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

<b>let </b><font color="Maroon">A</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K5">NAT</a> ;<br/>

<a NAME="E1:2"/>
 <a href="wellord2.html#K1">RelIncl</a> <font color="Maroon">A</font> is <a href="wellord1.html#V2">well-ordering</a>
 
<b>by </b><i><a class="ref" href="wellord2.html#T9">WELLORD2:9</a></i>;<br/>
<b>then consider </b><font color="Maroon">a</font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E3:2"/><i><font color="Green">E47</font></i>: 
 <a href="wellord2.html#K1">RelIncl</a> <font color="Maroon">A</font>, <a href="wellord2.html#K1">RelIncl</a> <font color="Maroon">a</font> <a href="wellord1.html#R4">are_isomorphic</a> 
 <b>by </b><i><a class="ref" href="wellord2.html#T14">WELLORD2:14</a></i>;<br/>
<a NAME="E4:2"/>
 <a href="wellord2.html#K1">RelIncl</a> <font color="Maroon">a</font>, <a href="wellord2.html#K1">RelIncl</a> <font color="Maroon">A</font> <a href="wellord1.html#R4">are_isomorphic</a> 
 
<b>by </b><i>, <a class="ref" href="wellord1.html#T50">WELLORD1:50</a></i>;<br/>
<b>then consider </b><font color="Maroon">F</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E6:2"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">F</font> <a href="wellord1.html#R3">is_isomorphism_of</a>  <a href="wellord2.html#K1">RelIncl</a> <font color="Maroon">a</font>, <a href="wellord2.html#K1">RelIncl</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="wellord1.html#D8">WELLORD1:def 8</a></i>;<br/>
<a NAME="E7:2"/><i><font color="Green">E49</font></i>: 
for <font color="Olive">m</font>, <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">m</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font> &amp; <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font> &amp; <font color="Olive">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">m</font> holds <br/><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">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">m</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>



<b>assume </b><a NAME="E1:2_1"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font> &amp; <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font> )
 ;<br/>

<a NAME="E2:2_1"/>
( <font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><a href="wellord2.html#K1">RelIncl</a> <font color="Maroon">a</font></span>)</span> &amp; <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><a href="wellord2.html#K1">RelIncl</a> <font color="Maroon">a</font></span>)</span> )
 
<b>by </b><i>, , <a class="ref" href="wellord1.html#D7">WELLORD1:def 7</a></i>;<br/>
<a NAME="E3:2_1"/><b>then </b><i><font color="Green">E51</font></i>: 
( <font color="Maroon">m</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font> &amp; <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font> )
 
<b>by </b><i><a class="ref" href="wellord2.html#D1">WELLORD2:def 1</a></i>;<br/>
<a NAME="E4:2_1"/>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">i</font> where <font color="Olive">i</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font> </span>}</span> 
 
<b>by </b><i/>;<br/>
<a NAME="E5:2_1"/><b>then </b>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">m</font>
 
<b>by </b><i><a class="ref" href="axioms.html#T30">AXIOMS:30</a></i>;<br/>
<a NAME="E6:2_1"/><b>then </b>
( <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">m</font> &amp; <font color="Maroon">n</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">m</font> )
 
<b>by </b><i><a class="ref" href="ordinal1.html#D2">ORDINAL1:def 2</a></i>;<br/>
<a NAME="E7:2_1"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">n</font>,<font color="Maroon">m</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="wellord2.html#K1">RelIncl</a> <font color="Maroon">a</font>
 
<b>by </b><i>, <a class="ref" href="wellord2.html#D1">WELLORD2:def 1</a></i>;<br/>
<a NAME="E8:2_1"/><b>then </b><i><font color="Green">E52</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="wellord2.html#K1">RelIncl</a> <font color="Maroon">A</font>
 
<b>by </b><i>, <a class="ref" href="wellord1.html#D7">WELLORD1:def 7</a></i>;<br/>
<a NAME="E9:2_1"/>
( <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font> &amp; <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font> )
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/>
<a NAME="E10:2_1"/><b>then </b>
( <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><a href="wellord2.html#K1">RelIncl</a> <font color="Maroon">A</font></span>)</span> &amp; <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><a href="wellord2.html#K1">RelIncl</a> <font color="Maroon">A</font></span>)</span> )
 
<b>by </b><i>, <a class="ref" href="wellord1.html#D7">WELLORD1:def 7</a></i>;<br/>
<a NAME="E11:2_1"/><b>then </b><i><font color="Green">E53</font></i>: 
( <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 
<b>by </b><i><a class="ref" href="wellord2.html#D1">WELLORD2:def 1</a></i>;<br/>
<a NAME="E12:2_1"/><b>then </b><i><font color="Green">E54</font></i>: 
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font>
 
<b>by </b><i>, <a class="ref" href="wellord2.html#D1">WELLORD2:def 1</a></i>;<br/>
<a NAME="E13:2_1"/>
<font color="Maroon">F</font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i>, <a class="ref" href="wellord1.html#D7">WELLORD1:def 7</a></i>;<br/>
<a NAME="E14:2_1"/><b>then </b>
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/>
<a NAME="E15:2_1"/><b>then </b><i><font color="Green">E55</font></i>: 
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="xboole_0.html#R2">c&lt;</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D8">XBOOLE_0:def 8</a></i>;<br/>
<b>reconsider </b><font color="Maroon">b</font> = <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">c</font> = <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/>
<a NAME="E18:2_1"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">c</font>
 
<b>by </b><i>, <a class="ref" href="ordinal1.html#T21">ORDINAL1:21</a></i>;<br/>
<b>hence </b><a NAME="E19:2_1"/>
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E8:2"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><a href="wellord2.html#K1">RelIncl</a> <font color="Maroon">A</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="wellord1.html#D7">WELLORD1:def 7</a></i>;<br/>
<a NAME="E9:2"/><b>then </b><i><font color="Green">E56</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="wellord2.html#D1">WELLORD2:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon">F1</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E11:2"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">F1</font> <a href="wellord1.html#R3">is_isomorphism_of</a>  <a href="wellord2.html#K1">RelIncl</a> <font color="Maroon">A</font>, <a href="wellord2.html#K1">RelIncl</a> <font color="Maroon">a</font>
 <b>by </b><i>, <a class="ref" href="wellord1.html#D8">WELLORD1:def 8</a></i>;<br/>
<a NAME="E12:2"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">F1</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><a href="wellord2.html#K1">RelIncl</a> <font color="Maroon">A</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="wellord1.html#D7">WELLORD1:def 7</a></i>;<br/>
<a NAME="E13:2"/><b>then </b><i><font color="Green">E58</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">F1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="wellord2.html#D1">WELLORD2:def 1</a></i>;<br/>
<a NAME="E14:2"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">F1</font> is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i><a class="ref" href="finset_1.html#T26">FINSET_1:26</a></i>;<br/>
<a NAME="E15:2"/><b>then </b>
 <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><a href="wellord2.html#K1">RelIncl</a> <font color="Maroon">a</font></span>)</span> is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i>, <a class="ref" href="wellord1.html#D7">WELLORD1:def 7</a></i>;<br/>
<a NAME="E16:2"/><b>then </b><i><font color="Green">E59</font></i>: 
<font color="Maroon">a</font> is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i><a class="ref" href="wellord2.html#D1">WELLORD2:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon">b</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E18:2"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F1</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<a NAME="E19:2"/>
<font color="Maroon">F1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F1</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/>
<a NAME="E20:2"/><b>then </b>
not  <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><a href="wellord2.html#K1">RelIncl</a> <font color="Maroon">a</font></span>)</span> is <a href="xboole_0.html#V1">empty</a>
 
<b>by </b><i>, <a class="ref" href="wellord1.html#D7">WELLORD1:def 7</a></i>;<br/>
<a NAME="E21:2"/><b>then </b>
not <font color="Maroon">a</font> is <a href="xboole_0.html#V1">empty</a>
 
<b>by </b><i><a class="ref" href="wellord2.html#D1">WELLORD2:def 1</a></i>;<br/>
<a NAME="E22:2"/><b>then </b>
not <font color="Maroon">a</font> <a href="ordinal1.html#R1">c=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/>
<a NAME="E23:2"/><b>then </b><i><font color="Green">E66</font></i>: 
 <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">a</font>
 
<b>by </b><i><a class="ref" href="ordinal1.html#T26">ORDINAL1:26</a></i>;<br/>
<div><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">a</font> is <a href="ordinal1.html#V4">being_limit_ordinal</a>
 ;<br/><a NAME="E2:2_2"/><b>then </b><i><font color="Green">E67</font></i>: 
 <a href="ordinal1.html#K5">omega</a>  <a href="ordinal1.html#R1">c=</a> <font color="Maroon">a</font>
 <b>by </b><i>, <a class="ref" href="ordinal2.html#T19">ORDINAL2:19</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">omega</a>  is <a href="finset_1.html#V1">finite</a>
 ;<br/><a NAME="E2:2_2_1"/><b>then </b>
 <a href="card_1.html#K1">Card</a> <a href="ordinal1.html#K5">omega</a>  <a href="card_1.html#NR2" target="_self">&lt;`</a>  <a href="card_1.html#K1">Card</a> <a href="ordinal1.html#K5">omega</a> 
 <b>by </b><i><a class="ref" href="card_3.html#T58">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="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/></div><b>end;</b></div>
<a NAME="E25:2"/><b>then </b><i><font color="Green">E68</font></i>: 
 ex <font color="Olive">b</font> being  <a href="ordinal1.html#NM2">Ordinal</a> st  <a href="ordinal1.html#K1">succ</a> <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 
<b>by </b><i><a class="ref" href="ordinal1.html#T42">ORDINAL1:42</a></i>;<br/>
<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"/>
not <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K5">omega</a> 
 ;<br/><a NAME="E2:2_3"/><b>then </b><i><font color="Green">E69</font></i>: 
 <a href="ordinal1.html#K5">omega</a>  <a href="ordinal1.html#R1">c=</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="ordinal1.html#T26">ORDINAL1:26</a></i>;<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">omega</a>  is <a href="finset_1.html#V1">finite</a>
 ;<br/><a NAME="E2:2_3_1"/><b>then </b>
 <a href="card_1.html#K1">Card</a> <a href="ordinal1.html#K5">omega</a>  <a href="card_1.html#NR2" target="_self">&lt;`</a>  <a href="card_1.html#K1">Card</a> <a href="ordinal1.html#K5">omega</a> 
 <b>by </b><i><a class="ref" href="card_3.html#T58">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="E4:2_3"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">a</font> = <font color="Maroon">a</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/>
<div><i><font color="Green">E70</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">b</font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/><b>assume </b><a NAME="E1:2_4"/>
 <a href="ordinal1.html#K1">succ</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 ;<br/><a NAME="E2:2_4"/><b>then </b>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="ordinal1.html#T10">ORDINAL1:10</a></i>;<br/><b>hence </b><a NAME="E3:2_4"/>
<font color="Maroon">b</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i><a class="ref" href="ordinal1.html#T19">ORDINAL1:19</a></i>;<br/></div><b>end;</b></div>
<a NAME="E29:2"/><i><font color="Green">E71</font></i>: 
 ex <font color="Olive">m</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st  <a href="ordinal1.html#K1">succ</a> <font color="Olive">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">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">b</font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E2:2_5"/><i><font color="Green">E72</font></i>: 
 <a href="ordinal1.html#K1">succ</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">b</font> = <font color="Maroon">b</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i>, </i>;<br/>
<b>take </b>
<font color="Maroon">b</font>
;<br/>

<b>thus </b><a NAME="E4:2_5"/>
 <a href="ordinal1.html#K1">succ</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E30:2"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><a href="wellord2.html#K1">RelIncl</a> <font color="Maroon">a</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="wellord1.html#D7">WELLORD1:def 7</a></i>;<br/>
<a NAME="E31:2"/><b>then </b><i><font color="Green">E73</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 
<b>by </b><i><a class="ref" href="wellord2.html#D1">WELLORD2:def 1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">F</font> = <font color="Maroon">F</font> as   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">a</font>,<font color="Maroon">A</font> <b>by </b><i>, <a class="ref" href="funct_2.html#T3">FUNCT_2:3</a></i>;<br/>
<b>consider </b><font color="Maroon">m</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E34:2"/><i><font color="Green">E74</font></i>: 
 <a href="ordinal1.html#K1">succ</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 <b>by </b><i/>;<br/>
<a NAME="E35:2"/>
(  ex <font color="Olive">m</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st  <a href="ordinal1.html#K1">succ</a> <font color="Olive">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">F</font> is <a href="funct_1.html#V2">one-to-one</a> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> &amp; ( for <font color="Olive">n</font>, <font color="Olive">m</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">m</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font> &amp; <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font> &amp; <font color="Olive">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">m</font> holds <br/><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">m</font> ) )
 
<b>by </b><i>, , , , <a class="ref" href="wellord1.html#D7">WELLORD1:def 7</a></i>;<br/>
<a NAME="E36:2"/><b>then </b><i><font color="Green">E75</font></i>: 
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">a</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">F</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E37:2"/><i><font color="Green">E76</font></i>: 
 <a href="tarski.html#K3">union</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font>
 
<b>by </b><i>, <a class="ref" href="ordinal2.html#T2">ORDINAL2:2</a></i>;<br/>
<a NAME="E38:2"/><b>then </b>
 <a href="tarski.html#K3">union</a> <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font>
 
<b>by </b><i>, , <a class="ref" href="ordinal1.html#T10">ORDINAL1:10</a></i>;<br/>
<b>hence </b><a NAME="E39:2"/>
 <a href="tarski.html#K3">union</a> <font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/>

<b>thus </b><a NAME="E40:2"/>
for <font color="Olive">b</font> being   <a href="hidden.html#M1">set</a>   holds <br/>(  not <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> or <font color="Olive">b</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">A</font> or <font color="Olive">b</font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">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">b</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:2_6"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">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">E78</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">A</font>
 ;<br/><b>consider </b><font color="Maroon">c</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:2_6_1"/><i><font color="Green">E79</font></i>: 
( <font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font> &amp; <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> )
 <b>by </b><i>, , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E4:2_6_1"/>
<font color="Maroon">c</font> is   <a href="ordinal1.html#NM2">Ordinal</a>
 <b>by </b><i>, <a class="ref" href="ordinal1.html#T23">ORDINAL1:23</a></i>;<br/><b>then reconsider </b><font color="Maroon">c</font> = <font color="Maroon">c</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i>, <a class="ref" href="ordinal1.html#T19">ORDINAL1:19</a></i>;<br/><a NAME="E6:2_6_1"/><i><font color="Green">E80</font></i>: 
( <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">m</font> or <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> )
 <b>by </b><i>, , <a class="ref" href="ordinal1.html#T13">ORDINAL1:13</a></i>;<br/><a NAME="E7:2_6_1"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font>
 <b>by </b><i>, , <a class="ref" href="ordinal1.html#T10">ORDINAL1:10</a></i>;<br/><a NAME="E8:2_6_1"/>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">k</font> where <font color="Olive">k</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font> </span>}</span> 
 <b>by </b><i>, , , , , , <a class="ref" href="axioms.html#T30">AXIOMS:30</a>, <a class="ref" href="ordinal2.html#T2">ORDINAL2:2</a></i>;<br/><a NAME="E9:2_6_1"/><b>then </b>
 ex <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font> )
 ;<br/><b>hence </b><a NAME="E10:2_6_1"/>
( <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">A</font> or <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">A</font> )
 <b>by </b><i>, , , , , </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:2_6"/>
( <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">A</font> or <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">A</font> )
 ;<br/>


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


</div>
