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<b>let </b><font color="Maroon" title="c1">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">A</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>;<br/>



<b>assume </b><a NAME="E1:17"/><i><font color="Green" title="E16">A1</font></i>: 
<font color="Maroon" title="c1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">A</font>
 ;<br/>

<a NAME="E2:17"/><b>then </b><i><font color="Green" title="E17">A2</font></i>: 
 <a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c1">X</font> is <a href="wellord1.html#V2" title="WELLORD1:attr.2">well-ordering</a>
 
<b>by </b><i><a class="ref" href="wellord2.html#T9" target="_self" title="WELLORD2:th.9">Th9</a></i>;<br/>
<a NAME="E3:17"/><i><font color="Green" title="E18">A3</font></i>: 
 <a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c2">A</font> is <a href="wellord1.html#V2" title="WELLORD1:attr.2">well-ordering</a>
 
<b>by </b><i><a class="ref" href="wellord2.html#T7" target="_self" title="WELLORD2:th.7">Th7</a></i>;<br/>
<a NAME="E4:17"/><i><font color="Green" title="E19">A4</font></i>: 
 <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">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font>
 
<b>by </b><i><a class="ref" href="wellord2.html#D1" target="_self" title="WELLORD2:def.1">Def1</a></i>;<br/>
<a NAME="E5:17"/><i><font color="Green" title="E20">A5</font></i>: 
<span class="p1">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="txt" href="wellord2.html#E1:17"><i><font color="Green" title="E16">A1</font></i></a>, <a class="ref" href="wellord2.html#T8" target="_self" title="WELLORD2:th.8">Th8</a></i>;<br/>
<div><i><font color="Green" title="E21">A6</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:17_1"/>
 <a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c2">A</font>,<span class="p1">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <font color="Maroon" title="c1">X</font> <a href="wellord1.html#R4" title="WELLORD1:pred.4">are_isomorphic</a> 
 ;<br/><a NAME="E2:17_1"/><b>then </b>
 <a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c1">X</font>, <a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c2">A</font> <a href="wellord1.html#R4" title="WELLORD1:pred.4">are_isomorphic</a> 
 <b>by </b><i><a class="txt" href="wellord2.html#E5:17"><i><font color="Green" title="E20">A5</font></i></a>, <a class="ref" href="wellord1.html#T50" title="WELLORD1:th.50">WELLORD1:50</a></i>;<br/><b>hence </b><a NAME="E3:17_1"/>
 <a href="wellord2.html#K2" title="WELLORD2:func.2">order_type_of</a> <span class="p1">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c2">A</font>
 <b>by </b><i><a class="txt" href="wellord2.html#E2:17"><i><font color="Green" title="E17">A2</font></i></a>, <a class="ref" href="wellord2.html#D2" target="_self" title="WELLORD2:def.2">Def2</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2"><b>now </b></a><div class="add"><b>given </b><font color="Maroon" title="c3">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><a NAME="E1:17_2"/><i><font color="Green" title="E22">A7</font></i>: 
( <font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <span class="p1">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c3">a</font></span>)</span>,<span class="p1">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <font color="Maroon" title="c1">X</font> <a href="wellord1.html#R4" title="WELLORD1:pred.4">are_isomorphic</a>  )
 ;<br/><b>reconsider </b><font color="Maroon" title="c4">a</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="txt" href="wellord2.html#E1:17_2"><i><font color="Green" title="E22">A7</font></i></a>, <a class="ref" href="ordinal1.html#T23" title="ORDINAL1:th.23">ORDINAL1:23</a></i>;<br/><a NAME="E3:17_2"/><i><font color="Green" title="E23">A8</font></i>: 
<span class="p1">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font>
 <b>by </b><i><a class="txt" href="wellord2.html#E1:17_2"><i><font color="Green" title="E22">A7</font></i></a>, <a class="ref" href="wellord2.html#T10" target="_self" title="WELLORD2:th.10">Th10</a></i>;<br/><a NAME="E4:17_2"/><i><font color="Green" title="E24">A9</font></i>: 
<font color="Maroon" title="c4">a</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c2">A</font>
 <b>by </b><i><a class="txt" href="wellord2.html#E1:17_2"><i><font color="Green" title="E22">A7</font></i></a>, <a class="ref" href="ordinal1.html#D2" title="ORDINAL1:def.2">ORDINAL1:def 2</a></i>;<br/><a NAME="E5:17_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c4">a</font>
 <b>by </b><i><a class="ref" href="wellord2.html#T8" target="_self" title="WELLORD2:th.8">Th8</a></i>;<br/><a NAME="E6:17_2"/><b>then </b>
 <a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c1">X</font>, <a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c4">a</font> <a href="wellord1.html#R4" title="WELLORD1:pred.4">are_isomorphic</a> 
 <b>by </b><i><a class="txt" href="wellord2.html#E5:17"><i><font color="Green" title="E20">A5</font></i></a>, <a class="txt" href="wellord2.html#E1:17_2"><i><font color="Green" title="E22">A7</font></i></a>, <a class="txt" href="wellord2.html#E3:17_2"><i><font color="Green" title="E23">A8</font></i></a>, <a class="ref" href="wellord1.html#T50" title="WELLORD1:th.50">WELLORD1:50</a></i>;<br/><b>hence </b><a NAME="E7:17_2"/>
 <a href="wellord2.html#K2" title="WELLORD2:func.2">order_type_of</a> <span class="p1">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c2">A</font>
 <b>by </b><i><a class="txt" href="wellord2.html#E2:17"><i><font color="Green" title="E17">A2</font></i></a>, <a class="txt" href="wellord2.html#E4:17_2"><i><font color="Green" title="E24">A9</font></i></a>, <a class="ref" href="wellord2.html#D2" target="_self" title="WELLORD2:def.2">Def2</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:17"/>
 <a href="wellord2.html#K2" title="WELLORD2:func.2">order_type_of</a> <span class="p1">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c2">A</font>
 <b>by </b><i><a class="txt" href="wellord2.html#E1:17"><i><font color="Green" title="E16">A1</font></i></a>, <a class="txt" href="wellord2.html#E3:17"><i><font color="Green" title="E18">A3</font></i></a>, <a class="txt" href="wellord2.html#E4:17"><i><font color="Green" title="E19">A4</font></i></a>, <a class="txt" href="wellord2.html#E6:17"><i><font color="Green" title="E21">A6</font></i></a>, <a class="ref" href="wellord1.html#T64" title="WELLORD1:th.64">WELLORD1:64</a></i>;<br/>


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