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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="quantal1.html#NM5">Girard-Quantale</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>



<a NAME="E1:64"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="quantal1.html#K5" target="_self">Bottom</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="quantal1.html#K5" target="_self">Bottom</a> <font color="Olive">b<sub>2</sub></font></span>)</span> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> </span>}</span>  </span>}</span>  <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></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="E1:64_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="quantal1.html#K5" target="_self">Bottom</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="quantal1.html#K5" target="_self">Bottom</a> <font color="Olive">b<sub>2</sub></font></span>)</span> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> </span>}</span>  </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:64_1"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="quantal1.html#K5" target="_self">Bottom</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="quantal1.html#K5" target="_self">Bottom</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> </span>}</span>  )
 ;<br/>
<b>consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E5:64_1"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="quantal1.html#K5" target="_self">Bottom</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="quantal1.html#E3:64_1"><i><font color="Green">E29</font></i></a></i>;<br/>
<b>thus </b><a NAME="E6:64_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="quantal1.html#E3:64_1"><i><font color="Green">E29</font></i></a>, <a class="txt" href="quantal1.html#E5:64_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="quantal1.html#T22" target="_self">Th22</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:64"/><b>then </b><i><font color="Green">E29</font></i>: 
 <a href="lattice3.html#K16">"/\"</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="lattices.html#R3">[=</a>  <a href="lattice3.html#K16">"/\"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="quantal1.html#K5" target="_self">Bottom</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="quantal1.html#K5" target="_self">Bottom</a> <font color="Olive">b<sub>2</sub></font></span>)</span> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> </span>}</span>  </span>}</span> ,<font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="lattice3.html#T46">LATTICE3:46</a></i>;<br/>
<a NAME="E3:64"/>
<font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#NR7" target="_self">is_great_than</a>  <a href="lattice3.html#K16">"/\"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="quantal1.html#K5" target="_self">Bottom</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="quantal1.html#K5" target="_self">Bottom</a> <font color="Olive">b<sub>2</sub></font></span>)</span> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> </span>}</span>  </span>}</span> ,<font color="Maroon">c<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D16">LATTICE3:def 16</a><br/>

<b>assume </b><a NAME="E1:64_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<a NAME="E2:64_2"/><b>then </b>
 <a href="quantal1.html#K5" target="_self">Bottom</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="quantal1.html#K5" target="_self">Bottom</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> </span>}</span> 
 
;<br/>
<a NAME="E3:64_2"/><b>then </b>
(  <a href="quantal1.html#K5" target="_self">Bottom</a> <span class="p1">(<span class="default"><a href="quantal1.html#K5" target="_self">Bottom</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="quantal1.html#K5" target="_self">Bottom</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="quantal1.html#K5" target="_self">Bottom</a> <font color="Olive">b<sub>2</sub></font></span>)</span> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> </span>}</span>  </span>}</span>  &amp;  <a href="quantal1.html#K5" target="_self">Bottom</a> <span class="p1">(<span class="default"><a href="quantal1.html#K5" target="_self">Bottom</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="ref" href="quantal1.html#T22" target="_self">Th22</a></i>;<br/>
<b>hence </b><a NAME="E4:64_2"/>
 <a href="lattice3.html#K16">"/\"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="quantal1.html#K5" target="_self">Bottom</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="quantal1.html#K5" target="_self">Bottom</a> <font color="Olive">b<sub>2</sub></font></span>)</span> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> </span>}</span>  </span>}</span> ,<font color="Maroon">c<sub>1</sub></font> <a href="lattices.html#R1">[=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="lattice3.html#T38">LATTICE3:38</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:64"/><b>then </b>
 <a href="lattice3.html#K16">"/\"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="quantal1.html#K5" target="_self">Bottom</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="quantal1.html#K5" target="_self">Bottom</a> <font color="Olive">b<sub>2</sub></font></span>)</span> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> </span>}</span>  </span>}</span> ,<font color="Maroon">c<sub>1</sub></font> <a href="lattices.html#R3">[=</a>  <a href="lattice3.html#K16">"/\"</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="lattice3.html#T40">LATTICE3:40</a></i>;<br/>
<a NAME="E5:64"/><b>then </b>
 <a href="lattice3.html#K16">"/\"</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="lattice3.html#K16">"/\"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="quantal1.html#K5" target="_self">Bottom</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="quantal1.html#K5" target="_self">Bottom</a> <font color="Olive">b<sub>2</sub></font></span>)</span> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> </span>}</span>  </span>}</span> ,<font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="quantal1.html#E2:64"><i><font color="Green">E29</font></i></a>, <a class="ref" href="lattices.html#T26">LATTICES:26</a></i>;<br/>
<b>hence </b> <a href="quantal1.html#K5" target="_self">Bottom</a> <span class="p1">(<span class="default"><a href="lattice3.html#K16">"/\"</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> = 
 <a href="quantal1.html#K5" target="_self">Bottom</a> <span class="p1">(<span class="default"><a href="quantal1.html#K5" target="_self">Bottom</a> <span class="p2">(<span class="default"><a href="lattice3.html#K15">"\/"</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><a href="quantal1.html#K5" target="_self">Bottom</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> </span>}</span> ,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="quantal1.html#T24" target="_self">Th24</a></i>
<br/>.= 
 <a href="lattice3.html#K15">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="quantal1.html#K5" target="_self">Bottom</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> </span>}</span> ,<font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="ref" href="quantal1.html#T22" target="_self">Th22</a></i>
;<br/><br/>


</div>
