<?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">a</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">b</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:23"/><i><font color="Green">E56</font></i>: 
( <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font> &amp; <font color="Maroon">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">D</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">A</font>,<font color="Maroon">C</font>;<br/>
<b>set </b><font color="Maroon">g</font> = <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="funct_4.html#K4">--&gt;</a> <font color="Maroon">B</font>,<font color="Maroon">D</font>;<br/>
<a NAME="E2:23"/><i><font color="Green">E57</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">A</font>,<font color="Maroon">C</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">a</font>,<font color="Maroon">b</font> <a href="funct_4.html#K4">--&gt;</a> <font color="Maroon">B</font>,<font color="Maroon">D</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> )
 
<b>by </b><i><a class="ref" href="funct_4.html#T65">FUNCT_4:65</a></i>;<br/>
<a NAME="E3:23"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">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">A</font>,<font color="Maroon">C</font></span>)</span> holds <br/><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">A</font>,<font color="Maroon">C</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="tarski.html#R1">c=</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">B</font>,<font color="Maroon">D</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1"><b>proof </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="E1:23_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">a</font>,<font color="Maroon">b</font> <a href="funct_4.html#K4">--&gt;</a> <font color="Maroon">A</font>,<font color="Maroon">C</font></span>)</span>
 ;<br/>

<a NAME="E2:23_1"/><b>then </b><i><font color="Green">E58</font></i>: 
( <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> )
 
<b>by </b><i>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:23_1_1"/>
( <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font> or <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_1"><b>suppose </b></a><a NAME="E1:23_1_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font>
 ;<br/><div class="add"><a NAME="E2:23_1_1_1"/><b>then </b>
( <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">A</font>,<font color="Maroon">C</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> &amp; <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">A</font>,<font color="Maroon">C</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font> &amp; <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">B</font>,<font color="Maroon">D</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> &amp; <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">B</font>,<font color="Maroon">D</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font> )
 <b>by </b><i><a class="ref" href="funct_4.html#T66">FUNCT_4:66</a></i>;<br/><b>hence </b><a NAME="E3:23_1_1_1"/>
<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">A</font>,<font color="Maroon">C</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="tarski.html#R1">c=</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">B</font>,<font color="Maroon">D</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_2"><b>suppose </b></a><a NAME="E1:23_1_1_2"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 ;<br/><div class="add"><a NAME="E2:23_1_1_2"/><b>then </b>
( <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="funct_4.html#K4">--&gt;</a> <font color="Maroon">A</font>,<font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">C</font> &amp; <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="funct_4.html#K4">--&gt;</a> <font color="Maroon">B</font>,<font color="Maroon">D</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">D</font> )
 <b>by </b><i><a class="ref" href="cqc_lang.html#T44">CQC_LANG:44</a></i>;<br/><a NAME="E3:23_1_1_2"/><b>then </b>
( <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">A</font>,<font color="Maroon">C</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font> &amp; <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">A</font>,<font color="Maroon">C</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font> &amp; <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">B</font>,<font color="Maroon">D</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font> &amp; <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">B</font>,<font color="Maroon">D</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font> )
 <b>by </b><i>, <a class="ref" href="cqc_lang.html#T6">CQC_LANG:6</a></i>;<br/><b>hence </b><a NAME="E4:23_1_1_2"/>
<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">A</font>,<font color="Maroon">C</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="tarski.html#R1">c=</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">B</font>,<font color="Maroon">D</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E4:23"/>
 <a href="card_3.html#K4">product</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">A</font>,<font color="Maroon">C</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="card_3.html#K4">product</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">B</font>,<font color="Maroon">D</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="card_3.html#T38">CARD_3:38</a></i>;<br/>


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