<?xml version="1.0"?>
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<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> ;<br/><b>let </b><font color="Maroon">I</font> be  <a href="scmpds_4.html#V3">shiftable</a> <a href="scmpds_5.html#V3">No-StopCode</a> <a href="scmpds_4.html#NM1">Program-block</a>;<br/><b>let </b><font color="Maroon">a</font> be    <a href="scmpds_2.html#M1">Int_position</a> ;<br/><b>let </b><font color="Maroon">i</font> be   <a href="int_1.html#NM1">Integer</a>, <font color="Maroon">c</font> be   <a href="int_1.html#NM1">Integer</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">g</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K4">INT</a> ;<br/><b>let </b><font color="Maroon">m</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> , <font color="Maroon">m1</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>











<b>set </b><font color="Maroon">b</font> =  <a href="scmpds_2.html#K3">DataLoc</a> <font color="Maroon">c</font>,<font color="Maroon">i</font>;<br/>
<b>assume </b><a NAME="E1:16"/><i><font color="Green">E42</font></i>: 
(  <a href="card_1.html#K4">card</a> <font color="Maroon">I</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> &amp;  <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">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> )
 ;<br/>

<b>assume </b><a NAME="E2:16"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">f</font> <a href="scpisort.html#R1">is_FinSequence_on</a> <font color="Maroon">s</font>,<font color="Maroon">m</font>
 ;<br/>

<b>assume </b><a NAME="E3:16"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">g</font> <a href="scpisort.html#R1">is_FinSequence_on</a>  <a href="scmpds_4.html#K9">IExec</a> <span class="p1">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span>,<font color="Maroon">s</font>,<font color="Maroon">m</font>
 ;<br/>

<b>assume </b><a NAME="E4:16"/><i><font color="Green">E60</font></i>: 
1 <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="scmpds_2.html#K3">DataLoc</a> <font color="Maroon">c</font>,<font color="Maroon">i</font></span>)</span>
 ;<br/>

<b>assume </b><a NAME="E5:16"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">m1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1 <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">m1</font></span>)</span> &amp; <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font> <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> <span class="p2">(<span class="default"><font color="Maroon">m1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> )
 ;<br/>

<b>assume </b><a NAME="E6:16"/><i><font color="Green">E62</font></i>: 
for <font color="Olive">t</font> being  <a href="ami_1.html#NM4">State</a> of <a href="scmpds_2.html#K1">SCMPDS</a> <br/> for <font color="Olive">f1</font>, <font color="Olive">f2</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K4">INT</a> <br/> for <font color="Olive">k1</font>, <font color="Olive">k2</font>, <font color="Olive">y1</font>, <font color="Olive">yn</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">t</font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> &amp; <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Olive">k1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Olive">t</font> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <font color="Maroon">c</font>,<font color="Maroon">i</font></span>)</span> &amp; <font color="Olive">k2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Olive">k1</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Olive">y1</font> <a href="hidden.html#R1">=</a> <font color="Olive">t</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">k2</font></span>)</span> &amp; <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Olive">yn</font> <a href="hidden.html#R1">=</a> <font color="Olive">t</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="Olive">k2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> &amp; ( ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">y1</font> &amp; <font color="Olive">yn</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> ) or <font color="Olive">y1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">yn</font> ) holds <br/>( <font color="Maroon">I</font> <a href="scmpds_6.html#R1">is_closed_on</a> <font color="Olive">t</font> &amp; <font color="Maroon">I</font> <a href="scmpds_6.html#R2">is_halting_on</a> <font color="Olive">t</font> &amp; <span class="p1">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <font color="Maroon">I</font>,<font color="Olive">t</font></span>)</span> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">t</font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">a</font> &amp; ( for <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">j</font> &amp; <font color="Olive">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Olive">k1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 holds <br/><span class="p1">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <font color="Maroon">I</font>,<font color="Olive">t</font></span>)</span> <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"><span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Olive">j</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">t</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"><span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Olive">j</font></span>)</span></span>)</span> ) &amp; ( <font color="Olive">y1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">yn</font> implies ( <span class="p1">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <font color="Maroon">I</font>,<font color="Olive">t</font></span>)</span> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <font color="Maroon">c</font>,<font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Olive">k1</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 &amp; ( for <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">j</font> &amp; <font color="Olive">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> holds <br/><span class="p1">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <font color="Maroon">I</font>,<font color="Olive">t</font></span>)</span> <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">m</font> <a href="nat_1.html#K1">+</a> <font color="Olive">j</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">t</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">m</font> <a href="nat_1.html#K1">+</a> <font color="Olive">j</font></span>)</span></span>)</span> ) ) ) &amp; ( <font color="Olive">y1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">yn</font> implies ( <span class="p1">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <font color="Maroon">I</font>,<font color="Olive">t</font></span>)</span> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <font color="Maroon">c</font>,<font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Olive">k1</font></span>)</span> <a href="nat_1.html#K1">+</a> 3 &amp; ( for <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st ( ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">j</font> &amp; <font color="Olive">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">y1</font> ) or ( <font color="Olive">yn</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">j</font> &amp; <font color="Olive">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> ) ) holds <br/><span class="p1">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <font color="Maroon">I</font>,<font color="Olive">t</font></span>)</span> <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">m</font> <a href="nat_1.html#K1">+</a> <font color="Olive">j</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">t</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">m</font> <a href="nat_1.html#K1">+</a> <font color="Olive">j</font></span>)</span></span>)</span> ) &amp;  ex <font color="Olive">ym</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">y1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">ym</font> &amp; <font color="Olive">ym</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">yn</font> &amp; <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Olive">y1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <font color="Maroon">I</font>,<font color="Olive">t</font></span>)</span> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1">intpos</a> <font color="Olive">k2</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Olive">ym</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <font color="Maroon">I</font>,<font color="Olive">t</font></span>)</span> <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="Olive">k2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Olive">ym</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <font color="Maroon">I</font>,<font color="Olive">t</font></span>)</span> <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="Olive">k2</font> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> &amp; <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Olive">yn</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <font color="Maroon">I</font>,<font color="Olive">t</font></span>)</span> <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="Olive">k2</font> <a href="nat_1.html#K1">+</a> 3</span>)</span></span>)</span> &amp; ( 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">y1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">j</font> &amp; <font color="Olive">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">ym</font> holds <br/><span class="p1">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <font color="Maroon">I</font>,<font color="Olive">t</font></span>)</span> <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">m</font> <a href="nat_1.html#K1">+</a> <font color="Olive">j</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <font color="Maroon">I</font>,<font color="Olive">t</font></span>)</span> <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">m</font> <a href="nat_1.html#K1">+</a> <font color="Olive">ym</font></span>)</span></span>)</span> ) &amp; ( 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">ym</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">j</font> &amp; <font color="Olive">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">yn</font> holds <br/><span class="p1">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <font color="Maroon">I</font>,<font color="Olive">t</font></span>)</span> <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">m</font> <a href="nat_1.html#K1">+</a> <font color="Olive">j</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <font color="Maroon">I</font>,<font color="Olive">t</font></span>)</span> <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">m</font> <a href="nat_1.html#K1">+</a> <font color="Olive">ym</font></span>)</span></span>)</span> ) ) ) ) &amp; ( <font color="Olive">f1</font> <a href="scpisort.html#R1">is_FinSequence_on</a> <font color="Olive">t</font>,<font color="Maroon">m</font> &amp; <font color="Olive">f2</font> <a href="scpisort.html#R1">is_FinSequence_on</a>  <a href="scmpds_4.html#K9">IExec</a> <font color="Maroon">I</font>,<font color="Olive">t</font>,<font color="Maroon">m</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Olive">f1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Olive">f2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> implies <font color="Olive">f1</font>,<font color="Olive">f2</font> <a href="rfinseq.html#R1">are_fiberwise_equipotent</a>  ) )
 ;<br/>

<b>set </b><font color="Maroon">WH</font> =  <a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font>;<br/>
<b>set </b><font color="Maroon">sWH</font> =  <a href="scmpds_4.html#K8">stop</a> <span class="p1">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">iWH</font> =  <a href="scmpds_4.html#K2">Initialized</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p2">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>for <font color="Olive">t</font> being  <a href="ami_1.html#NM4">State</a> of <a href="scmpds_2.html#K1">SCMPDS</a> <br/> for <font color="Olive">f1</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K4">INT</a> <br/> for <font color="Olive">k1</font>, <font color="Olive">k2</font>, <font color="Olive">y1</font>, <font color="Olive">yn</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">t</font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> &amp; <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Olive">k1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Olive">t</font> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <font color="Maroon">c</font>,<font color="Maroon">i</font></span>)</span> &amp; <font color="Olive">k2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Olive">k1</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 &amp; ( ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">y1</font> &amp; <font color="Olive">yn</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> ) or <font color="Olive">y1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">yn</font> ) &amp; <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Olive">y1</font> <a href="hidden.html#R1">=</a> <font color="Olive">t</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">k2</font></span>)</span> &amp; <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Olive">yn</font> <a href="hidden.html#R1">=</a> <font color="Olive">t</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="Olive">k2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> &amp; <font color="Olive">yn</font> <a href="xcmplx_0.html#K6">-</a> <font color="Olive">y1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Olive">f1</font> <a href="scpisort.html#R1">is_FinSequence_on</a> <font color="Olive">t</font>,<font color="Maroon">m</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Olive">f1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> holds <br/> ex <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  ex <font color="Olive">f2</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K4">INT</a>  st <br/>( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p3">(<span class="default"><font color="Olive">t</font> <a href="ami_5.html#K8">+*</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p0">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Olive">k</font></span>)</span> <a href="ami_5.html#K8">+*</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p3">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p2">(<span class="default"><font color="Olive">t</font> <a href="ami_5.html#K8">+*</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p5">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Olive">k</font> &amp; <font color="Olive">f2</font> <a href="scpisort.html#R1">is_FinSequence_on</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p2">(<span class="default"><font color="Olive">t</font> <a href="ami_5.html#K8">+*</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p5">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Olive">k</font>,<font color="Maroon">m</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Olive">f2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp; <font color="Olive">f1</font>,<font color="Olive">f2</font> <a href="rfinseq.html#R1">are_fiberwise_equipotent</a>  &amp; <font color="Olive">f2</font> <a href="sfmastr3.html#R1">is_non_decreasing_on</a> <font color="Olive">y1</font>,<font color="Olive">yn</font> &amp; ( 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">y1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">yn</font> &amp; ( ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">j</font> &amp; <font color="Olive">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">y1</font> ) or ( <font color="Olive">yn</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">j</font> &amp; <font color="Olive">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> ) ) holds <br/><font color="Olive">f2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font> <a href="hidden.html#R1">=</a> <font color="Olive">t</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">m</font> <a href="nat_1.html#K1">+</a> <font color="Olive">j</font></span>)</span></span>)</span> ) &amp; ( 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">y1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">yn</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">j</font> &amp; <font color="Olive">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> holds <br/><font color="Olive">f2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font> <a href="hidden.html#R1">=</a> <font color="Olive">t</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">m</font> <a href="nat_1.html#K1">+</a> <font color="Olive">j</font></span>)</span></span>)</span> ) &amp; ( for <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">j</font> &amp; <font color="Olive">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Olive">k1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p3">(<span class="default"><font color="Olive">t</font> <a href="ami_5.html#K8">+*</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p0">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Olive">k</font></span>)</span> <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"><span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Olive">j</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">t</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"><span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Olive">j</font></span>)</span></span>)</span> ) &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p3">(<span class="default"><font color="Olive">t</font> <a href="ami_5.html#K8">+*</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p0">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Olive">k</font></span>)</span> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <font color="Maroon">c</font>,<font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">t</font> <a href="scmpds_2.html#K2">.</a> <span class="p2">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <font color="Maroon">c</font>,<font color="Maroon">i</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p3">(<span class="default"><font color="Olive">t</font> <a href="ami_5.html#K8">+*</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p0">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Olive">k</font></span>)</span> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> );<br/>
<a NAME="E7:16"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">t</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">f1</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K4">INT</a> ;<br/><b>let </b><font color="Maroon">k1</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">k2</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">y1</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">yn</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>





<b>assume </b><a NAME="E1:16_1"/><i><font color="Green">E64</font></i>: 
( <font color="Maroon">t</font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> &amp; <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <font color="Maroon">c</font>,<font color="Maroon">i</font></span>)</span> &amp; <font color="Maroon">k2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k1</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 &amp; ( ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y1</font> &amp; <font color="Maroon">yn</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> ) or <font color="Maroon">y1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">yn</font> ) &amp; <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</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">k2</font></span>)</span> &amp; <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">yn</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</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">k2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon">yn</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">y1</font> <a href="xxreal_0.html#R1">&lt;=</a> 0 &amp; <font color="Maroon">f1</font> <a href="scpisort.html#R1">is_FinSequence_on</a> <font color="Maroon">t</font>,<font color="Maroon">m</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/>

<a NAME="E2:16_1"/><i><font color="Green">E65</font></i>: 
( <font color="Maroon">I</font> <a href="scmpds_6.html#R1">is_closed_on</a> <font color="Maroon">t</font> &amp; <font color="Maroon">I</font> <a href="scmpds_6.html#R2">is_halting_on</a> <font color="Maroon">t</font> &amp; <span class="p1">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <font color="Maroon">I</font>,<font color="Maroon">t</font></span>)</span> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">a</font> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E3:16_1"/><i><font color="Green">E66</font></i>: 
( <font color="Maroon">y1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">yn</font> implies ( <span class="p1">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <font color="Maroon">I</font>,<font color="Maroon">t</font></span>)</span> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <font color="Maroon">c</font>,<font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k1</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 &amp; ( for <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">j</font> &amp; <font color="Olive">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> holds <br/><span class="p1">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <font color="Maroon">I</font>,<font color="Maroon">t</font></span>)</span> <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">m</font> <a href="nat_1.html#K1">+</a> <font color="Olive">j</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">t</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">m</font> <a href="nat_1.html#K1">+</a> <font color="Olive">j</font></span>)</span></span>)</span> ) ) )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:16_1"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">y1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">yn</font>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T52">XREAL_1:52</a></i>;<br/>
<a NAME="E5:16_1"/><i><font color="Green">E68</font></i>: 
<span class="p1">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <font color="Maroon">I</font>,<font color="Maroon">t</font></span>)</span> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <font color="Maroon">c</font>,<font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="scmpds_2.html#K2">.</a> <span class="p2">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <font color="Maroon">c</font>,<font color="Maroon">i</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i>, , <a class="ref" href="xreal_1.html#T52">XREAL_1:52</a></i>;<br/>
<b>take </b><font color="Maroon">k</font> = <span class="p1">(<span class="default"><a href="scm_1.html#NK3" target="_self">LifeSpan</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2;<br/>

<b>set </b><font color="Maroon">tk</font> = <span class="p1">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p5">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">k</font>;<br/>
<a NAME="E6:16_1"/><i><font color="Green">E72</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p3">(<span class="default"><font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p0">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">k</font></span>)</span> <a href="relat_1.html#K7">|</a> <a href="ami_2.html#K2">SCM-Data-Loc</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <font color="Maroon">I</font>,<font color="Maroon">t</font></span>)</span> <a href="relat_1.html#K7">|</a> <a href="ami_2.html#K2">SCM-Data-Loc</a> 
 
<b>by </b><i>, , , </i>;<br/>
<b>consider </b><font color="Maroon">f2</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K4">INT</a> <b> such that </b><br/><a NAME="E8:16_1"/><i><font color="Green">E73</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp; ( for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f2</font> holds <br/><font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <font color="Maroon">I</font>,<font color="Maroon">t</font></span>)</span> <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">m</font> <a href="nat_1.html#K1">+</a> <font color="Olive">i</font></span>)</span></span>)</span> ) )
 <b>by </b><i><a class="ref" href="scpisort.html#T2">SCPISORT:2</a></i>;<br/>
<a NAME="E9:16_1"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">f2</font> <a href="scpisort.html#R1">is_FinSequence_on</a>  <a href="scmpds_4.html#K9">IExec</a> <font color="Maroon">I</font>,<font color="Maroon">t</font>,<font color="Maroon">m</font>
 
<b>by </b><i>, <a class="ref" href="scpisort.html#D1">SCPISORT:def 1</a></i>;<br/>
<b>take </b>
<font color="Maroon">f2</font>
;<br/>

<b>thus </b><a NAME="E10:16_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p3">(<span class="default"><font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p0">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">k</font></span>)</span> <a href="ami_5.html#K8">+*</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p3">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p5">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">k</font>
 <b>by </b><i>, , , </i>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_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="E1:16_1_1"/>
( 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>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f2</font> )
 ;<br/><b>hence </b><font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> = 
<span class="p1">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <font color="Maroon">I</font>,<font color="Maroon">t</font></span>)</span> <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">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p3">(<span class="default"><font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p0">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">k</font></span>)</span> <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">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="scmpds_4.html#T23">SCMPDS_4:23</a></i>
;<br/><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E12:16_1"/>
<font color="Maroon">f2</font> <a href="scpisort.html#R1">is_FinSequence_on</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p5">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">k</font>,<font color="Maroon">m</font>
 <b>by </b><i><a class="ref" href="scpisort.html#D1">SCPISORT:def 1</a></i>;<br/>

<b>thus </b><a NAME="E13:16_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">f2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 <b>by </b><i/>;<br/>

<b>thus </b><a NAME="E14:16_1"/>
<font color="Maroon">f1</font>,<font color="Maroon">f2</font> <a href="rfinseq.html#R1">are_fiberwise_equipotent</a> 
 <b>by </b><i>, , , </i>;<br/>

<b>thus </b><a NAME="E15:16_1"/>
<font color="Maroon">f2</font> <a href="sfmastr3.html#R1">is_non_decreasing_on</a> <font color="Maroon">y1</font>,<font color="Maroon">yn</font>
 <b>by </b><i>, <a class="ref" href="scpisort.html#T1">SCPISORT:1</a></i>;<br/>

<b>thus </b><a NAME="E16:16_1"/>
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="Maroon">y1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">yn</font> &amp; ( ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">j</font> &amp; <font color="Olive">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">y1</font> ) or ( <font color="Maroon">yn</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">j</font> &amp; <font color="Olive">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> ) ) holds <br/><font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</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">m</font> <a href="nat_1.html#K1">+</a> <font color="Olive">j</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T52">XREAL_1:52</a></i>;<br/>

<div><b>hereby </b>
<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="E1:16_1_2"/><i><font color="Green">E75</font></i>: 
( <font color="Maroon">y1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">yn</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> )
 ;<br/><b>hence </b><font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> = 
<span class="p1">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <font color="Maroon">I</font>,<font color="Maroon">t</font></span>)</span> <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">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">t</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">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font></span>)</span></span>)</span>
<b>by </b><i>, , <a class="txt" href="scpqsort.html#E1"><i><font color="Green">Lemma31</font></i></a></i>
;<br/><br/></div>
<b>end;</b></div>

<div><b>hereby </b>
<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="E1:16_1_3"/><i><font color="Green">E76</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p3">(<span class="default"><font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p0">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">k</font></span>)</span> <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"><span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <font color="Maroon">I</font>,<font color="Maroon">t</font></span>)</span> <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"><span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="scmpds_4.html#T23">SCMPDS_4:23</a></i>
<br/>.= 
<font color="Maroon">t</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"><span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font></span>)</span></span>)</span>
<b>by </b><i>, , <a class="txt" href="scpqsort.html#E1"><i><font color="Green">Lemma31</font></i></a></i>
;<br/></div>
<b>end;</b></div>

<b>thus </b><a NAME="E19:16_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p3">(<span class="default"><font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p0">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">k</font></span>)</span> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <font color="Maroon">c</font>,<font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="scmpds_2.html#K2">.</a> <span class="p2">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <font color="Maroon">c</font>,<font color="Maroon">i</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2
 <b>by </b><i>, , <a class="ref" href="scmpds_4.html#T23">SCMPDS_4:23</a></i>;<br/>

<b>thus </b><a NAME="E20:16_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p3">(<span class="default"><font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p0">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">k</font></span>)</span> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font>
 <b>by </b><i>, , , <a class="ref" href="scmpds_4.html#T23">SCMPDS_4:23</a></i>;<br/>


</div><b>end;</b></div>
<div><i><font color="Green">E77</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2"><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="E1:16_2"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>]
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">t</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">f1</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K4">INT</a> ;<br/><b>let </b><font color="Maroon">k1</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">k2</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">y1</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">ym</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">yn</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:16_2_1"/><i><font color="Green">E81</font></i>: 
( <font color="Maroon">t</font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> &amp; <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <font color="Maroon">c</font>,<font color="Maroon">i</font></span>)</span> &amp; <font color="Maroon">k2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k1</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 &amp; ( ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y1</font> &amp; <font color="Maroon">yn</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> ) or <font color="Maroon">y1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">yn</font> ) &amp; <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</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">k2</font></span>)</span> &amp; <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">yn</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</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">k2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon">yn</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">y1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">f1</font> <a href="scpisort.html#R1">is_FinSequence_on</a> <font color="Maroon">t</font>,<font color="Maroon">m</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/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:16_2_1_1"/>
( <font color="Maroon">yn</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">y1</font> <a href="xxreal_0.html#R1">&lt;=</a> 0 or <font color="Maroon">yn</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">y1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_2_1_1_1"><b>suppose </b></a><a NAME="E1:16_2_1_1_1"/>
<font color="Maroon">yn</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">y1</font> <a href="xxreal_0.html#R1">&lt;=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:16_2_1_1_1"/>
 ex <font color="Olive">kk</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  ex <font color="Olive">f2</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K4">INT</a>  st <br/>( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p3">(<span class="default"><font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p0">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Olive">kk</font></span>)</span> <a href="ami_5.html#K8">+*</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p3">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p5">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Olive">kk</font> &amp; <font color="Olive">f2</font> <a href="scpisort.html#R1">is_FinSequence_on</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p5">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Olive">kk</font>,<font color="Maroon">m</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Olive">f2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">f1</font>,<font color="Olive">f2</font> <a href="rfinseq.html#R1">are_fiberwise_equipotent</a>  &amp; <font color="Olive">f2</font> <a href="sfmastr3.html#R1">is_non_decreasing_on</a> <font color="Maroon">y1</font>,<font color="Maroon">yn</font> &amp; ( 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="Maroon">y1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">yn</font> &amp; ( ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">j</font> &amp; <font color="Olive">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">y1</font> ) or ( <font color="Maroon">yn</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">j</font> &amp; <font color="Olive">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> ) ) holds <br/><font color="Olive">f2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</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">m</font> <a href="nat_1.html#K1">+</a> <font color="Olive">j</font></span>)</span></span>)</span> ) &amp; ( 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="Maroon">y1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">yn</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">j</font> &amp; <font color="Olive">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> holds <br/><font color="Olive">f2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</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">m</font> <a href="nat_1.html#K1">+</a> <font color="Olive">j</font></span>)</span></span>)</span> ) &amp; ( for <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">j</font> &amp; <font color="Olive">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p3">(<span class="default"><font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p0">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Olive">kk</font></span>)</span> <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"><span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Olive">j</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">t</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"><span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Olive">j</font></span>)</span></span>)</span> ) &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p3">(<span class="default"><font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p0">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Olive">kk</font></span>)</span> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <font color="Maroon">c</font>,<font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="scmpds_2.html#K2">.</a> <span class="p2">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <font color="Maroon">c</font>,<font color="Maroon">i</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p3">(<span class="default"><font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p0">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Olive">kk</font></span>)</span> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> )
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_2_1_1_2"><b>suppose </b></a><a NAME="E1:16_2_1_1_2"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">yn</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">y1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:16_2_1_1_2"/><b>then </b><i><font color="Green">E84</font></i>: 
<font color="Maroon">yn</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">y1</font>
 <b>by </b><i><a class="ref" href="real_2.html#T106">REAL_2:106</a></i>;<br/><a NAME="E3:16_2_1_1_2"/><i><font color="Green">E85</font></i>: 
( <font color="Maroon">I</font> <a href="scmpds_6.html#R1">is_closed_on</a> <font color="Maroon">t</font> &amp; <font color="Maroon">I</font> <a href="scmpds_6.html#R2">is_halting_on</a> <font color="Maroon">t</font> &amp; <span class="p1">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <font color="Maroon">I</font>,<font color="Maroon">t</font></span>)</span> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">a</font> )
 <b>by </b><i>, </i>;<br/><b>set </b><font color="Maroon">It</font> =  <a href="scmpds_4.html#K9">IExec</a> <font color="Maroon">I</font>,<font color="Maroon">t</font>;<br/><a NAME="E4:16_2_1_1_2"/><i><font color="Green">E102</font></i>: 
( <font color="Maroon">y1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">yn</font> implies ( <span class="p1">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <font color="Maroon">I</font>,<font color="Maroon">t</font></span>)</span> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <font color="Maroon">c</font>,<font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k1</font></span>)</span> <a href="nat_1.html#K1">+</a> 3 &amp; ( for <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st ( ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">j</font> &amp; <font color="Olive">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">y1</font> ) or ( <font color="Maroon">yn</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">j</font> &amp; <font color="Olive">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> ) ) holds <br/><span class="p1">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <font color="Maroon">I</font>,<font color="Maroon">t</font></span>)</span> <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">m</font> <a href="nat_1.html#K1">+</a> <font color="Olive">j</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">t</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">m</font> <a href="nat_1.html#K1">+</a> <font color="Olive">j</font></span>)</span></span>)</span> ) &amp;  ex <font color="Olive">ym</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Maroon">y1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">ym</font> &amp; <font color="Olive">ym</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">yn</font> &amp; <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <font color="Maroon">I</font>,<font color="Maroon">t</font></span>)</span> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1">intpos</a> <font color="Maroon">k2</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Olive">ym</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <font color="Maroon">I</font>,<font color="Maroon">t</font></span>)</span> <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">k2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Olive">ym</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <font color="Maroon">I</font>,<font color="Maroon">t</font></span>)</span> <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">k2</font> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> &amp; <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">yn</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <font color="Maroon">I</font>,<font color="Maroon">t</font></span>)</span> <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">k2</font> <a href="nat_1.html#K1">+</a> 3</span>)</span></span>)</span> &amp; ( 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="Maroon">y1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">j</font> &amp; <font color="Olive">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">ym</font> holds <br/><span class="p1">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <font color="Maroon">I</font>,<font color="Maroon">t</font></span>)</span> <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">m</font> <a href="nat_1.html#K1">+</a> <font color="Olive">j</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <font color="Maroon">I</font>,<font color="Maroon">t</font></span>)</span> <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">m</font> <a href="nat_1.html#K1">+</a> <font color="Olive">ym</font></span>)</span></span>)</span> ) &amp; ( 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">ym</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">j</font> &amp; <font color="Olive">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">yn</font> holds <br/><span class="p1">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <font color="Maroon">I</font>,<font color="Maroon">t</font></span>)</span> <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">m</font> <a href="nat_1.html#K1">+</a> <font color="Olive">j</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <font color="Maroon">I</font>,<font color="Maroon">t</font></span>)</span> <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">m</font> <a href="nat_1.html#K1">+</a> <font color="Olive">ym</font></span>)</span></span>)</span> ) ) ) )
 <b>by </b><i>, </i>;<br/><b>set </b><font color="Maroon">m1</font> = <span class="p1">(<span class="default"><a href="scm_1.html#NK3" target="_self">LifeSpan</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2;<br/><b>set </b><font color="Maroon">t1</font> = <span class="p1">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p5">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scm_1.html#NK3" target="_self">LifeSpan</a> <span class="p3">(<span class="default"><font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span>;<br/><a NAME="E5:16_2_1_1_2"/><i><font color="Green">E135</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p3">(<span class="default"><font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p0">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scm_1.html#NK3" target="_self">LifeSpan</a> <span class="p4">(<span class="default"><font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p0">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> <a href="relat_1.html#K7">|</a> <a href="ami_2.html#K2">SCM-Data-Loc</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <font color="Maroon">I</font>,<font color="Maroon">t</font></span>)</span> <a href="relat_1.html#K7">|</a> <a href="ami_2.html#K2">SCM-Data-Loc</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p3">(<span class="default"><font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p0">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scm_1.html#NK3" target="_self">LifeSpan</a> <span class="p4">(<span class="default"><font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p0">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> <a href="ami_5.html#K8">+*</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p3">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p5">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scm_1.html#NK3" target="_self">LifeSpan</a> <span class="p3">(<span class="default"><font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> )
 <b>by </b><i>, , , </i>;<br/><b>consider </b><font color="Maroon">ym</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E7:16_2_1_1_2"/><i><font color="Green">E136</font></i>: 
( <font color="Maroon">y1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">ym</font> &amp; <font color="Maroon">ym</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">yn</font> &amp; <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <font color="Maroon">I</font>,<font color="Maroon">t</font></span>)</span> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1">intpos</a> <font color="Maroon">k2</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">ym</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <font color="Maroon">I</font>,<font color="Maroon">t</font></span>)</span> <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">k2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">ym</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <font color="Maroon">I</font>,<font color="Maroon">t</font></span>)</span> <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">k2</font> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> &amp; <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">yn</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <font color="Maroon">I</font>,<font color="Maroon">t</font></span>)</span> <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">k2</font> <a href="nat_1.html#K1">+</a> 3</span>)</span></span>)</span> &amp; ( 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="Maroon">y1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">j</font> &amp; <font color="Olive">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">ym</font> holds <br/><span class="p1">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <font color="Maroon">I</font>,<font color="Maroon">t</font></span>)</span> <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">m</font> <a href="nat_1.html#K1">+</a> <font color="Olive">j</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <font color="Maroon">I</font>,<font color="Maroon">t</font></span>)</span> <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">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">ym</font></span>)</span></span>)</span> ) &amp; ( 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="Maroon">ym</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">j</font> &amp; <font color="Olive">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">yn</font> holds <br/><span class="p1">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <font color="Maroon">I</font>,<font color="Maroon">t</font></span>)</span> <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">m</font> <a href="nat_1.html#K1">+</a> <font color="Olive">j</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <font color="Maroon">I</font>,<font color="Maroon">t</font></span>)</span> <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">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">ym</font></span>)</span></span>)</span> ) )
 <b>by </b><i>, , <a class="ref" href="real_2.html#T106">REAL_2:106</a></i>;<br/><b>set </b><font color="Maroon">k3</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><font color="Maroon">k1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1;<br/><b>set </b><font color="Maroon">yd</font> = <font color="Maroon">ym</font> <a href="nat_1.html#K1">+</a> 1;<br/><a NAME="E8:16_2_1_1_2"/><i><font color="Green">E137</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p3">(<span class="default"><font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p0">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scm_1.html#NK3" target="_self">LifeSpan</a> <span class="p4">(<span class="default"><font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p0">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font>
 <b>by </b><i>, , , <a class="ref" href="scmpds_4.html#T23">SCMPDS_4:23</a></i>;<br/><a NAME="E9:16_2_1_1_2"/><i><font color="Green">E138</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p3">(<span class="default"><font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p0">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scm_1.html#NK3" target="_self">LifeSpan</a> <span class="p4">(<span class="default"><font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p0">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <font color="Maroon">c</font>,<font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k1</font></span>)</span> <a href="nat_1.html#K1">+</a> 3
 <b>by </b><i>, , , <a class="ref" href="real_2.html#T106">REAL_2:106</a>, <a class="ref" href="scmpds_4.html#T23">SCMPDS_4:23</a></i>;<br/><a NAME="E10:16_2_1_1_2"/><i><font color="Green">E139</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p3">(<span class="default"><font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p0">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scm_1.html#NK3" target="_self">LifeSpan</a> <span class="p4">(<span class="default"><font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p0">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <font color="Maroon">c</font>,<font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">k1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, , , <a class="ref" href="real_2.html#T106">REAL_2:106</a>, <a class="ref" href="scmpds_4.html#T23">SCMPDS_4:23</a></i>;<br/><a NAME="E11:16_2_1_1_2"/><i><font color="Green">E140</font></i>: 
<font color="Maroon">ym</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">ym</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T31">XREAL_1:31</a></i>;<br/><a NAME="E12:16_2_1_1_2"/><b>then </b><i><font color="Green">E141</font></i>: 
<font color="Maroon">ym</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">y1</font>
 <b>by </b><i><a class="txt" href="scpqsort.html#E1"><i><font color="Green">Lemma31</font></i></a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E13:16_2_1_1_2"/><b>then </b><i><font color="Green">E142</font></i>: 
<font color="Maroon">ym</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>by </b><i>, , <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a>, <a class="ref" href="real_2.html#T106">REAL_2:106</a></i>;<br/><a NAME="E14:16_2_1_1_2"/><i><font color="Green">E143</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p3">(<span class="default"><font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p0">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scm_1.html#NK3" target="_self">LifeSpan</a> <span class="p4">(<span class="default"><font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p0">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> <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"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p5">(<span class="default"><font color="Maroon">k1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">ym</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, , <a class="ref" href="scmpds_4.html#T23">SCMPDS_4:23</a>, <a class="txt" href="scpqsort.html#E1"><i><font color="Green">Lemma31</font></i></a></i>;<br/><a NAME="E15:16_2_1_1_2"/><i><font color="Green">E144</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p3">(<span class="default"><font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p0">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scm_1.html#NK3" target="_self">LifeSpan</a> <span class="p4">(<span class="default"><font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p0">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> <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"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p5">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p0">(<span class="default"><font color="Maroon">k1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">yn</font>
 <b>by </b><i>, , <a class="txt" href="scpqsort.html#E1"><i><font color="Green">Lemma31</font></i></a>, <a class="ref" href="scmpds_4.html#T23">SCMPDS_4:23</a></i>;<br/><a NAME="E16:16_2_1_1_2"/><i><font color="Green">E145</font></i>: 
<font color="Maroon">ym</font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">y1</font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span>
 <b>by </b><i><a class="txt" href="scpqsort.html#E1"><i><font color="Green">Lemma31</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E17:16_2_1_1_2"/>
<font color="Maroon">yn</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y1</font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T22">XREAL_1:22</a></i>;<br/><a NAME="E18:16_2_1_1_2"/><b>then </b>
<font color="Maroon">yn</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">ym</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="scpqsort.html#T1" target="_self">Th1</a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E19:16_2_1_1_2"/><b>then </b><i><font color="Green">E146</font></i>: 
<font color="Maroon">yn</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">ym</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T22">XREAL_1:22</a></i>;<br/><b>consider </b><font color="Maroon">f2</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K4">INT</a> <b> such that </b><br/><a NAME="E21:16_2_1_1_2"/><i><font color="Green">E147</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp; ( for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f2</font> holds <br/><font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p3">(<span class="default"><font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p0">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scm_1.html#NK3" target="_self">LifeSpan</a> <span class="p4">(<span class="default"><font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p0">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> <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">m</font> <a href="nat_1.html#K1">+</a> <font color="Olive">i</font></span>)</span></span>)</span> ) )
 <b>by </b><i><a class="ref" href="scpisort.html#T2">SCPISORT:2</a></i>;<br/><a NAME="E22:16_2_1_1_2"/>
<font color="Maroon">f2</font> <a href="scpisort.html#R1">is_FinSequence_on</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p5">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scm_1.html#NK3" target="_self">LifeSpan</a> <span class="p3">(<span class="default"><font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span>,<font color="Maroon">m</font>
 <b>by </b><i><a class="txt" href="scpqsort.html#E1:16"><i><font color="Green">E42</font></i></a>, <a class="ref" href="scpisort.html#D1">SCPISORT:def 1</a></i>;<br/><b>then consider </b><font color="Maroon">kl</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">f3</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K4">INT</a> <b> such that </b><br/><a NAME="E24:16_2_1_1_2"/><i><font color="Green">E162</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p0">(<span class="default"><font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p3">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="scm_1.html#NK3" target="_self">LifeSpan</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> <a href="ami_5.html#K8">+*</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p0">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">kl</font></span>)</span> <a href="ami_5.html#K8">+*</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p3">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p5">(<span class="default"><font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <span class="p0">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p2">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="scm_1.html#NK3" target="_self">LifeSpan</a> <span class="p0">(<span class="default"><font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> <a href="ami_5.html#K8">+*</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p5">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">kl</font> &amp; <font color="Maroon">f3</font> <a href="scpisort.html#R1">is_FinSequence_on</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p5">(<span class="default"><font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <span class="p0">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p2">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="scm_1.html#NK3" target="_self">LifeSpan</a> <span class="p0">(<span class="default"><font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> <a href="ami_5.html#K8">+*</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p5">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">kl</font>,<font color="Maroon">m</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">f2</font>,<font color="Maroon">f3</font> <a href="rfinseq.html#R1">are_fiberwise_equipotent</a>  &amp; <font color="Maroon">f3</font> <a href="sfmastr3.html#R1">is_non_decreasing_on</a> <font color="Maroon">ym</font> <a href="nat_1.html#K1">+</a> 1,<font color="Maroon">yn</font> &amp; ( 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="Maroon">ym</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">yn</font> &amp; ( ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">j</font> &amp; <font color="Olive">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">ym</font> <a href="nat_1.html#K1">+</a> 1 ) or ( <font color="Maroon">yn</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">j</font> &amp; <font color="Olive">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> ) ) holds <br/><font color="Maroon">f3</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p3">(<span class="default"><font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p0">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scm_1.html#NK3" target="_self">LifeSpan</a> <span class="p4">(<span class="default"><font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p0">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> <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">m</font> <a href="nat_1.html#K1">+</a> <font color="Olive">j</font></span>)</span></span>)</span> ) &amp; ( 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="Maroon">ym</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">yn</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">j</font> &amp; <font color="Olive">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> holds <br/><font color="Maroon">f3</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p3">(<span class="default"><font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p0">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scm_1.html#NK3" target="_self">LifeSpan</a> <span class="p4">(<span class="default"><font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p0">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> <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">m</font> <a href="nat_1.html#K1">+</a> <font color="Olive">j</font></span>)</span></span>)</span> ) &amp; ( for <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">j</font> &amp; <font color="Olive">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">k1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p0">(<span class="default"><font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p3">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="scm_1.html#NK3" target="_self">LifeSpan</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> <a href="ami_5.html#K8">+*</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p0">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">kl</font></span>)</span> <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"><span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Olive">j</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p3">(<span class="default"><font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p0">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scm_1.html#NK3" target="_self">LifeSpan</a> <span class="p4">(<span class="default"><font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p0">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> <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"><span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Olive">j</font></span>)</span></span>)</span> ) &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p0">(<span class="default"><font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p3">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="scm_1.html#NK3" target="_self">LifeSpan</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> <a href="ami_5.html#K8">+*</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p0">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">kl</font></span>)</span> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <font color="Maroon">c</font>,<font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p4">(<span class="default"><font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p0">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p1">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="scm_1.html#NK3" target="_self">LifeSpan</a> <span class="p5">(<span class="default"><font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <span class="p0">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> <a href="scmpds_2.html#K2">.</a> <span class="p2">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <font color="Maroon">c</font>,<font color="Maroon">i</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p0">(<span class="default"><font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p3">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="scm_1.html#NK3" target="_self">LifeSpan</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> <a href="ami_5.html#K8">+*</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p0">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">kl</font></span>)</span> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> )
 <b>by </b><i>, , , , , , , , , <a class="txt" href="scpqsort.html#E1:16"><i><font color="Green">E42</font></i></a>, <a class="ref" href="real_2.html#T106">REAL_2:106</a></i>;<br/><b>set </b><font color="Maroon">t2</font> = <span class="p1">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p5">(<span class="default"><font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <span class="p0">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p2">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="scm_1.html#NK3" target="_self">LifeSpan</a> <span class="p0">(<span class="default"><font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> <a href="ami_5.html#K8">+*</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p5">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">kl</font>;<br/><a NAME="E25:16_2_1_1_2"/><i><font color="Green">E163</font></i>: 
<font color="Maroon">ym</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>by </b><i>, , <a class="txt" href="scpqsort.html#E1"><i><font color="Green">Lemma31</font></i></a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a>, <a class="ref" href="real_2.html#T106">REAL_2:106</a></i>;<br/><a NAME="E26:16_2_1_1_2"/><b>then </b>
<font color="Maroon">ym</font> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/><b>then reconsider </b><font color="Maroon">yc</font> = <font color="Maroon">ym</font> <a href="xcmplx_0.html#K6">-</a> 1 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="E28:16_2_1_1_2"/><i><font color="Green">E165</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p0">(<span class="default"><font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p3">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="scm_1.html#NK3" target="_self">LifeSpan</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> <a href="ami_5.html#K8">+*</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p0">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">kl</font></span>)</span> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <font color="Maroon">c</font>,<font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, </i>;<br/><a NAME="E29:16_2_1_1_2"/><i><font color="Green">E166</font></i>: 
<font color="Maroon">yc</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">ym</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T148">XREAL_1:148</a></i>;<br/><a NAME="E30:16_2_1_1_2"/><b>then </b><i><font color="Green">E167</font></i>: 
<font color="Maroon">yc</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">yn</font>
 <b>by </b><i><a class="txt" href="scpqsort.html#E1"><i><font color="Green">Lemma31</font></i></a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E31:16_2_1_1_2"/><b>then </b><i><font color="Green">E168</font></i>: 
<font color="Maroon">yc</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 <b>by </b><i>, , <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a>, <a class="ref" href="real_2.html#T106">REAL_2:106</a></i>;<br/><b>set </b><font color="Maroon">jj</font> = <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1;<br/><a NAME="E32:16_2_1_1_2"/><i><font color="Green">E170</font></i>: 
<font color="Maroon">k2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i/>;<br/><a NAME="E33:16_2_1_1_2"/><i><font color="Green">E171</font></i>: 
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E34:16_2_1_1_2"/><i><font color="Green">E172</font></i>: 
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k1</font></span>)</span> <a href="nat_1.html#K1">+</a> 3 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">k1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
 ;<br/><a NAME="E35:16_2_1_1_2"/><b>then </b>
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">k1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><b>then </b><i><font color="Green">E173</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p0">(<span class="default"><font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p3">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="scm_1.html#NK3" target="_self">LifeSpan</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> <a href="ami_5.html#K8">+*</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p0">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">kl</font></span>)</span> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1">intpos</a> <font color="Maroon">k2</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p3">(<span class="default"><font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p0">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scm_1.html#NK3" target="_self">LifeSpan</a> <span class="p4">(<span class="default"><font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p0">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> <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"><span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="scpqsort.html#T2" target="_self">Th2</a>, </i>
<br/>.= 
<font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">y1</font>
<b>by </b><i>, <a class="txt" href="scpqsort.html#E1"><i><font color="Green">Lemma31</font></i></a>, <a class="ref" href="scpqsort.html#T2" target="_self">Th2</a>, <a class="ref" href="scmpds_4.html#T23">SCMPDS_4:23</a></i>
;<br/><b>set </b><font color="Maroon">jj</font> = <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k1</font></span>)</span> <a href="nat_1.html#K1">+</a> 2;<br/><a NAME="E37:16_2_1_1_2"/>
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k1</font></span>)</span> <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E38:16_2_1_1_2"/><b>then </b><i><font color="Green">E174</font></i>: 
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k1</font></span>)</span> <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E39:16_2_1_1_2"/>
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k1</font></span>)</span> <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">k1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><b>then </b><i><font color="Green">E175</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p0">(<span class="default"><font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p3">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="scm_1.html#NK3" target="_self">LifeSpan</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> <a href="ami_5.html#K8">+*</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p0">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">kl</font></span>)</span> <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">k2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p3">(<span class="default"><font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p0">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scm_1.html#NK3" target="_self">LifeSpan</a> <span class="p4">(<span class="default"><font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p0">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> <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"><span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k1</font></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span></span>)</span>
<b>by </b><i>, , </i>
<br/>.= 
<font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">yc</font>
<b>by </b><i>, , <a class="txt" href="scpqsort.html#E1"><i><font color="Green">Lemma31</font></i></a>, <a class="ref" href="scmpds_4.html#T23">SCMPDS_4:23</a></i>
;<br/><a NAME="E41:16_2_1_1_2"/>
<font color="Maroon">yc</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">yn</font> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i><a class="txt" href="scpqsort.html#E1"><i><font color="Green">Lemma31</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E42:16_2_1_1_2"/><b>then </b><i><font color="Green">E176</font></i>: 
<font color="Maroon">yc</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">y1</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">yn</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">y1</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E43:16_2_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">yn</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">y1</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E44:16_2_1_1_2"/><b>then </b>
<font color="Maroon">yc</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">y1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><b>then consider </b><font color="Maroon">km</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">f4</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K4">INT</a> <b> such that </b><br/><a NAME="E46:16_2_1_1_2"/><i><font color="Green">E179</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p3">(<span class="default"><font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p0">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scm_1.html#NK3" target="_self">LifeSpan</a> <span class="p4">(<span class="default"><font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p0">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> <a href="ami_5.html#K8">+*</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p3">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">kl</font></span>)</span> <a href="ami_5.html#K8">+*</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p0">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">km</font></span>)</span> <a href="ami_5.html#K8">+*</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p3">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p5">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scm_1.html#NK3" target="_self">LifeSpan</a> <span class="p3">(<span class="default"><font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> <a href="ami_5.html#K8">+*</a> <span class="p0">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p2">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">kl</font></span>)</span> <a href="ami_5.html#K8">+*</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p5">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">km</font> &amp; <font color="Maroon">f4</font> <a href="scpisort.html#R1">is_FinSequence_on</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p5">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scm_1.html#NK3" target="_self">LifeSpan</a> <span class="p3">(<span class="default"><font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> <a href="ami_5.html#K8">+*</a> <span class="p0">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p2">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">kl</font></span>)</span> <a href="ami_5.html#K8">+*</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p5">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">km</font>,<font color="Maroon">m</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f4</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">f3</font>,<font color="Maroon">f4</font> <a href="rfinseq.html#R1">are_fiberwise_equipotent</a>  &amp; <font color="Maroon">f4</font> <a href="sfmastr3.html#R1">is_non_decreasing_on</a> <font color="Maroon">y1</font>,<font color="Maroon">yc</font> &amp; ( 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="Maroon">y1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">yc</font> &amp; ( ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">j</font> &amp; <font color="Olive">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">y1</font> ) or ( <font color="Maroon">yc</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">j</font> &amp; <font color="Olive">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> ) ) holds <br/><font color="Maroon">f4</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p0">(<span class="default"><font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p3">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="scm_1.html#NK3" target="_self">LifeSpan</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> <a href="ami_5.html#K8">+*</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p0">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">kl</font></span>)</span> <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">m</font> <a href="nat_1.html#K1">+</a> <font color="Olive">j</font></span>)</span></span>)</span> ) &amp; ( 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="Maroon">y1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">yc</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">j</font> &amp; <font color="Olive">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> holds <br/><font color="Maroon">f4</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p0">(<span class="default"><font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p3">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="scm_1.html#NK3" target="_self">LifeSpan</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> <a href="ami_5.html#K8">+*</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p0">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">kl</font></span>)</span> <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">m</font> <a href="nat_1.html#K1">+</a> <font color="Olive">j</font></span>)</span></span>)</span> ) &amp; ( for <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">j</font> &amp; <font color="Olive">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p3">(<span class="default"><font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p0">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scm_1.html#NK3" target="_self">LifeSpan</a> <span class="p4">(<span class="default"><font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p0">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> <a href="ami_5.html#K8">+*</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p3">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">kl</font></span>)</span> <a href="ami_5.html#K8">+*</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p0">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">km</font></span>)</span> <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"><span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Olive">j</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p0">(<span class="default"><font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p3">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="scm_1.html#NK3" target="_self">LifeSpan</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> <a href="ami_5.html#K8">+*</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p0">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">kl</font></span>)</span> <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"><span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Olive">j</font></span>)</span></span>)</span> ) &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p3">(<span class="default"><font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p0">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scm_1.html#NK3" target="_self">LifeSpan</a> <span class="p4">(<span class="default"><font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p0">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> <a href="ami_5.html#K8">+*</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p3">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">kl</font></span>)</span> <a href="ami_5.html#K8">+*</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p0">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">km</font></span>)</span> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <font color="Maroon">c</font>,<font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p4">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="scm_1.html#NK3" target="_self">LifeSpan</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> <a href="ami_5.html#K8">+*</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p0">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p1">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">kl</font></span>)</span> <a href="scmpds_2.html#K2">.</a> <span class="p2">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <font color="Maroon">c</font>,<font color="Maroon">i</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p3">(<span class="default"><font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p0">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scm_1.html#NK3" target="_self">LifeSpan</a> <span class="p4">(<span class="default"><font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p0">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> <a href="ami_5.html#K8">+*</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p3">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">kl</font></span>)</span> <a href="ami_5.html#K8">+*</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p0">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">km</font></span>)</span> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</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="real_2.html#T106">REAL_2:106</a></i>;<br/><b>take </b><font color="Maroon">mm</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scm_1.html#NK3" target="_self">LifeSpan</a> <span class="p3">(<span class="default"><font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">kl</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">km</font></span>)</span>;<br/><b>set </b><font color="Maroon">tm</font> = <span class="p1">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p5">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">mm</font>;<br/><b>take </b><font color="Maroon">f4</font> = <font color="Maroon">f4</font>;<br/><i><font color="Green">E182</font></i>: <span class="p1">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p5">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">mm</font> = 
<span class="p1">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p5">(<span class="default"><font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <span class="p0">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p2">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="scm_1.html#NK3" target="_self">LifeSpan</a> <span class="p0">(<span class="default"><font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> <a href="ami_5.html#K8">+*</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p5">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p1">(<span class="default"><font color="Maroon">kl</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">km</font></span>)</span>
<b>by </b><i>, <a class="ref" href="ami_1.html#T51">AMI_1:51</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p5">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scm_1.html#NK3" target="_self">LifeSpan</a> <span class="p3">(<span class="default"><font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> <a href="ami_5.html#K8">+*</a> <span class="p0">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p2">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">kl</font></span>)</span> <a href="ami_5.html#K8">+*</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p5">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">km</font>
<b>by </b><i>, <a class="ref" href="ami_1.html#T51">AMI_1:51</a></i>
;<br/><b>hence </b><a NAME="E48:16_2_1_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p3">(<span class="default"><font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p0">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">mm</font></span>)</span> <a href="ami_5.html#K8">+*</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p3">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p5">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">mm</font>
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E49:16_2_1_1_2"/>
<font color="Maroon">f4</font> <a href="scpisort.html#R1">is_FinSequence_on</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p5">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">mm</font>,<font color="Maroon">m</font>
 <b>by </b><i>, </i>;<br/><b>thus </b><a NAME="E50:16_2_1_1_2"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">f4</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 <b>by </b><i/>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_1_1_2_4"><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="E1:16_2_1_1_2_4"/>
( 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>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f2</font> )
 ;<br/><b>hence </b><font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p3">(<span class="default"><font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p0">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scm_1.html#NK3" target="_self">LifeSpan</a> <span class="p4">(<span class="default"><font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p0">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> <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">m</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#E1:16"><i><font color="Green">E42</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <font color="Maroon">I</font>,<font color="Maroon">t</font></span>)</span> <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">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="scmpds_4.html#T23">SCMPDS_4:23</a></i>
;<br/><br/></div><b>end;</b></div><a NAME="E52:16_2_1_1_2"/><b>then </b>
<font color="Maroon">f2</font> <a href="scpisort.html#R1">is_FinSequence_on</a>  <a href="scmpds_4.html#K9">IExec</a> <font color="Maroon">I</font>,<font color="Maroon">t</font>,<font color="Maroon">m</font>
 <b>by </b><i><a class="ref" href="scpisort.html#D1">SCPISORT:def 1</a></i>;<br/><a NAME="E53:16_2_1_1_2"/><b>then </b>
<font color="Maroon">f1</font>,<font color="Maroon">f2</font> <a href="rfinseq.html#R1">are_fiberwise_equipotent</a> 
 <b>by </b><i>, , <a class="txt" href="scpqsort.html#E1:16"><i><font color="Green">E42</font></i></a></i>;<br/><a NAME="E54:16_2_1_1_2"/><b>then </b>
<font color="Maroon">f1</font>,<font color="Maroon">f3</font> <a href="rfinseq.html#R1">are_fiberwise_equipotent</a> 
 <b>by </b><i>, <a class="ref" href="rfinseq.html#T2">RFINSEQ:2</a></i>;<br/><b>hence </b><a NAME="E55:16_2_1_1_2"/>
<font color="Maroon">f1</font>,<font color="Maroon">f4</font> <a href="rfinseq.html#R1">are_fiberwise_equipotent</a> 
 <b>by </b><i>, <a class="ref" href="rfinseq.html#T2">RFINSEQ:2</a></i>;<br/><a NAME="E56:16_2_1_1_2"/><i><font color="Green">E183</font></i>: 
<font color="Maroon">yc</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">ym</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, , <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><div><i><font color="Green">E184</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_1_1_2_5"><b>now </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="E1:16_2_1_1_2_5"/><i><font color="Green">E185</font></i>: 
( <font color="Maroon">ym</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">yn</font> )
 ;<br/><a NAME="E2:16_2_1_1_2_5"/><b>then </b><i><font color="Green">E186</font></i>: 
<font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 <b>by </b><i>, , <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a>, <a class="ref" href="real_2.html#T106">REAL_2:106</a></i>;<br/><a NAME="E3:16_2_1_1_2_5"/><i><font color="Green">E187</font></i>: 
<font color="Maroon">yc</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="txt" href="scpqsort.html#E2:16"><i><font color="Green">E58</font></i></a>, <a class="txt" href="scpqsort.html#E4:16"><i><font color="Green">E60</font></i></a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E4:16_2_1_1_2_5"/><i><font color="Green">E188</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font>
 <b>by </b><i>, <a class="txt" href="scpqsort.html#E4:16"><i><font color="Green">E60</font></i></a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_1_1_2_5_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_2_1_1_2_5_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:16_2_1_1_2_5_1_1"/>
( <font color="Maroon">y1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">yc</font> or <font color="Maroon">y1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">yc</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_2_1_1_2_5_1_1_1"><b>suppose </b></a><a NAME="E1:16_2_1_1_2_5_1_1_1"/>
<font color="Maroon">y1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">yc</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:16_2_1_1_2_5_1_1_1"/>
<font color="Maroon">f4</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p0">(<span class="default"><font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p3">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="scm_1.html#NK3" target="_self">LifeSpan</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> <a href="ami_5.html#K8">+*</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p0">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">kl</font></span>)</span> <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">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="txt" href="scpqsort.html#E5:16"><i><font color="Green">E61</font></i></a>, <a class="txt" href="scpqsort.html#E6:16"><i><font color="Green">E62</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_2_1_1_2_5_1_1_2"><b>suppose </b></a><a NAME="E1:16_2_1_1_2_5_1_1_2"/>
<font color="Maroon">y1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">yc</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:16_2_1_1_2_5_1_1_2"/>
<font color="Maroon">f4</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p0">(<span class="default"><font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p3">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="scm_1.html#NK3" target="_self">LifeSpan</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> <a href="ami_5.html#K8">+*</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p0">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">kl</font></span>)</span> <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">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="txt" href="scpqsort.html#E5:16"><i><font color="Green">E61</font></i></a>, <a class="txt" href="scpqsort.html#E7:16"><i><font color="Green">E63</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E6:16_2_1_1_2_5"/>
<font color="Maroon">f4</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f3</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 <b>by </b><i>, <a class="txt" href="scpqsort.html#E5:16"><i><font color="Green">E61</font></i></a>, <a class="txt" href="scpqsort.html#E7:16"><i><font color="Green">E63</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="hs2(this)" href="javascript:()" title="16_2_1_1_2_6"><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> , <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="E1:16_2_1_1_2_6"/><i><font color="Green">E189</font></i>: 
( <font color="Maroon">ym</font> <a href="nat_1.html#K1">+</a> 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">j</font> &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">yn</font> )
 ;<br/><a NAME="E2:16_2_1_1_2_6"/><b>then </b><i><font color="Green">E190</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">yn</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E3:16_2_1_1_2_6"/><i><font color="Green">E191</font></i>: 
<font color="Maroon">ym</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="txt" href="scpqsort.html#E4:16"><i><font color="Green">E60</font></i></a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E4:16_2_1_1_2_6"/><i><font color="Green">E192</font></i>: 
<font color="Maroon">f4</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f3</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="txt" href="scpqsort.html#E3:16"><i><font color="Green">E59</font></i></a>, <a class="txt" href="scpqsort.html#E4:16"><i><font color="Green">E60</font></i></a>, <a class="txt" href="scpqsort.html#E5:16"><i><font color="Green">E61</font></i></a></i>;<br/><a NAME="E5:16_2_1_1_2_6"/>
<font color="Maroon">f4</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f3</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="txt" href="scpqsort.html#E3:16"><i><font color="Green">E59</font></i></a>, <a class="txt" href="scpqsort.html#E4:16"><i><font color="Green">E60</font></i></a>, <a class="txt" href="scpqsort.html#E6:16"><i><font color="Green">E62</font></i></a></i>;<br/><b>hence </b><a NAME="E6:16_2_1_1_2_6"/>
<font color="Maroon">f4</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">f4</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 <b>by </b><i>, <a class="txt" href="scpqsort.html#E4:16"><i><font color="Green">E60</font></i></a>, <a class="txt" href="scpqsort.html#E7:16"><i><font color="Green">E63</font></i></a>, <a class="ref" href="sfmastr3.html#D4">SFMASTR3:def 4</a></i>;<br/></div><b>end;</b></div><a NAME="E59:16_2_1_1_2"/><b>then </b><i><font color="Green">E193</font></i>: 
<font color="Maroon">f4</font> <a href="sfmastr3.html#R1">is_non_decreasing_on</a> <font color="Maroon">ym</font> <a href="nat_1.html#K1">+</a> 1,<font color="Maroon">yn</font>
 <b>by </b><i><a class="ref" href="sfmastr3.html#D4">SFMASTR3:def 4</a></i>;<br/><a NAME="E60:16_2_1_1_2"/>
<font color="Maroon">y1</font> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i>, , <a class="ref" href="real_2.html#T106">REAL_2:106</a>, <a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/><b>then reconsider </b><font color="Maroon">y0</font> = <font color="Maroon">y1</font> <a href="xcmplx_0.html#K6">-</a> 1 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="E62:16_2_1_1_2"/><i><font color="Green">E195</font></i>: 
<font color="Maroon">y0</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">yc</font>
 <b>by </b><i><a class="txt" href="scpqsort.html#E1"><i><font color="Green">Lemma31</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><div><i><font color="Green">E196</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_1_1_2_7"><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="E1:16_2_1_1_2_7"/><i><font color="Green">E197</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">y0</font> )
 ;<br/><a NAME="E2:16_2_1_1_2_7"/>
<font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T148">XREAL_1:148</a></i>;<br/><a NAME="E3:16_2_1_1_2_7"/><b>then </b>
<font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">y1</font> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i><a class="txt" href="scpqsort.html#E7:16"><i><font color="Green">E63</font></i></a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E4:16_2_1_1_2_7"/><b>then </b><i><font color="Green">E198</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">y1</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E5:16_2_1_1_2_7"/>
<font color="Maroon">y1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 <b>by </b><i>, , <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E6:16_2_1_1_2_7"/><b>then </b><i><font color="Green">E199</font></i>: 
<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="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_1_1_2_7_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_2_1_1_2_7_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:16_2_1_1_2_7_1_1"/>
( <font color="Maroon">y1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">yc</font> or <font color="Maroon">y1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">yc</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_2_1_1_2_7_1_1_1"><b>suppose </b></a><a NAME="E1:16_2_1_1_2_7_1_1_1"/>
<font color="Maroon">y1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">yc</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:16_2_1_1_2_7_1_1_1"/>
<font color="Maroon">f4</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p0">(<span class="default"><font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p3">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="scm_1.html#NK3" target="_self">LifeSpan</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> <a href="ami_5.html#K8">+*</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p0">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">kl</font></span>)</span> <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">m</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#E7:16"><i><font color="Green">E63</font></i></a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_2_1_1_2_7_1_1_2"><b>suppose </b></a><a NAME="E1:16_2_1_1_2_7_1_1_2"/>
<font color="Maroon">y1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">yc</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:16_2_1_1_2_7_1_1_2"/>
<font color="Maroon">f4</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p0">(<span class="default"><font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p3">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="scm_1.html#NK3" target="_self">LifeSpan</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> <a href="ami_5.html#K8">+*</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p0">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">kl</font></span>)</span> <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">m</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#E7:16"><i><font color="Green">E63</font></i></a>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E8:16_2_1_1_2_7"/>
<font color="Maroon">f4</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f3</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, <a class="txt" href="scpqsort.html#E7:16"><i><font color="Green">E63</font></i></a>, , <a class="ref" href="scpisort.html#D1">SCPISORT:def 1</a></i>;<br/></div><b>end;</b></div><div><i><font color="Green">E200</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_1_1_2_8"><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="E1:16_2_1_1_2_8"/><i><font color="Green">E201</font></i>: 
( <font color="Maroon">yc</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f4</font> )
 ;<br/><a NAME="E2:16_2_1_1_2_8"/><i><font color="Green">E202</font></i>: 
1 <a href="nat_1.html#K1">+</a> 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font>
 <b>by </b><i>, <a class="ref" href="int_1.html#T20">INT_1:20</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_1_1_2_8_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_2_1_1_2_8_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:16_2_1_1_2_8_1_1"/>
( <font color="Maroon">y1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">yc</font> or <font color="Maroon">y1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">yc</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_2_1_1_2_8_1_1_1"><b>suppose </b></a><a NAME="E1:16_2_1_1_2_8_1_1_1"/>
<font color="Maroon">y1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">yc</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:16_2_1_1_2_8_1_1_1"/>
<font color="Maroon">f4</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p0">(<span class="default"><font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p3">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="scm_1.html#NK3" target="_self">LifeSpan</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> <a href="ami_5.html#K8">+*</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p0">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">kl</font></span>)</span> <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">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span></span>)</span>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_2_1_1_2_8_1_1_2"><b>suppose </b></a><a NAME="E1:16_2_1_1_2_8_1_1_2"/>
<font color="Maroon">y1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">yc</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:16_2_1_1_2_8_1_1_2"/>
<font color="Maroon">f4</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p0">(<span class="default"><font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p3">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="scm_1.html#NK3" target="_self">LifeSpan</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> <a href="ami_5.html#K8">+*</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p0">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">kl</font></span>)</span> <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">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span></span>)</span>
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:16_2_1_1_2_8"/>
<font color="Maroon">f4</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f3</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, , , , <a class="ref" href="scpisort.html#D1">SCPISORT:def 1</a></i>;<br/></div><b>end;</b></div><a NAME="E65:16_2_1_1_2"/><i><font color="Green">E203</font></i>: 
<font color="Maroon">ym</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 <b>by </b><i>, , <a class="txt" href="scpqsort.html#E1"><i><font color="Green">Lemma31</font></i></a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a>, <a class="ref" href="real_2.html#T106">REAL_2:106</a></i>;<br/><a NAME="E66:16_2_1_1_2"/><b>then </b><i><font color="Green">E204</font></i>: 
<font color="Maroon">f4</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">ym</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f3</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">ym</font>
 <b>by </b><i>, , <a class="txt" href="scpqsort.html#E7:16"><i><font color="Green">E63</font></i></a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_1_1_2_9"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_2_1_1_2_9_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:16_2_1_1_2_9_1"/>
( <font color="Maroon">ym</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">yn</font> or <font color="Maroon">ym</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">yn</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_2_1_1_2_9_1_1"><b>suppose </b></a><a NAME="E1:16_2_1_1_2_9_1_1"/>
<font color="Maroon">ym</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">yn</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:16_2_1_1_2_9_1_1"/>
<font color="Maroon">f3</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">ym</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p3">(<span class="default"><font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p0">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scm_1.html#NK3" target="_self">LifeSpan</a> <span class="p4">(<span class="default"><font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p0">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> <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">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">ym</font></span>)</span></span>)</span>
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_2_1_1_2_9_1_2"><b>suppose </b></a><a NAME="E1:16_2_1_1_2_9_1_2"/>
<font color="Maroon">ym</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">yn</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:16_2_1_1_2_9_1_2"/>
<font color="Maroon">f3</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">ym</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p3">(<span class="default"><font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p0">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scm_1.html#NK3" target="_self">LifeSpan</a> <span class="p4">(<span class="default"><font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p0">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> <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">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">ym</font></span>)</span></span>)</span>
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E68:16_2_1_1_2"/><b>then </b><i><font color="Green">E205</font></i>: 
<font color="Maroon">f4</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">ym</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <font color="Maroon">I</font>,<font color="Maroon">t</font></span>)</span> <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">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">ym</font></span>)</span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="scmpds_4.html#T23">SCMPDS_4:23</a></i>;<br/><div><i><font color="Green">E206</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_1_1_2_10"><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="E1:16_2_1_1_2_10"/><i><font color="Green">E207</font></i>: 
( <font color="Maroon">y1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">ym</font> )
 ;<br/><a NAME="E2:16_2_1_1_2_10"/>
<font color="Maroon">y0</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">y1</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T148">XREAL_1:148</a></i>;<br/><a NAME="E3:16_2_1_1_2_10"/><b>then </b><i><font color="Green">E208</font></i>: 
<font color="Maroon">y0</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E4:16_2_1_1_2_10"/>
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">ym</font>
 <b>by </b><i>, <a class="ref" href="int_1.html#T20">INT_1:20</a></i>;<br/><a NAME="E5:16_2_1_1_2_10"/><b>then </b>
<font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">yc</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><b>then consider </b><font color="Maroon">j</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E7:16_2_1_1_2_10"/><i><font color="Green">E209</font></i>: 
( <font color="Maroon">y0</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">yc</font> &amp; <font color="Maroon">f4</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f3</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> )
 <b>by </b><i><a class="txt" href="scpqsort.html#E4"><i><font color="Green">Lemma48</font></i></a>, , <a class="txt" href="scpqsort.html#E5:16"><i><font color="Green">E61</font></i></a>, <a class="txt" href="scpqsort.html#E6:16"><i><font color="Green">E62</font></i></a>, <a class="txt" href="scpqsort.html#E7:16"><i><font color="Green">E63</font></i></a>, , </i>;<br/><a NAME="E8:16_2_1_1_2_10"/><i><font color="Green">E210</font></i>: 
1 <a href="nat_1.html#K1">+</a> 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font>
 <b>by </b><i>, <a class="ref" href="int_1.html#T20">INT_1:20</a></i>;<br/><a NAME="E9:16_2_1_1_2_10"/><i><font color="Green">E211</font></i>: 
<font color="Maroon">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">ym</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="scpqsort.html#E2:16"><i><font color="Green">E58</font></i></a>, , <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E10:16_2_1_1_2_10"/><i><font color="Green">E212</font></i>: 
<font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="txt" href="scpqsort.html#E4"><i><font color="Green">Lemma48</font></i></a>, , <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E11:16_2_1_1_2_10"/><i><font color="Green">E213</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">y1</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font>
 <b>by </b><i>, <a class="ref" href="int_1.html#T20">INT_1:20</a></i>;<br/><a NAME="E12:16_2_1_1_2_10"/>
<font color="Maroon">yc</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">ym</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T148">XREAL_1:148</a></i>;<br/><a NAME="E13:16_2_1_1_2_10"/><b>then </b><i><font color="Green">E214</font></i>: 
<font color="Maroon">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">ym</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_1_1_2_10_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_2_1_1_2_10_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:16_2_1_1_2_10_1_1"/>
( <font color="Maroon">ym</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">yn</font> or <font color="Maroon">ym</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">yn</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_2_1_1_2_10_1_1_1"><b>suppose </b></a><a NAME="E1:16_2_1_1_2_10_1_1_1"/>
<font color="Maroon">ym</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">yn</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:16_2_1_1_2_10_1_1_1"/>
<font color="Maroon">f3</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p3">(<span class="default"><font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p0">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scm_1.html#NK3" target="_self">LifeSpan</a> <span class="p4">(<span class="default"><font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p0">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> <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">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font></span>)</span></span>)</span>
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_2_1_1_2_10_1_1_2"><b>suppose </b></a><a NAME="E1:16_2_1_1_2_10_1_1_2"/>
<font color="Maroon">ym</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">yn</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:16_2_1_1_2_10_1_1_2"/>
<font color="Maroon">f3</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p3">(<span class="default"><font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p0">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scm_1.html#NK3" target="_self">LifeSpan</a> <span class="p4">(<span class="default"><font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p0">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> <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">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font></span>)</span></span>)</span>
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E15:16_2_1_1_2_10"/><b>then </b>
<font color="Maroon">f4</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <font color="Maroon">I</font>,<font color="Maroon">t</font></span>)</span> <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">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font></span>)</span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="scmpds_4.html#T23">SCMPDS_4:23</a></i>;<br/><b>hence </b><a NAME="E16:16_2_1_1_2_10"/>
<font color="Maroon">f4</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">f4</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">ym</font>
 <b>by </b><i><a class="txt" href="scpqsort.html#E1"><i><font color="Green">Lemma31</font></i></a>, , , </i>;<br/></div><b>end;</b></div><div><i><font color="Green">E215</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_1_1_2_11"><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="E1:16_2_1_1_2_11"/><i><font color="Green">E216</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">ym</font> )
 ;<br/><a NAME="E2:16_2_1_1_2_11"/><b>then </b><i><font color="Green">E217</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">ym</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E3:16_2_1_1_2_11"/><i><font color="Green">E218</font></i>: 
<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="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_1_1_2_11_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_2_1_1_2_11_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:16_2_1_1_2_11_1_1"/>
( <font color="Maroon">ym</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">yn</font> or <font color="Maroon">ym</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">yn</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_2_1_1_2_11_1_1_1"><b>suppose </b></a><a NAME="E1:16_2_1_1_2_11_1_1_1"/>
<font color="Maroon">ym</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">yn</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:16_2_1_1_2_11_1_1_1"/>
<font color="Maroon">f3</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p3">(<span class="default"><font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p0">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scm_1.html#NK3" target="_self">LifeSpan</a> <span class="p4">(<span class="default"><font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p0">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> <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">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span></span>)</span>
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_2_1_1_2_11_1_1_2"><b>suppose </b></a><a NAME="E1:16_2_1_1_2_11_1_1_2"/>
<font color="Maroon">ym</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">yn</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:16_2_1_1_2_11_1_1_2"/>
<font color="Maroon">f3</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p3">(<span class="default"><font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p0">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scm_1.html#NK3" target="_self">LifeSpan</a> <span class="p4">(<span class="default"><font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p0">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> <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">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span></span>)</span>
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E5:16_2_1_1_2_11"/>
<font color="Maroon">f3</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="txt" href="scpqsort.html#E1:16"><i><font color="Green">E42</font></i></a>, , </i>;<br/></div><b>end;</b></div><div><i><font color="Green">E219</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_1_1_2_12"><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="E1:16_2_1_1_2_12"/><i><font color="Green">E220</font></i>: 
( <font color="Maroon">yn</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f3</font> )
 ;<br/><a NAME="E2:16_2_1_1_2_12"/><b>then </b><i><font color="Green">E221</font></i>: 
1 <a href="nat_1.html#K1">+</a> 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="int_1.html#T20">INT_1:20</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_1_1_2_12_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_2_1_1_2_12_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:16_2_1_1_2_12_1_1"/>
( <font color="Maroon">ym</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">yn</font> or <font color="Maroon">ym</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">yn</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_2_1_1_2_12_1_1_1"><b>suppose </b></a><a NAME="E1:16_2_1_1_2_12_1_1_1"/>
<font color="Maroon">ym</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">yn</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:16_2_1_1_2_12_1_1_1"/>
<font color="Maroon">f3</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p3">(<span class="default"><font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p0">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scm_1.html#NK3" target="_self">LifeSpan</a> <span class="p4">(<span class="default"><font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p0">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> <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">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span></span>)</span>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_2_1_1_2_12_1_1_2"><b>suppose </b></a><a NAME="E1:16_2_1_1_2_12_1_1_2"/>
<font color="Maroon">ym</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">yn</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:16_2_1_1_2_12_1_1_2"/>
<font color="Maroon">f3</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p3">(<span class="default"><font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p0">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scm_1.html#NK3" target="_self">LifeSpan</a> <span class="p4">(<span class="default"><font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p0">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> <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">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span></span>)</span>
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:16_2_1_1_2_12"/>
<font color="Maroon">f3</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="txt" href="scpqsort.html#E1:16"><i><font color="Green">E42</font></i></a>, , , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_1_1_2_13"><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="E1:16_2_1_1_2_13"/><i><font color="Green">E222</font></i>: 
( <font color="Maroon">ym</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">yn</font> )
 ;<br/><a NAME="E2:16_2_1_1_2_13"/><b>then </b><i><font color="Green">E223</font></i>: 
<font color="Maroon">yc</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E3:16_2_1_1_2_13"/><i><font color="Green">E224</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f4</font>
 <b>by </b><i>, , , , <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a>, <a class="ref" href="real_2.html#T106">REAL_2:106</a></i>;<br/><b>consider </b><font color="Maroon">j</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E5:16_2_1_1_2_13"/><i><font color="Green">E225</font></i>: 
( <font color="Maroon">ym</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">yn</font> &amp; <font color="Maroon">f3</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> )
 <b>by </b><i>, , <a class="txt" href="scpqsort.html#E1"><i><font color="Green">Lemma31</font></i></a>, , , , , , <a class="ref" href="real_2.html#T106">REAL_2:106</a></i>;<br/><a NAME="E6:16_2_1_1_2_13"/><i><font color="Green">E226</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font>
 <b>by </b><i>, , <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E7:16_2_1_1_2_13"/><i><font color="Green">E227</font></i>: 
<font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f2</font>
 <b>by </b><i>, , <a class="txt" href="scpqsort.html#E1:16"><i><font color="Green">E42</font></i></a>, , <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a>, <a class="ref" href="real_2.html#T106">REAL_2:106</a></i>;<br/><font color="Maroon">f4</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> = 
<font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
<b>by </b><i><a class="txt" href="scpqsort.html#E7:16"><i><font color="Green">E63</font></i></a>, , , </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p3">(<span class="default"><font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p0">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scm_1.html#NK3" target="_self">LifeSpan</a> <span class="p4">(<span class="default"><font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p0">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> <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">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="scpqsort.html#E1:16"><i><font color="Green">E42</font></i></a>, , </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <font color="Maroon">I</font>,<font color="Maroon">t</font></span>)</span> <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">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="scmpds_4.html#T23">SCMPDS_4:23</a></i>
;<br/><b>hence </b><a NAME="E9:16_2_1_1_2_13"/>
<font color="Maroon">f4</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">ym</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">f4</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="txt" href="scpqsort.html#E1"><i><font color="Green">Lemma31</font></i></a>, , </i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E73:16_2_1_1_2"/>
<font color="Maroon">f4</font> <a href="sfmastr3.html#R1">is_non_decreasing_on</a> <font color="Maroon">y1</font>,<font color="Maroon">yn</font>
 <b>by </b><i><a class="txt" href="scpqsort.html#E1"><i><font color="Green">Lemma31</font></i></a>, , <a class="txt" href="scpqsort.html#E4:16"><i><font color="Green">E60</font></i></a>, , </i>;<br/><b>thus </b><a NAME="E74:16_2_1_1_2"/>
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="Maroon">y1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">yn</font> &amp; ( ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">j</font> &amp; <font color="Olive">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">y1</font> ) or ( <font color="Maroon">yn</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">j</font> &amp; <font color="Olive">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> ) ) holds <br/><font color="Maroon">f4</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</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">m</font> <a href="nat_1.html#K1">+</a> <font color="Olive">j</font></span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_1_1_2_14"><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="E1:16_2_1_1_2_14"/><i><font color="Green">E228</font></i>: 
( <font color="Maroon">y1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">yn</font> &amp; ( ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">y1</font> ) or ( <font color="Maroon">yn</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> ) ) )
 ;<br/>

<a NAME="E2:16_2_1_1_2_14"/><i><font color="Green">E229</font></i>: 
( ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">y1</font> ) or ( <font color="Maroon">yc</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_1_1_2_14_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_2_1_1_2_14_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:16_2_1_1_2_14_1_1"/>
( ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">y1</font> ) or ( <font color="Maroon">yn</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> ) )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_2_1_1_2_14_1_1_1"><b>suppose </b></a><a NAME="E1:16_2_1_1_2_14_1_1_1"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">y1</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:16_2_1_1_2_14_1_1_1"/>
( ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">y1</font> ) or ( <font color="Maroon">yc</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> ) )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_2_1_1_2_14_1_1_2"><b>suppose </b></a><a NAME="E1:16_2_1_1_2_14_1_1_2"/>
( <font color="Maroon">yn</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:16_2_1_1_2_14_1_1_2"/>
( ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">y1</font> ) or ( <font color="Maroon">yc</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> ) )
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E3:16_2_1_1_2_14"/><i><font color="Green">E230</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_1_1_2_14_2"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_2_1_1_2_14_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:16_2_1_1_2_14_2_1"/>
( ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">y1</font> ) or ( <font color="Maroon">yn</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> ) )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_2_1_1_2_14_2_1_1"><b>suppose </b></a><a NAME="E1:16_2_1_1_2_14_2_1_1"/><i><font color="Green">E231</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">y1</font> )
 ;<br/><div class="add"><a NAME="E2:16_2_1_1_2_14_2_1_1"/><b>then </b>
<font color="Maroon">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">yn</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><b>hence </b><a NAME="E3:16_2_1_1_2_14_2_1_1"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> )
 <b>by </b><i>, , , <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a>, <a class="ref" href="real_2.html#T106">REAL_2:106</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_2_1_1_2_14_2_1_2"><b>suppose </b></a><a NAME="E1:16_2_1_1_2_14_2_1_2"/><i><font color="Green">E232</font></i>: 
( <font color="Maroon">yn</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> )
 ;<br/><div class="add"><a NAME="E2:16_2_1_1_2_14_2_1_2"/><b>then </b>
<font color="Maroon">y1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><b>hence </b><a NAME="E3:16_2_1_1_2_14_2_1_2"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> )
 <b>by </b><i>, , , <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a>, <a class="ref" href="real_2.html#T106">REAL_2:106</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E4:16_2_1_1_2_14"/><i><font color="Green">E233</font></i>: 
( ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">ym</font> <a href="nat_1.html#K1">+</a> 1 ) or ( <font color="Maroon">yn</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_1_1_2_14_3"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_2_1_1_2_14_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:16_2_1_1_2_14_3_1"/>
( ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">y1</font> ) or ( <font color="Maroon">yn</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> ) )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_2_1_1_2_14_3_1_1"><b>suppose </b></a><a NAME="E1:16_2_1_1_2_14_3_1_1"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">y1</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:16_2_1_1_2_14_3_1_1"/>
( ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">ym</font> <a href="nat_1.html#K1">+</a> 1 ) or ( <font color="Maroon">yn</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> ) )
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_2_1_1_2_14_3_1_2"><b>suppose </b></a><a NAME="E1:16_2_1_1_2_14_3_1_2"/>
( <font color="Maroon">yn</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:16_2_1_1_2_14_3_1_2"/>
( ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">ym</font> <a href="nat_1.html#K1">+</a> 1 ) or ( <font color="Maroon">yn</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> ) )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<div><i><font color="Green">E234</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_1_1_2_14_4"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_2_1_1_2_14_4_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:16_2_1_1_2_14_4_1"/>
( <font color="Maroon">ym</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">yn</font> or <font color="Maroon">ym</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">yn</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_2_1_1_2_14_4_1_1"><b>suppose </b></a><a NAME="E1:16_2_1_1_2_14_4_1_1"/>
<font color="Maroon">ym</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">yn</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:16_2_1_1_2_14_4_1_1"/>
<font color="Maroon">f3</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p3">(<span class="default"><font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p0">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scm_1.html#NK3" target="_self">LifeSpan</a> <span class="p4">(<span class="default"><font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p0">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> <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">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font></span>)</span></span>)</span>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_2_1_1_2_14_4_1_2"><b>suppose </b></a><a NAME="E1:16_2_1_1_2_14_4_1_2"/>
<font color="Maroon">ym</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">yn</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:16_2_1_1_2_14_4_1_2"/>
<font color="Maroon">f3</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p3">(<span class="default"><font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p0">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scm_1.html#NK3" target="_self">LifeSpan</a> <span class="p4">(<span class="default"><font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p0">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> <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">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font></span>)</span></span>)</span>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_1_1_2_14_5"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_2_1_1_2_14_5_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:16_2_1_1_2_14_5_1"/>
( <font color="Maroon">y1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">yc</font> or <font color="Maroon">y1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">yc</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_2_1_1_2_14_5_1_1"><b>suppose </b></a><a NAME="E1:16_2_1_1_2_14_5_1_1"/>
<font color="Maroon">y1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">yc</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:16_2_1_1_2_14_5_1_1"/>
<font color="Maroon">f4</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p0">(<span class="default"><font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p3">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="scm_1.html#NK3" target="_self">LifeSpan</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> <a href="ami_5.html#K8">+*</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p0">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">kl</font></span>)</span> <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">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font></span>)</span></span>)</span>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_2_1_1_2_14_5_1_2"><b>suppose </b></a><a NAME="E1:16_2_1_1_2_14_5_1_2"/>
<font color="Maroon">y1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">yc</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:16_2_1_1_2_14_5_1_2"/>
<font color="Maroon">f4</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p0">(<span class="default"><font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p3">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="scm_1.html#NK3" target="_self">LifeSpan</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> <a href="ami_5.html#K8">+*</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p0">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">kl</font></span>)</span> <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">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font></span>)</span></span>)</span>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><font color="Maroon">f4</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> = 
<font color="Maroon">f3</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
<b>by </b><i>, , <a class="ref" href="scpisort.html#D1">SCPISORT:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <font color="Maroon">I</font>,<font color="Maroon">t</font></span>)</span> <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">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font></span>)</span></span>)</span>
<b>by </b><i>, , <a class="ref" href="scmpds_4.html#T23">SCMPDS_4:23</a></i>
<br/>.= 
<font color="Maroon">t</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">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font></span>)</span></span>)</span>
<b>by </b><i>, , </i>
;<br/><br/>


</div><b>end;</b></div><b>thus </b><a NAME="E75:16_2_1_1_2"/>
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="Maroon">y1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">yn</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">j</font> &amp; <font color="Olive">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> holds <br/><font color="Maroon">f4</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</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">m</font> <a href="nat_1.html#K1">+</a> <font color="Olive">j</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="real_2.html#T106">REAL_2:106</a></i>;<br/><div><b>hereby </b>
<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="E1:16_2_1_1_2_15"/><i><font color="Green">E235</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/><a NAME="E2:16_2_1_1_2_15"/>
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">k1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E3:16_2_1_1_2_15"/><b>then </b><i><font color="Green">E236</font></i>: 
<font color="Maroon">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">k1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p3">(<span class="default"><font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p0">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">mm</font></span>)</span> <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"><span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p0">(<span class="default"><font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p3">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="scm_1.html#NK3" target="_self">LifeSpan</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> <a href="ami_5.html#K8">+*</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p0">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">kl</font></span>)</span> <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"><span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font></span>)</span></span>)</span>
<b>by </b><i>, , </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p3">(<span class="default"><font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p0">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scm_1.html#NK3" target="_self">LifeSpan</a> <span class="p4">(<span class="default"><font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p0">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> <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"><span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font></span>)</span></span>)</span>
<b>by </b><i>, , </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <font color="Maroon">I</font>,<font color="Maroon">t</font></span>)</span> <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"><span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="scmpds_4.html#T23">SCMPDS_4:23</a></i>
<br/>.= 
<font color="Maroon">t</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"><span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font></span>)</span></span>)</span>
<b>by </b><i>, , </i>
;<br/></div>
<b>end;</b></div><b>thus </b><a NAME="E77:16_2_1_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p3">(<span class="default"><font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p0">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">mm</font></span>)</span> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <font color="Maroon">c</font>,<font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="scmpds_2.html#K2">.</a> <span class="p2">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <font color="Maroon">c</font>,<font color="Maroon">i</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2
 <b>by </b><i>, , , , </i>;<br/><b>thus </b><a NAME="E78:16_2_1_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p3">(<span class="default"><font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p0">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">mm</font></span>)</span> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:16_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1]
 ;<br/></div><b>end;</b></div>
<a NAME="E9:16"/><i><font color="Green">E237</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );<br/></i>
<a NAME="E10:16"/><i><font color="Green">E238</font></i>: 
<font color="Maroon">m1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> 0</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i/>;<br/>
<a NAME="E11:16"/>
 ex <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  ex <font color="Olive">f2</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K4">INT</a>  st <br/>( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="ami_5.html#K8">+*</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p0">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Olive">k</font></span>)</span> <a href="ami_5.html#K8">+*</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p3">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="ami_5.html#K8">+*</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p5">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Olive">k</font> &amp; <font color="Olive">f2</font> <a href="scpisort.html#R1">is_FinSequence_on</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="ami_5.html#K8">+*</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p5">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Olive">k</font>,<font color="Maroon">m</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Olive">f2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">f</font>,<font color="Olive">f2</font> <a href="rfinseq.html#R1">are_fiberwise_equipotent</a>  &amp; <font color="Olive">f2</font> <a href="sfmastr3.html#R1">is_non_decreasing_on</a> 1,<font color="Maroon">n</font> &amp; ( for <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> &amp; ( ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">j</font> &amp; <font color="Olive">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 ) or ( <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">j</font> &amp; <font color="Olive">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> ) ) holds <br/><font color="Olive">f2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font> <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> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Olive">j</font></span>)</span></span>)</span> ) &amp; ( for <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">n</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">j</font> &amp; <font color="Olive">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> holds <br/><font color="Olive">f2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font> <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> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Olive">j</font></span>)</span></span>)</span> ) &amp; ( for <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">j</font> &amp; <font color="Olive">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> 0</span>)</span> <a href="nat_1.html#K1">+</a> 1 holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="ami_5.html#K8">+*</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p0">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Olive">k</font></span>)</span> <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"><span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Olive">j</font></span>)</span></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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Olive">j</font></span>)</span></span>)</span> ) &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="ami_5.html#K8">+*</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p0">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Olive">k</font></span>)</span> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <font color="Maroon">c</font>,<font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <span class="p2">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <font color="Maroon">c</font>,<font color="Maroon">i</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="ami_5.html#K8">+*</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p0">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Olive">k</font></span>)</span> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_3"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:16_3_1"/>
( <font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> 0 or <font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_3_1_1"><b>suppose </b></a><a NAME="E1:16_3_1_1"/>
<font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:16_3_1_1"/>
 ex <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  ex <font color="Olive">f2</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K4">INT</a>  st <br/>( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="ami_5.html#K8">+*</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p0">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Olive">k</font></span>)</span> <a href="ami_5.html#K8">+*</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p3">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="ami_5.html#K8">+*</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p5">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Olive">k</font> &amp; <font color="Olive">f2</font> <a href="scpisort.html#R1">is_FinSequence_on</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="ami_5.html#K8">+*</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p5">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Olive">k</font>,<font color="Maroon">m</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Olive">f2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">f</font>,<font color="Olive">f2</font> <a href="rfinseq.html#R1">are_fiberwise_equipotent</a>  &amp; <font color="Olive">f2</font> <a href="sfmastr3.html#R1">is_non_decreasing_on</a> 1,<font color="Maroon">n</font> &amp; ( for <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> &amp; ( ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">j</font> &amp; <font color="Olive">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 ) or ( <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">j</font> &amp; <font color="Olive">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> ) ) holds <br/><font color="Olive">f2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font> <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> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Olive">j</font></span>)</span></span>)</span> ) &amp; ( for <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">n</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">j</font> &amp; <font color="Olive">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> holds <br/><font color="Olive">f2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font> <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> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Olive">j</font></span>)</span></span>)</span> ) &amp; ( for <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">j</font> &amp; <font color="Olive">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> 0</span>)</span> <a href="nat_1.html#K1">+</a> 1 holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="ami_5.html#K8">+*</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p0">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Olive">k</font></span>)</span> <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"><span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Olive">j</font></span>)</span></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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Olive">j</font></span>)</span></span>)</span> ) &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="ami_5.html#K8">+*</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p0">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Olive">k</font></span>)</span> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <font color="Maroon">c</font>,<font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <span class="p2">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <font color="Maroon">c</font>,<font color="Maroon">i</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="ami_5.html#K8">+*</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p0">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Olive">k</font></span>)</span> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> )
 <b>by </b><i>, , , , , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_3_1_2"><b>suppose </b></a><a NAME="E1:16_3_1_2"/>
<font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">nn</font> = <font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> 1 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="E3:16_3_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">nn</font>]
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E4:16_3_1_2"/>
 ex <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  ex <font color="Olive">f2</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K4">INT</a>  st <br/>( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="ami_5.html#K8">+*</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p0">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Olive">k</font></span>)</span> <a href="ami_5.html#K8">+*</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p3">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="ami_5.html#K8">+*</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p5">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Olive">k</font> &amp; <font color="Olive">f2</font> <a href="scpisort.html#R1">is_FinSequence_on</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="ami_5.html#K8">+*</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p5">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Olive">k</font>,<font color="Maroon">m</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Olive">f2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">f</font>,<font color="Olive">f2</font> <a href="rfinseq.html#R1">are_fiberwise_equipotent</a>  &amp; <font color="Olive">f2</font> <a href="sfmastr3.html#R1">is_non_decreasing_on</a> 1,<font color="Maroon">n</font> &amp; ( for <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> &amp; ( ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">j</font> &amp; <font color="Olive">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 ) or ( <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">j</font> &amp; <font color="Olive">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> ) ) holds <br/><font color="Olive">f2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font> <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> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Olive">j</font></span>)</span></span>)</span> ) &amp; ( for <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">n</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">j</font> &amp; <font color="Olive">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> holds <br/><font color="Olive">f2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font> <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> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Olive">j</font></span>)</span></span>)</span> ) &amp; ( for <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">j</font> &amp; <font color="Olive">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> 0</span>)</span> <a href="nat_1.html#K1">+</a> 1 holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="ami_5.html#K8">+*</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p0">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Olive">k</font></span>)</span> <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"><span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Olive">j</font></span>)</span></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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Olive">j</font></span>)</span></span>)</span> ) &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="ami_5.html#K8">+*</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p0">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Olive">k</font></span>)</span> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <font color="Maroon">c</font>,<font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <span class="p2">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <font color="Maroon">c</font>,<font color="Maroon">i</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="ami_5.html#K8">+*</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p0">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Olive">k</font></span>)</span> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> )
 <b>by </b><i>, , , , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>then consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">f2</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K4">INT</a> <b> such that </b><br/><a NAME="E13:16"/><i><font color="Green">E240</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="ami_5.html#K8">+*</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p0">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">k</font></span>)</span> <a href="ami_5.html#K8">+*</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p3">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="ami_5.html#K8">+*</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p5">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">k</font> &amp; <font color="Maroon">f2</font> <a href="scpisort.html#R1">is_FinSequence_on</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="ami_5.html#K8">+*</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p5">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">k</font>,<font color="Maroon">m</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">f</font>,<font color="Maroon">f2</font> <a href="rfinseq.html#R1">are_fiberwise_equipotent</a>  &amp; <font color="Maroon">f2</font> <a href="sfmastr3.html#R1">is_non_decreasing_on</a> 1,<font color="Maroon">n</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="ami_5.html#K8">+*</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p0">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">k</font></span>)</span> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <font color="Maroon">c</font>,<font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <span class="p2">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <font color="Maroon">c</font>,<font color="Maroon">i</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="ami_5.html#K8">+*</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p0">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">k</font></span>)</span> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> )
 ;<br/>
<b>set </b><font color="Maroon">sk</font> = <span class="p1">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="ami_5.html#K8">+*</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p5">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">k</font>;<br/>
<b>set </b><font color="Maroon">s1</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="ami_5.html#K8">+*</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p0">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">k</font></span>)</span> <a href="ami_5.html#K8">+*</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p3">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon">s2</font> = <span class="p1">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p5">(<span class="default"><font color="Maroon">s</font> <a href="ami_5.html#K8">+*</a> <span class="p0">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p2">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">k</font></span>)</span> <a href="ami_5.html#K8">+*</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p5">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> 1;<br/>
<b>set </b><font color="Maroon">i1</font> = <font color="Maroon">a</font>,<font color="Maroon">i</font> <a href="scmpds_2.html#K9">&lt;=0_goto</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">I</font></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span>;<br/>
<b>set </b><font color="Maroon">i2</font> =  <a href="scmpds_2.html#K4">goto</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">I</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>;<br/>
<a NAME="E14:16"/><i><font color="Green">E242</font></i>: 
 <a href="ami_1.html#K6">IC</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p4">(<span class="default"><font color="Maroon">s</font> <a href="ami_5.html#K8">+*</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p0">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p1">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">k</font></span>)</span> <a href="ami_5.html#K8">+*</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p4">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="scmpds_3.html#K2">inspos</a> 0
 
<b>by </b><i><a class="ref" href="scmpds_6.html#T21">SCMPDS_6:21</a></i>;<br/>
<a NAME="E15:16"/>
 <a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">i</font> <a href="scmpds_2.html#K9">&lt;=0_goto</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">I</font></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> <a href="scmpds_4.html#K4">';'</a> <span class="p1">(<span class="default"><font color="Maroon">I</font> <a href="scmpds_4.html#K5">';'</a> <span class="p2">(<span class="default"><a href="scmpds_2.html#K4">goto</a> <span class="p3">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">I</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E16:16"/><b>then </b><i><font color="Green">E243</font></i>: 
 <a href="ami_1.html#K8">CurInstr</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p4">(<span class="default"><font color="Maroon">s</font> <a href="ami_5.html#K8">+*</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p0">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p1">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">k</font></span>)</span> <a href="ami_5.html#K8">+*</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p4">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>,<font color="Maroon">i</font> <a href="scmpds_2.html#K9">&lt;=0_goto</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">I</font></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span>
 
<b>by </b><i><a class="ref" href="scmpds_6.html#T22">SCMPDS_6:22</a></i>;<br/>
<i><font color="Green">E244</font></i>: <span class="p1">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p5">(<span class="default"><font color="Maroon">s</font> <a href="ami_5.html#K8">+*</a> <span class="p0">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p2">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">k</font></span>)</span> <a href="ami_5.html#K8">+*</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p5">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p1">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
 <a href="ami_1.html#K9">Following</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p0">(<span class="default"><font color="Maroon">s</font> <a href="ami_5.html#K8">+*</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p3">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">k</font></span>)</span> <a href="ami_5.html#K8">+*</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p0">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> 0</span>)</span>
<b>by </b><i><a class="ref" href="ami_1.html#D19">AMI_1:def 19</a></i>
<br/>.= 
 <a href="ami_1.html#K9">Following</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p4">(<span class="default"><font color="Maroon">s</font> <a href="ami_5.html#K8">+*</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p0">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p1">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">k</font></span>)</span> <a href="ami_5.html#K8">+*</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p4">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="ami_1.html#D19">AMI_1:def 19</a></i>
<br/>.= 
 <a href="ami_1.html#K4">Exec</a> <span class="p1">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">i</font> <a href="scmpds_2.html#K9">&lt;=0_goto</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">I</font></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p4">(<span class="default"><font color="Maroon">s</font> <a href="ami_5.html#K8">+*</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p0">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p1">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">k</font></span>)</span> <a href="ami_5.html#K8">+*</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p4">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="ami_1.html#D18">AMI_1:def 18</a></i>
;<br/>
<a NAME="E18:16"/>
 <a href="scmpds_4.html#K2">Initialized</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p2">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="ami_5.html#K8">+*</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p0">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">k</font></span>)</span> <a href="ami_5.html#K8">+*</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p3">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_4.html#T26">FUNCT_4:26</a></i>;<br/>
<a NAME="E19:16"/><b>then </b>
 <a href="scmpds_4.html#K8">stop</a> <span class="p1">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="ami_5.html#K8">+*</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p0">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">k</font></span>)</span> <a href="ami_5.html#K8">+*</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p3">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="scmpds_4.html#T57">SCMPDS_4:57</a></i>;<br/>
<a NAME="E20:16"/><b>then </b><i><font color="Green">E245</font></i>: 
 <a href="scmpds_4.html#K8">stop</a> <span class="p1">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p5">(<span class="default"><font color="Maroon">s</font> <a href="ami_5.html#K8">+*</a> <span class="p0">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p2">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">k</font></span>)</span> <a href="ami_5.html#K8">+*</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p5">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> 1
 
<b>by </b><i><a class="ref" href="ami_3.html#T38">AMI_3:38</a></i>;<br/>
<a NAME="E21:16"/><i><font color="Green">E246</font></i>: 
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">I</font></span>)</span> <a href="nat_1.html#K1">+</a> 2
 
<b>by </b><i><a class="ref" href="scmpds_8.html#T17">SCMPDS_8:17</a></i>;<br/>
<a NAME="E22:16"/><b>then </b><i><font color="Green">E247</font></i>: 
 <a href="scmpds_3.html#K2">inspos</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">I</font></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p2">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="scmpds_6.html#T25">SCMPDS_6:25</a></i>;<br/>
<i><font color="Green">E248</font></i>:  <a href="ami_1.html#K6">IC</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p0">(<span class="default"><font color="Maroon">s</font> <a href="ami_5.html#K8">+*</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p3">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">k</font></span>)</span> <a href="ami_5.html#K8">+*</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p0">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p0">(<span class="default"><font color="Maroon">s</font> <a href="ami_5.html#K8">+*</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p3">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">k</font></span>)</span> <a href="ami_5.html#K8">+*</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p0">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> 1</span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="ami_1.html#K2">IC</a> <a href="scmpds_2.html#K1">SCMPDS</a> </span>)</span>
<b>by </b><i><a class="ref" href="ami_1.html#D15">AMI_1:def 15</a></i>
<br/>.= 
 <a href="scmpds_2.html#K19">ICplusConst</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p4">(<span class="default"><font color="Maroon">s</font> <a href="ami_5.html#K8">+*</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p0">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p1">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">k</font></span>)</span> <a href="ami_5.html#K8">+*</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p4">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">I</font></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span>
<b>by </b><i>, , , <a class="ref" href="scmpds_2.html#T68">SCMPDS_2:68</a></i>
<br/>.= 
 <a href="scmpds_3.html#K2">inspos</a> <span class="p1">(<span class="default">0 <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">I</font></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="scmpds_6.html#T23">SCMPDS_6:23</a></i>
;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p0">(<span class="default"><font color="Maroon">s</font> <a href="ami_5.html#K8">+*</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p3">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">k</font></span>)</span> <a href="ami_5.html#K8">+*</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p0">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> 1</span>)</span> <a href="ami_1.html#K13">.</a> <span class="p1">(<span class="default"><a href="scmpds_3.html#K2">inspos</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">I</font></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p2">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="scmpds_3.html#K2">inspos</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">I</font></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span>
<b>by </b><i>, , <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>
<br/>.= 
 <a href="ami_1.html#K5">halt</a> <a href="scmpds_2.html#K1">SCMPDS</a> 
<b>by </b><i><a class="txt" href="scpqsort.html#E1"><i><font color="Green">Lemma31</font></i></a>, <a class="ref" href="scmpds_6.html#T25">SCMPDS_6:25</a></i>
;<br/>
<a NAME="E25:16"/><b>then </b><i><font color="Green">E249</font></i>: 
 <a href="ami_1.html#K8">CurInstr</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p0">(<span class="default"><font color="Maroon">s</font> <a href="ami_5.html#K8">+*</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p3">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">k</font></span>)</span> <a href="ami_5.html#K8">+*</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p0">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="ami_1.html#K5">halt</a> <a href="scmpds_2.html#K1">SCMPDS</a> 
 
<b>by </b><i>, <a class="ref" href="ami_1.html#D17">AMI_1:def 17</a></i>;<br/>
<a NAME="E26:16"/><i><font color="Green">E250</font></i>: 
<span class="p1">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p5">(<span class="default"><font color="Maroon">s</font> <a href="ami_5.html#K8">+*</a> <span class="p0">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p2">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">k</font></span>)</span> <a href="ami_5.html#K8">+*</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p5">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="ami_5.html#K8">+*</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p5">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i>, <a class="ref" href="ami_1.html#T51">AMI_1:51</a></i>;<br/>
<a NAME="E27:16"/><b>then </b><i><font color="Green">E251</font></i>: 
<font color="Maroon">s</font> <a href="ami_5.html#K8">+*</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p3">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span> is <a href="ami_1.html#V9">halting</a>
 
<b>by </b><i>, <a class="ref" href="ami_1.html#D20">AMI_1:def 20</a></i>;<br/>
<a NAME="E28:16"/><b>then </b><i><font color="Green">E252</font></i>: 
 <a href="ami_1.html#K12">Result</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="ami_5.html#K8">+*</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p4">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p5">(<span class="default"><font color="Maroon">s</font> <a href="ami_5.html#K8">+*</a> <span class="p0">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p2">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">k</font></span>)</span> <a href="ami_5.html#K8">+*</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p5">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> 1
 
<b>by </b><i>, , <a class="ref" href="ami_1.html#D22">AMI_1:def 22</a></i>;<br/>
<b>thus </b><a NAME="E29:16"/>
 <a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font> <a href="scmpds_6.html#R2">is_halting_on</a> <font color="Maroon">s</font>
 <b>by </b><i>, <a class="ref" href="scmpds_6.html#D3">SCMPDS_6:def 3</a></i>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_7"><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="E1:16_7"/>
<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="E2:16_7"/><b>then </b><i><font color="Green">E253</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="E3:16_7"/><i><font color="Green">E254</font></i>: 
 <a href="scmpds_4.html#K9">IExec</a> <span class="p1">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span>,<font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p0">(<span class="default"><font color="Maroon">s</font> <a href="ami_5.html#K8">+*</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p3">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">k</font></span>)</span> <a href="ami_5.html#K8">+*</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p0">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> 1</span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="relat_1.html#K7">|</a> the <a href="ami_1.html#U2">Instruction-Locations</a> of <a href="scmpds_2.html#K1">SCMPDS</a> </span>)</span>
 <b>by </b><i>, <a class="ref" href="scmpds_4.html#D8">SCMPDS_4:def 8</a></i>;<br/><b>set </b><font color="Maroon">xi</font> =  <a href="scmp_gcd.html#K1">intpos</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span>;<br/><a NAME="E4:16_7"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="relat_1.html#K7">|</a> the <a href="ami_1.html#U2">Instruction-Locations</a> of <a href="scmpds_2.html#K1">SCMPDS</a> </span>)</span> <a href="hidden.html#R1">=</a> the <a href="ami_1.html#U2">Instruction-Locations</a> of <a href="scmpds_2.html#K1">SCMPDS</a> 
 <b>by </b><i><a class="ref" href="scmpds_6.html#T1">SCMPDS_6:1</a></i>;<br/><a NAME="E5:16_7"/><b>then </b><i><font color="Green">E256</font></i>: 
not  <a href="scmp_gcd.html#K1">intpos</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="relat_1.html#K7">|</a> the <a href="ami_1.html#U2">Instruction-Locations</a> of <a href="scmpds_2.html#K1">SCMPDS</a> </span>)</span>
 <b>by </b><i><a class="ref" href="scmpds_2.html#T53">SCMPDS_2:53</a></i>;<br/><b>thus </b><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="ami_5.html#K8">+*</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p4">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">k</font></span>)</span> <a href="ami_5.html#K8">+*</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p0">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p1">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> 1</span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="relat_1.html#K7">|</a> the <a href="ami_1.html#U2">Instruction-Locations</a> of <a href="scmpds_2.html#K1">SCMPDS</a> </span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1">intpos</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span></span>)</span>
<b>by </b><i>, , , , <a class="ref" href="scpisort.html#D1">SCPISORT:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p0">(<span class="default"><font color="Maroon">s</font> <a href="ami_5.html#K8">+*</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p3">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">k</font></span>)</span> <a href="ami_5.html#K8">+*</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p0">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> 1</span>)</span> <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">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="scpqsort.html#T1" target="_self">Th1</a>, <a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p4">(<span class="default"><font color="Maroon">s</font> <a href="ami_5.html#K8">+*</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p0">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p1">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">k</font></span>)</span> <a href="ami_5.html#K8">+*</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <span class="p4">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <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">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="scmpds_2.html#T68">SCMPDS_2:68</a></i>
<br/>.= 
<font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</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="E31:16"/>
( <font color="Maroon">f</font>,<font color="Maroon">g</font> <a href="rfinseq.html#R1">are_fiberwise_equipotent</a>  &amp; <font color="Maroon">g</font> <a href="sfmastr3.html#R1">is_non_decreasing_on</a> 1,<font color="Maroon">n</font> )
 <b>by </b><i>, , <a class="ref" href="finseq_2.html#T10">FINSEQ_2:10</a></i>;<br/>


</div>
