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

<b>let </b><font color="Maroon" title="c1">A</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<a NAME="E1:2"/>
 <a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c1">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#T9" title="WELLORD2:th.9">WELLORD2:9</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c2">a</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><b> such that </b><br/><a NAME="E3:2"/><i><font color="Green" title="E3">A1</font></i>: 
 <a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c1">A</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="ref" href="wellord2.html#T14" title="WELLORD2:th.14">WELLORD2:14</a></i>;<br/>
<a NAME="E4:2"/>
 <a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c2">a</font>, <a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c1">A</font> <a href="wellord1.html#R4" title="WELLORD1:pred.4">are_isomorphic</a> 
 
<b>by </b><i><a class="txt" href="henmodel.html#E3:2"><i><font color="Green" title="E3">A1</font></i></a>, <a class="ref" href="wellord1.html#T50" title="WELLORD1:th.50">WELLORD1:50</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c3">F</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E6:2"/><i><font color="Green" title="E4">A2</font></i>: 
<font color="Maroon" title="c3">F</font> <a href="wellord1.html#R3" title="WELLORD1:pred.3">is_isomorphism_of</a>  <a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c2">a</font>, <a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c1">A</font>
 <b>by </b><i><a class="ref" href="wellord1.html#D8" title="WELLORD1:def.8">WELLORD1:def 8</a></i>;<br/>
<a NAME="E7:2"/><i><font color="Green" title="E5">A3</font></i>: 
for <font color="Olive" title="b1">m</font>, <font color="Olive" title="b2">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">F</font> &amp; <font color="Olive" title="b2">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">F</font> &amp; <font color="Olive" title="b2">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">m</font> holds <br/><font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">m</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">m</font>, <font color="Maroon" title="c5">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>



<b>assume </b><a NAME="E1:2_1"/><i><font color="Green" title="E6">A4</font></i>: 
( <font color="Maroon" title="c4">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">F</font> &amp; <font color="Maroon" title="c5">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">F</font> &amp; <font color="Maroon" title="c5">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">m</font> )
 ;<br/>

<a NAME="E2:2_1"/>
( <font color="Maroon" title="c4">m</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">a</font></span>)</span> &amp; <font color="Maroon" title="c5">n</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">a</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="henmodel.html#E6:2"><i><font color="Green" title="E4">A2</font></i></a>, <a class="txt" href="henmodel.html#E1:2_1"><i><font color="Green" title="E6">A4</font></i></a>, <a class="ref" href="wellord1.html#D7" title="WELLORD1:def.7">WELLORD1:def 7</a></i>;<br/>
<a NAME="E3:2_1"/><b>then </b><i><font color="Green" title="E7">A5</font></i>: 
( <font color="Maroon" title="c4">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">a</font> &amp; <font color="Maroon" title="c5">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">a</font> )
 
<b>by </b><i><a class="ref" href="wellord2.html#D1" title="WELLORD2:def.1">WELLORD2:def 1</a></i>;<br/>
<a NAME="E4:2_1"/>
<font color="Maroon" title="c5">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">i</font> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">m</font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="henmodel.html#E1:2_1"><i><font color="Green" title="E6">A4</font></i></a></i>;<br/>
<a NAME="E5:2_1"/><b>then </b>
<font color="Maroon" title="c5">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">m</font>
 
<b>by </b><i><a class="ref" href="axioms.html#T30" title="AXIOMS:th.30">AXIOMS:30</a></i>;<br/>
<a NAME="E6:2_1"/><b>then </b>
( <font color="Maroon" title="c5">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">m</font> &amp; <font color="Maroon" title="c5">n</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c4">m</font> )
 
<b>by </b><i><a class="ref" href="ordinal1.html#D2" title="ORDINAL1:def.2">ORDINAL1:def 2</a></i>;<br/>
<a NAME="E7:2_1"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c4">m</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</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">a</font>
 
<b>by </b><i><a class="txt" href="henmodel.html#E3:2_1"><i><font color="Green" title="E7">A5</font></i></a>, <a class="ref" href="wellord2.html#D1" title="WELLORD2:def.1">WELLORD2:def 1</a></i>;<br/>
<a NAME="E8:2_1"/><b>then </b><i><font color="Green" title="E8">A6</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">m</font></span>)</span></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="c1">A</font>
 
<b>by </b><i><a class="txt" href="henmodel.html#E6:2"><i><font color="Green" title="E4">A2</font></i></a>, <a class="ref" href="wellord1.html#D7" title="WELLORD1:def.7">WELLORD1:def 7</a></i>;<br/>
<a NAME="E9:2_1"/>
( <font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">F</font> &amp; <font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">F</font> )
 
<b>by </b><i><a class="txt" href="henmodel.html#E1:2_1"><i><font color="Green" title="E6">A4</font></i></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/>
<a NAME="E10:2_1"/><b>then </b>
( <font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n</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="c1">A</font></span>)</span> &amp; <font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">m</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="c1">A</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="henmodel.html#E6:2"><i><font color="Green" title="E4">A2</font></i></a>, <a class="ref" href="wellord1.html#D7" title="WELLORD1:def.7">WELLORD1:def 7</a></i>;<br/>
<a NAME="E11:2_1"/><b>then </b><i><font color="Green" title="E9">A7</font></i>: 
( <font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">m</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="wellord2.html#D1" title="WELLORD2:def.1">WELLORD2:def 1</a></i>;<br/>
<a NAME="E12:2_1"/><b>then </b><i><font color="Green" title="E10">A8</font></i>: 
<font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">m</font>
 
<b>by </b><i><a class="txt" href="henmodel.html#E8:2_1"><i><font color="Green" title="E8">A6</font></i></a>, <a class="ref" href="wellord2.html#D1" title="WELLORD2:def.1">WELLORD2:def 1</a></i>;<br/>
<a NAME="E13:2_1"/>
<font color="Maroon" title="c3">F</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 
<b>by </b><i><a class="txt" href="henmodel.html#E6:2"><i><font color="Green" title="E4">A2</font></i></a>, <a class="ref" href="wellord1.html#D7" title="WELLORD1:def.7">WELLORD1:def 7</a></i>;<br/>
<a NAME="E14:2_1"/><b>then </b>
<font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">m</font>
 
<b>by </b><i><a class="txt" href="henmodel.html#E1:2_1"><i><font color="Green" title="E6">A4</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/>
<a NAME="E15:2_1"/><b>then </b><i><font color="Green" title="E11">A9</font></i>: 
<font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n</font> <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> <font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">m</font>
 
<b>by </b><i><a class="txt" href="henmodel.html#E12:2_1"><i><font color="Green" title="E10">A8</font></i></a>, <a class="ref" href="xboole_0.html#D8" title="XBOOLE_0:def.8">XBOOLE_0:def 8</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">b</font> = <font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="henmodel.html#E11:2_1"><i><font color="Green" title="E9">A7</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">c</font> = <font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">m</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="henmodel.html#E11:2_1"><i><font color="Green" title="E9">A7</font></i></a></i>;<br/>
<a NAME="E18:2_1"/>
<font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">c</font>
 
<b>by </b><i><a class="txt" href="henmodel.html#E15:2_1"><i><font color="Green" title="E11">A9</font></i></a>, <a class="ref" href="ordinal1.html#T21" title="ORDINAL1:th.21">ORDINAL1:21</a></i>;<br/>
<b>hence </b><a NAME="E19:2_1"/>
<font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">m</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E8:2"/><i><font color="Green" title="E6">A10</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c1">A</font></span>)</span>
 
<b>by </b><i><a class="txt" href="henmodel.html#E6:2"><i><font color="Green" title="E4">A2</font></i></a>, <a class="ref" href="wellord1.html#D7" title="WELLORD1:def.7">WELLORD1:def 7</a></i>;<br/>
<a NAME="E9:2"/><b>then </b><i><font color="Green" title="E7">A11</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">A</font>
 
<b>by </b><i><a class="ref" href="wellord2.html#D1" title="WELLORD2:def.1">WELLORD2:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c4">F1</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E11:2"/><i><font color="Green" title="E8">A12</font></i>: 
<font color="Maroon" title="c4">F1</font> <a href="wellord1.html#R3" title="WELLORD1:pred.3">is_isomorphism_of</a>  <a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c1">A</font>, <a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c2">a</font>
 <b>by </b><i><a class="txt" href="henmodel.html#E3:2"><i><font color="Green" title="E3">A1</font></i></a>, <a class="ref" href="wellord1.html#D8" title="WELLORD1:def.8">WELLORD1:def 8</a></i>;<br/>
<a NAME="E12:2"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c1">A</font></span>)</span>
 
<b>by </b><i><a class="txt" href="henmodel.html#E11:2"><i><font color="Green" title="E8">A12</font></i></a>, <a class="ref" href="wellord1.html#D7" title="WELLORD1:def.7">WELLORD1:def 7</a></i>;<br/>
<a NAME="E13:2"/><b>then </b><i><font color="Green" title="E9">A13</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">A</font>
 
<b>by </b><i><a class="ref" href="wellord2.html#D1" title="WELLORD2:def.1">WELLORD2:def 1</a></i>;<br/>
<a NAME="E14:2"/><b>then </b>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">F1</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 
<b>by </b><i><a class="ref" href="finset_1.html#T26" title="FINSET_1:th.26">FINSET_1:26</a></i>;<br/>
<a NAME="E15:2"/><b>then </b>
 <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> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 
<b>by </b><i><a class="txt" href="henmodel.html#E11:2"><i><font color="Green" title="E8">A12</font></i></a>, <a class="ref" href="wellord1.html#D7" title="WELLORD1:def.7">WELLORD1:def 7</a></i>;<br/>
<a NAME="E16:2"/><b>then </b><i><font color="Green" title="E10">A14</font></i>: 
<font color="Maroon" title="c2">a</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 
<b>by </b><i><a class="ref" href="wellord2.html#D1" title="WELLORD2:def.1">WELLORD2:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c5">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E18:2"/><i><font color="Green" title="E11">A15</font></i>: 
<font color="Maroon" title="c5">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">F1</font>
 <b>by </b><i><a class="txt" href="henmodel.html#E13:2"><i><font color="Green" title="E9">A13</font></i></a>, <a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/>
<a NAME="E19:2"/>
<font color="Maroon" title="c4">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">F1</font>
 
<b>by </b><i><a class="txt" href="henmodel.html#E18:2"><i><font color="Green" title="E11">A15</font></i></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/>
<a NAME="E20:2"/><b>then </b>
not  <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> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 
<b>by </b><i><a class="txt" href="henmodel.html#E11:2"><i><font color="Green" title="E8">A12</font></i></a>, <a class="ref" href="wellord1.html#D7" title="WELLORD1:def.7">WELLORD1:def 7</a></i>;<br/>
<a NAME="E21:2"/><b>then </b>
not <font color="Maroon" title="c2">a</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 
<b>by </b><i><a class="ref" href="wellord2.html#D1" title="WELLORD2:def.1">WELLORD2:def 1</a></i>;<br/>
<a NAME="E22:2"/><b>then </b><i><font color="Green" title="E12">A16</font></i>: 
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">a</font>
 
<b>by </b><i><a class="ref" href="ordinal1.html#T26" title="ORDINAL1:th.26">ORDINAL1:26</a>, <a class="ref" href="xboole_1.html#T3" title="XBOOLE_1:th.3">XBOOLE_1:3</a></i>;<br/>
<div><i><font color="Green" title="E13">A17</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:2_2"/>
<font color="Maroon" title="c2">a</font> is <a href="ordinal1.html#V4" title="ORDINAL1:attr.4">being_limit_ordinal</a>
 ;<br/><a NAME="E2:2_2"/><b>then </b><i><font color="Green" title="E14">A18</font></i>: 
 <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  <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="henmodel.html#E22:2"><i><font color="Green" title="E12">A16</font></i></a>, <a class="ref" href="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:2_2_1"/>
 <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 ;<br/><a NAME="E2:2_2_1"/><b>then </b>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  <a href="card_1.html#NR2" title="CARD_1:NR.2">&lt;`</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> 
 <b>by </b><i><a class="ref" href="card_3.html#T58" title="CARD_3:th.58">CARD_3:58</a></i>;<br/><b>hence </b><a NAME="E3:2_2_1"/>
contradiction
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:2_2"/>
contradiction
 <b>by </b><i><a class="txt" href="henmodel.html#E16:2"><i><font color="Green" title="E10">A14</font></i></a>, <a class="txt" href="henmodel.html#E2:2_2"><i><font color="Green" title="E14">A18</font></i></a>, <a class="ref" href="finset_1.html#T13" title="FINSET_1:th.13">FINSET_1:13</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:2_3"/><i><font color="Green" title="E14">A19</font></i>: 
not <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> 
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_3_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:2_3_1"/>
 <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 ;<br/><a NAME="E2:2_3_1"/><b>then </b>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  <a href="card_1.html#NR2" title="CARD_1:NR.2">&lt;`</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> 
 <b>by </b><i><a class="ref" href="card_3.html#T58" title="CARD_3:th.58">CARD_3:58</a></i>;<br/><b>hence </b><a NAME="E3:2_3_1"/>
contradiction
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:2_3"/>
contradiction
 <b>by </b><i><a class="txt" href="henmodel.html#E16:2"><i><font color="Green" title="E10">A14</font></i></a>, <a class="txt" href="henmodel.html#E1:2_3"><i><font color="Green" title="E14">A19</font></i></a>, <a class="ref" href="finset_1.html#T13" title="FINSET_1:th.13">FINSET_1:13</a>, <a class="ref" href="ordinal1.html#T26" title="ORDINAL1:th.26">ORDINAL1:26</a></i>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c6">a</font> = <font color="Maroon" title="c2">a</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/>
<div><i><font color="Green" title="E14">A20</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="2_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">b</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>;<br/><b>assume </b><a NAME="E1:2_4"/>
 <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c7">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a</font>
 ;<br/><a NAME="E2:2_4"/><b>then </b>
<font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">a</font>
 <b>by </b><i><a class="ref" href="ordinal1.html#T10" title="ORDINAL1:th.10">ORDINAL1:10</a></i>;<br/><b>hence </b><a NAME="E3:2_4"/>
<font color="Maroon" title="c7">b</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>by </b><i><a class="ref" href="ordinal1.html#T19" title="ORDINAL1:th.19">ORDINAL1:19</a></i>;<br/></div><b>end;</b></div>
<a NAME="E27:2"/><i><font color="Green" title="E15">A21</font></i>: 
 ex <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st  <a href="card_1.html#K4" title="CARD_1:func.4">succ</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_5"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon" title="c7">b</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><b> such that </b><br/><a NAME="E2:2_5"/><i><font color="Green" title="E16">A22</font></i>: 
 <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c7">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a</font>
 <b>by </b><i><a class="txt" href="henmodel.html#E23:2"><i><font color="Green" title="E13">A17</font></i></a>, <a class="ref" href="ordinal1.html#T42" title="ORDINAL1:th.42">ORDINAL1:42</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">b</font> = <font color="Maroon" title="c7">b</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="henmodel.html#E26:2"><i><font color="Green" title="E14">A20</font></i></a>, <a class="txt" href="henmodel.html#E2:2_5"><i><font color="Green" title="E16">A22</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c8">b</font>
;<br/>

<b>thus </b><a NAME="E4:2_5"/>
 <a href="card_1.html#K4" title="CARD_1:func.4">succ</a> <font color="Maroon" title="c8">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a</font>
 <b>by </b><i><a class="txt" href="henmodel.html#E2:2_5"><i><font color="Green" title="E16">A22</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E28:2"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c6">a</font></span>)</span>
 
<b>by </b><i><a class="txt" href="henmodel.html#E6:2"><i><font color="Green" title="E4">A2</font></i></a>, <a class="ref" href="wellord1.html#D7" title="WELLORD1:def.7">WELLORD1:def 7</a></i>;<br/>
<a NAME="E29:2"/><b>then </b><i><font color="Green" title="E16">A23</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a</font>
 
<b>by </b><i><a class="ref" href="wellord2.html#D1" title="WELLORD2:def.1">WELLORD2:def 1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c7">F</font> = <font color="Maroon" title="c3">F</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c1">A</font> <b>by </b><i><a class="txt" href="henmodel.html#E9:2"><i><font color="Green" title="E7">A11</font></i></a>, <a class="ref" href="funct_2.html#T3" title="FUNCT_2:th.3">FUNCT_2:3</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">m</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E32:2"/><i><font color="Green" title="E17">A24</font></i>: 
 <a href="card_1.html#K4" title="CARD_1:func.4">succ</a> <font color="Maroon" title="c8">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a</font>
 <b>by </b><i><a class="txt" href="henmodel.html#E27:2"><i><font color="Green" title="E15">A21</font></i></a></i>;<br/>
<a NAME="E33:2"/>
(  ex <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st  <a href="card_1.html#K4" title="CARD_1:func.4">succ</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a</font> &amp; <font color="Maroon" title="c7">F</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c7">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">A</font> &amp; ( for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b2">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">F</font> &amp; <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">F</font> &amp; <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">m</font> holds <br/><font color="Maroon" title="c7">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">m</font> ) )
 
<b>by </b><i><a class="txt" href="henmodel.html#E6:2"><i><font color="Green" title="E4">A2</font></i></a>, <a class="txt" href="henmodel.html#E7:2"><i><font color="Green" title="E5">A3</font></i></a>, <a class="txt" href="henmodel.html#E8:2"><i><font color="Green" title="E6">A10</font></i></a>, <a class="txt" href="henmodel.html#E27:2"><i><font color="Green" title="E15">A21</font></i></a>, <a class="ref" href="wellord1.html#D7" title="WELLORD1:def.7">WELLORD1:def 7</a>, <a class="ref" href="wellord2.html#D1" title="WELLORD2:def.1">WELLORD2:def 1</a></i>;<br/>
<a NAME="E34:2"/><b>then </b><i><font color="Green" title="E18">A25</font></i>: 
<font color="Maroon" title="c7">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c6">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c7">F</font></span>)</span>
 
<b>by </b><i><a class="ref" href="henmodel.html#T1" target="_self" title="HENMODEL:th.1">Th1</a></i>;<br/>
<a NAME="E35:2"/><i><font color="Green" title="E19">A26</font></i>: 
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">m</font>
 
<b>by </b><i><a class="txt" href="henmodel.html#E32:2"><i><font color="Green" title="E17">A24</font></i></a>, <a class="ref" href="ordinal2.html#T2" title="ORDINAL2:th.2">ORDINAL2:2</a></i>;<br/>
<b>hence </b><a NAME="E36:2"/>
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c1">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="txt" href="henmodel.html#E9:2"><i><font color="Green" title="E7">A11</font></i></a>, <a class="txt" href="henmodel.html#E29:2"><i><font color="Green" title="E16">A23</font></i></a>, <a class="txt" href="henmodel.html#E32:2"><i><font color="Green" title="E17">A24</font></i></a>, <a class="txt" href="henmodel.html#E34:2"><i><font color="Green" title="E18">A25</font></i></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a>, <a class="ref" href="ordinal1.html#T10" title="ORDINAL1:th.10">ORDINAL1:10</a></i>;<br/>

<b>thus </b><a NAME="E37:2"/>
for <font color="Olive" title="b1">b</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> or <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c1">A</font> or <font color="Olive" title="b1">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c1">A</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">b</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:2_6"/><i><font color="Green" title="E20">A27</font></i>: 
<font color="Maroon" title="c9">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_6_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:2_6_1"/><i><font color="Green" title="E21">A28</font></i>: 
<font color="Maroon" title="c9">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c1">A</font>
 ;<br/><b>consider </b><font color="Maroon" title="c10">c</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:2_6_1"/><i><font color="Green" title="E22">A29</font></i>: 
( <font color="Maroon" title="c10">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">F</font> &amp; <font color="Maroon" title="c7">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">b</font> )
 <b>by </b><i><a class="txt" href="henmodel.html#E9:2"><i><font color="Green" title="E7">A11</font></i></a>, <a class="txt" href="henmodel.html#E1:2_6"><i><font color="Green" title="E20">A27</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E4:2_6_1"/>
<font color="Maroon" title="c10">c</font> is   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>
 <b>by </b><i><a class="txt" href="henmodel.html#E3:2_6_1"><i><font color="Green" title="E22">A29</font></i></a>, <a class="ref" href="ordinal1.html#T23" title="ORDINAL1:th.23">ORDINAL1:23</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c11">c</font> = <font color="Maroon" title="c10">c</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="henmodel.html#E3:2_6_1"><i><font color="Green" title="E22">A29</font></i></a>, <a class="ref" href="ordinal1.html#T19" title="ORDINAL1:th.19">ORDINAL1:19</a></i>;<br/><a NAME="E6:2_6_1"/><i><font color="Green" title="E23">A30</font></i>: 
( <font color="Maroon" title="c11">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">m</font> or <font color="Maroon" title="c11">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">m</font> )
 <b>by </b><i><a class="txt" href="henmodel.html#E32:2"><i><font color="Green" title="E17">A24</font></i></a>, <a class="txt" href="henmodel.html#E3:2_6_1"><i><font color="Green" title="E22">A29</font></i></a>, <a class="ref" href="ordinal1.html#T13" title="ORDINAL1:th.13">ORDINAL1:13</a></i>;<br/><a NAME="E7:2_6_1"/><i><font color="Green" title="E24">A31</font></i>: 
<font color="Maroon" title="c8">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">F</font>
 <b>by </b><i><a class="txt" href="henmodel.html#E29:2"><i><font color="Green" title="E16">A23</font></i></a>, <a class="txt" href="henmodel.html#E32:2"><i><font color="Green" title="E17">A24</font></i></a>, <a class="ref" href="ordinal1.html#T10" title="ORDINAL1:th.10">ORDINAL1:10</a></i>;<br/><a NAME="E8:2_6_1"/>
<font color="Maroon" title="c11">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">m</font> </span>}</span> 
 <b>by </b><i><a class="txt" href="henmodel.html#E9:2"><i><font color="Green" title="E7">A11</font></i></a>, <a class="txt" href="henmodel.html#E32:2"><i><font color="Green" title="E17">A24</font></i></a>, <a class="txt" href="henmodel.html#E34:2"><i><font color="Green" title="E18">A25</font></i></a>, <a class="txt" href="henmodel.html#E1:2_6_1"><i><font color="Green" title="E21">A28</font></i></a>, <a class="txt" href="henmodel.html#E3:2_6_1"><i><font color="Green" title="E22">A29</font></i></a>, <a class="txt" href="henmodel.html#E6:2_6_1"><i><font color="Green" title="E23">A30</font></i></a>, <a class="ref" href="axioms.html#T30" title="AXIOMS:th.30">AXIOMS:30</a>, <a class="ref" href="ordinal2.html#T2" title="ORDINAL2:th.2">ORDINAL2:2</a></i>;<br/><a NAME="E9:2_6_1"/><b>then </b>
 ex <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">c</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">m</font> )
 ;<br/><b>hence </b><a NAME="E10:2_6_1"/>
( <font color="Maroon" title="c9">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c1">A</font> or <font color="Maroon" title="c9">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c1">A</font> )
 <b>by </b><i><a class="txt" href="henmodel.html#E7:2"><i><font color="Green" title="E5">A3</font></i></a>, <a class="txt" href="henmodel.html#E9:2"><i><font color="Green" title="E7">A11</font></i></a>, <a class="txt" href="henmodel.html#E34:2"><i><font color="Green" title="E18">A25</font></i></a>, <a class="txt" href="henmodel.html#E35:2"><i><font color="Green" title="E19">A26</font></i></a>, <a class="txt" href="henmodel.html#E3:2_6_1"><i><font color="Green" title="E22">A29</font></i></a>, <a class="txt" href="henmodel.html#E7:2_6_1"><i><font color="Green" title="E24">A31</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:2_6"/>
( <font color="Maroon" title="c9">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c1">A</font> or <font color="Maroon" title="c9">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c1">A</font> )
 ;<br/>


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


</div>
