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

<b>let </b><font color="Maroon">f</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> ;<br/>

<b>assume </b><a NAME="E1:37"/><i><font color="Green">E27</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 ;<br/>

<a NAME="E2:37"/><b>then </b><i><font color="Green">E28</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
;<br/>
<a NAME="E3:37"/><i><font color="Green">E30</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="rfinseq2.html#K5" target="_self">sort_d</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font>
 
<b>by </b><i/>;<br/>
<a NAME="E4:37"/><b>then </b>
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="rfinseq2.html#K5" target="_self">sort_d</a> <font color="Maroon">f</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E5:37"/><b>then </b><i><font color="Green">E31</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="rfinseq2.html#K5" target="_self">sort_d</a> <font color="Maroon">f</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E6:37"/><i><font color="Green">E32</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <br/> for <font color="Olive">r1</font>, <font color="Olive">r2</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="rfinseq2.html#K5" target="_self">sort_d</a> <font color="Maroon">f</font></span>)</span> &amp; <font color="Olive">r1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rfinseq2.html#K5" target="_self">sort_d</a> <font color="Maroon">f</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">i</font> &amp; <font color="Olive">r2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rfinseq2.html#K5" target="_self">sort_d</a> <font color="Maroon">f</font></span>)</span> <a href="seq_1.html#K2">.</a> 1 holds <br/><font color="Olive">r1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">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">i</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">r1</font> be   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">r2</font> be   <a href="real_1.html#NM1">Real</a>;<br/>



<b>assume </b><a NAME="E1:37_1"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="rfinseq2.html#K5" target="_self">sort_d</a> <font color="Maroon">f</font></span>)</span> &amp; <font color="Maroon">r1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rfinseq2.html#K5" target="_self">sort_d</a> <font color="Maroon">f</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">r2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rfinseq2.html#K5" target="_self">sort_d</a> <font color="Maroon">f</font></span>)</span> <a href="seq_1.html#K2">.</a> 1 )
 ;<br/>

<a NAME="E2:37_1"/><b>then </b>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="rfinseq2.html#K5" target="_self">sort_d</a> <font color="Maroon">f</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E3:37_1"/><b>then </b><i><font color="Green">E36</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">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">=</a> <font color="Maroon">i</font> or 1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">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">=</a> <font color="Maroon">i</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:37_1_1_1_1"/>
<font color="Maroon">r1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r2</font>
 <b>by </b><i><a class="txt" href="rfinseq2.html#E2:37"><i><font color="Green">E28</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" target="_self">&lt;&gt;</a> <font color="Maroon">i</font>
 ;<br/><div class="add"><a NAME="E2:37_1_1_1_2"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">i</font>
 <b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><b>hence </b><a NAME="E3:37_1_1_1_2"/>
<font color="Maroon">r1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r2</font>
 <b>by </b><i>, <a class="txt" href="rfinseq2.html#E2:37"><i><font color="Green">E28</font></i></a>, <a class="ref" href="rfinseq.html#T32">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">r1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r2</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E7:37"/><i><font color="Green">E37</font></i>: 
for <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="rfinseq2.html#K5" target="_self">sort_d</a> <font color="Maroon">f</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="rfinseq2.html#K5" target="_self">sort_d</a> <font color="Maroon">f</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rfinseq2.html#K5" target="_self">sort_d</a> <font color="Maroon">f</font></span>)</span> <a href="seq_1.html#K2">.</a> 1 holds <br/>1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">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">j</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:37_2"/>
( <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="rfinseq2.html#K5" target="_self">sort_d</a> <font color="Maroon">f</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="rfinseq2.html#K5" target="_self">sort_d</a> <font color="Maroon">f</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rfinseq2.html#K5" target="_self">sort_d</a> <font color="Maroon">f</font></span>)</span> <a href="seq_1.html#K2">.</a> 1 )
 ;<br/>

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


</div><b>end;</b></div>
<a NAME="E8:37"/><b>then </b><i><font color="Green">E38</font></i>: 
 <a href="rfinseq2.html#K1" target="_self">max_p</a> <span class="p1">(<span class="default"><a href="rfinseq2.html#K5" target="_self">sort_d</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i>, , , <a class="txt" href="rfinseq2.html#E1:37"><i><font color="Green">E27</font></i></a>, </i>;<br/>
<a NAME="E9:37"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">f</font>, <a href="rfinseq2.html#K5" target="_self">sort_d</a> <font color="Maroon">f</font> <a href="rfinseq.html#R1">are_fiberwise_equipotent</a> 
 
<b>by </b><i/>;<br/>
<i><font color="Green">E40</font></i>: <span class="p1">(<span class="default"><a href="rfinseq2.html#K5" target="_self">sort_d</a> <font color="Maroon">f</font></span>)</span> <a href="seq_1.html#K2">.</a> 1 = 
 <a href="rfinseq2.html#K3" target="_self">max</a> <span class="p1">(<span class="default"><a href="rfinseq2.html#K5" target="_self">sort_d</a> <font color="Maroon">f</font></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="rfinseq2.html#K3" target="_self">max</a> <font color="Maroon">f</font>
<b>by </b><i><a class="txt" href="rfinseq2.html#E3:37"><i><font color="Green">E30</font></i></a>, </i>
;<br/>
<a NAME="E11:37"/><i><font color="Green">E41</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="rfinseq2.html#K6" target="_self">sort_a</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="ref" href="rfinseq2.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E12:37"/><b>then </b><i><font color="Green">E42</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="rfinseq2.html#K6" target="_self">sort_a</a> <font color="Maroon">f</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E13:37"/><i><font color="Green">E43</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <br/> for <font color="Olive">r1</font>, <font color="Olive">r2</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="rfinseq2.html#K6" target="_self">sort_a</a> <font color="Maroon">f</font></span>)</span> &amp; <font color="Olive">r1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rfinseq2.html#K6" target="_self">sort_a</a> <font color="Maroon">f</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">i</font> &amp; <font color="Olive">r2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rfinseq2.html#K6" target="_self">sort_a</a> <font color="Maroon">f</font></span>)</span> <a href="seq_1.html#K2">.</a> 1 holds <br/><font color="Olive">r1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">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">i</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">r1</font> be   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">r2</font> be   <a href="real_1.html#NM1">Real</a>;<br/>



<b>assume </b><a NAME="E1:37_4"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="rfinseq2.html#K6" target="_self">sort_a</a> <font color="Maroon">f</font></span>)</span> &amp; <font color="Maroon">r1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rfinseq2.html#K6" target="_self">sort_a</a> <font color="Maroon">f</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">r2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rfinseq2.html#K6" target="_self">sort_a</a> <font color="Maroon">f</font></span>)</span> <a href="seq_1.html#K2">.</a> 1 )
 ;<br/>

<a NAME="E2:37_4"/><b>then </b><i><font color="Green">E45</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27">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">=</a> <font color="Maroon">i</font> or 1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">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">=</a> <font color="Maroon">i</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:37_4_1_1"/>
<font color="Maroon">r1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">r2</font>
 <b>by </b><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" target="_self">&lt;&gt;</a> <font color="Maroon">i</font>
 ;<br/><div class="add"><a NAME="E2:37_4_1_2"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">i</font>
 <b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><b>hence </b><a NAME="E3:37_4_1_2"/>
<font color="Maroon">r1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">r2</font>
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E14:37"/><i><font color="Green">E46</font></i>: 
for <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="rfinseq2.html#K6" target="_self">sort_a</a> <font color="Maroon">f</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="rfinseq2.html#K6" target="_self">sort_a</a> <font color="Maroon">f</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rfinseq2.html#K6" target="_self">sort_a</a> <font color="Maroon">f</font></span>)</span> <a href="seq_1.html#K2">.</a> 1 holds <br/>1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">j</font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E15:37"/><b>then </b><i><font color="Green">E47</font></i>: 
 <a href="rfinseq2.html#K2" target="_self">min_p</a> <span class="p1">(<span class="default"><a href="rfinseq2.html#K6" target="_self">sort_a</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i>, <a class="txt" href="rfinseq2.html#E6:37"><i><font color="Green">E32</font></i></a>, , , </i>;<br/>
<a NAME="E16:37"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">f</font>, <a href="rfinseq2.html#K6" target="_self">sort_a</a> <font color="Maroon">f</font> <a href="rfinseq.html#R1">are_fiberwise_equipotent</a> 
 
<b>by </b><i/>;<br/>
<span class="p1">(<span class="default"><a href="rfinseq2.html#K6" target="_self">sort_a</a> <font color="Maroon">f</font></span>)</span> <a href="seq_1.html#K2">.</a> 1 = 
 <a href="rfinseq2.html#K4" target="_self">min</a> <span class="p1">(<span class="default"><a href="rfinseq2.html#K6" target="_self">sort_a</a> <font color="Maroon">f</font></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="rfinseq2.html#K4" target="_self">min</a> <font color="Maroon">f</font>
<b>by </b><i><a class="txt" href="rfinseq2.html#E7:37"><i><font color="Green">E37</font></i></a>, </i>
;<br/>
<b>hence </b><a NAME="E18:37"/>
(  <a href="rfinseq2.html#K1" target="_self">max_p</a> <span class="p1">(<span class="default"><a href="rfinseq2.html#K5" target="_self">sort_d</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a> 1 &amp;  <a href="rfinseq2.html#K2" target="_self">min_p</a> <span class="p1">(<span class="default"><a href="rfinseq2.html#K6" target="_self">sort_a</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a> 1 &amp; <span class="p1">(<span class="default"><a href="rfinseq2.html#K5" target="_self">sort_d</a> <font color="Maroon">f</font></span>)</span> <a href="seq_1.html#K2">.</a> 1 <a href="hidden.html#R1">=</a>  <a href="rfinseq2.html#K3" target="_self">max</a> <font color="Maroon">f</font> &amp; <span class="p1">(<span class="default"><a href="rfinseq2.html#K6" target="_self">sort_a</a> <font color="Maroon">f</font></span>)</span> <a href="seq_1.html#K2">.</a> 1 <a href="hidden.html#R1">=</a>  <a href="rfinseq2.html#K4" target="_self">min</a> <font color="Maroon">f</font> )
 <b>by </b><i>, , , <a class="txt" href="rfinseq2.html#E1:37"><i><font color="Green">E27</font></i></a>, <a class="txt" href="rfinseq2.html#E2:37"><i><font color="Green">E28</font></i></a>, <a class="txt" href="rfinseq2.html#E5:37"><i><font color="Green">E31</font></i></a>, <a class="txt" href="rfinseq2.html#E6:37"><i><font color="Green">E32</font></i></a>, , , , , </i>;<br/>


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