</span> st <br/>( <font color="Olive" title="b1">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</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> ) implies <font color="Maroon" title="c4">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="c2">a</font>,<font color="Maroon" title="c3">b</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> )</span>
<div class="add"><b>assume </b><a NAME="E1:73_1"/><i><font color="Green" title="E46">A1</font></i>: 
<font color="Maroon" title="c4">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="c2">a</font>,<font color="Maroon" title="c3">b</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">  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="c2">a</font></span>)</span>,<span class="p1">(<span class="default"><a href="yellow21.html#K4" title="YELLOW21:func.4">latt</a> <font color="Maroon" title="c3">b</font></span>)</span> st <br/>( <font color="Olive" title="b1">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</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> )</span><br/><a NAME="E2:73_1"/><i><font color="Green" title="E47">A2</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="c2">a</font>,<font color="Maroon" title="c3">b</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c5">a1</font>,<font color="Maroon" title="c6">b1</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_2.html#T32" title="ALTCAT_2:th.32">ALTCAT_2:32</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c7">g</font> = <font color="Maroon" title="c4">f</font> as   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c5">a1</font>,<font color="Maroon" title="c6">b1</font> <b>by </b><i><a class="txt" href="waybel34.html#E1:73_1"><i><font color="Green" title="E46">A1</font></i></a></i>;<br/><a NAME="E4:73_1"/>
( <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow21.html#K5" title="YELLOW21:func.5">@</a> <font color="Maroon" title="c7">g</font> &amp;  <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">g</font></span>)</span> is <a href="waybel_0.html#V22" title="WAYBEL_0:attr.22">directed-sups-preserving</a> &amp; <font color="Maroon" title="c4">f</font> is  <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="c5">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="c6">b1</font></span>)</span> )
 <b>by </b><i><a class="txt" href="waybel34.html#E1:73_1"><i><font color="Green" title="E46">A1</font></i></a>, <a class="txt" href="waybel34.html#E2:73_1"><i><font color="Green" title="E47">A2</font></i></a>, <a class="ref" href="waybel34.html#D11" target="_self" title="WAYBEL34:def.11">Def11</a>, <a class="ref" href="waybel34.html#T16" target="_self" title="WAYBEL34:th.16">Th16</a>, <a class="ref" href="yellow21.html#D7" title="YELLOW21:def.7">YELLOW21:def 7</a></i>;<br/><b>hence </b><a NAME="E5:73_1"/>
 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="c2">a</font></span>)</span>,<span class="p1">(<span class="default"><a href="yellow21.html#K4" title="YELLOW21:func.4">latt</a> <font color="Maroon" title="c3">b</font></span>)</span> st <br/>( <font color="Olive" title="b1">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</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> )
 ; <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>given </b><font color="Maroon" title="c7">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="c2">a</font></span>)</span>,<span class="p1">(<span class="default"><a href="yellow21.html#K4" title="YELLOW21:func.4">latt</a> <font color="Maroon" title="c3">b</font></span>)</span><b> such that </b><a NAME="E5:73"/><i><font color="Green" title="E46">A3</font></i>: 
( <font color="Maroon" title="c7">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> &amp;  <a href="waybel34.html#K2" title="WAYBEL34:func.2">UpperAdj</a> <font color="Maroon" title="c7">g</font> is <a href="waybel_0.html#V22" title="WAYBEL_0:attr.22">directed-sups-preserving</a> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">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="c2">a</font>,<font color="Maroon" title="c3">b</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span></span><br/>

<a NAME="E6:73"/><i><font color="Green" title="E47">A4</font></i>: 
<font color="Maroon" title="c4">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="c5">a1</font>,<font color="Maroon" title="c6">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="waybel34.html#E5:73"><i><font color="Green" title="E46">A3</font></i></a>, <a class="ref" href="waybel34.html#T16" target="_self" title="WAYBEL34:th.16">Th16</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">g</font> = <font color="Maroon" title="c4">f</font> as   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c5">a1</font>,<font color="Maroon" title="c6">b1</font> <b>by </b><i><a class="txt" href="waybel34.html#E5:73"><i><font color="Green" title="E46">A3</font></i></a>, <a class="ref" href="waybel34.html#T16" target="_self" title="WAYBEL34:th.16">Th16</a></i>;<br/>
<a NAME="E8:73"/>
<font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow21.html#K5" title="YELLOW21:func.5">@</a> <font color="Maroon" title="c8">g</font>
 
<b>by </b><i><a class="txt" href="waybel34.html#E6:73"><i><font color="Green" title="E47">A4</font></i></a>, <a class="ref" href="yellow21.html#D7" title="YELLOW21:def.7">YELLOW21:def 7</a></i>;<br/>
<b>hence </b><a NAME="E9:73"/>
<font color="Maroon" title="c4">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="c2">a</font>,<font color="Maroon" title="c3">b</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#E5:73"><i><font color="Green" title="E46">A3</font></i></a>, <a class="txt" href="waybel34.html#E6:73"><i><font color="Green" title="E47">A4</font></i></a>, <a class="ref" href="waybel34.html#D11" target="_self" title="WAYBEL34:def.11">Def11</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>
