<?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  <a href="relat_1.html#V2">V3</a>  <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">J</font>;<br/><b>let </b><font color="Maroon">F</font> be   <a href="waybel_5.html#NM2">DoubleIndexedSet</a> of <font color="Maroon">K</font>,<font color="Maroon">L</font>;<br/>







<a NAME="E1:60"/>
 <a href="yellow_2.html#NK7" target="_self">Inf</a>  <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="waybel_5.html#NK7" target="_self">Infs</a> </span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</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="E1:60_1"/>
<font color="Maroon">x</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#NK7" target="_self">Infs</a> </span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">a</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">J</font><b> such that </b><br/><a NAME="E3:60_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="yellow_2.html#NK7" target="_self">Inf</a> 
 <b>by </b><i><a class="ref" href="waybel_5.html#T14">WAYBEL_5:14</a></i>;<br/>
<a NAME="E4:60_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</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"><font color="Maroon">F</font> <a href="waybel_5.html#K1">.</a> <font color="Maroon">a</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow_7.html#T5" target="_self">Th5</a>, <a class="ref" href="yellow_2.html#D6">YELLOW_2:def 6</a></i>;<br/>
<a NAME="E5:60_1"/>
<font color="Maroon">x</font> <a href="lattice3.html#R1">is_&lt;=_than</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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</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="E1:60_1_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">y</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>reconsider </b><font color="Maroon">J'</font> =  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">F</font></span>)</span> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ;<br/>
<b>reconsider </b><font color="Maroon">K'</font> = <font color="Maroon">J'</font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">J</font> as    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">J'</font> ;<br/>
<b>reconsider </b><font color="Maroon">G</font> =  <a href="waybel_5.html#K2">Frege</a> <font color="Maroon">F</font> as   <a href="waybel_5.html#NM2">DoubleIndexedSet</a> of <font color="Maroon">K'</font>,<font color="Maroon">L</font> ;<br/>
<b>consider </b><font color="Maroon">f</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">J'</font><b> such that </b><br/><a NAME="E6:60_1_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="yellow_2.html#NK6" target="_self">Sup</a> 
 <b>by </b><i>, <a class="ref" href="waybel_5.html#T14">WAYBEL_5:14</a></i>;<br/>
<b>reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">F</font></span>)</span> ;<br/>
<a NAME="E8:60_1_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</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"><span class="p3">(<span class="default"><a href="waybel_5.html#K2">Frege</a> <font color="Maroon">F</font></span>)</span> <a href="waybel_5.html#K3">.</a> <font color="Maroon">f</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow_7.html#T7" target="_self">Th7</a>, <a class="ref" href="yellow_2.html#D5">YELLOW_2:def 5</a></i>;<br/>
<a NAME="E9:60_1_1"/><i><font color="Green">E44</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">J</font> &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="waybel_5.html#K2">Frege</a> <font color="Maroon">F</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">F</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E10:60_1_1"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="waybel_5.html#K1">.</a> <font color="Maroon">a</font></span>)</span>
 
<b>by </b><i><a class="ref" href="waybel_5.html#T9">WAYBEL_5:9</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">j</font> = <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> ;<br/>
<a NAME="E12:60_1_1"/>
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="waybel_5.html#K1">.</a> <font color="Maroon">a</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="waybel_5.html#T9">WAYBEL_5:9</a></i>;<br/>
<a NAME="E13:60_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="waybel_5.html#K1">.</a> <font color="Maroon">a</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="waybel_5.html#K1">.</a> <font color="Maroon">a</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="waybel_5.html#K1">.</a> <font color="Maroon">a</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">j</font> <a href="hidden.html#R2">in</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#K2">Frege</a> <font color="Maroon">F</font></span>)</span> <a href="waybel_5.html#K3">.</a> <font color="Maroon">f</font></span>)</span> )
 
<b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a>, <a class="ref" href="waybel_5.html#T9">WAYBEL_5:9</a></i>;<br/>
<a NAME="E14:60_1_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="waybel_5.html#K1">.</a> <font color="Maroon">a</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">j</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="waybel_5.html#K1">.</a> <font color="Maroon">a</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">j</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">y</font> )
 
<b>by </b><i><a class="ref" href="yellow_7.html#T6" target="_self">Th6</a>, <a class="ref" href="yellow_7.html#T8" target="_self">Th8</a>, <a class="ref" href="yellow_2.html#T24">YELLOW_2:24</a></i>;<br/>
<b>hence </b><a NAME="E15:60_1_1"/>
<font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="orders_2.html#T26">ORDERS_2:26</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:60_1"/><b>then </b>
<font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</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#NK8" target="_self">Sups</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T33">YELLOW_0:33</a></i>;<br/>
<b>hence </b><a NAME="E7:60_1"/>
<font color="Maroon">x</font> <a href="orders_2.html#R1">&lt;=</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="E2:60"/><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#NK7" target="_self">Infs</a> </span>)</span></span>)</span> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_2.html#NK7" target="_self">Inf</a> 
 
<b>by </b><i><a class="ref" href="yellow_0.html#T32">YELLOW_0:32</a></i>;<br/>
<b>hence </b><a NAME="E3:60"/>
 <a href="yellow_2.html#NK6" target="_self">Sup</a>  <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_2.html#NK7" target="_self">Inf</a> 
 <b>by </b><i><a class="ref" href="yellow_2.html#D5">YELLOW_2:def 5</a></i>;<br/>


</div>
