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

<span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">R</font> is  <a href="wellfnd1.html#V1" title="WELLFND1:attr.1">well_founded</a>  implies  <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>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:70"/><span class="lab"><font color="Green" title="E43">A1</font></span>: 
<font color="Maroon" title="c1">R</font> is  <a href="wellfnd1.html#V1" title="WELLFND1:attr.1">well_founded</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <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> </span><br/>

<span class="kw">set </span><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/>
<span class="kw">set </span><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/>
<span class="kw">set </span><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/>
<span class="kw">set </span><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/>
<span class="kw">assume </span><a NAME="E2:70"/>
 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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">then </span><span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E4:70"/><span class="lab"><font color="Green" title="E44">A2</font></span>: 
<font color="Maroon" title="c6">A</font> is  <a href="wellfnd1.html#V3" title="WELLFND1:attr.3">descending</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="wellfnd1.html#T14" title="WELLFND1:th.14">WELLFND1:14</a></span>;<br/>
<span class="kw">set </span><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="E5:70"/>
<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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:70"><span class="lab"><font color="Green" title="E44">A2</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><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/>
<span class="kw">set </span><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/>
<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span> 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="nat_1.html#NM1" title="NAT_1:NM.1">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="nat_1.html#NM1" title="NAT_1:NM.1">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="nat_1.html#K6" title="NAT_1:func.6">.</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="nat_1.html#K6" title="NAT_1:func.6">.</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> <font color="Olive" title="b1">Sn2</font> <a href="nat_1.html#K8" title="NAT_1:func.8">^\</a> <font color="Olive" title="b4">ix</font> &amp; (  for <font color="Olive" title="b7">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Olive" title="b6">bn</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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="nat_1.html#K6" title="NAT_1:func.6">.</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="nat_1.html#NM1" title="NAT_1:NM.1">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="nat_1.html#K6" title="NAT_1:func.6">.</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="E7:70"/><span class="lab"><font color="Green" title="E45">A3</font></span>: 
 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="70_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">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="b2">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="Maroon" title="c10">n</font>,<font color="Olive" title="b1">Sn</font>,<font color="Olive" title="b2">Sn1</font>]</span><br/><span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">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="Maroon" title="c10">n</font>,<font color="Maroon" title="c11">Sn</font>,<font color="Olive" title="b1">Sn1</font>]</span><br/>



<span class="kw">set </span><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="E1:70_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/>
<a NAME="E2:70_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A4</font></span>: 
 ex <font color="Olive" title="b1">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> st <br/>( <font color="Olive" title="b1">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="Olive" title="b1">z</font> is  <a href="wellfnd1.html#V3" title="WELLFND1:attr.3">descending</a>  )
 
;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c13">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> ;<br/>
<div><span class="lab"><font color="Green" title="E47">A5</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c14">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <font color="Maroon" title="c13">Sn2</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c14">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><br/><span class="kw">assume </span><a NAME="E1:70_1_1"/><span class="lab"><font color="Green" title="E48">A6</font></span>: 
<font color="Maroon" title="c13">Sn2</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c14">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:70_1_1"/><span class="lab"><font color="Green" title="E49">A7</font></span>: 
<font color="Maroon" title="c13">Sn2</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">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="c13">Sn2</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c14">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:70_1"><span class="lab"><font color="Green" title="E46">A4</font></span></a>, <a class="ref" href="wellfnd1.html#D6" title="WELLFND1:def.6">WELLFND1:def 6</a></span>;<br/><a NAME="E3:70_1_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="c13">Sn2</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">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="c13">Sn2</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c14">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:70_1"><span class="lab"><font color="Green" title="E46">A4</font></span></a>, <a class="ref" href="wellfnd1.html#D6" title="WELLFND1:def.6">WELLFND1:def 6</a></span>;<br/><span class="kw">hence </span><a NAME="E4:70_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:70_1_1"><span class="lab"><font color="Green" title="E48">A6</font></span></a>, <a class="txt" href="#E2:70_1_1"><span class="lab"><font color="Green" title="E49">A7</font></span></a>, <a class="ref" href="bagorder.html#T36" target="_self" title="BAGORDER:th.36">Th37</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">defpred </span><font color="Maroon">S<sub>2</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span> 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="c13">Sn2</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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="E5:70_1"/><span class="lab"><font color="Green" title="E48">A8</font></span>: 
 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="70_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c14">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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">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="Maroon" title="c14">i</font>,<font color="Olive" title="b1">y</font>]</span><br/>

<span class="kw">set </span><font color="Maroon" title="c15">Sn2i</font> = <font color="Maroon" title="c13">Sn2</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c14">i</font>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c16">Sn2i</font> = <font color="Maroon" title="c13">Sn2</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c14">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/>
<span class="kw">set </span><font color="Maroon" title="c17">y</font> =  <a href="bagorder.html#K11" title="BAGORDER:func.11">PosetMax</a> <font color="Maroon" title="c16">Sn2i</font>;<br/>
<span class="kw">take </span>
 <a href="bagorder.html#K11" title="BAGORDER:func.11">PosetMax</a> <font color="Maroon" title="c16">Sn2i</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c14">i</font>, <a href="bagorder.html#K11" title="BAGORDER:func.11">PosetMax</a> <font color="Maroon" title="c16">Sn2i</font>]</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c16">Sn2i</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c16">Sn2i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">Sn2</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c14">i</font> &amp; <font color="Maroon" title="c16">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;  <a href="bagorder.html#K11" title="BAGORDER:func.11">PosetMax</a> <font color="Maroon" title="c16">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="c16">Sn2i</font> )</span><br/>

<span class="kw">thus </span><a NAME="E2:70_1_2"/>
<font color="Maroon" title="c16">Sn2i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">Sn2</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c14">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c16">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;  <a href="bagorder.html#K11" title="BAGORDER:func.11">PosetMax</a> <font color="Maroon" title="c16">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="c16">Sn2i</font> )</span><br/>

<span class="kw">thus </span><a NAME="E3:70_1_2"/>
<font color="Maroon" title="c16">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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:70_1"><span class="lab"><font color="Green" title="E47">A5</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="bagorder.html#K11" title="BAGORDER:func.11">PosetMax</a> <font color="Maroon" title="c16">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="c16">Sn2i</font></span><br/>

<span class="kw">thus </span><a NAME="E4:70_1_2"/>
 <a href="bagorder.html#K11" title="BAGORDER:func.11">PosetMax</a> <font color="Maroon" title="c16">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="c16">Sn2i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">consider </span><font color="Maroon" title="c14">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><span class="kw"> such that </span><br/><a NAME="E7:70_1"/><span class="lab"><font color="Green" title="E49">A9</font></span>: 
 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="c14">Smax</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">i</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S3" title="FUNCT_2:sch.3">FUNCT_2:sch 3</a>(<a class="txt" href="#E5:70_1"><span class="lab"><font color="Green" title="E48">A8</font></span></a>);<br/></span>
<span class="kw">set </span><font color="Maroon" title="c15">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="c14">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font>);<br/>
<span class="kw">set </span><font color="Maroon" title="c16">ix</font> = <font color="Maroon" title="c14">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="c14">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font>)</span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c17">bnt</font> = <font color="Maroon" title="c13">Sn2</font> <a href="nat_1.html#K8" title="NAT_1:func.8">^\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">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="c14">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font>)</span>)</span></span>)</span>;<br/>
<span class="kw">defpred </span><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> ] <span class="kw">means </span> 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"><font color="Maroon" title="c13">Sn2</font> <a href="nat_1.html#K8" title="NAT_1:func.8">^\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">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="c14">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="c14">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="70_1_3"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c18">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">y</font>, <font color="Olive" title="b2">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="b2">bni</font> <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="c13">Sn2</font> <a href="nat_1.html#K8" title="NAT_1:func.8">^\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">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="c14">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font>)</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c18">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="c14">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="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">bni</font> )</span><br/><span class="kw">set </span><font color="Maroon" title="c19">bni</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">Sn2</font> <a href="nat_1.html#K8" title="NAT_1:func.8">^\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">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="c14">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font>)</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c18">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="c14">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/><span class="kw">reconsider </span><font color="Maroon" title="c20">k</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c13">Sn2</font> <a href="nat_1.html#K8" title="NAT_1:func.8">^\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">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="c14">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font>)</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c18">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> <span class="kw">by</span> <span class="lab"><a class="ref" href="finsub_1.html#D5" title="FINSUB_1:def.5">FINSUB_1:def 5</a></span>;<br/><a NAME="E2:70_1_3"/>
<font color="Maroon" title="c20">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="c14">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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finsub_1.html#D5" title="FINSUB_1:def.5">FINSUB_1:def 5</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c21">bni</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">Sn2</font> <a href="nat_1.html#K8" title="NAT_1:func.8">^\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">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="c14">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font>)</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c18">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="c14">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/><span class="kw">set </span><font color="Maroon" title="c22">y</font> = <font color="Maroon" title="c21">bni</font>;<br/><span class="kw">take </span><font color="Maroon" title="c23">y</font> = <font color="Maroon" title="c21">bni</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  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"><font color="Maroon" title="c13">Sn2</font> <a href="nat_1.html#K8" title="NAT_1:func.8">^\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">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="c14">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font>)</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c18">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="c14">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="c23">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">bni</font> )</span><br/><span class="kw">take </span><font color="Maroon" title="c24">bni</font> = <font color="Maroon" title="c21">bni</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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"><font color="Maroon" title="c13">Sn2</font> <a href="nat_1.html#K8" title="NAT_1:func.8">^\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">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="c14">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font>)</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c18">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="c14">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="c23">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">bni</font> )</span><br/><span class="kw">thus </span><a NAME="E4:70_1_3"/>
<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"><font color="Maroon" title="c13">Sn2</font> <a href="nat_1.html#K8" title="NAT_1:func.8">^\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">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="c14">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font>)</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c18">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="c14">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 class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c23">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">bni</font></span><br/><span class="kw">thus </span><a NAME="E5:70_1_3"/>
<font color="Maroon" title="c23">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">bni</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E9:70_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A10</font></span>: 
 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/>
<span class="kw">defpred </span><font color="Maroon">S<sub>4</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <span class="kw">means </span><font color="Maroon" title="c14">Smax</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">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="c14">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font>)</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> $1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="c14">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font>);<br/>
<a NAME="E10:70_1"/><span class="lab"><font color="Green" title="E51">A11</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c14">Smax</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E11:70_1"/><span class="lab"><font color="Green" title="E52">A12</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c14">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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#D19" title="RELAT_1:def.19">RELAT_1:def 19</a></span>;<br/>
<a NAME="E12:70_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A13</font></span>: 
 <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="c14">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="c14">Smax</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:70"><span class="lab"><font color="Green" title="E43">A1</font></span></a>, <a class="ref" href="bagorder.html#D17" target="_self" title="BAGORDER:def.17">Def19</a></span>;<br/>
<a NAME="E13:70_1"/><span class="lab"><font color="Green" title="E54">A14</font></span>: 
 <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="c14">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="c14">Smax</font>, the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:70"><span class="lab"><font color="Green" title="E43">A1</font></span></a>, <a class="txt" href="#E11:70_1"><span class="lab"><font color="Green" title="E52">A12</font></span></a>, <a class="ref" href="bagorder.html#D17" target="_self" title="BAGORDER:def.17">Def19</a></span>;<br/>
<a NAME="E14:70_1"/><span class="lab"><font color="Green" title="E55">A15</font></span>: 
<font color="Maroon">S<sub>4</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:70_1"><span class="lab"><font color="Green" title="E51">A11</font></span></a>, <a class="txt" href="#E12:70_1"><span class="lab"><font color="Green" title="E53">A13</font></span></a>, <a class="ref" href="dickson.html#D11" title="DICKSON:def.11">DICKSON:def 11</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E56">A16</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1_4"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c18">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c18">k</font>] implies <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/><span class="kw">assume </span><a NAME="E1:70_1_4"/><span class="lab"><font color="Green" title="E57">A17</font></span>: 
<font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c18">k</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/><span class="kw">set </span><font color="Maroon" title="c19">ixk</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c14">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="c14">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font>)</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c18">k</font>;<br/><span class="kw">set </span><font color="Maroon" title="c20">ixk1</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c14">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="c14">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font>)</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>;<br/><span class="kw">set </span><font color="Maroon" title="c21">ixk19</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">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="c14">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font>)</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c18">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1;<br/><span class="kw">consider </span><font color="Maroon" title="c22">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><span class="kw"> such that </span><br/><a NAME="E3:70_1_4"/><span class="lab"><font color="Green" title="E58">A18</font></span>: 
<font color="Maroon" title="c22">Sn2ixk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">Sn2</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">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="c14">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font>)</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c18">k</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:70_1_4"/>
<font color="Maroon" title="c22">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> 
 <span class="kw">and </span><br/><a NAME="E5:70_1_4"/><span class="lab"><font color="Green" title="E59">A19</font></span>: 
<font color="Maroon" title="c14">Smax</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">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="c14">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font>)</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c18">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="c22">Sn2ixk</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:70_1"><span class="lab"><font color="Green" title="E49">A9</font></span></a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c23">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><span class="kw"> such that </span><br/><a NAME="E7:70_1_4"/><span class="lab"><font color="Green" title="E60">A20</font></span>: 
<font color="Maroon" title="c23">Sn2ixk1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">Sn2</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">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="c14">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font>)</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E8:70_1_4"/><span class="lab"><font color="Green" title="E61">A21</font></span>: 
<font color="Maroon" title="c23">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> 
 <span class="kw">and </span><br/><a NAME="E9:70_1_4"/><span class="lab"><font color="Green" title="E62">A22</font></span>: 
<font color="Maroon" title="c14">Smax</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">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="c14">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font>)</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">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="c23">Sn2ixk1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:70_1"><span class="lab"><font color="Green" title="E49">A9</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c24">Sn2ixk9</font> = <font color="Maroon" title="c22">Sn2ixk</font>, <font color="Maroon" title="c25">Sn2ixk19</font> = <font color="Maroon" title="c23">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="E11:70_1_4"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c14">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="c14">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font>)</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">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="c14">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="c14">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font>)</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c18">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ;<br/><a NAME="E12:70_1_4"/><span class="kw">then </span>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c25">Sn2ixk19</font>,<font color="Maroon" title="c24">Sn2ixk9</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:70_1"><span class="lab"><font color="Green" title="E46">A4</font></span></a>, <a class="txt" href="#E3:70_1_4"><span class="lab"><font color="Green" title="E58">A18</font></span></a>, <a class="txt" href="#E7:70_1_4"><span class="lab"><font color="Green" title="E60">A20</font></span></a>, <a class="ref" href="wellfnd1.html#D6" title="WELLFND1:def.6">WELLFND1:def 6</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c26">x</font>, <font color="Maroon" title="c27">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><span class="kw"> such that </span><br/><a NAME="E14:70_1_4"/><span class="lab"><font color="Green" title="E63">A23</font></span>: 
<font color="Maroon" title="c23">Sn2ixk1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">x</font>
 <span class="kw">and </span><br/><a NAME="E15:70_1_4"/><span class="lab"><font color="Green" title="E64">A24</font></span>: 
<font color="Maroon" title="c22">Sn2ixk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">y</font>
 <span class="kw">and </span><br/><a NAME="E16:70_1_4"/><span class="lab"><font color="Green" title="E65">A25</font></span>: 
( <font color="Maroon" title="c26">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="c26">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="c27">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="c26">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="c27">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="c26">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="c27">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="c26">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="c27">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="c26">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="c27">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="c26">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="c26">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="c27">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="c27">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 class="kw">by</span> <span class="lab"><a class="ref" href="bagorder.html#T39" target="_self" title="BAGORDER:th.39">Th40</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="70_1_4_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:70_1_4_1"/>
( <font color="Maroon" title="c26">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="c26">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="c27">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="c26">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="c27">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="c26">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="c27">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="c26">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="c27">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="c26">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="c27">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="c26">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="c26">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="c27">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="c27">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><span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:70_1_4"><span class="lab"><font color="Green" title="E65">A25</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="70_1_4_1_1"><span class="kw">suppose </span></a><a NAME="E1:70_1_4_1_1"/>
<font color="Maroon" title="c26">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:70_1_4_1_1"/>
<font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c18">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:70_1_4"><span class="lab"><font color="Green" title="E61">A21</font></span></a>, <a class="txt" href="#E14:70_1_4"><span class="lab"><font color="Green" title="E63">A23</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="70_1_4_1_2"><span class="kw">suppose </span></a><a NAME="E1:70_1_4_1_2"/><span class="lab"><font color="Green" title="E66">A26</font></span>: 
( <font color="Maroon" title="c26">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="c27">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="c26">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="c27">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="c26">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="c27">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> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/><div class="add"><a NAME="E2:70_1_4_1_2"/>
<font color="Maroon" title="c14">Smax</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">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="c14">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font>)</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">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="c14">Smax</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:70_1"><span class="lab"><font color="Green" title="E51">A11</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/><span class="kw">hence </span><a NAME="E3:70_1_4_1_2"/>
<font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c18">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:70_1"><span class="lab"><font color="Green" title="E54">A14</font></span></a>, <a class="txt" href="#E1:70_1_4"><span class="lab"><font color="Green" title="E57">A17</font></span></a>, <a class="txt" href="#E5:70_1_4"><span class="lab"><font color="Green" title="E59">A19</font></span></a>, <a class="txt" href="#E9:70_1_4"><span class="lab"><font color="Green" title="E62">A22</font></span></a>, <a class="txt" href="#E14:70_1_4"><span class="lab"><font color="Green" title="E63">A23</font></span></a>, <a class="txt" href="#E15:70_1_4"><span class="lab"><font color="Green" title="E64">A24</font></span></a>, <a class="txt" href="#E1:70_1_4_1_2"><span class="lab"><font color="Green" title="E66">A26</font></span></a>, <a class="ref" href="waybel_4.html#D25" title="WAYBEL_4:def.25">WAYBEL_4:def 25</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="70_1_4_1_3"><span class="kw">suppose </span></a><a NAME="E1:70_1_4_1_3"/>
( <font color="Maroon" title="c26">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="c27">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="c26">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="c27">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="c26">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="c26">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="c27">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="c27">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> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:70_1_4_1_3"/>
<font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c18">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:70_1_4"><span class="lab"><font color="Green" title="E57">A17</font></span></a>, <a class="txt" href="#E5:70_1_4"><span class="lab"><font color="Green" title="E59">A19</font></span></a>, <a class="txt" href="#E9:70_1_4"><span class="lab"><font color="Green" title="E62">A22</font></span></a>, <a class="txt" href="#E14:70_1_4"><span class="lab"><font color="Green" title="E63">A23</font></span></a>, <a class="txt" href="#E15:70_1_4"><span class="lab"><font color="Green" title="E64">A24</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<a NAME="E16:70_1"/><span class="lab"><font color="Green" title="E57">A27</font></span>: 
 for <font color="Olive" title="b1">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">k</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S2" title="NAT_1:sch.2">NAT_1:sch 2</a>(<a class="txt" href="#E14:70_1"><span class="lab"><font color="Green" title="E55">A15</font></span></a>, <a class="txt" href="#E15:70_1"><span class="lab"><font color="Green" title="E56">A16</font></span></a>);<br/></span>
<div><span class="lab"><font color="Green" title="E58">A28</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1_5"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c18">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c14">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="c14">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="c18">i</font> implies <font color="Maroon" title="c14">Smax</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c18">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="c14">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font>) )</span><br/><span class="kw">assume </span><a NAME="E1:70_1_5"/>
<font color="Maroon" title="c14">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="c14">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="c18">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c14">Smax</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c18">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="c14">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font>)</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c19">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E3:70_1_5"/><span class="lab"><font color="Green" title="E59">A29</font></span>: 
<font color="Maroon" title="c18">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">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="c14">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font>)</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c19">k</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T10" title="NAT_1:th.10">NAT_1:10</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c20">k</font> = <font color="Maroon" title="c19">k</font> as   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> ;<br/><a NAME="E5:70_1_5"/>
<font color="Maroon" title="c18">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">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="c14">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font>)</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c20">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:70_1_5"><span class="lab"><font color="Green" title="E59">A29</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E6:70_1_5"/>
<font color="Maroon" title="c14">Smax</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c18">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="c14">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:70_1"><span class="lab"><font color="Green" title="E57">A27</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<div><span class="lab"><font color="Green" title="E59">A30</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1_6"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c18">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c14">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="c14">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="c18">i</font> implies  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="c13">Sn2</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c18">i</font> &amp;  <a href="bagorder.html#K11" title="BAGORDER:func.11">PosetMax</a> <font color="Olive" title="b1">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="c14">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font>) ) )</span><br/><span class="kw">assume </span><a NAME="E1:70_1_6"/><span class="lab"><font color="Green" title="E60">A31</font></span>: 
<font color="Maroon" title="c14">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="c14">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="c18">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  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="c13">Sn2</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c18">i</font> &amp;  <a href="bagorder.html#K11" title="BAGORDER:func.11">PosetMax</a> <font color="Olive" title="b1">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="c14">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font>) )</span><br/><span class="kw">reconsider </span><font color="Maroon" title="c19">i0</font> = <font color="Maroon" title="c18">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>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a></span>;<br/><span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E4:70_1_6"/><span class="lab"><font color="Green" title="E61">A32</font></span>: 
<font color="Maroon" title="c20">Sn2i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">Sn2</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c18">i</font>
 <span class="kw">and </span><br/><a NAME="E5:70_1_6"/>
<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> 
 <span class="kw">and </span><br/><a NAME="E6:70_1_6"/><span class="lab"><font color="Green" title="E62">A33</font></span>: 
<font color="Maroon" title="c14">Smax</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c19">i0</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:70_1"><span class="lab"><font color="Green" title="E49">A9</font></span></a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c21">Sn2i</font> = <font color="Maroon" title="c20">Sn2i</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c21">Sn2i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">Sn2</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c18">i</font> &amp;  <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="c14">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font>) )</span><br/><span class="kw">thus </span><a NAME="E7:70_1_6"/>
<font color="Maroon" title="c21">Sn2i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">Sn2</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c18">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:70_1_6"><span class="lab"><font color="Green" title="E61">A32</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <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="c14">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font>)</span><br/><span class="kw">thus </span><a NAME="E8:70_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="c14">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:70_1"><span class="lab"><font color="Green" title="E58">A28</font></span></a>, <a class="txt" href="#E1:70_1_6"><span class="lab"><font color="Green" title="E60">A31</font></span></a>, <a class="txt" href="#E6:70_1_6"><span class="lab"><font color="Green" title="E62">A33</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<div><span class="lab"><font color="Green" title="E60">A34</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1_7"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c18">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">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> st <br/>( <font color="Olive" title="b1">bnti</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">Sn2</font> <a href="nat_1.html#K8" title="NAT_1:func.8">^\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">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="c14">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font>)</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c18">i</font> &amp;  <a href="bagorder.html#K11" title="BAGORDER:func.11">PosetMax</a> <font color="Olive" title="b1">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="c14">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font>) )</span><br/><span class="kw">set </span><font color="Maroon" title="c19">bnti</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c13">Sn2</font> <a href="nat_1.html#K8" title="NAT_1:func.8">^\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">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="c14">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font>)</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c18">i</font>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c20">bnti</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c13">Sn2</font> <a href="nat_1.html#K8" title="NAT_1:func.8">^\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">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="c14">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font>)</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c18">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/><span class="kw">take </span><font color="Maroon" title="c21">bnti</font> = <font color="Maroon" title="c20">bnti</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c21">bnti</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">Sn2</font> <a href="nat_1.html#K8" title="NAT_1:func.8">^\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">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="c14">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font>)</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c18">i</font> &amp;  <a href="bagorder.html#K11" title="BAGORDER:func.11">PosetMax</a> <font color="Maroon" title="c21">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="c14">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font>) )</span><br/><span class="kw">thus </span><a NAME="E2:70_1_7"/>
<font color="Maroon" title="c21">bnti</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">Sn2</font> <a href="nat_1.html#K8" title="NAT_1:func.8">^\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">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="c14">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font>)</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c18">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="bagorder.html#K11" title="BAGORDER:func.11">PosetMax</a> <font color="Maroon" title="c21">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="c14">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font>)</span><br/><span class="kw">set </span><font color="Maroon" title="c22">iix</font> = <font color="Maroon" title="c18">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">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="c14">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font>)</span>)</span></span>)</span>;<br/><a NAME="E3:70_1_7"/>
 ex <font color="Olive" title="b1">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> st <br/>( <font color="Olive" title="b1">Sn2iix</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">Sn2</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">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="c14">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="Olive" title="b1">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="c14">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:70_1"><span class="lab"><font color="Green" title="E59">A30</font></span></a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/><span class="kw">hence </span><a NAME="E4:70_1_7"/>
 <a href="bagorder.html#K11" title="BAGORDER:func.11">PosetMax</a> <font color="Maroon" title="c21">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="c14">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#D3" title="NAT_1:def.3">NAT_1:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">consider </span><font color="Maroon" title="c18">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>,<span class="p1">(<span class="default"><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></span>)</span><span class="kw"> such that </span><br/><a NAME="E21:70_1"/><span class="lab"><font color="Green" title="E61">A35</font></span>: 
 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="c18">bn</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">i</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S3" title="FUNCT_2:sch.3">FUNCT_2:sch 3</a>(<a class="txt" href="#E9:70_1"><span class="lab"><font color="Green" title="E50">A10</font></span></a>);<br/></span>
<span class="kw">reconsider </span><font color="Maroon" title="c19">bn</font> = <font color="Maroon" title="c18">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/>
<span class="kw">set </span><font color="Maroon" title="c20">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="c14">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font>)</span>)</span>,<font color="Maroon" title="c19">bn</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>;<br/>
<a NAME="E23:70_1"/><span class="lab"><font color="Green" title="E62">A36</font></span>: 
<font color="Maroon" title="c13">Sn2</font> <a href="nat_1.html#K8" title="NAT_1:func.8">^\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">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="c14">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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:70_1"><span class="lab"><font color="Green" title="E46">A4</font></span></a>, <a class="ref" href="bagorder.html#T40" target="_self" title="BAGORDER:th.40">Th41</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1_8"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c21">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c19">bn</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">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="c19">bn</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c21">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="c19">bn</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></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="c19">bn</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive">b<sub>1</sub></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> )</span><br/><span class="kw">reconsider </span><font color="Maroon" title="c22">i0</font> = <font color="Maroon" title="c21">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>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a></span>;<br/><a NAME="E2:70_1_8"/><span class="lab"><font color="Green" title="E63">A37</font></span>: 
 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"><font color="Maroon" title="c13">Sn2</font> <a href="nat_1.html#K8" title="NAT_1:func.8">^\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">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="c14">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font>)</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c22">i0</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="c14">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="c19">bn</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c22">i0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">bni</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:70_1"><span class="lab"><font color="Green" title="E61">A35</font></span></a></span>;<br/><a NAME="E3:70_1_8"/><span class="lab"><font color="Green" title="E64">A38</font></span>: 
 ex <font color="Olive" title="b1">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> st <br/>( <font color="Olive" title="b1">bni1</font> <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="c13">Sn2</font> <a href="nat_1.html#K8" title="NAT_1:func.8">^\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">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="c14">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font>)</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">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="c14">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="c19">bn</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">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="Olive" title="b1">bni1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:70_1"><span class="lab"><font color="Green" title="E61">A35</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c23">bnti</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c13">Sn2</font> <a href="nat_1.html#K8" title="NAT_1:func.8">^\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">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="c14">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font>)</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c21">i</font>, <font color="Maroon" title="c24">bnti1</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c13">Sn2</font> <a href="nat_1.html#K8" title="NAT_1:func.8">^\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">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="c14">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font>)</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">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/><span class="kw">reconsider </span><font color="Maroon" title="c25">bnti9</font> = <font color="Maroon" title="c23">bnti</font>, <font color="Maroon" title="c26">bnti19</font> = <font color="Maroon" title="c24">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="E6:70_1_8"/><span class="lab"><font color="Green" title="E65">A39</font></span>: 
<font color="Maroon" title="c24">bnti1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c23">bnti</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:70_1"><span class="lab"><font color="Green" title="E62">A36</font></span></a>, <a class="ref" href="wellfnd1.html#D6" title="WELLFND1:def.6">WELLFND1:def 6</a></span>;<br/><a NAME="E7:70_1_8"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c24">bnti1</font>,<font color="Maroon" title="c23">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:70_1"><span class="lab"><font color="Green" title="E62">A36</font></span></a>, <a class="ref" href="wellfnd1.html#D6" title="WELLFND1:def.6">WELLFND1:def 6</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="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><span class="kw"> such that </span><br/><a NAME="E9:70_1_8"/><span class="lab"><font color="Green" title="E66">A40</font></span>: 
<font color="Maroon" title="c24">bnti1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">x</font>
 <span class="kw">and </span><br/><a NAME="E10:70_1_8"/><span class="lab"><font color="Green" title="E67">A41</font></span>: 
<font color="Maroon" title="c23">bnti</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">y</font>
 <span class="kw">and </span><br/><a NAME="E11:70_1_8"/><span class="lab"><font color="Green" title="E68">A42</font></span>: 
( <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 class="kw">by</span> <span class="lab"><a class="ref" href="bagorder.html#T39" target="_self" title="BAGORDER:th.39">Th40</a></span>;<br/><div><span class="lab"><font color="Green" title="E69">A43</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1_8_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c29">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c13">Sn2</font> <a href="nat_1.html#K8" title="NAT_1:func.8">^\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">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="c14">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font>)</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c29">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> </span><br/><a NAME="E1:70_1_8_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c13">Sn2</font> <a href="nat_1.html#K8" title="NAT_1:func.8">^\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">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="c14">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font>)</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c29">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">Sn2</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c29">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">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="c14">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font>)</span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#D3" title="NAT_1:def.3">NAT_1:def 3</a></span>;<br/><span class="kw">hence </span><a NAME="E2:70_1_8_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c13">Sn2</font> <a href="nat_1.html#K8" title="NAT_1:func.8">^\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">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="c14">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font>)</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c29">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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:70_1"><span class="lab"><font color="Green" title="E47">A5</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><span class="lab"><font color="Green" title="E70">A44</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1_8_2"><span class="kw">now </span></a><div class="add"><a NAME="E1:70_1_8_2"/><span class="lab"><font color="Green" title="E71">A45</font></span>: 
 ex <font color="Olive" title="b1">bnti99</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">bnti99</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">Sn2</font> <a href="nat_1.html#K8" title="NAT_1:func.8">^\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">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="c14">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font>)</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c21">i</font> &amp;  <a href="bagorder.html#K11" title="BAGORDER:func.11">PosetMax</a> <font color="Olive" title="b1">bnti99</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="c14">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:70_1"><span class="lab"><font color="Green" title="E60">A34</font></span></a></span>;<br/><a NAME="E2:70_1_8_2"/>
 ex <font color="Olive" title="b1">bnti199</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">bnti199</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">Sn2</font> <a href="nat_1.html#K8" title="NAT_1:func.8">^\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">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="c14">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font>)</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">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="Olive" title="b1">bnti199</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="c14">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:70_1"><span class="lab"><font color="Green" title="E60">A34</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:70_1_8_2"/>
(  <a href="bagorder.html#K11" title="BAGORDER:func.11">PosetMax</a> <font color="Maroon" title="c25">bnti9</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="c14">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="c26">bnti19</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="c14">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:70_1_8_2"><span class="lab"><font color="Green" title="E71">A45</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E14:70_1_8"/><span class="lab"><font color="Green" title="E71">A46</font></span>: 
<font color="Maroon" title="c25">bnti9</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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:70_1_8"><span class="lab"><font color="Green" title="E69">A43</font></span></a></span>;<br/><a NAME="E15:70_1_8"/><span class="lab"><font color="Green" title="E72">A47</font></span>: 
<font color="Maroon" title="c26">bnti19</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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:70_1_8"><span class="lab"><font color="Green" title="E69">A43</font></span></a></span>;<br/><a NAME="E16:70_1_8"/><span class="lab"><font color="Green" title="E73">A48</font></span>: 
 <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="c14">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="c23">bnti</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:70_1_8"><span class="lab"><font color="Green" title="E70">A44</font></span></a>, <a class="txt" href="#E14:70_1_8"><span class="lab"><font color="Green" title="E71">A46</font></span></a>, <a class="ref" href="bagorder.html#D13" target="_self" title="BAGORDER:def.13">Def15</a></span>;<br/><a NAME="E17:70_1_8"/>
 <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="c14">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="c24">bnti1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:70_1_8"><span class="lab"><font color="Green" title="E70">A44</font></span></a>, <a class="txt" href="#E15:70_1_8"><span class="lab"><font color="Green" title="E72">A47</font></span></a>, <a class="ref" href="bagorder.html#D13" target="_self" title="BAGORDER:def.13">Def15</a></span>;<br/><span class="kw">hence </span><a NAME="E18:70_1_8"/>
<font color="Maroon" title="c19">bn</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">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="c19">bn</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c21">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:70_1_8"><span class="lab"><font color="Green" title="E63">A37</font></span></a>, <a class="txt" href="#E3:70_1_8"><span class="lab"><font color="Green" title="E64">A38</font></span></a>, <a class="txt" href="#E6:70_1_8"><span class="lab"><font color="Green" title="E65">A39</font></span></a>, <a class="txt" href="#E16:70_1_8"><span class="lab"><font color="Green" title="E73">A48</font></span></a>, <a class="ref" href="bagorder.html#T1" target="_self" title="BAGORDER:th.1">Th1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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="c19">bn</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></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="c19">bn</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive">b<sub>1</sub></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></span><br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="70_1_8_3"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:70_1_8_3"/>
( <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><span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:70_1_8"><span class="lab"><font color="Green" title="E68">A42</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="70_1_8_3_1"><span class="kw">suppose </span></a><a NAME="E1:70_1_8_3_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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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="c19">bn</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></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="c19">bn</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive">b<sub>1</sub></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></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:70_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="c19">bn</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c21">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="c19">bn</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c21">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:70_1_8"><span class="lab"><font color="Green" title="E66">A40</font></span></a>, <a class="txt" href="#E12:70_1_8"><span class="lab"><font color="Green" title="E69">A43</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="70_1_8_3_2"><span class="kw">suppose </span></a><a NAME="E1:70_1_8_3_2"/>
( <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> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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="c19">bn</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></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="c19">bn</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive">b<sub>1</sub></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></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:70_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="c19">bn</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c21">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="c19">bn</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c21">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:70_1_8"><span class="lab"><font color="Green" title="E66">A40</font></span></a>, <a class="txt" href="#E10:70_1_8"><span class="lab"><font color="Green" title="E67">A41</font></span></a>, <a class="txt" href="#E13:70_1_8"><span class="lab"><font color="Green" title="E70">A44</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="70_1_8_3_3"><span class="kw">suppose </span></a><a NAME="E1:70_1_8_3_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> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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="c19">bn</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></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="c19">bn</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive">b<sub>1</sub></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></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:70_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="c19">bn</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c21">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="c19">bn</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c21">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:70_1_8"><span class="lab"><font color="Green" title="E63">A37</font></span></a>, <a class="txt" href="#E3:70_1_8"><span class="lab"><font color="Green" title="E64">A38</font></span></a>, <a class="txt" href="#E9:70_1_8"><span class="lab"><font color="Green" title="E66">A40</font></span></a>, <a class="txt" href="#E10:70_1_8"><span class="lab"><font color="Green" title="E67">A41</font></span></a>, <a class="txt" href="#E13:70_1_8"><span class="lab"><font color="Green" title="E70">A44</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<a NAME="E25:70_1"/><span class="kw">then </span>
<font color="Maroon" title="c19">bn</font> is  <a href="wellfnd1.html#V3" title="WELLFND1:attr.3">descending</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="wellfnd1.html#D6" title="WELLFND1:def.6">WELLFND1:def 6</a></span>;<br/>
<a NAME="E26:70_1"/><span class="kw">then </span>
<font color="Maroon" title="c19">bn</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">zz</font>
 
;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><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="c14">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font>)</span>)</span>,<font color="Maroon" title="c19">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> <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c21">Sn1</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c10">n</font>,<font color="Maroon" title="c11">Sn</font>,<font color="Maroon" title="c21">Sn1</font>]</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c13">Sn2</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">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="b2">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="b3">ix</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> ex <font color="Olive" title="b4">bnt</font>, <font color="Olive" title="b5">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="Maroon" title="c13">Sn2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">Sn</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  &amp; (  for <font color="Olive" title="b6">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  ex <font color="Olive" title="b7">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="b7">Sn2i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">Sn2</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b6">i</font> &amp; <font color="Olive" title="b7">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="b1">Smax</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b6">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="b7">Sn2i</font> ) ) &amp; <font color="Olive" title="b2">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="b1">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font>) &amp; <font color="Olive" title="b3">ix</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">Smax</font> <a href="dickson.html#K7" title="DICKSON:func.7">mindex</a> <font color="Olive" title="b2">an</font> &amp; <font color="Olive" title="b4">bnt</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c13">Sn2</font> <a href="nat_1.html#K8" title="NAT_1:func.8">^\</a> <font color="Olive" title="b3">ix</font> &amp; (  for <font color="Olive" title="b6">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Olive" title="b5">bn</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">bnt</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b6">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="b2">an</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> ) &amp; (  for <font color="Olive" title="b6">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b3">ix</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b6">i</font> holds <br/><font color="Olive" title="b1">Smax</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">an</font> ) &amp; <font color="Maroon" title="c21">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"><font color="Olive" title="b2">an</font>,<font color="Olive" title="b5">bn</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c14">Smax</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">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="b2">ix</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> ex <font color="Olive" title="b3">bnt</font>, <font color="Olive" title="b4">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="Maroon" title="c13">Sn2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">Sn</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  &amp; (  for <font color="Olive" title="b5">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  ex <font color="Olive" title="b6">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="b6">Sn2i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">Sn2</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b5">i</font> &amp; <font color="Olive" title="b6">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="c14">Smax</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b5">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="b6">Sn2i</font> ) ) &amp; <font color="Olive" title="b1">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="c14">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font>) &amp; <font color="Olive" title="b2">ix</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">Smax</font> <a href="dickson.html#K7" title="DICKSON:func.7">mindex</a> <font color="Olive" title="b1">an</font> &amp; <font color="Olive" title="b3">bnt</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c13">Sn2</font> <a href="nat_1.html#K8" title="NAT_1:func.8">^\</a> <font color="Olive" title="b2">ix</font> &amp; (  for <font color="Olive" title="b5">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Olive" title="b4">bn</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">bnt</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b5">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="b1">an</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> ) &amp; (  for <font color="Olive" title="b5">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b2">ix</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b5">i</font> holds <br/><font color="Maroon" title="c14">Smax</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">an</font> ) &amp; <font color="Maroon" title="c21">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"><font color="Olive" title="b1">an</font>,<font color="Olive" title="b4">bn</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> )</span><br/>

<span class="kw">take </span>
 <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="c14">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font>)
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">ix</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> ex <font color="Olive" title="b2">bnt</font>, <font color="Olive" title="b3">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="Maroon" title="c13">Sn2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">Sn</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  &amp; (  for <font color="Olive" title="b4">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  ex <font color="Olive" title="b5">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="b5">Sn2i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">Sn2</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b4">i</font> &amp; <font color="Olive" title="b5">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="c14">Smax</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b4">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="b5">Sn2i</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="c14">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="c14">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font>) &amp; <font color="Olive" title="b1">ix</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">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="c14">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font>)</span>)</span> &amp; <font color="Olive" title="b2">bnt</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c13">Sn2</font> <a href="nat_1.html#K8" title="NAT_1:func.8">^\</a> <font color="Olive" title="b1">ix</font> &amp; (  for <font color="Olive" title="b4">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Olive" title="b3">bn</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">bnt</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b4">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="c14">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; (  for <font color="Olive" title="b4">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">ix</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b4">i</font> holds <br/><font color="Maroon" title="c14">Smax</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b4">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="c14">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font>) ) &amp; <font color="Maroon" title="c21">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="c14">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font>)</span>)</span>,<font color="Olive" title="b3">bn</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c14">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="c14">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font>)</span>)</span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">bnt</font>, <font color="Olive" title="b2">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="Maroon" title="c13">Sn2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">Sn</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  &amp; (  for <font color="Olive" title="b3">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  ex <font color="Olive" title="b4">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="b4">Sn2i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">Sn2</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b3">i</font> &amp; <font color="Olive" title="b4">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="c14">Smax</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b3">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="b4">Sn2i</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="c14">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="c14">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font>) &amp; <font color="Maroon" title="c14">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="c14">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="c14">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="c14">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font>)</span>)</span> &amp; <font color="Olive" title="b1">bnt</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c13">Sn2</font> <a href="nat_1.html#K8" title="NAT_1:func.8">^\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">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="c14">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font>)</span>)</span></span>)</span> &amp; (  for <font color="Olive" title="b3">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Olive" title="b2">bn</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">bnt</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b3">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="c14">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; (  for <font color="Olive" title="b3">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon" title="c14">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="c14">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="b3">i</font> holds <br/><font color="Maroon" title="c14">Smax</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b3">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="c14">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font>) ) &amp; <font color="Maroon" title="c21">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="c14">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font>)</span>)</span>,<font color="Olive" title="b2">bn</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c13">Sn2</font> <a href="nat_1.html#K8" title="NAT_1:func.8">^\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">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="c14">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font>)</span>)</span></span>)</span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">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="Maroon" title="c13">Sn2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">Sn</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  &amp; (  for <font color="Olive" title="b2">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  ex <font color="Olive" title="b3">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="b3">Sn2i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">Sn2</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b2">i</font> &amp; <font color="Olive" title="b3">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="c14">Smax</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b2">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="b3">Sn2i</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="c14">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="c14">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font>) &amp; <font color="Maroon" title="c14">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="c14">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="c14">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="c14">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font>)</span>)</span> &amp; <font color="Maroon" title="c13">Sn2</font> <a href="nat_1.html#K8" title="NAT_1:func.8">^\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">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="c14">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> <font color="Maroon" title="c13">Sn2</font> <a href="nat_1.html#K8" title="NAT_1:func.8">^\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">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="c14">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font>)</span>)</span></span>)</span> &amp; (  for <font color="Olive" title="b2">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Olive" title="b1">bn</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b2">i</font> <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="c13">Sn2</font> <a href="nat_1.html#K8" title="NAT_1:func.8">^\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">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="c14">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font>)</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b2">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="c14">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; (  for <font color="Olive" title="b2">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon" title="c14">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="c14">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="b2">i</font> holds <br/><font color="Maroon" title="c14">Smax</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b2">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="c14">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font>) ) &amp; <font color="Maroon" title="c21">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="c14">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font>)</span>)</span>,<font color="Olive" title="b1">bn</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c19">bn</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c13">Sn2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">Sn</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  &amp; (  for <font color="Olive" title="b1">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">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="c13">Sn2</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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="c14">Smax</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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;  <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="c14">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="c14">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font>) &amp; <font color="Maroon" title="c14">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="c14">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="c14">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="c14">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font>)</span>)</span> &amp; <font color="Maroon" title="c13">Sn2</font> <a href="nat_1.html#K8" title="NAT_1:func.8">^\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">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="c14">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> <font color="Maroon" title="c13">Sn2</font> <a href="nat_1.html#K8" title="NAT_1:func.8">^\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">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="c14">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font>)</span>)</span></span>)</span> &amp; (  for <font color="Olive" title="b1">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon" title="c19">bn</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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"><font color="Maroon" title="c13">Sn2</font> <a href="nat_1.html#K8" title="NAT_1:func.8">^\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">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="c14">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font>)</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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="c14">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; (  for <font color="Olive" title="b1">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon" title="c14">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="c14">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="c14">Smax</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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="c14">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font>) ) &amp; <font color="Maroon" title="c21">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="c14">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font>)</span>)</span>,<font color="Maroon" title="c19">bn</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> )</span><br/>

<span class="kw">thus </span><a NAME="E28:70_1"/>
<font color="Maroon" title="c13">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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( (  for <font color="Olive" title="b1">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">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="c13">Sn2</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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="c14">Smax</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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;  <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="c14">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="c14">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font>) &amp; <font color="Maroon" title="c14">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="c14">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="c14">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="c14">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font>)</span>)</span> &amp; <font color="Maroon" title="c13">Sn2</font> <a href="nat_1.html#K8" title="NAT_1:func.8">^\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">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="c14">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> <font color="Maroon" title="c13">Sn2</font> <a href="nat_1.html#K8" title="NAT_1:func.8">^\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">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="c14">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font>)</span>)</span></span>)</span> &amp; (  for <font color="Olive" title="b1">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon" title="c19">bn</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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"><font color="Maroon" title="c13">Sn2</font> <a href="nat_1.html#K8" title="NAT_1:func.8">^\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">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="c14">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font>)</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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="c14">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; (  for <font color="Olive" title="b1">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon" title="c14">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="c14">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="c14">Smax</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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="c14">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font>) ) &amp; <font color="Maroon" title="c21">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="c14">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font>)</span>)</span>,<font color="Maroon" title="c19">bn</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> )</span><br/>

<span class="kw">thus </span><a NAME="E29:70_1"/>
 for <font color="Olive" title="b1">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">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="c13">Sn2</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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="c14">Smax</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <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="c14">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="c14">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font>) &amp; <font color="Maroon" title="c14">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="c14">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="c14">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="c14">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font>)</span>)</span> &amp; <font color="Maroon" title="c13">Sn2</font> <a href="nat_1.html#K8" title="NAT_1:func.8">^\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">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="c14">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> <font color="Maroon" title="c13">Sn2</font> <a href="nat_1.html#K8" title="NAT_1:func.8">^\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">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="c14">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font>)</span>)</span></span>)</span> &amp; (  for <font color="Olive" title="b1">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon" title="c19">bn</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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"><font color="Maroon" title="c13">Sn2</font> <a href="nat_1.html#K8" title="NAT_1:func.8">^\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">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="c14">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font>)</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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="c14">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; (  for <font color="Olive" title="b1">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon" title="c14">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="c14">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="c14">Smax</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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="c14">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font>) ) &amp; <font color="Maroon" title="c21">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="c14">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font>)</span>)</span>,<font color="Maroon" title="c19">bn</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1_9"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c22">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  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="c13">Sn2</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c22">i</font> &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; <font color="Maroon" title="c14">Smax</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c22">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="b1">Sn2i</font> )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c23">i0</font> = <font color="Maroon" title="c22">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>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a></span>;<br/>
<a NAME="E2:70_1_9"/>
 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="c13">Sn2</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c22">i</font> &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; <font color="Maroon" title="c14">Smax</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c23">i0</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="b1">Sn2i</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:70_1"><span class="lab"><font color="Green" title="E49">A9</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:70_1_9"/>
 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="c13">Sn2</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c22">i</font> &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; <font color="Maroon" title="c14">Smax</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c22">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="b1">Sn2i</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">thus </span><a NAME="E30:70_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="c14">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="c14">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c14">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="c14">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="c14">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="c14">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font>)</span>)</span> &amp; <font color="Maroon" title="c13">Sn2</font> <a href="nat_1.html#K8" title="NAT_1:func.8">^\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">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="c14">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> <font color="Maroon" title="c13">Sn2</font> <a href="nat_1.html#K8" title="NAT_1:func.8">^\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">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="c14">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font>)</span>)</span></span>)</span> &amp; (  for <font color="Olive" title="b1">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon" title="c19">bn</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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"><font color="Maroon" title="c13">Sn2</font> <a href="nat_1.html#K8" title="NAT_1:func.8">^\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">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="c14">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font>)</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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="c14">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; (  for <font color="Olive" title="b1">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon" title="c14">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="c14">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="c14">Smax</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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="c14">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font>) ) &amp; <font color="Maroon" title="c21">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="c14">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font>)</span>)</span>,<font color="Maroon" title="c19">bn</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> )</span><br/>

<span class="kw">thus </span><a NAME="E31:70_1"/>
<font color="Maroon" title="c14">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="c14">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="c14">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="c14">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font>)</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c13">Sn2</font> <a href="nat_1.html#K8" title="NAT_1:func.8">^\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">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="c14">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> <font color="Maroon" title="c13">Sn2</font> <a href="nat_1.html#K8" title="NAT_1:func.8">^\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">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="c14">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font>)</span>)</span></span>)</span> &amp; (  for <font color="Olive" title="b1">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon" title="c19">bn</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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"><font color="Maroon" title="c13">Sn2</font> <a href="nat_1.html#K8" title="NAT_1:func.8">^\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">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="c14">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font>)</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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="c14">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; (  for <font color="Olive" title="b1">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon" title="c14">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="c14">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="c14">Smax</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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="c14">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font>) ) &amp; <font color="Maroon" title="c21">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="c14">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font>)</span>)</span>,<font color="Maroon" title="c19">bn</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> )</span><br/>

<span class="kw">thus </span><a NAME="E32:70_1"/>
<font color="Maroon" title="c13">Sn2</font> <a href="nat_1.html#K8" title="NAT_1:func.8">^\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">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="c14">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> <font color="Maroon" title="c13">Sn2</font> <a href="nat_1.html#K8" title="NAT_1:func.8">^\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">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="c14">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font>)</span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( (  for <font color="Olive" title="b1">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon" title="c19">bn</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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"><font color="Maroon" title="c13">Sn2</font> <a href="nat_1.html#K8" title="NAT_1:func.8">^\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">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="c14">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font>)</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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="c14">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; (  for <font color="Olive" title="b1">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon" title="c14">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="c14">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="c14">Smax</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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="c14">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font>) ) &amp; <font color="Maroon" title="c21">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="c14">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font>)</span>)</span>,<font color="Maroon" title="c19">bn</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> )</span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1_10"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c22">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c19">bn</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c22">i</font> <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="c13">Sn2</font> <a href="nat_1.html#K8" title="NAT_1:func.8">^\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">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="c14">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font>)</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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="c14">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></span><br/><span class="kw">reconsider </span><font color="Maroon" title="c23">i0</font> = <font color="Maroon" title="c22">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>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a></span>;<br/><a NAME="E2:70_1_10"/>
 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"><font color="Maroon" title="c13">Sn2</font> <a href="nat_1.html#K8" title="NAT_1:func.8">^\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">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="c14">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font>)</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c23">i0</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="c14">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="c19">bn</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c23">i0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">bni</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:70_1"><span class="lab"><font color="Green" title="E61">A35</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:70_1_10"/>
<font color="Maroon" title="c19">bn</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c22">i</font> <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="c13">Sn2</font> <a href="nat_1.html#K8" title="NAT_1:func.8">^\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">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="c14">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font>)</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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="c14">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 class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E34:70_1"/>
 for <font color="Olive" title="b1">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon" title="c19">bn</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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"><font color="Maroon" title="c13">Sn2</font> <a href="nat_1.html#K8" title="NAT_1:func.8">^\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">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="c14">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font>)</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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="c14">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 class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( (  for <font color="Olive" title="b1">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon" title="c14">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="c14">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="c14">Smax</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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="c14">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font>) ) &amp; <font color="Maroon" title="c21">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="c14">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font>)</span>)</span>,<font color="Maroon" title="c19">bn</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> )</span><br/>

<span class="kw">thus </span><a NAME="E35:70_1"/>
 for <font color="Olive" title="b1">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon" title="c14">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="c14">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="c14">Smax</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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="c14">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:70_1"><span class="lab"><font color="Green" title="E58">A28</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c21">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="c14">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font>)</span>)</span>,<font color="Maroon" title="c19">bn</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><br/>

<span class="kw">thus </span><a NAME="E36:70_1"/>
<font color="Maroon" title="c21">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="c14">Smax</font></span>)</span>,<font color="Maroon" title="c1">R</font>)</span>)</span>,<font color="Maroon" title="c19">bn</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">set </span><font color="Maroon" title="c10">aStart</font> =  the   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font>;<br/>
<span class="kw">set </span><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"> the   <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="c6">A</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>;<br/>
<a NAME="E8:70"/>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:70"><span class="lab"><font color="Green" title="E44">A2</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><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"> the   <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="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> <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></span>;<br/>
<span class="kw">consider </span><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="nat_1.html#NM1" title="NAT_1:NM.1">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><span class="kw"> such that </span><br/><a NAME="E11:70"/>
<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> 
 <span class="kw">and </span><br/><a NAME="E12:70"/><span class="lab"><font color="Green" title="E46">A49</font></span>: 
 for <font color="Olive" title="b1">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">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="nat_1.html#K6" title="NAT_1:func.6">.</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="nat_1.html#K6" title="NAT_1:func.6">.</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> )
 <span class="kw">and </span><br/><a NAME="E13:70"/>
<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>)
 <span class="kw">and </span><br/><a NAME="E14:70"/>
<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>
 <span class="kw">and </span><br/><a NAME="E15:70"/><span class="lab"><font color="Green" title="E47">A50</font></span>: 
<font color="Maroon" title="c18">b0t</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c14">S02</font> <a href="nat_1.html#K8" title="NAT_1:func.8">^\</a> <font color="Maroon" title="c17">i0</font>
 <span class="kw">and </span><br/><a NAME="E16:70"/><span class="lab"><font color="Green" title="E48">A51</font></span>: 
 for <font color="Olive" title="b1">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon" title="c19">b0</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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="nat_1.html#K6" title="NAT_1:func.6">.</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>
 <span class="kw">and </span><br/><a NAME="E17:70"/><span class="lab"><font color="Green" title="E49">A52</font></span>: 
 for <font color="Olive" title="b1">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">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="nat_1.html#K6" title="NAT_1:func.6">.</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>
 <span class="kw">and </span><br/><a NAME="E18:70"/><span class="lab"><font color="Green" title="E50">A53</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:70"><span class="lab"><font color="Green" title="E45">A3</font></span></a></span>;<br/>
<span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E20:70"/><span class="lab"><font color="Green" title="E51">A54</font></span>: 
<font color="Maroon" title="c20">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">S01</font>
 <span class="kw">and </span><br/><a NAME="E21:70"/><span class="lab"><font color="Green" title="E52">A55</font></span>: 
 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="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">n</font>,<font color="Maroon" title="c20">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="recdef_1.html#S2" title="RECDEF_1:sch.2">RECDEF_1:sch 2</a>(<a class="txt" href="#E7:70"><span class="lab"><font color="Green" title="E45">A3</font></span></a>);<br/></span>
<a NAME="E22:70"/><span class="lab"><font color="Green" title="E53">A56</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">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="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">n</font>,<font color="Maroon" title="c20">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c21">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c21">n</font>,<font color="Maroon" title="c20">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c21">n</font>,<font color="Maroon" title="c20">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>]</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c22">n0</font> = <font color="Maroon" title="c21">n</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>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a></span>;<br/>
<a NAME="E2:70_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c22">n0</font>,<font color="Maroon" title="c20">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c22">n0</font>,<font color="Maroon" title="c20">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">n0</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:70"><span class="lab"><font color="Green" title="E52">A55</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:70_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c21">n</font>,<font color="Maroon" title="c20">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c21">n</font>,<font color="Maroon" title="c20">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">deffunc </span><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <span class="kw">-&gt; </span>   <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><span class="lab"><font color="Green" title="E54">A57</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="70_3"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c21">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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>  implies <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> )</span><br/><span class="kw">assume </span><a NAME="E1:70_3"/>
<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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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></span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c22">x9</font> = <font color="Maroon" title="c21">x</font> as   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c23">Sx</font> = <font color="Maroon" title="c20">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c22">x9</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="E4:70_3"/>
<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/><span class="kw">hence </span><a NAME="E5:70_3"/>
<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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E25:70"/><span class="lab"><font color="Green" title="E55">A58</font></span>: 
 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>)
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S2" title="FUNCT_2:sch.2">FUNCT_2:sch 2</a>(<a class="txt" href="#E23:70"><span class="lab"><font color="Green" title="E54">A57</font></span></a>);<br/></span>
<span class="kw">reconsider </span><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/>
<span class="kw">defpred </span><font color="Maroon">S<sub>2</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <span class="kw">means </span> for <font color="Olive" title="b1">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">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="nat_1.html#K6" title="NAT_1:func.6">.</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="nat_1.html#K6" title="NAT_1:func.6">.</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="nat_1.html#K6" title="NAT_1:func.6">.</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="nat_1.html#K6" title="NAT_1:func.6">.</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="E27:70"/><span class="lab"><font color="Green" title="E56">A59</font></span>: 
<font color="Maroon">S<sub>2</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c23">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  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> 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="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a>  &amp; (  for <font color="Olive" title="b2">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">b</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c23">i</font> holds <br/>( <font color="Olive" title="b2">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="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></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="b2">x</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c20">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></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> ) ) )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c24">b</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c20">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> ;<br/>
<a NAME="E1:70_4"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c20">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></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/>
<a NAME="E2:70_4"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">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> st <br/>( <font color="Olive" title="b1">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="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a>  &amp; <font color="Olive" title="b1">z</font> is  <a href="wellfnd1.html#V3" title="WELLFND1:attr.3">descending</a>  )
 
;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c25">b</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c20">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></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> ;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c25">b</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></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="c25">b</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c23">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="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></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="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></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> ) ) )</span><br/>

<span class="kw">thus </span><a NAME="E4:70_4"/>
<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="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  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">b</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c23">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="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></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="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></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> )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c26">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c26">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c25">b</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c23">i</font> implies ( <font color="Maroon" title="c26">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="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></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="Maroon" title="c26">x</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c20">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></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> ) )</span><br/>

<span class="kw">assume </span><a NAME="E5:70_4"/><span class="lab"><font color="Green" title="E57">A60</font></span>: 
<font color="Maroon" title="c26">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c25">b</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c23">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c26">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="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></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="Maroon" title="c26">x</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c20">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></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> )</span><br/>

<a NAME="E6:70_4"/><span class="lab"><font color="Green" title="E58">A61</font></span>: 
<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="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:70"><span class="lab"><font color="Green" title="E50">A53</font></span></a>, <a class="txt" href="#E20:70"><span class="lab"><font color="Green" title="E51">A54</font></span></a>, <a class="ref" href="mcart_1.html#D1" title="MCART_1:def.1">MCART_1:def 1</a></span>;<br/>
<a NAME="E7:70_4"/>
<font color="Maroon" title="c19">b0</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c25">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:70"><span class="lab"><font color="Green" title="E50">A53</font></span></a>, <a class="txt" href="#E20:70"><span class="lab"><font color="Green" title="E51">A54</font></span></a>, <a class="ref" href="mcart_1.html#D2" title="MCART_1:def.2">MCART_1:def 2</a></span>;<br/>
<a NAME="E8:70_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A62</font></span>: 
<font color="Maroon" title="c26">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="nat_1.html#K6" title="NAT_1:func.6">.</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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:70"><span class="lab"><font color="Green" title="E48">A51</font></span></a>, <a class="txt" href="#E5:70_4"><span class="lab"><font color="Green" title="E57">A60</font></span></a></span>;<br/>
<a NAME="E9:70_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A63</font></span>: 
<font color="Maroon" title="c26">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">b0t</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c23">i</font>
 
;<br/>
<a NAME="E10:70_4"/>
 not <font color="Maroon" title="c26">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:70_4"><span class="lab"><font color="Green" title="E59">A62</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:70_4"/><span class="lab"><font color="Green" title="E61">A64</font></span>: 
<font color="Maroon" title="c26">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="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:70_4"><span class="lab"><font color="Green" title="E58">A61</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c26">x</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c20">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></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></span><br/>

<a NAME="E12:70_4"/>
<font color="Maroon" title="c25">b</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finsub_1.html#D5" title="FINSUB_1:def.5">FINSUB_1:def 5</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c27">x9</font> = <font color="Maroon" title="c26">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:70_4"><span class="lab"><font color="Green" title="E57">A60</font></span></a></span>;<br/>
<a NAME="E14:70_4"/><span class="lab"><font color="Green" title="E62">A65</font></span>: 
<font color="Maroon" title="c26">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">S02</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c17">i0</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:70"><span class="lab"><font color="Green" title="E47">A50</font></span></a>, <a class="txt" href="#E9:70_4"><span class="lab"><font color="Green" title="E60">A63</font></span></a>, <a class="ref" href="nat_1.html#D3" title="NAT_1:def.3">NAT_1:def 3</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c28">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><span class="kw"> such that </span><br/><a NAME="E16:70_4"/><span class="lab"><font color="Green" title="E63">A66</font></span>: 
<font color="Maroon" title="c28">S02ib</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">S02</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c17">i0</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E17:70_4"/><span class="lab"><font color="Green" title="E64">A67</font></span>: 
<font color="Maroon" title="c28">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> 
 <span class="kw">and </span><br/><a NAME="E18:70_4"/><span class="lab"><font color="Green" title="E65">A68</font></span>: 
<font color="Maroon" title="c15">S0max</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</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="c28">S02ib</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:70"><span class="lab"><font color="Green" title="E46">A49</font></span></a></span>;<br/>
<a NAME="E19:70_4"/>
 <a href="bagorder.html#K11" title="BAGORDER:func.11">PosetMax</a> <font color="Maroon" title="c28">S02ib</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">a0</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:70"><span class="lab"><font color="Green" title="E49">A52</font></span></a>, <a class="txt" href="#E18:70_4"><span class="lab"><font color="Green" title="E65">A68</font></span></a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/>
<a NAME="E20:70_4"/><span class="kw">then </span>
<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="c28">S02ib</font>, the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:70_4"><span class="lab"><font color="Green" title="E64">A67</font></span></a>, <a class="ref" href="bagorder.html#D13" target="_self" title="BAGORDER:def.13">Def15</a></span>;<br/>
<a NAME="E21:70_4"/><span class="kw">then </span>
 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="c26">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:70_4"><span class="lab"><font color="Green" title="E58">A61</font></span></a>, <a class="txt" href="#E11:70_4"><span class="lab"><font color="Green" title="E61">A64</font></span></a>, <a class="txt" href="#E14:70_4"><span class="lab"><font color="Green" title="E62">A65</font></span></a>, <a class="txt" href="#E16:70_4"><span class="lab"><font color="Green" title="E63">A66</font></span></a>, <a class="ref" href="waybel_4.html#D23" title="WAYBEL_4:def.23">WAYBEL_4:def 23</a></span>;<br/>
<a NAME="E22:70_4"/><span class="kw">then </span>
 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="c27">x9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="orders_2.html#D5" title="ORDERS_2:def.5">ORDERS_2:def 5</a></span>;<br/>
<a NAME="E23:70_4"/><span class="kw">then </span>
<font color="Maroon" title="c27">x9</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c16">a0</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_0.html#D29" title="WAYBEL_0:def.29">WAYBEL_0:def 29</a></span>;<br/>
<span class="kw">hence </span><a NAME="E24:70_4"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c26">x</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c20">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:70_4"><span class="lab"><font color="Green" title="E58">A61</font></span></a>, <a class="ref" href="orders_2.html#D5" title="ORDERS_2:def.5">ORDERS_2:def 5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E28:70"/><span class="lab"><font color="Green" title="E57">A69</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">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="70_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c23">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c23">n</font>] implies <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:70_5"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c23">n</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/>

<span class="kw">let </span><font color="Maroon" title="c24">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  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> 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="nat_1.html#K6" title="NAT_1:func.6">.</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="b2">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">b</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c24">i</font> holds <br/>( <font color="Olive" title="b2">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="nat_1.html#K6" title="NAT_1:func.6">.</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="b2">x</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c20">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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> ) ) )</span><br/>

<span class="kw">set </span><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/>
<span class="kw">reconsider </span><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="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c27">b</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c20">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c26">n1</font></span>)</span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> ;<br/>
<span class="kw">consider </span><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="nat_1.html#NM1" title="NAT_1:NM.1">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><span class="kw"> such that </span><br/><a NAME="E4:70_5"/>
<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="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c23">n</font></span>)</span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> 
 <span class="kw">and </span><br/><a NAME="E5:70_5"/><span class="lab"><font color="Green" title="E58">A70</font></span>: 
 for <font color="Olive" title="b1">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">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="nat_1.html#K6" title="NAT_1:func.6">.</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="nat_1.html#K6" title="NAT_1:func.6">.</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> )
 <span class="kw">and </span><br/><a NAME="E6:70_5"/>
<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>)
 <span class="kw">and </span><br/><a NAME="E7:70_5"/>
<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>
 <span class="kw">and </span><br/><a NAME="E8:70_5"/><span class="lab"><font color="Green" title="E59">A71</font></span>: 
<font color="Maroon" title="c32">bnt</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c28">Sn2</font> <a href="nat_1.html#K8" title="NAT_1:func.8">^\</a> <font color="Maroon" title="c31">ix</font>
 <span class="kw">and </span><br/><a NAME="E9:70_5"/><span class="lab"><font color="Green" title="E60">A72</font></span>: 
 for <font color="Olive" title="b1">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon" title="c33">bn</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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="nat_1.html#K6" title="NAT_1:func.6">.</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>
 <span class="kw">and </span><br/><a NAME="E10:70_5"/><span class="lab"><font color="Green" title="E61">A73</font></span>: 
 for <font color="Olive" title="b1">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">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="nat_1.html#K6" title="NAT_1:func.6">.</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>
 <span class="kw">and </span><br/><a NAME="E11:70_5"/><span class="lab"><font color="Green" title="E62">A74</font></span>: 
<font color="Maroon" title="c20">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:70"><span class="lab"><font color="Green" title="E53">A56</font></span></a></span>;<br/>
<a NAME="E12:70_5"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c20">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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/>
<a NAME="E13:70_5"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">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> st <br/>( <font color="Olive" title="b1">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="nat_1.html#K6" title="NAT_1:func.6">.</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="Olive" title="b1">z</font> is  <a href="wellfnd1.html#V3" title="WELLFND1:attr.3">descending</a>  )
 
;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c34">b</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c20">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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> ;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c34">b</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c34">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="nat_1.html#K6" title="NAT_1:func.6">.</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="c34">b</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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="nat_1.html#K6" title="NAT_1:func.6">.</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="nat_1.html#K6" title="NAT_1:func.6">.</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> ) ) )</span><br/>

<span class="kw">thus </span><a NAME="E15:70_5"/>
<font color="Maroon" title="c34">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="nat_1.html#K6" title="NAT_1:func.6">.</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 class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  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="c34">b</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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="nat_1.html#K6" title="NAT_1:func.6">.</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="nat_1.html#K6" title="NAT_1:func.6">.</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> )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c35">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c35">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c34">b</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c24">i</font> implies ( <font color="Maroon" title="c35">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="nat_1.html#K6" title="NAT_1:func.6">.</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="Maroon" title="c35">x</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c20">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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> ) )</span><br/>

<span class="kw">assume </span><a NAME="E16:70_5"/><span class="lab"><font color="Green" title="E63">A75</font></span>: 
<font color="Maroon" title="c35">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c34">b</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c24">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c35">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="nat_1.html#K6" title="NAT_1:func.6">.</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="Maroon" title="c35">x</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c20">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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> )</span><br/>

<a NAME="E17:70_5"/><span class="lab"><font color="Green" title="E64">A76</font></span>: 
<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="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c26">n1</font></span>)</span> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:70_5"><span class="lab"><font color="Green" title="E62">A74</font></span></a>, <a class="ref" href="mcart_1.html#D1" title="MCART_1:def.1">MCART_1:def 1</a></span>;<br/>
<a NAME="E18:70_5"/>
<font color="Maroon" title="c33">bn</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c34">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:70_5"><span class="lab"><font color="Green" title="E62">A74</font></span></a>, <a class="ref" href="mcart_1.html#D2" title="MCART_1:def.2">MCART_1:def 2</a></span>;<br/>
<a NAME="E19:70_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A77</font></span>: 
<font color="Maroon" title="c35">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="nat_1.html#K6" title="NAT_1:func.6">.</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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:70_5"><span class="lab"><font color="Green" title="E60">A72</font></span></a>, <a class="txt" href="#E16:70_5"><span class="lab"><font color="Green" title="E63">A75</font></span></a></span>;<br/>
<a NAME="E20:70_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">A78</font></span>: 
<font color="Maroon" title="c35">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c32">bnt</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c24">i</font>
 
;<br/>
<a NAME="E21:70_5"/>
 not <font color="Maroon" title="c35">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:70_5"><span class="lab"><font color="Green" title="E65">A77</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E22:70_5"/><span class="lab"><font color="Green" title="E67">A79</font></span>: 
<font color="Maroon" title="c35">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="nat_1.html#K6" title="NAT_1:func.6">.</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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:70_5"><span class="lab"><font color="Green" title="E64">A76</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c35">x</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c20">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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></span><br/>

<a NAME="E23:70_5"/>
<font color="Maroon" title="c34">b</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finsub_1.html#D5" title="FINSUB_1:def.5">FINSUB_1:def 5</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c36">x9</font> = <font color="Maroon" title="c35">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:70_5"><span class="lab"><font color="Green" title="E63">A75</font></span></a></span>;<br/>
<a NAME="E25:70_5"/><span class="lab"><font color="Green" title="E68">A80</font></span>: 
<font color="Maroon" title="c35">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c28">Sn2</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c31">ix</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:70_5"><span class="lab"><font color="Green" title="E59">A71</font></span></a>, <a class="txt" href="#E20:70_5"><span class="lab"><font color="Green" title="E66">A78</font></span></a>, <a class="ref" href="nat_1.html#D3" title="NAT_1:def.3">NAT_1:def 3</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c37">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><span class="kw"> such that </span><br/><a NAME="E27:70_5"/><span class="lab"><font color="Green" title="E69">A81</font></span>: 
<font color="Maroon" title="c37">Sn2ib</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">Sn2</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c31">ix</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E28:70_5"/><span class="lab"><font color="Green" title="E70">A82</font></span>: 
<font color="Maroon" title="c37">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> 
 <span class="kw">and </span><br/><a NAME="E29:70_5"/><span class="lab"><font color="Green" title="E71">A83</font></span>: 
<font color="Maroon" title="c29">Smax</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</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="c37">Sn2ib</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:70_5"><span class="lab"><font color="Green" title="E58">A70</font></span></a></span>;<br/>
<a NAME="E30:70_5"/>
 <a href="bagorder.html#K11" title="BAGORDER:func.11">PosetMax</a> <font color="Maroon" title="c37">Sn2ib</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">an</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:70_5"><span class="lab"><font color="Green" title="E61">A73</font></span></a>, <a class="txt" href="#E29:70_5"><span class="lab"><font color="Green" title="E71">A83</font></span></a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/>
<a NAME="E31:70_5"/><span class="kw">then </span>
<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="c37">Sn2ib</font>, the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:70_5"><span class="lab"><font color="Green" title="E70">A82</font></span></a>, <a class="ref" href="bagorder.html#D13" target="_self" title="BAGORDER:def.13">Def15</a></span>;<br/>
<a NAME="E32:70_5"/><span class="kw">then </span>
 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="c35">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:70_5"><span class="lab"><font color="Green" title="E64">A76</font></span></a>, <a class="txt" href="#E22:70_5"><span class="lab"><font color="Green" title="E67">A79</font></span></a>, <a class="txt" href="#E25:70_5"><span class="lab"><font color="Green" title="E68">A80</font></span></a>, <a class="txt" href="#E27:70_5"><span class="lab"><font color="Green" title="E69">A81</font></span></a>, <a class="ref" href="waybel_4.html#D23" title="WAYBEL_4:def.23">WAYBEL_4:def 23</a></span>;<br/>
<a NAME="E33:70_5"/><span class="kw">then </span>
 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="c36">x9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="orders_2.html#D5" title="ORDERS_2:def.5">ORDERS_2:def 5</a></span>;<br/>
<a NAME="E34:70_5"/><span class="kw">then </span>
<font color="Maroon" title="c36">x9</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c30">an</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_0.html#D29" title="WAYBEL_0:def.29">WAYBEL_0:def 29</a></span>;<br/>
<span class="kw">hence </span><a NAME="E35:70_5"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c35">x</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c20">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:70_5"><span class="lab"><font color="Green" title="E64">A76</font></span></a>, <a class="ref" href="orders_2.html#D5" title="ORDERS_2:def.5">ORDERS_2:def 5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E29:70"/><span class="lab"><font color="Green" title="E58">A84</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">n</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S2" title="NAT_1:sch.2">NAT_1:sch 2</a>(<a class="txt" href="#E27:70"><span class="lab"><font color="Green" title="E56">A59</font></span></a>, <a class="txt" href="#E28:70"><span class="lab"><font color="Green" title="E57">A69</font></span></a>);<br/></span>
<span class="kw">defpred </span><font color="Maroon">S<sub>3</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <span class="kw">means </span> 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="nat_1.html#NM1" title="NAT_1:NM.1">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="nat_1.html#K6" title="NAT_1:func.6">.</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="nat_1.html#K6" title="NAT_1:func.6">.</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="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b2">i</font> );<br/>
<a NAME="E30:70"/><span class="lab"><font color="Green" title="E59">A85</font></span>: 
<font color="Maroon">S<sub>3</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_6"><span class="kw">proof </span></a><div class="add">

<span class="kw">set </span><font color="Maroon" title="c23">b</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c20">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> ;<br/>
<a NAME="E1:70_6"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c20">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></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/>
<a NAME="E2:70_6"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">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> st <br/>( <font color="Olive" title="b1">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="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a>  &amp; <font color="Olive" title="b1">z</font> is  <a href="wellfnd1.html#V3" title="WELLFND1:attr.3">descending</a>  )
 
;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c24">b</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c20">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></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> ;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c24">b</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <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="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></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="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <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="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">i</font> )</span><br/>

<a NAME="E4:70_6"/><span class="lab"><font color="Green" title="E60">A86</font></span>: 
<font color="Maroon" title="c22">a</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <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="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <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 class="kw">by</span> <span class="lab"><a class="txt" href="#E25:70"><span class="lab"><font color="Green" title="E55">A58</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c25">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="c26">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="c27">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="c28">i1</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>, <font color="Maroon" title="c29">b1t</font>, <font color="Maroon" title="c30">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><span class="kw"> such that </span><br/><a NAME="E6:70_6"/><span class="lab"><font color="Green" title="E61">A87</font></span>: 
<font color="Maroon" title="c25">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="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> 
 <span class="kw">and </span><br/><a NAME="E7:70_6"/><span class="lab"><font color="Green" title="E62">A88</font></span>: 
 for <font color="Olive" title="b1">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">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="c25">S12</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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="c26">S1max</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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> )
 <span class="kw">and </span><br/><a NAME="E8:70_6"/><span class="lab"><font color="Green" title="E63">A89</font></span>: 
<font color="Maroon" title="c27">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="c26">S1max</font></span>)</span>,<font color="Maroon" title="c1">R</font>)
 <span class="kw">and </span><br/><a NAME="E9:70_6"/>
<font color="Maroon" title="c28">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">S1max</font> <a href="dickson.html#K7" title="DICKSON:func.7">mindex</a> <font color="Maroon" title="c27">a1</font>
 <span class="kw">and </span><br/><a NAME="E10:70_6"/>
<font color="Maroon" title="c29">b1t</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c25">S12</font> <a href="nat_1.html#K8" title="NAT_1:func.8">^\</a> <font color="Maroon" title="c28">i1</font>
 <span class="kw">and </span><br/><a NAME="E11:70_6"/>
 for <font color="Olive" title="b1">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon" title="c30">b1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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="c29">b1t</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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="c27">a1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">and </span><br/><a NAME="E12:70_6"/>
 for <font color="Olive" title="b1">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon" title="c28">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="c26">S1max</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">a1</font>
 <span class="kw">and </span><br/><a NAME="E13:70_6"/><span class="lab"><font color="Green" title="E64">A90</font></span>: 
<font color="Maroon" title="c20">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <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="c27">a1</font>,<font color="Maroon" title="c30">b1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:70"><span class="lab"><font color="Green" title="E52">A55</font></span></a></span>;<br/>
<a NAME="E14:70_6"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c26">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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#D19" title="RELAT_1:def.19">RELAT_1:def 19</a></span>;<br/>
<a NAME="E15:70_6"/><span class="kw">then </span>
<font color="Maroon" title="c27">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="c26">S1max</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:70"><span class="lab"><font color="Green" title="E43">A1</font></span></a>, <a class="txt" href="#E8:70_6"><span class="lab"><font color="Green" title="E63">A89</font></span></a>, <a class="ref" href="bagorder.html#D17" target="_self" title="BAGORDER:def.17">Def19</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c31">i</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E17:70_6"/><span class="lab"><font color="Green" title="E65">A91</font></span>: 
<font color="Maroon" title="c31">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c26">S1max</font>
 <span class="kw">and </span><br/><a NAME="E18:70_6"/><span class="lab"><font color="Green" title="E66">A92</font></span>: 
<font color="Maroon" title="c26">S1max</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">a1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/>
<a NAME="E19:70_6"/><span class="lab"><font color="Green" title="E67">A93</font></span>: 
 ex <font color="Olive" title="b1">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="b1">S12i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">S12</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">i</font> &amp; <font color="Olive" title="b1">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="c26">S1max</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">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="b1">S12i</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:70_6"><span class="lab"><font color="Green" title="E62">A88</font></span></a>, <a class="txt" href="#E17:70_6"><span class="lab"><font color="Green" title="E65">A91</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c32">i</font> = <font color="Maroon" title="c31">i</font> as   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:70_6"><span class="lab"><font color="Green" title="E65">A91</font></span></a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c32">i</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></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="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <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="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c32">i</font> )</span><br/>

<span class="kw">thus </span><a NAME="E21:70_6"/>
<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="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c22">a</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <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="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c32">i</font></span><br/>

<a NAME="E22:70_6"/>
<font color="Maroon" title="c27">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c24">b</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c32">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:70_6"><span class="lab"><font color="Green" title="E61">A87</font></span></a>, <a class="txt" href="#E18:70_6"><span class="lab"><font color="Green" title="E66">A92</font></span></a>, <a class="txt" href="#E19:70_6"><span class="lab"><font color="Green" title="E67">A93</font></span></a>, <a class="ref" href="bagorder.html#D13" target="_self" title="BAGORDER:def.13">Def15</a></span>;<br/>
<span class="kw">hence </span><a NAME="E23:70_6"/>
<font color="Maroon" title="c22">a</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <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="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c32">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:70_6"><span class="lab"><font color="Green" title="E60">A86</font></span></a>, <a class="txt" href="#E13:70_6"><span class="lab"><font color="Green" title="E64">A90</font></span></a>, <a class="ref" href="mcart_1.html#D1" title="MCART_1:def.1">MCART_1:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E31:70"/><span class="lab"><font color="Green" title="E60">A94</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">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="70_7"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c23">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c23">n</font>] implies <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:70_7"/>
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c23">n</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/>

<span class="kw">set </span><font color="Maroon" title="c24">b</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c20">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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="E2:70_7"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c20">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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/>
<a NAME="E3:70_7"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">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> st <br/>( <font color="Olive" title="b1">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="nat_1.html#K6" title="NAT_1:func.6">.</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="Olive" title="b1">z</font> is  <a href="wellfnd1.html#V3" title="WELLFND1:attr.3">descending</a>  )
 
;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c25">b</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c20">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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> ;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c25">b</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <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="nat_1.html#K6" title="NAT_1:func.6">.</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="nat_1.html#K6" title="NAT_1:func.6">.</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="c25">b</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">i</font> )</span><br/>

<span class="kw">set </span><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;<br/>
<span class="kw">reconsider </span><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 as   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c28">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="c29">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="c30">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="c31">in1</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>, <font color="Maroon" title="c32">bn1t</font>, <font color="Maroon" title="c33">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><span class="kw"> such that </span><br/><a NAME="E7:70_7"/><span class="lab"><font color="Green" title="E61">A95</font></span>: 
<font color="Maroon" title="c28">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="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c27">n1</font></span>)</span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> 
 <span class="kw">and </span><br/><a NAME="E8:70_7"/><span class="lab"><font color="Green" title="E62">A96</font></span>: 
 for <font color="Olive" title="b1">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">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="c28">Sn12</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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="c29">Sn1max</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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> )
 <span class="kw">and </span><br/><a NAME="E9:70_7"/><span class="lab"><font color="Green" title="E63">A97</font></span>: 
<font color="Maroon" title="c30">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="c29">Sn1max</font></span>)</span>,<font color="Maroon" title="c1">R</font>)
 <span class="kw">and </span><br/><a NAME="E10:70_7"/>
<font color="Maroon" title="c31">in1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">Sn1max</font> <a href="dickson.html#K7" title="DICKSON:func.7">mindex</a> <font color="Maroon" title="c30">an1</font>
 <span class="kw">and </span><br/><a NAME="E11:70_7"/>
<font color="Maroon" title="c32">bn1t</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c28">Sn12</font> <a href="nat_1.html#K8" title="NAT_1:func.8">^\</a> <font color="Maroon" title="c31">in1</font>
 <span class="kw">and </span><br/><a NAME="E12:70_7"/>
 for <font color="Olive" title="b1">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon" title="c33">bn1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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">bn1t</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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">an1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">and </span><br/><a NAME="E13:70_7"/>
 for <font color="Olive" title="b1">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon" title="c31">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="c29">Sn1max</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">an1</font>
 <span class="kw">and </span><br/><a NAME="E14:70_7"/><span class="lab"><font color="Green" title="E64">A98</font></span>: 
<font color="Maroon" title="c20">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c27">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="c30">an1</font>,<font color="Maroon" title="c33">bn1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:70"><span class="lab"><font color="Green" title="E52">A55</font></span></a></span>;<br/>
<a NAME="E15:70_7"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c29">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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#D19" title="RELAT_1:def.19">RELAT_1:def 19</a></span>;<br/>
<a NAME="E16:70_7"/><span class="kw">then </span>
<font color="Maroon" title="c30">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="c29">Sn1max</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:70"><span class="lab"><font color="Green" title="E43">A1</font></span></a>, <a class="txt" href="#E9:70_7"><span class="lab"><font color="Green" title="E63">A97</font></span></a>, <a class="ref" href="bagorder.html#D17" target="_self" title="BAGORDER:def.17">Def19</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c34">i</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E18:70_7"/><span class="lab"><font color="Green" title="E65">A99</font></span>: 
<font color="Maroon" title="c34">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c29">Sn1max</font>
 <span class="kw">and </span><br/><a NAME="E19:70_7"/><span class="lab"><font color="Green" title="E66">A100</font></span>: 
<font color="Maroon" title="c29">Sn1max</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c34">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">an1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/>
<a NAME="E20:70_7"/><span class="lab"><font color="Green" title="E67">A101</font></span>: 
 ex <font color="Olive" title="b1">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="b1">Sn12i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">Sn12</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c34">i</font> &amp; <font color="Olive" title="b1">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="c29">Sn1max</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c34">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="b1">Sn12i</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:70_7"><span class="lab"><font color="Green" title="E62">A96</font></span></a>, <a class="txt" href="#E18:70_7"><span class="lab"><font color="Green" title="E65">A99</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c35">i</font> = <font color="Maroon" title="c34">i</font> as   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:70_7"><span class="lab"><font color="Green" title="E65">A99</font></span></a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c35">i</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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="nat_1.html#K6" title="NAT_1:func.6">.</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="nat_1.html#K6" title="NAT_1:func.6">.</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="c25">b</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c35">i</font> )</span><br/>

<span class="kw">thus </span><a NAME="E22:70_7"/>
<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="nat_1.html#K6" title="NAT_1:func.6">.</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 class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c22">a</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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="c25">b</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c35">i</font></span><br/>

<a NAME="E23:70_7"/><span class="lab"><font color="Green" title="E68">A102</font></span>: 
<font color="Maroon" title="c30">an1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c25">b</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c35">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:70_7"><span class="lab"><font color="Green" title="E61">A95</font></span></a>, <a class="txt" href="#E19:70_7"><span class="lab"><font color="Green" title="E66">A100</font></span></a>, <a class="txt" href="#E20:70_7"><span class="lab"><font color="Green" title="E67">A101</font></span></a>, <a class="ref" href="bagorder.html#D13" target="_self" title="BAGORDER:def.13">Def15</a></span>;<br/>
<a NAME="E24:70_7"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c20">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c27">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="c30">an1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:70_7"><span class="lab"><font color="Green" title="E64">A98</font></span></a>, <a class="ref" href="mcart_1.html#D1" title="MCART_1:def.1">MCART_1:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E25:70_7"/>
<font color="Maroon" title="c22">a</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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="c25">b</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c35">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:70"><span class="lab"><font color="Green" title="E55">A58</font></span></a>, <a class="txt" href="#E23:70_7"><span class="lab"><font color="Green" title="E68">A102</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E32:70"/><span class="lab"><font color="Green" title="E61">A103</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">n</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S2" title="NAT_1:sch.2">NAT_1:sch 2</a>(<a class="txt" href="#E30:70"><span class="lab"><font color="Green" title="E59">A85</font></span></a>, <a class="txt" href="#E31:70"><span class="lab"><font color="Green" title="E60">A94</font></span></a>);<br/></span>
<span class="kw">defpred </span><font color="Maroon">S<sub>4</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <span class="kw">means </span>( <font color="Maroon" title="c22">a</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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="nat_1.html#K6" title="NAT_1:func.6">.</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="nat_1.html#K6" title="NAT_1:func.6">.</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="nat_1.html#K6" title="NAT_1:func.6">.</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="E33:70"/><span class="lab"><font color="Green" title="E62">A104</font></span>: 
<font color="Maroon">S<sub>4</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_8"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:70_8"/><span class="lab"><font color="Green" title="E63">A105</font></span>: 
<font color="Maroon" title="c22">a</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <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="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:70"><span class="lab"><font color="Green" title="E55">A58</font></span></a></span>;<br/>
<span class="kw">consider </span><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="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E3:70_8"/><span class="lab"><font color="Green" title="E64">A106</font></span>: 
<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="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> 
 <span class="kw">and </span><br/><a NAME="E4:70_8"/><span class="lab"><font color="Green" title="E65">A107</font></span>: 
<font color="Maroon" title="c22">a</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <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="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c24">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:70"><span class="lab"><font color="Green" title="E59">A85</font></span></a></span>;<br/>
<a NAME="E5:70_8"/>
 ex <font color="Olive" title="b1">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> st <br/>( <font color="Olive" title="b1">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="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a>  &amp; (  for <font color="Olive" title="b2">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">bb</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c24">i</font> holds <br/>( <font color="Olive" title="b2">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="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></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="b2">x</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c20">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></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> ) ) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:70"><span class="lab"><font color="Green" title="E56">A59</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:70_8"/>
( <font color="Maroon" title="c22">a</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <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="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> &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="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p3">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <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="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></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> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:70_8"><span class="lab"><font color="Green" title="E63">A105</font></span></a>, <a class="txt" href="#E3:70_8"><span class="lab"><font color="Green" title="E64">A106</font></span></a>, <a class="txt" href="#E4:70_8"><span class="lab"><font color="Green" title="E65">A107</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E34:70"/><span class="lab"><font color="Green" title="E63">A108</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">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="70_9"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c23">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c23">n</font>] implies <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:70_9"/>
<font color="Maroon" title="c22">a</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c23">n</font>
 <span class="kw">and </span><br/><a NAME="E2:70_9"/>
<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="nat_1.html#K6" title="NAT_1:func.6">.</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="nat_1.html#K6" title="NAT_1:func.6">.</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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/>

<a NAME="E3:70_9"/><span class="lab"><font color="Green" title="E64">A109</font></span>: 
<font color="Maroon" title="c22">a</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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="nat_1.html#K6" title="NAT_1:func.6">.</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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:70"><span class="lab"><font color="Green" title="E55">A58</font></span></a></span>;<br/>
<span class="kw">consider </span><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="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E5:70_9"/><span class="lab"><font color="Green" title="E65">A110</font></span>: 
<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="nat_1.html#K6" title="NAT_1:func.6">.</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> 
 <span class="kw">and </span><br/><a NAME="E6:70_9"/><span class="lab"><font color="Green" title="E66">A111</font></span>: 
<font color="Maroon" title="c22">a</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c24">b</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c25">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:70"><span class="lab"><font color="Green" title="E61">A103</font></span></a></span>;<br/>
<a NAME="E7:70_9"/>
 ex <font color="Olive" title="b1">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> st <br/>( <font color="Olive" title="b1">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="nat_1.html#K6" title="NAT_1:func.6">.</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="b2">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">bb</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c25">i</font> holds <br/>( <font color="Olive" title="b2">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="nat_1.html#K6" title="NAT_1:func.6">.</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="b2">x</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c20">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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> ) ) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:70"><span class="lab"><font color="Green" title="E58">A84</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:70_9"/>
<font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:70_9"><span class="lab"><font color="Green" title="E64">A109</font></span></a>, <a class="txt" href="#E5:70_9"><span class="lab"><font color="Green" title="E65">A110</font></span></a>, <a class="txt" href="#E6:70_9"><span class="lab"><font color="Green" title="E66">A111</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E35:70"/>
 for <font color="Olive" title="b1">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">n</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S2" title="NAT_1:sch.2">NAT_1:sch 2</a>(<a class="txt" href="#E33:70"><span class="lab"><font color="Green" title="E62">A104</font></span></a>, <a class="txt" href="#E34:70"><span class="lab"><font color="Green" title="E63">A108</font></span></a>);<br/></span>
<a NAME="E36:70"/><span class="kw">then </span>
<font color="Maroon" title="c22">a</font> is  <a href="wellfnd1.html#V3" title="WELLFND1:attr.3">descending</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="wellfnd1.html#D6" title="WELLFND1:def.6">WELLFND1:def 6</a></span>;<br/>
<span class="kw">hence </span><a NAME="E37:70"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:70"><span class="lab"><font color="Green" title="E43">A1</font></span></a>, <a class="ref" href="wellfnd1.html#T14" title="WELLFND1:th.14">WELLFND1:14</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
