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

<b>let </b><font color="Maroon">L</font> be  <a href="yellow_0.html#V3">bounded</a> <a href="waybel_0.html#NM8">LATTICE</a>;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>,  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>] <b>means </b>for <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> holds <br/> ( <span class="p1">(<span class="default"><font color="Olive">y</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a<sub>2</sub></font></span>)</span> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">a<sub>1</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">y</font> &amp; <font color="Olive">y</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><font color="Olive">y</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a<sub>2</sub></font> );<br/>
<b>assume </b><a NAME="E1:141"/><i><font color="Green">E32</font></i>: 
for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>  ex <font color="Olive">x'</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">x'</font>]
 ;<br/>

<b>consider </b><font color="Maroon">g'</font> being   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">L</font>,<font color="Maroon">L</font><b> such that </b><br/><a NAME="E3:141"/><i><font color="Green">E35</font></i>: 
for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Maroon">g'</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3">FUNCT_2:sch 3</a>();<br/></i>
<div><i><font color="Green">E45</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="141_1"><b>now </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>;<br/><b>let </b><font color="Maroon">g</font> be   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">L</font>,<font color="Maroon">L</font>;<br/><b>assume </b><a NAME="E1:141_1"/><i><font color="Green">E48</font></i>: 
for <font color="Olive">z</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> holds  <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">g'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Olive">z</font>
 ;<br/><a NAME="E2:141_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">g</font> is <a href="orders_3.html#V5">monotone</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="141_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">z1</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Maroon">z2</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="waybel_1.html#D2" target="_self">WAYBEL_1:def 2</a><br/>

<b>assume </b><a NAME="E1:141_1_1"/>
<font color="Maroon">z1</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z2</font>
 ;<br/>

<a NAME="E2:141_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">g'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">z1</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z2</font> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">g'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E3:141_1_1"/><b>then </b>
<font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z1</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">g'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">z2</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E4:141_1_1"/>
<font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z1</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z2</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="141_1_2"><b>now </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>;<br/><b>let </b><font color="Maroon">z</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:141_1_2_1"/>
<font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font>
 ;<br/><a NAME="E2:141_1_2_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">g'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">z</font>
 <b>by </b><i/>;<br/><a NAME="E3:141_1_2_1"/><b>then </b><i><font color="Green">E50</font></i>: 
<font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">y</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">g'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">z</font></span>)</span> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">y</font>
 <b>by </b><i/>;<br/><a NAME="E4:141_1_2_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">g'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">z</font></span>)</span> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">y</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z</font>
 <b>by </b><i><a class="ref" href="waybel_1.html#T12" target="_self">Th12</a></i>;<br/><a NAME="E5:141_1_2_1"/><b>then </b>
<font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">y</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z</font>
 <b>by </b><i>, <a class="ref" href="orders_2.html#T26">ORDERS_2:26</a></i>;<br/><b>hence </b><a NAME="E6:141_1_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="waybel_1.html#K4" target="_self">"/\"</a> </span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z</font>
 <b>by </b><i><a class="txt" href="waybel_1.html#E7:141"><i><font color="Green">E60</font></i></a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:141_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="waybel_1.html#K4" target="_self">"/\"</a> </span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z</font>
 ;<br/><a NAME="E3:141_1_2"/><b>then </b>
<font color="Maroon">y</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z</font>
 <b>by </b><i><a class="txt" href="waybel_1.html#E7:141"><i><font color="Green">E60</font></i></a></i>;<br/><a NAME="E4:141_1_2"/><b>then </b><i><font color="Green">E55</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">y</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">g'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z</font> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">g'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E5:141_1_2"/>
<font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">y</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">g'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span>
 <b>by </b><i><a class="ref" href="waybel_1.html#T12" target="_self">Th12</a></i>;<br/><a NAME="E6:141_1_2"/><b>then </b>
<font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z</font> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">g'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="orders_2.html#T26">ORDERS_2:26</a></i>;<br/><b>hence </b><a NAME="E7:141_1_2"/>
<font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:141_1"/>
<span class="p1"><a href="waybel_1.html#K1" target="_self">[</a><span class="default"><font color="Maroon">g</font>,<span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="waybel_1.html#K4" target="_self">"/\"</a> </span>)</span></span><a href="waybel_1.html#K1" target="_self">]</a></span> is <a href="waybel_1.html#V3" target="_self">Galois</a>
 <b>by </b><i><a class="ref" href="waybel_1.html#T13" target="_self">Th13</a>, </i>;<br/></div><b>end;</b></div>
<b>thus </b><a NAME="E5:141"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">L</font> <a href="waybel_1.html#NR13" target="_self">is_a_Heyting_algebra</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="141_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:141_2"/>
<font color="Maroon">L</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" target="_self">WAYBEL_1:def 19</a><br/>

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

<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>) <b>-&gt; </b>  <a href="subset_1.html#M1">M2</a>(the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>) = <span class="p1">(<span class="default"><font color="Maroon">g'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<b>consider </b><font color="Maroon">g</font> being   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">L</font>,<font color="Maroon">L</font><b> such that </b><br/><a NAME="E3:141_2"/><i><font color="Green">E58</font></i>: 
for <font color="Olive">z</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> holds  <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">z</font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S4">FUNCT_2:sch 4</a>();<br/></i>
<a NAME="E4:141_2"/>
<span class="p1"><a href="waybel_1.html#K1" target="_self">[</a><span class="default"><font color="Maroon">g</font>,<span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="waybel_1.html#K4" target="_self">"/\"</a> </span>)</span></span><a href="waybel_1.html#K1" target="_self">]</a></span> is <a href="waybel_1.html#V3" target="_self">Galois</a>
 
<b>by </b><i>, <a class="ref" href="waybel_1.html#T13" target="_self">Th13</a></i>;<br/>
<b>hence </b><a NAME="E5:141_2"/>
<font color="Maroon">y</font> <a href="waybel_1.html#K4" target="_self">"/\"</a>  <a href="waybel_1.html#NR9" target="_self">has_an_upper_adjoint</a> 
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>

<a NAME="E6:141"/><b>then </b><i><font color="Green">E59</font></i>: 
<font color="Maroon">L</font> is <a href="waybel_1.html#V2" target="_self">distributive</a>
 
<b>by </b><i/>;<br/>
<div><i><font color="Green">E60</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="141_3"><b>now </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>;<br/><b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>) <b>-&gt; </b>  <a href="subset_1.html#M1">M2</a>(the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>) = <span class="p1">(<span class="default"><font color="Maroon">g'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a<sub>1</sub></font>;<br/><b>consider </b><font color="Maroon">g</font> being   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">L</font>,<font color="Maroon">L</font><b> such that </b><br/><a NAME="E2:141_3"/><i><font color="Green">E61</font></i>: 
for <font color="Olive">z</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> holds  <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">z</font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S4">FUNCT_2:sch 4</a>();<br/></i><a NAME="E3:141_3"/>
<span class="p1"><a href="waybel_1.html#K1" target="_self">[</a><span class="default"><font color="Maroon">g</font>,<span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="waybel_1.html#K4" target="_self">"/\"</a> </span>)</span></span><a href="waybel_1.html#K1" target="_self">]</a></span> is <a href="waybel_1.html#V3" target="_self">Galois</a>
 <b>by </b><i>, <a class="ref" href="waybel_1.html#T14" target="_self">Th14</a></i>;<br/><a NAME="E4:141_3"/><b>then </b><i><font color="Green">E62</font></i>: 
<font color="Maroon">g</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">x</font> <a href="waybel_1.html#K5" target="_self">=&gt;</a> 
 <b>by </b><i>, </i>;<br/><b>thus </b> <a href="waybel_1.html#K7" target="_self">'not'</a> <font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">g'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i>, <a class="ref" href="waybel_1.html#T14" target="_self">Th14</a></i>
<br/>.= 
<font color="Maroon">g'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
<b>by </b><i/>
;<br/></div><b>end;</b></div>
<div><i><font color="Green">E63</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="141_4"><b>now </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>;<br/><a NAME="E1:141_4"/>
 <a href="yellow_0.html#K4">Top</a> <font color="Maroon">L</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K4">Top</a> <font color="Maroon">L</font></span>)</span> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">x</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">g'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>
 <b>by </b><i><a class="ref" href="waybel_1.html#T12" target="_self">Th12</a></i>;<br/><a NAME="E2:141_4"/><b>then </b><i><font color="Green">E64</font></i>: 
 <a href="yellow_0.html#K4">Top</a> <font color="Maroon">L</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">x</font> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">g'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E3:141_4"/>
<font color="Maroon">x</font> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">g'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#K4">Top</a> <font color="Maroon">L</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T45">YELLOW_0:45</a></i>;<br/><b>hence </b> <a href="yellow_0.html#K4">Top</a> <font color="Maroon">L</font> = 
<font color="Maroon">x</font> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">g'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i>, <a class="ref" href="orders_2.html#T25">ORDERS_2:25</a></i>
<br/>.= 
<font color="Maroon">x</font> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><a href="waybel_1.html#K7" target="_self">'not'</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i/>
;<br/><br/></div><b>end;</b></div>
<div><i><font color="Green">E65</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="141_5"><b>now </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>;<br/><a NAME="E1:141_5"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon">g'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font>
 <b>by </b><i><a class="ref" href="waybel_1.html#T12" target="_self">Th12</a></i>;<br/><a NAME="E2:141_5"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <span class="p2">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font></span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <span class="p2">(<span class="default"><font color="Maroon">g'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font>
 <b>by </b><i><a class="ref" href="waybel_1.html#T13" target="_self">Th13</a>, </i>;<br/><a NAME="E3:141_5"/><b>then </b>
<span class="p1">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <span class="p2">(<span class="default"><font color="Maroon">g'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font>
 <b>by </b><i/>;<br/><a NAME="E4:141_5"/><b>then </b><i><font color="Green">E66</font></i>: 
<font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">g'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font>
 <b>by </b><i/>;<br/><a NAME="E5:141_5"/>
 <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">g'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>
 <b>by </b><i><a class="ref" href="yellow_0.html#T44">YELLOW_0:44</a></i>;<br/><b>hence </b> <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font> = 
<font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">g'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="ref" href="waybel_1.html#T15" target="_self">Th15</a>, <a class="ref" href="orders_2.html#T25">ORDERS_2:25</a></i>
<br/>.= 
<font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><a href="waybel_1.html#K7" target="_self">'not'</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i/>
;<br/><br/></div><b>end;</b></div>
<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/>

<a NAME="E10:141"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_1.html#K7" target="_self">'not'</a> <font color="Maroon">x</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p2">(<span class="default"><a href="waybel_1.html#K7" target="_self">'not'</a> <span class="p3">(<span class="default"><a href="waybel_1.html#K7" target="_self">'not'</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K4">Top</a> <font color="Maroon">L</font></span>)</span> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="ref" href="waybel_1.html#T14" target="_self">Th14</a></i>;<br/>
<b>then </b><font color="Maroon">x</font> = 
<font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_1.html#K7" target="_self">'not'</a> <font color="Maroon">x</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p2">(<span class="default"><a href="waybel_1.html#K7" target="_self">'not'</a> <span class="p3">(<span class="default"><a href="waybel_1.html#K7" target="_self">'not'</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <span class="p2">(<span class="default"><a href="waybel_1.html#K7" target="_self">'not'</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <span class="p2">(<span class="default"><a href="waybel_1.html#K7" target="_self">'not'</a> <span class="p3">(<span class="default"><a href="waybel_1.html#K7" target="_self">'not'</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="waybel_1.html#T13" target="_self">Th13</a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <span class="p2">(<span class="default"><a href="waybel_1.html#K7" target="_self">'not'</a> <span class="p3">(<span class="default"><a href="waybel_1.html#K7" target="_self">'not'</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><a href="waybel_1.html#K7" target="_self">'not'</a> <span class="p2">(<span class="default"><a href="waybel_1.html#K7" target="_self">'not'</a> <font color="Maroon">x</font></span>)</span></span>)</span>
<b>by </b><i/>
;<br/>
<a NAME="E12:141"/><b>then </b><i><font color="Green">E67</font></i>: 
<font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="waybel_1.html#K7" target="_self">'not'</a> <span class="p1">(<span class="default"><a href="waybel_1.html#K7" target="_self">'not'</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="E13:141"/>
<span class="p1">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_1.html#K7" target="_self">'not'</a> <font color="Maroon">x</font></span>)</span> <a href="lattice3.html#K12">"/\"</a> <span class="p2">(<span class="default"><a href="waybel_1.html#K7" target="_self">'not'</a> <span class="p3">(<span class="default"><a href="waybel_1.html#K7" target="_self">'not'</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">x</font>
 
<b>by </b><i/>;<br/>
<b>then </b><font color="Maroon">x</font> = 
<font color="Maroon">x</font> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_1.html#K7" target="_self">'not'</a> <font color="Maroon">x</font></span>)</span> <a href="lattice3.html#K12">"/\"</a> <span class="p2">(<span class="default"><a href="waybel_1.html#K7" target="_self">'not'</a> <span class="p3">(<span class="default"><a href="waybel_1.html#K7" target="_self">'not'</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="lattice3.html#K13">"\/"</a> <span class="p2">(<span class="default"><a href="waybel_1.html#K7" target="_self">'not'</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="lattice3.html#K13">"\/"</a> <span class="p2">(<span class="default"><a href="waybel_1.html#K7" target="_self">'not'</a> <span class="p3">(<span class="default"><a href="waybel_1.html#K7" target="_self">'not'</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="waybel_1.html#T13" target="_self">Th13</a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="yellow_0.html#K4">Top</a> <font color="Maroon">L</font></span>)</span> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="lattice3.html#K13">"\/"</a> <span class="p2">(<span class="default"><a href="waybel_1.html#K7" target="_self">'not'</a> <span class="p3">(<span class="default"><a href="waybel_1.html#K7" target="_self">'not'</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="waybel_1.html#T14" target="_self">Th14</a></i>
<br/>.= 
<font color="Maroon">x</font> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><a href="waybel_1.html#K7" target="_self">'not'</a> <span class="p2">(<span class="default"><a href="waybel_1.html#K7" target="_self">'not'</a> <font color="Maroon">x</font></span>)</span></span>)</span>
<b>by </b><i/>
;<br/>
<b>hence </b><a NAME="E15:141"/>
 <a href="waybel_1.html#K7" target="_self">'not'</a> <span class="p1">(<span class="default"><a href="waybel_1.html#K7" target="_self">'not'</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="waybel_1.html#T15" target="_self">Th15</a>, <a class="ref" href="yellow_0.html#T24">YELLOW_0:24</a></i>;<br/>


</div>
