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

<b>let </b><font color="Maroon" title="c1">R</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="waybel_0.html#V16" title="WAYBEL_0:attr.16">connected</a> <a href="orders_2.html#NM1" title="ORDERS_2:NM.1">Poset</a>;<br/>

<b>assume </b><a NAME="E1:73"/><i><font color="Green" title="E45">A1</font></i>: 
<font color="Maroon" title="c1">R</font> is <a href="wellfnd1.html#V1" title="WELLFND1:attr.1">well_founded</a>
 ;<br/>

<b>set </b><font color="Maroon" title="c2">IR</font> = the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font>;<br/>
<b>set </b><font color="Maroon" title="c3">CR</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>;<br/>
<b>set </b><font color="Maroon" title="c4">FIR</font> =  <a href="bagorder.html#K13" title="BAGORDER:func.13">FinOrd</a> <font color="Maroon" title="c1">R</font>;<br/>
<b>set </b><font color="Maroon" title="c5">FCR</font> =  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>;<br/>
<b>assume </b><a NAME="E2:73"/>
not  <a href="bagorder.html#K14" title="BAGORDER:func.14">FinPoset</a> <font color="Maroon" title="c1">R</font> is <a href="wellfnd1.html#V1" title="WELLFND1:attr.1">well_founded</a>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c6">A</font> being   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <span class="p1">(<span class="default"><a href="bagorder.html#K14" title="BAGORDER:func.14">FinPoset</a> <font color="Maroon" title="c1">R</font></span>)</span><b> such that </b><br/><a NAME="E4:73"/><i><font color="Green" title="E46">A2</font></i>: 
<font color="Maroon" title="c6">A</font> is <a href="wellfnd1.html#V3" title="WELLFND1:attr.3">descending</a>
 <b>by </b><i><a class="ref" href="wellfnd1.html#T15" title="WELLFND1:th.15">WELLFND1:15</a></i>;<br/>
<b>set </b><font color="Maroon" title="c7">zz</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">z</font> where <font color="Olive" title="b1">z</font> is   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <span class="p1">(<span class="default"><a href="bagorder.html#K14" title="BAGORDER:func.14">FinPoset</a> <font color="Maroon" title="c1">R</font></span>)</span> : <font color="Olive" title="b1">z</font> is <a href="wellfnd1.html#V3" title="WELLFND1:attr.3">descending</a> </span>}</span> ;<br/>
<a NAME="E5:73"/>
<font color="Maroon" title="c6">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">z</font> where <font color="Olive" title="b1">z</font> is   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <span class="p1">(<span class="default"><a href="bagorder.html#K14" title="BAGORDER:func.14">FinPoset</a> <font color="Maroon" title="c1">R</font></span>)</span> : <font color="Olive" title="b1">z</font> is <a href="wellfnd1.html#V3" title="WELLFND1:attr.3">descending</a> </span>}</span> 
 
<b>by </b><i><a class="txt" href="bagorder.html#E4:73"><i><font color="Green" title="E46">A2</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c8">zz</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">z</font> where <font color="Olive" title="b1">z</font> is   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <span class="p1">(<span class="default"><a href="bagorder.html#K14" title="BAGORDER:func.14">FinPoset</a> <font color="Maroon" title="c1">R</font></span>)</span> : <font color="Olive" title="b1">z</font> is <a href="wellfnd1.html#V3" title="WELLFND1:attr.3">descending</a> </span>}</span>  as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/>
<b>set </b><font color="Maroon" title="c9">Z</font> = <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>,<font color="Maroon" title="c8">zz</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>;<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> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> ex <font color="Olive" title="b1">Sn2</font> being  <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <span class="p1">(<span class="default"><a href="bagorder.html#K14" title="BAGORDER:func.14">FinPoset</a> <font color="Maroon" title="c1">R</font></span>)</span> ex <font color="Olive" title="b2">Smax</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font> ex <font color="Olive" title="b3">an</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> ex <font color="Olive" title="b4">ix</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>  ex <font color="Olive" title="b5">bnt</font>, <font color="Olive" title="b6">bn</font> being  <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <span class="p1">(<span class="default"><a href="bagorder.html#K14" title="BAGORDER:func.14">FinPoset</a> <font color="Maroon" title="c1">R</font></span>)</span> st <br/>( <font color="Olive" title="b1">Sn2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $2 <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  &amp; ( for <font color="Olive" title="b7">i</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>   ex <font color="Olive" title="b8">Sn2i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font> st <br/>( <font color="Olive" title="b8">Sn2i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">Sn2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b7">i</font> &amp; <font color="Olive" title="b8">Sn2i</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>  &amp; <font color="Olive" title="b2">Smax</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bagorder.html#K11" title="BAGORDER:func.11">PosetMax</a> <font color="Olive" title="b8">Sn2i</font> ) ) &amp; <font color="Olive" title="b3">an</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bagorder.html#K15" title="BAGORDER:func.15">MinElement</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b2">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font> &amp; <font color="Olive" title="b4">ix</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">Smax</font> <a href="dickson.html#K7" title="DICKSON:func.7">mindex</a> <font color="Olive" title="b3">an</font> &amp; <font color="Olive" title="b5">bnt</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="bagorder.html#K16" title="BAGORDER:func.16">SeqShift</a> <font color="Olive" title="b1">Sn2</font>,<font color="Olive" title="b4">ix</font> &amp; ( for <font color="Olive" title="b7">i</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>  holds  <font color="Olive" title="b6">bn</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b5">bnt</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b7">i</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b3">an</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> ) &amp; ( for <font color="Olive" title="b7">i</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="Olive" title="b4">ix</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b7">i</font> holds <br/><font color="Olive" title="b2">Smax</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">an</font> ) &amp; $3 <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="Olive" title="b3">an</font>,<font color="Olive" title="b6">bn</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> );<br/>
<a NAME="E7:73"/><i><font color="Green" title="E47">A3</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> <br/> for <font color="Olive" title="b2">Sn</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>,<font color="Maroon" title="c8">zz</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>  ex <font color="Olive" title="b3">Sn1</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>,<font color="Maroon" title="c8">zz</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>,<font color="Olive" title="b2">Sn</font>,<font color="Olive" title="b3">Sn1</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">n</font> be    <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> ;<br/><b>let </b><font color="Maroon" title="c11">Sn</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>,<font color="Maroon" title="c8">zz</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>;<br/>



<b>set </b><font color="Maroon" title="c12">Sn2</font> = <font color="Maroon" title="c11">Sn</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> ;<br/>
<a NAME="E1:73_1"/>
<font color="Maroon" title="c11">Sn</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">zz</font>
 
;<br/>
<b>then consider </b><font color="Maroon" title="c13">z</font> being   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <span class="p1">(<span class="default"><a href="bagorder.html#K14" title="BAGORDER:func.14">FinPoset</a> <font color="Maroon" title="c1">R</font></span>)</span><b> such that </b><br/><a NAME="E3:73_1"/><i><font color="Green" title="E48">A4</font></i>: 
( <font color="Maroon" title="c13">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">Sn</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a>  &amp; <font color="Maroon" title="c13">z</font> is <a href="wellfnd1.html#V3" title="WELLFND1:attr.3">descending</a> )
 ;<br/>
<b>reconsider </b><font color="Maroon" title="c14">Sn2</font> = <font color="Maroon" title="c11">Sn</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a>  as   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <span class="p1">(<span class="default"><a href="bagorder.html#K14" title="BAGORDER:func.14">FinPoset</a> <font color="Maroon" title="c1">R</font></span>)</span> <b>by </b><i><a class="txt" href="bagorder.html#E3:73_1"><i><font color="Green" title="E48">A4</font></i></a></i>;<br/>
<div><i><font color="Green" title="E49">A5</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c15">i</font> be    <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> ;<br/><b>reconsider </b><font color="Maroon" title="c16">Sn2i1</font> = <font color="Maroon" title="c14">Sn2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font> ;<br/><b>assume </b><a NAME="E2:73_1_1"/><i><font color="Green" title="E50">A6</font></i>: 
<font color="Maroon" title="c14">Sn2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><a NAME="E3:73_1_1"/>
( <font color="Maroon" title="c14">Sn2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c14">Sn2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">i</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">Sn2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c15">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">Sn2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">i</font></span>)</span></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="bagorder.html#K13" title="BAGORDER:func.13">FinOrd</a> <font color="Maroon" title="c1">R</font> )
 <b>by </b><i><a class="txt" href="bagorder.html#E3:73_1"><i><font color="Green" title="E48">A4</font></i></a>, <a class="ref" href="wellfnd1.html#D7" title="WELLFND1:def.7">WELLFND1:def 7</a></i>;<br/><b>hence </b><a NAME="E4:73_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="bagorder.html#E2:73_1_1"><i><font color="Green" title="E50">A6</font></i></a>, <a class="ref" href="bagorder.html#T37" target="_self" title="BAGORDER:th.37">Th37</a></i>;<br/></div><b>end;</b></div>
<b>defpred </b><font color="Maroon">S<sub>2</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> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> ex <font color="Olive" title="b1">Sn2i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font> st <br/>( <font color="Olive" title="b1">Sn2i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">Sn2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> $1 &amp; <font color="Olive" title="b1">Sn2i</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>  &amp; $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bagorder.html#K11" title="BAGORDER:func.11">PosetMax</a> <font color="Olive" title="b1">Sn2i</font> );<br/>
<a NAME="E6:73_1"/><i><font color="Green" title="E50">A7</font></i>: 
for <font color="Olive" title="b1">i</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>   ex <font color="Olive" title="b2">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font> st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">y</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c15">i</font> be    <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> ;<br/>

<b>set </b><font color="Maroon" title="c16">Sn2i</font> = <font color="Maroon" title="c14">Sn2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">i</font>;<br/>
<b>reconsider </b><font color="Maroon" title="c17">Sn2i</font> = <font color="Maroon" title="c14">Sn2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">i</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font> ;<br/>
<b>set </b><font color="Maroon" title="c18">y</font> =  <a href="bagorder.html#K11" title="BAGORDER:func.11">PosetMax</a> <font color="Maroon" title="c17">Sn2i</font>;<br/>
<b>take </b>
 <a href="bagorder.html#K11" title="BAGORDER:func.11">PosetMax</a> <font color="Maroon" title="c17">Sn2i</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c17">Sn2i</font>
;<br/>

<b>thus </b><a NAME="E2:73_1_2"/>
<font color="Maroon" title="c17">Sn2i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">Sn2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">i</font>
 ;<br/>

<b>thus </b><a NAME="E3:73_1_2"/>
<font color="Maroon" title="c17">Sn2i</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> 
 <b>by </b><i><a class="txt" href="bagorder.html#E5:73_1"><i><font color="Green" title="E49">A5</font></i></a></i>;<br/>

<b>thus </b><a NAME="E4:73_1_2"/>
 <a href="bagorder.html#K11" title="BAGORDER:func.11">PosetMax</a> <font color="Maroon" title="c17">Sn2i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bagorder.html#K11" title="BAGORDER:func.11">PosetMax</a> <font color="Maroon" title="c17">Sn2i</font>
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c15">Smax</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font><b> such that </b><br/><a NAME="E8:73_1"/><i><font color="Green" title="E51">A8</font></i>: 
for <font color="Olive" title="b1">i</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>  holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">i</font>,<font color="Maroon" title="c15">Smax</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3" title="FUNCT_2:sch.3">FUNCT_2:sch 3</a>(<a class="txt" href="bagorder.html#E6:73_1"><i><font color="Green" title="E50">A7</font></i></a>);<br/></i>
<b>set </b><font color="Maroon" title="c16">an</font> =  <a href="bagorder.html#K15" title="BAGORDER:func.15">MinElement</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c15">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font>;<br/>
<b>set </b><font color="Maroon" title="c17">ix</font> = <font color="Maroon" title="c15">Smax</font> <a href="dickson.html#K7" title="DICKSON:func.7">mindex</a> <span class="p1">(<span class="default"><a href="bagorder.html#K15" title="BAGORDER:func.15">MinElement</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c15">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c18">bnt</font> =  <a href="bagorder.html#K16" title="BAGORDER:func.16">SeqShift</a> <font color="Maroon" title="c14">Sn2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c15">Smax</font> <a href="dickson.html#K7" title="DICKSON:func.7">mindex</a> <span class="p2">(<span class="default"><a href="bagorder.html#K15" title="BAGORDER:func.15">MinElement</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c15">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font></span>)</span></span>)</span>;<br/>
<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> ex <font color="Olive" title="b1">bni</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font> st <br/>( <font color="Olive" title="b1">bni</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bagorder.html#K16" title="BAGORDER:func.16">SeqShift</a> <font color="Maroon" title="c14">Sn2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c15">Smax</font> <a href="dickson.html#K7" title="DICKSON:func.7">mindex</a> <span class="p4">(<span class="default"><a href="bagorder.html#K15" title="BAGORDER:func.15">MinElement</a> <span class="p5">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c15">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1</span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="bagorder.html#K15" title="BAGORDER:func.15">MinElement</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c15">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">bni</font> );<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c19">i</font> be    <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> ;<br/><b>set </b><font color="Maroon" title="c20">bni</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bagorder.html#K16" title="BAGORDER:func.16">SeqShift</a> <font color="Maroon" title="c14">Sn2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c15">Smax</font> <a href="dickson.html#K7" title="DICKSON:func.7">mindex</a> <span class="p4">(<span class="default"><a href="bagorder.html#K15" title="BAGORDER:func.15">MinElement</a> <span class="p5">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c15">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">i</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="bagorder.html#K15" title="BAGORDER:func.15">MinElement</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c15">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>;<br/><b>reconsider </b><font color="Maroon" title="c21">k</font> = <span class="p1">(<span class="default"><a href="bagorder.html#K16" title="BAGORDER:func.16">SeqShift</a> <font color="Maroon" title="c14">Sn2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c15">Smax</font> <a href="dickson.html#K7" title="DICKSON:func.7">mindex</a> <span class="p3">(<span class="default"><a href="bagorder.html#K15" title="BAGORDER:func.15">MinElement</a> <span class="p4">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c15">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">i</font> as  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font> <b>by </b><i><a class="ref" href="finsub_1.html#D5" title="FINSUB_1:def.5">FINSUB_1:def 5</a></i>;<br/><a NAME="E2:73_1_3"/>
<font color="Maroon" title="c21">k</font> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="bagorder.html#K15" title="BAGORDER:func.15">MinElement</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c15">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>
 <b>by </b><i><a class="ref" href="finsub_1.html#D5" title="FINSUB_1:def.5">FINSUB_1:def 5</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c22">bni</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bagorder.html#K16" title="BAGORDER:func.16">SeqShift</a> <font color="Maroon" title="c14">Sn2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c15">Smax</font> <a href="dickson.html#K7" title="DICKSON:func.7">mindex</a> <span class="p4">(<span class="default"><a href="bagorder.html#K15" title="BAGORDER:func.15">MinElement</a> <span class="p5">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c15">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">i</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="bagorder.html#K15" title="BAGORDER:func.15">MinElement</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c15">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font> ;<br/><b>set </b><font color="Maroon" title="c23">y</font> = <font color="Maroon" title="c22">bni</font>;<br/><b>take </b><font color="Maroon" title="c24">y</font> = <font color="Maroon" title="c22">bni</font>;<br/><b>take </b><font color="Maroon" title="c25">bni</font> = <font color="Maroon" title="c22">bni</font>;<br/><b>thus </b><a NAME="E4:73_1_3"/>
<font color="Maroon" title="c25">bni</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bagorder.html#K16" title="BAGORDER:func.16">SeqShift</a> <font color="Maroon" title="c14">Sn2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c15">Smax</font> <a href="dickson.html#K7" title="DICKSON:func.7">mindex</a> <span class="p4">(<span class="default"><a href="bagorder.html#K15" title="BAGORDER:func.15">MinElement</a> <span class="p5">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c15">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">i</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="bagorder.html#K15" title="BAGORDER:func.15">MinElement</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c15">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 ;<br/><b>thus </b><a NAME="E5:73_1_3"/>
<font color="Maroon" title="c24">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">bni</font>
 ;<br/></div><b>end;</b></div>
<a NAME="E10:73_1"/><b>then </b><i><font color="Green" title="E52">A9</font></i>: 
for <font color="Olive" title="b1">i</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>   ex <font color="Olive" title="b2">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font> st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">y</font>]
 
;<br/>
<b>defpred </b><font color="Maroon">S<sub>4</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" title="c15">Smax</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c15">Smax</font> <a href="dickson.html#K7" title="DICKSON:func.7">mindex</a> <span class="p3">(<span class="default"><a href="bagorder.html#K15" title="BAGORDER:func.15">MinElement</a> <span class="p4">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c15">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> $1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bagorder.html#K15" title="BAGORDER:func.15">MinElement</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c15">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font>;<br/>
<a NAME="E11:73_1"/><i><font color="Green" title="E53">A10</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c15">Smax</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/>
<a NAME="E12:73_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c15">Smax</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 <font color="Maroon" title="c1">R</font>
 
<b>by </b><i><a class="ref" href="relset_1.html#T12" title="RELSET_1:th.12">RELSET_1:12</a></i>;<br/>
<a NAME="E13:73_1"/><b>then </b><i><font color="Green" title="E54">A11</font></i>: 
(  <a href="bagorder.html#K15" title="BAGORDER:func.15">MinElement</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c15">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c15">Smax</font> &amp;  <a href="bagorder.html#K15" title="BAGORDER:func.15">MinElement</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c15">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font> <a href="waybel_4.html#R4" title="WAYBEL_4:pred.4">is_minimal_wrt</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c15">Smax</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font> )
 
<b>by </b><i><a class="txt" href="bagorder.html#E1:73"><i><font color="Green" title="E45">A1</font></i></a>, <a class="ref" href="bagorder.html#D19" target="_self" title="BAGORDER:def.19">Def19</a></i>;<br/>
<a NAME="E14:73_1"/><b>then </b><i><font color="Green" title="E55">A12</font></i>: 
<font color="Maroon">S<sub>4</sub></font>[0]
 
<b>by </b><i><a class="txt" href="bagorder.html#E11:73_1"><i><font color="Green" title="E53">A10</font></i></a>, <a class="ref" href="dickson.html#D11" title="DICKSON:def.11">DICKSON:def 11</a></i>;<br/>
<div><i><font color="Green" title="E56">A13</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c19">k</font> be    <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> ;<br/><b>assume </b><a NAME="E1:73_1_4"/><i><font color="Green" title="E57">A14</font></i>: 
<font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c19">k</font>]
 ;<br/><b>set </b><font color="Maroon" title="c20">ixk</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c15">Smax</font> <a href="dickson.html#K7" title="DICKSON:func.7">mindex</a> <span class="p2">(<span class="default"><a href="bagorder.html#K15" title="BAGORDER:func.15">MinElement</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c15">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c19">k</font>;<br/><b>set </b><font color="Maroon" title="c21">ixk1</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c15">Smax</font> <a href="dickson.html#K7" title="DICKSON:func.7">mindex</a> <span class="p2">(<span class="default"><a href="bagorder.html#K15" title="BAGORDER:func.15">MinElement</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c15">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>;<br/><b>set </b><font color="Maroon" title="c22">ixk1'</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c15">Smax</font> <a href="dickson.html#K7" title="DICKSON:func.7">mindex</a> <span class="p3">(<span class="default"><a href="bagorder.html#K15" title="BAGORDER:func.15">MinElement</a> <span class="p4">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c15">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c19">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1;<br/><b>consider </b><font color="Maroon" title="c23">Sn2ixk</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font><b> such that </b><br/><a NAME="E3:73_1_4"/><i><font color="Green" title="E58">A15</font></i>: 
( <font color="Maroon" title="c23">Sn2ixk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">Sn2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c15">Smax</font> <a href="dickson.html#K7" title="DICKSON:func.7">mindex</a> <span class="p3">(<span class="default"><a href="bagorder.html#K15" title="BAGORDER:func.15">MinElement</a> <span class="p4">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c15">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c19">k</font></span>)</span> &amp; <font color="Maroon" title="c23">Sn2ixk</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>  &amp; <font color="Maroon" title="c15">Smax</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c15">Smax</font> <a href="dickson.html#K7" title="DICKSON:func.7">mindex</a> <span class="p3">(<span class="default"><a href="bagorder.html#K15" title="BAGORDER:func.15">MinElement</a> <span class="p4">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c15">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c19">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bagorder.html#K11" title="BAGORDER:func.11">PosetMax</a> <font color="Maroon" title="c23">Sn2ixk</font> )
 <b>by </b><i><a class="txt" href="bagorder.html#E8:73_1"><i><font color="Green" title="E51">A8</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c24">Sn2ixk1</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font><b> such that </b><br/><a NAME="E5:73_1_4"/><i><font color="Green" title="E59">A16</font></i>: 
( <font color="Maroon" title="c24">Sn2ixk1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">Sn2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c15">Smax</font> <a href="dickson.html#K7" title="DICKSON:func.7">mindex</a> <span class="p3">(<span class="default"><a href="bagorder.html#K15" title="BAGORDER:func.15">MinElement</a> <span class="p4">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c15">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon" title="c24">Sn2ixk1</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>  &amp; <font color="Maroon" title="c15">Smax</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c15">Smax</font> <a href="dickson.html#K7" title="DICKSON:func.7">mindex</a> <span class="p3">(<span class="default"><a href="bagorder.html#K15" title="BAGORDER:func.15">MinElement</a> <span class="p4">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c15">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bagorder.html#K11" title="BAGORDER:func.11">PosetMax</a> <font color="Maroon" title="c24">Sn2ixk1</font> )
 <b>by </b><i><a class="txt" href="bagorder.html#E8:73_1"><i><font color="Green" title="E51">A8</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c25">Sn2ixk'</font> = <font color="Maroon" title="c23">Sn2ixk</font>, <font color="Maroon" title="c26">Sn2ixk1'</font> = <font color="Maroon" title="c24">Sn2ixk1</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="bagorder.html#K14" title="BAGORDER:func.14">FinPoset</a> <font color="Maroon" title="c1">R</font></span>)</span> ;<br/><a NAME="E7:73_1_4"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c15">Smax</font> <a href="dickson.html#K7" title="DICKSON:func.7">mindex</a> <span class="p2">(<span class="default"><a href="bagorder.html#K15" title="BAGORDER:func.15">MinElement</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c15">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c15">Smax</font> <a href="dickson.html#K7" title="DICKSON:func.7">mindex</a> <span class="p3">(<span class="default"><a href="bagorder.html#K15" title="BAGORDER:func.15">MinElement</a> <span class="p4">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c15">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c19">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ;<br/><a NAME="E8:73_1_4"/><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="c26">Sn2ixk1'</font>,<font color="Maroon" title="c25">Sn2ixk'</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="bagorder.html#K13" title="BAGORDER:func.13">FinOrd</a> <font color="Maroon" title="c1">R</font>
 <b>by </b><i><a class="txt" href="bagorder.html#E3:73_1"><i><font color="Green" title="E48">A4</font></i></a>, <a class="txt" href="bagorder.html#E3:73_1_4"><i><font color="Green" title="E58">A15</font></i></a>, <a class="txt" href="bagorder.html#E5:73_1_4"><i><font color="Green" title="E59">A16</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="c27">x</font>, <font color="Maroon" title="c28">y</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font><b> such that </b><br/><a NAME="E10:73_1_4"/><i><font color="Green" title="E60">A17</font></i>: 
( <font color="Maroon" title="c24">Sn2ixk1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">x</font> &amp; <font color="Maroon" title="c23">Sn2ixk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">y</font> )
 <b>and </b><br/><a NAME="E11:73_1_4"/><i><font color="Green" title="E61">A18</font></i>: 
( <font color="Maroon" title="c27">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or ( <font color="Maroon" title="c27">x</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>  &amp; <font color="Maroon" title="c28">y</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>  &amp;  <a href="bagorder.html#K11" title="BAGORDER:func.11">PosetMax</a> <font color="Maroon" title="c27">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="bagorder.html#K11" title="BAGORDER:func.11">PosetMax</a> <font color="Maroon" title="c28">y</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><a href="bagorder.html#K11" title="BAGORDER:func.11">PosetMax</a> <font color="Maroon" title="c27">x</font></span>)</span>,<span class="p2">(<span class="default"><a href="bagorder.html#K11" title="BAGORDER:func.11">PosetMax</a> <font color="Maroon" title="c28">y</font></span>)</span></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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font> ) or ( <font color="Maroon" title="c27">x</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>  &amp; <font color="Maroon" title="c28">y</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>  &amp;  <a href="bagorder.html#K11" title="BAGORDER:func.11">PosetMax</a> <font color="Maroon" title="c27">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bagorder.html#K11" title="BAGORDER:func.11">PosetMax</a> <font color="Maroon" title="c28">y</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c27">x</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p4">(<span class="default"><a href="bagorder.html#K11" title="BAGORDER:func.11">PosetMax</a> <font color="Maroon" title="c27">x</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c28">y</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p4">(<span class="default"><a href="bagorder.html#K11" title="BAGORDER:func.11">PosetMax</a> <font color="Maroon" title="c28">y</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></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="bagorder.html#K13" title="BAGORDER:func.13">FinOrd</a> <font color="Maroon" title="c1">R</font> ) )
 <b>by </b><i><a class="ref" href="bagorder.html#T40" target="_self" title="BAGORDER:th.40">Th40</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="73_1_4_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:73_1_4_1"/>
( <font color="Maroon" title="c27">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or ( <font color="Maroon" title="c27">x</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>  &amp; <font color="Maroon" title="c28">y</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>  &amp;  <a href="bagorder.html#K11" title="BAGORDER:func.11">PosetMax</a> <font color="Maroon" title="c27">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="bagorder.html#K11" title="BAGORDER:func.11">PosetMax</a> <font color="Maroon" title="c28">y</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><a href="bagorder.html#K11" title="BAGORDER:func.11">PosetMax</a> <font color="Maroon" title="c27">x</font></span>)</span>,<span class="p2">(<span class="default"><a href="bagorder.html#K11" title="BAGORDER:func.11">PosetMax</a> <font color="Maroon" title="c28">y</font></span>)</span></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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font> ) or ( <font color="Maroon" title="c27">x</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>  &amp; <font color="Maroon" title="c28">y</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>  &amp;  <a href="bagorder.html#K11" title="BAGORDER:func.11">PosetMax</a> <font color="Maroon" title="c27">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bagorder.html#K11" title="BAGORDER:func.11">PosetMax</a> <font color="Maroon" title="c28">y</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c27">x</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p4">(<span class="default"><a href="bagorder.html#K11" title="BAGORDER:func.11">PosetMax</a> <font color="Maroon" title="c27">x</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c28">y</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p4">(<span class="default"><a href="bagorder.html#K11" title="BAGORDER:func.11">PosetMax</a> <font color="Maroon" title="c28">y</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></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="bagorder.html#K13" title="BAGORDER:func.13">FinOrd</a> <font color="Maroon" title="c1">R</font> ) )
 </span><b>by </b><i><a class="txt" href="bagorder.html#E11:73_1_4"><i><font color="Green" title="E61">A18</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="73_1_4_1_1"><b>suppose </b></a><a NAME="E1:73_1_4_1_1"/>
<font color="Maroon" title="c27">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:73_1_4_1_1"/>
<font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c19">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 <b>by </b><i><a class="txt" href="bagorder.html#E5:73_1_4"><i><font color="Green" title="E59">A16</font></i></a>, <a class="txt" href="bagorder.html#E10:73_1_4"><i><font color="Green" title="E60">A17</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="73_1_4_1_2"><b>suppose </b></a><a NAME="E1:73_1_4_1_2"/><i><font color="Green" title="E62">A19</font></i>: 
( <font color="Maroon" title="c27">x</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>  &amp; <font color="Maroon" title="c28">y</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>  &amp;  <a href="bagorder.html#K11" title="BAGORDER:func.11">PosetMax</a> <font color="Maroon" title="c27">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="bagorder.html#K11" title="BAGORDER:func.11">PosetMax</a> <font color="Maroon" title="c28">y</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><a href="bagorder.html#K11" title="BAGORDER:func.11">PosetMax</a> <font color="Maroon" title="c27">x</font></span>)</span>,<span class="p2">(<span class="default"><a href="bagorder.html#K11" title="BAGORDER:func.11">PosetMax</a> <font color="Maroon" title="c28">y</font></span>)</span></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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font> )
 ;<br/><div class="add"><a NAME="E2:73_1_4_1_2"/>
<font color="Maroon" title="c15">Smax</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c15">Smax</font> <a href="dickson.html#K7" title="DICKSON:func.7">mindex</a> <span class="p3">(<span class="default"><a href="bagorder.html#K15" title="BAGORDER:func.15">MinElement</a> <span class="p4">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c15">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c15">Smax</font>
 <b>by </b><i><a class="txt" href="bagorder.html#E11:73_1"><i><font color="Green" title="E53">A10</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><b>hence </b><a NAME="E3:73_1_4_1_2"/>
<font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c19">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 <b>by </b><i><a class="txt" href="bagorder.html#E13:73_1"><i><font color="Green" title="E54">A11</font></i></a>, <a class="txt" href="bagorder.html#E1:73_1_4"><i><font color="Green" title="E57">A14</font></i></a>, <a class="txt" href="bagorder.html#E3:73_1_4"><i><font color="Green" title="E58">A15</font></i></a>, <a class="txt" href="bagorder.html#E5:73_1_4"><i><font color="Green" title="E59">A16</font></i></a>, <a class="txt" href="bagorder.html#E10:73_1_4"><i><font color="Green" title="E60">A17</font></i></a>, <a class="txt" href="bagorder.html#E1:73_1_4_1_2"><i><font color="Green" title="E62">A19</font></i></a>, <a class="ref" href="waybel_4.html#D26" title="WAYBEL_4:def.26">WAYBEL_4:def 26</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="73_1_4_1_3"><b>suppose </b></a><a NAME="E1:73_1_4_1_3"/>
( <font color="Maroon" title="c27">x</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>  &amp; <font color="Maroon" title="c28">y</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>  &amp;  <a href="bagorder.html#K11" title="BAGORDER:func.11">PosetMax</a> <font color="Maroon" title="c27">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bagorder.html#K11" title="BAGORDER:func.11">PosetMax</a> <font color="Maroon" title="c28">y</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c27">x</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p4">(<span class="default"><a href="bagorder.html#K11" title="BAGORDER:func.11">PosetMax</a> <font color="Maroon" title="c27">x</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c28">y</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p4">(<span class="default"><a href="bagorder.html#K11" title="BAGORDER:func.11">PosetMax</a> <font color="Maroon" title="c28">y</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></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="bagorder.html#K13" title="BAGORDER:func.13">FinOrd</a> <font color="Maroon" title="c1">R</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:73_1_4_1_3"/>
<font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c19">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 <b>by </b><i><a class="txt" href="bagorder.html#E1:73_1_4"><i><font color="Green" title="E57">A14</font></i></a>, <a class="txt" href="bagorder.html#E3:73_1_4"><i><font color="Green" title="E58">A15</font></i></a>, <a class="txt" href="bagorder.html#E5:73_1_4"><i><font color="Green" title="E59">A16</font></i></a>, <a class="txt" href="bagorder.html#E10:73_1_4"><i><font color="Green" title="E60">A17</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E16:73_1"/><i><font color="Green" title="E57">A20</font></i>: 
for <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>  holds  <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">k</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="bagorder.html#E14:73_1"><i><font color="Green" title="E55">A12</font></i></a>, <a class="txt" href="bagorder.html#E15:73_1"><i><font color="Green" title="E56">A13</font></i></a>);<br/></i>
<div><i><font color="Green" title="E58">A21</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c19">i</font> be    <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> ;<br/><b>assume </b><a NAME="E1:73_1_5"/><i><font color="Green" title="E59">A22</font></i>: 
<font color="Maroon" title="c15">Smax</font> <a href="dickson.html#K7" title="DICKSON:func.7">mindex</a> <span class="p1">(<span class="default"><a href="bagorder.html#K15" title="BAGORDER:func.15">MinElement</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c15">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">i</font>
 ;<br/><b>consider </b><font color="Maroon" title="c20">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E3:73_1_5"/><i><font color="Green" title="E60">A23</font></i>: 
<font color="Maroon" title="c19">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">Smax</font> <a href="dickson.html#K7" title="DICKSON:func.7">mindex</a> <span class="p2">(<span class="default"><a href="bagorder.html#K15" title="BAGORDER:func.15">MinElement</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c15">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c20">k</font>
 <b>by </b><i><a class="txt" href="bagorder.html#E1:73_1_5"><i><font color="Green" title="E59">A22</font></i></a>, <a class="ref" href="nat_1.html#T10" title="NAT_1:th.10">NAT_1:10</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c21">k</font> = <font color="Maroon" title="c20">k</font> as    <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>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><a NAME="E5:73_1_5"/>
<font color="Maroon" title="c19">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">Smax</font> <a href="dickson.html#K7" title="DICKSON:func.7">mindex</a> <span class="p2">(<span class="default"><a href="bagorder.html#K15" title="BAGORDER:func.15">MinElement</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c15">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c21">k</font>
 <b>by </b><i><a class="txt" href="bagorder.html#E3:73_1_5"><i><font color="Green" title="E60">A23</font></i></a></i>;<br/><b>hence </b><a NAME="E6:73_1_5"/>
<font color="Maroon" title="c15">Smax</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bagorder.html#K15" title="BAGORDER:func.15">MinElement</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c15">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font>
 <b>by </b><i><a class="txt" href="bagorder.html#E16:73_1"><i><font color="Green" title="E57">A20</font></i></a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green" title="E59">A24</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1_6"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c19">i</font> be    <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> ;<br/><b>assume </b><a NAME="E1:73_1_6"/><i><font color="Green" title="E60">A25</font></i>: 
<font color="Maroon" title="c15">Smax</font> <a href="dickson.html#K7" title="DICKSON:func.7">mindex</a> <span class="p1">(<span class="default"><a href="bagorder.html#K15" title="BAGORDER:func.15">MinElement</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c15">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">i</font>
 ;<br/><b>consider </b><font color="Maroon" title="c20">Sn2i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font><b> such that </b><br/><a NAME="E3:73_1_6"/><i><font color="Green" title="E61">A26</font></i>: 
( <font color="Maroon" title="c20">Sn2i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">Sn2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">i</font> &amp; <font color="Maroon" title="c20">Sn2i</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>  &amp; <font color="Maroon" title="c15">Smax</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bagorder.html#K11" title="BAGORDER:func.11">PosetMax</a> <font color="Maroon" title="c20">Sn2i</font> )
 <b>by </b><i><a class="txt" href="bagorder.html#E8:73_1"><i><font color="Green" title="E51">A8</font></i></a></i>;<br/><b>take </b><font color="Maroon" title="c21">Sn2i</font> = <font color="Maroon" title="c20">Sn2i</font>;<br/><b>thus </b><a NAME="E4:73_1_6"/>
<font color="Maroon" title="c21">Sn2i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">Sn2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">i</font>
 <b>by </b><i><a class="txt" href="bagorder.html#E3:73_1_6"><i><font color="Green" title="E61">A26</font></i></a></i>;<br/><b>thus </b><a NAME="E5:73_1_6"/>
 <a href="bagorder.html#K11" title="BAGORDER:func.11">PosetMax</a> <font color="Maroon" title="c21">Sn2i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bagorder.html#K15" title="BAGORDER:func.15">MinElement</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c15">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font>
 <b>by </b><i><a class="txt" href="bagorder.html#E17:73_1"><i><font color="Green" title="E58">A21</font></i></a>, <a class="txt" href="bagorder.html#E1:73_1_6"><i><font color="Green" title="E60">A25</font></i></a>, <a class="txt" href="bagorder.html#E3:73_1_6"><i><font color="Green" title="E61">A26</font></i></a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green" title="E60">A27</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1_7"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c19">i</font> be    <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> ;<br/><b>set </b><font color="Maroon" title="c20">bnti</font> = <span class="p1">(<span class="default"><a href="bagorder.html#K16" title="BAGORDER:func.16">SeqShift</a> <font color="Maroon" title="c14">Sn2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c15">Smax</font> <a href="dickson.html#K7" title="DICKSON:func.7">mindex</a> <span class="p3">(<span class="default"><a href="bagorder.html#K15" title="BAGORDER:func.15">MinElement</a> <span class="p4">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c15">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">i</font>;<br/><b>reconsider </b><font color="Maroon" title="c21">bnti</font> = <span class="p1">(<span class="default"><a href="bagorder.html#K16" title="BAGORDER:func.16">SeqShift</a> <font color="Maroon" title="c14">Sn2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c15">Smax</font> <a href="dickson.html#K7" title="DICKSON:func.7">mindex</a> <span class="p3">(<span class="default"><a href="bagorder.html#K15" title="BAGORDER:func.15">MinElement</a> <span class="p4">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c15">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">i</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font> ;<br/><b>take </b><font color="Maroon" title="c22">bnti</font> = <font color="Maroon" title="c21">bnti</font>;<br/><b>thus </b><a NAME="E2:73_1_7"/>
<font color="Maroon" title="c22">bnti</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="bagorder.html#K16" title="BAGORDER:func.16">SeqShift</a> <font color="Maroon" title="c14">Sn2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c15">Smax</font> <a href="dickson.html#K7" title="DICKSON:func.7">mindex</a> <span class="p3">(<span class="default"><a href="bagorder.html#K15" title="BAGORDER:func.15">MinElement</a> <span class="p4">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c15">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">i</font>
 ;<br/><b>set </b><font color="Maroon" title="c23">iix</font> = <font color="Maroon" title="c19">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">Smax</font> <a href="dickson.html#K7" title="DICKSON:func.7">mindex</a> <span class="p2">(<span class="default"><a href="bagorder.html#K15" title="BAGORDER:func.15">MinElement</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c15">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font></span>)</span></span>)</span>;<br/><a NAME="E3:73_1_7"/>
<font color="Maroon" title="c15">Smax</font> <a href="dickson.html#K7" title="DICKSON:func.7">mindex</a> <span class="p1">(<span class="default"><a href="bagorder.html#K15" title="BAGORDER:func.15">MinElement</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c15">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">Smax</font> <a href="dickson.html#K7" title="DICKSON:func.7">mindex</a> <span class="p2">(<span class="default"><a href="bagorder.html#K15" title="BAGORDER:func.15">MinElement</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c15">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/><b>then consider </b><font color="Maroon" title="c24">Sn2iix</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font><b> such that </b><br/><a NAME="E5:73_1_7"/><i><font color="Green" title="E61">A28</font></i>: 
( <font color="Maroon" title="c24">Sn2iix</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">Sn2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">Smax</font> <a href="dickson.html#K7" title="DICKSON:func.7">mindex</a> <span class="p3">(<span class="default"><a href="bagorder.html#K15" title="BAGORDER:func.15">MinElement</a> <span class="p4">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c15">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font></span>)</span></span>)</span></span>)</span> &amp;  <a href="bagorder.html#K11" title="BAGORDER:func.11">PosetMax</a> <font color="Maroon" title="c24">Sn2iix</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bagorder.html#K15" title="BAGORDER:func.15">MinElement</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c15">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font> )
 <b>by </b><i><a class="txt" href="bagorder.html#E18:73_1"><i><font color="Green" title="E59">A24</font></i></a></i>;<br/><b>thus </b><a NAME="E6:73_1_7"/>
 <a href="bagorder.html#K11" title="BAGORDER:func.11">PosetMax</a> <font color="Maroon" title="c22">bnti</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bagorder.html#K15" title="BAGORDER:func.15">MinElement</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c15">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font>
 <b>by </b><i><a class="txt" href="bagorder.html#E5:73_1_7"><i><font color="Green" title="E61">A28</font></i></a>, <a class="ref" href="bagorder.html#D20" target="_self" title="BAGORDER:def.20">Def20</a></i>;<br/></div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c19">bn</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> , <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font><b> such that </b><br/><a NAME="E21:73_1"/><i><font color="Green" title="E61">A29</font></i>: 
for <font color="Olive" title="b1">i</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>  holds  <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">i</font>,<font color="Maroon" title="c19">bn</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3" title="FUNCT_2:sch.3">FUNCT_2:sch 3</a>(<a class="txt" href="bagorder.html#E10:73_1"><i><font color="Green" title="E52">A9</font></i></a>);<br/></i>
<b>reconsider </b><font color="Maroon" title="c20">bn</font> = <font color="Maroon" title="c19">bn</font> as   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <span class="p1">(<span class="default"><a href="bagorder.html#K14" title="BAGORDER:func.14">FinPoset</a> <font color="Maroon" title="c1">R</font></span>)</span> ;<br/>
<b>set </b><font color="Maroon" title="c21">Sn1</font> = <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><a href="bagorder.html#K15" title="BAGORDER:func.15">MinElement</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c15">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font></span>)</span>,<font color="Maroon" title="c20">bn</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>;<br/>
<a NAME="E23:73_1"/><i><font color="Green" title="E62">A30</font></i>: 
 <a href="bagorder.html#K16" title="BAGORDER:func.16">SeqShift</a> <font color="Maroon" title="c14">Sn2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c15">Smax</font> <a href="dickson.html#K7" title="DICKSON:func.7">mindex</a> <span class="p2">(<span class="default"><a href="bagorder.html#K15" title="BAGORDER:func.15">MinElement</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c15">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font></span>)</span></span>)</span> is <a href="wellfnd1.html#V3" title="WELLFND1:attr.3">descending</a>
 
<b>by </b><i><a class="txt" href="bagorder.html#E3:73_1"><i><font color="Green" title="E48">A4</font></i></a>, <a class="ref" href="bagorder.html#T41" target="_self" title="BAGORDER:th.41">Th41</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1_8"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c22">i</font> be    <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> ;<br/><b>consider </b><font color="Maroon" title="c23">bni</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font><b> such that </b><br/><a NAME="E2:73_1_8"/><i><font color="Green" title="E63">A31</font></i>: 
( <font color="Maroon" title="c23">bni</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bagorder.html#K16" title="BAGORDER:func.16">SeqShift</a> <font color="Maroon" title="c14">Sn2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c15">Smax</font> <a href="dickson.html#K7" title="DICKSON:func.7">mindex</a> <span class="p4">(<span class="default"><a href="bagorder.html#K15" title="BAGORDER:func.15">MinElement</a> <span class="p5">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c15">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">i</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="bagorder.html#K15" title="BAGORDER:func.15">MinElement</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c15">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <font color="Maroon" title="c20">bn</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">bni</font> )
 <b>by </b><i><a class="txt" href="bagorder.html#E21:73_1"><i><font color="Green" title="E61">A29</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c24">bni1</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font><b> such that </b><br/><a NAME="E4:73_1_8"/><i><font color="Green" title="E64">A32</font></i>: 
( <font color="Maroon" title="c24">bni1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bagorder.html#K16" title="BAGORDER:func.16">SeqShift</a> <font color="Maroon" title="c14">Sn2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c15">Smax</font> <a href="dickson.html#K7" title="DICKSON:func.7">mindex</a> <span class="p4">(<span class="default"><a href="bagorder.html#K15" title="BAGORDER:func.15">MinElement</a> <span class="p5">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c15">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c22">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="bagorder.html#K15" title="BAGORDER:func.15">MinElement</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c15">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <font color="Maroon" title="c20">bn</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">bni1</font> )
 <b>by </b><i><a class="txt" href="bagorder.html#E21:73_1"><i><font color="Green" title="E61">A29</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c25">bnti</font> = <span class="p1">(<span class="default"><a href="bagorder.html#K16" title="BAGORDER:func.16">SeqShift</a> <font color="Maroon" title="c14">Sn2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c15">Smax</font> <a href="dickson.html#K7" title="DICKSON:func.7">mindex</a> <span class="p3">(<span class="default"><a href="bagorder.html#K15" title="BAGORDER:func.15">MinElement</a> <span class="p4">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c15">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">i</font>, <font color="Maroon" title="c26">bnti1</font> = <span class="p1">(<span class="default"><a href="bagorder.html#K16" title="BAGORDER:func.16">SeqShift</a> <font color="Maroon" title="c14">Sn2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c15">Smax</font> <a href="dickson.html#K7" title="DICKSON:func.7">mindex</a> <span class="p3">(<span class="default"><a href="bagorder.html#K15" title="BAGORDER:func.15">MinElement</a> <span class="p4">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c15">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="bagorder.html#K14" title="BAGORDER:func.14">FinPoset</a> <font color="Maroon" title="c1">R</font></span>)</span> ;<br/><b>reconsider </b><font color="Maroon" title="c27">bnti'</font> = <font color="Maroon" title="c25">bnti</font>, <font color="Maroon" title="c28">bnti1'</font> = <font color="Maroon" title="c26">bnti1</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font> ;<br/><a NAME="E7:73_1_8"/><i><font color="Green" title="E65">A33</font></i>: 
( <font color="Maroon" title="c26">bnti1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c25">bnti</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c26">bnti1</font>,<font color="Maroon" title="c25">bnti</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="bagorder.html#K13" title="BAGORDER:func.13">FinOrd</a> <font color="Maroon" title="c1">R</font> )
 <b>by </b><i><a class="txt" href="bagorder.html#E23:73_1"><i><font color="Green" title="E62">A30</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="c29">x</font>, <font color="Maroon" title="c30">y</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font><b> such that </b><br/><a NAME="E9:73_1_8"/><i><font color="Green" title="E66">A34</font></i>: 
( <font color="Maroon" title="c26">bnti1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">x</font> &amp; <font color="Maroon" title="c25">bnti</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">y</font> )
 <b>and </b><br/><a NAME="E10:73_1_8"/><i><font color="Green" title="E67">A35</font></i>: 
( <font color="Maroon" title="c29">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or ( <font color="Maroon" title="c29">x</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>  &amp; <font color="Maroon" title="c30">y</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>  &amp;  <a href="bagorder.html#K11" title="BAGORDER:func.11">PosetMax</a> <font color="Maroon" title="c29">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="bagorder.html#K11" title="BAGORDER:func.11">PosetMax</a> <font color="Maroon" title="c30">y</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><a href="bagorder.html#K11" title="BAGORDER:func.11">PosetMax</a> <font color="Maroon" title="c29">x</font></span>)</span>,<span class="p2">(<span class="default"><a href="bagorder.html#K11" title="BAGORDER:func.11">PosetMax</a> <font color="Maroon" title="c30">y</font></span>)</span></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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font> ) or ( <font color="Maroon" title="c29">x</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>  &amp; <font color="Maroon" title="c30">y</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>  &amp;  <a href="bagorder.html#K11" title="BAGORDER:func.11">PosetMax</a> <font color="Maroon" title="c29">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bagorder.html#K11" title="BAGORDER:func.11">PosetMax</a> <font color="Maroon" title="c30">y</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c29">x</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p4">(<span class="default"><a href="bagorder.html#K11" title="BAGORDER:func.11">PosetMax</a> <font color="Maroon" title="c29">x</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c30">y</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p4">(<span class="default"><a href="bagorder.html#K11" title="BAGORDER:func.11">PosetMax</a> <font color="Maroon" title="c30">y</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></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="bagorder.html#K13" title="BAGORDER:func.13">FinOrd</a> <font color="Maroon" title="c1">R</font> ) )
 <b>by </b><i><a class="ref" href="bagorder.html#T40" target="_self" title="BAGORDER:th.40">Th40</a></i>;<br/><div><i><font color="Green" title="E68">A36</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1_8_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c31">i</font> be    <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> ;<br/><a NAME="E1:73_1_8_1"/>
<span class="p1">(<span class="default"><a href="bagorder.html#K16" title="BAGORDER:func.16">SeqShift</a> <font color="Maroon" title="c14">Sn2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c15">Smax</font> <a href="dickson.html#K7" title="DICKSON:func.7">mindex</a> <span class="p3">(<span class="default"><a href="bagorder.html#K15" title="BAGORDER:func.15">MinElement</a> <span class="p4">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c15">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c31">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">Sn2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c31">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">Smax</font> <a href="dickson.html#K7" title="DICKSON:func.7">mindex</a> <span class="p3">(<span class="default"><a href="bagorder.html#K15" title="BAGORDER:func.15">MinElement</a> <span class="p4">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c15">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="bagorder.html#D20" target="_self" title="BAGORDER:def.20">Def20</a></i>;<br/><b>hence </b><a NAME="E2:73_1_8_1"/>
<span class="p1">(<span class="default"><a href="bagorder.html#K16" title="BAGORDER:func.16">SeqShift</a> <font color="Maroon" title="c14">Sn2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c15">Smax</font> <a href="dickson.html#K7" title="DICKSON:func.7">mindex</a> <span class="p3">(<span class="default"><a href="bagorder.html#K15" title="BAGORDER:func.15">MinElement</a> <span class="p4">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c15">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c31">i</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> 
 <b>by </b><i><a class="txt" href="bagorder.html#E5:73_1"><i><font color="Green" title="E49">A5</font></i></a></i>;<br/></div><b>end;</b></div><div><i><font color="Green" title="E69">A37</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1_8_2"><b>now </b></a><div class="add"><b>consider </b><font color="Maroon" title="c31">bnti''</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font><b> such that </b><br/><a NAME="E2:73_1_8_2"/><i><font color="Green" title="E70">A38</font></i>: 
( <font color="Maroon" title="c31">bnti''</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="bagorder.html#K16" title="BAGORDER:func.16">SeqShift</a> <font color="Maroon" title="c14">Sn2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c15">Smax</font> <a href="dickson.html#K7" title="DICKSON:func.7">mindex</a> <span class="p3">(<span class="default"><a href="bagorder.html#K15" title="BAGORDER:func.15">MinElement</a> <span class="p4">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c15">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">i</font> &amp;  <a href="bagorder.html#K11" title="BAGORDER:func.11">PosetMax</a> <font color="Maroon" title="c31">bnti''</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bagorder.html#K15" title="BAGORDER:func.15">MinElement</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c15">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font> )
 <b>by </b><i><a class="txt" href="bagorder.html#E19:73_1"><i><font color="Green" title="E60">A27</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c32">bnti1''</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font><b> such that </b><br/><a NAME="E4:73_1_8_2"/><i><font color="Green" title="E71">A39</font></i>: 
( <font color="Maroon" title="c32">bnti1''</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="bagorder.html#K16" title="BAGORDER:func.16">SeqShift</a> <font color="Maroon" title="c14">Sn2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c15">Smax</font> <a href="dickson.html#K7" title="DICKSON:func.7">mindex</a> <span class="p3">(<span class="default"><a href="bagorder.html#K15" title="BAGORDER:func.15">MinElement</a> <span class="p4">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c15">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp;  <a href="bagorder.html#K11" title="BAGORDER:func.11">PosetMax</a> <font color="Maroon" title="c32">bnti1''</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bagorder.html#K15" title="BAGORDER:func.15">MinElement</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c15">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font> )
 <b>by </b><i><a class="txt" href="bagorder.html#E19:73_1"><i><font color="Green" title="E60">A27</font></i></a></i>;<br/><b>thus </b><a NAME="E5:73_1_8_2"/>
(  <a href="bagorder.html#K11" title="BAGORDER:func.11">PosetMax</a> <font color="Maroon" title="c27">bnti'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bagorder.html#K15" title="BAGORDER:func.15">MinElement</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c15">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font> &amp;  <a href="bagorder.html#K11" title="BAGORDER:func.11">PosetMax</a> <font color="Maroon" title="c28">bnti1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bagorder.html#K15" title="BAGORDER:func.15">MinElement</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c15">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font> )
 <b>by </b><i><a class="txt" href="bagorder.html#E2:73_1_8_2"><i><font color="Green" title="E70">A38</font></i></a>, <a class="txt" href="bagorder.html#E4:73_1_8_2"><i><font color="Green" title="E71">A39</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E13:73_1_8"/>
( <font color="Maroon" title="c27">bnti'</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>  &amp; <font color="Maroon" title="c28">bnti1'</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>  )
 <b>by </b><i><a class="txt" href="bagorder.html#E11:73_1_8"><i><font color="Green" title="E68">A36</font></i></a></i>;<br/><a NAME="E14:73_1_8"/><b>then </b>
(  <a href="bagorder.html#K15" title="BAGORDER:func.15">MinElement</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c15">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c25">bnti</font> &amp;  <a href="bagorder.html#K15" title="BAGORDER:func.15">MinElement</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c15">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c26">bnti1</font> )
 <b>by </b><i><a class="txt" href="bagorder.html#E12:73_1_8"><i><font color="Green" title="E69">A37</font></i></a>, <a class="ref" href="bagorder.html#D15" target="_self" title="BAGORDER:def.15">Def15</a></i>;<br/><b>hence </b><a NAME="E15:73_1_8"/>
<font color="Maroon" title="c20">bn</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c20">bn</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">i</font>
 <b>by </b><i><a class="txt" href="bagorder.html#E2:73_1_8"><i><font color="Green" title="E63">A31</font></i></a>, <a class="txt" href="bagorder.html#E4:73_1_8"><i><font color="Green" title="E64">A32</font></i></a>, <a class="txt" href="bagorder.html#E7:73_1_8"><i><font color="Green" title="E65">A33</font></i></a>, <a class="ref" href="bagorder.html#T1" target="_self" title="BAGORDER:th.1">Th1</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="73_1_8_3"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:73_1_8_3"/>
( <font color="Maroon" title="c29">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or ( <font color="Maroon" title="c29">x</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>  &amp; <font color="Maroon" title="c30">y</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>  &amp;  <a href="bagorder.html#K11" title="BAGORDER:func.11">PosetMax</a> <font color="Maroon" title="c29">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="bagorder.html#K11" title="BAGORDER:func.11">PosetMax</a> <font color="Maroon" title="c30">y</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><a href="bagorder.html#K11" title="BAGORDER:func.11">PosetMax</a> <font color="Maroon" title="c29">x</font></span>)</span>,<span class="p2">(<span class="default"><a href="bagorder.html#K11" title="BAGORDER:func.11">PosetMax</a> <font color="Maroon" title="c30">y</font></span>)</span></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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font> ) or ( <font color="Maroon" title="c29">x</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>  &amp; <font color="Maroon" title="c30">y</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>  &amp;  <a href="bagorder.html#K11" title="BAGORDER:func.11">PosetMax</a> <font color="Maroon" title="c29">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bagorder.html#K11" title="BAGORDER:func.11">PosetMax</a> <font color="Maroon" title="c30">y</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c29">x</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p4">(<span class="default"><a href="bagorder.html#K11" title="BAGORDER:func.11">PosetMax</a> <font color="Maroon" title="c29">x</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c30">y</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p4">(<span class="default"><a href="bagorder.html#K11" title="BAGORDER:func.11">PosetMax</a> <font color="Maroon" title="c30">y</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></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="bagorder.html#K13" title="BAGORDER:func.13">FinOrd</a> <font color="Maroon" title="c1">R</font> ) )
 </span><b>by </b><i><a class="txt" href="bagorder.html#E10:73_1_8"><i><font color="Green" title="E67">A35</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="73_1_8_3_1"><b>suppose </b></a><a NAME="E1:73_1_8_3_1"/>
<font color="Maroon" title="c29">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:73_1_8_3_1"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c20">bn</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c22">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c20">bn</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">i</font></span>)</span></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="bagorder.html#K13" title="BAGORDER:func.13">FinOrd</a> <font color="Maroon" title="c1">R</font>
 <b>by </b><i><a class="txt" href="bagorder.html#E9:73_1_8"><i><font color="Green" title="E66">A34</font></i></a>, <a class="txt" href="bagorder.html#E11:73_1_8"><i><font color="Green" title="E68">A36</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="73_1_8_3_2"><b>suppose </b></a><a NAME="E1:73_1_8_3_2"/>
( <font color="Maroon" title="c29">x</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>  &amp; <font color="Maroon" title="c30">y</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>  &amp;  <a href="bagorder.html#K11" title="BAGORDER:func.11">PosetMax</a> <font color="Maroon" title="c29">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="bagorder.html#K11" title="BAGORDER:func.11">PosetMax</a> <font color="Maroon" title="c30">y</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><a href="bagorder.html#K11" title="BAGORDER:func.11">PosetMax</a> <font color="Maroon" title="c29">x</font></span>)</span>,<span class="p2">(<span class="default"><a href="bagorder.html#K11" title="BAGORDER:func.11">PosetMax</a> <font color="Maroon" title="c30">y</font></span>)</span></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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:73_1_8_3_2"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c20">bn</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c22">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c20">bn</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">i</font></span>)</span></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="bagorder.html#K13" title="BAGORDER:func.13">FinOrd</a> <font color="Maroon" title="c1">R</font>
 <b>by </b><i><a class="txt" href="bagorder.html#E9:73_1_8"><i><font color="Green" title="E66">A34</font></i></a>, <a class="txt" href="bagorder.html#E12:73_1_8"><i><font color="Green" title="E69">A37</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="73_1_8_3_3"><b>suppose </b></a><a NAME="E1:73_1_8_3_3"/>
( <font color="Maroon" title="c29">x</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>  &amp; <font color="Maroon" title="c30">y</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>  &amp;  <a href="bagorder.html#K11" title="BAGORDER:func.11">PosetMax</a> <font color="Maroon" title="c29">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bagorder.html#K11" title="BAGORDER:func.11">PosetMax</a> <font color="Maroon" title="c30">y</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c29">x</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p4">(<span class="default"><a href="bagorder.html#K11" title="BAGORDER:func.11">PosetMax</a> <font color="Maroon" title="c29">x</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c30">y</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p4">(<span class="default"><a href="bagorder.html#K11" title="BAGORDER:func.11">PosetMax</a> <font color="Maroon" title="c30">y</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></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="bagorder.html#K13" title="BAGORDER:func.13">FinOrd</a> <font color="Maroon" title="c1">R</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:73_1_8_3_3"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c20">bn</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c22">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c20">bn</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">i</font></span>)</span></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="bagorder.html#K13" title="BAGORDER:func.13">FinOrd</a> <font color="Maroon" title="c1">R</font>
 <b>by </b><i><a class="txt" href="bagorder.html#E2:73_1_8"><i><font color="Green" title="E63">A31</font></i></a>, <a class="txt" href="bagorder.html#E4:73_1_8"><i><font color="Green" title="E64">A32</font></i></a>, <a class="txt" href="bagorder.html#E9:73_1_8"><i><font color="Green" title="E66">A34</font></i></a>, <a class="txt" href="bagorder.html#E12:73_1_8"><i><font color="Green" title="E69">A37</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E25:73_1"/><b>then </b>
<font color="Maroon" title="c20">bn</font> is <a href="wellfnd1.html#V3" title="WELLFND1:attr.3">descending</a>
 
<b>by </b><i><a class="ref" href="wellfnd1.html#D7" title="WELLFND1:def.7">WELLFND1:def 7</a></i>;<br/>
<a NAME="E26:73_1"/><b>then </b>
(  <a href="bagorder.html#K15" title="BAGORDER:func.15">MinElement</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c15">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font> &amp; <font color="Maroon" title="c20">bn</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">zz</font> )
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c22">Sn1</font> = <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><a href="bagorder.html#K15" title="BAGORDER:func.15">MinElement</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c15">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font></span>)</span>,<font color="Maroon" title="c20">bn</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>,<font color="Maroon" title="c8">zz</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <b>by </b><i><a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c22">Sn1</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c14">Sn2</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c15">Smax</font>
;<br/>

<b>take </b>
 <a href="bagorder.html#K15" title="BAGORDER:func.15">MinElement</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c15">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c15">Smax</font> <a href="dickson.html#K7" title="DICKSON:func.7">mindex</a> <span class="p1">(<span class="default"><a href="bagorder.html#K15" title="BAGORDER:func.15">MinElement</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c15">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font></span>)</span>
;<br/>

<b>take </b>
 <a href="bagorder.html#K16" title="BAGORDER:func.16">SeqShift</a> <font color="Maroon" title="c14">Sn2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c15">Smax</font> <a href="dickson.html#K7" title="DICKSON:func.7">mindex</a> <span class="p2">(<span class="default"><a href="bagorder.html#K15" title="BAGORDER:func.15">MinElement</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c15">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font></span>)</span></span>)</span>
;<br/>

<b>take </b>
<font color="Maroon" title="c20">bn</font>
;<br/>

<b>thus </b><a NAME="E28:73_1"/>
<font color="Maroon" title="c14">Sn2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">Sn</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> 
 ;<br/>

<b>thus </b><a NAME="E29:73_1"/>
for <font color="Olive" title="b1">i</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>   ex <font color="Olive" title="b2">Sn2i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font> st <br/>( <font color="Olive" title="b2">Sn2i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">Sn2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b2">Sn2i</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>  &amp; <font color="Maroon" title="c15">Smax</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bagorder.html#K11" title="BAGORDER:func.11">PosetMax</a> <font color="Olive" title="b2">Sn2i</font> )
 <b>by </b><i><a class="txt" href="bagorder.html#E8:73_1"><i><font color="Green" title="E51">A8</font></i></a></i>;<br/>

<b>thus </b><a NAME="E30:73_1"/>
 <a href="bagorder.html#K15" title="BAGORDER:func.15">MinElement</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c15">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bagorder.html#K15" title="BAGORDER:func.15">MinElement</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c15">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font>
 ;<br/>

<b>thus </b><a NAME="E31:73_1"/>
<font color="Maroon" title="c15">Smax</font> <a href="dickson.html#K7" title="DICKSON:func.7">mindex</a> <span class="p1">(<span class="default"><a href="bagorder.html#K15" title="BAGORDER:func.15">MinElement</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c15">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">Smax</font> <a href="dickson.html#K7" title="DICKSON:func.7">mindex</a> <span class="p1">(<span class="default"><a href="bagorder.html#K15" title="BAGORDER:func.15">MinElement</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c15">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font></span>)</span>
 ;<br/>

<b>thus </b><a NAME="E32:73_1"/>
 <a href="bagorder.html#K16" title="BAGORDER:func.16">SeqShift</a> <font color="Maroon" title="c14">Sn2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c15">Smax</font> <a href="dickson.html#K7" title="DICKSON:func.7">mindex</a> <span class="p2">(<span class="default"><a href="bagorder.html#K15" title="BAGORDER:func.15">MinElement</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c15">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font></span>)</span></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="bagorder.html#K16" title="BAGORDER:func.16">SeqShift</a> <font color="Maroon" title="c14">Sn2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c15">Smax</font> <a href="dickson.html#K7" title="DICKSON:func.7">mindex</a> <span class="p2">(<span class="default"><a href="bagorder.html#K15" title="BAGORDER:func.15">MinElement</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c15">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font></span>)</span></span>)</span>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1_9"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c23">i</font> be    <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> ;<br/><b>consider </b><font color="Maroon" title="c24">bni</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font><b> such that </b><br/><a NAME="E2:73_1_9"/><i><font color="Green" title="E63">A40</font></i>: 
( <font color="Maroon" title="c24">bni</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bagorder.html#K16" title="BAGORDER:func.16">SeqShift</a> <font color="Maroon" title="c14">Sn2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c15">Smax</font> <a href="dickson.html#K7" title="DICKSON:func.7">mindex</a> <span class="p4">(<span class="default"><a href="bagorder.html#K15" title="BAGORDER:func.15">MinElement</a> <span class="p5">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c15">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c23">i</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="bagorder.html#K15" title="BAGORDER:func.15">MinElement</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c15">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <font color="Maroon" title="c20">bn</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c23">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">bni</font> )
 <b>by </b><i><a class="txt" href="bagorder.html#E21:73_1"><i><font color="Green" title="E61">A29</font></i></a></i>;<br/><b>thus </b><a NAME="E3:73_1_9"/>
<font color="Maroon" title="c20">bn</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c23">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bagorder.html#K16" title="BAGORDER:func.16">SeqShift</a> <font color="Maroon" title="c14">Sn2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c15">Smax</font> <a href="dickson.html#K7" title="DICKSON:func.7">mindex</a> <span class="p4">(<span class="default"><a href="bagorder.html#K15" title="BAGORDER:func.15">MinElement</a> <span class="p5">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c15">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c23">i</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="bagorder.html#K15" title="BAGORDER:func.15">MinElement</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c15">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="txt" href="bagorder.html#E2:73_1_9"><i><font color="Green" title="E63">A40</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E34:73_1"/>
for <font color="Olive" title="b1">i</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>  holds  <font color="Maroon" title="c20">bn</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bagorder.html#K16" title="BAGORDER:func.16">SeqShift</a> <font color="Maroon" title="c14">Sn2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c15">Smax</font> <a href="dickson.html#K7" title="DICKSON:func.7">mindex</a> <span class="p4">(<span class="default"><a href="bagorder.html#K15" title="BAGORDER:func.15">MinElement</a> <span class="p5">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c15">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="bagorder.html#K15" title="BAGORDER:func.15">MinElement</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c15">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 ;<br/>

<b>thus </b><a NAME="E35:73_1"/>
for <font color="Olive" title="b1">i</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" title="c15">Smax</font> <a href="dickson.html#K7" title="DICKSON:func.7">mindex</a> <span class="p1">(<span class="default"><a href="bagorder.html#K15" title="BAGORDER:func.15">MinElement</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c15">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> holds <br/><font color="Maroon" title="c15">Smax</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bagorder.html#K15" title="BAGORDER:func.15">MinElement</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c15">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font>
 <b>by </b><i><a class="txt" href="bagorder.html#E17:73_1"><i><font color="Green" title="E58">A21</font></i></a></i>;<br/>

<b>thus </b><a NAME="E36:73_1"/>
<font color="Maroon" title="c22">Sn1</font> <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"><span class="p2">(<span class="default"><a href="bagorder.html#K15" title="BAGORDER:func.15">MinElement</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c15">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font></span>)</span>,<font color="Maroon" title="c20">bn</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c10">aStart</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font>;<br/>
<b>set </b><font color="Maroon" title="c11">SS</font> = <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c10">aStart</font>,<font color="Maroon" title="c6">A</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>;<br/>
<a NAME="E9:73"/>
<font color="Maroon" title="c6">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">zz</font>
 
<b>by </b><i><a class="txt" href="bagorder.html#E4:73"><i><font color="Green" title="E46">A2</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c12">SS</font> = <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c10">aStart</font>,<font color="Maroon" title="c6">A</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>,<font color="Maroon" title="c8">zz</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <b>by </b><i><a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c13">S01</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>,<font color="Maroon" title="c8">zz</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>, <font color="Maroon" title="c14">S02</font> being   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <span class="p1">(<span class="default"><a href="bagorder.html#K14" title="BAGORDER:func.14">FinPoset</a> <font color="Maroon" title="c1">R</font></span>)</span>, <font color="Maroon" title="c15">S0max</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>, <font color="Maroon" title="c16">a0</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font>, <font color="Maroon" title="c17">i0</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> , <font color="Maroon" title="c18">b0t</font>, <font color="Maroon" title="c19">b0</font> being   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <span class="p1">(<span class="default"><a href="bagorder.html#K14" title="BAGORDER:func.14">FinPoset</a> <font color="Maroon" title="c1">R</font></span>)</span><b> such that </b><br/><a NAME="E12:73"/>
<font color="Maroon" title="c14">S02</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">SS</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> 
 <b>and </b><br/><a NAME="E13:73"/><i><font color="Green" title="E48">A41</font></i>: 
for <font color="Olive" title="b1">i</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>   ex <font color="Olive" title="b2">S02i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font> st <br/>( <font color="Olive" title="b2">S02i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">S02</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b2">S02i</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>  &amp; <font color="Maroon" title="c15">S0max</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bagorder.html#K11" title="BAGORDER:func.11">PosetMax</a> <font color="Olive" title="b2">S02i</font> )
 <b>and </b><br/><a NAME="E14:73"/>
<font color="Maroon" title="c16">a0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bagorder.html#K15" title="BAGORDER:func.15">MinElement</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c15">S0max</font></span>)</span>,<font color="Maroon" title="c1">R</font>
 <b>and </b><br/><a NAME="E15:73"/>
<font color="Maroon" title="c17">i0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">S0max</font> <a href="dickson.html#K7" title="DICKSON:func.7">mindex</a> <font color="Maroon" title="c16">a0</font>
 <b>and </b><br/><a NAME="E16:73"/><i><font color="Green" title="E49">A42</font></i>: 
<font color="Maroon" title="c18">b0t</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="bagorder.html#K16" title="BAGORDER:func.16">SeqShift</a> <font color="Maroon" title="c14">S02</font>,<font color="Maroon" title="c17">i0</font>
 <b>and </b><br/><a NAME="E17:73"/><i><font color="Green" title="E50">A43</font></i>: 
for <font color="Olive" title="b1">i</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>  holds  <font color="Maroon" title="c19">b0</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">b0t</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c16">a0</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>and </b><br/><a NAME="E18:73"/><i><font color="Green" title="E51">A44</font></i>: 
for <font color="Olive" title="b1">i</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" title="c17">i0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> holds <br/><font color="Maroon" title="c15">S0max</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">a0</font>
 <b>and </b><br/><a NAME="E19:73"/><i><font color="Green" title="E52">A45</font></i>: 
<font color="Maroon" title="c13">S01</font> <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="c16">a0</font>,<font color="Maroon" title="c19">b0</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 <b>by </b><i><a class="txt" href="bagorder.html#E7:73"><i><font color="Green" title="E47">A3</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c20">S</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>,<font color="Maroon" title="c8">zz</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span><b> such that </b><br/><a NAME="E21:73"/><i><font color="Green" title="E53">A46</font></i>: 
<font color="Maroon" title="c20">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">S01</font>
 <b>and </b><br/><a NAME="E22:73"/><i><font color="Green" title="E54">A47</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>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>,<font color="Maroon" title="c20">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font>,<font color="Maroon" title="c20">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>]
 <b>from </b><i><a class="ref" href="recdef_1.html#S2" title="RECDEF_1:sch.2">RECDEF_1:sch 2</a>(<a class="txt" href="bagorder.html#E7:73"><i><font color="Green" title="E47">A3</font></i></a>);<br/></i>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = <span class="p1">(<span class="default"><font color="Maroon" title="c20">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1</span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> ;<br/>
<div><i><font color="Green" title="E55">A48</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="73_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c21">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:73_2"/><i><font color="Green" title="E56">A49</font></i>: 
<font color="Maroon" title="c21">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 ;<br/><b>reconsider </b><font color="Maroon" title="c22">x'</font> = <font color="Maroon" title="c21">x</font> as    <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>by </b><i><a class="txt" href="bagorder.html#E1:73_2"><i><font color="Green" title="E56">A49</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c23">Sx</font> = <font color="Maroon" title="c20">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">x'</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>,<font color="Maroon" title="c8">zz</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> ;<br/><a NAME="E4:73_2"/>
<font color="Maroon" title="c23">Sx</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>
 ;<br/><b>hence </b><a NAME="E5:73_2"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c21">x</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>
 ;<br/></div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c21">a</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font><b> such that </b><br/><a NAME="E25:73"/><i><font color="Green" title="E56">A50</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/><font color="Maroon" title="c21">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">x</font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S2" title="FUNCT_2:sch.2">FUNCT_2:sch 2</a>(<a class="txt" href="bagorder.html#E23:73"><i><font color="Green" title="E55">A48</font></i></a>);<br/></i>
<b>reconsider </b><font color="Maroon" title="c22">a</font> = <font color="Maroon" title="c21">a</font> as   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Maroon" title="c1">R</font> ;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</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>for <font color="Olive" title="b1">i</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>   ex <font color="Olive" title="b2">b</font> being  <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <span class="p1">(<span class="default"><a href="bagorder.html#K14" title="BAGORDER:func.14">FinPoset</a> <font color="Maroon" title="c1">R</font></span>)</span> st <br/>( <font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> $1</span>)</span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a>  &amp; ( for <font color="Olive" title="b3">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">b</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font> holds <br/>( <font color="Olive" title="b3">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> $1</span>)</span> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a>  &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">x</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c20">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> $1</span>)</span> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span></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 <font color="Maroon" title="c1">R</font> ) ) );<br/>
<a NAME="E27:73"/><i><font color="Green" title="E57">A51</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c23">i</font> be    <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> ;<br/>

<b>set </b><font color="Maroon" title="c24">b</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c20">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 0</span>)</span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> ;<br/>
<a NAME="E1:73_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c20">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 0</span>)</span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">zz</font>
 
;<br/>
<b>then consider </b><font color="Maroon" title="c25">z</font> being   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <span class="p1">(<span class="default"><a href="bagorder.html#K14" title="BAGORDER:func.14">FinPoset</a> <font color="Maroon" title="c1">R</font></span>)</span><b> such that </b><br/><a NAME="E3:73_3"/><i><font color="Green" title="E58">A52</font></i>: 
( <font color="Maroon" title="c25">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 0</span>)</span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a>  &amp; <font color="Maroon" title="c25">z</font> is <a href="wellfnd1.html#V3" title="WELLFND1:attr.3">descending</a> )
 ;<br/>
<b>reconsider </b><font color="Maroon" title="c26">b</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c20">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 0</span>)</span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a>  as   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <span class="p1">(<span class="default"><a href="bagorder.html#K14" title="BAGORDER:func.14">FinPoset</a> <font color="Maroon" title="c1">R</font></span>)</span> <b>by </b><i><a class="txt" href="bagorder.html#E3:73_3"><i><font color="Green" title="E58">A52</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c26">b</font>
;<br/>

<b>thus </b><a NAME="E5:73_3"/>
<font color="Maroon" title="c26">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 0</span>)</span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> 
 ;<br/>

<b>let </b><font color="Maroon" title="c27">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E6:73_3"/><i><font color="Green" title="E59">A53</font></i>: 
<font color="Maroon" title="c27">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c26">b</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c23">i</font>
 ;<br/>

<a NAME="E7:73_3"/><i><font color="Green" title="E60">A54</font></i>: 
<font color="Maroon" title="c16">a0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 0</span>)</span> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> 
 
<b>by </b><i><a class="txt" href="bagorder.html#E19:73"><i><font color="Green" title="E52">A45</font></i></a>, <a class="txt" href="bagorder.html#E21:73"><i><font color="Green" title="E53">A46</font></i></a>, <a class="ref" href="mcart_1.html#D1" title="MCART_1:def.1">MCART_1:def 1</a></i>;<br/>
<a NAME="E8:73_3"/>
<font color="Maroon" title="c19">b0</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c26">b</font>
 
<b>by </b><i><a class="txt" href="bagorder.html#E19:73"><i><font color="Green" title="E52">A45</font></i></a>, <a class="txt" href="bagorder.html#E21:73"><i><font color="Green" title="E53">A46</font></i></a>, <a class="ref" href="mcart_1.html#D2" title="MCART_1:def.2">MCART_1:def 2</a></i>;<br/>
<a NAME="E9:73_3"/><b>then </b>
<font color="Maroon" title="c27">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">b0t</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c23">i</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c16">a0</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<b>by </b><i><a class="txt" href="bagorder.html#E17:73"><i><font color="Green" title="E50">A43</font></i></a>, <a class="txt" href="bagorder.html#E6:73_3"><i><font color="Green" title="E59">A53</font></i></a></i>;<br/>
<a NAME="E10:73_3"/><b>then </b><i><font color="Green" title="E61">A55</font></i>: 
( <font color="Maroon" title="c27">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">b0t</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c23">i</font> &amp; not <font color="Maroon" title="c27">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c16">a0</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )
 
<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>hence </b><a NAME="E11:73_3"/><i><font color="Green" title="E62">A56</font></i>: 
<font color="Maroon" title="c27">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 0</span>)</span> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> 
 <b>by </b><i><a class="txt" href="bagorder.html#E7:73_3"><i><font color="Green" title="E60">A54</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_3_1"><b>now </b></a><div class="add"><a NAME="E1:73_3_1"/>
( <font color="Maroon" title="c26">b</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c23">i</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 <font color="Maroon" title="c1">R</font> &amp; <font color="Maroon" title="c26">b</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c23">i</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> )
 <b>by </b><i><a class="ref" href="finsub_1.html#D5" title="FINSUB_1:def.5">FINSUB_1:def 5</a></i>;<br/><b>hence </b><a NAME="E2:73_3_1"/>
<font color="Maroon" title="c27">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>
 <b>by </b><i><a class="txt" href="bagorder.html#E6:73_3"><i><font color="Green" title="E59">A53</font></i></a></i>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c28">x'</font> = <font color="Maroon" title="c27">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> ;<br/>
<a NAME="E14:73_3"/><i><font color="Green" title="E63">A57</font></i>: 
<font color="Maroon" title="c27">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">S02</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c17">i0</font></span>)</span>
 
<b>by </b><i><a class="txt" href="bagorder.html#E16:73"><i><font color="Green" title="E49">A42</font></i></a>, <a class="txt" href="bagorder.html#E10:73_3"><i><font color="Green" title="E61">A55</font></i></a>, <a class="ref" href="bagorder.html#D20" target="_self" title="BAGORDER:def.20">Def20</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c29">S02ib</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font><b> such that </b><br/><a NAME="E16:73_3"/><i><font color="Green" title="E64">A58</font></i>: 
( <font color="Maroon" title="c29">S02ib</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">S02</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c17">i0</font></span>)</span> &amp; <font color="Maroon" title="c29">S02ib</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>  &amp; <font color="Maroon" title="c15">S0max</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c17">i0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bagorder.html#K11" title="BAGORDER:func.11">PosetMax</a> <font color="Maroon" title="c29">S02ib</font> )
 <b>by </b><i><a class="txt" href="bagorder.html#E13:73"><i><font color="Green" title="E48">A41</font></i></a></i>;<br/>
<a NAME="E17:73_3"/>
<font color="Maroon" title="c17">i0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c17">i0</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/>
<a NAME="E18:73_3"/><b>then </b>
 <a href="bagorder.html#K11" title="BAGORDER:func.11">PosetMax</a> <font color="Maroon" title="c29">S02ib</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">a0</font>
 
<b>by </b><i><a class="txt" href="bagorder.html#E18:73"><i><font color="Green" title="E51">A44</font></i></a>, <a class="txt" href="bagorder.html#E16:73_3"><i><font color="Green" title="E64">A58</font></i></a></i>;<br/>
<a NAME="E19:73_3"/><b>then </b>
<font color="Maroon" title="c16">a0</font> <a href="waybel_4.html#R2" title="WAYBEL_4:pred.2">is_maximal_wrt</a> <font color="Maroon" title="c29">S02ib</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font>
 
<b>by </b><i><a class="txt" href="bagorder.html#E16:73_3"><i><font color="Green" title="E64">A58</font></i></a>, <a class="ref" href="bagorder.html#D15" target="_self" title="BAGORDER:def.15">Def15</a></i>;<br/>
<a NAME="E20:73_3"/><b>then </b>
not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">a0</font>,<font color="Maroon" title="c27">x</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 <font color="Maroon" title="c1">R</font>
 
<b>by </b><i><a class="txt" href="bagorder.html#E7:73_3"><i><font color="Green" title="E60">A54</font></i></a>, <a class="txt" href="bagorder.html#E11:73_3"><i><font color="Green" title="E62">A56</font></i></a>, <a class="txt" href="bagorder.html#E14:73_3"><i><font color="Green" title="E63">A57</font></i></a>, <a class="txt" href="bagorder.html#E16:73_3"><i><font color="Green" title="E64">A58</font></i></a>, <a class="ref" href="waybel_4.html#D24" title="WAYBEL_4:def.24">WAYBEL_4:def 24</a></i>;<br/>
<a NAME="E21:73_3"/><b>then </b>
not <font color="Maroon" title="c16">a0</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c28">x'</font>
 
<b>by </b><i><a class="ref" href="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</a></i>;<br/>
<a NAME="E22:73_3"/><b>then </b>
<font color="Maroon" title="c28">x'</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c16">a0</font>
 
<b>by </b><i><a class="ref" href="waybel_0.html#D29" title="WAYBEL_0:def.29">WAYBEL_0:def 29</a></i>;<br/>
<b>hence </b><a NAME="E23:73_3"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c27">x</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c20">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 0</span>)</span> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span></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 <font color="Maroon" title="c1">R</font>
 <b>by </b><i><a class="txt" href="bagorder.html#E7:73_3"><i><font color="Green" title="E60">A54</font></i></a>, <a class="ref" href="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E28:73"/><i><font color="Green" title="E58">A59</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>2</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c23">n</font> be    <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> ;<br/>

<b>assume </b><a NAME="E1:73_4"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c23">n</font>]
 ;<br/>

<b>let </b><font color="Maroon" title="c24">i</font> be    <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> ;<br/>

<b>set </b><font color="Maroon" title="c25">n1</font> = <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1;<br/>
<b>reconsider </b><font color="Maroon" title="c26">n1</font> = <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 as    <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>  ;<br/>
<b>set </b><font color="Maroon" title="c27">b</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c20">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c26">n1</font></span>)</span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> ;<br/>
<b>consider </b><font color="Maroon" title="c28">Sn2</font> being   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <span class="p1">(<span class="default"><a href="bagorder.html#K14" title="BAGORDER:func.14">FinPoset</a> <font color="Maroon" title="c1">R</font></span>)</span>, <font color="Maroon" title="c29">Smax</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>, <font color="Maroon" title="c30">an</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font>, <font color="Maroon" title="c31">ix</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> , <font color="Maroon" title="c32">bnt</font>, <font color="Maroon" title="c33">bn</font> being   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <span class="p1">(<span class="default"><a href="bagorder.html#K14" title="BAGORDER:func.14">FinPoset</a> <font color="Maroon" title="c1">R</font></span>)</span><b> such that </b><br/><a NAME="E4:73_4"/><i><font color="Green" title="E59">A60</font></i>: 
( <font color="Maroon" title="c28">Sn2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c23">n</font></span>)</span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a>  &amp; ( for <font color="Olive" title="b1">i</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>   ex <font color="Olive" title="b2">Sn2i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font> st <br/>( <font color="Olive" title="b2">Sn2i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">Sn2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b2">Sn2i</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>  &amp; <font color="Maroon" title="c29">Smax</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bagorder.html#K11" title="BAGORDER:func.11">PosetMax</a> <font color="Olive" title="b2">Sn2i</font> ) ) &amp; <font color="Maroon" title="c30">an</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bagorder.html#K15" title="BAGORDER:func.15">MinElement</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c29">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font> &amp; <font color="Maroon" title="c31">ix</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">Smax</font> <a href="dickson.html#K7" title="DICKSON:func.7">mindex</a> <font color="Maroon" title="c30">an</font> &amp; <font color="Maroon" title="c32">bnt</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="bagorder.html#K16" title="BAGORDER:func.16">SeqShift</a> <font color="Maroon" title="c28">Sn2</font>,<font color="Maroon" title="c31">ix</font> &amp; ( for <font color="Olive" title="b1">i</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>  holds  <font color="Maroon" title="c33">bn</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c32">bnt</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c30">an</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> ) &amp; ( for <font color="Olive" title="b1">i</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" title="c31">ix</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> holds <br/><font color="Maroon" title="c29">Smax</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">an</font> ) )
 <b>and </b><br/><a NAME="E5:73_4"/><i><font color="Green" title="E60">A61</font></i>: 
<font color="Maroon" title="c20">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</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="c30">an</font>,<font color="Maroon" title="c33">bn</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 <b>by </b><i><a class="txt" href="bagorder.html#E22:73"><i><font color="Green" title="E54">A47</font></i></a></i>;<br/>
<a NAME="E6:73_4"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c20">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c26">n1</font></span>)</span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">zz</font>
 
;<br/>
<b>then consider </b><font color="Maroon" title="c34">z</font> being   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <span class="p1">(<span class="default"><a href="bagorder.html#K14" title="BAGORDER:func.14">FinPoset</a> <font color="Maroon" title="c1">R</font></span>)</span><b> such that </b><br/><a NAME="E8:73_4"/><i><font color="Green" title="E61">A62</font></i>: 
( <font color="Maroon" title="c34">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c26">n1</font></span>)</span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a>  &amp; <font color="Maroon" title="c34">z</font> is <a href="wellfnd1.html#V3" title="WELLFND1:attr.3">descending</a> )
 ;<br/>
<b>reconsider </b><font color="Maroon" title="c35">b</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c20">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c26">n1</font></span>)</span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a>  as   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <span class="p1">(<span class="default"><a href="bagorder.html#K14" title="BAGORDER:func.14">FinPoset</a> <font color="Maroon" title="c1">R</font></span>)</span> <b>by </b><i><a class="txt" href="bagorder.html#E8:73_4"><i><font color="Green" title="E61">A62</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c35">b</font>
;<br/>

<b>thus </b><a NAME="E10:73_4"/>
<font color="Maroon" title="c35">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> 
 ;<br/>

<b>let </b><font color="Maroon" title="c36">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E11:73_4"/><i><font color="Green" title="E62">A63</font></i>: 
<font color="Maroon" title="c36">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c35">b</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c24">i</font>
 ;<br/>

<a NAME="E12:73_4"/><i><font color="Green" title="E63">A64</font></i>: 
<font color="Maroon" title="c30">an</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c26">n1</font></span>)</span> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> 
 
<b>by </b><i><a class="txt" href="bagorder.html#E5:73_4"><i><font color="Green" title="E60">A61</font></i></a>, <a class="ref" href="mcart_1.html#D1" title="MCART_1:def.1">MCART_1:def 1</a></i>;<br/>
<a NAME="E13:73_4"/>
<font color="Maroon" title="c33">bn</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c35">b</font>
 
<b>by </b><i><a class="txt" href="bagorder.html#E5:73_4"><i><font color="Green" title="E60">A61</font></i></a>, <a class="ref" href="mcart_1.html#D2" title="MCART_1:def.2">MCART_1:def 2</a></i>;<br/>
<a NAME="E14:73_4"/><b>then </b>
<font color="Maroon" title="c36">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c32">bnt</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c24">i</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c30">an</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<b>by </b><i><a class="txt" href="bagorder.html#E4:73_4"><i><font color="Green" title="E59">A60</font></i></a>, <a class="txt" href="bagorder.html#E11:73_4"><i><font color="Green" title="E62">A63</font></i></a></i>;<br/>
<a NAME="E15:73_4"/><b>then </b><i><font color="Green" title="E64">A65</font></i>: 
( <font color="Maroon" title="c36">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c32">bnt</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c24">i</font> &amp; not <font color="Maroon" title="c36">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c30">an</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )
 
<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>hence </b><a NAME="E16:73_4"/><i><font color="Green" title="E65">A66</font></i>: 
<font color="Maroon" title="c36">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> 
 <b>by </b><i><a class="txt" href="bagorder.html#E12:73_4"><i><font color="Green" title="E63">A64</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_4_1"><b>now </b></a><div class="add"><a NAME="E1:73_4_1"/>
( <font color="Maroon" title="c35">b</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c24">i</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 <font color="Maroon" title="c1">R</font> &amp; <font color="Maroon" title="c35">b</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c24">i</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> )
 <b>by </b><i><a class="ref" href="finsub_1.html#D5" title="FINSUB_1:def.5">FINSUB_1:def 5</a></i>;<br/><b>hence </b><a NAME="E2:73_4_1"/>
<font color="Maroon" title="c36">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>
 <b>by </b><i><a class="txt" href="bagorder.html#E11:73_4"><i><font color="Green" title="E62">A63</font></i></a></i>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c37">x'</font> = <font color="Maroon" title="c36">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> ;<br/>
<a NAME="E19:73_4"/><i><font color="Green" title="E66">A67</font></i>: 
<font color="Maroon" title="c36">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c28">Sn2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c31">ix</font></span>)</span>
 
<b>by </b><i><a class="txt" href="bagorder.html#E4:73_4"><i><font color="Green" title="E59">A60</font></i></a>, <a class="txt" href="bagorder.html#E15:73_4"><i><font color="Green" title="E64">A65</font></i></a>, <a class="ref" href="bagorder.html#D20" target="_self" title="BAGORDER:def.20">Def20</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c38">Sn2ib</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font><b> such that </b><br/><a NAME="E21:73_4"/><i><font color="Green" title="E67">A68</font></i>: 
( <font color="Maroon" title="c38">Sn2ib</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">Sn2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c31">ix</font></span>)</span> &amp; <font color="Maroon" title="c38">Sn2ib</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>  &amp; <font color="Maroon" title="c29">Smax</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c31">ix</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bagorder.html#K11" title="BAGORDER:func.11">PosetMax</a> <font color="Maroon" title="c38">Sn2ib</font> )
 <b>by </b><i><a class="txt" href="bagorder.html#E4:73_4"><i><font color="Green" title="E59">A60</font></i></a></i>;<br/>
<a NAME="E22:73_4"/>
<font color="Maroon" title="c31">ix</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c31">ix</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/>
<a NAME="E23:73_4"/><b>then </b>
 <a href="bagorder.html#K11" title="BAGORDER:func.11">PosetMax</a> <font color="Maroon" title="c38">Sn2ib</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">an</font>
 
<b>by </b><i><a class="txt" href="bagorder.html#E4:73_4"><i><font color="Green" title="E59">A60</font></i></a>, <a class="txt" href="bagorder.html#E21:73_4"><i><font color="Green" title="E67">A68</font></i></a></i>;<br/>
<a NAME="E24:73_4"/><b>then </b>
<font color="Maroon" title="c30">an</font> <a href="waybel_4.html#R2" title="WAYBEL_4:pred.2">is_maximal_wrt</a> <font color="Maroon" title="c38">Sn2ib</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font>
 
<b>by </b><i><a class="txt" href="bagorder.html#E21:73_4"><i><font color="Green" title="E67">A68</font></i></a>, <a class="ref" href="bagorder.html#D15" target="_self" title="BAGORDER:def.15">Def15</a></i>;<br/>
<a NAME="E25:73_4"/><b>then </b>
not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c30">an</font>,<font color="Maroon" title="c36">x</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 <font color="Maroon" title="c1">R</font>
 
<b>by </b><i><a class="txt" href="bagorder.html#E12:73_4"><i><font color="Green" title="E63">A64</font></i></a>, <a class="txt" href="bagorder.html#E16:73_4"><i><font color="Green" title="E65">A66</font></i></a>, <a class="txt" href="bagorder.html#E19:73_4"><i><font color="Green" title="E66">A67</font></i></a>, <a class="txt" href="bagorder.html#E21:73_4"><i><font color="Green" title="E67">A68</font></i></a>, <a class="ref" href="waybel_4.html#D24" title="WAYBEL_4:def.24">WAYBEL_4:def 24</a></i>;<br/>
<a NAME="E26:73_4"/><b>then </b>
not <font color="Maroon" title="c30">an</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c37">x'</font>
 
<b>by </b><i><a class="ref" href="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</a></i>;<br/>
<a NAME="E27:73_4"/><b>then </b>
<font color="Maroon" title="c37">x'</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c30">an</font>
 
<b>by </b><i><a class="ref" href="waybel_0.html#D29" title="WAYBEL_0:def.29">WAYBEL_0:def 29</a></i>;<br/>
<b>hence </b><a NAME="E28:73_4"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c36">x</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c20">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span></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 <font color="Maroon" title="c1">R</font>
 <b>by </b><i><a class="txt" href="bagorder.html#E12:73_4"><i><font color="Green" title="E63">A64</font></i></a>, <a class="ref" href="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E29:73"/><i><font color="Green" title="E59">A69</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>  holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="bagorder.html#E27:73"><i><font color="Green" title="E57">A51</font></i></a>, <a class="txt" href="bagorder.html#E28:73"><i><font color="Green" title="E58">A59</font></i></a>);<br/></i>
<b>defpred </b><font color="Maroon">S<sub>3</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> ex <font color="Olive" title="b1">b</font> being  <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <span class="p1">(<span class="default"><a href="bagorder.html#K14" title="BAGORDER:func.14">FinPoset</a> <font color="Maroon" title="c1">R</font></span>)</span> ex <font color="Olive" title="b2">i</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 <br/>( <font color="Olive" title="b1">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> $1</span>)</span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a>  &amp; <font color="Maroon" title="c22">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default">$1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">b</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">i</font> );<br/>
<a NAME="E30:73"/><i><font color="Green" title="E60">A70</font></i>: 
<font color="Maroon">S<sub>3</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_5"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon" title="c23">b</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c20">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 0</span>)</span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> ;<br/>
<a NAME="E1:73_5"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c20">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 0</span>)</span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">zz</font>
 
;<br/>
<b>then consider </b><font color="Maroon" title="c24">z</font> being   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <span class="p1">(<span class="default"><a href="bagorder.html#K14" title="BAGORDER:func.14">FinPoset</a> <font color="Maroon" title="c1">R</font></span>)</span><b> such that </b><br/><a NAME="E3:73_5"/><i><font color="Green" title="E61">A71</font></i>: 
( <font color="Maroon" title="c24">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 0</span>)</span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a>  &amp; <font color="Maroon" title="c24">z</font> is <a href="wellfnd1.html#V3" title="WELLFND1:attr.3">descending</a> )
 ;<br/>
<b>reconsider </b><font color="Maroon" title="c25">b</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c20">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 0</span>)</span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a>  as   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <span class="p1">(<span class="default"><a href="bagorder.html#K14" title="BAGORDER:func.14">FinPoset</a> <font color="Maroon" title="c1">R</font></span>)</span> <b>by </b><i><a class="txt" href="bagorder.html#E3:73_5"><i><font color="Green" title="E61">A71</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c25">b</font>
;<br/>

<a NAME="E5:73_5"/><i><font color="Green" title="E62">A72</font></i>: 
<font color="Maroon" title="c22">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default">0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default">0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> 
 
<b>by </b><i><a class="txt" href="bagorder.html#E25:73"><i><font color="Green" title="E56">A50</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c26">S12</font> being   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <span class="p1">(<span class="default"><a href="bagorder.html#K14" title="BAGORDER:func.14">FinPoset</a> <font color="Maroon" title="c1">R</font></span>)</span>, <font color="Maroon" title="c27">S1max</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>, <font color="Maroon" title="c28">a1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font>, <font color="Maroon" title="c29">i1</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> , <font color="Maroon" title="c30">b1t</font>, <font color="Maroon" title="c31">b1</font> being   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <span class="p1">(<span class="default"><a href="bagorder.html#K14" title="BAGORDER:func.14">FinPoset</a> <font color="Maroon" title="c1">R</font></span>)</span><b> such that </b><br/><a NAME="E7:73_5"/><i><font color="Green" title="E63">A73</font></i>: 
<font color="Maroon" title="c26">S12</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 0</span>)</span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> 
 <b>and </b><br/><a NAME="E8:73_5"/><i><font color="Green" title="E64">A74</font></i>: 
for <font color="Olive" title="b1">i</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>   ex <font color="Olive" title="b2">S12i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font> st <br/>( <font color="Olive" title="b2">S12i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">S12</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b2">S12i</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>  &amp; <font color="Maroon" title="c27">S1max</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bagorder.html#K11" title="BAGORDER:func.11">PosetMax</a> <font color="Olive" title="b2">S12i</font> )
 <b>and </b><br/><a NAME="E9:73_5"/><i><font color="Green" title="E65">A75</font></i>: 
<font color="Maroon" title="c28">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bagorder.html#K15" title="BAGORDER:func.15">MinElement</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c27">S1max</font></span>)</span>,<font color="Maroon" title="c1">R</font>
 <b>and </b><br/><a NAME="E10:73_5"/>
<font color="Maroon" title="c29">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">S1max</font> <a href="dickson.html#K7" title="DICKSON:func.7">mindex</a> <font color="Maroon" title="c28">a1</font>
 <b>and </b><br/><a NAME="E11:73_5"/>
<font color="Maroon" title="c30">b1t</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="bagorder.html#K16" title="BAGORDER:func.16">SeqShift</a> <font color="Maroon" title="c26">S12</font>,<font color="Maroon" title="c29">i1</font>
 <b>and </b><br/><a NAME="E12:73_5"/>
( ( for <font color="Olive" title="b1">i</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>  holds  <font color="Maroon" title="c31">b1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c30">b1t</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c28">a1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> ) &amp; ( for <font color="Olive" title="b1">i</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" title="c29">i1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> holds <br/><font color="Maroon" title="c27">S1max</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">a1</font> ) )
 <b>and </b><br/><a NAME="E13:73_5"/><i><font color="Green" title="E66">A76</font></i>: 
<font color="Maroon" title="c20">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default">0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</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="c28">a1</font>,<font color="Maroon" title="c31">b1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 <b>by </b><i><a class="txt" href="bagorder.html#E22:73"><i><font color="Green" title="E54">A47</font></i></a></i>;<br/>
<a NAME="E14:73_5"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c27">S1max</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 <font color="Maroon" title="c1">R</font>
 
<b>by </b><i><a class="ref" href="relset_1.html#T12" title="RELSET_1:th.12">RELSET_1:12</a></i>;<br/>
<a NAME="E15:73_5"/><b>then </b>
<font color="Maroon" title="c28">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c27">S1max</font>
 
<b>by </b><i><a class="txt" href="bagorder.html#E1:73"><i><font color="Green" title="E45">A1</font></i></a>, <a class="txt" href="bagorder.html#E9:73_5"><i><font color="Green" title="E65">A75</font></i></a>, <a class="ref" href="bagorder.html#D19" target="_self" title="BAGORDER:def.19">Def19</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c32">i</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E17:73_5"/><i><font color="Green" title="E67">A77</font></i>: 
( <font color="Maroon" title="c32">i</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="c27">S1max</font> &amp; <font color="Maroon" title="c27">S1max</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c32">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">a1</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>consider </b><font color="Maroon" title="c33">S12i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font><b> such that </b><br/><a NAME="E19:73_5"/><i><font color="Green" title="E68">A78</font></i>: 
( <font color="Maroon" title="c33">S12i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">S12</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c32">i</font> &amp; <font color="Maroon" title="c33">S12i</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>  &amp; <font color="Maroon" title="c27">S1max</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c32">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bagorder.html#K11" title="BAGORDER:func.11">PosetMax</a> <font color="Maroon" title="c33">S12i</font> )
 <b>by </b><i><a class="txt" href="bagorder.html#E8:73_5"><i><font color="Green" title="E64">A74</font></i></a>, <a class="txt" href="bagorder.html#E17:73_5"><i><font color="Green" title="E67">A77</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c34">i</font> = <font color="Maroon" title="c32">i</font> as    <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>by </b><i><a class="txt" href="bagorder.html#E17:73_5"><i><font color="Green" title="E67">A77</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c34">i</font>
;<br/>

<b>thus </b><a NAME="E21:73_5"/>
<font color="Maroon" title="c25">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 0</span>)</span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> 
 ;<br/>

<a NAME="E22:73_5"/>
<font color="Maroon" title="c28">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c25">b</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c34">i</font>
 
<b>by </b><i><a class="txt" href="bagorder.html#E7:73_5"><i><font color="Green" title="E63">A73</font></i></a>, <a class="txt" href="bagorder.html#E17:73_5"><i><font color="Green" title="E67">A77</font></i></a>, <a class="txt" href="bagorder.html#E19:73_5"><i><font color="Green" title="E68">A78</font></i></a>, <a class="ref" href="bagorder.html#D15" target="_self" title="BAGORDER:def.15">Def15</a></i>;<br/>
<b>hence </b><a NAME="E23:73_5"/>
<font color="Maroon" title="c22">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default">0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c25">b</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c34">i</font>
 <b>by </b><i><a class="txt" href="bagorder.html#E5:73_5"><i><font color="Green" title="E62">A72</font></i></a>, <a class="txt" href="bagorder.html#E13:73_5"><i><font color="Green" title="E66">A76</font></i></a>, <a class="ref" href="mcart_1.html#D1" title="MCART_1:def.1">MCART_1:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E31:73"/><i><font color="Green" title="E61">A79</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>3</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c23">n</font> be    <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> ;<br/>

<b>assume </b><a NAME="E1:73_6"/>
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c23">n</font>]
 ;<br/>

<b>set </b><font color="Maroon" title="c24">b</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c20">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> ;<br/>
<a NAME="E2:73_6"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c20">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">zz</font>
 
;<br/>
<b>then consider </b><font color="Maroon" title="c25">z</font> being   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <span class="p1">(<span class="default"><a href="bagorder.html#K14" title="BAGORDER:func.14">FinPoset</a> <font color="Maroon" title="c1">R</font></span>)</span><b> such that </b><br/><a NAME="E4:73_6"/><i><font color="Green" title="E62">A80</font></i>: 
( <font color="Maroon" title="c25">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a>  &amp; <font color="Maroon" title="c25">z</font> is <a href="wellfnd1.html#V3" title="WELLFND1:attr.3">descending</a> )
 ;<br/>
<b>reconsider </b><font color="Maroon" title="c26">b</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c20">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a>  as   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <span class="p1">(<span class="default"><a href="bagorder.html#K14" title="BAGORDER:func.14">FinPoset</a> <font color="Maroon" title="c1">R</font></span>)</span> <b>by </b><i><a class="txt" href="bagorder.html#E4:73_6"><i><font color="Green" title="E62">A80</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c26">b</font>
;<br/>

<b>set </b><font color="Maroon" title="c27">n1</font> = <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1;<br/>
<b>reconsider </b><font color="Maroon" title="c28">n1</font> = <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 as    <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>  ;<br/>
<b>consider </b><font color="Maroon" title="c29">Sn12</font> being   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <span class="p1">(<span class="default"><a href="bagorder.html#K14" title="BAGORDER:func.14">FinPoset</a> <font color="Maroon" title="c1">R</font></span>)</span>, <font color="Maroon" title="c30">Sn1max</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>, <font color="Maroon" title="c31">an1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font>, <font color="Maroon" title="c32">in1</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> , <font color="Maroon" title="c33">bn1t</font>, <font color="Maroon" title="c34">bn1</font> being   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <span class="p1">(<span class="default"><a href="bagorder.html#K14" title="BAGORDER:func.14">FinPoset</a> <font color="Maroon" title="c1">R</font></span>)</span><b> such that </b><br/><a NAME="E8:73_6"/><i><font color="Green" title="E63">A81</font></i>: 
<font color="Maroon" title="c29">Sn12</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c28">n1</font></span>)</span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> 
 <b>and </b><br/><a NAME="E9:73_6"/><i><font color="Green" title="E64">A82</font></i>: 
for <font color="Olive" title="b1">i</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>   ex <font color="Olive" title="b2">Sn12i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font> st <br/>( <font color="Olive" title="b2">Sn12i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">Sn12</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b2">Sn12i</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>  &amp; <font color="Maroon" title="c30">Sn1max</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bagorder.html#K11" title="BAGORDER:func.11">PosetMax</a> <font color="Olive" title="b2">Sn12i</font> )
 <b>and </b><br/><a NAME="E10:73_6"/><i><font color="Green" title="E65">A83</font></i>: 
<font color="Maroon" title="c31">an1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bagorder.html#K15" title="BAGORDER:func.15">MinElement</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c30">Sn1max</font></span>)</span>,<font color="Maroon" title="c1">R</font>
 <b>and </b><br/><a NAME="E11:73_6"/>
<font color="Maroon" title="c32">in1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">Sn1max</font> <a href="dickson.html#K7" title="DICKSON:func.7">mindex</a> <font color="Maroon" title="c31">an1</font>
 <b>and </b><br/><a NAME="E12:73_6"/>
<font color="Maroon" title="c33">bn1t</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="bagorder.html#K16" title="BAGORDER:func.16">SeqShift</a> <font color="Maroon" title="c29">Sn12</font>,<font color="Maroon" title="c32">in1</font>
 <b>and </b><br/><a NAME="E13:73_6"/>
( ( for <font color="Olive" title="b1">i</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>  holds  <font color="Maroon" title="c34">bn1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c33">bn1t</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c31">an1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> ) &amp; ( for <font color="Olive" title="b1">i</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" title="c32">in1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> holds <br/><font color="Maroon" title="c30">Sn1max</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c31">an1</font> ) )
 <b>and </b><br/><a NAME="E14:73_6"/><i><font color="Green" title="E66">A84</font></i>: 
<font color="Maroon" title="c20">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c28">n1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</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="c31">an1</font>,<font color="Maroon" title="c34">bn1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 <b>by </b><i><a class="txt" href="bagorder.html#E22:73"><i><font color="Green" title="E54">A47</font></i></a></i>;<br/>
<a NAME="E15:73_6"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c30">Sn1max</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 <font color="Maroon" title="c1">R</font>
 
<b>by </b><i><a class="ref" href="relset_1.html#T12" title="RELSET_1:th.12">RELSET_1:12</a></i>;<br/>
<a NAME="E16:73_6"/><b>then </b>
<font color="Maroon" title="c31">an1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c30">Sn1max</font>
 
<b>by </b><i><a class="txt" href="bagorder.html#E1:73"><i><font color="Green" title="E45">A1</font></i></a>, <a class="txt" href="bagorder.html#E10:73_6"><i><font color="Green" title="E65">A83</font></i></a>, <a class="ref" href="bagorder.html#D19" target="_self" title="BAGORDER:def.19">Def19</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c35">i</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E18:73_6"/><i><font color="Green" title="E67">A85</font></i>: 
( <font color="Maroon" title="c35">i</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="c30">Sn1max</font> &amp; <font color="Maroon" title="c30">Sn1max</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c35">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c31">an1</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>consider </b><font color="Maroon" title="c36">Sn12i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font><b> such that </b><br/><a NAME="E20:73_6"/><i><font color="Green" title="E68">A86</font></i>: 
( <font color="Maroon" title="c36">Sn12i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">Sn12</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c35">i</font> &amp; <font color="Maroon" title="c36">Sn12i</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>  &amp; <font color="Maroon" title="c30">Sn1max</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c35">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bagorder.html#K11" title="BAGORDER:func.11">PosetMax</a> <font color="Maroon" title="c36">Sn12i</font> )
 <b>by </b><i><a class="txt" href="bagorder.html#E9:73_6"><i><font color="Green" title="E64">A82</font></i></a>, <a class="txt" href="bagorder.html#E18:73_6"><i><font color="Green" title="E67">A85</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c37">i</font> = <font color="Maroon" title="c35">i</font> as    <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>by </b><i><a class="txt" href="bagorder.html#E18:73_6"><i><font color="Green" title="E67">A85</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c37">i</font>
;<br/>

<b>thus </b><a NAME="E22:73_6"/>
<font color="Maroon" title="c26">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> 
 ;<br/>

<a NAME="E23:73_6"/><i><font color="Green" title="E69">A87</font></i>: 
<font color="Maroon" title="c31">an1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c26">b</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c37">i</font>
 
<b>by </b><i><a class="txt" href="bagorder.html#E8:73_6"><i><font color="Green" title="E63">A81</font></i></a>, <a class="txt" href="bagorder.html#E18:73_6"><i><font color="Green" title="E67">A85</font></i></a>, <a class="txt" href="bagorder.html#E20:73_6"><i><font color="Green" title="E68">A86</font></i></a>, <a class="ref" href="bagorder.html#D15" target="_self" title="BAGORDER:def.15">Def15</a></i>;<br/>
<a NAME="E24:73_6"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c20">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c28">n1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c31">an1</font>
 
<b>by </b><i><a class="txt" href="bagorder.html#E14:73_6"><i><font color="Green" title="E66">A84</font></i></a>, <a class="ref" href="mcart_1.html#D1" title="MCART_1:def.1">MCART_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E25:73_6"/>
<font color="Maroon" title="c22">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c26">b</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c37">i</font>
 <b>by </b><i><a class="txt" href="bagorder.html#E25:73"><i><font color="Green" title="E56">A50</font></i></a>, <a class="txt" href="bagorder.html#E23:73_6"><i><font color="Green" title="E69">A87</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E32:73"/><i><font color="Green" title="E62">A88</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>  holds  <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="bagorder.html#E30:73"><i><font color="Green" title="E60">A70</font></i></a>, <a class="txt" href="bagorder.html#E31:73"><i><font color="Green" title="E61">A79</font></i></a>);<br/></i>
<b>defpred </b><font color="Maroon">S<sub>4</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" title="c22">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default">$1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c22">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> $1 &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c22">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default">$1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> $1</span>)</span></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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font> );<br/>
<a NAME="E33:73"/><i><font color="Green" title="E63">A89</font></i>: 
<font color="Maroon">S<sub>4</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_7"><b>proof </b></a><div class="add">

<a NAME="E1:73_7"/><i><font color="Green" title="E64">A90</font></i>: 
<font color="Maroon" title="c22">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 0</span>)</span> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> 
 
<b>by </b><i><a class="txt" href="bagorder.html#E25:73"><i><font color="Green" title="E56">A50</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c23">b</font> being   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <span class="p1">(<span class="default"><a href="bagorder.html#K14" title="BAGORDER:func.14">FinPoset</a> <font color="Maroon" title="c1">R</font></span>)</span>, <font color="Maroon" title="c24">i</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="E3:73_7"/><i><font color="Green" title="E65">A91</font></i>: 
( <font color="Maroon" title="c23">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 0</span>)</span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a>  &amp; <font color="Maroon" title="c22">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default">0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c23">b</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c24">i</font> )
 <b>by </b><i><a class="txt" href="bagorder.html#E30:73"><i><font color="Green" title="E60">A70</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c25">bb</font> being   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <span class="p1">(<span class="default"><a href="bagorder.html#K14" title="BAGORDER:func.14">FinPoset</a> <font color="Maroon" title="c1">R</font></span>)</span><b> such that </b><br/><a NAME="E5:73_7"/><i><font color="Green" title="E66">A92</font></i>: 
<font color="Maroon" title="c25">bb</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 0</span>)</span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> 
 <b>and </b><br/><a NAME="E6:73_7"/><i><font color="Green" title="E67">A93</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c25">bb</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c24">i</font> holds <br/>( <font color="Olive" title="b1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 0</span>)</span> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a>  &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c20">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 0</span>)</span> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span></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 <font color="Maroon" title="c1">R</font> )
 <b>by </b><i><a class="txt" href="bagorder.html#E27:73"><i><font color="Green" title="E57">A51</font></i></a></i>;<br/>
<b>thus </b><a NAME="E7:73_7"/>
( <font color="Maroon" title="c22">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default">0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c22">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 0 &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c22">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default">0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 0</span>)</span></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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font> )
 <b>by </b><i><a class="txt" href="bagorder.html#E1:73_7"><i><font color="Green" title="E64">A90</font></i></a>, <a class="txt" href="bagorder.html#E3:73_7"><i><font color="Green" title="E65">A91</font></i></a>, <a class="txt" href="bagorder.html#E5:73_7"><i><font color="Green" title="E66">A92</font></i></a>, <a class="txt" href="bagorder.html#E6:73_7"><i><font color="Green" title="E67">A93</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E34:73"/><i><font color="Green" title="E64">A94</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>4</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_8"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c23">n</font> be    <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> ;<br/>

<b>assume </b><a NAME="E1:73_8"/>
( <font color="Maroon" title="c22">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c22">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c23">n</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c22">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c23">n</font></span>)</span></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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font> )
 ;<br/>

<a NAME="E2:73_8"/><i><font color="Green" title="E65">A95</font></i>: 
<font color="Maroon" title="c22">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> 
 
<b>by </b><i><a class="txt" href="bagorder.html#E25:73"><i><font color="Green" title="E56">A50</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c24">b</font> being   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <span class="p1">(<span class="default"><a href="bagorder.html#K14" title="BAGORDER:func.14">FinPoset</a> <font color="Maroon" title="c1">R</font></span>)</span>, <font color="Maroon" title="c25">i</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="E4:73_8"/><i><font color="Green" title="E66">A96</font></i>: 
( <font color="Maroon" title="c24">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a>  &amp; <font color="Maroon" title="c22">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c24">b</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c25">i</font> )
 <b>by </b><i><a class="txt" href="bagorder.html#E32:73"><i><font color="Green" title="E62">A88</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c26">bb</font> being   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <span class="p1">(<span class="default"><a href="bagorder.html#K14" title="BAGORDER:func.14">FinPoset</a> <font color="Maroon" title="c1">R</font></span>)</span><b> such that </b><br/><a NAME="E6:73_8"/><i><font color="Green" title="E67">A97</font></i>: 
( <font color="Maroon" title="c26">bb</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a>  &amp; ( for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c26">bb</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c25">i</font> holds <br/>( <font color="Olive" title="b1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a>  &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c20">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span></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 <font color="Maroon" title="c1">R</font> ) ) )
 <b>by </b><i><a class="txt" href="bagorder.html#E29:73"><i><font color="Green" title="E59">A69</font></i></a></i>;<br/>
<b>thus </b><a NAME="E7:73_8"/>
( <font color="Maroon" title="c22">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c22">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c22">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font> )
 <b>by </b><i><a class="txt" href="bagorder.html#E2:73_8"><i><font color="Green" title="E65">A95</font></i></a>, <a class="txt" href="bagorder.html#E4:73_8"><i><font color="Green" title="E66">A96</font></i></a>, <a class="txt" href="bagorder.html#E6:73_8"><i><font color="Green" title="E67">A97</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E35:73"/>
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>  holds  <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="bagorder.html#E33:73"><i><font color="Green" title="E63">A89</font></i></a>, <a class="txt" href="bagorder.html#E34:73"><i><font color="Green" title="E64">A94</font></i></a>);<br/></i>
<a NAME="E36:73"/><b>then </b>
<font color="Maroon" title="c22">a</font> is <a href="wellfnd1.html#V3" title="WELLFND1:attr.3">descending</a>
 
<b>by </b><i><a class="ref" href="wellfnd1.html#D7" title="WELLFND1:def.7">WELLFND1:def 7</a></i>;<br/>
<b>hence </b><a NAME="E37:73"/>
contradiction
 <b>by </b><i><a class="txt" href="bagorder.html#E1:73"><i><font color="Green" title="E45">A1</font></i></a>, <a class="ref" href="wellfnd1.html#T15" title="WELLFND1:th.15">WELLFND1:15</a></i>;<br/>


</div>
