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

<b>let </b><font color="Maroon" title="c1">f</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/>

<b>assume </b><a NAME="E1:37"/><i><font color="Green" title="E23">A1</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 ;<br/>

<a NAME="E2:37"/><i><font color="Green" title="E24">A2</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="rfinseq2.html#K5" title="RFINSEQ2:func.5">sort_d</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font>
 
<b>by </b><i><a class="ref" href="rfinseq2.html#T30" target="_self" title="RFINSEQ2:th.30">Th30</a></i>;<br/>
<a NAME="E3:37"/><b>then </b>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="rfinseq2.html#K5" title="RFINSEQ2:func.5">sort_d</a> <font color="Maroon" title="c1">f</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="rfinseq2.html#E1:37"><i><font color="Green" title="E23">A1</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E4:37"/><b>then </b><i><font color="Green" title="E25">A3</font></i>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="rfinseq2.html#K5" title="RFINSEQ2:func.5">sort_d</a> <font color="Maroon" title="c1">f</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E5:37"/><i><font color="Green" title="E26">A4</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/> for <font color="Olive" title="b2">r1</font>, <font color="Olive" title="b3">r2</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="rfinseq2.html#K5" title="RFINSEQ2:func.5">sort_d</a> <font color="Maroon" title="c1">f</font></span>)</span> &amp; <font color="Olive" title="b2">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rfinseq2.html#K5" title="RFINSEQ2:func.5">sort_d</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b3">r2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rfinseq2.html#K5" title="RFINSEQ2:func.5">sort_d</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 1 holds <br/><font color="Olive" title="b2">r1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">r2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c3">r1</font>, <font color="Maroon" title="c4">r2</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/>



<b>assume </b><a NAME="E1:37_1"/><i><font color="Green" title="E27">A5</font></i>: 
( <font color="Maroon" title="c2">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="rfinseq2.html#K5" title="RFINSEQ2:func.5">sort_d</a> <font color="Maroon" title="c1">f</font></span>)</span> &amp; <font color="Maroon" title="c3">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rfinseq2.html#K5" title="RFINSEQ2:func.5">sort_d</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c2">i</font> &amp; <font color="Maroon" title="c4">r2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rfinseq2.html#K5" title="RFINSEQ2:func.5">sort_d</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 1 )
 ;<br/>

<a NAME="E2:37_1"/><b>then </b>
<font color="Maroon" title="c2">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="rfinseq2.html#K5" title="RFINSEQ2:func.5">sort_d</a> <font color="Maroon" title="c1">f</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E3:37_1"/><b>then </b><i><font color="Green" title="E28">A6</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">i</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:37_1_1_1"/>
( 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">i</font> or 1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">i</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_1_1"><b>case </b></a><a NAME="E1:37_1_1_1_1"/>
1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">i</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:37_1_1_1_1"/>
<font color="Maroon" title="c3">r1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">r2</font>
 <b>by </b><i><a class="txt" href="rfinseq2.html#E1:37_1"><i><font color="Green" title="E27">A5</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_1_2"><b>case </b></a><a NAME="E1:37_1_1_1_2"/>
1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">i</font>
 ;<br/><div class="add"><a NAME="E2:37_1_1_1_2"/><b>then </b>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c2">i</font>
 <b>by </b><i><a class="txt" href="rfinseq2.html#E3:37_1"><i><font color="Green" title="E28">A6</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><b>hence </b><a NAME="E3:37_1_1_1_2"/>
<font color="Maroon" title="c3">r1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">r2</font>
 <b>by </b><i><a class="txt" href="rfinseq2.html#E4:37"><i><font color="Green" title="E25">A3</font></i></a>, <a class="txt" href="rfinseq2.html#E1:37_1"><i><font color="Green" title="E27">A5</font></i></a>, <a class="ref" href="rfinseq.html#T32" title="RFINSEQ:th.32">RFINSEQ:32</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E5:37_1"/>
<font color="Maroon" title="c3">r1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">r2</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E6:37"/><i><font color="Green" title="E27">A7</font></i>: 
for <font color="Olive" title="b1">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="rfinseq2.html#K5" title="RFINSEQ2:func.5">sort_d</a> <font color="Maroon" title="c1">f</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="rfinseq2.html#K5" title="RFINSEQ2:func.5">sort_d</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rfinseq2.html#K5" title="RFINSEQ2:func.5">sort_d</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 1 holds <br/>1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">j</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">j</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:37_2"/>
( <font color="Maroon" title="c2">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="rfinseq2.html#K5" title="RFINSEQ2:func.5">sort_d</a> <font color="Maroon" title="c1">f</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="rfinseq2.html#K5" title="RFINSEQ2:func.5">sort_d</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rfinseq2.html#K5" title="RFINSEQ2:func.5">sort_d</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 1 )
 ;<br/>

<a NAME="E2:37_2"/><b>then </b>
<font color="Maroon" title="c2">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="rfinseq2.html#K5" title="RFINSEQ2:func.5">sort_d</a> <font color="Maroon" title="c1">f</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<b>hence </b><a NAME="E3:37_2"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">j</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:37"/><i><font color="Green" title="E28">A8</font></i>: 
<font color="Maroon" title="c1">f</font>, <a href="rfinseq2.html#K5" title="RFINSEQ2:func.5">sort_d</a> <font color="Maroon" title="c1">f</font> <a href="rfinseq.html#R1" title="RFINSEQ:pred.1">are_fiberwise_equipotent</a> 
 
<b>by </b><i><a class="ref" href="rfinseq2.html#D5" target="_self" title="RFINSEQ2:def.5">Def5</a></i>;<br/>
<i><font color="Green" title="E29">A9</font></i>: <span class="p1">(<span class="default"><a href="rfinseq2.html#K5" title="RFINSEQ2:func.5">sort_d</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 1 = 
 <a href="rfinseq2.html#K3" title="RFINSEQ2:func.3">max</a> <span class="p1">(<span class="default"><a href="rfinseq2.html#K5" title="RFINSEQ2:func.5">sort_d</a> <font color="Maroon" title="c1">f</font></span>)</span>
<b>by </b><i><a class="txt" href="rfinseq2.html#E1:37"><i><font color="Green" title="E23">A1</font></i></a>, <a class="txt" href="rfinseq2.html#E2:37"><i><font color="Green" title="E24">A2</font></i></a>, <a class="txt" href="rfinseq2.html#E4:37"><i><font color="Green" title="E25">A3</font></i></a>, <a class="txt" href="rfinseq2.html#E5:37"><i><font color="Green" title="E26">A4</font></i></a>, <a class="txt" href="rfinseq2.html#E6:37"><i><font color="Green" title="E27">A7</font></i></a>, <a class="ref" href="rfinseq2.html#D1" target="_self" title="RFINSEQ2:def.1">Def1</a></i>
<br/>.= 
 <a href="rfinseq2.html#K3" title="RFINSEQ2:func.3">max</a> <font color="Maroon" title="c1">f</font>
<b>by </b><i><a class="txt" href="rfinseq2.html#E7:37"><i><font color="Green" title="E28">A8</font></i></a>, <a class="ref" href="rfinseq2.html#T14" target="_self" title="RFINSEQ2:th.14">Th14</a></i>
;<br/>
<a NAME="E9:37"/><i><font color="Green" title="E30">A10</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="rfinseq2.html#K6" title="RFINSEQ2:func.6">sort_a</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font>
 
<b>by </b><i><a class="ref" href="rfinseq2.html#T31" target="_self" title="RFINSEQ2:th.31">Th31</a></i>;<br/>
<a NAME="E10:37"/><b>then </b><i><font color="Green" title="E31">A11</font></i>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="rfinseq2.html#K6" title="RFINSEQ2:func.6">sort_a</a> <font color="Maroon" title="c1">f</font></span>)</span>
 
<b>by </b><i><a class="txt" href="rfinseq2.html#E1:37"><i><font color="Green" title="E23">A1</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E11:37"/><i><font color="Green" title="E32">A12</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/> for <font color="Olive" title="b2">r1</font>, <font color="Olive" title="b3">r2</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="rfinseq2.html#K6" title="RFINSEQ2:func.6">sort_a</a> <font color="Maroon" title="c1">f</font></span>)</span> &amp; <font color="Olive" title="b2">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rfinseq2.html#K6" title="RFINSEQ2:func.6">sort_a</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b3">r2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rfinseq2.html#K6" title="RFINSEQ2:func.6">sort_a</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 1 holds <br/><font color="Olive" title="b2">r1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b3">r2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c3">r1</font>, <font color="Maroon" title="c4">r2</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/>



<b>assume </b><a NAME="E1:37_4"/><i><font color="Green" title="E33">A13</font></i>: 
( <font color="Maroon" title="c2">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="rfinseq2.html#K6" title="RFINSEQ2:func.6">sort_a</a> <font color="Maroon" title="c1">f</font></span>)</span> &amp; <font color="Maroon" title="c3">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rfinseq2.html#K6" title="RFINSEQ2:func.6">sort_a</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c2">i</font> &amp; <font color="Maroon" title="c4">r2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rfinseq2.html#K6" title="RFINSEQ2:func.6">sort_a</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 1 )
 ;<br/>

<a NAME="E2:37_4"/><b>then </b><i><font color="Green" title="E34">A14</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">i</font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_4_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:37_4_1"/>
( 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">i</font> or 1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">i</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_4_1_1"><b>suppose </b></a><a NAME="E1:37_4_1_1"/>
1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">i</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:37_4_1_1"/>
<font color="Maroon" title="c3">r1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c4">r2</font>
 <b>by </b><i><a class="txt" href="rfinseq2.html#E1:37_4"><i><font color="Green" title="E33">A13</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_4_1_2"><b>suppose </b></a><a NAME="E1:37_4_1_2"/>
1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">i</font>
 ;<br/><div class="add"><a NAME="E2:37_4_1_2"/><b>then </b>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c2">i</font>
 <b>by </b><i><a class="txt" href="rfinseq2.html#E2:37_4"><i><font color="Green" title="E34">A14</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><b>hence </b><a NAME="E3:37_4_1_2"/>
<font color="Maroon" title="c3">r1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c4">r2</font>
 <b>by </b><i><a class="txt" href="rfinseq2.html#E10:37"><i><font color="Green" title="E31">A11</font></i></a>, <a class="txt" href="rfinseq2.html#E1:37_4"><i><font color="Green" title="E33">A13</font></i></a>, <a class="ref" href="rfinseq2.html#T17" target="_self" title="RFINSEQ2:th.17">Th17</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E12:37"/><i><font color="Green" title="E33">A15</font></i>: 
for <font color="Olive" title="b1">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="rfinseq2.html#K6" title="RFINSEQ2:func.6">sort_a</a> <font color="Maroon" title="c1">f</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="rfinseq2.html#K6" title="RFINSEQ2:func.6">sort_a</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rfinseq2.html#K6" title="RFINSEQ2:func.6">sort_a</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 1 holds <br/>1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">j</font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E13:37"/><i><font color="Green" title="E34">A16</font></i>: 
<font color="Maroon" title="c1">f</font>, <a href="rfinseq2.html#K6" title="RFINSEQ2:func.6">sort_a</a> <font color="Maroon" title="c1">f</font> <a href="rfinseq.html#R1" title="RFINSEQ:pred.1">are_fiberwise_equipotent</a> 
 
<b>by </b><i><a class="ref" href="rfinseq2.html#D6" target="_self" title="RFINSEQ2:def.6">Def6</a></i>;<br/>
<span class="p1">(<span class="default"><a href="rfinseq2.html#K6" title="RFINSEQ2:func.6">sort_a</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 1 = 
 <a href="rfinseq2.html#K4" title="RFINSEQ2:func.4">min</a> <span class="p1">(<span class="default"><a href="rfinseq2.html#K6" title="RFINSEQ2:func.6">sort_a</a> <font color="Maroon" title="c1">f</font></span>)</span>
<b>by </b><i><a class="txt" href="rfinseq2.html#E1:37"><i><font color="Green" title="E23">A1</font></i></a>, <a class="txt" href="rfinseq2.html#E9:37"><i><font color="Green" title="E30">A10</font></i></a>, <a class="txt" href="rfinseq2.html#E10:37"><i><font color="Green" title="E31">A11</font></i></a>, <a class="txt" href="rfinseq2.html#E11:37"><i><font color="Green" title="E32">A12</font></i></a>, <a class="txt" href="rfinseq2.html#E12:37"><i><font color="Green" title="E33">A15</font></i></a>, <a class="ref" href="rfinseq2.html#D2" target="_self" title="RFINSEQ2:def.2">Def2</a></i>
<br/>.= 
 <a href="rfinseq2.html#K4" title="RFINSEQ2:func.4">min</a> <font color="Maroon" title="c1">f</font>
<b>by </b><i><a class="txt" href="rfinseq2.html#E13:37"><i><font color="Green" title="E34">A16</font></i></a>, <a class="ref" href="rfinseq2.html#T15" target="_self" title="RFINSEQ2:th.15">Th15</a></i>
;<br/>
<b>hence </b><a NAME="E15:37"/>
(  <a href="rfinseq2.html#K1" title="RFINSEQ2:func.1">max_p</a> <span class="p1">(<span class="default"><a href="rfinseq2.html#K5" title="RFINSEQ2:func.5">sort_d</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  <a href="rfinseq2.html#K2" title="RFINSEQ2:func.2">min_p</a> <span class="p1">(<span class="default"><a href="rfinseq2.html#K6" title="RFINSEQ2:func.6">sort_a</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <span class="p1">(<span class="default"><a href="rfinseq2.html#K5" title="RFINSEQ2:func.5">sort_d</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rfinseq2.html#K3" title="RFINSEQ2:func.3">max</a> <font color="Maroon" title="c1">f</font> &amp; <span class="p1">(<span class="default"><a href="rfinseq2.html#K6" title="RFINSEQ2:func.6">sort_a</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rfinseq2.html#K4" title="RFINSEQ2:func.4">min</a> <font color="Maroon" title="c1">f</font> )
 <b>by </b><i><a class="txt" href="rfinseq2.html#E1:37"><i><font color="Green" title="E23">A1</font></i></a>, <a class="txt" href="rfinseq2.html#E2:37"><i><font color="Green" title="E24">A2</font></i></a>, <a class="txt" href="rfinseq2.html#E4:37"><i><font color="Green" title="E25">A3</font></i></a>, <a class="txt" href="rfinseq2.html#E5:37"><i><font color="Green" title="E26">A4</font></i></a>, <a class="txt" href="rfinseq2.html#E6:37"><i><font color="Green" title="E27">A7</font></i></a>, <a class="txt" href="rfinseq2.html#E8:37"><i><font color="Green" title="E29">A9</font></i></a>, <a class="txt" href="rfinseq2.html#E9:37"><i><font color="Green" title="E30">A10</font></i></a>, <a class="txt" href="rfinseq2.html#E10:37"><i><font color="Green" title="E31">A11</font></i></a>, <a class="txt" href="rfinseq2.html#E11:37"><i><font color="Green" title="E32">A12</font></i></a>, <a class="txt" href="rfinseq2.html#E12:37"><i><font color="Green" title="E33">A15</font></i></a>, <a class="ref" href="rfinseq2.html#D1" target="_self" title="RFINSEQ2:def.1">Def1</a>, <a class="ref" href="rfinseq2.html#D2" target="_self" title="RFINSEQ2:def.2">Def2</a></i>;<br/>


</div>
