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<b>let </b><font color="Maroon" title="c1">R</font> be   <a href="relat_1.html#NM1" title="RELAT_1:NM.1">Relation</a>;<br/>

<b>assume </b><a NAME="E1:13"/><i><font color="Green" title="E13">A1</font></i>: 
( <font color="Maroon" title="c1">R</font> is <a href="wellord1.html#V2" title="WELLORD1:attr.2">well-ordering</a> &amp;  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c1">R</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 ;<br/>

<a NAME="E2:13"/><b>then </b><i><font color="Green" title="E14">A2</font></i>: 
( <font color="Maroon" title="c1">R</font> is <a href="relat_2.html#V6" title="RELAT_2:attr.6">connected</a> &amp; <font color="Maroon" title="c1">R</font> is <a href="relat_2.html#V4" title="RELAT_2:attr.4">antisymmetric</a> &amp; <font color="Maroon" title="c1">R</font> is <a href="relat_2.html#V1" title="RELAT_2:attr.1">reflexive</a> &amp; ( for <font color="Olive" title="b1">Y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c1">R</font> &amp; <font color="Olive" title="b1">Y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  holds <br/> ex <font color="Olive" title="b2">a</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Y</font> &amp; ( for <font color="Olive" title="b3">b</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Y</font> holds <br/><span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">a</font>,<font color="Olive" title="b3">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">R</font> ) ) ) )
 
<b>by </b><i><a class="ref" href="wellord1.html#D4" target="_self" title="WELLORD1:def.4">Def4</a>, <a class="ref" href="wellord1.html#T10" target="_self" title="WELLORD1:th.10">Th10</a></i>;<br/>
<b>let </b><font color="Maroon" title="c2">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E3:13"/><i><font color="Green" title="E15">A3</font></i>: 
<font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c1">R</font>
 ;<br/>

<b>given </b><font color="Maroon" title="c3">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><a NAME="E5:13"/><i><font color="Green" title="E16">A4</font></i>: 
( <font color="Maroon" title="c3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c1">R</font> &amp; not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c2">a</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">R</font> )
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">$1,<font color="Maroon" title="c2">a</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">R</font>;<br/>
<b>consider </b><font color="Maroon" title="c4">Z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E7:13"/><i><font color="Green" title="E17">A5</font></i>: 
for <font color="Olive" title="b1">c</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Z</font> iff ( <font color="Olive" title="b1">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c1">R</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">c</font>] ) )
 <b>from </b><i><a class="ref" href="xboole_0.html#S1" title="XBOOLE_0:sch.1">XBOOLE_0:sch 1</a>();<br/></i>
<a NAME="E8:13"/>
for <font color="Olive" title="b1">b</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Z</font> holds <br/><font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c1">R</font>
 
<b>by </b><i><a class="txt" href="wellord1.html#E7:13"><i><font color="Green" title="E17">A5</font></i></a></i>;<br/>
<a NAME="E9:13"/><b>then </b><i><font color="Green" title="E18">A6</font></i>: 
<font color="Maroon" title="c4">Z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c1">R</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<a NAME="E10:13"/>
<font color="Maroon" title="c4">Z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="txt" href="wellord1.html#E5:13"><i><font color="Green" title="E16">A4</font></i></a>, <a class="txt" href="wellord1.html#E7:13"><i><font color="Green" title="E17">A5</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c5">d</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E12:13"/><i><font color="Green" title="E19">A7</font></i>: 
( <font color="Maroon" title="c5">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Z</font> &amp; ( for <font color="Olive" title="b1">c</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Z</font> holds <br/><span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">d</font>,<font color="Olive" title="b1">c</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">R</font> ) )
 <b>by </b><i><a class="txt" href="wellord1.html#E1:13"><i><font color="Green" title="E13">A1</font></i></a>, <a class="txt" href="wellord1.html#E9:13"><i><font color="Green" title="E18">A6</font></i></a>, <a class="ref" href="wellord1.html#T10" target="_self" title="WELLORD1:th.10">Th10</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c5">d</font>
;<br/>

<b>thus </b><a NAME="E13:13"/><i><font color="Green" title="E20">A8</font></i>: 
<font color="Maroon" title="c5">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c1">R</font>
 <b>by </b><i><a class="txt" href="wellord1.html#E7:13"><i><font color="Green" title="E17">A5</font></i></a>, <a class="txt" href="wellord1.html#E12:13"><i><font color="Green" title="E19">A7</font></i></a></i>;<br/>

<a NAME="E14:13"/><i><font color="Green" title="E21">A9</font></i>: 
not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c2">a</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">R</font>
 
<b>by </b><i><a class="txt" href="wellord1.html#E7:13"><i><font color="Green" title="E17">A5</font></i></a>, <a class="txt" href="wellord1.html#E12:13"><i><font color="Green" title="E19">A7</font></i></a></i>;<br/>
<a NAME="E15:13"/><b>then </b>
<font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">d</font>
 
<b>by </b><i><a class="txt" href="wellord1.html#E12:13"><i><font color="Green" title="E19">A7</font></i></a></i>;<br/>
<b>hence </b><a NAME="E16:13"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">d</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">R</font>
 <b>by </b><i><a class="txt" href="wellord1.html#E2:13"><i><font color="Green" title="E14">A2</font></i></a>, <a class="txt" href="wellord1.html#E3:13"><i><font color="Green" title="E15">A3</font></i></a>, <a class="txt" href="wellord1.html#E13:13"><i><font color="Green" title="E20">A8</font></i></a>, <a class="txt" href="wellord1.html#E14:13"><i><font color="Green" title="E21">A9</font></i></a>, <a class="txt" href="wellord1.html#E4"><i><font color="Green" title="E4">Lm4</font></i></a></i>;<br/>

<b>let </b><font color="Maroon" title="c6">c</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E17:13"/><i><font color="Green" title="E22">A10</font></i>: 
( <font color="Maroon" title="c6">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c1">R</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c6">c</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">R</font> )
 ;<br/>

<b>assume </b><a NAME="E18:13"/>
<font color="Maroon" title="c6">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">a</font>
 ;<br/>

<a NAME="E19:13"/><b>then </b>
not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c2">a</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">R</font>
 
<b>by </b><i><a class="txt" href="wellord1.html#E2:13"><i><font color="Green" title="E14">A2</font></i></a>, <a class="txt" href="wellord1.html#E17:13"><i><font color="Green" title="E22">A10</font></i></a>, <a class="txt" href="wellord1.html#E3"><i><font color="Green" title="E3">Lm3</font></i></a></i>;<br/>
<a NAME="E20:13"/><b>then </b>
<font color="Maroon" title="c6">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Z</font>
 
<b>by </b><i><a class="txt" href="wellord1.html#E7:13"><i><font color="Green" title="E17">A5</font></i></a>, <a class="txt" href="wellord1.html#E17:13"><i><font color="Green" title="E22">A10</font></i></a></i>;<br/>
<b>hence </b><a NAME="E21:13"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c6">c</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">R</font>
 <b>by </b><i><a class="txt" href="wellord1.html#E12:13"><i><font color="Green" title="E19">A7</font></i></a></i>;<br/>


</div>
