<?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">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>;<br/><b>let </b><font color="Maroon">K</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K8">Normal_forms_on</a> <font color="Maroon">A</font>;<br/>





<b>assume </b><a NAME="E1:31"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">K</font> <a href="normform.html#K10">^</a> <span class="p1"><a href="heyting1.html#K2" target="_self">{</a><span class="default"><font color="Maroon">a</font></span><a href="heyting1.html#K2" target="_self">}</a></span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:31_1_1"/>
(  not <font color="Maroon">A</font> is <a href="xboole_0.html#V1">empty</a> or <font color="Maroon">A</font> is <a href="xboole_0.html#V1">empty</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_1_1_1"><b>suppose </b></a><a NAME="E1:31_1_1_1"/>
not <font color="Maroon">A</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><a NAME="E2:31_1_1_1"/><b>then </b><i><font color="Green">E23</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="heyting1.html#K1" target="_self">[</a><span class="default"><font color="Maroon">A</font></span><a href="heyting1.html#K1" target="_self">]</a></span>
 <b>by </b><i/>;<br/><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b><font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="mcart_1.html#K1">`1</a> </span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="domain_1.html#K3">`2</a> </span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="mcart_1.html#K2">`2</a> </span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="domain_1.html#K2">`1</a> </span>)</span></span>)</span>;<br/><div><i><font color="Green">E52</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">s</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><a NAME="E1:31_1_1_1_1"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font>
 ;<br/><a NAME="E2:31_1_1_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="heyting1.html#K2" target="_self">{</a><span class="default"><font color="Maroon">a</font></span><a href="heyting1.html#K2" target="_self">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E3:31_1_1_1_1"/><b>then </b>
not <font color="Maroon">s</font> <a href="normform.html#K1">\/</a> <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">A</font>
 <b>by </b><i>, , <a class="ref" href="normform.html#T56">NORMFORM:56</a></i>;<br/><b>then consider </b><font color="Maroon">x</font> being    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="heyting1.html#K1" target="_self">[</a><span class="default"><font color="Maroon">A</font></span><a href="heyting1.html#K1" target="_self">]</a></span><b> such that </b><br/><a NAME="E5:31_1_1_1_1"/><i><font color="Green">E59</font></i>: 
( ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">s</font> <a href="domain_1.html#K2">`1</a>  &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font> <a href="domain_1.html#K3">`2</a>  ) or ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font> <a href="domain_1.html#K2">`1</a>  &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">s</font> <a href="domain_1.html#K3">`2</a>  ) )
 <b>by </b><i>, <a class="ref" href="normform.html#T47">NORMFORM:47</a></i>;<br/><b>take </b><font color="Maroon">x</font> = <font color="Maroon">x</font>;<br/><a NAME="E6:31_1_1_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="domain_1.html#K2">`1</a> </span>)</span> <a href="finsub_1.html#K3">/\</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="domain_1.html#K3">`2</a> </span>)</span> or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="domain_1.html#K3">`2</a> </span>)</span> <a href="finsub_1.html#K3">/\</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="domain_1.html#K2">`1</a> </span>)</span> )
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>hence </b><a NAME="E7:31_1_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">s</font>,<font color="Maroon">x</font>]
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div><b>consider </b><font color="Maroon">g</font> being    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <span class="p1">(<span class="default"><a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">A</font></span>)</span>,<span class="p1"><a href="heyting1.html#K1" target="_self">[</a><span class="default"><font color="Maroon">A</font></span><a href="heyting1.html#K1" target="_self">]</a></span><b> such that </b><br/><a NAME="E5:31_1_1_1"/><i><font color="Green">E60</font></i>: 
for <font color="Olive">s</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">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">s</font>,<font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">s</font>]
 <b>from </b><i><a class="ref" href="fraenkel.html#S27">FRAENKEL:sch 27</a>();<br/></i><div><i><font color="Green">E61</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">s</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><a NAME="E1:31_1_1_1_2"/>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font>
 ;<br/><a NAME="E2:31_1_1_1_2"/><b>then </b>
<font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="domain_1.html#K2">`1</a> </span>)</span> <a href="finsub_1.html#K3">/\</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="domain_1.html#K3">`2</a> </span>)</span></span>)</span> <a href="setwiseo.html#K5">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="domain_1.html#K3">`2</a> </span>)</span> <a href="finsub_1.html#K3">/\</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="domain_1.html#K2">`1</a> </span>)</span></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E3:31_1_1_1_2"/><b>then </b>
( <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="domain_1.html#K2">`1</a> </span>)</span> <a href="finsub_1.html#K3">/\</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="domain_1.html#K3">`2</a> </span>)</span> or <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="domain_1.html#K3">`2</a> </span>)</span> <a href="finsub_1.html#K3">/\</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="domain_1.html#K2">`1</a> </span>)</span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E4:31_1_1_1_2"/><b>then </b>
( ( <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">s</font> <a href="domain_1.html#K2">`1</a>  &amp; <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font> <a href="domain_1.html#K3">`2</a>  ) or ( <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">s</font> <a href="domain_1.html#K3">`2</a>  &amp; <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font> <a href="domain_1.html#K2">`1</a>  ) )
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>hence </b><a NAME="E5:31_1_1_1_2"/>
<font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="domain_1.html#K2">`1</a> </span>)</span> <a href="setwiseo.html#K5">\/</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="domain_1.html#K3">`2</a> </span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div><b>set </b><font color="Maroon">c1</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">t1</font></span>)</span> where <font color="Olive">t1</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">A</font> : ( <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">t1</font> <a href="hidden.html#R2">in</a> <font color="Olive">t1</font> <a href="domain_1.html#K3">`2</a>  &amp; <font color="Olive">t1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> ) </span>}</span> ;<br/><b>set </b><font color="Maroon">c2</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">t2</font></span>)</span> where <font color="Olive">t2</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">A</font> : ( <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">t2</font> <a href="hidden.html#R2">in</a> <font color="Olive">t2</font> <a href="domain_1.html#K2">`1</a>  &amp; <font color="Olive">t2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> ) </span>}</span> ;<br/><a NAME="E7:31_1_1_1"/><i><font color="Green">E62</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">t1</font></span>)</span> where <font color="Olive">t1</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">A</font> : ( <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">t1</font> <a href="hidden.html#R2">in</a> <font color="Olive">t1</font> <a href="domain_1.html#K3">`2</a>  &amp; <font color="Olive">t1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> ) </span>}</span>  <a href="tarski.html#R1">c=</a> <font color="Maroon">a</font> <a href="domain_1.html#K2">`1</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_1_1_3"><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="E1:31_1_1_1_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">t1</font></span>)</span> where <font color="Olive">t1</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">A</font> : ( <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">t1</font> <a href="hidden.html#R2">in</a> <font color="Olive">t1</font> <a href="domain_1.html#K3">`2</a>  &amp; <font color="Olive">t1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">t</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="E3:31_1_1_1_3"/><i><font color="Green">E64</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">t</font> &amp; <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">t</font> <a href="domain_1.html#K3">`2</a>  )
 <b>and </b><br/><a NAME="E4:31_1_1_1_3"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font>
 ;<br/>
<a NAME="E5:31_1_1_1_3"/>
<font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="domain_1.html#K2">`1</a> </span>)</span> <a href="finsub_1.html#K3">/\</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="domain_1.html#K3">`2</a> </span>)</span></span>)</span> <a href="setwiseo.html#K5">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="domain_1.html#K3">`2</a> </span>)</span> <a href="finsub_1.html#K3">/\</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="domain_1.html#K2">`1</a> </span>)</span></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E6:31_1_1_1_3"/><b>then </b>
( <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="domain_1.html#K2">`1</a> </span>)</span> <a href="finsub_1.html#K3">/\</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="domain_1.html#K3">`2</a> </span>)</span> or <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="domain_1.html#K3">`2</a> </span>)</span> <a href="finsub_1.html#K3">/\</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="domain_1.html#K2">`1</a> </span>)</span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E7:31_1_1_1_3"/><b>then </b>
( ( <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">t</font> <a href="domain_1.html#K2">`1</a>  &amp; <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font> <a href="domain_1.html#K3">`2</a>  ) or ( <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">t</font> <a href="domain_1.html#K3">`2</a>  &amp; <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font> <a href="domain_1.html#K2">`1</a>  ) )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E8:31_1_1_1_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font> <a href="domain_1.html#K2">`1</a> 
 <b>by </b><i>, <a class="ref" href="normform.html#T46">NORMFORM:46</a></i>;<br/>


</div><b>end;</b></div><a NAME="E8:31_1_1_1"/><i><font color="Green">E66</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">t2</font></span>)</span> where <font color="Olive">t2</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">A</font> : ( <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">t2</font> <a href="hidden.html#R2">in</a> <font color="Olive">t2</font> <a href="domain_1.html#K2">`1</a>  &amp; <font color="Olive">t2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> ) </span>}</span>  <a href="tarski.html#R1">c=</a> <font color="Maroon">a</font> <a href="domain_1.html#K3">`2</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_1_1_4"><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="E1:31_1_1_1_4"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">t2</font></span>)</span> where <font color="Olive">t2</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">A</font> : ( <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">t2</font> <a href="hidden.html#R2">in</a> <font color="Olive">t2</font> <a href="domain_1.html#K2">`1</a>  &amp; <font color="Olive">t2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">t</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="E3:31_1_1_1_4"/><i><font color="Green">E67</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">t</font> &amp; <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">t</font> <a href="domain_1.html#K2">`1</a>  )
 <b>and </b><br/><a NAME="E4:31_1_1_1_4"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font>
 ;<br/>
<a NAME="E5:31_1_1_1_4"/>
<font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="domain_1.html#K2">`1</a> </span>)</span> <a href="finsub_1.html#K3">/\</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="domain_1.html#K3">`2</a> </span>)</span></span>)</span> <a href="setwiseo.html#K5">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="domain_1.html#K3">`2</a> </span>)</span> <a href="finsub_1.html#K3">/\</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="domain_1.html#K2">`1</a> </span>)</span></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E6:31_1_1_1_4"/><b>then </b>
( <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="domain_1.html#K2">`1</a> </span>)</span> <a href="finsub_1.html#K3">/\</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="domain_1.html#K3">`2</a> </span>)</span> or <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="domain_1.html#K3">`2</a> </span>)</span> <a href="finsub_1.html#K3">/\</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="domain_1.html#K2">`1</a> </span>)</span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E7:31_1_1_1_4"/><b>then </b>
( ( <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">t</font> <a href="domain_1.html#K2">`1</a>  &amp; <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font> <a href="domain_1.html#K3">`2</a>  ) or ( <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">t</font> <a href="domain_1.html#K3">`2</a>  &amp; <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font> <a href="domain_1.html#K2">`1</a>  ) )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E8:31_1_1_1_4"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font> <a href="domain_1.html#K3">`2</a> 
 <b>by </b><i>, <a class="ref" href="normform.html#T46">NORMFORM:46</a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">c</font> = <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">{<span class="default"> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">t1</font></span>)</span> where <font color="Olive">t1</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">A</font> : ( <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">t1</font> <a href="hidden.html#R2">in</a> <font color="Olive">t1</font> <a href="domain_1.html#K3">`2</a>  &amp; <font color="Olive">t1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> ) </span>}</span> ,<span class="p2">{<span class="default"> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">t2</font></span>)</span> where <font color="Olive">t2</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">A</font> : ( <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">t2</font> <a href="hidden.html#R2">in</a> <font color="Olive">t2</font> <a href="domain_1.html#K2">`1</a>  &amp; <font color="Olive">t2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> ) </span>}</span> </span><a href="tarski.html#K4">]</a></span> as    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">A</font> <b>by </b><i>, <a class="ref" href="normform.html#T50">NORMFORM:50</a></i>;<br/><b>take </b><font color="Maroon">c</font> = <font color="Maroon">c</font>;<br/><a NAME="E10:31_1_1_1"/>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">{<span class="default"> <span class="p4">(<span class="default"><font color="Olive">h</font> <a href="funct_2.html#K3">.</a> <font color="Olive">t1</font></span>)</span> where <font color="Olive">t1</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">A</font> : ( <font color="Olive">h</font> <a href="funct_2.html#K3">.</a> <font color="Olive">t1</font> <a href="hidden.html#R2">in</a> <font color="Olive">t1</font> <a href="domain_1.html#K3">`2</a>  &amp; <font color="Olive">t1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> ) </span>}</span> ,<span class="p3">{<span class="default"> <span class="p4">(<span class="default"><font color="Olive">h</font> <a href="funct_2.html#K3">.</a> <font color="Olive">t2</font></span>)</span> where <font color="Olive">t2</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">A</font> : ( <font color="Olive">h</font> <a href="funct_2.html#K3">.</a> <font color="Olive">t2</font> <a href="hidden.html#R2">in</a> <font color="Olive">t2</font> <a href="domain_1.html#K2">`1</a>  &amp; <font color="Olive">t2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> ) </span>}</span> </span><a href="tarski.html#K4">]</a></span> where <font color="Olive">h</font> is    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <span class="p1">(<span class="default"><a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">A</font></span>)</span>,<span class="p1"><a href="heyting1.html#K1" target="_self">[</a><span class="default"><font color="Maroon">A</font></span><a href="heyting1.html#K1" target="_self">]</a></span> : for <font color="Olive">s</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">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> holds <br/><font color="Olive">h</font> <a href="funct_2.html#K3">.</a> <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Olive">s</font> <a href="domain_1.html#K2">`1</a> </span>)</span> <a href="setwiseo.html#K5">\/</a> <span class="p1">(<span class="default"><font color="Olive">s</font> <a href="domain_1.html#K3">`2</a> </span>)</span> </span>}</span> 
 <b>by </b><i/>;<br/><a NAME="E11:31_1_1_1"/><b>then </b>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">A</font></span>)</span> <a href="subset_1.html#K8">/\</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">{<span class="default"> <span class="p4">(<span class="default"><font color="Olive">h</font> <a href="funct_2.html#K3">.</a> <font color="Olive">t1</font></span>)</span> where <font color="Olive">t1</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">A</font> : ( <font color="Olive">h</font> <a href="funct_2.html#K3">.</a> <font color="Olive">t1</font> <a href="hidden.html#R2">in</a> <font color="Olive">t1</font> <a href="domain_1.html#K3">`2</a>  &amp; <font color="Olive">t1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> ) </span>}</span> ,<span class="p3">{<span class="default"> <span class="p4">(<span class="default"><font color="Olive">h</font> <a href="funct_2.html#K3">.</a> <font color="Olive">t2</font></span>)</span> where <font color="Olive">t2</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">A</font> : ( <font color="Olive">h</font> <a href="funct_2.html#K3">.</a> <font color="Olive">t2</font> <a href="hidden.html#R2">in</a> <font color="Olive">t2</font> <a href="domain_1.html#K2">`1</a>  &amp; <font color="Olive">t2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> ) </span>}</span> </span><a href="tarski.html#K4">]</a></span> where <font color="Olive">h</font> is    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <span class="p1">(<span class="default"><a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">A</font></span>)</span>,<span class="p1"><a href="heyting1.html#K1" target="_self">[</a><span class="default"><font color="Maroon">A</font></span><a href="heyting1.html#K1" target="_self">]</a></span> : for <font color="Olive">s</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">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> holds <br/><font color="Olive">h</font> <a href="funct_2.html#K3">.</a> <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Olive">s</font> <a href="domain_1.html#K2">`1</a> </span>)</span> <a href="setwiseo.html#K5">\/</a> <span class="p1">(<span class="default"><font color="Olive">s</font> <a href="domain_1.html#K3">`2</a> </span>)</span> </span>}</span> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>hence </b><a NAME="E12:31_1_1_1"/>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="heyting1.html#K6" target="_self">-</a> <font color="Maroon">K</font>
 ;<br/><a NAME="E13:31_1_1_1"/>
( <font color="Maroon">c</font> <a href="domain_1.html#K2">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">t1</font></span>)</span> where <font color="Olive">t1</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">A</font> : ( <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">t1</font> <a href="hidden.html#R2">in</a> <font color="Olive">t1</font> <a href="domain_1.html#K3">`2</a>  &amp; <font color="Olive">t1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> ) </span>}</span>  &amp; <font color="Maroon">c</font> <a href="domain_1.html#K3">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">t2</font></span>)</span> where <font color="Olive">t2</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">A</font> : ( <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">t2</font> <a href="hidden.html#R2">in</a> <font color="Olive">t2</font> <a href="domain_1.html#K2">`1</a>  &amp; <font color="Olive">t2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> ) </span>}</span>  )
 <b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/><b>hence </b><a NAME="E14:31_1_1_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="normform.html#D1">NORMFORM:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_1_1_2"><b>suppose </b></a><a NAME="E1:31_1_1_2"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">A</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><a NAME="E2:31_1_1_2"/><b>then </b><i><font color="Green">E70</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><a href="xboole_0.html#K1">{}</a> ,<a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i/>;<br/><b>take </b><font color="Maroon">b</font> = <font color="Maroon">a</font>;<br/><b>reconsider </b><font color="Maroon">Z</font> = <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><a href="xboole_0.html#K1">{}</a> ,<a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span> as    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K8">Normal_forms_on</a> <a href="xboole_0.html#K1">{}</a>  <b>by </b><i/>;<br/><a NAME="E4:31_1_1_2"/><i><font color="Green">E71</font></i>: 
 <a href="normform.html#K9">mi</a> <span class="p1">(<span class="default"><font color="Maroon">Z</font> <a href="normform.html#K10">^</a> <font color="Maroon">Z</font></span>)</span> <a href="heyting1.html#R1" target="_self">c=</a> <font color="Maroon">Z</font> <a href="normform.html#K10">^</a> <font color="Maroon">Z</font>
 <b>by </b><i><a class="ref" href="normform.html#T64">NORMFORM:64</a></i>;<br/><a NAME="E5:31_1_1_2"/>
 <a href="normform.html#K9">mi</a> <span class="p1">(<span class="default"><font color="Maroon">Z</font> <a href="normform.html#K10">^</a> <font color="Maroon">Z</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i/>;<br/><a NAME="E6:31_1_1_2"/><b>then </b>
<font color="Maroon">K</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><a href="xboole_0.html#K1">{}</a> ,<a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, , , , <a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/><a NAME="E7:31_1_1_2"/><b>then </b>
<font color="Maroon">K</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, <a class="ref" href="heyting1.html#D2" target="_self">Def2</a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><a NAME="E8:31_1_1_2"/><b>then </b>
 <a href="heyting1.html#K6" target="_self">-</a> <font color="Maroon">K</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><a href="xboole_0.html#K1">{}</a> ,<a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E9:31_1_1_2"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="heyting1.html#K6" target="_self">-</a> <font color="Maroon">K</font>
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>thus </b><a NAME="E10:31_1_1_2"/>
<font color="Maroon">b</font> <a href="normform.html#R1">c=</a> <font color="Maroon">a</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:31"/>
 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>  <a href="heyting1.html#K6" target="_self">-</a> <font color="Maroon">K</font> &amp; <font color="Olive">b</font> <a href="normform.html#R1">c=</a> <font color="Maroon">a</font> )
 ;<br/>


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