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<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 <a href="dickson.html#K11" title="DICKSON:func.11">OrderedNAT</a> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_6_1_1"><b>now </b></a><div class="add"><b>given </b><font color="Maroon" title="c2">f</font> being   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <a href="dickson.html#K11" title="DICKSON:func.11">OrderedNAT</a> <b> such that </b><a NAME="E2:72_6_1_1"/><i><font color="Green" title="E51">A13</font></i>: 
<font color="Maroon" title="c2">f</font> is <a href="wellfnd1.html#V3" title="WELLFND1:attr.3">descending</a>
 ;<br/><a NAME="E3:72_6_1_1"/><i><font color="Green" title="E52">A14</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c3">rf</font> =  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">f</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/><b>set </b><font color="Maroon" title="c4">m</font> =  <a href="seq_4.html#K6" title="SEQ_4:func.6">min</a> <font color="Maroon" title="c3">rf</font>;<br/><a NAME="E5:72_6_1_1"/>
 <a href="seq_4.html#K6" title="SEQ_4:func.6">min</a> <font color="Maroon" title="c3">rf</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">f</font>
 <b>by </b><i><a class="ref" href="seq_4.html#D6" title="SEQ_4:def.6">SEQ_4:def 6</a></i>;<br/><b>then consider </b><font color="Maroon" title="c5">d</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E7:72_6_1_1"/><i><font color="Green" title="E53">A15</font></i>: 
( <font color="Maroon" title="c5">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">f</font> &amp;  <a href="seq_4.html#K6" title="SEQ_4:func.6">min</a> <font color="Maroon" title="c3">rf</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">d</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c6">d</font> = <font color="Maroon" title="c5">d</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="dickson.html#E7:72_6_1_1"><i><font color="Green" title="E53">A15</font></i></a></i>;<br/><a NAME="E9:72_6_1_1"/><i><font color="Green" title="E54">A16</font></i>: 
( <font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">d</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">d</font> &amp; <span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">d</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">d</font></span>)</span></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</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 <a href="dickson.html#K11" title="DICKSON:func.11">OrderedNAT</a>  )
 <b>by </b><i><a class="txt" href="dickson.html#E2:72_6_1_1"><i><font color="Green" title="E51">A13</font></i></a>, <a class="ref" href="wellfnd1.html#D7" title="WELLFND1:def.7">WELLFND1:def 7</a></i>;<br/><b>then consider </b><font color="Maroon" title="c7">x</font>, <font color="Maroon" title="c8">y</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E11:72_6_1_1"/><i><font color="Green" title="E55">A17</font></i>: 
<span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">d</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">d</font></span>)</span></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</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="c7">x</font>,<font color="Maroon" title="c8">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 <b>and </b><br/><a NAME="E12:72_6_1_1"/><i><font color="Green" title="E56">A18</font></i>: 
<font color="Maroon" title="c7">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">y</font>
 ;<br/><b>reconsider </b><font color="Maroon" title="c9">fd1</font> = <font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">d</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>, <font color="Maroon" title="c10">fd</font> = <font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">d</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/><a NAME="E14:72_6_1_1"/>
( <font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">d</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x</font> &amp; <font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">y</font> )
 <b>by </b><i><a class="txt" href="dickson.html#E11:72_6_1_1"><i><font color="Green" title="E55">A17</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/><a NAME="E15:72_6_1_1"/><b>then </b><i><font color="Green" title="E57">A19</font></i>: 
<font color="Maroon" title="c9">fd1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">fd</font>
 <b>by </b><i><a class="txt" href="dickson.html#E9:72_6_1_1"><i><font color="Green" title="E54">A16</font></i></a>, <a class="txt" href="dickson.html#E12:72_6_1_1"><i><font color="Green" title="E56">A18</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><a NAME="E16:72_6_1_1"/>
<font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">d</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">rf</font>
 <b>by </b><i><a class="txt" href="dickson.html#E3:72_6_1_1"><i><font color="Green" title="E52">A14</font></i></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/><b>hence </b><a NAME="E17:72_6_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="dickson.html#E7:72_6_1_1"><i><font color="Green" title="E53">A15</font></i></a>, <a class="txt" href="dickson.html#E15:72_6_1_1"><i><font color="Green" title="E57">A19</font></i></a>, <a class="ref" href="seq_4.html#D6" title="SEQ_4:def.6">SEQ_4:def 6</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:72_6_1"/>
 <a href="dickson.html#K11" title="DICKSON:func.11">OrderedNAT</a>  is <a href="wellfnd1.html#V1" title="WELLFND1:attr.1">well_founded</a>
 <b>by </b><i><a class="ref" href="wellfnd1.html#T15" title="WELLFND1:th.15">WELLFND1:15</a></i>;<br/>


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