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

<b>let </b><font color="Maroon">A</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">u</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="normform.html#K12">NormForm</a> <font color="Maroon">A</font></span>)</span>;<br/><b>let </b><font color="Maroon">v</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="normform.html#K12">NormForm</a> <font color="Maroon">A</font></span>)</span>;<br/>





<b>assume </b><a NAME="E1:19"/><i><font color="Green">E21</font></i>: 
for <font color="Olive">a</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">A</font>  st <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">u</font> holds <br/> ex <font color="Olive">b</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">A</font> st <br/>( <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">v</font> &amp; <font color="Olive">b</font> <a href="normform.html#R1">c=</a> <font color="Olive">a</font> )
 ;<br/>

<a NAME="E2:19"/><i><font color="Green">E23</font></i>: 
 <a href="normform.html#K9">mi</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="heyting1.html#K3" target="_self">@</a> <font color="Maroon">u</font></span>)</span> <a href="setwiseo.html#K5">\/</a> <span class="p2">(<span class="default"><a href="heyting1.html#K3" target="_self">@</a> <font color="Maroon">v</font></span>)</span></span>)</span> <a href="heyting1.html#R1" target="_self">c=</a>  <a href="heyting1.html#K3" target="_self">@</a> <font color="Maroon">v</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">A</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="heyting1.html#D1" target="_self">HEYTING1:def 1</a><br/>

<b>assume </b><a NAME="E1:19_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="normform.html#K9">mi</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="heyting1.html#K3" target="_self">@</a> <font color="Maroon">u</font></span>)</span> <a href="setwiseo.html#K5">\/</a> <span class="p2">(<span class="default"><a href="heyting1.html#K3" target="_self">@</a> <font color="Maroon">v</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:19_1"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">u</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">v</font>
 
<b>by </b><i><a class="ref" href="normform.html#T58">NORMFORM:58</a></i>;<br/>
<a NAME="E3:19_1"/><b>then </b>
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">u</font> or <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">v</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>then consider </b><font color="Maroon">b</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">A</font><b> such that </b><br/><a NAME="E5:19_1"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">v</font>
 <b>and </b><br/><a NAME="E6:19_1"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">b</font> <a href="normform.html#R1">c=</a> <font color="Maroon">a</font>
 <b>by </b><i/>;<br/>
<a NAME="E7:19_1"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">u</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">v</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>hence </b><a NAME="E8:19_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="heyting1.html#K3" target="_self">@</a> <font color="Maroon">v</font>
 <b>by </b><i>, , <a class="txt" href="heyting1.html#E1:19"><i><font color="Green">E21</font></i></a>, <a class="ref" href="normform.html#T60">NORMFORM:60</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:19"/><i><font color="Green">E60</font></i>: 
 <a href="heyting1.html#K3" target="_self">@</a> <font color="Maroon">v</font> <a href="heyting1.html#R1" target="_self">c=</a>  <a href="normform.html#K9">mi</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="heyting1.html#K3" target="_self">@</a> <font color="Maroon">u</font></span>)</span> <a href="setwiseo.html#K5">\/</a> <span class="p2">(<span class="default"><a href="heyting1.html#K3" target="_self">@</a> <font color="Maroon">v</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">A</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="heyting1.html#D1" target="_self">HEYTING1:def 1</a><br/>

<b>assume </b><a NAME="E1:19_2"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="heyting1.html#K3" target="_self">@</a> <font color="Maroon">v</font>
 ;<br/>

<a NAME="E2:19_2"/><b>then </b><i><font color="Green">E62</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="heyting1.html#K3" target="_self">@</a> <font color="Maroon">u</font></span>)</span> <a href="setwiseo.html#K5">\/</a> <span class="p1">(<span class="default"><a href="heyting1.html#K3" target="_self">@</a> <font color="Maroon">v</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E3:19_2"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="normform.html#K9">mi</a> <span class="p1">(<span class="default"><a href="heyting1.html#K3" target="_self">@</a> <font color="Maroon">v</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="normform.html#T66">NORMFORM:66</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">b</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">A</font>;<br/><b>assume </b><b>that </b><br/><a NAME="E1:19_2_1"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">u</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">v</font>
 <b>and </b><br/><a NAME="E2:19_2_1"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">b</font> <a href="normform.html#R1">c=</a> <font color="Maroon">a</font>
 ;<br/><a NAME="E3:19_2_1"/>
( <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">u</font> or <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">v</font> )
 <b>by </b><i><a class="txt" href="heyting1.html#E2:19"><i><font color="Green">E23</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>then consider </b><font color="Maroon">c</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">A</font><b> such that </b><br/><a NAME="E5:19_2_1"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">v</font>
 <b>and </b><br/><a NAME="E6:19_2_1"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">c</font> <a href="normform.html#R1">c=</a> <font color="Maroon">b</font>
 <b>by </b><i/>;<br/><a NAME="E7:19_2_1"/>
<font color="Maroon">c</font> <a href="normform.html#R1">c=</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="heyting1.html#D2" target="_self">Def2</a>, , <a class="ref" href="normform.html#T5">NORMFORM:5</a></i>;<br/><a NAME="E8:19_2_1"/><b>then </b>
<font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 <b>by </b><i>, , <a class="ref" href="normform.html#T60">NORMFORM:60</a></i>;<br/><b>hence </b><a NAME="E9:19_2_1"/>
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="heyting1.html#D2" target="_self">Def2</a>, , <a class="ref" href="normform.html#T4">NORMFORM:4</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:19_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="normform.html#K9">mi</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="heyting1.html#K3" target="_self">@</a> <font color="Maroon">u</font></span>)</span> <a href="setwiseo.html#K5">\/</a> <span class="p2">(<span class="default"><a href="heyting1.html#K3" target="_self">@</a> <font color="Maroon">v</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="heyting1.html#E1:19"><i><font color="Green">E21</font></i></a>, <a class="ref" href="normform.html#T61">NORMFORM:61</a></i>;<br/>


</div><b>end;</b></div>
<font color="Maroon">u</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">v</font> = 
the <a href="lattices.html#U2">L_join</a> of <span class="p1">(<span class="default"><a href="normform.html#K12">NormForm</a> <font color="Maroon">A</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">u</font>,<font color="Maroon">v</font>
<b>by </b><i><a class="ref" href="lattices.html#D1">LATTICES:def 1</a></i>
<br/>.= 
 <a href="normform.html#K9">mi</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="heyting1.html#K3" target="_self">@</a> <font color="Maroon">u</font></span>)</span> <a href="setwiseo.html#K5">\/</a> <span class="p2">(<span class="default"><a href="heyting1.html#K3" target="_self">@</a> <font color="Maroon">v</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">v</font>
<b>by </b><i>, , <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>
;<br/>
<b>hence </b><a NAME="E5:19"/>
<font color="Maroon">u</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">v</font>
 <b>by </b><i><a class="ref" href="lattices.html#D3">LATTICES:def 3</a></i>;<br/>


</div>
