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

<b>consider </b><font color="Maroon">c<sub>2</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>
<a NAME="E2:41_1_1"/><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="tarski.html#K1">}</a></span>, <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="wellord2.html#R2">are_equipotent</a> 
 
<b>by </b><i><a class="ref" href="card_1.html#D5">CARD_1:def 5</a></i>;<br/>
<a NAME="E3:41_1_1"/><i><font color="Green">E15</font></i>: 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="tarski.html#K1">}</a></span>,<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K1">Card</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K6">\</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="wellord2.html#R2">are_equipotent</a> 
 
<b>by </b><i><a class="ref" href="card_1.html#T48">CARD_1:48</a></i>;<br/>
<a NAME="E4:41_1_1"/><i><font color="Green">E16</font></i>: 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T79">XBOOLE_1:79</a></i>;<br/>
<a NAME="E5:41_1_1"/>
not  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span>
 
;<br/>
<a NAME="E6:41_1_1"/><b>then </b><i><font color="Green">E17</font></i>: 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K1">Card</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K6">\</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#R1">misses</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T56">ZFMISC_1:56</a></i>;<br/>
<a NAME="E7:41_1_1"/><i><font color="Green">E18</font></i>: 
(  <a href="ordinal1.html#K1">succ</a> <span class="p1">(<span class="default"><a href="card_1.html#K1">Card</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K1">Card</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K1">Card</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K6">\</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i><a class="txt" href="yellow21.html#E15"><i><font color="Green">Lemma11</font></i></a>, <a class="ref" href="ordinal1.html#D1">ORDINAL1:def 1</a></i>;<br/>
<a NAME="E8:41_1_1"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font>, <a href="ordinal1.html#K1">succ</a> <span class="p1">(<span class="default"><a href="card_1.html#K1">Card</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="wellord2.html#R2">are_equipotent</a> 
 
<b>by </b><i><a class="txt" href="yellow21.html#E2:41_1_1"><i><font color="Green">E14</font></i></a>, <a class="txt" href="yellow21.html#E3:41_1_1"><i><font color="Green">E15</font></i></a>, <a class="txt" href="yellow21.html#E4:41_1_1"><i><font color="Green">E16</font></i></a>, <a class="txt" href="yellow21.html#E6:41_1_1"><i><font color="Green">E17</font></i></a>, <a class="ref" href="card_1.html#T58">CARD_1:58</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>3</sub></font> being   <a href="orders_1.html#NM2">Order</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E10:41_1_1"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="wellord1.html#R2">well_orders</a> <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E11:41_1_1"/><i><font color="Green">E20</font></i>: 
 <a href="wellord2.html#K2">order_type_of</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="ordinal1.html#K1">succ</a> <span class="p1">(<span class="default"><a href="card_1.html#K1">Card</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="yellow21.html#T7" target="_self">Th7</a></i>;<br/>
<a NAME="E12:41_1_1"/><i><font color="Green">E21</font></i>: 
 <a href="relat_1.html#K3">field</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#T100">ORDERS_1:100</a></i>;<br/>
<a NAME="E13:41_1_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> is <a href="wellord1.html#V2">well-ordering</a>
 
<b>by </b><i><a class="txt" href="yellow21.html#E10:41_1_1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="wellord1.html#T8">WELLORD1:8</a></i>;<br/>
<a NAME="E14:41_1_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font>, <a href="yellow_1.html#K1">RelIncl</a> <span class="p1">(<span class="default"><a href="wellord2.html#K2">order_type_of</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="wellord1.html#R4">are_isomorphic</a> 
 
<b>by </b><i><a class="ref" href="wellord2.html#D2">WELLORD2:def 2</a></i>;<br/>
<a NAME="E15:41_1_1"/><b>then </b>
 <a href="yellow_1.html#K1">RelIncl</a> <span class="p1">(<span class="default"><a href="wellord2.html#K2">order_type_of</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></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">c<sub>4</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E17:41_1_1"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="wellord1.html#R3">is_isomorphism_of</a>  <a href="yellow_1.html#K1">RelIncl</a> <span class="p1">(<span class="default"><a href="wellord2.html#K2">order_type_of</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="wellord1.html#D8">WELLORD1:def 8</a></i>;<br/>
<a NAME="E18:41_1_1"/>
 <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><a href="yellow_1.html#K1">RelIncl</a> <span class="p2">(<span class="default"><a href="wellord2.html#K2">order_type_of</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="wellord2.html#K2">order_type_of</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="wellord2.html#D1">WELLORD2:def 1</a></i>;<br/>
<a NAME="E19:41_1_1"/><b>then </b><i><font color="Green">E23</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> is <a href="funct_1.html#V2">one-to-one</a> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="wellord2.html#K2">order_type_of</a> <font color="Maroon">c<sub>3</sub></font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="txt" href="yellow21.html#E12:41_1_1"><i><font color="Green">E21</font></i></a>, <a class="txt" href="yellow21.html#E17:41_1_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="wellord1.html#D7">WELLORD1:def 7</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>3</sub></font>
;<br/>

<b>thus </b><a NAME="E20:41_1_1"/>
<font color="Maroon">c<sub>3</sub></font> is <a href="yellow21.html#V5" target="_self">upper-bounded</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_1_1"><b>proof </b></a><div class="add">

<b>take </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="card_1.html#K1">Card</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow21.html#D9" target="_self">YELLOW21:def 9</a><br/>

<b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:41_1_1_1"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<a NAME="E2:41_1_1_1"/><b>then </b><i><font color="Green">E25</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="wellord2.html#K2">order_type_of</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="yellow21.html#E12:41_1_1"><i><font color="Green">E21</font></i></a>, <a class="txt" href="yellow21.html#E19:41_1_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="funct_1.html#T54">FUNCT_1:54</a></i>;<br/>
<a NAME="E3:41_1_1_1"/><b>then </b><i><font color="Green">E26</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span> or <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span> )
 
<b>by </b><i><a class="txt" href="yellow21.html#E7:41_1_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="yellow21.html#E11:41_1_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="setwiseo.html#T6">SETWISEO:6</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> as   <a href="ordinal1.html#NM2">Ordinal</a> <b>by </b><i><a class="txt" href="yellow21.html#E2:41_1_1_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="ordinal1.html#T23">ORDINAL1:23</a></i>;<br/>
<a NAME="E5:41_1_1_1"/>
( <font color="Maroon">c<sub>7</sub></font> <a href="ordinal1.html#R1">c=</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span> &amp;  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="wellord2.html#K2">order_type_of</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="txt" href="yellow21.html#E7:41_1_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="yellow21.html#E11:41_1_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="yellow21.html#E3:41_1_1_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="ordinal1.html#D2">ORDINAL1:def 2</a>, <a class="ref" href="setwiseo.html#T6">SETWISEO:6</a></i>;<br/>
<a NAME="E6:41_1_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<span class="p2">(<span class="default"><a href="card_1.html#K1">Card</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K6">\</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="yellow_1.html#K1">RelIncl</a> <span class="p1">(<span class="default"><a href="wellord2.html#K2">order_type_of</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="yellow21.html#E2:41_1_1_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="wellord2.html#D1">WELLORD2:def 1</a></i>;<br/>
<a NAME="E7:41_1_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="yellow21.html#E17:41_1_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="wellord1.html#D7">WELLORD1:def 7</a></i>;<br/>
<b>hence </b><a NAME="E8:41_1_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="yellow21.html#E12:41_1_1"><i><font color="Green">E21</font></i></a>, <a class="txt" href="yellow21.html#E19:41_1_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="yellow21.html#E1:41_1_1_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="funct_1.html#T57">FUNCT_1:57</a></i>;<br/>


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

<b>thus </b><a NAME="E21:41_1_1"/>
<font color="Maroon">c<sub>3</sub></font> is <a href="wellord1.html#V2">well-ordering</a>
 <b>by </b><i><a class="txt" href="yellow21.html#E10:41_1_1"><i><font color="Green">E19</font></i></a>, <a class="txt" href="yellow21.html#E12:41_1_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="wellord1.html#T8">WELLORD1:8</a></i>;<br/>


</div>
