<?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">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/>



<a NAME="E1:24"/><i><font color="Green">E21</font></i>: 
<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> is    <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/>;<br/>
<b>assume </b><a NAME="E2:24"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">K</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<a NAME="E3:24"/><i><font color="Green">E52</font></i>: 
<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">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="Olive">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="p3">{<span class="default"> <span class="p4">(<span class="default"><font color="Olive">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="Olive">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> where <font color="Olive">g</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">g</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>  <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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:24_1"/>
<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">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="Olive">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="p3">{<span class="default"> <span class="p4">(<span class="default"><font color="Olive">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="Olive">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> where <font color="Olive">g</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">g</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>  <a href="tarski.html#R1">c=</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>
  <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="24_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="E1:24_1_1"/>
<font color="Maroon">x</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">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="Olive">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="p3">{<span class="default"> <span class="p4">(<span class="default"><font color="Olive">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="Olive">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> where <font color="Olive">g</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">g</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> 
 ;<br/>

<b>then 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="E3:24_1_1"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <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>
 <b>and </b><br/><a NAME="E4:24_1_1"/>
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">g</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>
 ;<br/>
<a NAME="E5:24_1_1"/><i><font color="Green">E59</font></i>: 
( <font color="Maroon">x</font> <a href="mcart_1.html#K1">`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">x</font> <a href="mcart_1.html#K2">`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="heyting1.html#D2" target="_self">Def2</a>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/>
<div><i><font color="Green">E60</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1_1"><b>now </b></a><div class="add"><b>consider </b><font color="Maroon">y</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">x</font> <a href="mcart_1.html#K1">`1</a> ;<br/><b>assume </b><a NAME="E2:24_1_1_1"/>
<font color="Maroon">x</font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><a NAME="E3:24_1_1_1"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font> <a href="mcart_1.html#K1">`1</a> 
 ;<br/><a NAME="E4:24_1_1_1"/><b>then </b>
 ex <font color="Olive">t1</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="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">t1</font> &amp; <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> )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E5:24_1_1_1"/>
contradiction
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1_2"><b>now </b></a><div class="add"><b>consider </b><font color="Maroon">y</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">x</font> <a href="mcart_1.html#K2">`2</a> ;<br/><b>assume </b><a NAME="E2:24_1_1_2"/>
<font color="Maroon">x</font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><a NAME="E3:24_1_1_2"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font> <a href="mcart_1.html#K2">`2</a> 
 ;<br/><a NAME="E4:24_1_1_2"/><b>then </b>
 ex <font color="Olive">t1</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="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">t1</font> &amp; <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#K2">`1</a>  &amp; <font color="Olive">t1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E5:24_1_1_2"/>
contradiction
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<a NAME="E8:24_1_1"/><b>then </b>
<font color="Maroon">x</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><a class="ref" href="heyting1.html#D2" target="_self">Def2</a>, , <a class="ref" href="mcart_1.html#T8">MCART_1:8</a></i>;<br/>
<b>hence </b><a NAME="E9:24_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</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="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


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

<b>thus </b><a NAME="E2:24_1"/>
<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> <a href="tarski.html#R1">c=</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">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="Olive">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="p3">{<span class="default"> <span class="p4">(<span class="default"><font color="Olive">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="Olive">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> where <font color="Olive">g</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">g</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> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_2"><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="E1:24_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</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>
 ;<br/>

<a NAME="E2:24_1_2"/><b>then </b><i><font color="Green">E61</font></i>: 
<font color="Maroon">x</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><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<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>;<br/>
<div><i><font color="Green">E62</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_2_1"><b>now </b></a><div class="add"><b>consider </b><font color="Maroon">y</font> being    <a href="subset_1.html#M1">Element</a> of <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>assume </b><a NAME="E2:24_1_2_1"/>
<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="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><a NAME="E3:24_1_2_1"/><b>then </b>
<font color="Maroon">y</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/><a NAME="E4:24_1_2_1"/><b>then </b>
 ex <font color="Olive">t1</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="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">t1</font> &amp; <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> )
 ;<br/><b>hence </b><a NAME="E5:24_1_2_1"/>
contradiction
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E64</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_2_2"><b>now </b></a><div class="add"><b>consider </b><font color="Maroon">y</font> being    <a href="subset_1.html#M1">Element</a> of <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>assume </b><a NAME="E2:24_1_2_2"/>
<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="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><a NAME="E3:24_1_2_2"/><b>then </b>
<font color="Maroon">y</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/><a NAME="E4:24_1_2_2"/><b>then </b>
 ex <font color="Olive">t1</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="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">t1</font> &amp; <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#K2">`1</a>  &amp; <font color="Olive">t1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> )
 ;<br/><b>hence </b><a NAME="E5:24_1_2_2"/>
contradiction
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<a NAME="E6:24_1_2"/>
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">g</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>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E7:24_1_2"/>
<font color="Maroon">x</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><a class="ref" href="heyting1.html#D2" target="_self">Def2</a>, , </i>;<br/>


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


</div><b>end;</b></div>
<b>thus </b><a NAME="E4:24"/>
 <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><a class="txt" href="heyting1.html#E1:24"><i><font color="Green">E21</font></i></a>, <a class="txt" href="heyting1.html#E2:24"><i><font color="Green">E23</font></i></a>, <a class="ref" href="zfmisc_1.html#T52">ZFMISC_1:52</a></i>;<br/>


</div>
