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

<b>let </b><font color="Maroon">c<sub>47</sub></font>, <font color="Maroon">c<sub>48</sub></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">c<sub>49</sub></font>, <font color="Maroon">c<sub>50</sub></font>, <font color="Maroon">c<sub>51</sub></font>, <font color="Maroon">c<sub>52</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>53</sub></font>, <font color="Maroon">c<sub>54</sub></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">E37</font></i>: 
( <font color="Maroon">c<sub>53</sub></font> <a href="scpisort.html#R1">is_FinSequence_on</a> <font color="Maroon">c<sub>47</sub></font>,<font color="Maroon">c<sub>49</sub></font> &amp; <font color="Maroon">c<sub>54</sub></font> <a href="scpisort.html#R1">is_FinSequence_on</a> <font color="Maroon">c<sub>48</sub></font>,<font color="Maroon">c<sub>49</sub></font> )
 ;<br/>

<b>assume </b><a NAME="E2:39"/><i><font color="Green">E38</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>53</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>52</sub></font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>54</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>52</sub></font> )
 ;<br/>

<b>assume </b><a NAME="E3:39"/><i><font color="Green">E39</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>49</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>50</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>51</sub></font> holds <br/><font color="Maroon">c<sub>48</sub></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">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>47</sub></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">b<sub>1</sub></font></span>)</span>
 ;<br/>

<b>assume </b><a NAME="E4:39"/><i><font color="Green">E40</font></i>: 
( ( <font color="Maroon">c<sub>48</sub></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">c<sub>50</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>47</sub></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">c<sub>50</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>48</sub></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">c<sub>51</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>47</sub></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">c<sub>51</sub></font></span>)</span> ) or ( <font color="Maroon">c<sub>50</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>49</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>51</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>49</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>50</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>49</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>52</sub></font> &amp; <font color="Maroon">c<sub>51</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>49</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>52</sub></font> &amp; <font color="Maroon">c<sub>48</sub></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">c<sub>50</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>47</sub></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">c<sub>51</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>48</sub></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">c<sub>51</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>47</sub></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">c<sub>50</sub></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">c<sub>48</sub></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">c<sub>50</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>47</sub></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">c<sub>50</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>48</sub></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">c<sub>51</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>47</sub></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">c<sub>51</sub></font></span>)</span> ) or ( <font color="Maroon">c<sub>50</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>49</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>51</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>49</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>50</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>49</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>52</sub></font> &amp; <font color="Maroon">c<sub>51</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>49</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>52</sub></font> &amp; <font color="Maroon">c<sub>48</sub></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">c<sub>50</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>47</sub></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">c<sub>51</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>48</sub></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">c<sub>51</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>47</sub></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">c<sub>50</sub></font></span>)</span> ) )
 </span><b>by </b><i><a class="txt" href="scpqsort.html#E4:39"><i><font color="Green">E40</font></i></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">E41</font></i>: 
( <font color="Maroon">c<sub>48</sub></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">c<sub>50</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>47</sub></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">c<sub>50</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>48</sub></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">c<sub>51</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>47</sub></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">c<sub>51</sub></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">c<sub>55</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:39_1_1_1"/>
<font color="Maroon">c<sub>55</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>52</sub></font>
 ;<br/><a NAME="E2:39_1_1_1"/><b>then </b><i><font color="Green">E42</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>55</sub></font> &amp; <font color="Maroon">c<sub>55</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>52</sub></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">E43</font></i>: 
( <font color="Maroon">c<sub>49</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>49</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>55</sub></font> &amp; <font color="Maroon">c<sub>49</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>55</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>49</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>52</sub></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">c<sub>49</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>55</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>50</sub></font> &amp; <font color="Maroon">c<sub>49</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>55</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>51</sub></font> ) or  not <font color="Maroon">c<sub>49</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>55</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>50</sub></font> or  not <font color="Maroon">c<sub>49</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>55</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>51</sub></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">E44</font></i>: 
( <font color="Maroon">c<sub>49</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>55</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>50</sub></font> &amp; <font color="Maroon">c<sub>49</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>55</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>51</sub></font> )
 ;<br/><div class="add"><b>thus </b><font color="Maroon">c<sub>54</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>55</sub></font> = 
<font color="Maroon">c<sub>48</sub></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">c<sub>49</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>55</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="scpqsort.html#E1:39"><i><font color="Green">E37</font></i></a>, <a class="txt" href="scpqsort.html#E2:39"><i><font color="Green">E38</font></i></a>, <a class="txt" href="scpqsort.html#E2:39_1_1_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="scpisort.html#D1">SCPISORT:def 1</a></i>
<br/>.= 
<font color="Maroon">c<sub>47</sub></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">c<sub>49</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>55</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="scpqsort.html#E3:39"><i><font color="Green">E39</font></i></a>, <a class="txt" href="scpqsort.html#E3:39_1_1_1"><i><font color="Green">E43</font></i></a>, <a class="txt" href="scpqsort.html#E1:39_1_1_1_1_1"><i><font color="Green">E44</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>53</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>55</sub></font>
<b>by </b><i><a class="txt" href="scpqsort.html#E1:39"><i><font color="Green">E37</font></i></a>, <a class="txt" href="scpqsort.html#E2:39"><i><font color="Green">E38</font></i></a>, <a class="txt" href="scpqsort.html#E2:39_1_1_1"><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">E44</font></i>: 
(  not <font color="Maroon">c<sub>49</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>55</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>50</sub></font> or  not <font color="Maroon">c<sub>49</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>55</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>51</sub></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">c<sub>49</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>55</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>50</sub></font> or <font color="Maroon">c<sub>49</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>55</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>51</sub></font> )
 </span><b>by </b><i><a class="txt" href="scpqsort.html#E1:39_1_1_1_1_2"><i><font color="Green">E44</font></i></a></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">c<sub>49</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>55</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>50</sub></font>
 ;<br/><div class="add"><b>hence </b><font color="Maroon">c<sub>54</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>55</sub></font> = 
<font color="Maroon">c<sub>47</sub></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">c<sub>49</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>55</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="scpqsort.html#E1:39"><i><font color="Green">E37</font></i></a>, <a class="txt" href="scpqsort.html#E2:39"><i><font color="Green">E38</font></i></a>, <a class="txt" href="scpqsort.html#E1:39_1_1"><i><font color="Green">E41</font></i></a>, <a class="txt" href="scpqsort.html#E2:39_1_1_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="scpisort.html#D1">SCPISORT:def 1</a></i>
<br/>.= 
<font color="Maroon">c<sub>53</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>55</sub></font>
<b>by </b><i><a class="txt" href="scpqsort.html#E1:39"><i><font color="Green">E37</font></i></a>, <a class="txt" href="scpqsort.html#E2:39"><i><font color="Green">E38</font></i></a>, <a class="txt" href="scpqsort.html#E2:39_1_1_1"><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">c<sub>49</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>55</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>51</sub></font>
 ;<br/><div class="add"><b>hence </b><font color="Maroon">c<sub>54</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>55</sub></font> = 
<font color="Maroon">c<sub>47</sub></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">c<sub>49</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>55</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="scpqsort.html#E1:39"><i><font color="Green">E37</font></i></a>, <a class="txt" href="scpqsort.html#E2:39"><i><font color="Green">E38</font></i></a>, <a class="txt" href="scpqsort.html#E1:39_1_1"><i><font color="Green">E41</font></i></a>, <a class="txt" href="scpqsort.html#E2:39_1_1_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="scpisort.html#D1">SCPISORT:def 1</a></i>
<br/>.= 
<font color="Maroon">c<sub>53</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>55</sub></font>
<b>by </b><i><a class="txt" href="scpqsort.html#E1:39"><i><font color="Green">E37</font></i></a>, <a class="txt" href="scpqsort.html#E2:39"><i><font color="Green">E38</font></i></a>, <a class="txt" href="scpqsort.html#E2:39_1_1_1"><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">c<sub>53</sub></font>,<font color="Maroon">c<sub>54</sub></font> <a href="rfinseq.html#R1">are_fiberwise_equipotent</a> 
 <b>by </b><i><a class="txt" href="scpqsort.html#E2:39"><i><font color="Green">E38</font></i></a>, <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">E41</font></i>: 
( <font color="Maroon">c<sub>50</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>49</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>51</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>49</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>50</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>49</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>52</sub></font> &amp; <font color="Maroon">c<sub>51</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>49</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>52</sub></font> &amp; <font color="Maroon">c<sub>48</sub></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">c<sub>50</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>47</sub></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">c<sub>51</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>48</sub></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">c<sub>51</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>47</sub></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">c<sub>50</sub></font></span>)</span> )
 ;<br/><div class="add"><a NAME="E2:39_1_2"/><b>then </b><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>50</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>49</sub></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">c<sub>50</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>49</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>55</sub></font> = <font color="Maroon">c<sub>50</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>49</sub></font> as   <a href="nat_1.html#NM1">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">E43</font></i>: 
<font color="Maroon">c<sub>51</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>49</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>by </b><i><a class="txt" href="scpqsort.html#E1:39_1_2"><i><font color="Green">E41</font></i></a>, <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">c<sub>51</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>49</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>56</sub></font> = <font color="Maroon">c<sub>51</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>49</sub></font> as   <a href="nat_1.html#NM1">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">E44</font></i>: 
<font color="Maroon">c<sub>55</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>53</sub></font>
 <b>by </b><i><a class="txt" href="scpqsort.html#E2:39"><i><font color="Green">E38</font></i></a>, <a class="txt" href="scpqsort.html#E1:39_1_2"><i><font color="Green">E41</font></i></a>, <a class="ref" href="xreal_1.html#T22">XREAL_1:22</a></i>;<br/><a NAME="E9:39_1_2"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">c<sub>56</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>54</sub></font>
 <b>by </b><i><a class="txt" href="scpqsort.html#E2:39"><i><font color="Green">E38</font></i></a>, <a class="txt" href="scpqsort.html#E1:39_1_2"><i><font color="Green">E41</font></i></a>, <a class="ref" href="xreal_1.html#T22">XREAL_1:22</a></i>;<br/><a NAME="E10:39_1_2"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>56</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>53</sub></font>
 <b>by </b><i><a class="txt" href="scpqsort.html#E2:39"><i><font color="Green">E38</font></i></a>, <a class="txt" href="scpqsort.html#E1:39_1_2"><i><font color="Green">E41</font></i></a>, <a class="ref" href="xreal_1.html#T22">XREAL_1:22</a></i>;<br/><a NAME="E11:39_1_2"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">c<sub>50</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>55</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>49</sub></font>
 ;<br/><a NAME="E12:39_1_2"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>51</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>56</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>49</sub></font>
 ;<br/><i><font color="Green">E49</font></i>: <font color="Maroon">c<sub>53</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>55</sub></font> = 
<font color="Maroon">c<sub>48</sub></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">c<sub>51</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="scpqsort.html#E1:39"><i><font color="Green">E37</font></i></a>, <a class="txt" href="scpqsort.html#E1:39_1_2"><i><font color="Green">E41</font></i></a>, <a class="txt" href="scpqsort.html#E2:39_1_2"><i><font color="Green">E42</font></i></a>, <a class="txt" href="scpqsort.html#E8:39_1_2"><i><font color="Green">E44</font></i></a>, <a class="txt" href="scpqsort.html#E11:39_1_2"><i><font color="Green">E47</font></i></a>, <a class="ref" href="scpisort.html#D1">SCPISORT:def 1</a></i>
<br/>.= 
<font color="Maroon">c<sub>54</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>56</sub></font>
<b>by </b><i><a class="txt" href="scpqsort.html#E1:39"><i><font color="Green">E37</font></i></a>, <a class="txt" href="scpqsort.html#E5:39_1_2"><i><font color="Green">E43</font></i></a>, <a class="txt" href="scpqsort.html#E9:39_1_2"><i><font color="Green">E45</font></i></a>, <a class="txt" href="scpqsort.html#E12:39_1_2"><i><font color="Green">E48</font></i></a>, <a class="ref" href="scpisort.html#D1">SCPISORT:def 1</a></i>
;<br/><i><font color="Green">E50</font></i>: <font color="Maroon">c<sub>53</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>56</sub></font> = 
<font color="Maroon">c<sub>48</sub></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">c<sub>50</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="scpqsort.html#E1:39"><i><font color="Green">E37</font></i></a>, <a class="txt" href="scpqsort.html#E1:39_1_2"><i><font color="Green">E41</font></i></a>, <a class="txt" href="scpqsort.html#E5:39_1_2"><i><font color="Green">E43</font></i></a>, <a class="txt" href="scpqsort.html#E10:39_1_2"><i><font color="Green">E46</font></i></a>, <a class="txt" href="scpqsort.html#E12:39_1_2"><i><font color="Green">E48</font></i></a>, <a class="ref" href="scpisort.html#D1">SCPISORT:def 1</a></i>
<br/>.= 
<font color="Maroon">c<sub>54</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>55</sub></font>
<b>by </b><i><a class="txt" href="scpqsort.html#E1:39"><i><font color="Green">E37</font></i></a>, <a class="txt" href="scpqsort.html#E2:39"><i><font color="Green">E38</font></i></a>, <a class="txt" href="scpqsort.html#E2:39_1_2"><i><font color="Green">E42</font></i></a>, <a class="txt" href="scpqsort.html#E8:39_1_2"><i><font color="Green">E44</font></i></a>, <a class="txt" href="scpqsort.html#E11:39_1_2"><i><font color="Green">E47</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">c<sub>57</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:39_1_2_3"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">c<sub>57</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>55</sub></font> &amp; <font color="Maroon">c<sub>57</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>56</sub></font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>57</sub></font> &amp; <font color="Maroon">c<sub>57</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>53</sub></font> )
 ;<br/><a NAME="E2:39_1_2_3"/><b>then </b><i><font color="Green">E52</font></i>: 
( <font color="Maroon">c<sub>57</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>49</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>55</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>49</sub></font> &amp; <font color="Maroon">c<sub>57</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>49</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>56</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>49</sub></font> )
 ;<br/><a NAME="E3:39_1_2_3"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">c<sub>49</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>49</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>57</sub></font>
 <b>by </b><i><a class="txt" href="scpqsort.html#E1:39_1_2_3"><i><font color="Green">E51</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><b>thus </b><font color="Maroon">c<sub>53</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>57</sub></font> = 
<font color="Maroon">c<sub>47</sub></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">c<sub>57</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>49</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="scpqsort.html#E1:39"><i><font color="Green">E37</font></i></a>, <a class="txt" href="scpqsort.html#E1:39_1_2_3"><i><font color="Green">E51</font></i></a>, <a class="ref" href="scpisort.html#D1">SCPISORT:def 1</a></i>
<br/>.= 
<font color="Maroon">c<sub>48</sub></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">c<sub>57</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>49</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="scpqsort.html#E3:39"><i><font color="Green">E39</font></i></a>, <a class="txt" href="scpqsort.html#E2:39_1_2_3"><i><font color="Green">E52</font></i></a>, <a class="txt" href="scpqsort.html#E3:39_1_2_3"><i><font color="Green">E53</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>54</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>57</sub></font>
<b>by </b><i><a class="txt" href="scpqsort.html#E1:39"><i><font color="Green">E37</font></i></a>, <a class="txt" href="scpqsort.html#E2:39"><i><font color="Green">E38</font></i></a>, <a class="txt" href="scpqsort.html#E1:39_1_2_3"><i><font color="Green">E51</font></i></a>, <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">c<sub>53</sub></font>,<font color="Maroon">c<sub>54</sub></font> <a href="rfinseq.html#R1">are_fiberwise_equipotent</a> 
 <b>by </b><i><a class="txt" href="scpqsort.html#E2:39"><i><font color="Green">E38</font></i></a>, <a class="txt" href="scpqsort.html#E2:39_1_2"><i><font color="Green">E42</font></i></a>, <a class="txt" href="scpqsort.html#E5:39_1_2"><i><font color="Green">E43</font></i></a>, <a class="txt" href="scpqsort.html#E8:39_1_2"><i><font color="Green">E44</font></i></a>, <a class="txt" href="scpqsort.html#E10:39_1_2"><i><font color="Green">E46</font></i></a>, <a class="txt" href="scpqsort.html#E13:39_1_2"><i><font color="Green">E49</font></i></a>, <a class="txt" href="scpqsort.html#E14:39_1_2"><i><font color="Green">E50</font></i></a>, <a class="ref" href="scpisort.html#T4">SCPISORT:4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
