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

<b>consider </b><font color="Maroon">c<sub>2</sub></font> being   <a href="relat_1.html#NM1">Relation</a><b> such that </b><br/><a NAME="E2:11_1_1"/><i><font color="Green">E8</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="wellord1.html#V2">well-ordering</a>
 <b>and </b><br/><a NAME="E3:11_1_1"/><i><font color="Green">E9</font></i>: 
 <a href="relat_1.html#K3">field</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="wellset1.html#T9">WELLSET1:9</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">c<sub>2</sub></font> as   <a href="relset_1.html#NM3">Relation</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="polynom1.html#E3:11_1_1"><i><font color="Green">E9</font></i></a>, <a class="ref" href="polynom1.html#T4" target="_self">Th4</a></i>;<br/>
<a NAME="E5:11_1_1"/><i><font color="Green">E10</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> is <a href="relat_2.html#V1">reflexive</a> &amp; <font color="Maroon">c<sub>3</sub></font> is <a href="relat_2.html#V8">transitive</a> &amp; <font color="Maroon">c<sub>3</sub></font> is <a href="relat_2.html#V4">antisymmetric</a> &amp; <font color="Maroon">c<sub>3</sub></font> is <a href="relat_2.html#V6">connected</a> &amp; <font color="Maroon">c<sub>3</sub></font> is <a href="wellord1.html#V1">well_founded</a> )
 
<b>by </b><i><a class="txt" href="polynom1.html#E2:11_1_1"><i><font color="Green">E8</font></i></a>, <a class="ref" href="wellord1.html#D4">WELLORD1:def 4</a></i>;<br/>
<a NAME="E6:11_1_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="relat_2.html#R1">is_reflexive_in</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="polynom1.html#E3:11_1_1"><i><font color="Green">E9</font></i></a>, <a class="ref" href="relat_2.html#D9">RELAT_2:def 9</a></i>;<br/>
<a NAME="E7:11_1_1"/><b>then </b>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="orders_1.html#T98">ORDERS_1:98</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> as   <a href="orders_1.html#NM2">Order</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="polynom1.html#E2:11_1_1"><i><font color="Green">E8</font></i></a>, <a class="ref" href="partfun1.html#D4">PARTFUN1:def 4</a>, <a class="ref" href="wellord1.html#D4">WELLORD1:def 4</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>4</sub></font>
;<br/>

<b>thus </b><a NAME="E9:11_1_1"/>
( <font color="Maroon">c<sub>4</sub></font> is <a href="orders_1.html#V3">being_linear-order</a> &amp; <font color="Maroon">c<sub>4</sub></font> is <a href="wellord1.html#V2">well-ordering</a> )
 <b>by </b><i><a class="txt" href="polynom1.html#E2:11_1_1"><i><font color="Green">E8</font></i></a>, <a class="txt" href="polynom1.html#E5:11_1_1"><i><font color="Green">E10</font></i></a>, <a class="ref" href="orders_1.html#D5">ORDERS_1:def 5</a></i>;<br/>


</div>
