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

<b>let </b><font color="Maroon">s</font> be   <a href="ami_1.html#NM4">State</a> of <a href="scmpds_2.html#K1">SCMPDS</a> , <font color="Maroon">s1</font> be   <a href="ami_1.html#NM4">State</a> of <a href="scmpds_2.html#K1">SCMPDS</a> ;<br/><b>let </b><font color="Maroon">n0</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">n1</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">n2</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">n</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">f</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K4">INT</a> , <font color="Maroon">f1</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K4">INT</a> ;<br/>





<b>assume </b><a NAME="E1:39"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">f</font> <a href="scpisort.html#R1">is_FinSequence_on</a> <font color="Maroon">s</font>,<font color="Maroon">n0</font> &amp; <font color="Maroon">f1</font> <a href="scpisort.html#R1">is_FinSequence_on</a> <font color="Maroon">s1</font>,<font color="Maroon">n0</font> )
 ;<br/>

<b>assume </b><a NAME="E2:39"/><i><font color="Green">E58</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> )
 ;<br/>

<b>assume </b><a NAME="E3:39"/><i><font color="Green">E59</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>   st <font color="Olive">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">n0</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Olive">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">n1</font> &amp; <font color="Olive">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">n2</font> holds <br/><font color="Maroon">s1</font> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1">intpos</a> <font color="Olive">i</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1">intpos</a> <font color="Olive">i</font></span>)</span>
 ;<br/>

<b>assume </b><a NAME="E4:39"/><i><font color="Green">E60</font></i>: 
( ( <font color="Maroon">s1</font> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1">intpos</a> <font color="Maroon">n1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1">intpos</a> <font color="Maroon">n1</font></span>)</span> &amp; <font color="Maroon">s1</font> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1">intpos</a> <font color="Maroon">n2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1">intpos</a> <font color="Maroon">n2</font></span>)</span> ) or ( <font color="Maroon">n1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">n0</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">n2</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">n0</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">n1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n0</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n0</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font> &amp; <font color="Maroon">s1</font> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1">intpos</a> <font color="Maroon">n1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1">intpos</a> <font color="Maroon">n2</font></span>)</span> &amp; <font color="Maroon">s1</font> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1">intpos</a> <font color="Maroon">n2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1">intpos</a> <font color="Maroon">n1</font></span>)</span> ) )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:39_1"/>
( ( <font color="Maroon">s1</font> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1">intpos</a> <font color="Maroon">n1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1">intpos</a> <font color="Maroon">n1</font></span>)</span> &amp; <font color="Maroon">s1</font> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1">intpos</a> <font color="Maroon">n2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1">intpos</a> <font color="Maroon">n2</font></span>)</span> ) or ( <font color="Maroon">n1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">n0</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">n2</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">n0</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">n1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n0</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n0</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font> &amp; <font color="Maroon">s1</font> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1">intpos</a> <font color="Maroon">n1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1">intpos</a> <font color="Maroon">n2</font></span>)</span> &amp; <font color="Maroon">s1</font> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1">intpos</a> <font color="Maroon">n2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1">intpos</a> <font color="Maroon">n1</font></span>)</span> ) )
 </span><b>by </b><i><a class="ref" href="scpqsort.html#T1" target="_self">Th1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1"><b>suppose </b></a><a NAME="E1:39_1_1"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">s1</font> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1">intpos</a> <font color="Maroon">n1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1">intpos</a> <font color="Maroon">n1</font></span>)</span> &amp; <font color="Maroon">s1</font> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1">intpos</a> <font color="Maroon">n2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1">intpos</a> <font color="Maroon">n2</font></span>)</span> )
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1_1"><b>now </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>assume </b><a NAME="E1:39_1_1_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 ;<br/><a NAME="E2:39_1_1_1"/><b>then </b><i><font color="Green">E62</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> )
 <b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E3:39_1_1_1"/><b>then </b><i><font color="Green">E63</font></i>: 
( <font color="Maroon">n0</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n0</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font> &amp; <font color="Maroon">n0</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n0</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font> )
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:39_1_1_1_1"/>
( ( <font color="Maroon">n0</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">n1</font> &amp; <font color="Maroon">n0</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">n2</font> ) or  not <font color="Maroon">n0</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">n1</font> or  not <font color="Maroon">n0</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">n2</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:39_1_1_1_1_1"/><i><font color="Green">E64</font></i>: 
( <font color="Maroon">n0</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">n1</font> &amp; <font color="Maroon">n0</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">n2</font> )
 ;<br/><div class="add"><b>thus </b><font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> = 
<font color="Maroon">s1</font> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1">intpos</a> <span class="p2">(<span class="default"><font color="Maroon">n0</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span></span>)</span>
<b>by </b><i>, , <a class="txt" href="scpqsort.html#E1:39"><i><font color="Green">E42</font></i></a>, <a class="ref" href="scpisort.html#D1">SCPISORT:def 1</a></i>
<br/>.= 
<font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1">intpos</a> <span class="p2">(<span class="default"><font color="Maroon">n0</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span></span>)</span>
<b>by </b><i>, , </i>
<br/>.= 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
<b>by </b><i>, , <a class="txt" href="scpqsort.html#E1:39"><i><font color="Green">E42</font></i></a>, <a class="ref" href="scpisort.html#D1">SCPISORT:def 1</a></i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:39_1_1_1_1_2"/><i><font color="Green">E65</font></i>: 
(  not <font color="Maroon">n0</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">n1</font> or  not <font color="Maroon">n0</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">n2</font> )
 ;<br/><div class="add"><div><b>hereby </b>
<div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:39_1_1_1_1_2_1_1"/>
( <font color="Maroon">n0</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n1</font> or <font color="Maroon">n0</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n2</font> )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:39_1_1_1_1_2_1_1_1"/>
<font color="Maroon">n0</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n1</font>
 ;<br/><div class="add"><b>hence </b><font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> = 
<font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1">intpos</a> <span class="p2">(<span class="default"><font color="Maroon">n0</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span></span>)</span>
<b>by </b><i>, , , <a class="txt" href="scpqsort.html#E1:39"><i><font color="Green">E42</font></i></a>, <a class="ref" href="scpisort.html#D1">SCPISORT:def 1</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
<b>by </b><i>, , <a class="txt" href="scpqsort.html#E1:39"><i><font color="Green">E42</font></i></a>, <a class="ref" href="scpisort.html#D1">SCPISORT:def 1</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:39_1_1_1_1_2_1_1_2"/>
<font color="Maroon">n0</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n2</font>
 ;<br/><div class="add"><b>hence </b><font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> = 
<font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1">intpos</a> <span class="p2">(<span class="default"><font color="Maroon">n0</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span></span>)</span>
<b>by </b><i>, , , <a class="txt" href="scpqsort.html#E1:39"><i><font color="Green">E42</font></i></a>, <a class="ref" href="scpisort.html#D1">SCPISORT:def 1</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
<b>by </b><i>, , <a class="txt" href="scpqsort.html#E1:39"><i><font color="Green">E42</font></i></a>, <a class="ref" href="scpisort.html#D1">SCPISORT:def 1</a></i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div></div>
<b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:39_1_1"/>
<font color="Maroon">f</font>,<font color="Maroon">f1</font> <a href="rfinseq.html#R1">are_fiberwise_equipotent</a> 
 <b>by </b><i>, <a class="ref" href="finseq_2.html#T10">FINSEQ_2:10</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_2"><b>suppose </b></a><a NAME="E1:39_1_2"/><i><font color="Green">E66</font></i>: 
( <font color="Maroon">n1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">n0</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">n2</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">n0</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">n1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n0</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n0</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font> &amp; <font color="Maroon">s1</font> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1">intpos</a> <font color="Maroon">n1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1">intpos</a> <font color="Maroon">n2</font></span>)</span> &amp; <font color="Maroon">s1</font> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1">intpos</a> <font color="Maroon">n2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1">intpos</a> <font color="Maroon">n1</font></span>)</span> )
 ;<br/><div class="add"><a NAME="E2:39_1_2"/><b>then </b><i><font color="Green">E67</font></i>: 
<font color="Maroon">n1</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">n0</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><a NAME="E3:39_1_2"/><b>then </b>
<font color="Maroon">n1</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">n0</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><b>then reconsider </b><font color="Maroon">m1</font> = <font color="Maroon">n1</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">n0</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T16">INT_1:16</a></i>;<br/><a NAME="E5:39_1_2"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">n2</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">n0</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><a NAME="E6:39_1_2"/><b>then </b>
<font color="Maroon">n2</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">n0</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><b>then reconsider </b><font color="Maroon">m2</font> = <font color="Maroon">n2</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">n0</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T16">INT_1:16</a></i>;<br/><a NAME="E8:39_1_2"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">m1</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font>
 <b>by </b><i>, , <a class="ref" href="xreal_1.html#T22">XREAL_1:22</a></i>;<br/><a NAME="E9:39_1_2"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">m2</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f1</font>
 <b>by </b><i>, , <a class="ref" href="xreal_1.html#T22">XREAL_1:22</a></i>;<br/><a NAME="E10:39_1_2"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">m2</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font>
 <b>by </b><i>, , <a class="ref" href="xreal_1.html#T22">XREAL_1:22</a></i>;<br/><a NAME="E11:39_1_2"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">n1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n0</font>
 ;<br/><a NAME="E12:39_1_2"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">n2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m2</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n0</font>
 ;<br/><i><font color="Green">E77</font></i>: <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m1</font> = 
<font color="Maroon">s1</font> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1">intpos</a> <font color="Maroon">n2</font></span>)</span>
<b>by </b><i>, , <a class="txt" href="scpqsort.html#E1:39"><i><font color="Green">E42</font></i></a>, , , <a class="ref" href="scpisort.html#D1">SCPISORT:def 1</a></i>
<br/>.= 
<font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m2</font>
<b>by </b><i>, , <a class="txt" href="scpqsort.html#E3"><i><font color="Green">Lemma45</font></i></a>, <a class="txt" href="scpqsort.html#E4"><i><font color="Green">Lemma48</font></i></a>, <a class="ref" href="scpisort.html#D1">SCPISORT:def 1</a></i>
;<br/><i><font color="Green">E80</font></i>: <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m2</font> = 
<font color="Maroon">s1</font> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1">intpos</a> <font color="Maroon">n1</font></span>)</span>
<b>by </b><i>, , , , <a class="txt" href="scpqsort.html#E4"><i><font color="Green">Lemma48</font></i></a>, <a class="ref" href="scpisort.html#D1">SCPISORT:def 1</a></i>
<br/>.= 
<font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m1</font>
<b>by </b><i>, , <a class="txt" href="scpqsort.html#E1:39"><i><font color="Green">E42</font></i></a>, , , <a class="ref" href="scpisort.html#D1">SCPISORT:def 1</a></i>
;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_2_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:39_1_2_3"/><i><font color="Green">E81</font></i>: 
( <font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">m1</font> &amp; <font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">m2</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> )
 ;<br/><a NAME="E2:39_1_2_3"/><b>then </b><i><font color="Green">E82</font></i>: 
( <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n0</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">m1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n0</font> &amp; <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n0</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">m2</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n0</font> )
 ;<br/><a NAME="E3:39_1_2_3"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">n0</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n0</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><b>thus </b><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> = 
<font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1">intpos</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n0</font></span>)</span></span>)</span>
<b>by </b><i>, , <a class="ref" href="scpisort.html#D1">SCPISORT:def 1</a></i>
<br/>.= 
<font color="Maroon">s1</font> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1">intpos</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n0</font></span>)</span></span>)</span>
<b>by </b><i>, , </i>
<br/>.= 
<font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
<b>by </b><i>, , , <a class="ref" href="scpisort.html#D1">SCPISORT:def 1</a></i>
;<br/></div><b>end;</b></div><b>hence </b><a NAME="E16:39_1_2"/>
<font color="Maroon">f</font>,<font color="Maroon">f1</font> <a href="rfinseq.html#R1">are_fiberwise_equipotent</a> 
 <b>by </b><i>, <a class="txt" href="scpqsort.html#E1:39"><i><font color="Green">E42</font></i></a>, , , , <a class="ref" href="scpqsort.html#T2" target="_self">Th2</a>, , <a class="ref" href="scpisort.html#T4">SCPISORT:4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
