<?xml version="1.0"?>
<div class="add">

<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">J</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">K</font> be  non <a href="xboole_0.html#V1">empty</a>  <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 <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">J</font>,<font color="Maroon">K</font></span><a href="zfmisc_1.html#K2">:]</a></span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>;<br/><b>let </b><font color="Maroon">X</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font>;<br/>









<b>assume </b><a NAME="E1:43"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> :  ex <font color="Olive">f</font> being <a href="relat_1.html#V2">V7</a>  <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">J</font> st <br/>( <font color="Olive">f</font> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">J</font>,<span class="p1">(<span class="default"><a href="finsub_1.html#K5">Fin</a> <font color="Maroon">K</font></span>)</span> &amp;  ex <font color="Olive">G</font> being  <a href="waybel_5.html#NM2">DoubleIndexedSet</a> of <font color="Olive">f</font>,<font color="Maroon">L</font> st <br/>( ( for <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">J</font><br/> for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font> holds <br/><span class="p1">(<span class="default"><font color="Olive">G</font> <a href="waybel_5.html#K1" target="_self">.</a> <font color="Olive">j</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="binop_1.html#K1">.</a> <font color="Olive">j</font>,<font color="Olive">x</font> ) &amp; <font color="Olive">a</font> <a href="hidden.html#R1">=</a>  <a href="yellow_2.html#NK7" target="_self">Inf</a>  ) ) </span>}</span> 
 ;<br/>

<a NAME="E2:43"/><i><font color="Green">E36</font></i>: 
for <font color="Olive">f</font> being <a href="relat_1.html#V2">V7</a>  <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">J</font>  st <font color="Olive">f</font> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">J</font>,<span class="p1">(<span class="default"><a href="finsub_1.html#K5">Fin</a> <font color="Maroon">K</font></span>)</span> holds <br/>for <font color="Olive">G</font> being  <a href="waybel_5.html#NM2">DoubleIndexedSet</a> of <font color="Olive">f</font>,<font color="Maroon">L</font>  st ( for <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">J</font><br/> for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font> holds <br/><span class="p1">(<span class="default"><font color="Olive">G</font> <a href="waybel_5.html#K1" target="_self">.</a> <font color="Olive">j</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="binop_1.html#K1">.</a> <font color="Olive">j</font>,<font color="Olive">x</font> ) holds <br/>for <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">J</font> holds   <a href="yellow_2.html#NK6" target="_self">Sup</a>  <a href="orders_2.html#NR2" target="_self">&gt;=</a>  <a href="yellow_2.html#NK6" target="_self">Sup</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">f</font> be  <a href="relat_1.html#V2">V7</a>  <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">J</font>;<br/>

<b>assume </b><a NAME="E1:43_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">J</font>,<span class="p1">(<span class="default"><a href="finsub_1.html#K5">Fin</a> <font color="Maroon">K</font></span>)</span>
 ;<br/>

<b>let </b><font color="Maroon">G</font> be   <a href="waybel_5.html#NM2">DoubleIndexedSet</a> of <font color="Maroon">f</font>,<font color="Maroon">L</font>;<br/>

<b>assume </b><a NAME="E2:43_1"/><i><font color="Green">E39</font></i>: 
for <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">J</font><br/> for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="waybel_5.html#K1" target="_self">.</a> <font color="Olive">j</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="binop_1.html#K1">.</a> <font color="Olive">j</font>,<font color="Olive">x</font>
 ;<br/>

<b>let </b><font color="Maroon">j</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">J</font>;<br/>

<a NAME="E3:43_1"/><i><font color="Green">E43</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"><span class="p2">(<span class="default"><a href="waybel_5.html#K6" target="_self">curry</a> <font color="Maroon">F</font></span>)</span> <a href="waybel_5.html#K1" target="_self">.</a> <font color="Maroon">j</font></span>)</span>,<font color="Maroon">L</font> &amp;  <a href="yellow_0.html#R1">ex_sup_of</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="waybel_5.html#K1" target="_self">.</a> <font color="Maroon">j</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="E4:43_1"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="waybel_5.html#K1" target="_self">.</a> <font color="Maroon">j</font></span>)</span> is   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel_5.html#K6" target="_self">curry</a> <font color="Maroon">F</font></span>)</span> <a href="waybel_5.html#K1" target="_self">.</a> <font color="Maroon">j</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel_5.html#E1"><i><font color="Green">Lemma32</font></i></a>, , </i>;<br/>
<a NAME="E5:43_1"/><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"><span class="p3">(<span class="default"><a href="waybel_5.html#K6" target="_self">curry</a> <font color="Maroon">F</font></span>)</span> <a href="waybel_5.html#K1" target="_self">.</a> <font color="Maroon">j</font></span>)</span></span>)</span> <a href="orders_2.html#NR2" target="_self">&gt;=</a>  <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"><font color="Maroon">G</font> <a href="waybel_5.html#K1" target="_self">.</a> <font color="Maroon">j</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel_5.html#E1:43"><i><font color="Green">E34</font></i></a>, <a class="ref" href="yellow_0.html#T34">YELLOW_0:34</a></i>;<br/>
<a NAME="E6:43_1"/><b>then </b>
 <a href="yellow_2.html#NK6" target="_self">Sup</a>  <a href="orders_2.html#NR2" target="_self">&gt;=</a>  <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"><font color="Maroon">G</font> <a href="waybel_5.html#K1" target="_self">.</a> <font color="Maroon">j</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow_2.html#D5">YELLOW_2:def 5</a></i>;<br/>
<b>hence </b><a NAME="E7:43_1"/>
 <a href="yellow_2.html#NK6" target="_self">Sup</a>  <a href="orders_2.html#NR2" target="_self">&gt;=</a>  <a href="yellow_2.html#NK6" target="_self">Sup</a> 
 <b>by </b><i><a class="ref" href="yellow_2.html#D5">YELLOW_2:def 5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:43"/><i><font color="Green">E44</font></i>: 
for <font color="Olive">f</font> being <a href="relat_1.html#V2">V7</a>  <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">J</font>  st <font color="Olive">f</font> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">J</font>,<span class="p1">(<span class="default"><a href="finsub_1.html#K5">Fin</a> <font color="Maroon">K</font></span>)</span> holds <br/>for <font color="Olive">G</font> being  <a href="waybel_5.html#NM2">DoubleIndexedSet</a> of <font color="Olive">f</font>,<font color="Maroon">L</font>  st ( for <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">J</font><br/> for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font> holds <br/><span class="p1">(<span class="default"><font color="Olive">G</font> <a href="waybel_5.html#K1" target="_self">.</a> <font color="Olive">j</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="binop_1.html#K1">.</a> <font color="Olive">j</font>,<font color="Olive">x</font> ) holds <br/> <a href="yellow_2.html#NK7" target="_self">Inf</a>  <a href="orders_2.html#NR2" target="_self">&gt;=</a>  <a href="yellow_2.html#NK7" target="_self">Inf</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">f</font> be  <a href="relat_1.html#V2">V7</a>  <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">J</font>;<br/>

<b>assume </b><a NAME="E1:43_2"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">J</font>,<span class="p1">(<span class="default"><a href="finsub_1.html#K5">Fin</a> <font color="Maroon">K</font></span>)</span>
 ;<br/>

<b>let </b><font color="Maroon">G</font> be   <a href="waybel_5.html#NM2">DoubleIndexedSet</a> of <font color="Maroon">f</font>,<font color="Maroon">L</font>;<br/>

<b>assume </b><a NAME="E2:43_2"/><i><font color="Green">E55</font></i>: 
for <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">J</font><br/> for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="waybel_5.html#K1" target="_self">.</a> <font color="Olive">j</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="binop_1.html#K1">.</a> <font color="Olive">j</font>,<font color="Olive">x</font>
 ;<br/>

<a NAME="E3:43_2"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="waybel_5.html#NK8" target="_self">Sups</a> </span>)</span> <a href="lattice3.html#NR3" target="_self">is_&gt;=_than</a>  <a href="yellow_2.html#NK7" target="_self">Inf</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</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#D8">LATTICE3:def 8</a><br/>

<b>assume </b><a NAME="E1:43_2_1"/>
<font color="Maroon">a</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#NK8" target="_self">Sups</a> </span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">j</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">J</font><b> such that </b><br/><a NAME="E3:43_2_1"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="yellow_2.html#NK6" target="_self">Sup</a> 
 <b>by </b><i/>;<br/>
<a NAME="E4:43_2_1"/><i><font color="Green">E77</font></i>: 
 <a href="yellow_2.html#NK6" target="_self">Sup</a>  <a href="orders_2.html#NR2" target="_self">&gt;=</a>  <a href="yellow_2.html#NK6" target="_self">Sup</a> 
 
<b>by </b><i>, , <a class="txt" href="waybel_5.html#E1:43"><i><font color="Green">E34</font></i></a></i>;<br/>
<a NAME="E5:43_2_1"/>
 <a href="yellow_2.html#NK6" target="_self">Sup</a>  <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#NK6" target="_self">Sups</a> </span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E6:43_2_1"/><b>then </b>
 <a href="yellow_2.html#NK6" target="_self">Sup</a>  <a href="orders_2.html#NR2" target="_self">&gt;=</a>  <a href="yellow_0.html#NK4" target="_self">inf</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#NK6" target="_self">Sups</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow_2.html#T24">YELLOW_2:24</a></i>;<br/>
<a NAME="E7:43_2_1"/><b>then </b>
 <a href="yellow_2.html#NK6" target="_self">Sup</a>  <a href="orders_2.html#NR2" target="_self">&gt;=</a>  <a href="yellow_2.html#NK7" target="_self">Inf</a> 
 
<b>by </b><i><a class="ref" href="yellow_2.html#D6">YELLOW_2:def 6</a></i>;<br/>
<b>hence </b><a NAME="E8:43_2_1"/>
<font color="Maroon">a</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a>  <a href="yellow_2.html#NK7" target="_self">Inf</a> 
 <b>by </b><i>, <a class="txt" href="waybel_5.html#E2:43"><i><font color="Green">E36</font></i></a>, <a class="ref" href="orders_2.html#T26">ORDERS_2:26</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:43_2"/><b>then </b>
 <a href="yellow_0.html#NK4" target="_self">inf</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#NK8" target="_self">Sups</a> </span>)</span></span>)</span> <a href="orders_2.html#NR2" target="_self">&gt;=</a>  <a href="yellow_2.html#NK7" target="_self">Inf</a> 
 
<b>by </b><i><a class="ref" href="yellow_0.html#T33">YELLOW_0:33</a></i>;<br/>
<b>hence </b><a NAME="E5:43_2"/>
 <a href="yellow_2.html#NK7" target="_self">Inf</a>  <a href="orders_2.html#NR2" target="_self">&gt;=</a>  <a href="yellow_2.html#NK7" target="_self">Inf</a> 
 <b>by </b><i><a class="ref" href="yellow_2.html#D6">YELLOW_2:def 6</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:43"/>
 <a href="yellow_2.html#NK7" target="_self">Inf</a>  <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a> <font color="Maroon">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</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="E1:43_3"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<b>then consider </b><font color="Maroon">a'</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E3:43_3"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">a'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 <b>and </b><br/><a NAME="E4:43_3"/><i><font color="Green">E79</font></i>: 
 ex <font color="Olive">f</font> being <a href="relat_1.html#V2">V7</a>  <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">J</font> st <br/>( <font color="Olive">f</font> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">J</font>,<span class="p1">(<span class="default"><a href="finsub_1.html#K5">Fin</a> <font color="Maroon">K</font></span>)</span> &amp;  ex <font color="Olive">G</font> being  <a href="waybel_5.html#NM2">DoubleIndexedSet</a> of <font color="Olive">f</font>,<font color="Maroon">L</font> st <br/>( ( for <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">J</font><br/> for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font> holds <br/><span class="p1">(<span class="default"><font color="Olive">G</font> <a href="waybel_5.html#K1" target="_self">.</a> <font color="Olive">j</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="binop_1.html#K1">.</a> <font color="Olive">j</font>,<font color="Olive">x</font> ) &amp; <font color="Maroon">a'</font> <a href="hidden.html#R1">=</a>  <a href="yellow_2.html#NK7" target="_self">Inf</a>  ) )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">f</font> being  <a href="relat_1.html#V2">V7</a>  <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">J</font><b> such that </b><br/><a NAME="E6:43_3"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">J</font>,<span class="p1">(<span class="default"><a href="finsub_1.html#K5">Fin</a> <font color="Maroon">K</font></span>)</span>
 <b>and </b><br/><a NAME="E7:43_3"/><i><font color="Green">E81</font></i>: 
 ex <font color="Olive">G</font> being  <a href="waybel_5.html#NM2">DoubleIndexedSet</a> of <font color="Maroon">f</font>,<font color="Maroon">L</font> st <br/>( ( for <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">J</font><br/> for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font> holds <br/><span class="p1">(<span class="default"><font color="Olive">G</font> <a href="waybel_5.html#K1" target="_self">.</a> <font color="Olive">j</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="binop_1.html#K1">.</a> <font color="Olive">j</font>,<font color="Olive">x</font> ) &amp; <font color="Maroon">a'</font> <a href="hidden.html#R1">=</a>  <a href="yellow_2.html#NK7" target="_self">Inf</a>  )
 <b>by </b><i><a class="txt" href="waybel_5.html#E1:43"><i><font color="Green">E34</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">G</font> being   <a href="waybel_5.html#NM2">DoubleIndexedSet</a> of <font color="Maroon">f</font>,<font color="Maroon">L</font><b> such that </b><br/><a NAME="E9:43_3"/><i><font color="Green">E82</font></i>: 
for <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">J</font><br/> for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="waybel_5.html#K1" target="_self">.</a> <font color="Olive">j</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="binop_1.html#K1">.</a> <font color="Olive">j</font>,<font color="Olive">x</font>
 <b>and </b><br/><a NAME="E10:43_3"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">a'</font> <a href="hidden.html#R1">=</a>  <a href="yellow_2.html#NK7" target="_self">Inf</a> 
 <b>by </b><i><a class="txt" href="waybel_5.html#E2:43"><i><font color="Green">E36</font></i></a></i>;<br/>
<b>thus </b><a NAME="E11:43_3"/>
 <a href="yellow_2.html#NK7" target="_self">Inf</a>  <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="txt" href="waybel_5.html#E1"><i><font color="Green">Lemma32</font></i></a>, , , <a class="txt" href="waybel_5.html#E2"><i><font color="Green">Lemma37</font></i></a>, </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:43"/>
 <a href="yellow_2.html#NK7" target="_self">Inf</a>  <a href="orders_2.html#NR2" target="_self">&gt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T32">YELLOW_0:32</a></i>;<br/>


</div>
