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

<b>let </b><font color="Maroon">Q</font> be   <a href="quantal1.html#NM5">Girard-Quantale</a>;<br/><b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Q</font>;<br/><b>let </b><font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Q</font>;<br/>





<a NAME="E1:65"/><i><font color="Green">E26</font></i>: 
( <font color="Maroon">a</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="lattice3.html#NK17" target="_self">"\/"</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K7">}</a></span> &amp; <span class="p1">(<span class="default"><a href="quantal1.html#K5" target="_self">Bottom</a> <font color="Maroon">a</font></span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><a href="quantal1.html#K5" target="_self">Bottom</a> <font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="lattice3.html#NK17" target="_self">"\/"</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2">(<span class="default"><a href="quantal1.html#K5" target="_self">Bottom</a> <font color="Maroon">a</font></span>)</span>,<span class="p2">(<span class="default"><a href="quantal1.html#K5" target="_self">Bottom</a> <font color="Maroon">b</font></span>)</span></span><a href="domain_1.html#K7">}</a></span> &amp; <font color="Maroon">a</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="lattice3.html#NK18" target="_self">"/\"</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K7">}</a></span> &amp; <span class="p1">(<span class="default"><a href="quantal1.html#K5" target="_self">Bottom</a> <font color="Maroon">a</font></span>)</span> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><a href="quantal1.html#K5" target="_self">Bottom</a> <font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="lattice3.html#NK18" target="_self">"/\"</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2">(<span class="default"><a href="quantal1.html#K5" target="_self">Bottom</a> <font color="Maroon">a</font></span>)</span>,<span class="p2">(<span class="default"><a href="quantal1.html#K5" target="_self">Bottom</a> <font color="Maroon">b</font></span>)</span></span><a href="domain_1.html#K7">}</a></span> )
 
<b>by </b><i><a class="ref" href="lattice3.html#T44">LATTICE3:44</a></i>;<br/>
<a NAME="E2:65"/><i><font color="Green">E27</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="quantal1.html#K5" target="_self">Bottom</a> <font color="Olive">c</font></span>)</span> where <font color="Olive">c</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Q</font> : <font color="Olive">c</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K7">}</a></span> </span>}</span>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2">(<span class="default"><a href="quantal1.html#K5" target="_self">Bottom</a> <font color="Maroon">a</font></span>)</span>,<span class="p2">(<span class="default"><a href="quantal1.html#K5" target="_self">Bottom</a> <font color="Maroon">b</font></span>)</span></span><a href="domain_1.html#K7">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:65_1"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="quantal1.html#K5" target="_self">Bottom</a> <font color="Olive">c</font></span>)</span> where <font color="Olive">c</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Q</font> : <font color="Olive">c</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K7">}</a></span> </span>}</span>  <a href="tarski.html#R1">c=</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2">(<span class="default"><a href="quantal1.html#K5" target="_self">Bottom</a> <font color="Maroon">a</font></span>)</span>,<span class="p2">(<span class="default"><a href="quantal1.html#K5" target="_self">Bottom</a> <font color="Maroon">b</font></span>)</span></span><a href="domain_1.html#K7">}</a></span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</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="E1:65_1_1"/>
<font color="Maroon">x</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">c</font></span>)</span> where <font color="Olive">c</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Q</font> : <font color="Olive">c</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K7">}</a></span> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Q</font><b> such that </b><br/><a NAME="E3:65_1_1"/><i><font color="Green">E28</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="quantal1.html#K5" target="_self">Bottom</a> <font color="Maroon">c</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K7">}</a></span> )
 ;<br/>
<a NAME="E4:65_1_1"/>
( <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> or <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> )
 
<b>by </b><i>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<b>hence </b><a NAME="E5:65_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2">(<span class="default"><a href="quantal1.html#K5" target="_self">Bottom</a> <font color="Maroon">a</font></span>)</span>,<span class="p2">(<span class="default"><a href="quantal1.html#K5" target="_self">Bottom</a> <font color="Maroon">b</font></span>)</span></span><a href="domain_1.html#K7">}</a></span>
 <b>by </b><i>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>


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

<b>let </b><font color="Maroon">x</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="E2:65_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2">(<span class="default"><a href="quantal1.html#K5" target="_self">Bottom</a> <font color="Maroon">a</font></span>)</span>,<span class="p2">(<span class="default"><a href="quantal1.html#K5" target="_self">Bottom</a> <font color="Maroon">b</font></span>)</span></span><a href="domain_1.html#K7">}</a></span>
 ;<br/>

<a NAME="E3:65_1"/><b>then </b>
( ( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="quantal1.html#K5" target="_self">Bottom</a> <font color="Maroon">a</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K7">}</a></span> ) or ( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="quantal1.html#K5" target="_self">Bottom</a> <font color="Maroon">b</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K7">}</a></span> ) )
 
<b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<b>hence </b><a NAME="E4:65_1"/>
<font color="Maroon">x</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">c</font></span>)</span> where <font color="Olive">c</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Q</font> : <font color="Olive">c</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K7">}</a></span> </span>}</span> 
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:65"/>
 <a href="quantal1.html#K5" target="_self">Bottom</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="quantal1.html#K5" target="_self">Bottom</a> <font color="Maroon">a</font></span>)</span> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><a href="quantal1.html#K5" target="_self">Bottom</a> <font color="Maroon">b</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="quantal1.html#E2:65"><i><font color="Green">E27</font></i></a></i>;<br/>

<b>thus </b><a NAME="E4:65"/>
 <a href="quantal1.html#K5" target="_self">Bottom</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="quantal1.html#K5" target="_self">Bottom</a> <font color="Maroon">a</font></span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><a href="quantal1.html#K5" target="_self">Bottom</a> <font color="Maroon">b</font></span>)</span>
 <b>by </b><i>, , </i>;<br/>


</div>
