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<b>let </b><font color="Maroon">L</font> be  non <a href="struct_0.html#V3">empty</a> <a href="orders_2.html#V2">reflexive</a> <a href="orders_2.html#V4">antisymmetric</a> <a href="lattice3.html#V3">complete</a>  <a href="orders_2.html#L1">RelStr</a> ;<br/>

<b>assume </b><a NAME="E1:70"/><i><font color="Green">E29</font></i>: 
for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><br/> for <font color="Olive">N</font> being  <a href="waybel_0.html#NM1">prenet</a> of <font color="Maroon">L</font>  st <font color="Olive">N</font> is <a href="waybel_0.html#V10">eventually-directed</a> holds <br/><font color="Olive">x</font> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><a href="waybel_2.html#K1" target="_self">sup</a> <font color="Olive">N</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">x</font></span><a href="domain_1.html#K6">}</a></span> <a href="yellow_4.html#K4">"/\"</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><a href="waybel_0.html#K1">netmap</a> <font color="Olive">N</font>,<font color="Maroon">L</font></span>)</span></span>)</span></span>)</span>
 ;<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>let </b><font color="Maroon">J</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">f</font> be   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">J</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>;<br/>





<b>set </b><font color="Maroon">F</font> =  <a href="waybel_2.html#K2" target="_self">FinSups</a> <font color="Maroon">f</font>;<br/>
<a NAME="E2:70"/><i><font color="Green">E32</font></i>: 
 <a href="yellow_0.html#R1">ex_sup_of</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font>,<font color="Maroon">L</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T17">YELLOW_0:17</a></i>;<br/>
<a NAME="E3:70"/><i><font color="Green">E34</font></i>: 
 <a href="yellow_0.html#R1">ex_sup_of</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K1">netmap</a> <span class="p2">(<span class="default"><a href="waybel_2.html#K2" target="_self">FinSups</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">L</font></span>)</span>,<font color="Maroon">L</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T17">YELLOW_0:17</a></i>;<br/>
<a NAME="E4:70"/><i><font color="Green">E35</font></i>: 
for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> <font color="Maroon">J</font> holds   <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Olive">x</font>,<font color="Maroon">L</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T17">YELLOW_0:17</a></i>;<br/>
<b>consider </b><font color="Maroon">g</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="finsub_1.html#K5">Fin</a> <font color="Maroon">J</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>;<br/>
<b>thus </b><font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><a href="yellow_2.html#NK6" target="_self">Sup</a> </span>)</span> = 
<font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><a href="waybel_2.html#K1" target="_self">sup</a> <span class="p2">(<span class="default"><a href="waybel_2.html#K2" target="_self">FinSups</a> <font color="Maroon">f</font></span>)</span></span>)</span>
<b>by </b><i>, , , </i>
<br/>.= 
 <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span> <a href="yellow_4.html#K4">"/\"</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><a href="waybel_0.html#K1">netmap</a> <span class="p4">(<span class="default"><a href="waybel_2.html#K2" target="_self">FinSups</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">L</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="waybel_2.html#T5" target="_self">Th5</a></i>
<br/>.= 
 <a href="yellow_0.html#K1">"\/"</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> the <a href="waybel_0.html#U1">mapping</a> of <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="waybel_2.html#K3" target="_self">"/\"</a> <span class="p3">(<span class="default"><a href="waybel_2.html#K2" target="_self">FinSups</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">L</font>
<b>by </b><i/>
<br/>.= 
 <a href="waybel_2.html#K1" target="_self">sup</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="waybel_2.html#K3" target="_self">"/\"</a> <span class="p2">(<span class="default"><a href="waybel_2.html#K2" target="_self">FinSups</a> <font color="Maroon">f</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="yellow_2.html#D5">YELLOW_2:def 5</a></i>
;<br/>


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