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<span class="kw">let </span><font color="Maroon" title="c1">L</font> be   <a href="waybel_0.html#NM7" title="WAYBEL_0:NM.7">sup-Semilattice</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">S</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="waybel_0.html#V12" title="WAYBEL_0:attr.12">lower</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font> holds <br/> ( <font color="Olive" title="b1">S</font> is   <a href="waybel_0.html#NM3" title="WAYBEL_0:NM.3">Ideal</a> of <font color="Maroon" title="c1">L</font> iff <font color="Olive" title="b1">S</font> is  <a href="waybel23.html#V2" title="WAYBEL23:attr.2">join-closed</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">S</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="waybel_0.html#V12" title="WAYBEL_0:attr.12">lower</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">S</font> is   <a href="waybel_0.html#NM3" title="WAYBEL_0:NM.3">Ideal</a> of <font color="Maroon" title="c1">L</font> iff <font color="Maroon" title="c2">S</font> is  <a href="waybel23.html#V2" title="WAYBEL23:attr.2">join-closed</a>  )</span><br/>



<a NAME="E1:44"/>
( <font color="Maroon" title="c2">S</font> is   <a href="waybel_0.html#NM3" title="WAYBEL_0:NM.3">Ideal</a> of <font color="Maroon" title="c1">L</font> iff  <a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c2">S</font> is  <a href="yellow_0.html#V6" title="YELLOW_0:attr.6">join-inheriting</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_0.html#T64" title="WAYBEL_0:th.64">WAYBEL_0:64</a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:44"/>
( <font color="Maroon" title="c2">S</font> is   <a href="waybel_0.html#NM3" title="WAYBEL_0:NM.3">Ideal</a> of <font color="Maroon" title="c1">L</font> iff <font color="Maroon" title="c2">S</font> is  <a href="waybel23.html#V2" title="WAYBEL23:attr.2">join-closed</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel23.html#D2" target="_self" title="WAYBEL23:def.2">Def2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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