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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="struct_0.html#V3">empty</a> <a href="yellow_0.html#V2">upper-bounded</a> <a href="orders_2.html#NM1">Poset</a>;<br/>

<b>set </b><font color="Maroon">c<sub>2</sub></font> = the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>3</sub></font> =  <a href="wellord2.html#K2">order_type_of</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>assume </b><a NAME="E1:44"/><i><font color="Green">E17</font></i>: 
the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font> is <a href="wellord1.html#V2">well-ordering</a>
 ;<br/>

<a NAME="E2:44"/><b>then </b>
the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</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> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</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/>
<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="E4:44"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="wellord1.html#R3">is_isomorphism_of</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</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> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="wellord1.html#D8">WELLORD1:def 8</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>1</sub></font> as  <a href="lattice3.html#V3">complete</a> <a href="waybel_0.html#NM5">Chain</a> <b>by </b><i><a class="txt" href="yellow21.html#E1:44"><i><font color="Green">E17</font></i></a>, <a class="ref" href="yellow21.html#T9" target="_self">Th9</a></i>;<br/>
<b>let </b><font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><a NAME="E6:44"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="orders_2.html#R2">&lt;</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/>

<a NAME="E7:44"/><b>then </b>
( <font color="Maroon">c<sub>7</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>6</sub></font> )
 
<b>by </b><i><a class="ref" href="orders_2.html#D10">ORDERS_2:def 10</a></i>;<br/>
<a NAME="E8:44"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/>
<a NAME="E9:44"/><b>then </b><i><font color="Green">E20</font></i>: 
( <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>,<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>6</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> <span class="p1">(<span class="default"><a href="wellord2.html#K2">order_type_of</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> )
 
<b>by </b><i><a class="txt" href="yellow21.html#E4:44"><i><font color="Green">E18</font></i></a>, <a class="txt" href="yellow21.html#E6:44"><i><font color="Green">E19</font></i></a>, <a class="ref" href="wellord1.html#T45">WELLORD1:45</a></i>;<br/>
<a NAME="E10:44"/><i><font color="Green">E21</font></i>: 
 <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> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="wellord2.html#K2">order_type_of</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="wellord2.html#D1">WELLORD2:def 1</a></i>;<br/>
<a NAME="E11:44"/><b>then </b><i><font color="Green">E22</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <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> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="wellord2.html#K2">order_type_of</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="txt" href="yellow21.html#E9:44"><i><font color="Green">E20</font></i></a>, <a class="ref" href="relat_1.html#T30">RELAT_1:30</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> as   <a href="ordinal1.html#NM2">Ordinal</a> <b>by </b><i><a class="ref" href="ordinal1.html#T23">ORDINAL1:23</a></i>;<br/>
<a NAME="E13:44"/>
<font color="Maroon">c<sub>9</sub></font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="yellow21.html#E9:44"><i><font color="Green">E20</font></i></a>, <a class="txt" href="yellow21.html#E11:44"><i><font color="Green">E22</font></i></a>, <a class="ref" href="wellord2.html#D1">WELLORD2:def 1</a></i>;<br/>
<a NAME="E14:44"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="xboole_0.html#R2">c&lt;</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="yellow21.html#E9:44"><i><font color="Green">E20</font></i></a>, <a class="ref" href="xboole_0.html#D8">XBOOLE_0:def 8</a></i>;<br/>
<a NAME="E15:44"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="ref" href="ordinal1.html#T21">ORDINAL1:21</a></i>;<br/>
<a NAME="E16:44"/><b>then </b><i><font color="Green">E23</font></i>: 
(  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>9</sub></font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="ordinal1.html#R1">c=</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>9</sub></font> )
 
<b>by </b><i><a class="ref" href="ordinal1.html#T33">ORDINAL1:33</a>, <a class="ref" href="ordinal3.html#T1">ORDINAL3:1</a></i>;<br/>
<a NAME="E17:44"/><b>then </b><i><font color="Green">E24</font></i>: 
 <a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="wellord2.html#K2">order_type_of</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="yellow21.html#E11:44"><i><font color="Green">E22</font></i></a>, <a class="ref" href="ordinal1.html#T22">ORDINAL1:22</a></i>;<br/>
<a NAME="E18:44"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="wellord2.html#K2">order_type_of</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="yellow21.html#E4:44"><i><font color="Green">E18</font></i></a>, <a class="txt" href="yellow21.html#E10:44"><i><font color="Green">E21</font></i></a>, <a class="ref" href="wellord1.html#D7">WELLORD1:def 7</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>10</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E20:44"/><i><font color="Green">E25</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>4</sub></font> &amp;  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>9</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> )
 <b>by </b><i><a class="txt" href="yellow21.html#E17:44"><i><font color="Green">E24</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E21:44"/><i><font color="Green">E26</font></i>: 
 <a href="relat_1.html#K3">field</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <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="E22:44"/><i><font color="Green">E27</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K3">field</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="yellow21.html#E4:44"><i><font color="Green">E18</font></i></a>, <a class="ref" href="wellord1.html#D7">WELLORD1:def 7</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>10</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="yellow21.html#E4:44"><i><font color="Green">E18</font></i></a>, <a class="txt" href="yellow21.html#E20:44"><i><font color="Green">E25</font></i></a>, <a class="txt" href="yellow21.html#E21:44"><i><font color="Green">E26</font></i></a>, <a class="ref" href="wellord1.html#D7">WELLORD1:def 7</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>11</sub></font>
;<br/>

<b>thus </b><a NAME="E24:44"/>
<font color="Maroon">c<sub>11</sub></font> is <a href="waybel_3.html#V1">compact</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>12</sub></font> be  non <a href="xboole_0.html#V1">empty</a> <a href="waybel_0.html#V1">directed</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_3.html#D1">WAYBEL_3:def 1</a>,<a class="ref" href="waybel_3.html#D2">WAYBEL_3:def 2</a><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:44_1"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">c<sub>12</sub></font>
 <b>and </b><br/><a NAME="E2:44_1"/><i><font color="Green">E29</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font> holds <br/> not <font color="Maroon">c<sub>11</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">b<sub>1</sub></font>
 ;<br/>

<a NAME="E3:44_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>5</sub></font> is <a href="lattice3.html#V3">complete</a>
 
;<br/>
<a NAME="E4:44_1"/>
<font color="Maroon">c<sub>12</sub></font> <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Maroon">c<sub>7</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>13</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D9">LATTICE3:def 9</a><br/>

<b>assume </b><a NAME="E1:44_1_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font>
 ;<br/>

<a NAME="E2:44_1_1"/><b>then </b>
( not <font color="Maroon">c<sub>11</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>13</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>11</sub></font> )
 
<b>by </b><i><a class="txt" href="yellow21.html#E2:44_1"><i><font color="Green">E29</font></i></a></i>;<br/>
<a NAME="E3:44_1_1"/><b>then </b>
<font color="Maroon">c<sub>11</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>13</sub></font>
 
<b>by </b><i><a class="txt" href="yellow21.html#E3:44_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="waybel_0.html#D29">WAYBEL_0:def 29</a></i>;<br/>
<a NAME="E4:44_1_1"/><b>then </b>
<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>11</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/>
<a NAME="E5:44_1_1"/><b>then </b><i><font color="Green">E32</font></i>: 
<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>13</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>9</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> <span class="p1">(<span class="default"><a href="wellord2.html#K2">order_type_of</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="yellow21.html#E4:44"><i><font color="Green">E18</font></i></a>, <a class="txt" href="yellow21.html#E20:44"><i><font color="Green">E25</font></i></a>, <a class="ref" href="wellord1.html#D7">WELLORD1:def 7</a></i>;<br/>
<a NAME="E6:44_1_1"/><b>then </b><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="wellord2.html#K2">order_type_of</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="yellow21.html#E10:44"><i><font color="Green">E21</font></i></a>, <a class="ref" href="relat_1.html#T30">RELAT_1:30</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font> as   <a href="ordinal1.html#NM2">Ordinal</a> <b>by </b><i><a class="ref" href="ordinal1.html#T23">ORDINAL1:23</a></i>;<br/>
<a NAME="E8:44_1_1"/>
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>11</sub></font> &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#E4:44"><i><font color="Green">E18</font></i></a>, <a class="txt" href="yellow21.html#E2:44_1"><i><font color="Green">E29</font></i></a>, <a class="txt" href="yellow21.html#E1:44_1_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="wellord1.html#D7">WELLORD1:def 7</a></i>;<br/>
<a NAME="E9:44_1_1"/><b>then </b>
( <font color="Maroon">c<sub>14</sub></font> <a href="ordinal1.html#R1">c=</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>9</sub></font> )
 
<b>by </b><i><a class="txt" href="yellow21.html#E17:44"><i><font color="Green">E24</font></i></a>, <a class="txt" href="yellow21.html#E20:44"><i><font color="Green">E25</font></i></a>, <a class="txt" href="yellow21.html#E21:44"><i><font color="Green">E26</font></i></a>, <a class="txt" href="yellow21.html#E22:44"><i><font color="Green">E27</font></i></a>, <a class="txt" href="yellow21.html#E5:44_1_1"><i><font color="Green">E32</font></i></a>, <a class="txt" href="yellow21.html#E6:44_1_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a>, <a class="ref" href="wellord2.html#D1">WELLORD2:def 1</a></i>;<br/>
<a NAME="E10:44_1_1"/><b>then </b>
<font color="Maroon">c<sub>14</sub></font> <a href="xboole_0.html#R2">c&lt;</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D8">XBOOLE_0:def 8</a></i>;<br/>
<a NAME="E11:44_1_1"/><b>then </b>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="ref" href="ordinal1.html#T21">ORDINAL1:21</a></i>;<br/>
<a NAME="E12:44_1_1"/><b>then </b>
<font color="Maroon">c<sub>14</sub></font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="ref" href="ordinal1.html#T34">ORDINAL1:34</a></i>;<br/>
<a NAME="E13:44_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>9</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> <span class="p1">(<span class="default"><a href="wellord2.html#K2">order_type_of</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="yellow21.html#E11:44"><i><font color="Green">E22</font></i></a>, <a class="txt" href="yellow21.html#E6:44_1_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="wellord2.html#D1">WELLORD2:def 1</a></i>;<br/>
<b>hence </b><a NAME="E14:44_1_1"/>
<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>7</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="yellow21.html#E4:44"><i><font color="Green">E18</font></i></a>, <a class="txt" href="yellow21.html#E21:44"><i><font color="Green">E26</font></i></a>, <a class="ref" href="wellord1.html#D7">WELLORD1:def 7</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a><br/>


</div><b>end;</b></div>
<a NAME="E5:44_1"/><b>then </b>
 <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">c<sub>12</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="yellow21.html#E3:44_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="yellow_0.html#T32">YELLOW_0:32</a></i>;<br/>
<a NAME="E6:44_1"/><b>then </b>
<font color="Maroon">c<sub>11</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="yellow21.html#E1:44_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="yellow_0.html#D2">YELLOW_0:def 2</a></i>;<br/>
<a NAME="E7:44_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/>
<a NAME="E8:44_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>,<font color="Maroon">c<sub>9</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> <span class="p1">(<span class="default"><a href="wellord2.html#K2">order_type_of</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="yellow21.html#E4:44"><i><font color="Green">E18</font></i></a>, <a class="txt" href="yellow21.html#E20:44"><i><font color="Green">E25</font></i></a>, <a class="ref" href="wellord1.html#D7">WELLORD1:def 7</a></i>;<br/>
<a NAME="E9:44_1"/><b>then </b>
 <a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>9</sub></font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="yellow21.html#E11:44"><i><font color="Green">E22</font></i></a>, <a class="txt" href="yellow21.html#E17:44"><i><font color="Green">E24</font></i></a>, <a class="ref" href="wellord2.html#D1">WELLORD2:def 1</a></i>;<br/>
<a NAME="E10:44_1"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="yellow21.html#E16:44"><i><font color="Green">E23</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<b>hence </b><a NAME="E11:44_1"/>
contradiction
 <b>by </b><i><a class="ref" href="ordinal1.html#T14">ORDINAL1:14</a></i>;<br/>


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

<a NAME="E25:44"/>
( <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>,<span class="p2">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>9</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> <span class="p1">(<span class="default"><a href="wellord2.html#K2">order_type_of</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>9</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>6</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> <span class="p1">(<span class="default"><a href="wellord2.html#K2">order_type_of</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="yellow21.html#E11:44"><i><font color="Green">E22</font></i></a>, <a class="txt" href="yellow21.html#E16:44"><i><font color="Green">E23</font></i></a>, <a class="txt" href="yellow21.html#E17:44"><i><font color="Green">E24</font></i></a>, <a class="ref" href="wellord2.html#D1">WELLORD2:def 1</a></i>;<br/>
<b>hence </b><a NAME="E26:44"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>7</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> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="yellow21.html#E4:44"><i><font color="Green">E18</font></i></a>, <a class="txt" href="yellow21.html#E20:44"><i><font color="Green">E25</font></i></a>, <a class="txt" href="yellow21.html#E21:44"><i><font color="Green">E26</font></i></a>, <a class="ref" href="wellord1.html#D7">WELLORD1:def 7</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a><br/>


</div>
