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<b>let </b><font color="Maroon">a</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> ;<br/><b>let </b><font color="Maroon">b</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> ;<br/><b>let </b><font color="Maroon">c</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> ;<br/><b>let </b><font color="Maroon">d</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> ;<br/>







<b>set </b><font color="Maroon">f</font> = 0,1,2,3 <a href="quaterni.html#K3" target="_self">--&gt;</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font>;<br/>
<a NAME="E1:23"/><i><font color="Green">E37</font></i>: 
0,1,2,3 <a href="quaterni.html#K3" target="_self">--&gt;</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="enumset1.html#K2">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">a</font></span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">1,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">2,<font color="Maroon">c</font></span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">3,<font color="Maroon">d</font></span><a href="tarski.html#K4">]</a></span></span><a href="enumset1.html#K2">}</a></span>
 
<b>by </b><i>, </i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:23_1"/>
0,1,2,3 <a href="quaterni.html#K3" target="_self">--&gt;</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">i</font>,<font color="Olive">j</font></span><a href="tarski.html#K4">]</a></span> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Olive">j</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Olive">i</font>,<font color="Olive">j</font> <a href="arytm_3.html#R1">are_relative_prime</a>  &amp; <font color="Olive">j</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="arytm_3.html#K13">{}</a>  ) </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">i</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:23_1"/><i><font color="Green">E40</font></i>: 
0,1,2,3 <a href="quaterni.html#K3" target="_self">--&gt;</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font></span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E4:23_1"/>
<font color="Maroon">i</font>,<font color="Maroon">j</font> <a href="arytm_3.html#R1">are_relative_prime</a> 
 <b>and </b><br/><a NAME="E5:23_1"/>
<font color="Maroon">j</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="arytm_3.html#K13">{}</a> 
 ;<br/><a NAME="E6:23_1"/><i><font color="Green">E41</font></i>: 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">i</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R2">in</a> 0,1,2,3 <a href="quaterni.html#K3" target="_self">--&gt;</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font>
 <b>by </b><i>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><a NAME="E7:23_1"/><i><font color="Green">E42</font></i>: 
<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font></span><a href="tarski.html#K2">}</a></span> <a href="hidden.html#R2">in</a> 0,1,2,3 <a href="quaterni.html#K3" target="_self">--&gt;</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font>
 <b>by </b><i>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><div><i><font color="Green">E43</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:23_1_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font>
 ;<br/><a NAME="E2:23_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">i</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font></span><a href="tarski.html#K2">}</a></span>
 <b>by </b><i><a class="ref" href="enumset1.html#T69">ENUMSET1:69</a></i>;<br/><a NAME="E3:23_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">i</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="enumset1.html#T69">ENUMSET1:69</a></i>;<br/><a NAME="E4:23_1_1"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">a</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">i</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">i</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default">2,<font color="Maroon">c</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">i</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default">3,<font color="Maroon">d</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">i</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i>, , <a class="ref" href="enumset1.html#D2">ENUMSET1:def 2</a></i>;<br/><a NAME="E5:23_1_1"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">a</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">i</font></span><a href="tarski.html#K1">}</a></span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">i</font></span><a href="tarski.html#K1">}</a></span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default">2,<font color="Maroon">c</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">i</font></span><a href="tarski.html#K1">}</a></span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default">3,<font color="Maroon">d</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">i</font></span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E6:23_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:23_1_2"/>
( ( <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">i</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">a</font></span><a href="tarski.html#K4">]</a></span> &amp; <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font></span><a href="tarski.html#K2">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">a</font></span><a href="tarski.html#K4">]</a></span> ) or ( <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">i</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">a</font></span><a href="tarski.html#K4">]</a></span> &amp; <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font></span><a href="tarski.html#K2">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> ) or ( <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">i</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">a</font></span><a href="tarski.html#K4">]</a></span> &amp; <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font></span><a href="tarski.html#K2">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">2,<font color="Maroon">c</font></span><a href="tarski.html#K4">]</a></span> ) or ( <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">i</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">a</font></span><a href="tarski.html#K4">]</a></span> &amp; <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font></span><a href="tarski.html#K2">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">3,<font color="Maroon">d</font></span><a href="tarski.html#K4">]</a></span> ) or ( <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">i</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> &amp; <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font></span><a href="tarski.html#K2">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">a</font></span><a href="tarski.html#K4">]</a></span> ) or ( <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">i</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> &amp; <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font></span><a href="tarski.html#K2">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> ) or ( <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">i</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> &amp; <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font></span><a href="tarski.html#K2">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">2,<font color="Maroon">c</font></span><a href="tarski.html#K4">]</a></span> ) or ( <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">i</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> &amp; <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font></span><a href="tarski.html#K2">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">3,<font color="Maroon">d</font></span><a href="tarski.html#K4">]</a></span> ) or ( <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">i</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">2,<font color="Maroon">c</font></span><a href="tarski.html#K4">]</a></span> &amp; <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font></span><a href="tarski.html#K2">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">a</font></span><a href="tarski.html#K4">]</a></span> ) or ( <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">i</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">2,<font color="Maroon">c</font></span><a href="tarski.html#K4">]</a></span> &amp; <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font></span><a href="tarski.html#K2">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> ) or ( <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">i</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">2,<font color="Maroon">c</font></span><a href="tarski.html#K4">]</a></span> &amp; <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font></span><a href="tarski.html#K2">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">2,<font color="Maroon">c</font></span><a href="tarski.html#K4">]</a></span> ) or ( <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">i</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">2,<font color="Maroon">c</font></span><a href="tarski.html#K4">]</a></span> &amp; <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font></span><a href="tarski.html#K2">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">3,<font color="Maroon">d</font></span><a href="tarski.html#K4">]</a></span> ) or ( <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">i</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">3,<font color="Maroon">d</font></span><a href="tarski.html#K4">]</a></span> &amp; <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font></span><a href="tarski.html#K2">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">a</font></span><a href="tarski.html#K4">]</a></span> ) or ( <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">i</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">3,<font color="Maroon">d</font></span><a href="tarski.html#K4">]</a></span> &amp; <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font></span><a href="tarski.html#K2">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> ) or ( <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">i</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">3,<font color="Maroon">d</font></span><a href="tarski.html#K4">]</a></span> &amp; <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font></span><a href="tarski.html#K2">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">2,<font color="Maroon">c</font></span><a href="tarski.html#K4">]</a></span> ) or ( <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">i</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">3,<font color="Maroon">d</font></span><a href="tarski.html#K4">]</a></span> &amp; <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font></span><a href="tarski.html#K2">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">3,<font color="Maroon">d</font></span><a href="tarski.html#K4">]</a></span> ) )
 </span><b>by </b><i>, , , <a class="ref" href="enumset1.html#D2">ENUMSET1:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_2_1"><b>suppose </b></a><a NAME="E1:23_1_2_1"/>
( <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">i</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">a</font></span><a href="tarski.html#K4">]</a></span> &amp; <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font></span><a href="tarski.html#K2">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">a</font></span><a href="tarski.html#K4">]</a></span> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:23_1_2_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T9">ZFMISC_1:9</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_2_2"><b>suppose </b></a><b>that </b><a NAME="E1:23_1_2_2"/><i><font color="Green">E44</font></i>: 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">i</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">a</font></span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E2:23_1_2_2"/><i><font color="Green">E68</font></i>: 
<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font></span><a href="tarski.html#K2">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span>
 ;<br/><div class="add"><a NAME="E3:23_1_2_2"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, </i>;<br/><a NAME="E4:23_1_2_2"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><a NAME="E5:23_1_2_2"/><b>then </b>
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">1,<font color="Maroon">b</font></span><a href="tarski.html#K2">}</a></span> or <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><a NAME="E6:23_1_2_2"/><b>then </b>
1 <a href="hidden.html#R2">in</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><b>hence </b><a NAME="E7:23_1_2_2"/>
contradiction
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_2_3"><b>suppose </b></a><b>that </b><a NAME="E1:23_1_2_3"/><i><font color="Green">E70</font></i>: 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">i</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">a</font></span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E2:23_1_2_3"/><i><font color="Green">E71</font></i>: 
<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font></span><a href="tarski.html#K2">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">2,<font color="Maroon">c</font></span><a href="tarski.html#K4">]</a></span>
 ;<br/><div class="add"><a NAME="E3:23_1_2_3"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, </i>;<br/><a NAME="E4:23_1_2_3"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">2,<font color="Maroon">c</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><a NAME="E5:23_1_2_3"/><b>then </b>
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">2,<font color="Maroon">c</font></span><a href="tarski.html#K2">}</a></span> or <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">2</span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><a NAME="E6:23_1_2_3"/><b>then </b>
2 <a href="hidden.html#R2">in</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><b>hence </b><a NAME="E7:23_1_2_3"/>
contradiction
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_2_4"><b>suppose </b></a><b>that </b><a NAME="E1:23_1_2_4"/><i><font color="Green">E73</font></i>: 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">i</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">a</font></span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E2:23_1_2_4"/><i><font color="Green">E74</font></i>: 
<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font></span><a href="tarski.html#K2">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">3,<font color="Maroon">d</font></span><a href="tarski.html#K4">]</a></span>
 ;<br/><div class="add"><a NAME="E3:23_1_2_4"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, </i>;<br/><a NAME="E4:23_1_2_4"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">3,<font color="Maroon">d</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><a NAME="E5:23_1_2_4"/><b>then </b>
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">3,<font color="Maroon">d</font></span><a href="tarski.html#K2">}</a></span> or <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">3</span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><a NAME="E6:23_1_2_4"/><b>then </b>
3 <a href="hidden.html#R2">in</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><b>hence </b><a NAME="E7:23_1_2_4"/>
contradiction
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_2_5"><b>suppose </b></a><b>that </b><a NAME="E1:23_1_2_5"/><i><font color="Green">E77</font></i>: 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">i</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E2:23_1_2_5"/><i><font color="Green">E78</font></i>: 
<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font></span><a href="tarski.html#K2">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">a</font></span><a href="tarski.html#K4">]</a></span>
 ;<br/><div class="add"><a NAME="E3:23_1_2_5"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, </i>;<br/><a NAME="E4:23_1_2_5"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">a</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><a NAME="E5:23_1_2_5"/><b>then </b>
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">0,<font color="Maroon">a</font></span><a href="tarski.html#K2">}</a></span> or <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><a NAME="E6:23_1_2_5"/><b>then </b>
0 <a href="hidden.html#R2">in</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><b>hence </b><a NAME="E7:23_1_2_5"/>
contradiction
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_2_6"><b>suppose </b></a><a NAME="E1:23_1_2_6"/>
( <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">i</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> &amp; <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font></span><a href="tarski.html#K2">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:23_1_2_6"/>
contradiction
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T9">ZFMISC_1:9</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_2_7"><b>suppose </b></a><b>that </b><a NAME="E1:23_1_2_7"/><i><font color="Green">E80</font></i>: 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">i</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E2:23_1_2_7"/><i><font color="Green">E81</font></i>: 
<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font></span><a href="tarski.html#K2">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">2,<font color="Maroon">c</font></span><a href="tarski.html#K4">]</a></span>
 ;<br/><div class="add"><a NAME="E3:23_1_2_7"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, </i>;<br/><a NAME="E4:23_1_2_7"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">2,<font color="Maroon">c</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><a NAME="E5:23_1_2_7"/><b>then </b>
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">2,<font color="Maroon">c</font></span><a href="tarski.html#K2">}</a></span> or <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">2</span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><a NAME="E6:23_1_2_7"/><b>then </b>
2 <a href="hidden.html#R2">in</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><b>hence </b><a NAME="E7:23_1_2_7"/>
contradiction
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_2_8"><b>suppose </b></a><b>that </b><a NAME="E1:23_1_2_8"/><i><font color="Green">E83</font></i>: 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">i</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E2:23_1_2_8"/><i><font color="Green">E84</font></i>: 
<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font></span><a href="tarski.html#K2">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">3,<font color="Maroon">d</font></span><a href="tarski.html#K4">]</a></span>
 ;<br/><div class="add"><a NAME="E3:23_1_2_8"/><i><font color="Green">E85</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, </i>;<br/><a NAME="E4:23_1_2_8"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">3,<font color="Maroon">d</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><a NAME="E5:23_1_2_8"/><b>then </b>
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">3,<font color="Maroon">d</font></span><a href="tarski.html#K2">}</a></span> or <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">3</span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><a NAME="E6:23_1_2_8"/><b>then </b>
3 <a href="hidden.html#R2">in</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><b>hence </b><a NAME="E7:23_1_2_8"/>
contradiction
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_2_9"><b>suppose </b></a><b>that </b><a NAME="E1:23_1_2_9"/><i><font color="Green">E86</font></i>: 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">i</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">2,<font color="Maroon">c</font></span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E2:23_1_2_9"/><i><font color="Green">E87</font></i>: 
<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font></span><a href="tarski.html#K2">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">a</font></span><a href="tarski.html#K4">]</a></span>
 ;<br/><div class="add"><a NAME="E3:23_1_2_9"/><i><font color="Green">E88</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">2</span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, </i>;<br/><a NAME="E4:23_1_2_9"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">a</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><a NAME="E5:23_1_2_9"/><b>then </b>
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">0,<font color="Maroon">a</font></span><a href="tarski.html#K2">}</a></span> or <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><a NAME="E6:23_1_2_9"/><b>then </b>
0 <a href="hidden.html#R2">in</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><b>hence </b><a NAME="E7:23_1_2_9"/>
contradiction
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_2_10"><b>suppose </b></a><b>that </b><a NAME="E1:23_1_2_10"/><i><font color="Green">E89</font></i>: 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">i</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">2,<font color="Maroon">c</font></span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E2:23_1_2_10"/><i><font color="Green">E90</font></i>: 
<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font></span><a href="tarski.html#K2">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span>
 ;<br/><div class="add"><a NAME="E3:23_1_2_10"/><i><font color="Green">E91</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">2</span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, </i>;<br/><a NAME="E4:23_1_2_10"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><a NAME="E5:23_1_2_10"/><b>then </b>
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">1,<font color="Maroon">b</font></span><a href="tarski.html#K2">}</a></span> or <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><a NAME="E6:23_1_2_10"/><b>then </b>
1 <a href="hidden.html#R2">in</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><b>hence </b><a NAME="E7:23_1_2_10"/>
contradiction
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_2_11"><b>suppose </b></a><a NAME="E1:23_1_2_11"/>
( <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">i</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">2,<font color="Maroon">c</font></span><a href="tarski.html#K4">]</a></span> &amp; <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font></span><a href="tarski.html#K2">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">2,<font color="Maroon">c</font></span><a href="tarski.html#K4">]</a></span> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:23_1_2_11"/>
contradiction
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T9">ZFMISC_1:9</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_2_12"><b>suppose </b></a><b>that </b><a NAME="E1:23_1_2_12"/><i><font color="Green">E92</font></i>: 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">i</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">2,<font color="Maroon">c</font></span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E2:23_1_2_12"/><i><font color="Green">E93</font></i>: 
<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font></span><a href="tarski.html#K2">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">3,<font color="Maroon">d</font></span><a href="tarski.html#K4">]</a></span>
 ;<br/><div class="add"><a NAME="E3:23_1_2_12"/><i><font color="Green">E94</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">2</span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, </i>;<br/><a NAME="E4:23_1_2_12"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">3,<font color="Maroon">d</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><a NAME="E5:23_1_2_12"/><b>then </b>
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">3,<font color="Maroon">d</font></span><a href="tarski.html#K2">}</a></span> or <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">3</span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><a NAME="E6:23_1_2_12"/><b>then </b>
3 <a href="hidden.html#R2">in</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><b>hence </b><a NAME="E7:23_1_2_12"/>
contradiction
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_2_13"><b>suppose </b></a><b>that </b><a NAME="E1:23_1_2_13"/><i><font color="Green">E95</font></i>: 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">i</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">3,<font color="Maroon">d</font></span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E2:23_1_2_13"/><i><font color="Green">E96</font></i>: 
<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font></span><a href="tarski.html#K2">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">a</font></span><a href="tarski.html#K4">]</a></span>
 ;<br/><div class="add"><a NAME="E3:23_1_2_13"/><i><font color="Green">E97</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">3</span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, </i>;<br/><a NAME="E4:23_1_2_13"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">a</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><a NAME="E5:23_1_2_13"/><b>then </b>
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">0,<font color="Maroon">a</font></span><a href="tarski.html#K2">}</a></span> or <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><a NAME="E6:23_1_2_13"/><b>then </b>
0 <a href="hidden.html#R2">in</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><b>hence </b><a NAME="E7:23_1_2_13"/>
contradiction
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_2_14"><b>suppose </b></a><b>that </b><a NAME="E1:23_1_2_14"/><i><font color="Green">E98</font></i>: 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">i</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">3,<font color="Maroon">d</font></span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E2:23_1_2_14"/><i><font color="Green">E99</font></i>: 
<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font></span><a href="tarski.html#K2">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span>
 ;<br/><div class="add"><a NAME="E3:23_1_2_14"/><i><font color="Green">E100</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">3</span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, </i>;<br/><a NAME="E4:23_1_2_14"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><a NAME="E5:23_1_2_14"/><b>then </b>
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">1,<font color="Maroon">b</font></span><a href="tarski.html#K2">}</a></span> or <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><a NAME="E6:23_1_2_14"/><b>then </b>
1 <a href="hidden.html#R2">in</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><b>hence </b><a NAME="E7:23_1_2_14"/>
contradiction
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_2_15"><b>suppose </b></a><b>that </b><a NAME="E1:23_1_2_15"/><i><font color="Green">E101</font></i>: 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">i</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">3,<font color="Maroon">d</font></span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E2:23_1_2_15"/><i><font color="Green">E102</font></i>: 
<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font></span><a href="tarski.html#K2">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">2,<font color="Maroon">c</font></span><a href="tarski.html#K4">]</a></span>
 ;<br/><div class="add"><a NAME="E3:23_1_2_15"/><i><font color="Green">E103</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">3</span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, </i>;<br/><a NAME="E4:23_1_2_15"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">2,<font color="Maroon">c</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><a NAME="E5:23_1_2_15"/><b>then </b>
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">2,<font color="Maroon">c</font></span><a href="tarski.html#K2">}</a></span> or <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">2</span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><a NAME="E6:23_1_2_15"/><b>then </b>
2 <a href="hidden.html#R2">in</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><b>hence </b><a NAME="E7:23_1_2_15"/>
contradiction
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_2_16"><b>suppose </b></a><a NAME="E1:23_1_2_16"/>
( <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">i</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">3,<font color="Maroon">d</font></span><a href="tarski.html#K4">]</a></span> &amp; <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font></span><a href="tarski.html#K2">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">3,<font color="Maroon">d</font></span><a href="tarski.html#K4">]</a></span> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:23_1_2_16"/>
contradiction
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T9">ZFMISC_1:9</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E3:23"/><b>then </b><i><font color="Green">E104</font></i>: 
not 0,1,2,3 <a href="quaterni.html#K3" target="_self">--&gt;</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">i</font>,<font color="Olive">j</font></span><a href="tarski.html#K4">]</a></span> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Olive">j</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Olive">i</font>,<font color="Olive">j</font> <a href="arytm_3.html#R1">are_relative_prime</a>  &amp; <font color="Olive">j</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="arytm_3.html#K13">{}</a>  ) </span>}</span>  <a href="subset_1.html#K6">\</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">k</font>,1</span><a href="tarski.html#K4">]</a></span> where <font color="Olive">k</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : verum </span>}</span> 
 
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:23_2"/>
0,1,2,3 <a href="quaterni.html#K3" target="_self">--&gt;</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K5">omega</a> 
 ;<br/><a NAME="E2:23_2"/><b>then </b>
0,1,2,3 <a href="quaterni.html#K3" target="_self">--&gt;</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font> is <a href="ordinal1.html#V3">ordinal</a>
 <b>by </b><i><a class="ref" href="ordinal1.html#T23">ORDINAL1:23</a></i>;<br/><a NAME="E3:23_2"/><b>then </b>
 <a href="arytm_3.html#K13">{}</a>  <a href="hidden.html#R2">in</a> 0,1,2,3 <a href="quaterni.html#K3" target="_self">--&gt;</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font>
 <b>by </b><i><a class="ref" href="ordinal3.html#T10">ORDINAL3:10</a></i>;<br/><b>hence </b><a NAME="E4:23_2"/>
contradiction
 <b>by </b><i>, <a class="ref" href="enumset1.html#D2">ENUMSET1:def 2</a></i>;<br/></div><b>end;</b></div>
<a NAME="E5:23"/><b>then </b><i><font color="Green">E105</font></i>: 
not 0,1,2,3 <a href="quaterni.html#K3" target="_self">--&gt;</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font> <a href="hidden.html#R2">in</a>  <a href="arytm_3.html#K7">RAT+</a> 
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>set </b><font color="Maroon">IR</font> = <span class="p1">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is   <a href="subset_1.html#NM2">Subset</a> of <a href="arytm_3.html#K7">RAT+</a>  : for <font color="Olive">r</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>   st <font color="Olive">r</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> holds <br/>( ( for <font color="Olive">s</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>   st <font color="Olive">s</font> <a href="arytm_3.html#R3">&lt;='</a> <font color="Olive">r</font> holds <br/><font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> ) &amp;  ex <font color="Olive">s</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  st <br/>( <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> &amp; <font color="Olive">r</font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Olive">s</font> ) ) </span>}</span> ;<br/>
<a NAME="E6:23"/>
for <font color="Olive">x</font>, <font color="Olive">y</font>, <font color="Olive">z</font>, <font color="Olive">w</font> being   <a href="hidden.html#M1">set</a>   holds not <span class="p1"><a href="enumset1.html#K2">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Olive">x</font></span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">1,<font color="Olive">y</font></span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">2,<font color="Olive">z</font></span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">3,<font color="Olive">w</font></span><a href="tarski.html#K4">]</a></span></span><a href="enumset1.html#K2">}</a></span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is   <a href="subset_1.html#NM2">Subset</a> of <a href="arytm_3.html#K7">RAT+</a>  : for <font color="Olive">r</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>   st <font color="Olive">r</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> holds <br/>( ( for <font color="Olive">s</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>   st <font color="Olive">s</font> <a href="arytm_3.html#R3">&lt;='</a> <font color="Olive">r</font> holds <br/><font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> ) &amp;  ex <font color="Olive">s</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  st <br/>( <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> &amp; <font color="Olive">r</font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Olive">s</font> ) ) </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_3"><b>proof </b></a><div class="add">

<b>given </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">z</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">w</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><a NAME="E1:23_3"/><i><font color="Green">E107</font></i>: 
<span class="p1"><a href="enumset1.html#K2">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">1,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">2,<font color="Maroon">z</font></span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">3,<font color="Maroon">w</font></span><a href="tarski.html#K4">]</a></span></span><a href="enumset1.html#K2">}</a></span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is   <a href="subset_1.html#NM2">Subset</a> of <a href="arytm_3.html#K7">RAT+</a>  : for <font color="Olive">r</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>   st <font color="Olive">r</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> holds <br/>( ( for <font color="Olive">s</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>   st <font color="Olive">s</font> <a href="arytm_3.html#R3">&lt;='</a> <font color="Olive">r</font> holds <br/><font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> ) &amp;  ex <font color="Olive">s</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  st <br/>( <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> &amp; <font color="Olive">r</font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Olive">s</font> ) ) </span>}</span> 
 ;<br/>

<b>consider </b><font color="Maroon">A</font> being   <a href="subset_1.html#NM2">Subset</a> of <a href="arytm_3.html#K7">RAT+</a> <b> such that </b><br/><a NAME="E3:23_3"/><i><font color="Green">E108</font></i>: 
<span class="p1"><a href="enumset1.html#K2">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">1,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">2,<font color="Maroon">z</font></span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">3,<font color="Maroon">w</font></span><a href="tarski.html#K4">]</a></span></span><a href="enumset1.html#K2">}</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font>
 <b>and </b><br/><a NAME="E4:23_3"/><i><font color="Green">E109</font></i>: 
for <font color="Olive">r</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>   st <font color="Olive">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> holds <br/>( ( for <font color="Olive">s</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>   st <font color="Olive">s</font> <a href="arytm_3.html#R3">&lt;='</a> <font color="Olive">r</font> holds <br/><font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> ) &amp;  ex <font color="Olive">s</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  st <br/>( <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Olive">r</font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Olive">s</font> ) )
 <b>by </b><i/>;<br/>
<a NAME="E5:23_3"/><i><font color="Green">E110</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 
<b>by </b><i>, <a class="ref" href="enumset1.html#D2">ENUMSET1:def 2</a></i>;<br/>
<a NAME="E6:23_3"/>
for <font color="Olive">r</font>, <font color="Olive">s</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>   st <font color="Olive">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Olive">s</font> <a href="arytm_3.html#R3">&lt;='</a> <font color="Olive">r</font> holds <br/><font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">r1</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a> , <font color="Maroon">r2</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a> , <font color="Maroon">r3</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a> , <font color="Maroon">r4</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a> , <font color="Maroon">r5</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a> <b> such that </b><br/><a NAME="E8:23_3"/><i><font color="Green">E111</font></i>: 
( <font color="Maroon">r1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">r2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">r3</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">r4</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">r5</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 <b>and </b><br/><a NAME="E9:23_3"/><i><font color="Green">E112</font></i>: 
( <font color="Maroon">r1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">r2</font> &amp; <font color="Maroon">r1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">r3</font> &amp; <font color="Maroon">r1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">r4</font> &amp; <font color="Maroon">r1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">r5</font> &amp; <font color="Maroon">r2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">r3</font> &amp; <font color="Maroon">r2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">r4</font> &amp; <font color="Maroon">r2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">r5</font> &amp; <font color="Maroon">r3</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">r4</font> &amp; <font color="Maroon">r3</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">r5</font> &amp; <font color="Maroon">r4</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">r5</font> )
 <b>by </b><i>, </i>;<br/>
<a NAME="E10:23_3"/>
( ( <font color="Maroon">r1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> or <font color="Maroon">r1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> or <font color="Maroon">r1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">2,<font color="Maroon">z</font></span><a href="tarski.html#K4">]</a></span> or <font color="Maroon">r1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">3,<font color="Maroon">w</font></span><a href="tarski.html#K4">]</a></span> ) &amp; ( <font color="Maroon">r2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> or <font color="Maroon">r2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> or <font color="Maroon">r2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">2,<font color="Maroon">z</font></span><a href="tarski.html#K4">]</a></span> or <font color="Maroon">r2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">3,<font color="Maroon">w</font></span><a href="tarski.html#K4">]</a></span> ) &amp; ( <font color="Maroon">r3</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> or <font color="Maroon">r3</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> or <font color="Maroon">r3</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">2,<font color="Maroon">z</font></span><a href="tarski.html#K4">]</a></span> or <font color="Maroon">r3</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">3,<font color="Maroon">w</font></span><a href="tarski.html#K4">]</a></span> ) &amp; ( <font color="Maroon">r4</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> or <font color="Maroon">r4</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> or <font color="Maroon">r4</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">2,<font color="Maroon">z</font></span><a href="tarski.html#K4">]</a></span> or <font color="Maroon">r4</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">3,<font color="Maroon">w</font></span><a href="tarski.html#K4">]</a></span> ) &amp; ( <font color="Maroon">r5</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> or <font color="Maroon">r5</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> or <font color="Maroon">r5</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">2,<font color="Maroon">z</font></span><a href="tarski.html#K4">]</a></span> or <font color="Maroon">r5</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">3,<font color="Maroon">w</font></span><a href="tarski.html#K4">]</a></span> ) )
 
<b>by </b><i>, , <a class="ref" href="enumset1.html#D2">ENUMSET1:def 2</a></i>;<br/>
<b>hence </b><a NAME="E11:23_3"/>
contradiction
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E7:23"/><b>then </b>
not 0,1,2,3 <a href="quaterni.html#K3" target="_self">--&gt;</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K1">DEDEKIND_CUTS</a> 
 
<b>by </b><i>, <a class="ref" href="arytm_2.html#D1">ARYTM_2:def 1</a></i>;<br/>
<a NAME="E8:23"/><b>then </b><i><font color="Green">E113</font></i>: 
not 0,1,2,3 <a href="quaterni.html#K3" target="_self">--&gt;</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a> 
 
<b>by </b><i><a class="ref" href="arytm_2.html#D2">ARYTM_2:def 2</a>, , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:23_4"/>
0,1,2,3 <a href="quaterni.html#K3" target="_self">--&gt;</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><a href="arytm_3.html#K13">{}</a> </span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/><b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:23_4"/><i><font color="Green">E114</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="arytm_3.html#K13">{}</a> </span><a href="tarski.html#K1">}</a></span>
 <b>and </b><br/><a NAME="E4:23_4"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a> 
 <b>and </b><br/><a NAME="E5:23_4"/><i><font color="Green">E115</font></i>: 
0,1,2,3 <a href="quaterni.html#K3" target="_self">--&gt;</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T103">ZFMISC_1:103</a></i>;<br/><a NAME="E6:23_4"/><i><font color="Green">E116</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E7:23_4"/>
0,1,2,3 <a href="quaterni.html#K3" target="_self">--&gt;</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="enumset1.html#K2">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">a</font></span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">1,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">2,<font color="Maroon">c</font></span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">3,<font color="Maroon">d</font></span><a href="tarski.html#K4">]</a></span></span><a href="enumset1.html#K2">}</a></span>
 <b>by </b><i>, </i>;<br/><a NAME="E8:23_4"/><b>then </b><i><font color="Green">E117</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> 0,1,2,3 <a href="quaterni.html#K3" target="_self">--&gt;</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font>
 <b>by </b><i><a class="ref" href="enumset1.html#D2">ENUMSET1:def 2</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_4_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:23_4_1"/>
( <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2"><a href="tarski.html#K2">{</a><span class="default">1,<font color="Maroon">b</font></span><a href="tarski.html#K2">}</a></span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K2">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span> or <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2"><a href="tarski.html#K2">{</a><span class="default">1,<font color="Maroon">b</font></span><a href="tarski.html#K2">}</a></span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K2">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">0,<font color="Maroon">y</font></span><a href="tarski.html#K2">}</a></span> )
 </span><b>by </b><i>, , , <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_4_1_1"><b>suppose </b></a><a NAME="E1:23_4_1_1"/>
<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2"><a href="tarski.html#K2">{</a><span class="default">1,<font color="Maroon">b</font></span><a href="tarski.html#K2">}</a></span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K2">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>
 ;<br/><div class="add"><a NAME="E2:23_4_1_1"/><b>then </b>
0 <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2"><a href="tarski.html#K2">{</a><span class="default">1,<font color="Maroon">b</font></span><a href="tarski.html#K2">}</a></span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K2">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:23_4_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_4_1_2"><b>suppose </b></a><a NAME="E1:23_4_1_2"/>
<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2"><a href="tarski.html#K2">{</a><span class="default">1,<font color="Maroon">b</font></span><a href="tarski.html#K2">}</a></span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K2">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">0,<font color="Maroon">y</font></span><a href="tarski.html#K2">}</a></span>
 ;<br/><div class="add"><a NAME="E2:23_4_1_2"/><b>then </b>
0 <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2"><a href="tarski.html#K2">{</a><span class="default">1,<font color="Maroon">b</font></span><a href="tarski.html#K2">}</a></span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K2">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><b>hence </b><a NAME="E3:23_4_1_2"/>
contradiction
 <b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E10:23"/>
not 0,1,2,3 <a href="quaterni.html#K3" target="_self">--&gt;</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a> 
 <b>by </b><i><a class="ref" href="numbers.html#D1">NUMBERS:def 1</a>, , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


</div>
