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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#V5" target="_self">auxiliary</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="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_2.html#D1">YELLOW_2:def 1</a><br/>





<b>assume </b><a NAME="E1:51"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> ;<br/>
<a NAME="E3:51"/><i><font color="Green">E31</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">AR</font> <a href="waybel_4.html#K7" target="_self">-below</a> </span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">AR</font> <a href="waybel_4.html#K5" target="_self">-below</a> <font color="Maroon">x'</font>
 
<b>by </b><i/>;<br/>
<a NAME="E4:51"/>
<span class="p1">(<span class="default"><a href="yellow_2.html#K3">IdsMap</a> <font color="Maroon">L</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x'</font> <a href="hidden.html#R1">=</a>  <a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">x'</font>
 
<b>by </b><i><a class="ref" href="yellow_2.html#D4">YELLOW_2:def 4</a></i>;<br/>
<a NAME="E5:51"/><b>then </b><i><font color="Green">E32</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">AR</font> <a href="waybel_4.html#K7" target="_self">-below</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="yellow_2.html#K3">IdsMap</a> <font color="Maroon">L</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 
<b>by </b><i>, </i>;<br/>
<b>reconsider </b><font color="Maroon">a</font> = <span class="p1">(<span class="default"><font color="Maroon">AR</font> <a href="waybel_4.html#K7" target="_self">-below</a> </span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x'</font>, <font color="Maroon">b</font> = <span class="p1">(<span class="default"><a href="yellow_2.html#K3">IdsMap</a> <font color="Maroon">L</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x'</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K7">Ids</a> <font color="Maroon">L</font></span>)</span></span>)</span> ;<br/>
<b>take </b>
<font color="Maroon">a</font>
;<br/>

<b>take </b>
<font color="Maroon">b</font>
;<br/>

<b>thus </b><a NAME="E7:51"/>
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">AR</font> <a href="waybel_4.html#K7" target="_self">-below</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="yellow_2.html#K3">IdsMap</a> <font color="Maroon">L</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> &amp; <font color="Maroon">a</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">b</font> )
 <b>by </b><i>, <a class="ref" href="yellow_1.html#T3">YELLOW_1:3</a></i>;<br/>


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