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<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:5_1_1_1"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="wellord2.html#K1">RelIncl</a> <font color="Maroon">X</font>
 ;<br/><b>then consider </b><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> <b> such that </b><br/><a NAME="E3:5_1_1_1"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="relat_1.html#D1">RELAT_1:def 1</a></i>;<br/><a NAME="E4:5_1_1_1"/>
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><a href="wellord2.html#K1">RelIncl</a> <font color="Maroon">X</font></span>)</span> &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><a href="wellord2.html#K1">RelIncl</a> <font color="Maroon">X</font></span>)</span> )
 <b>by </b><i>, , <a class="ref" href="relat_1.html#T30">RELAT_1:30</a></i>;<br/><a NAME="E5:5_1_1_1"/><b>then </b>
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 <b>by </b><i><a class="ref" href="wellord2.html#D1">WELLORD2:def 1</a></i>;<br/><b>hence </b><a NAME="E6:5_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/></div><b>end;</b></div>
<a NAME="E2:5_1_1"/><b>then </b>
 <a href="wellord2.html#K1">RelIncl</a> <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">R</font> =  <a href="wellord2.html#K1">RelIncl</a> <font color="Maroon">X</font> as   <a href="relset_1.html#NM3">Relation</a> of <font color="Maroon">X</font> <b>by </b><i><a class="ref" href="relset_1.html#D1">RELSET_1:def 1</a></i>;<br/>
<a NAME="E4:5_1_1"/><i><font color="Green">E25</font></i>: 
 <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="wellord2.html#D1">WELLORD2:def 1</a></i>;<br/>
<a NAME="E5:5_1_1"/>
<font color="Maroon">R</font> is <a href="relat_2.html#V1">reflexive</a>
 
<b>by </b><i><a class="ref" href="wellord2.html#T2">WELLORD2:2</a></i>;<br/>
<a NAME="E6:5_1_1"/><b>then </b>
<font color="Maroon">R</font> <a href="relat_2.html#R1">is_reflexive_in</a> <font color="Maroon">X</font>
 
<b>by </b><i>, <a class="ref" href="relat_2.html#D9">RELAT_2:def 9</a></i>;<br/>
<a NAME="E7:5_1_1"/><b>then </b>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">R</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="orders_1.html#T98">ORDERS_1:98</a></i>;<br/>
<b>hence </b><a NAME="E8:5_1_1"/>
 <a href="wellord2.html#K1">RelIncl</a> <font color="Maroon">X</font> is   <a href="orders_1.html#NM2">Order</a> of <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="partfun1.html#D4">PARTFUN1:def 4</a>, <a class="ref" href="wellord2.html#T2">WELLORD2:2</a>, <a class="ref" href="wellord2.html#T3">WELLORD2:3</a>, <a class="ref" href="wellord2.html#T5">WELLORD2:5</a></i>;<br/>


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