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<b>let </b><font color="Maroon">L</font> be  <a href="yellow_0.html#V1">lower-bounded</a> <a href="waybel_0.html#NM7">sup-Semilattice</a>;<br/><b>let </b><font color="Maroon">AR</font> be  <a href="waybel_4.html#V2" target="_self">auxiliary(ii)</a> <a href="waybel_4.html#V3" target="_self">auxiliary(iii)</a> <a href="alg_1.html#NM1">Relation</a> of <font color="Maroon">L</font>;<br/><b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <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>, <font color="Maroon">u</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/>





<b>assume </b><a NAME="E1:8"/><i><font color="Green">E31</font></i>: 
( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">z</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">AR</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">u</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">AR</font> )
 ;<br/>

<a NAME="E2:8"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">y</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">y</font> )
 
;<br/>
<a NAME="E3:8"/>
( <font color="Maroon">z</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">u</font> &amp; <font color="Maroon">u</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">u</font> )
 
<b>by </b><i><a class="ref" href="yellow_0.html#T22">YELLOW_0:22</a></i>;<br/>
<a NAME="E4:8"/><b>then </b>
( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">u</font></span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">AR</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">y</font>,<span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">u</font></span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">AR</font> )
 
<b>by </b><i>, , </i>;<br/>
<b>hence </b><a NAME="E5:8"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">y</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">u</font></span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">AR</font>
 <b>by </b><i/>;<br/>


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