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<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_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="E1:21_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K4">[</a><span class="default"><font color="Olive">i</font>,<font color="Olive">j</font>,<font color="Olive">k</font></span><a href="domain_1.html#K4">]</a></span> where <font color="Olive">i</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Olive">j</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Olive">k</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( ( <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Olive">j</font> or <font color="Olive">j</font> <a href="hidden.html#R1">=</a> <font color="Olive">k</font> or <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <font color="Olive">i</font> ) &amp; <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K8">{</a><span class="default">1,2,3</span><a href="domain_1.html#K8">}</a></span> &amp; <font color="Olive">j</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K8">{</a><span class="default">1,2,3</span><a href="domain_1.html#K8">}</a></span> &amp; <font color="Olive">k</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K8">{</a><span class="default">1,2,3</span><a href="domain_1.html#K8">}</a></span> ) </span>}</span> 
 ;<br/><a NAME="E2:21_1"/><b>then </b>
 ex <font color="Olive">i</font>, <font color="Olive">j</font>, <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K4">[</a><span class="default"><font color="Olive">i</font>,<font color="Olive">j</font>,<font color="Olive">k</font></span><a href="domain_1.html#K4">]</a></span> &amp; ( <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Olive">j</font> or <font color="Olive">j</font> <a href="hidden.html#R1">=</a> <font color="Olive">k</font> or <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <font color="Olive">i</font> ) &amp; <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K8">{</a><span class="default">1,2,3</span><a href="domain_1.html#K8">}</a></span> &amp; <font color="Olive">j</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K8">{</a><span class="default">1,2,3</span><a href="domain_1.html#K8">}</a></span> &amp; <font color="Olive">k</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K8">{</a><span class="default">1,2,3</span><a href="domain_1.html#K8">}</a></span> )
 ;<br/><b>hence </b><a NAME="E3:21_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K3">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K8">{</a><span class="default">1,2,3</span><a href="domain_1.html#K8">}</a></span>,<span class="p2"><a href="domain_1.html#K8">{</a><span class="default">1,2,3</span><a href="domain_1.html#K8">}</a></span>,<span class="p2"><a href="domain_1.html#K8">{</a><span class="default">1,2,3</span><a href="domain_1.html#K8">}</a></span></span><a href="zfmisc_1.html#K3">:]</a></span>
 <b>by </b><i><a class="ref" href="mcart_1.html#T73">MCART_1:73</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:21"/>
<span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K4">[</a><span class="default"><font color="Olive">i</font>,<font color="Olive">j</font>,<font color="Olive">k</font></span><a href="domain_1.html#K4">]</a></span> where <font color="Olive">i</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Olive">j</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Olive">k</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( ( <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Olive">j</font> or <font color="Olive">j</font> <a href="hidden.html#R1">=</a> <font color="Olive">k</font> or <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <font color="Olive">i</font> ) &amp; <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K8">{</a><span class="default">1,2,3</span><a href="domain_1.html#K8">}</a></span> &amp; <font color="Olive">j</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K8">{</a><span class="default">1,2,3</span><a href="domain_1.html#K8">}</a></span> &amp; <font color="Olive">k</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K8">{</a><span class="default">1,2,3</span><a href="domain_1.html#K8">}</a></span> ) </span>}</span>  <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K3">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K8">{</a><span class="default">1,2,3</span><a href="domain_1.html#K8">}</a></span>,<span class="p2"><a href="domain_1.html#K8">{</a><span class="default">1,2,3</span><a href="domain_1.html#K8">}</a></span>,<span class="p2"><a href="domain_1.html#K8">{</a><span class="default">1,2,3</span><a href="domain_1.html#K8">}</a></span></span><a href="zfmisc_1.html#K3">:]</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>


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