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

<b>set </b><font color="Maroon" title="c1">IR'</font> = the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="dickson.html#K11" title="DICKSON:func.11">OrderedNAT</a>  <a href="dickson.html#K5" title="DICKSON:func.5">\~</a> </span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c2">CR'</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="dickson.html#K11" title="DICKSON:func.11">OrderedNAT</a>  <a href="dickson.html#K5" title="DICKSON:func.5">\~</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" title="c3">N</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><b>that </b><br/><a NAME="E1:72_2_1_1"/><i><font color="Green" title="E51">A1</font></i>: 
<font color="Maroon" title="c3">N</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="dickson.html#K11" title="DICKSON:func.11">OrderedNAT</a>  <a href="dickson.html#K5" title="DICKSON:func.5">\~</a> </span>)</span>
 <b>and </b><br/><a NAME="E2:72_2_1_1"/><i><font color="Green" title="E52">A2</font></i>: 
<font color="Maroon" title="c3">N</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><b>consider </b><font color="Maroon" title="c4">k</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:72_2_1_1"/><i><font color="Green" title="E53">A3</font></i>: 
<font color="Maroon" title="c4">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">N</font>
 <b>by </b><i><a class="txt" href="dickson.html#E2:72_2_1_1"><i><font color="Green" title="E52">A2</font></i></a>, <a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <b>means </b><font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">N</font>;<br/><a NAME="E5:72_2_1_1"/><i><font color="Green" title="E54">A4</font></i>: 
 ex <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 <b>by </b><i><a class="txt" href="dickson.html#E1:72_2_1_1"><i><font color="Green" title="E51">A1</font></i></a>, <a class="txt" href="dickson.html#E4:72_2_1_1"><i><font color="Green" title="E53">A3</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c5">m</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E7:72_2_1_1"/><i><font color="Green" title="E55">A5</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">m</font>]
 <b>and </b><br/><a NAME="E8:72_2_1_1"/><i><font color="Green" title="E56">A6</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon" title="c5">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font>
 <b>from </b><i><a class="ref" href="nat_1.html#S5" title="NAT_1:sch.5">NAT_1:sch 5</a>(<a class="txt" href="dickson.html#E5:72_2_1_1"><i><font color="Green" title="E54">A4</font></i></a>);<br/></i><b>reconsider </b><font color="Maroon" title="c6">m</font> = <font color="Maroon" title="c5">m</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="dickson.html#K11" title="DICKSON:func.11">OrderedNAT</a>  ;<br/><b>thus </b><a NAME="E10:72_2_1_1"/>
 ex <font color="Olive" title="b1">m</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">N</font> &amp; the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="dickson.html#K11" title="DICKSON:func.11">OrderedNAT</a>  <a href="dickson.html#K5" title="DICKSON:func.5">\~</a> </span>)</span> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Olive" title="b1">m</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">N</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" title="c6">m</font>
;<br/>

<b>thus </b><a NAME="E1:72_2_1_1_1"/>
<font color="Maroon" title="c6">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">N</font>
 <b>by </b><i><a class="txt" href="dickson.html#E7:72_2_1_1"><i><font color="Green" title="E55">A5</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" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p2">(<span class="default"><a href="dickson.html#K11" title="DICKSON:func.11">OrderedNAT</a>  <a href="dickson.html#K5" title="DICKSON:func.5">\~</a> </span>)</span> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c6">m</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c3">N</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><b>then consider </b><font color="Maroon" title="c7">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:72_2_1_1_1_1"/><i><font color="Green" title="E57">A7</font></i>: 
<font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default">the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p2">(<span class="default"><a href="dickson.html#K11" title="DICKSON:func.11">OrderedNAT</a>  <a href="dickson.html#K5" title="DICKSON:func.5">\~</a> </span>)</span> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c6">m</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c3">N</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/><a NAME="E4:72_2_1_1_1_1"/><i><font color="Green" title="E58">A8</font></i>: 
( <font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="dickson.html#K11" title="DICKSON:func.11">OrderedNAT</a>  <a href="dickson.html#K5" title="DICKSON:func.5">\~</a> </span>)</span> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c6">m</font> &amp; <font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">N</font> )
 <b>by </b><i><a class="txt" href="dickson.html#E3:72_2_1_1_1_1"><i><font color="Green" title="E57">A7</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E5:72_2_1_1_1_1"/><b>then </b><i><font color="Green" title="E59">A9</font></i>: 
( <font color="Maroon" title="c7">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">m</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">z</font>,<font color="Maroon" title="c6">m</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="dickson.html#K11" title="DICKSON:func.11">OrderedNAT</a>  <a href="dickson.html#K5" title="DICKSON:func.5">\~</a> </span>)</span> )
 <b>by </b><i><a class="ref" href="wellord1.html#D1" title="WELLORD1:def.1">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" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">z</font>,<font color="Maroon" title="c6">m</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="dickson.html#K10" title="DICKSON:func.10">NATOrd</a>  <a href="relset_1.html#K3" title="RELSET_1:func.3">\</a> <span class="p1">(<span class="default"><a href="dickson.html#K10" title="DICKSON:func.10">NATOrd</a>  <a href="relset_1.html#K6" title="RELSET_1:func.6">~</a> </span>)</span>
 ;<br/><a NAME="E7:72_2_1_1_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">z</font>,<font color="Maroon" title="c6">m</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="dickson.html#K10" title="DICKSON:func.10">NATOrd</a> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><b>then consider </b><font color="Maroon" title="c8">x</font>, <font color="Maroon" title="c9">y</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E9:72_2_1_1_1_1"/><i><font color="Green" title="E60">A10</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">z</font>,<font color="Maroon" title="c6">m</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 <b>and </b><br/><a NAME="E10:72_2_1_1_1_1"/>
<font color="Maroon" title="c8">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">y</font>
 ;<br/><a NAME="E11:72_2_1_1_1_1"/><i><font color="Green" title="E61">A11</font></i>: 
( <font color="Maroon" title="c7">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">x</font> &amp; <font color="Maroon" title="c6">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y</font> )
 <b>by </b><i><a class="txt" href="dickson.html#E9:72_2_1_1_1_1"><i><font color="Green" title="E60">A10</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/><a NAME="E12:72_2_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c9">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">x</font>
 <b>by </b><i><a class="txt" href="dickson.html#E8:72_2_1_1"><i><font color="Green" title="E56">A6</font></i></a>, <a class="txt" href="dickson.html#E4:72_2_1_1_1_1"><i><font color="Green" title="E58">A8</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" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c9">y</font>,<font color="Maroon" title="c8">x</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="dickson.html#K10" title="DICKSON:func.10">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" title="E59">A9</font></i></a>, <a class="txt" href="dickson.html#E11:72_2_1_1_1_1"><i><font color="Green" title="E61">A11</font></i></a>, <a class="ref" href="dickson.html#T45" target="_self" title="DICKSON:th.45">Th45</a>, <a class="ref" href="relat_2.html#D4" title="RELAT_2:def.4">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" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="dickson.html#K11" title="DICKSON:func.11">OrderedNAT</a>  <a href="dickson.html#K5" title="DICKSON:func.5">\~</a> </span>)</span> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c6">m</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">N</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">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" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="dickson.html#K11" title="DICKSON:func.11">OrderedNAT</a>  <a href="dickson.html#K5" title="DICKSON:func.5">\~</a> </span>)</span> <a href="wellord1.html#R1" title="WELLORD1:pred.1">is_well_founded_in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="dickson.html#K11" title="DICKSON:func.11">OrderedNAT</a>  <a href="dickson.html#K5" title="DICKSON:func.5">\~</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="wellord1.html#D3" title="WELLORD1:def.3">WELLORD1:def 3</a></i>;<br/>
<a NAME="E3:72_2_1"/><b>then </b>
<a href="dickson.html#K11" title="DICKSON:func.11">OrderedNAT</a>  <a href="dickson.html#K5" title="DICKSON:func.5">\~</a>  is <a href="wellfnd1.html#V1" title="WELLFND1:attr.1">well_founded</a>
 
<b>by </b><i><a class="ref" href="wellfnd1.html#D2" title="WELLFND1:def.2">WELLFND1:def 2</a></i>;<br/>
<b>hence </b><a NAME="E4:72_2_1"/>
 <a href="dickson.html#K11" title="DICKSON:func.11">OrderedNAT</a>  is <a href="dickson.html#V4" title="DICKSON:attr.4">Dickson</a>
 <b>by </b><i><a class="ref" href="dickson.html#T28" target="_self" title="DICKSON:th.28">Th28</a></i>;<br/>


</div>
