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<div><span class="kw">theorem </span><span class="lab"><font color="Green" title="E26">Th19</font></span>: <a NAME="T19"><span class="comment"><font color="firebrick">:: YELLOW21:19</font></span><br/></a><div class="add"> for <font color="Olive" title="b1">W</font> being   <a href="setfam_1.html#NV3" title="SETFAM_1:NV.3">with_non-empty_element</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <br/>  for <font color="Olive" title="b2">a</font>, <font color="Olive" title="b3">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="Olive" title="b1">W</font> <a href="yellow21.html#K8" title="YELLOW21:func.8">-ALG_category</a></span>)</span><br/>  for <font color="Olive" title="b4">f</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  holds <br/> ( <font color="Olive" title="b4">f</font> <a href="tarski.html#R2" title="TARSKI: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">a</font>,<font color="Olive" title="b3">b</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> iff <font color="Olive" title="b4">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="Olive" title="b2">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="b3">b</font></span>)</span> )</div></div>
