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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="struct_0.html#V3">empty</a>  <a href="orders_2.html#L1">RelStr</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>



<a NAME="E1:60"/>
( ( <font color="Maroon">c<sub>2</sub></font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="waybel10.html#K5" target="_self">ClosureSystems</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> implies <font color="Maroon">c<sub>2</sub></font> is  <a href="orders_2.html#V1">strict</a> <a href="yellow_0.html#V7">closure</a> <a href="waybel10.html#NM1">System</a> of <font color="Maroon">c<sub>1</sub></font> ) &amp; ( <font color="Maroon">c<sub>2</sub></font> is  <a href="orders_2.html#V1">strict</a> <a href="yellow_0.html#V7">closure</a> <a href="waybel10.html#NM1">System</a> of <font color="Maroon">c<sub>1</sub></font> implies <font color="Maroon">c<sub>2</sub></font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="waybel10.html#K5" target="_self">ClosureSystems</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> ) &amp; ( <font color="Maroon">c<sub>2</sub></font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="waybel10.html#K9" target="_self">Subalgebras</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> implies <font color="Maroon">c<sub>2</sub></font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="waybel10.html#K5" target="_self">ClosureSystems</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> ) )
 
<b>by </b><i><a class="ref" href="waybel10.html#T17" target="_self">Th17</a>, <a class="ref" href="yellow_0.html#T59">YELLOW_0:59</a></i>;<br/>
<b>hence </b><a NAME="E2:60"/>
( <font color="Maroon">c<sub>2</sub></font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="waybel10.html#K9" target="_self">Subalgebras</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> iff <font color="Maroon">c<sub>2</sub></font> is  <a href="orders_2.html#V1">strict</a> <a href="yellow_0.html#V7">closure</a> <a href="waybel_0.html#V4">directed-sups-inheriting</a> <a href="waybel10.html#NM1">System</a> of <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="ref" href="waybel10.html#D8" target="_self">Def8</a></i>;<br/>


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