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<b>let </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be   <a href="funct_1.html#NM1">Function</a>;<br/>

<b>assume </b><a NAME="E1:10"/><i><font color="Green">E11</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<a NAME="E2:10"/><i><font color="Green">E12</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="mcart_1.html#K16">pr2</a> <span class="p2"><a href="funct_3.html#K13">&lt;:</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="funct_3.html#K13">:&gt;</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1"><a href="funct_3.html#K13">&lt;:</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="funct_3.html#K13">:&gt;</a></span>
 
<b>by </b><i><a class="ref" href="mcart_1.html#D13">MCART_1:def 13</a></i>;<br/>
<i><font color="Green">E13</font></i>:  <a href="relat_1.html#K1">dom</a> <span class="p1"><a href="funct_3.html#K13">&lt;:</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="funct_3.html#K13">:&gt;</a></span> = 
<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="funct_3.html#D8">FUNCT_3:def 8</a></i>
<br/>.= 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="txt" href="topalg_4.html#E1:10"><i><font color="Green">E11</font></i></a></i>
;<br/>
<a NAME="E4:10"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="mcart_1.html#K16">pr2</a> <span class="p2"><a href="funct_3.html#K13">&lt;:</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="funct_3.html#K13">:&gt;</a></span></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="mcart_1.html#K16">pr2</a> <span class="p2"><a href="funct_3.html#K13">&lt;:</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="funct_3.html#K13">:&gt;</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:10_2"/><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="mcart_1.html#K16">pr2</a> <span class="p2"><a href="funct_3.html#K13">&lt;:</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="funct_3.html#K13">:&gt;</a></span></span>)</span>
 ;<br/>

<b>thus </b><span class="p1">(<span class="default"><a href="mcart_1.html#K16">pr2</a> <span class="p2"><a href="funct_3.html#K13">&lt;:</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="funct_3.html#K13">:&gt;</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> = 
<span class="p1">(<span class="default"><span class="p2"><a href="funct_3.html#K13">&lt;:</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="funct_3.html#K13">:&gt;</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="mcart_1.html#K2">`2</a> 
<b>by </b><i><a class="txt" href="topalg_4.html#E2:10"><i><font color="Green">E12</font></i></a>, <a class="txt" href="topalg_4.html#E1:10_2"><i><font color="Green">E14</font></i></a>, <a class="ref" href="mcart_1.html#D13">MCART_1:def 13</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K2">`2</a> 
<b>by </b><i><a class="txt" href="topalg_4.html#E2:10"><i><font color="Green">E12</font></i></a>, <a class="txt" href="topalg_4.html#E1:10_2"><i><font color="Green">E14</font></i></a>, <a class="ref" href="funct_3.html#D8">FUNCT_3:def 8</a></i>
<br/>.= 
<font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="ref" href="mcart_1.html#D2">MCART_1:def 2</a></i>
;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:10"/>
 <a href="mcart_1.html#K16">pr2</a> <span class="p1"><a href="funct_3.html#K13">&lt;:</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="funct_3.html#K13">:&gt;</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="topalg_4.html#E2:10"><i><font color="Green">E12</font></i></a>, <a class="txt" href="topalg_4.html#E3:10"><i><font color="Green">E13</font></i></a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>


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