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<b>let </b><font color="Maroon">R</font> be   <a href="relat_1.html#NM1">Relation</a>;<br/>

<b>assume </b><a NAME="E1:40"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">R</font> is <a href="wellord1.html#V2" target="_self">well-ordering</a>
 ;<br/>

<a NAME="E2:40"/><b>then </b><i><font color="Green">E20</font></i>: 
<font color="Maroon">R</font> is <a href="relat_2.html#V4">antisymmetric</a>
 
<b>by </b><i/>;<br/>
<b>let </b><font color="Maroon">Z</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E3:40"/><i><font color="Green">E27</font></i>: 
for <font color="Olive">a</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font> &amp; <font color="Maroon">R</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Olive">a</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Z</font> holds <br/><font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 ;<br/>

<div><i><font color="Green">E28</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:40_1"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font> &amp; ( for <font color="Olive">b</font> being   <a href="hidden.html#M1">set</a>   st <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b</font>,<font color="Maroon">a</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b</font> holds <br/><font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> ) )
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">b</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:40_1_1"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">a</font>
 ;<br/><a NAME="E2:40_1_1"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">a</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a</font> )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:40_1_1"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><a NAME="E3:40_1"/><b>then </b>
<font color="Maroon">R</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">a</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Z</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><b>hence </b><a NAME="E4:40_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<b>given </b><font color="Maroon">a</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><a NAME="E5:40"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font> &amp; not <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<a NAME="E6:40"/>
<span class="p1">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">R</font></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">Z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>then consider </b><font color="Maroon">a</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E8:40"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">R</font></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">Z</font> &amp; ( for <font color="Olive">b</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">R</font></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">Z</font> holds <br/><span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Olive">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> ) )
 <b>by </b><i><a class="txt" href="wellord1.html#E8:40"><i><font color="Green">E38</font></i></a>, </i>;<br/>
<a NAME="E9:40"/>
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font> &amp; not <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> )
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>then consider </b><font color="Maroon">b</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E11:40"/><i><font color="Green">E39</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">a</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a</font> &amp; not <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> )
 <b>by </b><i/>;<br/>
<a NAME="E12:40"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font>
 
<b>by </b><i><a class="ref" href="wellord1.html#T13" target="_self">Th13</a>, <a class="ref" href="relat_1.html#T30">RELAT_1:30</a></i>;<br/>
<a NAME="E13:40"/><b>then </b>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">R</font></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">Z</font>
 
<b>by </b><i><a class="ref" href="wellord1.html#T13" target="_self">Th13</a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E14:40"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E15:40"/>
contradiction
 <b>by </b><i><a class="txt" href="wellord1.html#E11:40"><i><font color="Green">E39</font></i></a>, <a class="ref" href="wellord1.html#T13" target="_self">Th13</a>, </i>;<br/>


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