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

<b>let </b><font color="Maroon">R</font> be  non <a href="struct_0.html#V3">empty</a> <a href="lfuzzy_0.html#V2" target="_self">interval</a>  <a href="orders_2.html#L1">RelStr</a> ;<br/>

<b>thus </b><a NAME="E1:27_1_1"/>
<font color="Maroon">R</font> is   <a href="waybel_0.html#NM8">LATTICE</a>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_1.html#D19">WAYBEL_1:def 19</a><br/>

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>;<br/>

<b>set </b><font color="Maroon">f</font> = <font color="Maroon">x</font> <a href="waybel_1.html#K4">"/\"</a> ;<br/>
<a NAME="E2:27_1_1"/>
<font color="Maroon">x</font> <a href="waybel_1.html#K4">"/\"</a>  is <a href="waybel_0.html#V18">sups-preserving</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">X</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">R</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D33">WAYBEL_0:def 33</a><br/>

<b>assume </b><a NAME="E1:27_1_1_1"/>
 <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">X</font>,<font color="Maroon">R</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D31">WAYBEL_0:def 31</a><br/>

<b>thus </b><a NAME="E2:27_1_1_1"/>
 <a href="yellow_0.html#R1">ex_sup_of</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="waybel_1.html#K4">"/\"</a> </span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font>,<font color="Maroon">R</font>
 <b>by </b><i/>;<br/>

<a NAME="E3:27_1_1_1"/><i><font color="Green">E41</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="waybel_1.html#K4">"/\"</a> </span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">X</font></span>)</span> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="waybel_1.html#K4">"/\"</a> </span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1_1_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">R</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="E1:27_1_1_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="waybel_1.html#K4">"/\"</a> </span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font>
 ;<br/>

<b>then consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:27_1_1_1_1"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="waybel_1.html#K4">"/\"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> )
 <b>by </b><i><a class="ref" href="funct_2.html#T115">FUNCT_2:115</a></i>;<br/>
<b>reconsider </b><font color="Maroon">y</font> = <font color="Maroon">y</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> <b>by </b><i/>;<br/>
<a NAME="E5:27_1_1_1_1"/>
( <font color="Maroon">y</font> <a href="lfuzzy_0.html#NR1" target="_self">&lt;&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">X</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">y</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="waybel_1.html#K4">"/\"</a> </span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">X</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">X</font></span>)</span> &amp; <font color="Maroon">x</font> <a href="lfuzzy_0.html#NR1" target="_self">&lt;&lt;=</a> <font color="Maroon">x</font> )
 
<b>by </b><i>, <a class="ref" href="waybel_1.html#D18">WAYBEL_1:def 18</a>, <a class="ref" href="yellow_0.html#D1">YELLOW_0:def 1</a>, <a class="ref" href="yellow_2.html#T24">YELLOW_2:24</a></i>;<br/>
<b>hence </b><a NAME="E6:27_1_1_1_1"/>
<font color="Maroon">a</font> <a href="lfuzzy_0.html#NR1" target="_self">&lt;&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="waybel_1.html#K4">"/\"</a> </span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">X</font></span>)</span>
 <b>by </b><i><a class="ref" href="yellow_3.html#T2">YELLOW_3:2</a></i>;<br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>;<br/><b>assume </b><a NAME="E1:27_1_1_1_2"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">b</font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="waybel_1.html#K4">"/\"</a> </span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font>
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_1_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:27_1_1_1_2_1"/>
( <font color="Maroon">x</font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a> <font color="Maroon">X</font> or  not <font color="Maroon">x</font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a> <font color="Maroon">X</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_1_1_2_1_1"><b>suppose </b></a><a NAME="E1:27_1_1_1_2_1_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">x</font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a> <font color="Maroon">X</font>
 ;<br/><div class="add"><a NAME="E2:27_1_1_1_2_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="lfuzzy_0.html#NR2" target="_self">&gt;&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/><a NAME="E3:27_1_1_1_2_1_1"/><b>then </b><i><font color="Green">E55</font></i>: 
<font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">X</font></span>)</span> <a href="hidden.html#R1">=</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#T25">YELLOW_0:25</a></i>;<br/><a NAME="E4:27_1_1_1_2_1_1"/>
<font color="Maroon">b</font> <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="27_1_1_1_2_1_1_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">R</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="E1:27_1_1_1_2_1_1_1"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<a NAME="E2:27_1_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="lfuzzy_0.html#NR2" target="_self">&gt;&gt;=</a> <font color="Maroon">a</font>
 
<b>by </b><i>, <a class="ref" href="lattice3.html#D9">LATTICE3:def 9</a></i>;<br/>
<a NAME="E3:27_1_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T25">YELLOW_0:25</a></i>;<br/>
<a NAME="E4:27_1_1_1_2_1_1_1"/><b>then </b><i><font color="Green">E67</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="waybel_1.html#K4">"/\"</a> </span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font>
 
<b>by </b><i><a class="ref" href="waybel_1.html#D18">WAYBEL_1:def 18</a></i>;<br/>
<a NAME="E5:27_1_1_1_2_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="waybel_1.html#K4">"/\"</a> </span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="waybel_1.html#K4">"/\"</a> </span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font>
 
<b>by </b><i>, <a class="ref" href="funct_2.html#T43">FUNCT_2:43</a></i>;<br/>
<b>hence </b><a NAME="E6:27_1_1_1_2_1_1_1"/>
<font color="Maroon">a</font> <a href="lfuzzy_0.html#NR1" target="_self">&lt;&lt;=</a> <font color="Maroon">b</font>
 <b>by </b><i>, , <a class="ref" href="lattice3.html#D9">LATTICE3:def 9</a></i>;<br/>


</div><b>end;</b></div><a NAME="E5:27_1_1_1_2_1_1"/><b>then </b>
<font color="Maroon">b</font> <a href="lfuzzy_0.html#NR2" target="_self">&gt;&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/><b>hence </b><a NAME="E6:27_1_1_1_2_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="waybel_1.html#K4">"/\"</a> </span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">X</font></span>)</span> <a href="lfuzzy_0.html#NR1" target="_self">&lt;&lt;=</a> <font color="Maroon">b</font>
 <b>by </b><i>, <a class="ref" href="waybel_1.html#D18">WAYBEL_1:def 18</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_1_1_2_1_2"><b>suppose </b></a><a NAME="E1:27_1_1_1_2_1_2"/>
not <font color="Maroon">x</font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a> <font color="Maroon">X</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">a</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font><b> such that </b><br/><a NAME="E3:27_1_1_1_2_1_2"/><i><font color="Green">E69</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; not <font color="Maroon">x</font> <a href="lfuzzy_0.html#NR2" target="_self">&gt;&gt;=</a> <font color="Maroon">a</font> )
 <b>by </b><i><a class="ref" href="lattice3.html#D9">LATTICE3:def 9</a></i>;<br/><a NAME="E4:27_1_1_1_2_1_2"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">x</font> <a href="lfuzzy_0.html#NR1" target="_self">&lt;&lt;=</a> <font color="Maroon">a</font>
 <b>by </b><i>, <a class="ref" href="waybel_0.html#D29">WAYBEL_0:def 29</a></i>;<br/><b>then </b><i><font color="Green">E71</font></i>: <font color="Maroon">x</font> = 
<font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">a</font>
<b>by </b><i><a class="ref" href="yellow_0.html#T25">YELLOW_0:25</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="waybel_1.html#K4">"/\"</a> </span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font>
<b>by </b><i><a class="ref" href="waybel_1.html#D18">WAYBEL_1:def 18</a></i>
;<br/><a NAME="E6:27_1_1_1_2_1_2"/>
<font color="Maroon">a</font> <a href="lfuzzy_0.html#NR1" target="_self">&lt;&lt;=</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_2.html#T24">YELLOW_2:24</a></i>;<br/><a NAME="E7:27_1_1_1_2_1_2"/><b>then </b>
<font color="Maroon">x</font> <a href="lfuzzy_0.html#NR1" target="_self">&lt;&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">X</font>
 <b>by </b><i>, <a class="ref" href="orders_2.html#T26">ORDERS_2:26</a></i>;<br/><a NAME="E8:27_1_1_1_2_1_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">X</font></span>)</span>
 <b>by </b><i><a class="ref" href="yellow_0.html#T25">YELLOW_0:25</a></i>;<br/><a NAME="E9:27_1_1_1_2_1_2"/><b>then </b><i><font color="Green">E72</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="waybel_1.html#K4">"/\"</a> </span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">X</font></span>)</span>
 <b>by </b><i><a class="ref" href="waybel_1.html#D18">WAYBEL_1:def 18</a></i>;<br/><a NAME="E10:27_1_1_1_2_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="waybel_1.html#K4">"/\"</a> </span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="waybel_1.html#K4">"/\"</a> </span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font>
 <b>by </b><i>, <a class="ref" href="funct_2.html#T43">FUNCT_2:43</a></i>;<br/><b>hence </b><a NAME="E11:27_1_1_1_2_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="waybel_1.html#K4">"/\"</a> </span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">X</font></span>)</span> <a href="lfuzzy_0.html#NR1" target="_self">&lt;&lt;=</a> <font color="Maroon">b</font>
 <b>by </b><i>, , , <a class="ref" href="lattice3.html#D9">LATTICE3:def 9</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E5:27_1_1_1"/>
 <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="waybel_1.html#K4">"/\"</a> </span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="waybel_1.html#K4">"/\"</a> </span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">X</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="yellow_0.html#T30">YELLOW_0:30</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:27_1_1"/>
<font color="Maroon">x</font> <a href="waybel_1.html#K4">"/\"</a>  is <a href="waybel_1.html#V5">lower_adjoint</a>
 <b>by </b><i><a class="ref" href="waybel_1.html#T18">WAYBEL_1:18</a></i>;<br/>


</div>
