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<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> ;<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="yellow21.html#K7" title="YELLOW21:func.7">-CONT_category</a> </span>)</span>;<br/><b>let </b><font color="Maroon" title="c4">f</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>





<b>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="yellow21.html#K6" title="YELLOW21:func.6">-UPS_category</a> </span>)</span> <b>by </b><i><a class="ref" href="altcat_2.html#T30" title="ALTCAT_2:th.30">ALTCAT_2:30</a></i>;<br/>
<a NAME="E2:66"/>
<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="hidden.html#R1" title="HIDDEN:pred.1">=</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#T29" title="ALTCAT_2:th.29">ALTCAT_2:29</a></i>;<br/>
<b>hence </b><a NAME="E3:66"/>
( <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 <font color="Maroon" title="c4">f</font> is  <a href="waybel_0.html#V22" title="WAYBEL_0:attr.22">directed-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>by </b><i><a class="ref" href="yellow21.html#T15" target="_self" title="YELLOW21:th.15">Th15</a></i>;<br/>


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