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<b>let </b><font color="Maroon">L</font> be   <a href="waybel_0.html#NM7">sup-Semilattice</a>;<br/><b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/>



<b>reconsider </b><font color="Maroon">x1</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Maroon">z</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/><b>assume </b><b>that </b><br/><a NAME="E1:49_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">x</font></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E2:49_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">x</font></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E3:49_1"/>
 <a href="yellow_0.html#R1">ex_sup_of</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="domain_1.html#K7">}</a></span>,<font color="Maroon">L</font>
 ;<br/><a NAME="E4:49_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">x</font>
 <b>by </b><i>, <a class="ref" href="yellow_0.html#D15">YELLOW_0:def 15</a></i>;<br/><a NAME="E5:49_1"/><b>then </b><i><font color="Green">E33</font></i>: 
<font color="Maroon">y</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">x1</font>
 <b>by </b><i><a class="ref" href="waybel_0.html#T18">WAYBEL_0:18</a></i>;<br/><a NAME="E6:49_1"/>
<font color="Maroon">y</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">z</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T22">YELLOW_0:22</a></i>;<br/><a NAME="E7:49_1"/><b>then </b>
<font color="Maroon">y</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">z</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">x1</font>
 <b>by </b><i><a class="ref" href="waybel23.html#T1" target="_self">Th1</a>, <a class="ref" href="yellow_0.html#D2">YELLOW_0:def 2</a></i>;<br/><a NAME="E8:49_1"/><b>then </b>
<font color="Maroon">y</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="waybel_0.html#T18">WAYBEL_0:18</a></i>;<br/><a NAME="E9:49_1"/><b>then </b>
 <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="domain_1.html#K7">}</a></span> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T41">YELLOW_0:41</a></i>;<br/><b>hence </b><a NAME="E10:49_1"/>
 <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="domain_1.html#K7">}</a></span> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">x</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="yellow_0.html#D15">YELLOW_0:def 15</a></i>;<br/></div><b>end;</b></div>
<a NAME="E3:49"/><b>then </b>
 <a href="yellow_0.html#K5">subrelstr</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">x</font></span>)</span> is <a href="yellow_0.html#V6">join-inheriting</a>
 
<b>by </b><i><a class="ref" href="yellow_0.html#D17">YELLOW_0:def 17</a></i>;<br/>
<b>hence </b><a NAME="E4:49"/>
 <a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">x</font> is <a href="waybel23.html#V2" target="_self">join-closed</a>
 <b>by </b><i/>;<br/>


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