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<b>let </b><font color="Maroon">W</font> be  <a href="setfam_1.html#V2">with_non-empty_element</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">a</font> be   <a href="altcat_1.html#NM1">object</a> of <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="waybel34.html#K9" target="_self">-SUP(SO)_category</a> </span>)</span>, <font color="Maroon">b</font> be   <a href="altcat_1.html#NM1">object</a> of <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="waybel34.html#K9" target="_self">-SUP(SO)_category</a> </span>)</span>;<br/><b>let </b><font color="Maroon">f</font> be    <a href="hidden.html#M1">set</a> ;<br/>





<a NAME="E1:73"/>
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="waybel34.html#K9" target="_self">-SUP(SO)_category</a> </span>)</span> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="waybel34.html#K9" target="_self">-SUP(SO)_category</a> </span>)</span> &amp; the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="waybel34.html#K9" target="_self">-SUP(SO)_category</a> </span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="waybel34.html#K5" target="_self">-SUP_category</a> </span>)</span> )
 
<b>by </b><i><a class="ref" href="altcat_2.html#D11">ALTCAT_2:def 11</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">a1</font> = <font color="Maroon">a</font>, <font color="Maroon">b1</font> = <font color="Maroon">b</font> as   <a href="altcat_1.html#NM1">object</a> of <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="waybel34.html#K5" target="_self">-SUP_category</a> </span>)</span> ;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:73_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="altcat_1.html#K1">^&gt;</a></span>
 ;<br/><a NAME="E2:73_1"/><i><font color="Green">E45</font></i>: 
<span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="altcat_1.html#K1">^&gt;</a></span>
 <b>by </b><i><a class="ref" href="altcat_2.html#T32">ALTCAT_2:32</a></i>;<br/><b>then reconsider </b><font color="Maroon">g</font> = <font color="Maroon">f</font> as   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">a1</font>,<font color="Maroon">b1</font> <b>by </b><i><a class="txt" href="waybel34.html#E5:73"><i><font color="Green">E47</font></i></a></i>;<br/><a NAME="E4:73_1"/>
( <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="yellow21.html#K5">@</a> <font color="Maroon">g</font> &amp;  <a href="waybel34.html#K2" target="_self">UpperAdj</a> <span class="p1">(<span class="default"><a href="yellow21.html#K5">@</a> <font color="Maroon">g</font></span>)</span> is <a href="waybel_0.html#V22">directed-sups-preserving</a> &amp; <font color="Maroon">f</font> is  <a href="waybel_0.html#V18">sups-preserving</a> <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="yellow21.html#K4">latt</a> <font color="Maroon">a1</font></span>)</span>,<span class="p1">(<span class="default"><a href="yellow21.html#K4">latt</a> <font color="Maroon">b1</font></span>)</span> )
 <b>by </b><i><a class="txt" href="waybel34.html#E5:73"><i><font color="Green">E47</font></i></a>, , , , <a class="ref" href="yellow21.html#D7">YELLOW21:def 7</a></i>;<br/><b>hence </b><a NAME="E5:73_1"/>
 ex <font color="Olive">g</font> being <a href="waybel_0.html#V18">sups-preserving</a> <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="yellow21.html#K4">latt</a> <font color="Maroon">a</font></span>)</span>,<span class="p1">(<span class="default"><a href="yellow21.html#K4">latt</a> <font color="Maroon">b</font></span>)</span> st <br/>( <font color="Olive">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> &amp;  <a href="waybel34.html#K2" target="_self">UpperAdj</a> <font color="Olive">g</font> is <a href="waybel_0.html#V22">directed-sups-preserving</a> )
 ;<br/></div>
<b>end;</b></div>

<b>given </b><font color="Maroon">g</font> being  <a href="waybel_0.html#V18">sups-preserving</a> <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="yellow21.html#K4">latt</a> <font color="Maroon">a</font></span>)</span>,<span class="p1">(<span class="default"><a href="yellow21.html#K4">latt</a> <font color="Maroon">b</font></span>)</span><b> such that </b><a NAME="E4:73"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> &amp;  <a href="waybel34.html#K2" target="_self">UpperAdj</a> <font color="Maroon">g</font> is <a href="waybel_0.html#V22">directed-sups-preserving</a> )
 ;<br/>

<a NAME="E5:73"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="altcat_1.html#K1">^&gt;</a></span>
 
<b>by </b><i><a class="txt" href="waybel34.html#E5:73"><i><font color="Green">E47</font></i></a>, </i>;<br/>
<b>reconsider </b><font color="Maroon">g</font> = <font color="Maroon">f</font> as   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">a1</font>,<font color="Maroon">b1</font> <b>by </b><i><a class="txt" href="waybel34.html#E5:73"><i><font color="Green">E47</font></i></a>, </i>;<br/>
<a NAME="E7:73"/>
<font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="yellow21.html#K5">@</a> <font color="Maroon">g</font>
 
<b>by </b><i>, <a class="ref" href="yellow21.html#D7">YELLOW21:def 7</a></i>;<br/>
<b>hence </b><a NAME="E8:73"/>
<font color="Maroon">f</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="altcat_1.html#K1">^&gt;</a></span>
 <b>by </b><i><a class="txt" href="waybel34.html#E5:73"><i><font color="Green">E47</font></i></a>, , </i>;<br/>


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