<?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="E1:63"/>
 <a href="quantal1.html#K5" target="_self">Bottom</a> <span class="p1">(<span class="default"><a href="lattice3.html#K15">"\/"</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="lattice3.html#R3">is_less_than</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> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1"><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="E1:63_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> <font color="Maroon">c<sub>2</sub></font> </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="E3:63_1"/><i><font color="Green">E28</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> <font color="Maroon">c<sub>2</sub></font> )
 ;<br/>
<a NAME="E4:63_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="lattices.html#R3">[=</a>  <a href="lattice3.html#K15">"\/"</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="quantal1.html#E3:63_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="lattice3.html#T38">LATTICE3:38</a></i>;<br/>
<b>hence </b><a NAME="E5:63_1"/>
 <a href="quantal1.html#K5" target="_self">Bottom</a> <span class="p1">(<span class="default"><a href="lattice3.html#K15">"\/"</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="lattices.html#R1">[=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="quantal1.html#E3:63_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="quantal1.html#T23" target="_self">Th23</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:63"/><b>then </b><i><font color="Green">E28</font></i>: 
 <a href="quantal1.html#K5" target="_self">Bottom</a> <span class="p1">(<span class="default"><a href="lattice3.html#K15">"\/"</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <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> <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="lattice3.html#T40">LATTICE3:40</a></i>;<br/>
<a NAME="E3:63"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lattice3.html#K16">"/\"</a> <span class="p4">{<span class="default"> <span class="p5">(<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> ,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="monoid_0.html#NK3" target="_self">[*]</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>  <a href="lattice3.html#R4">is_less_than</a>  <a href="quantal1.html#K3" target="_self">Bottom</a> <font color="Maroon">c<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_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#D17">LATTICE3:def 17</a><br/>

<b>assume </b><a NAME="E1:63_2"/>
<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"><span class="p3">(<span class="default"><a href="lattice3.html#K16">"/\"</a> <span class="p4">{<span class="default"> <span class="p5">(<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> ,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="monoid_0.html#NK3" target="_self">[*]</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>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="E3:63_2"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="lattice3.html#K16">"/\"</a> <span class="p2">{<span class="default"> <span class="p3">(<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> <a href="monoid_0.html#NK3" target="_self">[*]</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> <font color="Maroon">c<sub>2</sub></font> )
 ;<br/>
<a NAME="E4:63_2"/>
 <a href="quantal1.html#K5" target="_self">Bottom</a> <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> 
 
<b>by </b><i><a class="txt" href="quantal1.html#E3:63_2"><i><font color="Green">E29</font></i></a></i>;<br/>
<a NAME="E5:63_2"/><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> <font color="Maroon">c<sub>2</sub></font> </span>}</span> ,<font color="Maroon">c<sub>1</sub></font> <a href="lattices.html#R3">[=</a>  <a href="quantal1.html#K5" target="_self">Bottom</a> <font color="Maroon">c<sub>4</sub></font> &amp;  <a href="quantal1.html#K5" target="_self">Bottom</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="quantal1.html#K1" target="_self">-r&gt;</a> <span class="p1">(<span class="default"><a href="quantal1.html#K3" target="_self">Bottom</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> )
 
<b>by </b><i><a class="ref" href="lattice3.html#T38">LATTICE3:38</a></i>;<br/>
<b>hence </b><a NAME="E6:63_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="lattices.html#R3">[=</a>  <a href="quantal1.html#K3" target="_self">Bottom</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="quantal1.html#E3:63_2"><i><font color="Green">E29</font></i></a>, <a class="ref" href="quantal1.html#T12" target="_self">Th12</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:63"/><b>then </b>
 <a href="lattice3.html#K15">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lattice3.html#K16">"/\"</a> <span class="p4">{<span class="default"> <span class="p5">(<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> ,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="monoid_0.html#NK3" target="_self">[*]</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> <a href="lattices.html#R3">[=</a>  <a href="quantal1.html#K3" target="_self">Bottom</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="lattice3.html#D21">LATTICE3:def 21</a></i>;<br/>
<a NAME="E5:63"/><b>then </b>
( <span class="p1">(<span class="default"><a href="lattice3.html#K16">"/\"</a> <span class="p2">{<span class="default"> <span class="p3">(<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> <a href="monoid_0.html#NK3" target="_self">[*]</a> <span class="p1">(<span class="default"><a href="lattice3.html#K15">"\/"</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="lattices.html#R3">[=</a>  <a href="quantal1.html#K3" target="_self">Bottom</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="quantal1.html#K5" target="_self">Bottom</a> <span class="p1">(<span class="default"><a href="lattice3.html#K15">"\/"</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="lattice3.html#K15">"\/"</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="quantal1.html#K1" target="_self">-r&gt;</a> <span class="p1">(<span class="default"><a href="quantal1.html#K3" target="_self">Bottom</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> )
 
<b>by </b><i><a class="ref" href="quantal1.html#D5" target="_self">Def5</a></i>;<br/>
<a NAME="E6:63"/><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> <font color="Maroon">c<sub>2</sub></font> </span>}</span> ,<font color="Maroon">c<sub>1</sub></font> <a href="lattices.html#R3">[=</a>  <a href="quantal1.html#K5" target="_self">Bottom</a> <span class="p1">(<span class="default"><a href="lattice3.html#K15">"\/"</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="quantal1.html#T12" target="_self">Th12</a></i>;<br/>
<b>hence </b><a NAME="E7:63"/>
 <a href="quantal1.html#K5" target="_self">Bottom</a> <span class="p1">(<span class="default"><a href="lattice3.html#K15">"\/"</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <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> <font color="Maroon">c<sub>2</sub></font> </span>}</span> ,<font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="quantal1.html#E2:63"><i><font color="Green">E28</font></i></a>, <a class="ref" href="lattices.html#T26">LATTICES:26</a></i>;<br/>


</div>
