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

<b>set </b><font color="Maroon">c<sub>1</sub></font> = the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="dickson.html#K11" target="_self">OrderedNAT</a>  <a href="dickson.html#K5" target="_self">\~</a> </span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>2</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="dickson.html#K11" target="_self">OrderedNAT</a>  <a href="dickson.html#K5" target="_self">\~</a> </span>)</span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_2_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><b>that </b><br/><a NAME="E1:72_2_1_1"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="dickson.html#K11" target="_self">OrderedNAT</a>  <a href="dickson.html#K5" target="_self">\~</a> </span>)</span>
 <b>and </b><br/><a NAME="E2:72_2_1_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><b>consider </b><font color="Maroon">c<sub>4</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:72_2_1_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="dickson.html#E2:72_2_1_1"><i><font color="Green">E52</font></i></a>, <a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>;<br/><a NAME="E5:72_2_1_1"/><i><font color="Green">E54</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 <b>by </b><i><a class="txt" href="dickson.html#E1:72_2_1_1"><i><font color="Green">E51</font></i></a>, <a class="txt" href="dickson.html#E4:72_2_1_1"><i><font color="Green">E53</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E7:72_2_1_1"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>5</sub></font>]
 <b>and </b><br/><a NAME="E8:72_2_1_1"/><i><font color="Green">E56</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font>
 <b>from </b><i><a class="ref" href="nat_1.html#S5">NAT_1:sch 5</a>(<a class="txt" href="dickson.html#E5:72_2_1_1"><i><font color="Green">E54</font></i></a>);<br/></i><b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <a href="dickson.html#K11" target="_self">OrderedNAT</a>  ;<br/><b>thus </b><a NAME="E10:72_2_1_1"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="dickson.html#K11" target="_self">OrderedNAT</a>  <a href="dickson.html#K5" target="_self">\~</a> </span>)</span> <a href="wellord1.html#K1">-Seg</a> <font color="Olive">b<sub>1</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>3</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_2_1_1_1"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon">c<sub>6</sub></font>
;<br/>

<b>thus </b><a NAME="E1:72_2_1_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="dickson.html#E7:72_2_1_1"><i><font color="Green">E55</font></i></a></i>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_2_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:72_2_1_1_1_1"/>
<span class="p1">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <span class="p2">(<span class="default"><a href="dickson.html#K11" target="_self">OrderedNAT</a>  <a href="dickson.html#K5" target="_self">\~</a> </span>)</span> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><b>then consider </b><font color="Maroon">c<sub>7</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:72_2_1_1_1_1"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <span class="p2">(<span class="default"><a href="dickson.html#K11" target="_self">OrderedNAT</a>  <a href="dickson.html#K5" target="_self">\~</a> </span>)</span> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/><a NAME="E4:72_2_1_1_1_1"/><i><font color="Green">E58</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="dickson.html#K11" target="_self">OrderedNAT</a>  <a href="dickson.html#K5" target="_self">\~</a> </span>)</span> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="dickson.html#E3:72_2_1_1_1_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E5:72_2_1_1_1_1"/><b>then </b><i><font color="Green">E59</font></i>: 
( <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> &amp; <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 <span class="p1">(<span class="default"><a href="dickson.html#K11" target="_self">OrderedNAT</a>  <a href="dickson.html#K5" target="_self">\~</a> </span>)</span> )
 <b>by </b><i><a class="ref" href="wellord1.html#D1">WELLORD1:def 1</a></i>;<br/><a NAME="E6:72_2_1_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>,<font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <a href="dickson.html#K10" target="_self">NATOrd</a>  <a href="relset_1.html#K3">\</a> <span class="p1">(<span class="default"><a href="dickson.html#K10" target="_self">NATOrd</a>  <a href="relset_1.html#K6">~</a> </span>)</span>
 ;<br/><a NAME="E7:72_2_1_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>,<font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="dickson.html#K10" target="_self">NATOrd</a> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>9</sub></font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E9:72_2_1_1_1_1"/><i><font color="Green">E60</font></i>: 
<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#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="domain_1.html#K1">]</a></span>
 <b>and </b><br/><a NAME="E10:72_2_1_1_1_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/><a NAME="E11:72_2_1_1_1_1"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> )
 <b>by </b><i><a class="txt" href="dickson.html#E9:72_2_1_1_1_1"><i><font color="Green">E60</font></i></a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><a NAME="E12:72_2_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="dickson.html#E8:72_2_1_1"><i><font color="Green">E56</font></i></a>, <a class="txt" href="dickson.html#E4:72_2_1_1_1_1"><i><font color="Green">E58</font></i></a></i>;<br/><a NAME="E13:72_2_1_1_1_1"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="dickson.html#K10" target="_self">NATOrd</a> 
 ;<br/><b>hence </b><a NAME="E14:72_2_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="dickson.html#E5:72_2_1_1_1_1"><i><font color="Green">E59</font></i></a>, <a class="txt" href="dickson.html#E11:72_2_1_1_1_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="dickson.html#T45" target="_self">Th45</a>, <a class="ref" href="relat_2.html#D4">RELAT_2:def 4</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:72_2_1_1_1"/>
the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="dickson.html#K11" target="_self">OrderedNAT</a>  <a href="dickson.html#K5" target="_self">\~</a> </span>)</span> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">c<sub>6</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E2:72_2_1"/><b>then </b>
the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="dickson.html#K11" target="_self">OrderedNAT</a>  <a href="dickson.html#K5" target="_self">\~</a> </span>)</span> <a href="wellord1.html#R1">is_well_founded_in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="dickson.html#K11" target="_self">OrderedNAT</a>  <a href="dickson.html#K5" target="_self">\~</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="wellord1.html#D3">WELLORD1:def 3</a></i>;<br/>
<a NAME="E3:72_2_1"/><b>then </b>
<a href="dickson.html#K11" target="_self">OrderedNAT</a>  <a href="dickson.html#K5" target="_self">\~</a>  is <a href="wellfnd1.html#V1">well_founded</a>
 
<b>by </b><i><a class="ref" href="wellfnd1.html#D2">WELLFND1:def 2</a></i>;<br/>
<b>hence </b><a NAME="E4:72_2_1"/>
 <a href="dickson.html#K11" target="_self">OrderedNAT</a>  is <a href="dickson.html#V4" target="_self">Dickson</a>
 <b>by </b><i><a class="ref" href="dickson.html#T28" target="_self">Th28</a></i>;<br/>


</div>
