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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#L1">RelStr</a> ;<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>assume </b><a NAME="E1:37"/><i><font color="Green">E29</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  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>
 ;<br/>

<b>set </b><font color="Maroon">h</font> =  <a href="waybel_0.html#K1">netmap</a> <span class="p1">(<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>;<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><b> such that </b><br/><a NAME="E3:37"/><i><font color="Green">E32</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 <br/> ( <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Olive">x</font></span>)</span> &amp;  <a href="waybel_2.html#K2" target="_self">FinSups</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="waybel_0.html#G1">NetStr</a>(# <span class="p1">(<span class="default"><a href="finsub_1.html#K5">Fin</a> <font color="Maroon">J</font></span>)</span>,<span class="p1">(<span class="default"><a href="yellow_1.html#K1">RelIncl</a> <span class="p2">(<span class="default"><a href="finsub_1.html#K5">Fin</a> <font color="Maroon">J</font></span>)</span></span>)</span>,<font color="Maroon">g</font> #) )
 <b>by </b><i/>;<br/>
<b>let </b><font color="Maroon">q</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E4:37"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</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>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:37"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</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> &amp; <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> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> <font color="Maroon">J</font> <b>by </b><i>, </i>;<br/>
<a NAME="E8:37"/><i><font color="Green">E35</font></i>: 
<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> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">x</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E9:37"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">x</font> is  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T144">RELAT_1:144</a></i>;<br/>
<a NAME="E10:37"/>
 <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">x</font>,<font color="Maroon">L</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E11:37"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K11">finsups</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i>, , </i>;<br/>


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