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

<b>let </b><font color="Maroon">H</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="orders_2.html#L1">RelStr</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:131"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">H</font> <a href="waybel_1.html#NR13" target="_self">is_a_Heyting_algebra</a> 
 <b>and </b><br/><a NAME="E2:131"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">H</font> is <a href="yellow_0.html#V1">lower-bounded</a>
 ;<br/>

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

<a NAME="E3:131"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">H</font> is   <a href="waybel_0.html#NM8">LATTICE</a>
 
<b>by </b><i>, </i>;<br/>
<b>set </b><font color="Maroon">X</font> = <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">H</font> : <font color="Maroon">a</font> <a href="lattice3.html#K11">"/\"</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">H</font> </span>}</span> ;<br/>
<b>set </b><font color="Maroon">Y</font> = <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">H</font> : <font color="Maroon">a</font> <a href="lattice3.html#K11">"/\"</a> <font color="Olive">x</font> <a href="orders_2.html#R1">&lt;=</a>  <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">H</font> </span>}</span> ;<br/>
<a NAME="E4:131"/><i><font color="Green">E48</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">H</font> : <font color="Maroon">a</font> <a href="lattice3.html#K11">"/\"</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">H</font> </span>}</span>  <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">H</font> : <font color="Maroon">a</font> <a href="lattice3.html#K11">"/\"</a> <font color="Olive">x</font> <a href="orders_2.html#R1">&lt;=</a>  <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">H</font> </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="131_1"><b>proof </b></a><div class="add">

<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a>,<a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a>
<div class="add"><b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:131_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">H</font> : <font color="Maroon">a</font> <a href="lattice3.html#K11">"/\"</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">H</font> </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">x</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">H</font><b> such that </b><br/><a NAME="E3:131_1_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>and </b><br/><a NAME="E4:131_1_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">a</font> <a href="lattice3.html#K11">"/\"</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">H</font>
 ;<br/><a NAME="E5:131_1_1"/>
<font color="Maroon">a</font> <a href="lattice3.html#K11">"/\"</a> <font color="Maroon">x</font> <a href="orders_2.html#R1">&lt;=</a>  <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">H</font>
 <b>by </b><i><a class="ref" href="waybel_1.html#D11" target="_self">Def11</a>, <a class="ref" href="waybel_1.html#T11" target="_self">Th11</a>, <a class="ref" href="orders_2.html#T24">ORDERS_2:24</a></i>;<br/><b>hence </b><a NAME="E6:131_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">H</font> : <font color="Maroon">a</font> <a href="lattice3.html#K11">"/\"</a> <font color="Olive">x</font> <a href="orders_2.html#R1">&lt;=</a>  <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">H</font> </span>}</span> 
 <b>by </b><i><a class="ref" href="waybel_1.html#T10" target="_self">Th10</a></i>;<br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:131_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">H</font> : <font color="Maroon">a</font> <a href="lattice3.html#K11">"/\"</a> <font color="Olive">x</font> <a href="orders_2.html#R1">&lt;=</a>  <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">H</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">H</font><b> such that </b><br/><a NAME="E4:131_1"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>and </b><br/><a NAME="E5:131_1"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">a</font> <a href="lattice3.html#K11">"/\"</a> <font color="Maroon">x</font> <a href="orders_2.html#R1">&lt;=</a>  <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">H</font>
 ;<br/>
<a NAME="E6:131_1"/>
 <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">H</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">a</font> <a href="lattice3.html#K11">"/\"</a> <font color="Maroon">x</font>
 
<b>by </b><i>, <a class="ref" href="waybel_1.html#D11" target="_self">Def11</a>, <a class="ref" href="yellow_0.html#T44">YELLOW_0:44</a></i>;<br/>
<a NAME="E7:131_1"/><b>then </b>
 <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="lattice3.html#K11">"/\"</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="ref" href="waybel_1.html#D11" target="_self">Def11</a>, <a class="ref" href="waybel_1.html#T11" target="_self">Th11</a>, <a class="ref" href="orders_2.html#T25">ORDERS_2:25</a></i>;<br/>
<b>hence </b><a NAME="E8:131_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">H</font> : <font color="Maroon">a</font> <a href="lattice3.html#K11">"/\"</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">H</font> </span>}</span> 
 <b>by </b><i><a class="ref" href="waybel_1.html#T10" target="_self">Th10</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:131"/>
for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">H</font> holds  <font color="Olive">x</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>, </i>;<br/>
<a NAME="E6:131"/><b>then </b><i><font color="Green">E58</font></i>: 
 <a href="waybel_1.html#R2" target="_self">ex_max_of</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">H</font> : <font color="Maroon">a</font> <a href="lattice3.html#K11">"/\"</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">H</font> </span>}</span> ,<font color="Maroon">H</font>
 
<b>by </b><i><a class="ref" href="waybel_1.html#D11" target="_self">Def11</a>, <a class="ref" href="waybel_1.html#D12" target="_self">Def12</a>, </i>;<br/>
<b>hence </b><a NAME="E7:131"/>
 <a href="yellow_0.html#R1">ex_sup_of</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">H</font> : <font color="Maroon">a</font> <a href="lattice3.html#K11">"/\"</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">H</font> </span>}</span> ,<font color="Maroon">H</font>
 <b>by </b><i/>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_1.html#D7" target="_self">WAYBEL_1:def 7</a><br/>

<a NAME="E8:131"/><i><font color="Green">E59</font></i>: 
 <a href="waybel_1.html#K7" target="_self">'not'</a> <font color="Maroon">a</font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">H</font> : <font color="Maroon">a</font> <a href="lattice3.html#K11">"/\"</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">H</font> </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="131_2"><b>proof </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">H</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="E1:131_2"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">H</font> : <font color="Maroon">a</font> <a href="lattice3.html#K11">"/\"</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">H</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">H</font><b> such that </b><br/><a NAME="E3:131_2"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 <b>and </b><br/><a NAME="E4:131_2"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">a</font> <a href="lattice3.html#K11">"/\"</a> <font color="Maroon">x</font> <a href="orders_2.html#R1">&lt;=</a>  <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">H</font>
 <b>by </b><i><a class="ref" href="waybel_1.html#D12" target="_self">Def12</a></i>;<br/>
<a NAME="E5:131_2"/>
<font color="Maroon">a</font> <a href="waybel_1.html#K6" target="_self">=&gt;</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">H</font></span>)</span> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">x</font>
 
<b>by </b><i>, , </i>;<br/>
<b>hence </b><a NAME="E6:131_2"/>
 <a href="waybel_1.html#K7" target="_self">'not'</a> <font color="Maroon">a</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">b</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="131_3"><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">H</font>;<br/><b>assume </b><a NAME="E1:131_3"/><i><font color="Green">E62</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="Olive">x</font> where <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">H</font> : <font color="Maroon">a</font> <a href="lattice3.html#K11">"/\"</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">H</font> </span>}</span> 
 ;<br/><a NAME="E2:131_3"/>
<font color="Maroon">a</font> <a href="waybel_1.html#K6" target="_self">=&gt;</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">H</font></span>)</span> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">a</font> <a href="waybel_1.html#K6" target="_self">=&gt;</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">H</font></span>)</span>
 <b>by </b><i><a class="ref" href="waybel_1.html#D11" target="_self">Def11</a>, <a class="ref" href="orders_2.html#T24">ORDERS_2:24</a></i>;<br/><a NAME="E3:131_3"/><b>then </b>
 <a href="waybel_1.html#K7" target="_self">'not'</a> <font color="Maroon">a</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">a</font> <a href="waybel_1.html#K6" target="_self">=&gt;</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">H</font></span>)</span>
 ;<br/><a NAME="E4:131_3"/><b>then </b>
<font color="Maroon">a</font> <a href="lattice3.html#K11">"/\"</a> <span class="p1">(<span class="default"><a href="waybel_1.html#K7" target="_self">'not'</a> <font color="Maroon">a</font></span>)</span> <a href="orders_2.html#R1">&lt;=</a>  <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">H</font>
 <b>by </b><i>, </i>;<br/><a NAME="E5:131_3"/><b>then </b>
 <a href="waybel_1.html#K7" target="_self">'not'</a> <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">H</font> : <font color="Maroon">a</font> <a href="lattice3.html#K11">"/\"</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">H</font> </span>}</span> 
 <b>by </b><i><a class="ref" href="waybel_1.html#D12" target="_self">Def12</a></i>;<br/><b>hence </b><a NAME="E6:131_3"/>
 <a href="waybel_1.html#K7" target="_self">'not'</a> <font color="Maroon">a</font> <a href="orders_2.html#R1">&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>
<b>hence </b><a NAME="E10:131"/>
 <a href="waybel_1.html#K7" target="_self">'not'</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K1">"\/"</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">H</font> : <font color="Maroon">a</font> <a href="lattice3.html#K11">"/\"</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">H</font> </span>}</span> ,<font color="Maroon">H</font>
 <b>by </b><i><a class="ref" href="waybel_1.html#D11" target="_self">Def11</a>, <a class="ref" href="waybel_1.html#T11" target="_self">Th11</a>, <a class="ref" href="yellow_0.html#T30">YELLOW_0:30</a></i>;<br/>

<b>thus </b><a NAME="E11:131"/>
 <a href="yellow_0.html#K1">"\/"</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">H</font> : <font color="Maroon">a</font> <a href="lattice3.html#K11">"/\"</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">H</font> </span>}</span> ,<font color="Maroon">H</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">H</font> : <font color="Maroon">a</font> <a href="lattice3.html#K11">"/\"</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">H</font> </span>}</span> 
 <b>by </b><i><a class="ref" href="waybel_1.html#T10" target="_self">Th10</a>, </i>;<br/>


</div>
