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<b>let </b><font color="Maroon" title="c1">b</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">X1</font>, <font color="Maroon" title="c3">X2</font>, <font color="Maroon" title="c4">X3</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c5">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">[:</a><span class="default"><font color="Maroon" title="c2">X1</font>,<font color="Maroon" title="c3">X2</font>,<font color="Maroon" title="c4">X3</font></span><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">:]</a></span>;<br/>









<b>thus </b><a NAME="E1:9"/>
( <font color="Maroon" title="c1">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x</font> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a>  implies for <font color="Olive" title="b1">x1</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X1</font><br/> for <font color="Olive" title="b2">x2</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">X2</font><br/> for <font color="Olive" title="b3">x3</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">X3</font>  st <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Olive" title="b1">x1</font>,<font color="Olive" title="b2">x2</font>,<font color="Olive" title="b3">x3</font></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span> holds <br/><font color="Maroon" title="c1">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">x2</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:9_1"/><i><font color="Green" title="E3">A1</font></i>: 
<font color="Maroon" title="c1">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x</font> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a> 
 ;<br/>

<b>let </b><font color="Maroon" title="c6">x1</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X1</font>;<br/><b>let </b><font color="Maroon" title="c7">x2</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">X2</font>;<br/><b>let </b><font color="Maroon" title="c8">x3</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">X3</font>;<br/>





<b>assume </b><a NAME="E2:9_1"/><i><font color="Green" title="E4">A2</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c6">x1</font>,<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c8">x3</font></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span>
 ;<br/>

<a NAME="E3:9_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a> </span>)</span></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span>
 
<b>by </b><i><a class="ref" href="mcart_1.html#T48" title="MCART_1:th.48">MCART_1:48</a></i>;<br/>
<b>hence </b><a NAME="E4:9_1"/>
<font color="Maroon" title="c1">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font>
 <b>by </b><i><a class="txt" href="domain_1.html#E1:9_1"><i><font color="Green" title="E3">A1</font></i></a>, <a class="txt" href="domain_1.html#E2:9_1"><i><font color="Green" title="E4">A2</font></i></a>, <a class="ref" href="mcart_1.html#T28" title="MCART_1:th.28">MCART_1:28</a></i>;<br/>


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

<b>thus </b><a NAME="E2:9"/>
( ( for <font color="Olive" title="b1">x1</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X1</font><br/> for <font color="Olive" title="b2">x2</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">X2</font><br/> for <font color="Olive" title="b3">x3</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">X3</font>  st <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Olive" title="b1">x1</font>,<font color="Olive" title="b2">x2</font>,<font color="Olive" title="b3">x3</font></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span> holds <br/><font color="Maroon" title="c1">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">x2</font> ) implies <font color="Maroon" title="c1">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x</font> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a>  )
 <b>by </b><i><a class="ref" href="mcart_1.html#T70" title="MCART_1:th.70">MCART_1:70</a></i>;<br/>


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