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<b>thus </b><a NAME="E1:24_1_1"/>
 <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font> is <a href="waybel11.html#V2" target="_self">closed_under_directed_sups</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">D</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="waybel_0.html#V1">directed</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel11.html#D2" target="_self">WAYBEL11:def 2</a><br/>

<b>assume </b><a NAME="E1:24_1_1_1"/>
<font color="Maroon">D</font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font>
 ;<br/>

<b>thus </b><a NAME="E2:24_1_1_1"/>
 <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">D</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font>
 ;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon">D</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="waybel_0.html#V1">directed</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel11.html#D1" target="_self">WAYBEL11:def 1</a><br/>

<b>assume </b><a NAME="E2:24_1_1"/>
 <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">D</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font>
 ;<br/>

<b>consider </b><font color="Maroon">p</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E4:24_1_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 <b>by </b><i><a class="ref" href="subset_1.html#T10">SUBSET_1:10</a></i>;<br/>
<b>thus </b><a NAME="E5:24_1_1"/>
<font color="Maroon">D</font> <a href="subset_1.html#R2">meets</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>


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