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<b>let </b><font color="Maroon">L</font> be  <a href="lattice3.html#V3">complete</a> <a href="waybel_0.html#NM8">LATTICE</a>;<br/><b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/><b>let </b><font color="Maroon">F</font> be  <a href="funcop_1.html#V1">Function-yielding</a> <a href="funct_1.html#NM1">Function</a>;<br/>





<b>assume </b><a NAME="E1:32"/><i><font color="Green">E34</font></i>: 
for <font color="Olive">f</font> being  <a href="funct_1.html#NM1">Function</a>  st <font color="Olive">f</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K3">Frege</a> <font color="Maroon">F</font></span>)</span> holds <br/> <a href="yellow_2.html#K5">//\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_2.html#K3">Frege</a> <font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">f</font></span>)</span>,<font color="Maroon">L</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">a</font>
 ;<br/>

<a NAME="E2:32"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="waybel_5.html#K5" target="_self">/\\</a> <span class="p2">(<span class="default"><a href="pralg_2.html#K3">Frege</a> <font color="Maroon">F</font></span>)</span>,<font color="Maroon">L</font></span>)</span> <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Maroon">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D9">LATTICE3:def 9</a><br/>

<b>assume </b><a NAME="E1:32_1"/>
<font color="Maroon">c</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_5.html#K5" target="_self">/\\</a> <span class="p2">(<span class="default"><a href="pralg_2.html#K3">Frege</a> <font color="Maroon">F</font></span>)</span>,<font color="Maroon">L</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">f</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:32_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K3">Frege</a> <font color="Maroon">F</font></span>)</span>
 <b>and </b><br/><a NAME="E4:32_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c</font> <a href="hidden.html#R1">=</a>  <a href="yellow_2.html#K5">//\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_2.html#K3">Frege</a> <font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">L</font>
 <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font> as   <a href="funct_1.html#NM1">Function</a> <b>by </b><i>, </i>;<br/>
<a NAME="E6:32_1"/>
<font color="Maroon">f</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K3">Frege</a> <font color="Maroon">F</font></span>)</span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E7:32_1"/>
<font color="Maroon">c</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">a</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:32"/><b>then </b>
 <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><a href="waybel_5.html#K5" target="_self">/\\</a> <span class="p3">(<span class="default"><a href="pralg_2.html#K3">Frege</a> <font color="Maroon">F</font></span>)</span>,<font color="Maroon">L</font></span>)</span></span>)</span> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">a</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T32">YELLOW_0:32</a></i>;<br/>
<b>hence </b><a NAME="E4:32"/>
 <a href="yellow_2.html#NK6" target="_self">Sup</a>  <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="yellow_2.html#D5">YELLOW_2:def 5</a></i>;<br/>


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