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<b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">b</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">R</font> be   <a href="relat_1.html#NM1">Relation</a>;<br/>





<b>assume </b><a NAME="E1:34"/><i><font color="Green">E19</font></i>: 
( <font color="Maroon">R</font> is <a href="wellord1.html#V2" target="_self">well-ordering</a> &amp; <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; <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="E2:34"/><b>then </b><i><font color="Green">E20</font></i>: 
( <font color="Maroon">R</font> is <a href="relat_2.html#V8">transitive</a> &amp; <font color="Maroon">R</font> is <a href="relat_2.html#V1">reflexive</a> &amp; <font color="Maroon">R</font> is <a href="relat_2.html#V4">antisymmetric</a> )
 
<b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">S</font> = <font color="Maroon">R</font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">a</font></span>)</span>;<br/>
<div><i><font color="Green">E27</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:34_1"/>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p2">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">b</font>
 ;<br/><a NAME="E2:34_1"/><b>then </b><i><font color="Green">E28</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c</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> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">a</font></span>)</span> &amp; <font color="Maroon">c</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font> )
 <b>by </b><i/>;<br/><a NAME="E3:34_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c</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="E4:34_1"/>
<font color="Maroon">c</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">b</font>
 <b>by </b><i><a class="ref" href="wellord1.html#T10" target="_self">Th10</a>, </i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:34_2"/>
<font color="Maroon">c</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">b</font>
 ;<br/><a NAME="E2:34_2"/><b>then </b><i><font color="Green">E36</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c</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> &amp; <font color="Maroon">c</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font> )
 <b>by </b><i/>;<br/><a NAME="E3:34_2"/><i><font color="Green">E37</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> )
 <b>by </b><i><a class="ref" href="wellord1.html#T5" target="_self">Th5</a>, </i>;<br/><a NAME="E4:34_2"/><b>then </b><i><font color="Green">E38</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c</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>
 <b>by </b><i><a class="ref" href="wellord1.html#D4" target="_self">Def4</a>, <a class="ref" href="wellord1.html#T10" target="_self">Th10</a>, </i>;<br/><a NAME="E5:34_2"/>
<font color="Maroon">c</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="wellord1.html#D4" target="_self">Def4</a>, <a class="ref" href="wellord1.html#T10" target="_self">Th10</a>, <a class="txt" href="wellord1.html#E2:34_2"><i><font color="Green">E36</font></i></a>, </i>;<br/><a NAME="E6:34_2"/><b>then </b>
<font color="Maroon">c</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>
 <b>by </b><i><a class="txt" href="wellord1.html#E3:34_2"><i><font color="Green">E37</font></i></a>, </i>;<br/><a NAME="E7:34_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c</font>,<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="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">a</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">a</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="ref" href="wellord1.html#T5" target="_self">Th5</a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E8:34_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c</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> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">a</font></span>)</span>
 <b>by </b><i><a class="ref" href="wellord1.html#T10" target="_self">Th10</a>, </i>;<br/><b>hence </b><a NAME="E9:34_2"/>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p2">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="wellord1.html#T10" target="_self">Th10</a>, </i>;<br/></div><b>end;</b></div>
<a NAME="E5:34"/><b>then </b>
( <font color="Maroon">R</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">b</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p2">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">b</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p2">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">b</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">R</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">b</font> )
 
<b>by </b><i><a class="ref" href="wellord1.html#D5" target="_self">Def5</a>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>hence </b><a NAME="E6:34"/>
<span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p2">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">R</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


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