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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/>



<b>given </b><font color="Maroon">c<sub>3</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><a NAME="E1:40"/><i><font color="Green">E14</font></i>: 
( <font color="Maroon">c<sub>3</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>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="wellord2.html#D4">WELLORD2:def 4</a><br/>

<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> as   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="yellow21.html#E1:40"><i><font color="Green">E14</font></i></a>, <a class="ref" href="funct_2.html#T4">FUNCT_2:4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> as  <a href="funct_1.html#V2">one-to-one</a> <a href="funct_1.html#NM1">Function</a> <b>by </b><i><a class="txt" href="yellow21.html#E1:40"><i><font color="Green">E14</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K2">"</a>  as   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="yellow21.html#E1:40"><i><font color="Green">E14</font></i></a>, <a class="ref" href="funct_2.html#T31">FUNCT_2:31</a></i>;<br/>
<a NAME="E5:40"/><i><font color="Green">E15</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>6</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#E1:40"><i><font color="Green">E14</font></i></a>, <a class="ref" href="funct_1.html#T55">FUNCT_1:55</a></i>;<br/>
<b>set </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="relset_1.html#K7">*</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K1">RelIncl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="relset_1.html#K7">*</a> <font color="Maroon">c<sub>6</sub></font>;<br/>
<a NAME="E6:40"/>
(  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="yellow_1.html#K1">RelIncl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="yellow_1.html#K1">RelIncl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="ref" href="orders_1.html#T99">ORDERS_1:99</a></i>;<br/>
<a NAME="E7:40"/><b>then </b>
(  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K7">*</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K1">RelIncl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K7">*</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K1">RelIncl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="txt" href="yellow21.html#E1:40"><i><font color="Green">E14</font></i></a>, <a class="ref" href="relat_1.html#T46">RELAT_1:46</a>, <a class="ref" href="relat_1.html#T47">RELAT_1:47</a></i>;<br/>
<a NAME="E8:40"/><b>then </b><i><font color="Green">E16</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K7">*</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K1">RelIncl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="relset_1.html#K7">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K7">*</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K1">RelIncl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="relset_1.html#K7">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="txt" href="yellow21.html#E5:40"><i><font color="Green">E15</font></i></a>, <a class="ref" href="relat_1.html#T46">RELAT_1:46</a>, <a class="ref" href="relat_1.html#T47">RELAT_1:47</a></i>;<br/>
<b>then </b><i><font color="Green">E17</font></i>:  <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K7">*</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K1">RelIncl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="relset_1.html#K7">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> = 
<font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="ref" href="relat_1.html#D6">RELAT_1:def 6</a></i>
<br/>.= 
<font color="Maroon">c<sub>1</sub></font>

;<br/>
<a NAME="E10:40"/><b>then </b><i><font color="Green">E18</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="relat_1.html#K5">*</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K1">RelIncl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="relat_1.html#K5">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="relat_2.html#R1">is_reflexive_in</a> <font color="Maroon">c<sub>1</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="relat_1.html#K5">*</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K1">RelIncl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="relat_1.html#K5">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="relat_2.html#R4">is_antisymmetric_in</a> <font color="Maroon">c<sub>1</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="relat_1.html#K5">*</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K1">RelIncl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="relat_1.html#K5">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="relat_2.html#R8">is_transitive_in</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="ref" href="relat_2.html#D9">RELAT_2:def 9</a>, <a class="ref" href="relat_2.html#D12">RELAT_2:def 12</a>, <a class="ref" href="relat_2.html#D16">RELAT_2:def 16</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K7">*</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K1">RelIncl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="relset_1.html#K7">*</a> <font color="Maroon">c<sub>6</sub></font> as   <a href="relset_1.html#NM3">Relation</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>8</sub></font> as   <a href="orders_1.html#NM2">Order</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="yellow21.html#E8:40"><i><font color="Green">E16</font></i></a>, <a class="txt" href="yellow21.html#E10:40"><i><font color="Green">E18</font></i></a>, <a class="ref" href="partfun1.html#D4">PARTFUN1:def 4</a></i>;<br/>
<a NAME="E13:40"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="wellord1.html#R3">is_isomorphism_of</a> <font color="Maroon">c<sub>9</sub></font>, <a href="yellow_1.html#K1">RelIncl</a> <font color="Maroon">c<sub>2</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:40_2"/>
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">c<sub>9</sub></font> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><a href="yellow_1.html#K1">RelIncl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>4</sub></font> is <a href="funct_1.html#V2">one-to-one</a> )
 <b>by </b><i><a class="txt" href="yellow21.html#E1:40"><i><font color="Green">E14</font></i></a>, <a class="txt" href="yellow21.html#E9:40"><i><font color="Green">E17</font></i></a>, <a class="ref" href="wellord2.html#D1">WELLORD2:def 1</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="wellord1.html#D7">WELLORD1:def 7</a><br/>

<b>let </b><font color="Maroon">c<sub>10</sub></font>, <font color="Maroon">c<sub>11</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:40_2_1"/><i><font color="Green">E20</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/><b>hence </b><a NAME="E2:40_2_1"/>
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">c<sub>9</sub></font> )
 <b>by </b><i><a class="ref" href="relat_1.html#T30">RELAT_1:30</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>12</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:40_2_1"/><i><font color="Green">E21</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K7">*</a> <span class="p1">(<span class="default"><a href="yellow_1.html#K1">RelIncl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="yellow21.html#E1:40_2_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="relat_1.html#D8">RELAT_1:def 8</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>13</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:40_2_1"/><i><font color="Green">E22</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>13</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="yellow_1.html#K1">RelIncl</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="yellow21.html#E4:40_2_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="relat_1.html#D8">RELAT_1:def 8</a></i>;<br/><a NAME="E7:40_2_1"/>
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="yellow21.html#E4:40_2_1"><i><font color="Green">E21</font></i></a>, <a class="txt" href="yellow21.html#E6:40_2_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/><b>hence </b><a NAME="E8:40_2_1"/>
<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>10</sub></font></span>)</span>,<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>11</sub></font></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> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="yellow21.html#E1:40"><i><font color="Green">E14</font></i></a>, <a class="txt" href="yellow21.html#E6:40_2_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="funct_1.html#T57">FUNCT_1:57</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E3:40_2"/><i><font color="Green">E20</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">c<sub>9</sub></font> &amp; <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>10</sub></font></span>)</span>,<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>11</sub></font></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> <font color="Maroon">c<sub>2</sub></font> )
 ;<br/>

<a NAME="E4:40_2"/><b>then </b>
( <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> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="yellow21.html#E1:40"><i><font color="Green">E14</font></i></a>, <a class="txt" href="yellow21.html#E9:40"><i><font color="Green">E17</font></i></a>, <a class="ref" href="funct_1.html#T56">FUNCT_1:56</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E5:40_2"/><b>then </b><i><font color="Green">E21</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>10</sub></font>,<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>10</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> &amp; <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>11</sub></font></span>)</span>,<font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> )
 
<b>by </b><i><a class="txt" href="yellow21.html#E1:40"><i><font color="Green">E14</font></i></a>, <a class="txt" href="yellow21.html#E5:40"><i><font color="Green">E15</font></i></a>, <a class="txt" href="yellow21.html#E9:40"><i><font color="Green">E17</font></i></a>, <a class="txt" href="yellow21.html#E3:40_2"><i><font color="Green">E20</font></i></a>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/>
<a NAME="E6:40_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>10</sub></font>,<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>11</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K7">*</a> <span class="p1">(<span class="default"><a href="yellow_1.html#K1">RelIncl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="yellow21.html#E3:40_2"><i><font color="Green">E20</font></i></a>, <a class="ref" href="relat_1.html#D8">RELAT_1:def 8</a></i>;<br/>
<b>hence </b><a NAME="E7:40_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="yellow21.html#E5:40_2"><i><font color="Green">E21</font></i></a>, <a class="ref" href="relat_1.html#D8">RELAT_1:def 8</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E14:40"/><b>then </b><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K2">"</a>  <a href="wellord1.html#R3">is_isomorphism_of</a>  <a href="yellow_1.html#K1">RelIncl</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="ref" href="wellord1.html#T49">WELLORD1:49</a></i>;<br/>
<a NAME="E15:40"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>9</sub></font>, <a href="yellow_1.html#K1">RelIncl</a> <font color="Maroon">c<sub>2</sub></font> <a href="wellord1.html#R4">are_isomorphic</a> 
 
<b>by </b><i><a class="txt" href="yellow21.html#E13:40"><i><font color="Green">E19</font></i></a>, <a class="ref" href="wellord1.html#D8">WELLORD1:def 8</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>9</sub></font>
;<br/>

<b>thus </b><a NAME="E16:40"/>
<font color="Maroon">c<sub>9</sub></font> <a href="wellord1.html#R2">well_orders</a> <font color="Maroon">c<sub>1</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_3"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:40_3"/>
( <font color="Maroon">c<sub>9</sub></font> <a href="relat_2.html#R1">is_reflexive_in</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="relat_2.html#R8">is_transitive_in</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="relat_2.html#R4">is_antisymmetric_in</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="yellow21.html#E9:40"><i><font color="Green">E17</font></i></a>, <a class="ref" href="relat_2.html#D9">RELAT_2:def 9</a>, <a class="ref" href="relat_2.html#D12">RELAT_2:def 12</a>, <a class="ref" href="relat_2.html#D16">RELAT_2:def 16</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="wellord1.html#D5">WELLORD1:def 5</a><br/>

<a NAME="E2:40_3"/>
(  <a href="yellow_1.html#K1">RelIncl</a> <font color="Maroon">c<sub>2</sub></font> is <a href="relat_2.html#V6">connected</a> &amp;  <a href="yellow_1.html#K1">RelIncl</a> <font color="Maroon">c<sub>2</sub></font> is <a href="wellord1.html#V1">well_founded</a> )
 
<b>by </b><i><a class="ref" href="wellord2.html#T4">WELLORD2:4</a>, <a class="ref" href="wellord2.html#T6">WELLORD2:6</a></i>;<br/>
<a NAME="E3:40_3"/><b>then </b>
( <font color="Maroon">c<sub>9</sub></font> is <a href="relat_2.html#V6">connected</a> &amp; <font color="Maroon">c<sub>9</sub></font> is <a href="wellord1.html#V1">well_founded</a> )
 
<b>by </b><i><a class="txt" href="yellow21.html#E14:40"><i><font color="Green">E20</font></i></a>, <a class="ref" href="wellord1.html#T53">WELLORD1:53</a></i>;<br/>
<b>hence </b><a NAME="E4:40_3"/>
( <font color="Maroon">c<sub>9</sub></font> <a href="relat_2.html#R6">is_connected_in</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="wellord1.html#R1">is_well_founded_in</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="yellow21.html#E9:40"><i><font color="Green">E17</font></i></a>, <a class="ref" href="relat_2.html#D14">RELAT_2:def 14</a>, <a class="ref" href="wellord1.html#T5">WELLORD1:5</a></i>;<br/>


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

<a NAME="E17:40"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> is <a href="wellord1.html#V2">well-ordering</a>
 
<b>by </b><i><a class="txt" href="yellow21.html#E9:40"><i><font color="Green">E17</font></i></a>, <a class="ref" href="wellord1.html#T8">WELLORD1:8</a></i>;<br/>
<b>hence </b><a NAME="E18:40"/>
 <a href="wellord2.html#K2">order_type_of</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="yellow21.html#E15:40"><i><font color="Green">E21</font></i></a>, <a class="ref" href="wellord2.html#D2">WELLORD2:def 2</a></i>;<br/>


</div>
