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<div class="add">

<b>let </b><font color="Maroon" title="c1">A</font>, <font color="Maroon" title="c2">B</font>, <font color="Maroon" title="c3">C</font>, <font color="Maroon" title="c4">D</font>, <font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c6">b</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:23"/><i><font color="Green" title="E16">A1</font></i>: 
( <font color="Maroon" title="c1">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">B</font> &amp; <font color="Maroon" title="c3">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">D</font> )
 ;<br/>

<b>set </b><font color="Maroon" title="c7">f</font> = <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c3">C</font>;<br/>
<b>set </b><font color="Maroon" title="c8">g</font> = <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> <font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c4">D</font>;<br/>
<a NAME="E2:23"/><i><font color="Green" title="E17">A2</font></i>: 
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c3">C</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> <font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c4">D</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> )
 
<b>by </b><i><a class="ref" href="funct_4.html#T65" title="FUNCT_4:th.65">FUNCT_4:65</a></i>;<br/>
<a NAME="E3:23"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c3">C</font></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c3">C</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> <font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c4">D</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">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" title="c9">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:23_1"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c3">C</font></span>)</span>
 ;<br/>

<a NAME="E2:23_1"/><b>then </b><i><font color="Green" title="E18">A3</font></i>: 
( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> or <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font> )
 
<b>by </b><i><a class="txt" href="topreal6.html#E2:23"><i><font color="Green" title="E17">A2</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">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" title="c5">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">b</font> or <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">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" title="c5">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">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" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c3">C</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">A</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c3">C</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">C</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> <font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c4">D</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">B</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> <font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c4">D</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">D</font> )
 <b>by </b><i><a class="ref" href="funct_4.html#T66" title="FUNCT_4:th.66">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" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c3">C</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> <font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c4">D</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font>
 <b>by </b><i><a class="txt" href="topreal6.html#E1:23"><i><font color="Green" title="E16">A1</font></i></a>, <a class="txt" href="topreal6.html#E2:23_1"><i><font color="Green" title="E18">A3</font></i></a></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" title="E19">A4</font></i>: 
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font>
 ;<br/><div class="add"><a NAME="E2:23_1_1_2"/><b>then </b>
( <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c3">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">.--&gt;</a> <font color="Maroon" title="c3">C</font> &amp; <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> <font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c4">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">.--&gt;</a> <font color="Maroon" title="c4">D</font> )
 <b>by </b><i><a class="ref" href="cqc_lang.html#T44" title="CQC_LANG:th.44">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" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c3">C</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">C</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c3">C</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">C</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> <font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c4">D</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">D</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> <font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c4">D</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">D</font> )
 <b>by </b><i><a class="txt" href="topreal6.html#E1:23_1_1_2"><i><font color="Green" title="E19">A4</font></i></a>, <a class="ref" href="cqc_lang.html#T6" title="CQC_LANG:th.6">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" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c3">C</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> <font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c4">D</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font>
 <b>by </b><i><a class="txt" href="topreal6.html#E1:23"><i><font color="Green" title="E16">A1</font></i></a>, <a class="txt" href="topreal6.html#E2:23_1"><i><font color="Green" title="E18">A3</font></i></a>, <a class="txt" href="topreal6.html#E1:23_1_1_2"><i><font color="Green" title="E19">A4</font></i></a></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" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c3">C</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> <font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c4">D</font></span>)</span>
 <b>by </b><i><a class="txt" href="topreal6.html#E2:23"><i><font color="Green" title="E17">A2</font></i></a>, <a class="ref" href="card_3.html#T38" title="CARD_3:th.38">CARD_3:38</a></i>;<br/>


</div>
