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<span class="kw">set </span><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"><span class="kw">now </span></a><div class="add"><span class="kw">given </span><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><span class="kw"> such that </span><a NAME="E2:72_6_1_1"/><span class="lab"><font color="Green" title="E51">A16</font></span>: 
<font color="Maroon" title="c2">f</font> is  <a href="wellfnd1.html#V3" title="WELLFND1:attr.3">descending</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E3:72_6_1_1"/><span class="lab"><font color="Green" title="E52">A17</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">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> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c3">rf</font> =  <a href="relset_1.html#K2" title="RELSET_1:func.2">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/><span class="kw">set </span><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#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">f</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_2.html#D7" title="XXREAL_2:def.7">XXREAL_2:def 7</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">d</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E7:72_6_1_1"/><span class="lab"><font color="Green" title="E53">A18</font></span>: 
<font color="Maroon" title="c5">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">f</font>
 <span class="kw">and </span><br/><a NAME="E8:72_6_1_1"/><span class="lab"><font color="Green" title="E54">A19</font></span>: 
 <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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/><span class="kw">reconsider </span><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>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:72_6_1_1"><span class="lab"><font color="Green" title="E53">A18</font></span></a></span>;<br/><a NAME="E10:72_6_1_1"/><span class="lab"><font color="Green" title="E55">A20</font></span>: 
<font color="Maroon" title="c2">f</font> <a href="yellow_6.html#K2" title="YELLOW_6:func.2">.</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="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <font color="Maroon" title="c6">d</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:72_6_1_1"><span class="lab"><font color="Green" title="E51">A16</font></span></a>, <a class="ref" href="wellfnd1.html#D6" title="WELLFND1:def.6">WELLFND1:def 6</a></span>;<br/><a NAME="E11:72_6_1_1"/>
<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="yellow_6.html#K2" title="YELLOW_6:func.2">.</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="yellow_6.html#K2" title="YELLOW_6:func.2">.</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:72_6_1_1"><span class="lab"><font color="Green" title="E51">A16</font></span></a>, <a class="ref" href="wellfnd1.html#D6" title="WELLFND1:def.6">WELLFND1:def 6</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><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> <span class="kw"> such that </span><br/><a NAME="E13:72_6_1_1"/><span class="lab"><font color="Green" title="E56">A21</font></span>: 
<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="yellow_6.html#K2" title="YELLOW_6:func.2">.</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="yellow_6.html#K2" title="YELLOW_6:func.2">.</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>
 <span class="kw">and </span><br/><a NAME="E14:72_6_1_1"/><span class="lab"><font color="Green" title="E57">A22</font></span>: 
<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/><span class="kw">reconsider </span><font color="Maroon" title="c9">fd1</font> = <font color="Maroon" title="c2">f</font> <a href="yellow_6.html#K2" title="YELLOW_6:func.2">.</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="yellow_6.html#K2" title="YELLOW_6:func.2">.</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="E16:72_6_1_1"/><span class="lab"><font color="Green" title="E58">A23</font></span>: 
<font color="Maroon" title="c2">f</font> <a href="yellow_6.html#K2" title="YELLOW_6:func.2">.</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:72_6_1_1"><span class="lab"><font color="Green" title="E56">A21</font></span></a>, <a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</a></span>;<br/><a NAME="E17:72_6_1_1"/>
<font color="Maroon" title="c2">f</font> <a href="yellow_6.html#K2" title="YELLOW_6:func.2">.</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:72_6_1_1"><span class="lab"><font color="Green" title="E56">A21</font></span></a>, <a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</a></span>;<br/><a NAME="E18:72_6_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A24</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:72_6_1_1"><span class="lab"><font color="Green" title="E55">A20</font></span></a>, <a class="txt" href="#E14:72_6_1_1"><span class="lab"><font color="Green" title="E57">A22</font></span></a>, <a class="txt" href="#E16:72_6_1_1"><span class="lab"><font color="Green" title="E58">A23</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E19:72_6_1_1"/>
<font color="Maroon" title="c2">f</font> <a href="yellow_6.html#K2" title="YELLOW_6:func.2">.</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:72_6_1_1"><span class="lab"><font color="Green" title="E52">A17</font></span></a>, <a class="ref" href="funct_1.html#T3" title="FUNCT_1:th.3">FUNCT_1:3</a></span>;<br/><span class="kw">hence </span><a NAME="E20:72_6_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:72_6_1_1"><span class="lab"><font color="Green" title="E54">A19</font></span></a>, <a class="txt" href="#E18:72_6_1_1"><span class="lab"><font color="Green" title="E59">A24</font></span></a>, <a class="ref" href="xxreal_2.html#D7" title="XXREAL_2:def.7">XXREAL_2:def 7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><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> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="wellfnd1.html#T14" title="WELLFND1:th.14">WELLFND1:14</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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