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<b>set </b><font color="Maroon">ir</font> = the <a href="orders_2.html#U1">InternalRel</a> of <a href="dickson.html#K11" target="_self">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">f</font> being   <a href="normsp_1.html#NM2">sequence</a> of <a href="dickson.html#K11" target="_self">OrderedNAT</a> <b> such that </b><a NAME="E1:72_6_1_1"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">f</font> is <a href="wellfnd1.html#V3">descending</a>
 ;<br/><a NAME="E2:72_6_1_1"/><i><font color="Green">E73</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><b>reconsider </b><font color="Maroon">rf</font> =  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K5">NAT</a>  ;<br/><b>set </b><font color="Maroon">m</font> =  <a href="cqc_sim1.html#K10">min</a> <font color="Maroon">rf</font>;<br/><a NAME="E4:72_6_1_1"/>
 <a href="cqc_sim1.html#K10">min</a> <font color="Maroon">rf</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="cqc_sim1.html#D8">CQC_SIM1:def 8</a></i>;<br/><b>then consider </b><font color="Maroon">d</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:72_6_1_1"/><i><font color="Green">E74</font></i>: 
( <font color="Maroon">d</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp;  <a href="cqc_sim1.html#K10">min</a> <font color="Maroon">rf</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">d</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon">d</font> = <font color="Maroon">d</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="dickson.html#T5" target="_self">Th5</a></i>;<br/><a NAME="E8:72_6_1_1"/><i><font color="Green">E75</font></i>: 
( <font color="Maroon">f</font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">f</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">d</font> &amp; <span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="normsp_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">d</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">d</font></span>)</span></span><a href="yellow_3.html#K7">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <a href="dickson.html#K11" target="_self">OrderedNAT</a>  )
 <b>by </b><i>, <a class="ref" href="wellfnd1.html#D7">WELLFND1:def 7</a></i>;<br/><b>then consider </b><font color="Maroon">x</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">y</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="E10:72_6_1_1"/><i><font color="Green">E89</font></i>: 
<span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="normsp_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">d</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">d</font></span>)</span></span><a href="yellow_3.html#K7">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K1">]</a></span>
 <b>and </b><br/><a NAME="E11:72_6_1_1"/><i><font color="Green">E90</font></i>: 
<font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y</font>
 ;<br/><b>reconsider </b><font color="Maroon">fd1</font> = <font color="Maroon">f</font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>, <font color="Maroon">fd</font> = <font color="Maroon">f</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">d</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/><a NAME="E13:72_6_1_1"/>
( <font color="Maroon">f</font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">f</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> )
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><a NAME="E14:72_6_1_1"/><b>then </b><i><font color="Green">E91</font></i>: 
<font color="Maroon">fd1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">fd</font>
 <b>by </b><i>, , <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E15:72_6_1_1"/>
<font color="Maroon">f</font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">rf</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/><b>hence </b><a NAME="E16:72_6_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="dickson.html#T5" target="_self">Th5</a>, , <a class="ref" href="cqc_sim1.html#D8">CQC_SIM1:def 8</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:72_6_1"/>
 <a href="dickson.html#K11" target="_self">OrderedNAT</a>  is <a href="wellfnd1.html#V1">well_founded</a>
 <b>by </b><i><a class="ref" href="wellfnd1.html#T15">WELLFND1:15</a></i>;<br/>


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