<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c1">A</font>, <font color="Maroon" title="c2">B</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>;<br/>



<b>assume </b><a NAME="E1:10"/><i><font color="Green" title="E11">A1</font></i>: 
<font color="Maroon" title="c1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font>
 ;<br/>

<b>thus </b><a NAME="E2:10"/>
for <font color="Olive" title="b1">a</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> holds <br/><font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c2">B</font></span>)</span> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c1">A</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a>,<a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:10_1"/><i><font color="Green" title="E12">A2</font></i>: 
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c4">C</font> = <font color="Maroon" title="c3">a</font> as   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> <b>by </b><i><a class="ref" href="ordinal1.html#T23" title="ORDINAL1:th.23">ORDINAL1:23</a></i>;<br/>
<a NAME="E3:10_1"/><i><font color="Green" title="E13">A3</font></i>: 
<font color="Maroon" title="c4">C</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c1">A</font>
 
<b>by </b><i><a class="txt" href="wellord2.html#E1:10_1"><i><font color="Green" title="E12">A2</font></i></a>, <a class="ref" href="ordinal1.html#D2" title="ORDINAL1:def.2">ORDINAL1:def 2</a></i>;<br/>
<a NAME="E4:10_1"/>
<font color="Maroon" title="c1">A</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c2">B</font>
 
<b>by </b><i><a class="txt" href="wellord2.html#E1:10"><i><font color="Green" title="E11">A1</font></i></a>, <a class="ref" href="ordinal1.html#D2" title="ORDINAL1:def.2">ORDINAL1:def 2</a></i>;<br/>
<a NAME="E5:10_1"/><b>then </b><i><font color="Green" title="E14">A4</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">C</font>,<font color="Maroon" title="c1">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>  <a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c2">B</font>
 
<b>by </b><i><a class="txt" href="wellord2.html#E1:10"><i><font color="Green" title="E11">A1</font></i></a>, <a class="txt" href="wellord2.html#E1:10_1"><i><font color="Green" title="E12">A2</font></i></a>, <a class="txt" href="wellord2.html#E3:10_1"><i><font color="Green" title="E13">A3</font></i></a>, <a class="ref" href="wellord2.html#D1" target="_self" title="WELLORD2:def.1">Def1</a></i>;<br/>
<a NAME="E6:10_1"/>
<font color="Maroon" title="c3">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">A</font>
 
<b>by </b><i><a class="txt" href="wellord2.html#E1:10_1"><i><font color="Green" title="E12">A2</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:10_1"/>
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c2">B</font></span>)</span> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c1">A</font>
 <b>by </b><i><a class="txt" href="wellord2.html#E5:10_1"><i><font color="Green" title="E14">A4</font></i></a>, <a class="ref" href="wellord1.html#D1" title="WELLORD1:def.1">WELLORD1:def 1</a></i>;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon" title="c3">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E3:10"/>
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c2">B</font></span>)</span> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c1">A</font>
 ;<br/>

<a NAME="E4:10"/><b>then </b><i><font color="Green" title="E12">A5</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c1">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>  <a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c2">B</font> &amp; <font color="Maroon" title="c3">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">A</font> )
 
<b>by </b><i><a class="ref" href="wellord1.html#D1" title="WELLORD1:def.1">WELLORD1:def 1</a></i>;<br/>
<a NAME="E5:10"/><b>then </b><i><font color="Green" title="E13">A6</font></i>: 
( <font color="Maroon" title="c3">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> <span class="p1">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c2">B</font></span>)</span> &amp;  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c2">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">B</font> )
 
<b>by </b><i><a class="ref" href="wellord2.html#D1" target="_self" title="WELLORD2:def.1">Def1</a>, <a class="ref" href="relat_1.html#T30" title="RELAT_1:th.30">RELAT_1:30</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c4">C</font> = <font color="Maroon" title="c3">a</font> as   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> <b>by </b><i><a class="ref" href="ordinal1.html#T23" title="ORDINAL1:th.23">ORDINAL1:23</a></i>;<br/>
<a NAME="E7:10"/>
<font color="Maroon" title="c4">C</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c1">A</font>
 
<b>by </b><i><a class="txt" href="wellord2.html#E1:10"><i><font color="Green" title="E11">A1</font></i></a>, <a class="txt" href="wellord2.html#E4:10"><i><font color="Green" title="E12">A5</font></i></a>, <a class="txt" href="wellord2.html#E5:10"><i><font color="Green" title="E13">A6</font></i></a>, <a class="ref" href="wellord2.html#D1" target="_self" title="WELLORD2:def.1">Def1</a></i>;<br/>
<a NAME="E8:10"/><b>then </b>
<font color="Maroon" title="c4">C</font> <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> <font color="Maroon" title="c1">A</font>
 
<b>by </b><i><a class="txt" href="wellord2.html#E4:10"><i><font color="Green" title="E12">A5</font></i></a>, <a class="ref" href="xboole_0.html#D8" title="XBOOLE_0:def.8">XBOOLE_0:def 8</a></i>;<br/>
<b>hence </b><a NAME="E9:10"/>
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font>
 <b>by </b><i><a class="ref" href="ordinal1.html#T21" title="ORDINAL1:th.21">ORDINAL1:21</a></i>;<br/>


</div>
