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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#K8" target="_self">-INF(SC)_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#K8" target="_self">-INF(SC)_category</a> </span>)</span>;<br/><b>let </b><font color="Maroon">f</font> be    <a href="hidden.html#M1">set</a> ;<br/>





<a NAME="E1:71"/>
( <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#K8" target="_self">-INF(SC)_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#K8" target="_self">-INF(SC)_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#K8" target="_self">-INF(SC)_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#K4" target="_self">-INF_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#K4" target="_self">-INF_category</a> </span>)</span> ;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:71_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="E2:71_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#E6:71"><i><font color="Green">E46</font></i></a></i>;<br/><a NAME="E4:71_1"/>
<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="txt" href="waybel34.html#E6:71"><i><font color="Green">E46</font></i></a>, , <a class="ref" href="yellow21.html#D7">YELLOW21:def 7</a></i>;<br/><b>hence </b><a NAME="E5:71_1"/>
<font color="Maroon">f</font> is  <a href="waybel_0.html#V17">infs-preserving</a> <a href="waybel_0.html#V22">directed-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>by </b><i><a class="txt" href="waybel34.html#E6:71"><i><font color="Green">E46</font></i></a>, , <a class="ref" href="waybel34.html#T1" target="_self">Th1</a>, </i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E4:71"/>
<font color="Maroon">f</font> is  <a href="waybel_0.html#V17">infs-preserving</a> <a href="waybel_0.html#V22">directed-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>
 ;<br/>

<b>then reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font> as  <a href="waybel_0.html#V17">infs-preserving</a> <a href="waybel_0.html#V22">directed-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> ;<br/>
<a NAME="E6:71"/><i><font color="Green">E46</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/>;<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/>;<br/>
<a NAME="E8:71"/>
<font color="Maroon">f</font> <a href="funct_2.html#R1">=</a>  <a href="yellow21.html#K5">@</a> <font color="Maroon">g</font>
 
<b>by </b><i><a class="txt" href="waybel34.html#E6:71"><i><font color="Green">E46</font></i></a>, <a class="ref" href="yellow21.html#D7">YELLOW21:def 7</a></i>;<br/>
<b>hence </b><a NAME="E9:71"/>
<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#E6:71"><i><font color="Green">E46</font></i></a>, <a class="ref" href="waybel34.html#T1" target="_self">Th1</a></i>;<br/>


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