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

<b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">b</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">c</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">d</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">e</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">i</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">j</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">k</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">h</font> be   <a href="funct_1.html#NM1">Function</a>;<br/>



<b>assume </b><a NAME="E1:18"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">d</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="enumset1.html#K2">{</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font></span><a href="enumset1.html#K2">}</a></span> &amp; <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e</font> &amp; <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> )
 ;<br/>

<b>set </b><font color="Maroon">f</font> = <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="funct_4.html#K4">--&gt;</a> <font color="Maroon">e</font>,<font color="Maroon">i</font>;<br/>
<b>set </b><font color="Maroon">g</font> = <font color="Maroon">c</font>,<font color="Maroon">d</font> <a href="funct_4.html#K4">--&gt;</a> <font color="Maroon">j</font>,<font color="Maroon">k</font>;<br/>
<a NAME="E2:18"/><i><font color="Green">E40</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="funct_4.html#K4">--&gt;</a> <font color="Maroon">e</font>,<font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K2">}</a></span> &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c</font>,<font color="Maroon">d</font> <a href="funct_4.html#K4">--&gt;</a> <font color="Maroon">j</font>,<font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">d</font></span><a href="tarski.html#K2">}</a></span> )
 
<b>by </b><i><a class="ref" href="funct_4.html#T65">FUNCT_4:65</a></i>;<br/>
<a NAME="E3:18"/><b>then </b><i><font color="Green">E41</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="funct_4.html#K4">--&gt;</a> <font color="Maroon">e</font>,<font color="Maroon">i</font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><font color="Maroon">c</font>,<font color="Maroon">d</font> <a href="funct_4.html#K4">--&gt;</a> <font color="Maroon">j</font>,<font color="Maroon">k</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="enumset1.html#T45">ENUMSET1:45</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:18_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="funct_4.html#K4">--&gt;</a> <font color="Maroon">e</font>,<font color="Maroon">i</font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><font color="Maroon">c</font>,<font color="Maroon">d</font> <a href="funct_4.html#K4">--&gt;</a> <font color="Maroon">j</font>,<font color="Maroon">k</font></span>)</span></span>)</span>
 ;<br/><b>thus </b><a NAME="E2:18_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c</font>,<font color="Maroon">d</font> <a href="funct_4.html#K4">--&gt;</a> <font color="Maroon">j</font>,<font color="Maroon">k</font></span>)</span> implies <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c</font>,<font color="Maroon">d</font> <a href="funct_4.html#K4">--&gt;</a> <font color="Maroon">j</font>,<font color="Maroon">k</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:18_1_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c</font>,<font color="Maroon">d</font> <a href="funct_4.html#K4">--&gt;</a> <font color="Maroon">j</font>,<font color="Maroon">k</font></span>)</span>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:18_1_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> or <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font> )
 </span><b>by </b><i>, , <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_1_1_1"><b>suppose </b></a><a NAME="E1:18_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:18_1_1_1_1"/>
<font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c</font>,<font color="Maroon">d</font> <a href="funct_4.html#K4">--&gt;</a> <font color="Maroon">j</font>,<font color="Maroon">k</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, <a class="ref" href="funct_4.html#T66">FUNCT_4:66</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_1_1_2"><b>suppose </b></a><a NAME="E1:18_1_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:18_1_1_1_2"/>
<font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c</font>,<font color="Maroon">d</font> <a href="funct_4.html#K4">--&gt;</a> <font color="Maroon">j</font>,<font color="Maroon">k</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, <a class="ref" href="funct_4.html#T66">FUNCT_4:66</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>thus </b><a NAME="E3:18_1"/>
( not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c</font>,<font color="Maroon">d</font> <a href="funct_4.html#K4">--&gt;</a> <font color="Maroon">j</font>,<font color="Maroon">k</font></span>)</span> implies <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="funct_4.html#K4">--&gt;</a> <font color="Maroon">e</font>,<font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:18_1_2"/>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c</font>,<font color="Maroon">d</font> <a href="funct_4.html#K4">--&gt;</a> <font color="Maroon">j</font>,<font color="Maroon">k</font></span>)</span>
 ;<br/>

<a NAME="E2:18_1_2"/><b>then </b><i><font color="Green">E44</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="funct_4.html#K4">--&gt;</a> <font color="Maroon">e</font>,<font color="Maroon">i</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:18_1_2_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> or <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> )
 </span><b>by </b><i>, , <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_2_1_1"><b>suppose </b></a><a NAME="E1:18_1_2_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:18_1_2_1_1"/>
<font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="funct_4.html#K4">--&gt;</a> <font color="Maroon">e</font>,<font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, <a class="ref" href="funct_4.html#T66">FUNCT_4:66</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_2_1_2"><b>suppose </b></a><a NAME="E1:18_1_2_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:18_1_2_1_2"/>
<font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="funct_4.html#K4">--&gt;</a> <font color="Maroon">e</font>,<font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, <a class="ref" href="funct_4.html#T66">FUNCT_4:66</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E5:18"/>
<font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font> <a href="quaterni.html#K2" target="_self">--&gt;</a> <font color="Maroon">e</font>,<font color="Maroon">i</font>,<font color="Maroon">j</font>,<font color="Maroon">k</font>
 <b>by </b><i>, <a class="ref" href="funct_4.html#D1">FUNCT_4:def 1</a></i>;<br/>


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